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Odisha State Board Elements of Mathematics Class 11 CHSE Odisha Solutions Chapter 4 Trigonometric Functions Ex 4(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 4 Trigonometric Functions Exercise 4(c)

Question 1.

Fill in the blanks choosing the correct answer from the brackets.

(i) The number of solutions of  2 sin θ – 1 = 0 is__________. (one, two, infinite)
Solution:
Infinite

(ii) If cos α = cos β, then α + β = ____________. (0, π, 2π)
Solution:
2π

(iii) The number of solution(s) of 2 sin θ + 1 = 0 is__________. (zero, two, infinite)
Solution:
Zero

(iv) If tan θ = tan α and 90° < α < 180°, then θ can be in ____________quadrant. (1st, 3rd, 4th)
Solution:
4th

(v) If tan x. tan 2x. tan 7x = tan x + tan 2x + tan 7x, then x = _____________. (\(\frac{\pi}{4}, \frac{\pi}{5}, \frac{\pi}{10}\))
Solution:
\(\frac{\pi}{10}\)

(vi) For_____________value of θ, sin θ + cos θ = √2. (\(\frac{\pi}{4}, \frac{\pi}{2}, \frac{\pi}{3}\))
Solution:
\(\frac{\pi}{4}\)

(vii) The number of values of x for which cos² x = 1 and x² ≤ 4 is______________. (1, 2, 3)
Solution:
1

(viii) In the 1st quadrant the solution of tan² θ = 3 is_____________. (\(\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}\))
Solution:
\(\frac{\pi}{3}\)

(ix) The least positive value of θ for which 1 + tan θ = 0 and √2 cos θ + 1 = 0 is___________. (\(\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}\))
Solution:
\(\frac{3 \pi}{4}\)

(x) the least positive value of x for which tan 3x = tan x is______________. (\(\frac{\pi}{2}, \frac{\pi}{3}, \pi\))
Solution:
\(\frac{\pi}{2}\)

Question 2.
Find the principal solution of the following equations:
(i) sin θ = sin 2θ
Solution:
sin θ = sin 2θ
or, 2θ = nπ + (-1)ⁿ θ
or, 2π – (-1)ⁿ θ = nπ
or, θ = \(\frac{n \pi}{2-(-1)^n}\)
when n = 0, θ = 0
when n = 1, θ = \(\frac{\pi}{3}\)
when n = 2, θ = 2π
when n = 3, θ = π
when n = 4, θ = 4π
when n = 5, θ = \(\frac{5 \pi}{3}\)
∴ The principal solution are 0, \(\frac{\pi}{3}\), π, \(\frac{5 \pi}{3}\)

(ii) √3 sin θ – cos θ = 2
Solution:
√3 sin θ – cos θ = 2
or, \(\frac{\sqrt{3}}{2}\) sin θ – 1/2 cos θ = 1

which is the only principal solution.

(iii) cos² θ + sin θ + 1 = 0
Solution:
cos² θ + sin θ + 1 = 0
or, 1 – sin² θ + sin θ + 1 = 0
or,  sin² θ – sin θ + 2 = 0
or, sin² θ – 2 sin θ + sin θ – 2 = 0
or, sinθ(sinθ – 2) + (sinθ – 2) = 0
or, (sinθ – 2) (sinθ + 1) = 0
∴ sinθ = 2, sinθ = – 1
= sin \(\left(-\frac{3 \pi}{2}\right)\) or, θ = – \(\frac{\pi}{2}\), \(\frac{3 \pi}{2}\)
∴ The principal solution is \(\frac{3 \pi}{2}\).

(iv) sin 4x + sin 2x = 0
Solution:

(v) sin x + cos x = \(\frac{1}{\sqrt{2}}\)
Solution:

Question 3.
Find the general solution of the following equations:
(i) cos 2x = θ
Solution:
cos 2x = θ
or, 2x = (2n + 1)\(\frac{\pi}{2}\)
or, x = (2n + 1)\(\frac{\pi}{4}\), n∈Z

(ii) sin(x° + 40°) = \(\frac{1}{\sqrt{2}}\)
Solution:

(iii) sin 5θ = sin 3θ
Solution:
sin 5θ = sin 3θ
or, 5θ = nπ + (-1)ⁿ 3θ
or, 5θ – (-1)ⁿ 3θ = nπ
or, θ[5 – (-1)ⁿ3] = nπ
or, θ = \(\frac{n \pi}{5-(-1)^n 3}\)

(iv) tan ax = cot bx
Solution:

(v) tan² 3θ = 3
Solution:

Question 4.
Solve the following:
(Hints : cos x ≠ 0 and sin² x- sin x + 1/2 = 0)
(i) tan² x + sec² x = 3
Solution:
tan² x + sec² x = 3
or, tan² x + 1 + tan2 x = 3
or, 2tan² x = 2
or, tan² x = 1
or, tan x = ± 1 = tan \(\left(\pm \frac{\pi}{4}\right)\)
∴ x = nπ ± \(\frac{\pi}{4}\)

(ii) 4 sin² x + 6 cos² x = 5
Solution:
4 sin² x + 6 cos² x = 5
or, 4 sin² x + 6(1 – sin² x) = 5
or, 4 sin² x + 6 – 6 sin² x = 5
or, 6- 2 sin² x = 5
or, 2 sin² x = 1
or, sin² x = 1/2
or, sin x = ± \(\frac{1}{\sqrt{2}}\) = sin \(\left(\pm \frac{\pi}{4}\right)\)
or, x = nπ + (-1)ⁿ \(\left(\pm \frac{\pi}{4}\right)\)

(iii) 3 sin x + 4 cos x = 5
Solution:

(iv) 3 tan x + cot x = 5 cosec x
Solution:

(v) cos x + √3 sin x = √2
Solution:

(vi) sin 3x – 2 cos² x = 0
Solution :
sin 2x – 2 cos² x = 0
or, 2 sin x cos x – 2 cos² x = 0
or, 2 cos x(sin x – cos x) = 0
∴ cos x = 0, sin x = cos x
∴ x = (2n + 1)\(\frac{\pi}{2}\), tan x = 1 = tan \(\frac{\pi}{4}\)
or, x = nπ + \(\frac{\pi}{4}\)

(vii) sec θ + tan θ = √3
Solution:

(viii) cos 2θ – cos θ = sin θ – sin 20
Solution:

(ix) sin θ + sin 2θ + sin 3θ + sin 4θ = 0
Solution:

(x) cos 2x° + cos x° – 2 = 0
Solution:
cos 2x° + cos x° – 2 = 0
or, 2 cos² x° – 1 + cos x° – 2 = 0
or, 2 cos² x° + cos x° – 3 = 0
or 2 cos² + 3cos x° – 2cos x°- 3 = 0
or, cos x°(2 cos x° + 3) – 1(2 cos x° + 3) = 0
or, (2 cos x° + 3)(cos x° – 1) = 0
∴ cos x° = 1 = cos 0°
∴ x° = 2nπ ± 0 = 2nπ
or, \(\frac{\pi x}{180}\) = 2nπ
or, x = 360 n
Again 2 cos x° + 3 = 0
⇒ cos x° = – 3/2 which has no solution.
Hence x = 360 n.

(xi) tan θ + tan 2θ = tan 3θ
Solution:

(xii) tan θ + tan (\(\theta+\frac{\pi}{3}\)) + tan (\(\theta+\frac{2\pi}{3}\)) = 3
Solution:

(xiii) cot² θ – tan² θ = 4 cot 2θ
Solution:

(xiv) cos 2θ = \((\sqrt{2}+1)\left(\cos \theta-\frac{1}{\sqrt{2}}\right)\)
Solution:

(xv) sec θ – 1 = \((\sqrt{2}-1)\) tan θ
Solution:

⇒ θ = 2nπ + \(\frac{\pi}{4}\)

(xvi) 3cot² θ – 2 sin θ = 0
Solution:

(xvii) 4 cos x. cos 2x . cos 3x = 1
Solution:
4 cos x cos 2x cos 3x = 1
⇒ 2 cos x cos 2x. 2 cos 3x = 1
⇒ (cos 3x + cos x) 2 cos 3x = 1
⇒ 2 cos² 3x + 2 cos 3x cos x = 1
⇒ 2 cos² 3x – 1 + cos 4x + cos 2x = 0
⇒ cos 6x + cos 4x + cos 2x = 0
⇒ cos 6x + cos 2x + cos 4x = 0
⇒ 2 cos 4x cos 2x + cos 4x = 0
⇒ cos 4x (2 cos 2x + 1) = 0
⇒ cos 4x = 0, cos 2x = – 1/2
cos 4x = 0 ⇒ 4x = (2n + 1) \(\frac{\pi}{2}\)

(xviii) cos 3x – cos 2x = sin 3x
Solution:

⇒ 1 – 2 sin x cos x = y²
∴ Equation (1) reduces to
1 – 2(1 – y²) + y = 0
⇒ 2y² + y – 1 = 0
⇒ (2y- 1) (y + 1) = 0

(xix) cos x + sin x = cos 2x + sin 2x
Solution:
cos x + sin x – cos 2x + sin 2x
[Refer (viii)]

(xx) tan x + tan 4x + tan 7x = tan x. tan 4x. tan 7x
Solution:
tan x + tan 4x + tan 7x = tan x tan 4x tan 7x
or, tan x + tan 4x
= – tan 7x + tan x tan 4x tan 7x
= – tan 7x (1 – tan x tan 4x)
or, \(\frac{\tan x+\tan 4 x}{1-\tan x \tan 4 x}\) = – tan 7x
or, tan (x + 4x) = tan (π – 7x)
or, tan 5x = tan (π – 7x)
or, 5x = nπ + π – 7x
or, 12x = π(n + 1)
or, x = \(\frac{\pi(n+1)}{12}\), n∈Z

(xxi) 2(sec² θ + sin² θ) = 5
Solution:

(xxii) \((\cos x)^{\sin ^2 x-\frac{3}{2} \sin x+\frac{1}{2}}=1\)
Solution:
\((\cos x)^{\sin ^2 x-\frac{3}{2} \sin x+\frac{1}{2}}=0\)
As cos x ≠ 0.
we have sin² – \(\frac{3}{2}\) sin x + \(\frac{1}{2}\) = 0
∴ 2 sin² x – 3 sin x + 1 = 0
or, 2 sin² x – 2 sin x – sin x + 1 = 0
or, (2sin x – 1)(sin x – 1) = 0
∴ sin x = \(\frac{1}{2}\) or, sin x = 1
But as cos x ≠ 0, we have sin x ≠ 1
∴ sin x = \(\frac{1}{2}\) = sin \(\frac{\pi}{6}\)
∴ x = nπ + (-1)ⁿ \(\frac{\pi}{6}\), n∈Z

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-3.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-3

Long Questions With Answers

Question 1.
What are the different types of psychotherapy? On what basis are they classified?
Answer:
Different types of psychotherapy are:

• Psychodynamic therapy
• Behaviour therapy
• Humanistic-existential therapy
• Biomedical therapy

Also, there are many alternative therapies such as yoga, meditation, acupuncture, herbal remedies etc.
Basis of classification of psychotherapy:

On the cause which has led to the problem:
Psychodynamic therapy is of the view that intrapsychic conflicts, i.e. the conflicts that are present within the psyche of the person, are the source of psychological problems.

On how did the cause come into existence:
The psychodynamic therapy, unfulfilled desires of childhood and unresolved childhood fears lead to intrapsychic conflicts.

What is the chief method of treatment?
Psychodynamic therapy uses the methods of free association and reporting of dreams to elicit the thoughts and feelings of the client.

What is the nature of the therapeutic relationship between the client and the therapist?
Psychodynamic therapy assumes that the therapist understands the client’s intrapsychic conflicts better than the client and hence it is the therapist who interprets the thoughts and feelings of the client to her/him so that s/he gains an understanding of the same.

What is the chief benefit to the client?
Psychodynamic therapy values emotional insight as the important benefit that the client derives from the treatment. Emotional insight is present when the client understands her/his conflicts intellectually; is able to accept the same emotionally, and is able to change her/his emotions towards the conflicts.

On the duration of treatment:
Hie duration of classical psychoanalysis may continue for several years. However, several recent versions of psychodynamic therapies are completed in 10—15 sessions.

Question 2.
A therapist asks the client to reveal all her/his thoughts including early childhood experiences. Describe the technique and type of therapy being used.
Answer:
In this case psychodynamic, therapy is used in the treatment of the client. Since the psychoanalytic approach views intrapsychic conflicts to be the cause of the psychological disorder. The first step in the treatment is to elicit this intrapsychic conflict. Psychoanalysis has invented free association and dream interpretation as two important methods for eliciting intrapsychic conflicts.

The free association method is the main method for understanding the client’s problems. Once a therapeutic relationship is established, and the client feels comfortable, the therapist makes her/him lie down on the couch, close her/his eyes and asks her/him to speak whatever comes to mind without censoring it in any way. The client is encouraged to freely associate one thought with another, and this method is called the method of free association.

The censoring superego and the watchful ego are kept in abeyance as the client speaks whatever comes to mind in an atmosphere that is relaxed and trusting. As the therapist does not interrupt, the free flow of ideas, desires and conflicts of the unconscious, which had been suppressed by the ego, emerges into the conscious mind. This free uncensored verbal narrative of the client is a window into the client’s unconscious to which the therapist gains access.

Along with this technique, the client is asked to write down her/his dreams upon waking up. Psychoanalysts look upon dreams as symbols of the unfulfilled desires present in the unconscious. The images of dead dreams are symbols which signify intrapsychic forces. Dreams use symbols because they are indirect expressions and hence would not alert the ego.

If the unfulfilled desires are expressed directly, the ever-vigilant ego would suppress them and that would leads to anxiety. These symbols are interpreted according to an accepted convention of translation as indicators of unfulfilled desires and conflicts.

Question 3.
Discuss the various techniques used in behaviour therapy.
Answer:
Various techniques used in behaviour therapy:
A range of techniques is available for changing behaviour. The principles of these techniques are to reduce the arousal level of the client, alter behaviour through classical conditioning or operant conditioning with different contingencies of reinforcements, as well as to use vicarious learning procedures, if necessary. Negative reinforcement and aversive conditioning are the two major techniques of behaviour modification.

Negative reinforcement refers to following an undesired response with an outcome that is gainful or not liked. For example, one learns to put on woollen clothes, bum firewood or use electric heaters to avoid the unpleasant cold weather. One learns to move away from dangerous stimuli because they provide negative reinforcement.

Aversive conditioning refers to the repeated association of an undesired response with an aversive consequence. For example, an alcoholic is given a mild electric shock and asked to smell the alcohol. With repeated pairings, the smell of alcohol is aversive as the pain of the shock is associated with it and the person will give up alcohol.

Positive reinforcement is given to increase the deficit if adaptive behaviour occurs rarely. For example, if a child does not do homework regularly, positive reinforcement may be used by the child’s mother by preparing the child’s favourite dish whenever s/he does homework at the appointed time. The positive reinforcement of food will increase the behaviour of doing homework at the appointed time.

The token economy in which persons with behavioural problems can be given a token as a reward every time a wanted behaviour occurs. The tokens are collected and exchanged for a reward such as an outing for the patient or a treat for the child. Unwanted behaviour can be reduced and waited behaviour can be increased simultaneously through differential reinforcement.

Positive reinforcement for the wanted behaviour and negative reinforcement for the unwanted behaviour attempted together may be one such method. The other method is to positively reinforce the wanted behaviour and ignore the unwanted behaviour. The latter method is less painful and equally effective. For example, let us consider the case of a girl who sulks and cries when she is not taken to the cinema when she asks.

The parent is instructed to take her to the cinema if she does not cry and sulk but not to take her if she does. Further, the parent is instructed to ignore the girl when she cries and sulks. The wanted behaviour of politely asking to be taken to the cinema increases and the unwanted behaviour of crying and sulking decreases.

Question 4.
Explain with the help of an example how cognitive distortions take place.
Answer:
Cognitive distortions are ways of thinking which are general in nature but which distort reality in a negative manner. These patterns of thought are called dysfunctional cognitive structures. They lead to errors of cognition about social reality. Aaron Beck’s theory of psychological distress states that childhood experiences provided by the family and society develop core, schemas or systems, which include beliefs and action patterns in the individual.

Thus, a client, who was neglected by the parents as a child, develops the core schema of “I am not wanted”. During the course of their life, a critical incident occurs in her/his life. S/he is publicly ridiculed by the teacher in school. This critical incident triggers the core schema of “I am not wanted” leading to the development of negative automatic thoughts. Negative thoughts are persistent irrational thoughts such as “nobody loves me”, “I am ugly”, “l am stupid”, “I will not succeed”, etc.

Such negative automatic thoughts are characterised by cognitive distortions. Repeated occurrence of these thoughts leads to the development of feelings of anxiety and depression. The therapist uses questioning, which is a gentle, non-threatening disputation of the client’s beliefs and thoughts. Examples of such questions would be, “Why should everyone love you?”, “What does it mean to you to succeed?” etc.

Question 5.
Which therapy encourages the client to seek personal growth and actualise their potential? Write about the therapies which are based on this principle.
Answer:
Humanistic-existential therapy encourages the client to seek personal growth and actualise their potential. It states that psychological distress arises from feelings of loneliness, alienation, and an inability to find meaning and genuine fulfilment in life.
The therapies which are based on this principle are:

Existential therapy:
There is a spiritual unconscious, which is the storehouse of love, aesthetic awareness, and values of life. Neurotic anxieties arise when the problems of life are attached t6 the physical, psychological or spiritual aspects of one’s existence. Frankl emphasised the role of spiritual anxieties in leading to meaninglessness and hence it may be called existential anxiety, i.e. neurotic anxiety of spiritual origin.

Client-centred therapy:
Client-centred therapy was given by Carl Rogers. He combined scientific rigour with the individualised practice of client-centred psychotherapy. Rogers brought into psychotherapy the concept of self, with freedom and choice as the core of one’s being. The therapy provides a warm relationship in which the client can reconnect with her/his disintegrated feelings. The therapist shows empathy, i.e. understanding the client’s experience as if it were her/his own, is warm and has unconditional positive regard, i.e. total acceptance of the client as s/he is. Empathy sets up an emotional resonance between the therapist and the client.

Gestalt therapy:
The German word gestalt means ‘whole’. This therapy was given by Frederick (Fritz) Peris together with his wife Laura Peris. The goal of gestalt therapy is to increase an individual’s self-awareness and self-acceptance. The client is taught to recognise the bodily processes and die emotions that are being blocked out from awareness. The therapist does this by encouraging the client to act out fantasies about feelings and conflicts. This therapy can also be used in group settings.

Question 6.
What are the factors that contribute to healing in psychotherapy? Enumerate some of the alternative therapies.
Answer:
Factors Contributing to Healing in Psychotherapy are:

A major factor in healing is the techniques adopted by the therapist and the implementation of the same with the patient/client. If the behavioural system and the CBT school are adopted to heal an anxious client, the relaxation procedures and the cognitive restructuring largely contribute to the healing.

The therapeutic alliance, which is formed between the therapist and the patient/ client, has healing properties, because of the regular availability of the therapist and the warmth and empathy provided by the therapist.

At the outset of therapy, while the patient/client is being interviewed in the initial sessions to understand the nature of the problem, s/he unburdens the emotional problems being faced. This process of emotional unburdening is known as catharsis and it has healing properties.

There are several non-specific factors associated with psychotherapy. Some of these factors are attributed to the patient/client and some to the therapist. These factors are called non-specific because they occur across different systems of psychotherapy and across .different clients/patients and different therapists. Non-specific factors attributable to the client/patient are the motivation for change, the expectation of improvement due to the treatment, etc.

These are called patient variables. Non-specific factors attributable to the therapist are positive nature, absence of unresolved emotional conflicts, presence of good mental health, etc. These are called therapist variables. Some of the alternative therapies are Yoga, meditation, acupuncture, herbal remedies etc.

Question 7.
What are the techniques used in the rehabilitation of the mentally ill?
Answer:
The treatment of psychological disorders has two components, i.e. reduction of symptoms, and improving the level of functioning or quality of life. In the case of milder disorders such as generalised anxiety, reactive depression or phobia, reduction of symptoms is associated with an improvement in the quality of life. However, in the case of severe mental disorders such as schizophrenia, reduction of symptoms may not be associated with an improvement in the quality of life.

Many patients suffer from negative symptoms such as disinterest and lack of motivation to do work or interact with people. The aim of rehabilitation is to empower the patient to become a productive member of society to the extent possible. In rehabilitation, the patients are given occupational therapy, social skills training, and vocational therapy. In occupational therapy, the patients are taught skills such as candle making, paper bag making and weaving to help them to form a work discipline.

Social skills. training helps the patients to develop interpersonal skills through role play, imitation and instruction. The objective is to teach the patient to function in a Social group. Cognitive retraining is given to improve the basic cognitive functions of attention, memory and executive functions. After the patient improves sufficiently, vocational training is given wherein the patient is helped to gain the skills necessary to undertake productive employment.

Question 8.
How would a social learning theorist account for a phobic fear of lizards/ cockroaches? How would a psychoanalyst account for the same phobia?
Answer:
Systematic desensitisation is a technique introduced by Wolpe for treating phobias or irrational fears. The client is interviewed to elicit fear-provoking situations and together with the client, the therapist prepares a hierarchy of anxiety-provoking stimuli with the least anxiety-provoking stimuli at the bottom of the hierarchy. The therapist relaxes the client and asks the client to think about the least anxiety-provoking situation.

The client is asked to stop thinking of the fearful situation if the slightest tension is felt. Over sessions, the client is able to imagine more severe fear-provoking situations while maintaining relaxation. The client gets systematically desensitised to the fear.

Question 9.
Should Electroconvulsive Therapy (ECT) be used in the treatment of mental disorders?
Answer:
Yes, Electro-convulsive Therapy (ECT) can be used in the treatment of mental disorders. Electroconvulsive Therapy (ECT) is another form of biomedical therapy. Mild electric shock is given via electrodes to the brain of the patient to induce convulsions. The shock is given by the psychiatrist only when it is necessary for the improvement of the patient. ECT is not a routine treatment and is given only when drugs are not effective in controlling, the symptoms of the patient.

Question 10.
What kind of problems is cognitive behaviour therapy best suited for?
Answer:
Cognitive behaviour treatment best suited for a wide range of psychological disorders such as anxiety, depression, panic attacks, borderline personality, etc. adopts a bio-CBT psychosocial approach to the delineation of psychopathology. It combines cognitive therapy with behavioural techniques.

Question 11.
What is the nature and process of therapeutic approaches?
Answer:
Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats the therapist. The purpose of the relationship is to help the client to solve the psychological problems faces by her or him. The relationship is conducive to building the trust of the client so that problems may be freely discussed.

Psychotherapies aim at changing maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment. The inadequate marital, occupational and social adjustment also requires that major changes be made in an individual’s personal environment. All psychotherapeutic approaches have the following characteristics:

• there is the systematic application of principles underlying the different theories of therapy.
• persons who have received practical training under expert supervision can practice psychotherapy and not everybody. An untrained person may unintentionally cause more harm than good.
• the therapeutic situation involves a therapist and a client who seeks and receives help for her/his emotional problems (this person is the focus of attention in the therapeutic process).
• the interaction of these two persons — the therapist and the client— results in the consolidation/formation of the therapeutic relationship. This is a confidential, interpersonal and dynamic relationship.

This human relationship is central to any sort of psychological therapy and is the vehicle for change. All psychotherapies aim at a few or all of the following goals :

• Reinforcing the client’s resolve for betterment.
• Lessening emotional pressure.
• Unfolding the potential for positive growth.
• Modifying habits.
• Changing thinking patterns
• Increasing self-awareness.
• Improving interpersonal relations and communication.
• Facilitating decision-making.
• Becoming aware of one’s choices in life.
• Relating to one’s social environment in a more creative and self-aware manner.

Question 12.
What is the relationship between the client and therapist?
Answer:
Therapeutic Relationship :
The special relationship between the client and the therapist is known as the therapeutic relationship or alliance. It is neither a passing acquaintance nor a permanent and lasting relationship. There are two major components of a therapeutic alliance. The first component is the contractual nature Of the relationship in which two willing individuals, the client and the therapist, enter into a partnership which aims at helping the client overcome her/his problems.

The second component of the therapeutic alliance is the limited duration of the therapy. This alliance lasts until the client becomes able to deal with her/his problems and take control of her/ his life. This relationship has several unique properties. It is a trusting and confiding relationship. The high level of trust enables the client to unburden herself/himself to the therapist and confide her/his psychological and personal problems to the latter.

The therapist encourages this by being accepting, empathic, genuine and warm to the client. The therapist conveys by her/his words and behaviours that s/he is not judging the client and will continue to show the same positive feelings towards the client even if the client is rude or confides in all the ‘wrong’ things that s/he may have done or thought about. This is the unconditional positive regard that the therapist has for the client. The therapist has empathy for the client.

Empathy:
Empathy is different from sympathy and intellectual understanding of another person’s situation. Iii sympathy, one has compassion and pity towards, the .suffering of another but is not able to feel like the other person. Intellectual understanding is cold in the sense that the person is unable to feel like the other person and does not feel sympathy either. On the other hand, empathy is present when one is able to understand the plight of another person and feel like the other person.

It means understanding things from the other person’s perspective, i.e. putting oneself in the other person’s shoes. Empathy enriches the therapeutic relationship and transforms it into a healing relationship. The therapeutic alliance also requires that the therapist must keep strict confidentiality of the experiences, events, feelings or thoughts disclosed by the client. The therapist must not exploit the trust and confidence of the client in any way. Finally, it is a professional relationship and must remain so.

Question 13.
Write the types of therapies.
Answer:
Though all psychotherapies aim at removing human distress and fostering effective behaviour, they differ greatly in concepts, methods, and techniques. Psychotherapies may be classified into three broad groups, viz. the psychodynamic, behaviour sad existential psychotherapies. In terms of chronological order, psychodynamic therapy emerged first followed by behaviour therapy while existential therapies which are also called the third force, emerged last. The classification of psychotherapies is based on the following parameters:

What is the cause, which has led to the problem?
Psychodynamic therapy is of the view that intrapsychic conflicts, i.e. the conflicts that are present within the psyche of the person, are the source of psychological problems. According to behaviour therapies, psychological problems arise due to faulty learning of behaviours and cognitions. Existential therapies postulate that questions about the meaning of one’s life and existence are the cause of psychological problems.

How did the cause come into existence?
In psychodynamic therapy, unfulfilled desires of childhood and unresolved childhood fears lead to intrapsychic conflicts. Behaviour therapy postulates that faulty conditioning patterns, faulty learning, and faulty thinking and beliefs lead to maladaptive behaviours that, in turn, lead to psychological problems. Existential therapy places importance on the present. It is the current feelings of loneliness, alienation, a sense of the futility of one’s existence, etc., which cause psychological problems.

What is the chief method of treatment?
Psychodynamic therapy uses the methods of free association and reporting of dreams to elicit the thoughts and feelings of the client. This material is interpreted to the client to help her/him to confront and resolve the conflicts and thus overcome problems. Behaviour therapy identifies faulty conditioning patterns and sets up alternate behavioural contingencies to improve behaviour.

The cognitive methods employed in this type of therapy challenge the faulty thinking patterns of the client to help her/him overcome psychological distress. Existential therapy provides a therapeutic environment which is positive, accepting and non-judgmental. The client is able to talk about the problems and the therapist acts as a facilitator. The client arrives at the solutions through a process of personal growth.

What is the nature of the therapeutic relationship between the client and the therapist?
Psychodynamic therapy assumes that the therapist understands the client’s intrapsychic conflicts better than the client and hence it is the therapist who interprets the. thoughts and feelings of the client to her/him so that s/he gains an understanding of the same. Behaviour therapy assumes that the therapist is able to discern the faulty behaviour and thought patterns of the client.

It further assumes that the therapist is capable of finding out the correct behaviour and thought patterns, which would be adaptive for the client. Both psychodynamic and behaviour therapies assume that the therapist is capable of arriving at solutions to the client’s problems. In contrast to these therapies, existential therapies emphasise that the therapist merely provides a warm, empathic relationship in . which the client feels secure to explore the nature and causes of her/his problems by herself/ himself.

What is the chief benefit to the client?
Psychodynamic therapy values emotional insight as the important benefit that the client derives from the treatment. Emotional insight is present when the client understands her/his conflicts intellectually; is able to accept the same emotionally and is able to change her/his emotions towards the conflicts. The client’s symptoms and distresses reduce as a consequence of this emotional insight.

Behaviour therapy considers changing faulty behaviour and thought patterns to adaptive ones as the chief benefit of the treatment. Instituting adaptive or healthy behaviour and thought patterns ensures the reduction of distress and the removal of symptoms. Humanistic therapy values personal growth as the chief benefit. Personal growth is the process of gaining an increasing understanding of oneself and one’s aspirations, emotions and motives.

What is the duration of treatment?
The duration of classical psychoanalysis may continue for several years. However, several recent versions of psychodynamic therapies are completed in 10-15 sessions. Behaviour and cognitive behaviour therapies as well as existential therapies are shorter and are completed in a few months. Thus, different types of psychotherapies differ on multiple parameters.

However, they all share the common method of providing treatment for psychological distress’ through psychological means. The therapist, the therapeutic relationship, and the process of therapy become the agents of change in the client leading to the alleviation of psychological distress. The process of psychotherapy begins by formulating the client’s problem.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-2.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-2

Long Questions With Answers

Question 1.
Write the classification of biological disorders.
Answer:
In order to understand psychological disorders, we need to begin by classifying them. A classification of such disorders consists of a list of categories of specific psychological disorders grouped into various classes on the basis of some shared characteristics. Classifications are useful because they enable users like psychologists, psychiatrists and social workers to communicate with each other about the disorder and help in understanding the causes of psychological disorders and the processes involved in their development and maintenance.

The American Psychiatric Association (APA) has published an official manual describing and classifying various kinds of psychological disorders. The current version of it, the Diagnostic and Statistical Manual of Mental Disorders, IV Edition (DSM-IV), evaluates the patient on five axes or dimensions rather than just one broad aspect of ‘mental disorder’. These dimensions relate to biological, psychological, social and other aspects.

The classification scheme officially used in India and elsewhere is the tenth revision of the International Classification of Diseases (ICD-10), which is known as the ICD-10 Classification of Behavioural and Mental Disorders. It was prepared by the World Health Organisation (WHO). For each disorder, a description of the main clinical features or symptoms and of other associated features including diagnostic guidelines is provided in this scheme.

Question 2.
What are the approaches to understanding abnormal behaviour?
Answer:
In order to understand something as complex as abnormal behaviour, psychologists use different approaches. Each approach in use today emphasises a different aspect of human behaviour and explains and treats abnormality in line with that aspect. These approaches also emphasise the role of different factors such as biological, psychological and interpersonal and socio-cultUral factors.

We will examine some of the approaches which are currently being used to explain abnormal behaviour. Biological factors influence all aspects of our behaviour. A wide range of biological factors such as faulty genes, endocrine imbalances, malnutrition, injuries and other conditions may interfere with the normal development and functioning of the human body. These factors may be potential causes of abnormal behaviour. We have already come across the biological model.

According to this model, abnormal behaviour has a biochemical or physiological basis. Biological researchers have found that psychological disorders are often related to problems in the transmission of messages from one neuron to another. You have studied in Class XI, that a tiny space called a synapse separates one neuron from the next and the message must move across that space.

When an electrical impulse reaches a neuron’s ending, the nerve ending is stimulated to release a chemical, called a neurotransmitter. Studies indicate that abnormal activity by certain neurotransmitters can lead to specific psychological disorders. Anxiety disorders have been linked to low activity of the neurotransmitter gamma-aminobutyric acid (GABA) schizophrenia to the excess activity of dopamine, and depression to low activity of serotonin.

Genetic factors have been linked to mood disorders, schizophrenia, mental retardation and other psychological disorders. Researchers have not, however, been able to identify the specific genes that are the culprits. It appears that in most cases, no single gene is responsible for a particular behaviour or a psychological disorder. In fact, many genes combine to help bring about our various behaviours and emotional reactions, both functional and dysfunctional.

Although there is sound evidence to believe that genetic/ biochemical factors are involved in mental disorders as diverse as schizophrenia, depression, anxiety, etc. and biology alone cannot account for most mental disorders. There are several psychological models which provide a psychological explanation of mental disorders. These models maintain that psychological and interpersonal factors have a significant role to play in abnormal behaviour.

These factors include maternal deprivation (separation from the mother, or lack of warmth and stimulation during early years of life), faulty parent-child relationships (rejection, overprotection, over permissiveness, faulty discipline, etc.), maladaptive family structures (inadequate or disturbed family) and severe stress. The psychological models include the psychodynamic, behavioural, cognitive and humanistic-existential models.

The psychodynamic model is the oldest and most famous of the modern psychological models. You have already read about this model in Chapter 2 on Self and Personality. Psychodynamic theorists believe that behaviour, whether normal or abnormal, is determined by psychological forces within the person of which s/he is not consciously aware. These internal forces are considered dynamic, i.e. they interact with one another and their interaction gives shape to behaviour, thoughts and emotions.

Abnormal symptoms are viewed as the result of conflicts between these forces. This model was first formulated by Freud who believed that three central forces shape personality — instinctual needs, drives and impulses (id), rational thinking (ego), and moral standards (superego). Freud stated that abnormal behaviour is a symbolic expression of unconscious mental conflicts that can be generally traced to early childhood or infancy.

Another model that emphasises the role of psychological factors is the behavioural model. This model states that both normal and abnormal behaviours are learned and psychological disorders are the result of learning maladaptive ways of behaving. The model concentrates on behaviours that are learned through conditioning and propose that what has been learned can be unlearned.

Learning can take place by classical conditioning (temporal association in which two events repeatedly occur close together in time), operant conditioning (behaviour is followed by a reward), and social learning (learning by imitating others’ behaviour). These three types of conditioning account for behaviour, whether adaptive or maladaptive. Psychological factors are also emphasised by the cognitive model. This model states that abnormal functioning can result from cognitive problems.

People may hold assumptions and attitudes about themselves that are irrational and inaccurate. People may also repeatedly think in illogical ways and makeover generalisations, that is, – they may draw broad, negative conclusions on the basis of a single insignificant event. Another psychological model is the humanistic-existential model which focuses on broader aspects of human existence.

Humanists believe that human beings are born with a natural tendency to be friendly, cooperative and constructive, and are driven to self-actualise, i.e. to fulfil this potential for goodness and growth. Existentialists believe that from birth we have total freedom to give meaning to our existence or to avoid that responsibility. Those who shirk from this responsibility would live empty, inauthentic and dysfunctional lives.

In addition to the biological and psychosocial factors, socio-cultural factors such as war and violence, group prejudice and discrimination, economic and employment problems and rapid social change, put stress on most of us and cafes also lead to psychological problems in some individuals. According to the sociocultural model, abnormal behaviour is best understood in light of the social and cultural forces that influence an individual.

As behaviour is shaped by societal forces, factors such as family structure and communication, social networks, societal conditions and societal labels and roles become more important. It has been found that certain family systems are likely to produce abnormal functioning in individual members. Some families have an enmeshed structure in which the members are over involved in each other’s activities, thoughts and feelings.

Children from this kind of family may have difficulty in becoming independent in life. The broader social networks in which people operate include their social and professional relationships. Studies have shown that people who are isolated and lack social support, i.e. strong and fulfilling interpersonal relationships in their lives are likely to become more depressed and remain depressed longer than those who have good friendships.

Socio-cultural theorists also believe that abnormal functioning is influenced by the societal labels and roles assigned to troubled people. When people break the norms of their society, they are called deviant and ‘mentally ill’. Such labels tend to stick so that the person may be viewed as ‘crazy’ and encouraged to act sick. The person gradually learns to accept and play the sick role, and functions in a disturbed manner.

In addition to these models, one of the most widely accepted explanations of abnormal behaviour has been provided by the diathesis-stress model. This model states that psychological disorders develop when a diathesis (biological predisposition to the disorder) is set off a stressful situation. This model has three components. The first is the diathesis or the presence of some biological aberration which may be inherited.

The second component is that the diathesis may carry a vulnerability to developing a psychological disorder. This means that the person is ‘at risk’ or ‘predisposed’ to develop the disorder. The third component is the presence of pathogenic stressors, i.e. factors/stressors that may lead to psychopathology. If such “at risk” persons are exposed to these stressors, their predisposition may actually evolve into a disorder. This model has been applied to several disorders including anxiety, depression, and schizophrenia.

Question 3.
What are the major psychological disorders?
Answer:
Anxiety Disorders:
One day while driving home, Deb felt his heart beating rapidly, he started sweating profusely and even felt short of breath. He was so scared that he stopped the car and stepped out. In the next few months, these attacks increased and now he was hesitant to drive for fear of being caught in traffic during an attack. Deb started feeling that he had gone crazy and would die. Soon he remained indoors and refused to move out of the house.

We experience anxiety when we are waiting to take an examination or visit a dentist, or even give a solo performance. This is normal and expected and even motivates us to do our tasks well. On the other hand, high levels of anxiety that are distressing and interfere with effective functioning indicate the presence of an anxiety disorder— the most common category of psychological disorders. Everyone has worries and fears.

The term anxiety is usually defined as a diffuse, vague, very unpleasant feeling of fear and apprehension. The anxious individual also shows combinations of the following symptoms: rapid heart rate, shortness of breath, diarrhoea, loss of appetite, fainting, dizziness, sweating, sleeplessness, frequent urination and tremors. There are many types of anxiety disorders (see Table 4.2).

They include generalised anxiety disorder, which consists of prolonged, vague, unexplained and intense fears that are not attached to any particular object. The symptoms include worry and apprehensive feelings about the future; hypervigilance, which involves constantly scanning the environment for dangers. It is marked by motor tension, as a result of which the person is unable to relax, is restless and visibly shaky and tense.

Another type of anxiety disorder is panic disorder, which consists of recurrent anxiety attacks in which the person experiences intense terror. A panic attack denotes an abrupt surge of intense anxiety rising to a peak when thoughts of particular stimuli are present. Such thoughts occur in an unpredictable manner. The clinical features include shortness of breath, dizziness, trembling, palpitations, choking, nausea, chest pain or discomfort, fear of going crazy, losing control or dying.

You might have met of heard of someone who was afraid to travel in a lift or climb to the tenth floor of a building or refused to enter a room if s/he saw a lizard. You may have also felt it yourself or seen a friend unable to speak a word of a well-memorised and rehearsed speech before an audience. These kinds of fears are termed as phobias. People who have phobias have irrational fears related to specific objects, people, or situations. Phobias often develop gradually or begin with a generalised anxiety disorder. Phobias can be grouped into three main types, i.e. specific phobias, social phobias and agoraphobia.

Specific phobias:
Specific phobias are the most commonly occurring type of phobia. This group includes irrational fears such as intense fear of a certain type of animal, or of being in an enclosed space. Intense and incapacitating fear and embarrassment when dealing with others characterises social phobias.

Agoraphobia:
Agoraphobia is the term used when people develop a fear of entering unfamiliar situations. Many agoraphobics are afraid of leaving their homes. So their ability to carry out normal life activities is severely limited. Have you ever noticed someone washing their hands every time they touch something, or washing even things like coins, or stepping only within the patterns on the floor or road while walking.

People affected by the obsessive-compulsive disorder are unable to control their preoccupation with specific ideas or are unable to prevent themselves from repeatedly carrying out a particular act or series of acts that affect their ability to carry out normal activities.

Obsessive behaviour:
Obsessive behaviour is the inability to stop thinking about a particular idea or topic. The person involved/often finds these thoughts to be unpleasant and shameful.

Compulsive behaviour:
Compulsive behaviour is the need to perform certain behaviours over and over again. Many compulsions deal with counting, ordering, checking, touching and washing. Very often people who have been caught in a natural disaster (such as a tsunami) or have been victims of bomb blasts by terrorists, or been in a serious accident or in a war-related situation, experience posttraumatic stress disorder (PTSD). PTSD symptoms vary widely but may include recurrent dreams, flashbacks, impaired concentration and emotional numbing.

Somatoform Disorders:
These are conditions in which there are physical symptoms in the absence of physical disease. In somatoform disorders, the individual has psychological difficulties and complains of physical symptoms, for which there is no biological cause. Somatoform disorders include pain disorders, somatisation disorders, conversion disorders, and hypochondriasis.

Pain disorders:
Pain disorders involve reports of extreme and incapacitating pain, either without any identifiable biological symptoms or greatly in excess of what might be expected to accompany biological symptoms. How people interpret pain influences their overall adjustment. Some pain sufferers can learn to use active coping, i.e. remaining active and ignoring the pain. Others engage in passive coping, which leads to reduced activity and social withdrawal.

Patients with somatisation disorders have multiple recurrent or chronic bodily complaints. These complaints are likely to be presented in a dramatic and exaggerated way. Common complaints are headaches, fatigue, heart palpitations, fainting spells, vomiting, and allergies. Patients with this disorder believe that they are sick, provide long and detailed histories of their illness and take large quantities of medicine.

The symptoms of conversion disorders are the reported loss of part or all of some basic body functions. Paralysis, blindness, deafness and difficulty in walking are generally among the symptoms reported. These symptoms often occur after a stressful experience and may be quite sudden.

Hypochondriasis:
Hypochondriasis is diagnosed if a person has a persistent belief that s/he has a serious illness, despite medical reassurance, lack of physical findings, and failure to develop the disease. Hypochondriacs have an obsessive preoccupation and concern with the condition of their bodily organs, and they continually worry about, their health.

Question 4.
Write the major anxiety disorders.
Answer:
Generalised Anxiety Disorder:
prolonged, vague, unexplained and intense fears that have no object, accompanied by hypervigilance and motor tension.

Panic Disorder:
frequent anxiety attacks characterised by feelings of intense terror arid dread; unpredictable ‘panic attacks’ along with physiological symptoms like breathlessness, palpitations, trembling, dizziness, and a sense of losing control or even dying.

Phobias :
irrational fears related to specific objects, interactions with others, and unfamiliar situations.

Obsessive-compulsive Disorder :
being preoccupied with certain thoughts that are viewed by the person to be embarrassing or shameful, and being unable to check the impulse to repeatedly carry out certain acts like checking, washing, counting, etc.

Post-traumatic Stress Disorder (PTSD) :
recurrent dreams, flashbacks, impaired concentration and emotional numbing followed by a traumatic or stressful event like a natural disaster, serious accident, etc.

Question 5.
What is dissociative disorders?
Answer:
Dissociative Disorders: Dissociation can be viewed as a severance of the connections between ideas and emotions. Dissociation involves feelings of unreality, estrangement, depersonalisation, and sometimes a loss or shift of identity. Sudden temporary alterations of consciousness that blot out painful experiences are a defining characteristic of dissociative disorders.

Four conditions are included in this group: dissociative amnesia, dissociative fugue, dissociative identity disorder, and depersonalisation. Salient features of somatoform and dissociative disorders are given.

Salient Features of Somatoform and Dissociative Disorders
Dissociative Disorders

Dissociative amnesia:
The person is unable to recall important, personal information often related to a stressful and traumatic report. The extent of forgetting is beyond normal.

Dissociative fugue:
The person suffers from a rare disorder that combines amnesia with travelling away from a stressful environment.

Dissociative identity (multiple personalities) :
The person exhibits two or more separate and contrasting personalities associated with a history of physical abuse.

Somatoform Disorders
Hypochondriasis:
A person interprets insignificant symptoms as signs of a serious illness despite repeated medical evaluations that point to no pathology disease.

Somatisation :
A person exhibits vague and recurring physical/bodily symptoms such as pain, acidity, etc., without any organic cause.

Conversion :
The person suffers from a loss or impairment of motor or sensory function (e.g., paralysis, blindness, etc.) that has no physical cause but may be a response to stress and psychological problems.

Dissociative amnesia:
Dissociative amnesia is characterised by extensive but selective memory loss that has no known organic cause (e.g., head injury). Some people cannot remember anything about their past. Others can no longer recall specific events, people, places, Or objects, while their memory for other events remains intact. This disorder is often associated with overwhelming stress.

Dissociative fugue:
Dissociative fugue has, as its essential feature, an unexpected travel away from home and the workplace, the assumption of a new identity, and the inability to recall the previous identity. The fugue usually ends when the person suddenly ‘wakes up’ with no memory of the events that occurred during the fugue.

Dissociative identity disorder:
Dissociative identity disorder often referred to as multiple personalities, is the most dramatic of the dissociative disorders. It is often associated with traumatic experiences in childhood. In this disorder, the person assumes alternate personalities that may or may not be aware of each other.

Depersonalisation:
Depersonalisation involves a dreamlike state in which the person has a sense of being separated both from self and from reality. In depersonalisation, there is a change of self-perception, and the person’s sense of reality is temporarily lost or changed.

Question 6.
What is mood disorders?
Answer:
Mood disorders are characterised by disturbances in mood or prolonged emotional state. The most common mood disorder is depression, which covers a variety of negative moods and behavioural changes. Depression can refer to a symptom Oi a disorder. In day-to-day life, we often use the term depression to refer to normal feelings after a significant loss, such as the break-up of a relationship, or the failure to attain a significant goal. The main types of mood disorders include depressive, manic dead bipolar disorders.

Major depressive disorder:
Major depressive disorder is defined as a period of depressed mood and/or loss of interest or pleasure in most activities, together with other symptoms which may include a change in body weight, constant sleep problems, tiredness, inability to think clearly, agitation, greatly slowed behaviour and thoughts of death and suicide. Other symptoms include excessive guilt or feelings of worthlessness.

Factors Predisposing towards Depression :
Genetic makeup or heredity is an important risk factor for major depression and bipolar disorders. Age is also a risk factor. For instance, women are particularly at risk during young adulthood, while for men the risk is highest in early middle age. Similarly, gender also plays a great role in this differential risk addition. For example, women in comparison to men are more likely to report a depressive disorder.

Other risk factors are experiencing negative life events and a lack of social support. Another less common mood disorder is mania. People suffering from mania become euphoric (‘high’), extremely active, excessively talkative, and easily distractible. Manic episodes rarely appear by themselves; they usually alternate with depression. Such a mood disorder, in which both mania and depression are alternately present, is sometimes interrupted by periods of normal mood.

This is known as a bipolar mood disorder. Bipolar mood disorders were earlier referred to as manic-depressive disorders. Among the mood disorders, the lifetime risk of a suicide attempt is highest in case of bipolar mood disorders. Several risk factors in addition to the mental health status of a person predict the likelihood of suicide. These include age, gender, ethnicity, or race and recent occurrence of serious life events. Teenagers and young adults are as much at high risk for suicide, as those who are over 70 years.

Gender is also an influencing factor, i.e. men have a higher rate of contemplated suicide than women. Other factors that affect suicide rates are cultural attitudes toward suicide. In Japan, for instance, suicide is the culturally appropriate way to deal with feelings of shame and disgrace. Negative expectations, hopelessness, setting unrealistically high standards and being over-critical in self-evaluation are important themes for those who have suicidal, preoccupations.

Suicide can be prevented by being alert to some of the symptoms which include :

• changes in eating and sleeping habits
• withdrawal from friends, family and regular activities
• violent actions, rebellious behaviour, running away
• drug and alcohol abuse.
• marked personality change
• persistent boredom
• difficulty in concentration.
• complaints about physical symptoms, and
• loss of interest in pleasurable activities.
  However, seeking timely help from a professional counsellor/psychologist can help to prevent the likelihood of suicide.

Question 7.
What is Schizophrenic Disorders and state its symptoms?
Answer:
Schizophrenia is the descriptive term for a group of psychotic disorders in which personal, social and occupational functioning deteriorate as a result of disturbed thought processes, strange perceptions, unusual emotional states, and motor abnormalities. It is a debilitating disorder. The social and psychological costs of schizophrenia are tremendous, both to patients as well as to their families and society.

Symptoms of Schizophrenia:
The symptoms of schizophrenia can be grouped into three categories, viz. positive symptoms (i.e. excesses of thought, emotion and behaviour), negative symptoms (i.e. deficits of thought, emotion, and behaviour) and psychomotor symptoms.

Positive symptoms:
Positive symptoms are ‘pathological excesses’ or ‘bizarre addition?’ to a person’s behaviour. Delusions, disorganised thinking and speech, heightened perception and hallucinations, and inappropriate effects are the ones most often found in schizophrenia. Many people with schizophrenia develop delusions. A delusion is a false belief that is firmly held on inadequate grounds. It is not affected by rational argument and has no basis in reality.

Delusions of persecution:
Delusions of persecution are the most common in schizophrenia. People with this delusion believe that they are being plotted against, spied on, slandered, threatened, attacked Or deliberately victimised. People with schizophrenia may also experience delusions of reference in which they attach special and personal meaning to the actions of others or to objects and events.

Delusions of grandeur:
In delusions of grandeur, people believe themselves to be specially empowered persons and in delusions of control, they believe that their feelings, thoughts and actions are controlled by others. People with schizophrenia may not be able to think logically and may speak in peculiar ways. These formal thought disorders can make communication extremely difficult.

These include rapidly shifting from one topic to another so that the normal structure of thinking is muddled and becomes illogical (loosening of associations, derailment), inventing new words or phrases (neologisms), and persistent aid inappropriate repetition of the same thoughts (perseveration). Schizophrenics may have hallucinations, i. e. perceptions that occur in the absence of external stimuli.

Auditory hallucinations:
Auditory hallucinations are most common in schizophrenia. Patients hear sounds or voices that speak words, phrases and sentences directly to the patient (second-person hallucination) or talk to one another referring/to the patient as s/he (third-person hallucination). Hallucinations can also involve the other senses.

These include tactile hallucinations (i.e. forms of tingling, burning), somatic hallucinations (i.e. something happening inside the body such as a snake crawling inside one’s stomach), visual hallucinations (i.e. vague perceptions of colour or distinct visions of people or objects), gustatory hallucinations (i.e. food or drink taste strange), and olfactory hallucinations (i.e. smell of poison or smoke). People with schizophrenia also show inappropriate effects, i.e’. emotions that are unsuited to the situation.

Negative symptoms:
Negative symptoms are ‘pathological deficits’ and include poverty of speech, blunted and flat affect, loss of volition, and social withdrawal. People with schizophrenia show alogia or poverty of speech, i.e. a reduction in speech and speech content, felony people with schizophrenia show less anger, sadness, joy, and other feelings than most people do. Thus they have blunted effect Some show no emotions at all, a condition is known as flat affect. Also, patients with schizophrenia experience avolition or apathy and an inability to start or complete a course of action.

People with this disorder may withdraw socially and become totally focused on their own ideas and fantasies. People with schizophrenia also show psychomotor Symptoms. They move less spontaneously or make odd grimaces and gestures. These symptoms may take extreme forms known as catatonia. People in a catatonic stupor remain motionless and silent for long stretches of time. Some show catatonic rigidity, i.e. maintaining a rigid, upright posture for hours. Others exhibit catatonic posturing, i.e. assuming awkward, bizarre positions for long periods.

Question 8.
Write the: Sub-types of Schizophrenia.
Answer:
According to DSM-IV-TR, the sub-types of schizophrenia and their characteristics are:

• Paranoid type :
  Preoccupation with delusions or auditory hallucinations; no disorganised speech or behaviour or inappropriate affect.
• Disorganised type:
  Disorganised speech and behaviour; inappropriate or flat affect; no catatonic symptoms.
• Catatonic type :
  Extreme motor immobility; excessive motor inactivity; extreme negativism (i.e. resistance to instructions) or mutism (i.e. refusing to speak).
• Undifferentiated type :
  Does not fit any of the sub-types but meets symptom criteria.
• Residua] type:
  Has experienced at least one episode of schizophrenia; no positive symptoms but shows negative symptoms.

Question 9.
What is Behavioural and Developmental Disorders?
Answer:
There are certain disorders that are specific to children and if neglected can lead to serious consequences later in life. Children have less self-understanding and they have not yet developed a stable sense of identity nor do they have an adequate frame of reference regarding reality, possibility and value. As a result, they are unable to cope with stressful events which might be reflected in behavioural and emotional problems.

On the other hand, although their inexperience and lack of self-sufficiency make them easily upset by problems that seem minor to an adult, children typically bounce back more quickly. We will now discuss several disorders of childhood like Attention-deficit Hyperactivity Disorder (ADHD), Conduct Disorder, and Separation Anxiety Disorder. These disorders, if not attended to, can lead to more serious and chronic disorders as the child moves into adulthood.

Classification of children’s disorders has followed a different path than that of adult disorders. Achenbach has identified two factors, i.e. extermination and internalisation, which include the majority of childhood behaviour problems. The externalising disorders, or under-controlled problems, include behaviours that are disruptive and often aggressive and aversive to others in the child’s environment.

Internalising disorders, or overcontrolled problems, are those conditions where the child experiences depression, anxiety, and discomfort that may not be evident to others. There are several disorders in which children display disruptive or externalising behaviours. We will now focus on three prominent disorders, viz. Attention-deficit Hyperactivity Disorder (ADHD), Oppositional Defiant Disorder (ODD), and Conduct Disorder.

The two main features of (ADHD) are inattention and hyperactivity-impulsivity. Children who are inattentive find it difficult to sustain mental effort during work or play. They have a hard time keeping their minds on any one thing or in following instructions. Common complaints are that the child does not listen, cannot concentrate, does not follow instructions, is disorganised, easily distracted, forgetful, does not finish assignments and is quick, to lose interest in boring activities.

Children who are impulsive seem unable to control their immediate reactions or to think before they act. They find it difficult to wait or take turns and have difficulty resisting immediate temptations or delaying gratification. Minor mishaps such as knocking things over are common whereas more serious accidents and injuries can also occur. Hyperactivity also takes many forms. Children with (ADHD) are in constant motion. Sitting still through a lesson is impossible for them.

The child may fidget, squirm, climb and run around the room aimlessly. Parents and teachers describe them as ‘driven by a motor’, always on the go, and talking incessantly. Boys are four times more likely to be given this diagnosis than girls. Children with Oppositional Defiant Disorder (ODD) display age-inappropriate amounts of stubbornness, are irritable, defiant, disobedient, and behave in a hostile manner. Unlike ADHD, the rates of ODD in boys and girls are not very different.

The terms Conduct Disorder and Antisocial Behaviour refer to age-inappropriate actions and attitudes that violate family expectations, societal norms, and the personal or property rights of others. The behaviours typical of conduct disorder include aggressive actions that cause or threaten harm to people or animals, non-aggressive conduct that causes property damage, major deceitfulness or theft, and serious rule violations.

Children show many different types of aggressive behaviour, such as verbal aggression (i.e. name-calling, swearing), physical aggression (i.e. hitting, fighting), hostile aggression (i.e. directed at inflicting injury to others) and proactive aggression (i.e. dominating and bullying others without provocation). Internalising disorders include Separation Anxiety Disorder (SAD) and Depression. Separation anxiety disorder is an internalising disorder unique to children.

Its most prominent symptom is excessive anxiety or even panic experienced by children at being separated from their parents. Children with SAD may have difficulty being in a room by themselves, going to school alone, are fearful of entering hew situations, and cling to and shadow their parents’ every move. To avoid separation, children with SAD may fuss, scream, throw severe tantrums, or make suicidal gestures.

The ways in which children express and experience depression are related to their level of physical, emotional, and cognitive development. An infant may show sadness by being passive and unresponsive; a pre¬schooler may appear withdrawn and inhibited; a school-age child may be argumentative and combative, and a teenager may express feelings of guilt and hopelessness. Children may also have more serious disorders called Pervasive Developmental Disorders.

These disorders are characterised by severe and widespread impairments in social interaction and communication skills, and stereotyped patterns of behaviours, interests and activities. Autistic disorder or autism is one of the most common of these disorders. Children with autistic disorder have marked difficulties in social interaction and communication a restricted range of interests, and a strong desire for routine.

About 70 per cent of children with autism are also mentally retarded. Children with autism experience profound difficulties in relating to other people. They are unable to initiate social behaviour and seem unresponsive to other people’s feelings. They are unable to share experiences or emotions with others. They also show serious abnormalities in communication and language that persist over time.

Many autistic children never develop speech and those who do, have repetitive and deviant speech patterns. Children with autism often show narrow patterns of interest and repetitive behaviours such as lining up objects or stereotyped body movements such as rocking. These motor movements may be self-stimulatory such as hand flapping or self-injurious such as banging their head against the wall.

Question 10.
What is Substance-use Disorders?
Answer:
Addictive behaviour, whether it involves excessive intake of high-calorie food resulting in extreme obesity or involving the abuse of substances such as alcohol or cocaine, is one of the most severe problems being faced by society today. Disorders relating to maladaptive behaviours resulting from regular and consistent use of the substance involved are called substance abuse disorders.

These disorders include problems associated with using and abusing Such drugs as alcohol, cocaine and heroin, which alter the way people think, feel and behave. There are two sub-groups of substance-use disorders, i.e. those related to substance dependence and those related to substance abuse.

Insubstance dependence:
In substance dependence, there is an intense craving for the substance to which the person is addicted, and the person shows tolerance, withdrawal symptoms and compulsive drug-taking. Tolerance means that the person has to use more and more of a substance to get the same effect. Withdrawal refers to physical symptoms that occur when a person stops or cuts down on the use of a psychoactive substance, i.e. a substance that has the ability to change an individual’s consciousness, mood and thinking processes.

Insubstance abuse:
In substance abuse, there are recurrent and significant adverse consequences related to the use of substances. People who regularly ingest drugs damage their family and social relationships, perform poorly at work and create physical hazards. We will now focus on the three most common forms of substance abuse, viz. alcohol abuse and dependence, heroin abuse and dependence and cocaine abuse and dependence.

Alcohol Abuse and Dependence People who abuse alcohol drink large amounts regularly and rely on it to help Heroin Abuse and Dependence Heroin intake significantly interferes with social and occupational functioning. Most abusers further develop a dependence on heroin, revolving their lives around the substance, building up a tolerance for it and experiencing a withdrawal reaction when they stop taking it.

The most direct and stopping it results in feelings of depression, fatigue, sleep problems, irritability and anxiety. Cocaine poses serious dangers. It has dangerous effects on psychological functioning and physical well-being.

Question 11.
Describe the nature and scope of psychotherapy. Highlight the importance of therapeutic relationships in psychotherapy.
Answer:
Nature and scope of psychotherapy: Psychotherapy is a voluntary relationship between the one seeking treatment or the client and the one who treats or the therapist. The purpose of the relationship is to help the client to solve the psychological problems faces by her or him. The relationship is conducive for building the trust of the client so that problems may be freely discussed.

Psychotherapies aim at changing maladaptive behaviours, decreasing the sense of personal distress and helping the client to adapt better to her/his environment. The inadequate marital, occupational and social adjustment also requires that major changes be made in an individual’s personal environment.

AH, psychotherapies aim at a few or all of the following goals :

• Reinforcing the client’s resolve for betterment.
• Lessening emotional pressure.
• Unfolding the potential for positive growth.
• Modifying habits,
• Changing thinking patterns.
• Increasing self-awareness.
• Improving interpersonal relations and communication.
• Facilitating decision-making.
• Becoming aware of one’s choices in life.

Relating to one’s social environment in a more creative and self-aware manner. The special relationship between the client and the therapist is known as the
therapeutic relationship or alliance.

There are two major components of a therapeutic alliance:

• The first component is the contractual nature of the relationship in which two willing individuals, the client and the therapist, enter into a partnership that aims at helping the client overcome her/his problems.
• The second component of the therapeutic alliance is the limited duration of the therapy. This alliance lasts until the client becomes able to deal with her/his problems and take control of her/his life.

This relationship has several unique properties. It is a trusting and confiding relationship. The high level of trust enables the client to unburden herself/himself to the therapist and confide her/his psychological and personal problems to the latter. The therapist encourages this by being accepting, empathic, genuine, and warm to the client.

The therapist conveys by her/his words and behaviours that she is not judging the client and will continue to show the same positive feelings towards the client even if the client is rude or confides in all the wrong things that she may have done or thought about. The therapeutic alliance also requires that the therapist must keep strict confidentiality of the experiences, events, feelings, or thoughts disclosed by the client. The therapist must not exploit the trust and confidence of the Client in any way.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Long Answer Questions Part-1.

CHSE Odisha 12th Class Psychology Unit 4 Long Answer Questions Part-1

Long Questions With Answers

Question 1.
Identify the symptoms associated with depression and mania.
Answer:
Symptoms associated with depression change in body weight, constant sleep problems, tiredness, inability to think clearly, agitation, greatly slowed behaviour, and thoughts of death and suicide. Other symptoms include excessive guilt or feelings of worthlessness. Symptoms associated with mania are people become euphoric (‘high’), extremely active, excessively talkative and easily distractible.

Question 2.
Describe the characteristics of hyperactive children.
Answer:
Hyperactive children are suffering from Attention-deficit Hyperactivity Disorder (ADHD) which can lead to more serious and chronic disorders as the child moves into adulthood if not attended. Children display disruptive or externalising behaviours. The two main features of ADHD are inattention and hyperactivity-impulsivity. Children who are inattentive find it difficult to sustain mental effort during work or play.

They have a hard time keeping their minds on any one thing or following instructions. Common complaints are that the child does not listen, cannot concentrate, does not follow instructions, is disorganised, easily distracted, forgetful, does not finish assignments and is quick to lose interest in boring activities. Children who are impulsive seem unable to control their immediate reactions or to think before they act. They find it difficult to wait or take turns, and have difficulty resisting immediate temptations or delaying gratification.

Minor mishaps such as knocking things over are common whereas more serious accidents and injuries can also occur. Hyperactivity also takes many forms. Children with ADHD are in constant motion. Sitting still through a lesson is impossible for them. The child may fidget, squirm, climb and run around the room aimlessly. Parents and teachers describe them as ‘driven by a motor’, always on the go and talk incessantly. Boys are four times more likely to be given this diagnosis than girls.

Question 3.
What do you understand by substance abuse and dependence?
Answer:
In substance abuse, there are recurrent and significant adverse consequences related to the use of substances. People who regularly ingest drugs damage their family and social relationships, perform poorly at work, and create physical hazards. In substance dependence, there is an intense craving for the substance to which the person is addicted, and the person shows tolerance, withdrawal symptoms and compulsive drug-taking.

Tolerance means that the person has to use more and more of a substance to get the same effect. Withdrawal refers to physical symptoms that occur when a person stops or cuts down bn the use of a psychoactive substance, i.e. a substance that has the ability to change an individual’s consciousness, mood and thinking processes.

Question 4.
Can a distorted body image lead to eating disorders? Classify the various forms of it.
Answer:
Yes, distorted body image can lead to eating disorders. The various forms of eating disorders are anorexia nervosa, bulimia nervosa, and binge eating.
Anorexia nervosa:
In this eating disorder, the individual has a distorted body image that leads her/him to see herself/himself as overweight. Often refusing to’ eat, exercising compulsively and developing unusual habits such as refusing to eat in front of others, the anorexic may lose large amounts of weight and even starve herself/himself to death.

Bulimia nervosa:
In this disorder, the individual may eat excessive amounts of food, then purge her/his body of food by using medicines such as laxatives or diuretics or by vomiting. The person often feels disgusted and ashamed when s/he binges and is relieved of tension and negative emotions after purging.

Binge eating:
In this disorder, there are frequent episodes of out-of-control eating.

Question 5.
“Physicians make diagnosis looking at a person’s physical symptoms”. How are psychological disorders diagnosed?
Answer:
Psychological disorders can be diagnosed by observations, interviews, counselling etc. In ancient days, abnormal behaviour can be explained by the operation of supernatural and magical forces such as evil spirits (bhoot-pret) or the devil (shaitan). In many Societies, the shaman, or medicine man (Ojha) is a person who is believed to have contact with supernatural forces and is the medium through which spirits communicate with human beings.

Through the shaman, an afflicted person can learn which spirits are responsible for her/his problems and what needs to be done to appease them. A recurring theme in the history of abnormal psychology is the belief that individuals behave strangely because their bodies and their brains are not working properly. This is the biological or organic approach. In the modem era, there is evidence that body and brain processes have been linked to many types of maladaptive behaviour. For certain types of disorders, correcting these defective biological processes results in improved functioning. Another approach is the psychological approach.

According to this point of view, psychological problems are caused by inadequacies in the way an individual thinks, feels, or perceives the world. The American Psychiatric Association (APA) has published an official manual describing and classifying various kinds of psychological disorders. The current version of it, the Diagnostic and Statistical Manual of Mental Disorders, TV Edition (DSM-IV), evaluates the patient on five axes or dimensions rather than just one broad aspect of ‘mental disorder’.

These dimensions relate to biological, psychological, social and other aspects. The classification scheme officially used in India and elsewhere is the tenth revision of the International Classification of Diseases (ICD-10), which is known as the ICD-10 Classification of Behavioural and Mental Disorders. It was prepared by the World Health Organisation (WHO). For each disorder, a description of the main clinical features or symptoms, and of other associated features including diagnostic guidelines is provided in this scheme.

Question 6.
Distinguish between obsessions and compulsions.
Answer:
Obsessions is the inability to stop thinking about a particular idea or topic. The person involved, often finds these thoughts to be unpleasant and shameful while Compulsions is the need to perform certain behaviours Over and over again. Many compulsions deal with counting, ordering, checking, touching and washing.

Question 7.
Can a long-standing pattern of deviant behaviour be considered abnormal? Elaborate.
Answer:
The first approach views abnormal behaviour as a deviation from social norms. Many psychologists have stated that ‘abnormal’ is simply a label that is given to a behaviour which is deviant from social expectations. Abnormal behaviour, thoughts and emotions are those that differ markedly from a society’s ideas of proper functioning. Each society has norms, which are stated or unstated rules for proper conduct.

Behaviours, thoughts and emotions that break societal norms are called abnormal. A society’s norms grow from its particular cultural history, values, institutions, habits, skills, technology and arts. Thus, a society whose culture values competition and assertiveness may accept aggressive behaviour, whereas one that emphasises cooperation and family values (such as in India) may consider aggressive behaviour as unacceptable Or even abnormal.

A society’s values may change over time, causing its views of what is psychologically abnormal to change as well. Serious questions have been raised about this definition. It is based on the assumption that socially accepted behaviour is not abnormal, and that normality is nothing more than conformity to social norms. The second approach views abnormal behaviour as maladaptive.

Many psychologists believe that the best criterion for determining the normality of behaviour is not whether society accepts it but whether it fosters the well-being of the individual and eventually of the group to which s/he belongs. Well-being is not simply maintenance and survival but also includes growth and fulfilment, i.e. the actualisation of potential, which you must have studied in Maslow’s need hierarchy theory.

According to this criterion, conforming behaviour can be seen as abnormal if it is maladaptive, i.e. if it interferes with optimal functioning and growth. For example, a student in the class prefers to remain silent even when s/he has questions in her/his mind. Describing behaviour as maladaptive implies that a problem exists; it also suggests that vulnerability in the individual, inability to cope, or exceptional stress in the environment have led to problems in life.

Question 8.
While speaking in public the patient changes topics frequently, is this a positive or a negative symptom of schizophrenia? Describe the other symptoms and sub-types of schizophrenia.
Answer:
Positive symptoms:
These are ‘pathological excesses’ or ‘bizarre additions’ to a person’s behaviour. Delusions, disorganised thinking and speech, heightened perception and hallucinations, and inappropriate effects are the ones most often found in schizophrenia.

Negative symptoms:
These are ‘pathological deficits’ and include poverty of speech, blunted and flat affect, loss of volition, and social withdrawal. People with schizophrenia show alogia or poverty of speech, i.e. a reduction in speech and speech content. Many people with schizophrenia show less anger, sadness, joy and other feelings than most people do. Thus they have blunted effect.

Some show no emotions at all, a condition is known as flat affect. Also patients with schizophrenia experience avolition, apathy and an inability to start or complete a course of action. People with this disorder may withdraw socially and become totally focused on their own ideas and fantasies.

Sub-types of Schizophrenia: According to DSM-IV-TR, the sub-types of schizophrenia and their characteristics are:

Paranoid type:
Preoccupation with delusions or auditory hallucinations; no disorganised speech or behaviour or inappropriate affect.

Disorganised type:
Disorganised speech and behaviour; inappropriate or flat affect; no catatonic symptoms.

Catatonic type:
Extreme motor immobility; excessive motor inactivity; extreme negativism (i.e. resistance to instructions) or mutism (i.e. refusing to speak).

Undifferentiated type:
Does not fit any of the sub-types but meets symptom criteria.

Residual type:
Has experienced at least one episode of schizophrenia; no positive symptoms but shows negative symptoms.

Question 9.
What do you understand by the term ‘dissociation’? Discuss its various forms.
Answer:
Dissociation can be viewed as a severance of the connections between ideas and emotions. Dissociation involves feelings of unreality, estrangement, depersonalisation, and sometimes a loss or shift of identity. Sudden temporary alterations of consciousness that blot out painful experiences are a defining characteristic of dissociative disorders. Four conditions are included in this group: dissociative amnesia, dissociative fugue, dissociative identity disorder, and depersonalisation.

Various forms of dissociation are as follows:

Dissociative amnesia:
It is characterised by extensive but selective memory loss that has no known organic cause (e.g. head injury). Some people cannot remember anything about their past. Others can no longer recall specific events, people, places, or objects, while their memory for other events remains intact. This disorder is often associated with overwhelming stress.

Dissociative fugue:
It has, as its essential feature, an unexpected travel away from home and workplace, the assumption of a new identity, and the inability to recall the previous identity. The fugue usually ends when the person suddenly ‘wakes up’ with no memory of the events that occurred during the fugue.

Dissociative identity disorder:
It is often referred to as multiple personalities, is the most dramatic of the dissociative disorders. It is often associated with traumatic experiences in childhood. In this disorder, the person assumes alternate personalities that may or may not be aware of each other.

Depersonalisation:
It involves a dreamlike state in which the person His a sense of being separated both from self and from reality. In depersonalisation, there is a change of self-perception, and the person’s sense of reality is temporarily lost or changed.

Question 10.
What are phobias? If someone had an intense fear of snakes, could this simple phobia be a result of faulty learning? Analyse how this phobia could have developed.
Answer:
Phobias are irrational fears related to specific objects, interactions with others, and unfamiliar situations. If someone had an intense fear of snakes, this simple phobia cannot be a result of faulty learning. It is a. specific phobia which is most common. This group includes irrational fears such as intense fear of a certain type of animal, or of being in an enclosed space. This phobia often develops gradually or begins with generalised anxiety disorders.

Question 11.
Anxiety has been called the “butterflies in the stomach feeling”. At what stage does anxiety become a disorder? Discuss its types.
Answer:
Everyone has worries and fears. The term anxiety is usually defined as a diffuse, vague, very unpleasant feeling of fear and apprehension. The anxious individual also shows combinations of the following symptoms: rapid heart rate, shortness of breath, diarrhoea, loss of appetite, fainting, dizziness, sweating, sleeplessness, frequent urination and tremors.

Different types of anxiety disorders and their symptoms are as follows:

Generalised anxiety disorder:
This disorder consists of prolonged, vague, unexplained and intense fears that are not attached to any particular object. The symptoms include worry and apprehensive feelings about the future; hypervigilance, which involves constantly scanning the environment for dangers.

Panic disorder:
This disorder consists of recurrent anxiety attacks in which the person experiences intense terror. The clinical symptoms include shortness of breath, dizziness, trembling, palpitations, choking, nausea, chest pain or discomfort, fear of going crazy, losing control or dying.

Obsessive-compulsive disorder:
People are unable to control their preoccupation with specific ideas or are unable to prevent themselves from repeatedly carrying out a particular actor series of acts that affect their ability to carry out normal activities. Obsessive behaviour is the inability to stop thinking about a particular idea or topic. The person involved, often finds these thoughts to be unpleasant and shameful. Compulsive behaviour is the need to perform certain behaviours over and over again. Many compulsions deal with counting, ordering, checking, touching and washing.

Phobias:
These are irrational fears related to specific objects, interactions with others, and unfamiliar situations.

Question 12.
What is the concept of abnormality and psychological disorders?
Answer:
Although many definitions of abnormality have been used over the years, none has won universal acceptance. Still, most definitions have certain common features, often called the ‘four Ds’: deviance, distress, dysfunction and danger. That is, psychological disorders are deviant (different, extreme, unusual, even bizarre), distressing (unpleasant and upsetting to the person and to others), dysfunctional (interfering with the person’s ability to carry out daily activities in a constructive You must have come across people who are unhappy, troubled and dissatisfied.

Their minds and hearts are filled with Sorrow, unrest and tension and they feel that they are unable to move ahead in their lives; they feel life is a painful, uphill struggle, sometimes, not worth living. Famous analytical psychologist Carl Jung has quite remarkably said, “How can I be substantial without casting a shadow? I must have a dark side, too, if I am to be whole and by becoming conscious of my shadow, I remember once more that I am a human being like any other”.

At times, some of you may have felt nervous before an important examination, tense and concerned about your future career or anxious when someone close to you was unwell. All of us face major problems at some point in our lives. However, some people have an extreme reaction to the problems and stresses of life. In this chapter, we will try to understand what goes wrong when people develop psychological problems, what are the causes and factors which lead to abnormal behaviour, and the various signs and symptoms associated with different types of psychological disorders.

The study of psychological disorders has intrigued and mystified all cultures for more than 2,500 years. Psychological disorders or mental disorders (as they are commonly referred to), like anything unusual, may make us uncomfortable and even a little frightened. Unhappiness, discomfort, anxiety and unrealised potential are seen all over the world. These failures in living are due mainly to failures in adaptation to life challenges.

As you must have studied in the previous chapters, adaptation refers to the person’s ability to modify her/his behaviour in response to changing environmental requirements. When the behaviour cannot be modified according to the needs of the situation, it is said to be maladaptive. Abnormal Psychology is the area within psychology that is focused on maladaptive behaviour – its causes, consequences, and treatment way), and possibly dangerous (to the person or to others).

This definition is a useful starting point from which we can explore psychological abnormality. Since the word ‘abnormal’ literally means “away from the normal”, it implies deviation from some clearly defined norms or standards. In psychology, we have no ‘ideal model’ or even ‘normal model’ of human behaviour to use as a base for comparison. Various approaches have been used in distinguishing between normal and abnormal behaviours.

From these approaches, there emerge two basic and conflicting views: The first approach views abnormal behaviour as a deviation from social norms. Many psychologists have stated that ‘abnormal’ is simply a label that is given to a behaviour which is deviant from social expectations. Abnormal behaviour, thoughts and emotions are those that differ markedly from a society’s ideas of proper functioning. Each society has norms, which are stated or unstated rules for proper conduct.

Behaviours, thoughts and emotions that break societal norms are called abnormal. A society’s norms grow from its particular culture — its history, values, institutions, habits, skills, technology and arts. Thus, a society whose culture values competition and assertiveness may accept aggressive behaviour, whereas one that emphasises cooperation and family values (such as in India) may consider aggressive behaviour as unacceptable or even abnormal.

A society’s values may change over time, causing its views of what is psychologically abnormal to change as well. Serious questions have been raised about this definition. It is based on the assumption that socially accepted behaviour is not abnormal and that normality is nothing more than conformity to social norms. The second approach views abnormal behaviour as maladaptive.

Many psychologists believe that the best criterion for determining the normality of behaviour is not whether society accepts it but whether it fosters the well-being of the individual and eventually of the group to Which s/he belongs. Well-being is not simply maintenance and survival but also includes growth and fulfilment, i.e. the actualisation of potential, which you must have studied in Maslow’s need hierarchy theory.

According to this criterion, conforming behaviour can be seen as abnormal if it is maladaptive, i.e. if it interferes with optimal functioning and growth. For example, a student in the class prefers to remain silent even when s/he has questions in her/his mind. Describing behaviour as maladaptive implies that a problem exists; it also suggests that vulnerability in the individual, inability to cope, or exceptional stress in the environment have led to problems in life.

If you talk to people around, you will see that they have vague ideas about psychological disorders that are characterised by superstition, ignorance and fear. Again it is commonly believed that psychological disorder is something to be ashamed of. the stigma attached to mental illness means that people are hesitant to consult a doctor or psychologist because they are ashamed of their problems. Actually, a psychological disorder which indicates a failure in adaptation should be viewed as any other illness.

Question 13.
Write the historical view of psychological disorders.
Answer:
Historical Background :
To understand psychological disorders, we would require a brief historical account of how these disorders have been viewed over the ages. When we study the history of abnormal psychology, we find that certain theories have occurred over and over again. One ancient theory that is still encountered today holds that abnormal behaviour can be explained by the operation of supernatural and magical forces such as evil spirits (bhoot-pret), or the devil (shaitan).

Exorcism, i.e. removing the evil that resides in the individual through countermagic and prayer, is still commonly used. In many societies, the shaman, or medicine man (Ojha) is a person who is believed to have contact with supernatural forces and is the medium through which spirits communicate with human beings. Through the shaman, an afflicted person can learn which spirits are responsible for her/his problems and what needs to be done to appease them.

A recurring theme in the history of abnormal psychology is the belief that individuals behave strangely because their bodies and their brains are not working properly. This is the biological or organic approach. In the modem era, there is evidence that body and brain processes have been linked to many types of maladaptive behaviour. For certain types of disorders, correcting these defective biological processes results in improved functioning.

Another approach is the psychological approach. According to this point of view, psychological problems are caused by inadequacies in the way an individual thinks, feels, or perceives the world. All three of these perspectives—supernatural, biological or organic, and psychological — have recurred throughout the history of Western civilisation.

In the ancient Western world, it was philosopher physicians of ancient Greece such as Hippocrates, Socrates, and in particular Plato who developed the organismic approach and viewed disturbed behaviour as arising out of conflicts between emotion and reason. Galen elaborated on the role of the four senses of humour in personal character and temperament. According to him, the material world was made up of four elements, viz. earth, air, fire and water which combined to form four essential body fluids, viz. blood, black bile, yellow bile and phlegm.

Each of these fluids was seen to be responsible for a different temperament. Imbalances among the humour were believed to cause various disorders. This is similar to the Indian notion of the three doshas of Vata, Pitta, and Kapha which were mentioned in the Atharva Veda and Ayurvedic texts. You have already read about it in Chapter 2. In the Middle Ages, demonology and superstition gained renewed importance in the explanation of abnormal behaviour.

Demonology related to a belief that people with mental problems were evil and there are numerous instances of ‘witch-hunts’ during this period. During the early Middle Ages, the Christian spirit of charity prevailed and St. Augustine wrote extensively about feelings, mental anguish, and conflict. This laid the groundwork for modem psychodynamic theories of abnormal behaviour. The Renaissance Period was marked by increased humanism and curiosity about behaviour.

Johann Weyer emphasized psychological conflict and disturbed interpersonal relationships as causes of psychological disorders. He also insisted that ‘ witches’ were mentally disturbed and required medical, not theological; treatment. The seventeenth and eighteenth centuries were known as the Age of Reason and Enlightenment, as the scientific method replaced faith and dogma as ways of understanding abnormal behaviour.

The growth of a scientific attitude towards psychological disorders in the eighteenth century contributed to the Reform Movement and to increased compassion for people who suffered from these disorders. Reforms of asylums were initiated in both Europe and America. One aspect of the reform movement was the new inclination for deinstitutionalization which placed emphasis on providing community Care for recovered mentally ill individuals.

In recent years, there has been a convergence of these approaches, which has resulted in an interactional, or biopsychosocial approach. From this perspective, all three factors, i.e. biological, psychological and social play important roles in influencing the expression and outcome of psychological disorders.

Odisha State Board CHSE Odisha Class 12 Psychology Solutions Unit 4 Objective & Short Answer Type Questions.

CHSE Odisha Class 12 Psychology Unit 4 Objective & Short Answer Type Questions

Multiple Choice Questions With Answers

Question 1.
Abnormality is also caused the _____.
(a) four’Es’
(b) four ‘Fs’
(c) four ‘Ds”
(d) none of the above
Answer:
(c) four ‘Ds”

Question 2.
The four Ds are:
(a) deviance
(b) distress
(c) dysfunctions and danger
(d) all the above
Answer:
(d) all the above

Question 3.
Approaches of abnormal behavior:
(a) deviation from social norms
(b) maladaptive
(c) only (b)
(d) both (a) and (b)
Answer:
(d) both (a) and (b)

Question 4.
When an electrical impulse reaches a neuron’s ending, the nerve ending is stimulated to release a chemical that is called ______.
(a) transmitter
(b) neuro
(c) neurotransmitter
(d) none of the above
Answer:
(c) neurotransmitter

Question 5.
Anxiety disorders have been linked to low activity of the neurotransmitter that aid called _____.
(a) GABA
(b) GBAA
(c) GABB
(d) GAAB
Answer:
(a) GABA

Question 6.
_____ is the excess activity of dopamine.
(a) depression
(b) anxiety disorder
(c) abnormality
(d) schizophrenia
Answer:
(d) schizophrenia

Question 7.
Depression to low activity of _____.
(a) dopamine
(b) serotonin
(c) genetic
(d) none of the above
Answer:
(b) serotonin

Question 8.
Genetic factors have been linked to ______.
(a) mood disorders
(b) schizophrenia
(c) mental retardation
(d) all the above
Answer:
(d) all the above

Question 9.
_____ is the oldest and most famous of the modern psychological models.
(a) psychodynamic
(b) humanistic
(c) cognitive
(d) behavioral
Answer:
(a) psychodynamic

Question 10.
Who stated that abnormal behavior is a symbolic expression of unconscious mental conflicts that can be traced to early childhood or infancy.
(a) Freud
(b) Teekman
(c) Kolo
(d) none of them
Answer:
(a) Freud

Question 11.
______ term is usually defined as a diffuse, vague, very unpleasant feeling of fear and apprehension.
(a) anxiety
(b) psychological
(c) mental retaradation
(d) none of the above
Answer:
(a) anxiety

Question 12.
Phobias can be grouped into three types. They are:
(a) specific phobias
(b) social phobias
(c) agoraphobia
(d) all the above
Answer:
(d) all the above

Question 13.
The most commonly occurring types of phobia is called _____.
(a) social
(b) specific
(c) agoraphobia
(d) only (a)
Answer:
(b) specific

Question 14.
_____ is the term used when people develop a fear of entering unfamiliar situations.
(a) social phobias
(b) agoraphobia
(c) specific phobias
(d) none of the above
Answer:
(b) agoraphobia

Question 15.
In which disorders people are unable to control their preoccupation with specific ideas or are unable to prevent.
(a) obsessive-compulsive
(b) anxiety
(c) schizophrenia
(d) none of the above
Answer:
(a) obsessive-compulsive

Question 16.
The symptoms of conversion disorders :
(a) paralysis
(b) blindness
(c) deafness
(d) all the above
Answer:
(a) paralysis

Question 17.
The symptoms of post-tramumative stress disorder:
(a) recurrent dreams
(b) flashbacks
(c) impaired concentration
(d) all the above
Answer:
(d) all the above

Question 18.
_____ can be viewed as a severance of connection between ideas and emotions,
(a) anxiety disorder
(b) post-traumatic stress disorder
(c) dissociative disorder
(d) none of the above
Answer:
(c) dissociative disorder

Question 19.
Four conditions are included in group of dissociative orders. They are :
(a) dissociative amnesia
(b) dissociative fugue
(c) dissociative identity disorder
(d) all the above
Answer:
(d) all the above

Question 20.
The person is unable to recall important, personal information often related to a stressful and traumatic report that is called ______.
(a) somato form disorder
(b) dissociative fugue
(c) dissociative amnesia
(d) none of the above
Answer:
(c) dissociative amnesia

Question 21.
The most common mood disorder is _____.
(a) depression
(b) abnormality
(c) anxiety disorder
(d) none of the above.
Answer:
(a) depression

Question 22.
_______ can refers to a symptom area disorder.
(a) depression
(b) anxiety
(c) schizophrenia
(d) none of the above
Answer:
(a) depression

Question 23.
Types of mood disorder:
(a) depressive
(b) manic
(c) bipolar disorder
(d) all the above
Answer:
(d) all the above

Question 24.
The symptoms of mood disorders are :
(a) sleep problems
(b) trideness
(c) inability to think clearly
(d) all the above
Answer:
(d) all the above

Question 25.
Mania is called ______ disorder.
(a) schizophrenia
(b) mood disorders
(c) only (a)
(d) none of the above
Answer:
(b) mood disorders

Question 26.
The symptoms of schizophrenia are divided in 3 categories. They are:
(a) positive
(b) negative
(c) psychomotor
(d) all the above
Answer:
(d) all the above

Question 27.
The positive symptoms of schizophrenia :
(a) excessness of thought
(b) emotion
(c) behavior
(d) all the above
Answer:
(d) all the above

Question 28.
Schizophrenia develop ______.
(a) illusion
(b) hallucination
(c) delusions
(d) only (a)
Answer:
(c) delusions

Question 29.
The two main features of ADHD (Attention Deficit Hyperactivity Disorder) are:
(a) inattention
(b) hyperactivity impulsivity
(c) only (a)
(d) both (a) and (b)
Answer:
(d) both (a) and (b)

Question 30.
Psychotherapies are classified in 3 broad groups. These are:
(a) psychodynamic
(b) behavior
(c) existential
(d) all the above
Answer:
(d) all the above

True/False Questions

Question 1.
Behaviour is a not group of psychotherapies
Answer:
False

Question 2.
Psychodynamic therapy is pioneered by Sigmund Freud.
Answer:
True

Question 3.
Psychoanalysis treatment are 4 stages.
Answer:
False

Question 4.
Positive reinforcement is given to increase the deficit.
Answer:
True

Question 5.
Freud formulated the RET (Rational Emotive Therapy).
Ans.
False

Question 6.
Existential therapy is called logotherapy.
Answer:
True

Question 7.
Client-centered therapy was given by Carl Rogers.
Answer:
True

Question 8.
Agoraphobia is the term used when people developed a fear of entering unfamiliar situations.
Answer:
True

Question 9.
Specific phobias are the most owned type of phobia.
Answer:
True

Question 10.
Biological factors influence all aspects of our behavior.
Answer:
True

Question 11.
Abnormality is also called the four P’s
Answer:
False

Question 12.
Anxiety disorders have not been linked to GABA aid.
Answer:
False

Question 13.
Depression is the excess activity of dopamine.
Answer:
False

Question 14.
Genetic factors have been linked with mood disorders only.
Answer:
False

Question 15.
Psychodynamics is the oldest and most famous modern psychological model.
Answer:
True

Question 16
Paralysis is not a symptom of conversion disorders.
Answer:
False

Question 17.
Flashbacks are not symptoms of post-traumatic stress disorder.
Answer:
False

Question 18.
Dissociative can be viewed as a severance of connection between ideas and emotions.
Answer:
True

Question 19.
Dissociative fugue are conditions group of dissociative orders.
Answer:
True

Question 20.
The most common mood disorder is depression.
Answer:
True

Very Short-Answer Type Questions

Question 1.
What is a neurotransmitter?
Answer:
Synapse separates one neuron from the next and the message must move across that space. When an electrical impulse reaches a neuron’s ending, the nerve ending is stimulated to release a chemical, called a neurotransmitter.

Question 2.
What is Anxiety Disorder?
Answer:
One day while driving home, Deb felt his heart beating rapidly, he started sweating profusely and even felt short of breath. He was so scared that he stopped the car and stepped out. In the next few months, these attacks increased and now he was hesitant to drive for fear of being caught in traffic during an attack. Deb started feeling that he had gone crazy and would die. Soon he remained indoors and refused to move out of the house.

Question 3.
What is hypochondriasis?
Answer:
Hypochondriasis is diagnosed if a person has a persistent belief that s/he has a serious illness, despite medical reassurance, lack of physical findings and failure to develop the disease. Hypochondriacs have an obsessive preoccupation and concern with the condition of their bodily organs, and they continually worn7 about their health.

Question 4.
Define two features of dissociative order.
Answer:
Dissociative amnesia :
The person is unable to recall important, personal information often related to a stressful and traumatic report. The extent of forgetting is beyond normal.

Dissociative fugue:
The person suffers from a rare disorder that combines amnesia with traveling away from a stressful environment. The person exhibits two or more separate, and contrasting personalities associated.

Question 5.
What is dissociative fugue?
Answer:
Dissociative fugue has, as its essential feature, an unexpected travel away from home and workplace, the assumption of a new identity and the inability to recall the previous identity. The fugue usually ends when the person suddenly ‘wakes up’ with no memory of the events that occurred during the fugue.

Question 6.
What is mood disorders?
Answer:
Mood disorders are characterized by disturbances in mood or prolonged emotional state. The most common mood disorder is depression, which covers a variety of negative moods and behavioral changes. Depression can refer to a symptom or a disorder. In day-to-day life, we often use the term depression to refer to normal feelings after a significant loss, such as the break-up of a relationship, or the failure to attain a significant goal.

Question 7.
Symptoms of Schizophrenia.
Answer:
The symptoms of schizophrenia can be grouped into three categories, viz. positive symptoms (i.e. excesses of thought, emotion, and behavior), negative symptoms, (i.e. deficits of thought, emotion, and behavior), and psychomotor symptoms. Positive symptoms are ‘pathological excesses’ or ‘bizarre additions’ to a person’s behavior.

Question 8.
What is PDD?
Answer:
The ways in which children express and experience depression are related to their level of physical, emotional and cognitive development. An infant may show sadness by being passive and unresponsive; a pre-schooler may appear withdrawn and inhibited; a school-age child may be argumentative and combative, and a teenager may express guilt and hopelessness. Children may also have more serious disorders called Pervasive Developmental Disorders.

Question 9.
What is a Therapeutic Relationship?
Answer:
The special relationship between the client and the therapist is known as the therapeutic relationship or alliance. It is neither a passing acquaintance nor a permanent and lasting relationship. There are two major components of a therapeutic alliance.

Question 10.
What is transference neurosis?
Answer:
The therapist encourages this process because it helps her/him in understanding the unconscious conflicts of the client. The client acts out her/his frustrations, anger, fear, and depression that s/he harbored towards that person in the past, but could not express at that time. The therapist becomes a substitute for that person in the present. This stage is called transference neurosis.

Question 11.
Logotherapy.
Answer:
Victor Frankl, a psychiatrist and neurologist propounded the Logotherapy. Logos is the Greek word for soul and Logotherapy means treatment for the soul. Frankl calls this process of finding meaning even in life-threatening circumstances as the process of meaning-making. The basis of meaning-making is a person’s quest for finding the spiritual truth of one’s existence. Just as there is an unconscious, which is the repository of instincts (see Chapter 2), there is a spiritual unconscious, which is the storehouse of love, aesthetic awareness, and values of life.

Question 12.
What is Gestalt Therapy?
Answer:
The German word gestalt means ‘whole’. This therapy was given by Frederick (Fritz) Peris together with his wife Laura Peris. The goal of gestalt therapy is to increase an individual’s self-awareness and self-acceptance. The client is taught to recognize the bodily processes and the emotions that are being blocked out from awareness. The therapist does this by encouraging the client to act out fantasies about feelings and conflicts. This therapy can also be used in group settings.

Question 13.
Ethics in Psychotherapy
Answer:
Some of the ethical standards that need to be practiced by professional psychotherapists are:

•  Informed consent needs to be taken.
• The confidentiality of the client should be maintained.
• Alleviating personal distress and suffering should be the goal of all attempts of the therapist.
• The integrity of the practitioner-client relationship is important.
• Respect for human rights and dignity.
• Professional competence and skills are essential.

Question 14.
What is CBT?
Answer:
CBT is a short and efficacious treatment for a wide range of psychological disorders such as anxiety, depression, panic attacks, borderline personality, etc. CBT adopts a biopsychosocial approach to the delineation of psychopathology. It combines cognitive therapy with behavioral techniques.

Question 15.
Dysfunctional cognitive structure?
Answer:
Negative thoughts are persistent irrational thoughts such as “nobody loves me”, “I am ugly”, “I am stupid”, “I will not succeed”, etc. Such negative automatic thoughts are characterized by cognitive distortions. Cognitive distortions are ways of thinking which are general in nature but which distort reality in a negative manner. These patterns of thought are called dysfunctional cognitive structures.

Short Questions With Answers 

Question 1.
What is Cognitive Behaviour Therapy (CBT)?
Answer:
Research into the outcome and effectiveness of psychotherapy has conclusively established CBT to be a short and efficacious treatment for a wide range of psychological disorders such as anxiety, depression, panic attacks, borderline personality, etc. CBT adopts a biopsychosocial approach to the delineation of psychopathology. It combines cognitive therapy with behavioral techniques.

The rationale is that the client’s distress has its origins in the biological, psychological, and social realms. Hence,c addressing the physical aspects through relaxation procedures, the psychological ones through behavior therapy and cognitive therapy techniques, and the social ones with environmental manipulations makes CBT a comprehensive technique that is easy to use, applicable to a variety of disorders, and has proven efficacy.

Question 2.
What is Humanistic-existential Therapy?
Answer:
The humanistic-existential therapies postulate that psychological distress arises from feelings of loneliness, alienation, and an inability to find meaning and genuine fulfillment in life. Human beings are motivated by the desire for personal growth and self-actualization, and an innate need to grow emotionally. When these needs are curbed by society and family, human beings experience psychological distress.

Self-actualization is defined as an innate or inborn force that moves the person to become more complex, balanced, and integrated, i.e. achieving complexity and balance without being fragmented. Integrated means a sense of the whole, being a complete person, being, in essence, the same person in spite of the variety of experiences that one is subjected to. Just as lack of food or water causes distress, the frustration of self-actualization also causes distress.

Question 3.
What is Existential Therapy?
Answer:
Victor Frankl, a psychiatrist, and neurologist propounded the Logotherapy. Logos is the Greek word for soul and Logotherapy means treatment for the soul. Frankl calls this process of finding meaning even in life-threatening circumstances as the process of meaning-making. The basis of meaning-making is a person’s quest for finding the Spiritual truth of one’s existence.

Just as there is an unconscious, which is the repository of instincts (see Chapter 2), there is a spiritual unconscious, which is the storehouse of love, aesthetic awareness, and values of life. Neurotic anxieties arise when the problems of life are attached to the physical, psychological or spiritual aspects of one’s existence. Frankl emphasized the role of spiritual anxieties in leading to meaninglessness and hence it may be called an
existential anxiety.

Question 4.
What is Client-centred Therapy?
Answer:
Client-centered therapy was given by Carl Rogers. Rogers combined scientific rigor with the individualized practice of client-centered psychotherapy. Rogers brought into psychotherapy the concept of self, with freedom and choice as the core of one’s being. The therapy provides a warm relationship in which the client can reconnect with her/his disintegrated feelings.

The therapist shows empathy, i.e. understanding the client’s experience as if it were her/his own, is warm, and has unconditional positive regard, i.e. total acceptance of the client as s/he is. Empathy sets up an emotional resonance between the therapist and the client. Unconditional positive regard indicates that the positive Warmth of the therapist is not dependent on what the client reveals or does in the therapy sessions. This unique unconditional warmth ensures that the client feels secure and can trust the therapist.

Question 5.
What is Biomedical Therapy?
Answer:
Medicines may be prescribed to treat psychological disorders. Prescription of medicines for the treatment of mental disorders is done by qualified medical professionals known as psychiatrists. They are medical doctors who have specialized in the understanding, diagnosis, and treatment of mental disorders. The nature of medicines used depends on the nature of the disorders.

Severe mental disorders such as schizophrenia or bipolar disorder require antipsychotic drugs. Common mental disorders such as generalized anxiety or reactive depression may also require milder drugs. The medicines prescribed to treat mental disorders can cause side effects that need to be understood and monitored. Hence, it is essential that medication is given under proper medical supervision.

Question 6.
Three factors Contributing to Healing in Psychotherapy!
Answer:
As we have read, psychotherapy is a treatment of psychological distress. There are several factors that contribute to the healing process. Some of these factors are as follows:

• A major factor in healing is the techniques adopted by the therapist and the implementation of the same with the patient/client. If the behavioral system and the CBT school are adopted to heal an anxious client, the relaxation procedures and the cognitive restructuring largely contribute to the healing.
• The therapeutic alliance, which is formed between the therapist and the patient/ client, has healing properties, because of the regular availability and the therapist, and the warmth and empathy provided by the therapist.
• At the outset of therapy, while the patient/client is being interviewed^ in the initial sessions to understand the nature of the problem, s/he unburdens the emotional problems being faced.

Question 7.
Rehabilitation of the mentally ill.
Answer:
The treatment of psychological disorders has two components, i.e. reduction of symptoms and improving the level of functioning or quality of life. In the case of milder disorders such as generalized anxiety, reactive depression, or phobia, reduction of symptoms is associated with an improvement in the quality of life. However, in the case of severe mental disorders such as schizophrenia, reduction of symptoms may not be associated with an improvement in the quality of life.

Many patients suffer from negative symptoms such as disinterest and lack of motivation to do work or to interact with people. Rehabilitation is required to help such patients become self-sufficient. The aim of rehabilitation is to empower the patient to become a productive member of society to the extent possible.

Question 8.
What are Alternative Therapies available for treatment?
Answer:
Alternative therapies are so-called because they are alternative treatment possibilities to conventional drug treatment or psychotherapy. There are many alternative therapies such as yoga, meditation, acupuncture, herbal remedies, and so on. In the past 25 years, yoga and meditation have gained popularity as treatment programs for psychological distress.

Yoga is an ancient Indian technique detailed in the Ashtanga Yoga of Patanjali’s Yoga Sutras. Yoga as it is commonly called today either refers to only the asanas or body posture component or to breathing practices Or pranayama, or to a combination of the two. Meditation refers to the practice of focusing attention on the breath or on an object or thought or mantra.

Question 9.
What is Cognitive Therapy?
Answer:
Cognitive therapies locate the cause of psychological distress in irrational thoughts and beliefs. Albert Ellis formulated Rational Emotive Therapy (RET). The central thesis of this therapy is that irrational beliefs mediate between antecedent events and their consequence. The first step in RET is the antecedent belief- consequence {ABC) analysis. Antecedent events, which caused psychological distress, are noted.

The client is also interviewed to find the irrational beliefs, which are distorting the present reality. Irrational beliefs may not be supported by empirical evidence in the environment. These beliefs are characterized by thoughts with ‘musts’ and ‘shoulds’, i.e. things ‘must’ and ‘should’ be in a particular manner.

Question 10.
Behavioral Techniques
Answer:
A range of techniques is available for changing behavior. The principles of these techniques are to reduce the arousal level of the client, alter behavior through classical conditioning or operant conditioning with different contingencies of info: elements, as well as to use vicarious learning procedures, if necessary. Negative reinforcement and aversive conditioning are the two major techniques of behavior modification.

As you have already studied in Class XI, Negative reinforcement refers to following an undesired response with an outcome that is painful or not liked. For example, the teacher reprimands a child who shouts in class. This is negative reinforcement. Aversive conditioning refers to the repeated association of an undesired response with an aversive consequence.

Odisha State Board Elements of Mathematics Class 11 CHSE Odisha Solutions Chapter 4 Trigonometric Functions Ex 4(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 4 Trigonometric Functions Exercise 4(b)

Question 1.
In the following questions, write ‘T’ for true and ‘F’ for false statements.
(i) If tan x + tan y = 5 and tan x, tan y = 1/2  then cot (x + y) = 10
Solution:
False

(ii) √3 (1 + tan 15°) = 1 – tan 15°
Solution:
False

(iii) If θ lies in 3rd quadrant, then cos \(\frac{\theta}{2}\) + sin \(\frac{\theta}{2}\) is positive.
Solution:
True

(iv) 2 sin 105°. sin 15° = 1/2.
Solution:
True

(v) If cos A = cos B = 1 then tan\(\frac{A+B}{2}\). tan\(\frac{A+B}{2}\) = 1
Solution:
False

(vi) cos 15° cos\(7 \frac{1}{2}^{\circ}\). sin \(7 \frac{1}{2}^{\circ}\) = 1
Solution:
False

(vii) sin 20° (3 – 4 cos² 70°) = \(\frac{\sqrt{3}}{2}\)
Solution:
True

(viii) √3 (3 tan 10° – tan³ 10°) = 1 – 3tan² 10°
Solution:
True

(ix) \(\frac{2 \tan 7 \frac{1^{\circ}}{2}\left(1-\tan ^2 7 \frac{1^{\circ}}{2}\right)}{\left(1+\tan ^2 7 \frac{1^{\circ}}{2}\right)^2}\) = 1
Solution:
False

(x) The minimum value of sin θ. cos θ is (-1)².
Solution:
False

Question 2.
In the following questions, fill in the gaps with correct answers choice from the brackets.
(i) If α and β lie in 1st and 2nd quadrants respectively, and if sin α = 1/2, sin β = 1/3, then sin (α + β) = _______. \(\left(\frac{1}{2 \sqrt{3}}+\frac{\sqrt{2}}{3}, \frac{1}{2 \sqrt{3}}-\frac{\sqrt{2}}{3}, \frac{-1}{2 \sqrt{3}}+\frac{\sqrt{2}}{3}\right)\)
Solution:
\(\frac{1}{2 \sqrt{3}}+\frac{\sqrt{2}}{3}\)

(ii) if tan α = 1/2, tan β = 1/3, then α + β = ______      \(\left(\frac{\pi}{2}, \frac{\pi}{4}, \frac{\pi}{3}\right)\)
Solution:
\(\frac{\pi}{4}\)

(iii) The value of \(\frac{\cos 15^{\circ}+\sin 15^{\circ}}{\cos 15^{\circ}-\sin 15^{\circ}}\) = ______ \(\left(\frac{\sqrt{3}}{2}, \sqrt{3}, \frac{1}{\sqrt{3}}\right)\)
Solution:
√3

(iv) if \(\frac{1+\sin A}{\cos A}\) = √2 + 1, then the value of \(\frac{1-\sin A}{\cos A}\) is_________, \(\left(\frac{1}{\sqrt{2}-1}, \sqrt{2}-1, \sqrt{2}+1\right)\)
Solution:
√2 – 1

(v) sin 105°. cos 105° = \(\left(\frac{1}{2},-\frac{1}{4},-\frac{1}{2}\right)\)
Solution:
– 1/4

(vi) 2 sin\(67 \frac{1}{2}^{\circ}\) cos\(22 \frac{1}{2}^{\circ}\) = ___ \(\left(1-\frac{1}{\sqrt{3}}, 1+\frac{1}{\sqrt{2}},-1+\frac{1}{\sqrt{2}}\right)\)
Solution:
1 + \(\frac{1}{\sqrt{2}}\)

(vii) sin 35° + cos 5° =____ (2 cos 25°, √3 cos 25°, √3 sin 25°)
Solution:
√3 cos 25°

(viii) sin² 24° – sin² 26° =_____ \(\left(\frac{\sqrt{5}+1}{8}, \frac{\sqrt{5}-1}{8}, \frac{\sqrt{5}-1}{4}\right)\)
Solution:
\(\frac{\sqrt{5}-1}{8}\)

(ix) sin 70° (4 cos² 20° – 3) =_____ \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}, \sqrt{3}\right)\)
Solution:
1/2

(x) cos 3θ + sin 3θ is maximum if θ =_____ (60°, 15°, 45°)
Solution:
15°

(xi) sin 15° – cos 15° = _____ (1/2, 0, positive, negative)
Solution:
Negative

(xii) If θ lies in the third quadrant and tan θ = 2, then the value of sin θ is ____. \(\left(\frac{1}{\sqrt{5}},-\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}, \frac{-2}{\sqrt{5}}\right)\)
Solution:
\(\frac{-2}{\sqrt{5}}\)

(xiii) The correct expression is. (sin 1° > sin 1, sin 1° < sin 1, sin 1° = sin 1, sin 1° = \(\frac{\pi}{180^{\circ}}\) sin 1)
Solution:
sin 1° < sin 1

(xiv) The correct expression is —. (tan 1 > tan 2, tan 1 < tan 2, tan 1 =  1/2 tan 2, tan 1 < 0)
Solution:
tan 1 > tan 2

Question 3.
Prove the following
(i) sin A. sin (B – C) + sin B sin (C – A) + sin C. sin(A – B) = 0
Solution:
L. H. S
= sin A sin (B – C) + sin B sin (C – A) + sin C sin (A – B)
= sin A (sin B cos C – cos B sin C) + sin B (sin C cos A – cos C sin A) + sin C (sin A cos B – cos A sin B)
= sin A sin B cos C – sin A cos B sin C + cos A sin B sin C – sin A sin B cos C + sin A cos B sin C – cos A sin B sin C
= 0 = R. H. S

(ii) cos A. sin (B – C) + sin B sin (C – A) + cos C. sin(A – B) = 0
Solution:
L.H.S.
= cos A sin (B – C) + cos B sin (C – A) + cos C sin (A – B)
= cos A (sin B cos C – cos B sin C) + cos B (sin C cos A – cos C sin A) + cos C (sin A cos B – cos A sin B)
= cos A sin B cos C – cos A cos B sin C + cos A cos B sin C- sin A cos B cos C + sin A cos B cos C – cos A sin B cos C = 0 = R. H. S.

(iii) \(\frac{\sin (B-C)}{\sin B \cdot \sin C}\) + \(\frac{\sin (C-A)}{\sin C \cdot \sin A}\) + \(\frac{\sin (A-B)}{\sin A \cdot \sin B}\) = 0
Solution:
L. H. S.

(iv) tan² A – tan² B = \(\frac{\sin (\mathbf{A}+\mathbf{B}) \cdot \sin (\mathbf{A}-\mathbf{B})}{\cos ^2 \mathbf{A} \cdot \cos ^2 \mathbf{B}}\)
Solution:

= tan² A sec² B – tan² B sec² A
= tan² A (1 + tan² B) – tan² B (1 + tan² A)
= tan² A + tan² A tan² B – tan² B tan²A tan² B
= tan² A – tan² B = L. H. S

Question 4.
Prove the following :
Solution:
(i) tan 75° + cot 75° = 4
Solution:
L.H.S = tan 75° + cot 75° = tan 75° + \(\frac{1}{\tan 75^{\circ}}\)

(ii) sin² 18° + cos² 36° = 3/4
Solution:

(iii) sin 18°. cos 36° = 1/4
Solution:
sin 18° cos 36°
\(=\left(\frac{\sqrt{5}-1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right)=\frac{5-1}{16}=\frac{1}{4}\)

(iv) sin 15° = \(\frac{\sqrt{3}-1}{2 \sqrt{2}}\)
Solution:
sin 15° = sin (45° – 30°)
= sin 45° cos 30° – cos 45° sin 30°
\(=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \cdot \frac{1}{2}=\frac{\sqrt{3}-1}{2 \sqrt{2}}\)

(v) cot \(\frac{\pi}{8}\) – tan \(\frac{\pi}{8}\) = 2
Solution:

(vi) \(\frac{\cos 9^{\circ}+\sin 9^{\circ}}{\cos 9^{\circ}-\sin 9^{\circ}}\) = tan 54°
Solution:

(vii) tan 10° + tan 35° + tan 10°. tan 35 = 1
Solution:

Question 5.
Prove the following:
(i) cot 2A = \(\frac{\cot ^2 A-1}{2 \cot A}\)
Solution:

(ii) \(\frac{\sin B}{\sin A}=\frac{\sin (2 A+B)}{\sin A}\) – 2cos(A + B)
Solution:

= \(\frac{\sin B}{\sin A}\) = L.H.S

(iii) \(\frac{\sin 2 A+\sin 2 B}{\sin 2 A-\sin 2 B}=\frac{\tan (A+B)}{\tan (A-B)}\)
Solution:

(iv) \(\frac{\cot A-\tan A}{\cot A+\tan A}\) = cos2A
Solution:

(v) \(\frac{\sin 2 A+\sin 5 A-\sin A}{\cos 2 A+\cos 5 A+\cos A}\) = tan 2A
Solution:

(vi) cot A – tan A = 2 cot 2A
Solution:

(vii) cot A – cosec 2A = cot 2A
Solution:

(viii) \(\frac{\cos A-\sin A}{\cos A+\sin A}\) = sec 2A – tan 2A
Solution:

(ix) tan θ (1 + sec 2θ) = tan 2θ
Solution:

(x) \(\frac{\sin A+\sin B}{\sin A-\sin B}\) = tan \(\frac{A+B}{2}\) . cot \(\frac{A-B}{2}\)
Solution:

(xi) sin 50° –  sin 70° + sin 10° = 0
Solution:
L. H. S. = sin 50° – sin 70° + sin 10°
= sin (60° – 10°) – sin (60° + 10°) + sin 10°
= – 2 cos 60° sin 10° + sin 10°
= – 2 × 1/2 sin 10° + sin 10°
= – sin 10° + sin 10° = 0 = R. H. S.

(xii) cos 80° + cos 40° – cos 20° = 0
Solution:
L. H. S. = cos 80° + cos 40° – cos 20°
= cos(60° + 20°) + cos (60° – 20°) – cos 20°
= 2 cos 60° cos 20° – cos 20° = 0

(xiii) 8 sin 10°. sin 50°. sin 70° = 1
Solution:
L. H. S. = 8 sin 10° sin 50° sin 70°
= 8 sin 10° sin (60° – 10°) sin (60° + 10°)
= 8 sin 10° (sin² 60° – sin² 10°)
= 8 sin 10°(3/4 – sin² 10°)
= 6 sin 10° – 8 sin³ 10°)
= 2 (3 sin 10° – 4 sin³ 10°)
= 2 sin (3 × 10°)
= 2 sin 30° = 2 × 1/2 = 1 = R.H.S

(xiv) 4 sin A sin (60° – A) sin (60° + A) – sin 3A = 0
Solution:
L. H. S. = 4 sin A sin (60° – A)
sin (60° + A) – sin 3A
= 4 sin A (sin² 60°- sin² A) – sin 3A
= 4 sin A. 3/4 – 4 sin³ A – sin 3A
= (3 sin A – 4 sin³A) – sin 3A
= sin 3A – sin 3A = 0

(xv) tan 3A – tan 2A – tan A = tan 3A tan 2A tan A
Solution:
We have tan 3A = tan (2A + A)
or, tan 3A = \(\frac{\tan 2 A+\tan A}{1-\tan 2 A \tan A}\)
or, tan 3A (1 – tan 2A tan A) = tan 2A + tan A
or, tan 3A – tan 3A tan 2A tan A = tan 2A + tan A
or, tan 3A – tan 2A – tan A = tan 3A tan 2A tan A (Proved)

Question 6.
Prove the following
(i) tan\(\frac{A}{2}\) = \(\sqrt{\frac{1-\cos A}{1+\cos A}}\)
Solution:

(ii) \(\sqrt{\frac{1+\sin A}{1-\sin A}}\) = tan\(\left(\frac{\pi}{4}+\frac{A}{2}\right)\)
Solution:

(iii) \(\frac{1+\tan \frac{A}{2}}{1-\tan \frac{A}{2}}\) = sec A + tan A
Solution:

(iv) sec θ + tan θ = tan\(\left(\frac{\pi}{4}+\frac{θ}{2}\right)\)
Solution:

(v) cot\(\frac{A}{2}\) = \(\frac{\sin A}{1-\cos A}\)
Solution:
R.H.S = \(\frac{\sin A}{1-\cos A}=\frac{2 \sin \frac{A}{2} \cos \frac{A}{2}}{2 \sin ^2 \frac{A}{2}}\)
= cot\(\frac{A}{2}\) R.H.S

Question 7.
Find the maximum value of the following.
(i) 5 sin x + 12 cos x
Solution:
5 sin x + 12 cos x
Let 5 = r cos θ, 12 = r sin θ 52
∴ 5² = r² cos² θ, 12² = r² sin² θ
∴ 5² + 12² = r² (cos² θ + sin² θ) = r²
∴ r = \(\sqrt{25+144}\) = 13
∴ The maximum value of 5 sin x + 12 cos x is 13.

(ii) 24 sin x – 7 cos x
Solution:
The maximum value of 24 sin x – 7 cos x is
\(\sqrt{(24)^2+(-7)^2}\)
= \(\sqrt{576+49}=\sqrt{625}\) = 25

(iii) 2 + 3 sin x + 4 cos x
Solution:
The maximum value of 3 sin x + 4 cos x is
\(\sqrt{3^2+4^2}\) = 5
∴ Maximum value of 2 + 3 sin x + 4 cos x is 2 + 5 = 7

(iv) 8 cos x – 15 sin x – 2
Solution:
Maximum value of 8 cos x- 15 sin x is \(\sqrt{(8)^2+(-15)^2}\)
= \(\sqrt{64+225}=\sqrt{289}\) = 17
∴ Maximum value of 8 cos x- 15 sin x – 2 is 17 – 2 = 15

Question 8.
Answer the following:
(i) If tan A = \(\frac{13}{27}\), tan B = \(\frac{7}{20}\) and A, B are acute, show that A + B = 45°.
Solution:

(ii) If tan θ = \(\frac{b}{a}\), find the value of a cos 2θ + b sin 2θ.
Solution:

(iii) If sec A – tan A = \(\frac{1}{2}\) and 0< A < 90° then show that sec A = \(\frac{5}{4}\)
Solution:
If sec A – tan A = \(\frac{1}{2}\),       ….(1)
0< A < 90°
⇒ sec A – tan A = \(\frac{\sec ^2 A-\tan ^2 A}{2}\)
= \(\frac{(\sec \mathrm{A}+\tan \mathrm{A})(\sec \mathrm{A}-\tan \mathrm{A})}{2}\)
or, sec A + tan A = 2    ……(2)
Now adding eqn. (1) and (2), We have
2 sec A = \(\frac{1}{2}\) + 2 = \(\frac{5}{2}\)
or, sec A = \(\frac{5}{4}\)

(iv) If sin θ + sin Φ = a and cos θ + cos Φ = b the show that tan \(\frac{1}{2}\) (θ + Φ) \(\frac{a}{b}\)
Solution:

(v) If tan θ = \(\frac{a \sin x+b \sin y}{a \cos x+b \cos y}\) then show that a sin (θ – x) + b sin (θ – y) = 0
Solution:
tan θ = \(\frac{a \sin x+b \sin y}{a \cos x+b \cos y}\)
or, \(\frac{\sin \theta}{\cos \theta}=\frac{a \sin x+b \sin y}{a \cos x+b \cos y}\)
or, a sin θ cos x + b sin θ cos y = a cos θ sin x + b cos θ sin y
or, a (sin θ cos x – cos θ sin x) + b (sin θ cos y – cos θ sin y) = 0
or, a sin (θ – x) + b sin (θ – y) = 0

(vi) If A + C = B, show that tan A. tan B. tan C = tan B – tan A – tan C.
Solution:
A + C = B
or, tan (A + C) = tan B
or, \(\frac{\tan A+\tan C}{1-\tan A \tan C}\)
or, tan A + tan C = tan B – tan A tan B tan C
or, tan A tan B tan C = tan B – tan A – tan C

(vii) If tan A = \(\frac{1}{5}\), tan B = \(\frac{2}{3}\) then show that cos 2A = sin 2B.
Solution:
tan A = \(\frac{1}{5}\), tan B = \(\frac{2}{3}\)
∴ cos 2A = \(\frac{1-\tan ^2 \mathrm{~A}}{1+\tan ^2 \mathrm{~A}}\)

(viii) If cos 2A = tan² B, then show that cos 2B = tan² A In Δ ABC, prove that.
Solution:

(ix) tan\(\frac{B+C}{2}\) = cot\(\frac{A}{2}\)
Solution:

(x) cos (A + B) + sin C = sin (A + B) – cos C
If A + B + C = π and cos A = cos B. cos C show that (x_i  and x_ii)
Solution:
We have A + B = π – C or, cos (A + B)
or, cos (A + B)
= cos (π – C) = – cos C
and sin (A + B) = sin (π – C) = sin C
∴ cos (A + B) + sin C
= – cos C + sin (A + B)
= sin (A + B) – cos C.

(xi) tan B + tan C = tan A
Solution:
[∴ A + B + C = π ⇒ B + C = π – A
⇒ sin (B + C) = sin (π – A) = sin A]
L. H. S. = tan B + tan C

(xii) 2 cot B. cot C = 1
Solution:
We have A + B + C = π
⇒ B + C = π – A
⇒ cos (B + C) = cos (π – A) = – cosA
⇒ cos B cos C – sin B sin C = – cos B cos C
(∵ cos B cos C = cos A)
⇒ 2 cos B cos C = sin B sin C
⇒ \(\frac{2 \cos B \cos C}{\sin B \sin C}\) = 1
⇒ 2 cot B cot C = 1

Question 9.
Prove the following:
(i) cos (A – D) sin (B – C) + cos (B – D) sin (C – A) + cos (C- D) sin (A – B) = 0
Solution:
L. H. S. = cos (A – D) sin (B – C) + cos (B – D) sin (C – A) + cos (C – D) sin (A- B)
= \(\frac{1}{2}\) [2 cos (A- D) sin (B – C) + 2 cos (B – D) sin (C – A) + 2 cos (C – D) sin (A – B)]
= \(\frac{1}{2}\) [ sin (A- D + B – C) – sin (A – D – B + C) + sin (B – D + C – A)- sin (B- D- C + A) + sin ( C – D + A – B)- sin (C – D- A + B)]
= \(\frac{1}{2}\) × 0 = 0 = R.H.S

(ii) sin 2A + sin 2B + sin 2 (A – B) = 4 sin A. cos B. cos (A – B)
Solution:
L. H. S. = sin 2A + sin 2B + sin 2 (A – B)
= 2 sin \(\frac{2 A+2 B}{2}\) cos \(\frac{2 A-2 B}{2}\) + 2 sin (A – B) cos (A – B)
= 2 sin (A + B) cos (A- B) + 2 sin (A – B) cos (A – B)
= 2 cos (A – B)[ sin (A-+ B) + sin(A – B)]
= 2 cos (A – B) × 2 sin A cos B
= 4 sin A cos B cos (A – B) = R. H. S.

(iii) cos 2A + cos 2B + cos 2 (A – B) + 1 = 4 cos A. cos B. cos (A – B)
Solution:
cos 2A + cos 2B + cos 2 (A- B) + 1
= 2 cos \(\frac{2 A+2 B}{2}\) cos \(\frac{2 A-2 B}{2}\) + 2 cos² (A – B)
= 2 cos (A + B) cos (A- B) + 2 cos2 (A – B)
= 2 cos (A – B) [cos (A + B) + cos (A – B)]
= 2 cos (A – B) x 2 cos A cos B
= 4 cos A cos B cos (A – B)
= R. H. S

(iv) sin 2A + sin 2B + sin 2C- sin 2 (A + B + C)
Solution:
L. H. S. = sin 2A + sin 2B + sin 2C- sin 2 (A + B + C)
= 2 sin \(\frac{2 A+2 B}{2}\) cos \(\frac{2 A-2 B}{2}\) = 2 cos \(\frac{2 \mathrm{C}+2(\mathrm{~A}+\mathrm{B}+\mathrm{C})}{2}\) × \(\frac{2 \mathrm{C}-2(\mathrm{~A}+\mathrm{B}+\mathrm{C})}{2}\)
= 2 sin (A + B) cos (A – B) + 2 cos (A + B + 2C) sin (A – B)
= 2 sin (A + B) [cos (A- B) – cos (A + B + 2C)]
= 2 sin (A + B) × 2 sin \(\frac{A-B+A+B+2 C}{2}\) sin \(\frac{A+B+2 C-A+B}{2}\)
= 4 sin (A + B) sin (C + A) sin (B + C) = R. H. S.

(v) sin A + sin 3A + sin 5A = sin 3A (1 + 2 cos 2A)
Solution:
L. H. S. = sin A + sin 3A + sin 5A
= sin 3A + sin 5A + sin A
= sin 3A + 2 sin \(\frac{5 \mathrm{~A}+\mathrm{A}}{2} cos \frac{5 \mathrm{~A}-\mathrm{A}}{2}\)
= sin 3A + 2 sin 3A cos 2A
= sin 3A (1 + 2 cos 2A) = R. H. S

(vi) sin A – sin 3A + sin 5A = sin 3A (2 cos 2A – 1)
Solution:
L. H. S. = sin A – sin 3A + sin 5A
= sin A + sin 5A – sin 3A
= sin 5A + sin A – sin 3A
= 2 sin \(\frac{5 \mathrm{~A}+\mathrm{A}}{2} cos \frac{5 \mathrm{~A}-\mathrm{A}}{2}\) – sin 3A
= 2 sin 3A cos 2A – sin 3A
= sin 3A (2 cos 2A – 1) = R. H. S.

(vii) cos (A + B) + sin (A – B) = 2 sin (45° + A) cos (45° + B)
Solution:
R. H. S. = 2 sin (45° + A) cos (45° + B)
= sin (45° + A + 45° + B) + sin (45° + A – 45° – B)
= sin (90° + A + B) sin (A – B)
= cos (A + B) sin (A – B) = L. H. S.

(viii) cos (120° + A) cos (120° – A) + cos (120° + A) cos A + cos A cos (120° – A) + \(\frac{3}{4}\) = 0
Solution:
L. H. S.
= cos (120° + A) cos (120° – A) + cos (120° + A) cos A + cos A cos (120° – A) + \(\frac{3}{4}\)
= cos² A – sin² 120° + cos A [ cos (120° + A) + cos (120° – A)] + \(\frac{3}{4}\)
= cos² A – \(\frac{3}{4}\) + cos A ( 2 cos 120°. cos A) + \(\frac{3}{4}\)
= cos² A + 2 cos 120°. cos² A
= cos² A + 2 \(\left(-\frac{1}{2}\right)\) . cos² A
= cos² A – cos² A = 0

(ix) cos 4A – cos 4B = 8 (cos A – cos B) (cos A + cos B) (cos A – sin B) (cos A + sin B)
Solution:
R. H. S. = 8 (cos A- cos B) (cos A + cos B)
(cos A – sin B) (cos A + sin B)
=  8 (cos² A – cos² B) (cos² A – sin² B)
= – 8(cos² B – cos² A)(cos² A – sin² B)
= – 8 sin (A +B) sin (A – B) cos (A + B) cos (A – B)
= – 2 x [ 2 sin (A + B) cos (A + B)] [2 sin (A – B) cos (A – B)]
= – 2 sin (2A + 2B) sin (2A – 2B)
= – [cos (2A + 2B – 2A + 2B) – cos (2A + 2B + 2A – 2B)]
= – cos 4B + cos 4A
= cos 4A – cos 4B = L. H. S.

Question 10.
Prove the following:
(i) \(\frac{1-\tan ^2\left(45^{\circ}-A\right)}{1+\tan ^2\left(45^{\circ}-A\right)}\) = sin 2A
Solution:
L.H.S \(\frac{1-\tan ^2\left(45^{\circ}-A\right)}{1+\tan ^2\left(45^{\circ}-A\right)}\) = sin 2A = cos 2(45° – A)
= cos (90° – 2A) = sin 2A = R. H. S.

(ii) \(\frac{\cos A+\sin A}{\cos A-\sin A}-\frac{\cos A-\sin A}{\cos A+\sin A}\) = 2 tan 2A
Solution:

(iii) \(\frac{1-\cos 2 A+\sin 2 A}{1+\cos 2 A+\sin 2 A}\) = tan A
Solution:

(iv) \(\frac{\sin (\mathbf{A}+\mathbf{B})+\cos (\mathbf{A}-\mathbf{B})}{\sin (\mathbf{A}-\mathbf{B})+\cos (\mathbf{A}+\mathbf{B})}\) = sec 2B + tan 2B
Solution:

(v) \(\frac{\cos 7 \alpha+\cos 3 \alpha-\cos 5 \alpha-\cos \alpha}{\sin 7 \alpha-\sin 3 \alpha-\sin 5 \alpha+\sin \alpha}\) = cot 2α
Solution:
L.H.S = \(\begin{array}{r}
\cos 7 \alpha+\cos 3 \alpha \\
-\cos 5 \alpha-\cos 3 \alpha \\
\hline \sin 7 \alpha-\sin 3 \alpha \\
-\sin 5 \alpha+\sin \alpha
\end{array}\)

(vi) \(\frac{\sin \theta+\sin 3 \theta+\sin 5 \theta+\sin 7 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta+\cos 7 \theta}\) = tan 4θ
Solution:

Question 11.
Prove the following:
(i) Express 4 cos A. cos B. cos C as the sum of four cosines.
Solution:
4 cos A cos B cos C
= 2 (2cos A cos B) cos C
= 2 [ cos (A + B) + cos (A – B)] cos C
= 2 cos (A + B) cos C + 2 cos (A – B) cos C
= cos (A + B + C) + cos (A + B – C) + cos (A – B + C) + cos (A – B – C)

(ii) Express cos 2A + cos 2B + cos 2C + cos 2 (A + B + C) as the product of three cosines.
Solution:
cos 2A + cos 2B + cos 2C + cos 2(A + B + C)
= 2 cos \(\frac{2 \mathrm{~A}+2 \mathrm{~B}}{2}\) cos \(\frac{2 \mathrm{~A}-2 \mathrm{~B}}{2}\) = 2 cos \(\frac{2 C+2(A+B+C)}{2}\) × cos \(\frac{2 C-2(A+B+C)}{2}\)
= 2 cos (A + B) cos (A- B) + 2 cos (A + B + 2C) cos (A + B)
= 2 cos (A + B) [cos (A- B) + cos (A + B + 2C)]
= 2 cos (A + B) × 2 cos \(\frac{(A-B+A+B+2 C)}{2}\) cos \(\frac{(A-B-A-B-2 C)}{2}\)
= 4 cos (A + B) cos (C + A) cos (B + C)

Question 12.
Prove the following:
(i) cos⁶ A – sin⁶ A = cos 2A(1 – \(\frac{1}{4}\) sin² 2A)
Solution:
L.H.S = cos⁶ A – sin⁶ A
= (cos² A)³– (sin² A)³ = (cos² A – sin² A)³ + 3 cos² A sin² A (cos² A – sin² A)
= cos³ 2A + \(\frac{3}{4}\) sin² 2A cos 2A
= cos² A (cos² 2A + \(\frac{3}{4}\) sin² 2A)
= cos 2A (1 – sin² 2A + \(\frac{3}{4}\) sin² 2A)
= cos 2A (1 – \(\frac{1}{4}\) sin² 2A) = R.H.S

(ii) cos⁶A + sin⁶ A = \(\frac{1}{4}\) (1 + 3 cos²2A)
Solution:
L.H.S = cos⁶ A + sin⁶ A
= (cos² A)³ + (sin² A)³
= (cos² A)³ + (sin² A)³ – 3 cos² A. sin² A (cos² A + sin² A)
= 1 – \(\frac{3}{4}\) sin² 2A = 1 – \(\frac{3}{4}\) (1 – cos² 2A)
= 1 – \(\frac{3}{4}\) + \(\frac{3}{4}\) cos² 2A
= \(\frac{1}{4}\) + \(\frac{3}{4}\) cos² 2A = \(\frac{1}{4}\) (1 + 3cos² 2A) = R.H.S.

(iii) cos³ A. cos 3A + sin³ A sin³ A = cos³ 2A
Solution:
L.H.S = cos³ A cos 3A + sin³ A sin 3A
= cos³ A (4 cos³ A- 3 cos A) + sin³ A (3 sin A- 4 sin³ A)
= 4 cos⁶A- 3 cos⁴A + 3 sin⁴A- 4 sin⁶A
= 4 (cos⁶A- sin⁶A) – 3(cos⁴A- sin⁴A)
= 4 {(cos²A)³– (sin²A)³} – 3 {(cos²A)²– (sin²A)²}
= 4 (cos²A- sin²A) {(cos²A)² + cos²A sin²A + (sin²A)²} – 3 (cos²A- sin²A) (cos²A + sin²A)
= (cos²A- sin²A) [4 {(cos²A + sin²A)²-2 cos²A sin²A + cos²A sin²A} -3×1]
= cos 2A ( 4- 4 sin²A cos²A- 3)
= cos 2A ( 1 – 4 sin²A cos²A)
= cos 2A (1 – sin²2A)
= cos 2A cos²2A = cos³2A = R. H. S.

(iv) sin⁴ θ = \(\frac{3}{8}\) – \(\frac{1}{2}\) cos 2θ + \(\frac{1}{8}\) cos 4θ
Solution:
R.H.S = \(\frac{3}{8}\) – \(\frac{1}{2}\) cos 2θ + \(\frac{1}{8}\) cos 4θ
= \(\frac{3}{8}\) – \(\frac{1}{2}\) (1 – 2 sin² θ) + \(\frac{1}{8}\) (1 – 2 sin² θ)
= \(\frac{3}{8}\) – \(\frac{1}{2}\) + sin² θ + \(\frac{1}{8}\) – \(\frac{1}{4}\)  sin² 2θ
= sin² θ – \(\frac{1}{4}\) × 4 sin² θ cos² θ
= sin² θ (1 – cos² θ)
= sin² θ sin² θ = sin⁴ = L.H.S

(v) cot 3A = \(\frac{\cot ^3 A-3 \cot A}{3 \cot ^2 A-1}\)
Solution:

(vi) tan 4θ = \(\frac{4 \tan \theta-4 \tan ^3 \theta}{1-6 \tan ^2 \theta+\tan ^4 \theta}\)
Solution:

(vii) \(\frac{1}{\tan 3 A-\tan A}-\frac{1}{\cot 3 A-\cot A}\) = cot 2A
Solution:

(viii) \(\frac{\cot A}{\cot A-\cot 3 A}-\frac{\tan A}{\tan 3 A-\tan A}\) = 1
Solution:

Question 13.
Find the value of
sin 3°,cos 3°, 2 sin \(\frac{\pi}{32}\)
Solution:
sin 3° = sin (18° – 15°)
= sin 18° cos 15° – cos 18° sin 15°

Question 14.
If sin A + sin B = a and cos A + cos B = b, then show that
(i) tan(A + B) = \(\frac{2 a b}{b^2-a^2}\)
Solution:

(ii) sin (A + B) = \(\frac{2 a b}{b^2-a^2}\)

(iii) cos (A + B) = \(\frac{b^2-a^2}{b^2+a^2}\)

Question 15.
Prove the following:
(i) \(\frac{1+\sin A-\cos A}{1+\sin A+\cos A}\) = tan \(\frac{A}{2}\)
Solution:

(ii) \(8 \sin ^4 \frac{1}{2} \theta-8 \sin ^2 \frac{1}{2} \theta\) + 1 = cos 2θ
Solution:

(iii) \(\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{5 \pi}{8}\) \(+\cos ^4 \frac{7 \pi}{8}=\frac{3}{2}\)
Solution:

(iv) cos² \(\frac{\alpha}{2}\) (1- 2cos α)² + sin² α(1+ 2 cos α)² =1
Solution:

Question 16.
Prove the following:
(i) sin 20°. sin 40°. sin 60°. sin 80° = \(\frac{3}{16}\)
Solution:

(ii) cos 36°. cos 72°. cos 108°. cos144° = \(\frac{7}{16}\)
Solution:

(iii) cos 10°. cos 30°. cos 50°. cos 70° = \(\frac{3}{16}\)
Solution:

(iv) cos 20°. cos 40°. cos 60°. cos 80° = \(\frac{1}{16}\)
Solution:

(v) tan 6°. tan 42°.tan 66°. tan 78° = 1 [Hints: Use the identity tan 3A = tan A tan (60° – A) tan (60° + A)]
Solution:
We have tan 3A
= tan A tan (60° – A) tan (60° + A)
Now putting A = 6° and 18°.
in (1) we have
tan 18° = tan 6° tan 54° tan 66°     …(2)
and tan 54°
= tan 18° tan 42° tan 78°            ……(3)
Multiplying (2) and (3) we have
tan 18° tan 54°
= tan 6° tan 54° tan 66° tan 18° . tan 42°. tan 78°
or, 1 = tan 6° tan 42° tan 66° tan 78°.

Question 17.
Prove the following:
(i) cos 7 \(\frac{1}{2}\)° = √6 + √3 + √2 + 2
Solution:

(ii) cot 22 \(\frac{1}{2}\)° = √2 + 1
Solution:

(iii) cot 37 \(\frac{1}{2}\)° = √6 – √3 – √2 + 2
Solution:

(iv) tan 37 \(\frac{1}{2}\)° = √6 + √3 – √2 + 2
Solution:

(v) cos \(\frac{\pi}{16}\) = 1/2 \(\sqrt{2+\sqrt{2+\sqrt{2}}}\)
∴ 2 cos \(\frac{\pi}{16}\)

Question 18.
(i) If sin A = K sin B, prove that tan 1/2(A – B) = \(\frac{K-1}{K+1}\) tan 1/2(A – B)
Solution:

(ii) Ifa cos (x + α) = b cos (x – α) show that (a + b) tan x = (a – b) cot α.
Solution:

or, cot x cot α = \(\frac{a+b}{a-b}\)
or, (a – b) cot α = \(\frac{a+b}{\cot x}\) = (a + b) tan x
or, (a + b) tan x =(a-b) cot α

(iii) An angle 0 is divided into two parts α, β such that tan α: tan β = x : y.
Prove that sin (α – β) = \(\frac{x-y}{x+y}\) sin θ.
Solution:

(iv) If sin θ + sin Φ = a, cos θ + cos Φ = b, show that
\(\frac{\sin \frac{\theta+\phi}{2}}{a}=\frac{\cos \frac{\theta+\phi}{2}}{b}=2 \frac{\cos \frac{\theta-\phi}{2}}{a^2-b^2}\)
Solution:
sin θ + sin Φ = a
cos θ + cos Φ = b
We have
a2 + b2 = (sin θ + sin Φ)² + (sin θ + sin Φ)²
= sin² θ + sin² Φ + 2 sin θ. sin Φ + cos² θ +cos² Φ + 2 cos θ. cos Φ
= 2 + 2 (cos θ cos Φ + sin θ sin Φ)
= 2 + 2 cos (θ – Φ)
= 2 [1+ cos (θ – Φ)]

(v) If a cos α + b sin α = c = a cos β + b sin α then prove that
\(\frac{a}{\cos \frac{1}{2}(\alpha+\beta)}=\frac{b}{\sin \frac{1}{2}(\alpha+\beta)}\) \(=\frac{c}{\cos \frac{1}{2}(\alpha-\beta)}\)
Solution:

(vi) Prove that \(\left(\frac{\cos A+\cos B}{\sin A-\sin B}\right)^n+\left(\frac{\sin A+\sin B}{\cos A-\cos B}\right)^n\) = 2 cosⁿ \(\frac{A-B}{2}\) or zero according as n is even or odd.
Solution:

Question 19.
(i) If (1 – e) tan² \(\frac{\boldsymbol{\beta}}{2}\) = (1 + e) tan² \(\frac{\boldsymbol{\alpha}}{2}\), prove that cos β = \(\frac{\cos \alpha-e}{1-e \cos \alpha}\)
Solution:

(ii) If cos θ = \(\frac{\cos \mathbf{A}-\cos \mathbf{B}}{1-\cos \mathbf{A} \cdot \cos \mathbf{B}}\) prove that one of the values of
tan \(\frac{θ}{2}\) is tan \(\frac{A}{2}\) . tan \(\frac{B}{2}\).
Solution:

(iii) If tan θ = \(\frac{\sin x \cdot \sin y}{\cos x+\cos y}\) then prove that one of the values of tan \(\frac{1}{2}\) θ tan \(\frac{1}{2}\) x and tan \(\frac{1}{2}\) y.
Solution:

(iv) If sec (Φ + a) + sec (Φ – α) = 2 sec Φ. show that cos Φ = ± √2 cos \(\frac{α}{2}\)
Solution:

(v) If tan A + tan B = a and cot A + cot B = b. then show that cot (A + B) = \(\frac{1}{a}\) – \(\frac{1}{b}\).
Solution:

(vi) If cot θ = cos (x + y) and cot Φ = cos (x – y) show that tan (θ = Φ) = \(\frac{2 \sin x \cdot \sin y}{\cos ^2 x+\cos ^2 y}\)
Solution:

(vii) If tan β = \(\frac{n^2 \sin \alpha \cdot \cos \alpha}{1-n^2 \sin ^2 \alpha}\), then show that \(\frac{\tan (\alpha-\beta)}{\tan \alpha}\) = 1 – n²
Solution:

(viii)If 2 tan α = 3 tan β, then prove that tan(α – β) = \(\frac{\sin 2 \beta}{5-\cos 2 \beta}\)
Solution:

(ix) If α, β are acute angles and cos 2α = \(\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}\) then prove that tan α = √2 tan β.
Solution:

Question 20.
If A + B + C = π, then prove the following.
(i) cos 2A + cos 2B + cos 2C + 1 + 4 cos A. cos B. cos C = 0
Solution:
A + B + C = π  or, A + B = π – C
or, cos (A + B) = cos (π – C) = – cos C
∴ cos 2A + cos 2B + cos 2C
= 2 cos \(\frac{2 \mathrm{~A}+2 \mathrm{~B}}{2}\) cos \(\frac{2 \mathrm{~A}-2 \mathrm{~B}}{2}\)
= 2 cos (A + B) cos (A- B) + 2 cos² C – 1
= – 2 cos C cos (A – B) + 2 cos² C –  1
= – 1 – 2 cos C cos (A – B) – cos C
= – 1 – 2 cos C [cos (A- B) + cos (A + B)]
= – 1 – 2 cos C × 2 cos A cos B
= – 1 – 4 cos A cos B cos C
∴ L. H. S. = cos 2A + cos 2B + cos 2C + 1 + 4 cos A cos B cos C
= – 1 – 4 cos A cos B cos C + 1 + 4 cos A cos B cos C = 0 = R. H. S.

(ii) sin 2A + sin 2B – sin 2C = 4 cos A. cos B. sin C
Solution:
L. H. S. = sin 2A + sin 2B – sin 2C
= 2 sin \(\frac{2 A+2 B}{2}\) cos \(\frac{2 A-2 B}{2}\) – sin 2C
= 2 sin(A + B) cos (A – B) -2 sinC cos C
= 2 sin C cos (A- B) – 2 sin C cos C [ ∵ A + B= π – C]
or, sin (A + B) = sin (n- C) = sin C]
= 2 sin C [cos (A – B)- cos C]
= 2 sin C [cos (A – B) – cos (A + B)]
= 2 sin C. 2 cos A. cos B
= 4 cos A cos B sin C = R. H. S.

(iii) cos A + cos B + cos C = 1 + 4 sin \(\frac{1}{2}\) A. sin \(\frac{1}{2}\) B. sin \(\frac{1}{2}\) C.
Solution:

(iv) sin A + cos B- sin C = 4 sin \(\frac{1}{2}\) A. sin \(\frac{1}{2}\) B. cos \(\frac{1}{2}\) C.
Solution:

(v) cos² A + cos² B + 2cos A. cos B. cos C = sin² C
Solution:

(vi) sin² \(\frac{A}{2}\) + sin² \(\frac{B}{2}\) + sin² \(\frac{C}{2}\) = 1 – 2 sin \(\frac{A}{2}\). sin \(\frac{B}{2}\). sin \(\frac{C}{2}\)
Solution:

(vii) sin \(\frac{A}{2}\) + sin \(\frac{B}{2}\) + sin \(\frac{c}{2}\) = 4 sin \(\frac{\pi-A}{4}\) sin \(\frac{\pi-B}{4}\). sin \(\frac{\pi-C}{4}\) + 1
Solution:

(viii) cos² \(\frac{A}{2}\) + cos² \(\frac{B}{2}\) – cos² \(\frac{C}{2}\) = 2 cos \(\frac{A}{2}\). cos \(\frac{B}{2}\). sin \(\frac{C}{2}\)
Solution:

(ix) sin (B + 2C) + sin (C + 2A) + sin (A + 2B) = 4 sin \(\frac{B-C}{2}\). sin \(\frac{C-A}{2}\). sin \(\frac{A-B}{2}\)
Solution:

Question 21.
(i) Show that (2 cos θ – 1) (2 cos 2θ – 1) (2 cos² 2θ – 1) ….. (2 cos 2^n-1 θ – 1) \(=\frac{2 \cos 2^n \theta+1}{2 \cos \theta+1}\)
Solution:
We have
(2 cos θ + 1) (2 cos θ – 1)
= 4 cos² θ – 1=4 cos² θ – 2+1
= 1 + 2(2 cos² θ – 1)
= 1+2 cos 2θ = 2 cos 2θ+1
And, (2 cos 2θ + 1) (2 cos 2θ – 1)
= 4 cos² 2θ – 1 = 2 ( 2 cos² 2θ – 1) + 1
= 2 cos 4 θ + 1 = 2 cos 2² θ + 1
Similarly, we can prove,
( 2 cos 2² θ + 1) ( 2 cos 2² θ- 1)
= 2 cos 2³ θ + 1
Proceeding in this way, we can prove,
(2 cos 2θ + 1) (2 cos 2θ – 1)
(2 cos 2θ- 1) (2 cos 2² θ – 1) ……  (2 cos 2^n-1 θ-1)
= 2 cos 2ⁿ θ + 1
or, (2 cos θ-1) (2 cos 2θ-1) (2 cos 2² θ- 1) (2 cos 2^n-1 θ-1)
\(=\frac{2 \cos 2^n \theta+1}{2 \cos \theta+1}\)

(ii) Show that 2ⁿ cos θ. cos 2θ. cos2² θ ……… 2^n-1 θ = 1 If θ = \(\frac{\pi}{2^n+1}\)
Solution:
we have, θ = \(\frac{\pi}{2^n+1}\)
or, 2ⁿ θ + θ = π
or, 2ⁿ θ  = π – θ
or, 2ⁿ θ = sin (π – θ) = sin θ
or, \(\frac{\sin 2^n \theta}{\sin \theta}\) = 1   …….(1)
Again, sin 2ⁿ θ = 2 sin 2^n-1 θ cos 2^n-1 θ
= 2 × 2 sin 2^n-2 θ cos 2^n-2 θ cos 2^n-1 θ
= 2² × 2 sin 2^n-3 θ cos 2^n-3 θ  cos 2^n-2 θ cos 2^n-v θ
= 2³ sin 2^n-3 θ cos 2^n-3 θ cos 2^n-2 θ cos 2^n-1 θ

………………………..

………………………..

………………………..

= 2ⁿ sin θ cos θ cos 2θ cos 2² θ …….. cos 2^n-2 θ cos 2^n-1 θ

or,  2ⁿ cos θ cos 2θ cos 2² θ …… cos 2^n-1 θ = 1

(iii) Prove that \(\frac{\tan 2^n \theta}{\tan \theta}\) = = (1 + sec 2θ) (1 + sec2² θ) …. (1 + sec2ⁿ θ)
Solution:

Question 22.
If x + y + z = xyz, prove that
(i) \(\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2}\) = =\(\frac{4 x y z}{\left(1-x^2\right)\left(1-y^2\right)\left(1-z^2\right)}\)
Solution:
x + y + z = xyz   (Given)
Let x = tan α, y = tan β, z = tan γ
∴ tan α + tan β + tan γ = tan α .tan β .tan γ or; tan α + tan β
= – tan γ + tan α tan β tan γ
= – tan γ(1 – tan α tan β)
or, \(\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}\) = – tan γ
or, tan (α + β) = tan (π- γ)
or, α + β = π-γ
or, 2α + 2β = 2π- 2γ
or, tan (2α + 2β = tan (2π- 2γ)
or, tan (2α + 2β) = tan (2π- 2γ)

(ii) \(\frac{3 x-x^3}{1-3 x^2}+\frac{3 y-y^3}{1-3 y^2}+\frac{3 z-z^3}{1-3 z^2}\) = \(\frac{3 x-x^3}{1-3 x^2} \cdot \frac{3 y-y^3}{1-3 y^2} \cdot \frac{3 z-z^3}{1-3 z^2}\)
Solution:

Question 23.
If \(\text { If } \frac{\sin ^4 \alpha}{a}+\frac{\cos ^4 \alpha}{b}=\frac{1}{a+b}\) show that \(\frac{\sin ^8 \alpha}{a^3}+\frac{\cos ^8 \alpha}{b^3}=\frac{1}{(a+b)^3}\)
Solution:

Odisha State Board Elements of Mathematics Class 11 CHSE Odisha Solutions Chapter 7 Linear Inequalities Ex 7(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 7 Linear Inequalities Exercise 7(b)

Solve graphically
Question 1.
x < y
Solution:
x < y
Step – 1: Let us draw the dotted line x = y

X	0	1
y	0	1

Step – 2: Let us take a point say (1, 0) which is not on the line. Putting x = 1, y = 0 in the equation we get 1 < 0 (false).
⇒ (1, 0) does not satisfy the inequality.
⇒ The solution is the half-plane that does not contain (1, 0)
Step – 3: The shaded region is the solution region.

Question 2.
3x + 4y ≥ 12
Solution:
3x + 4y ≥ 12
Step – 1: Let us draw the line 3x + 4y = 12

X	4	0
y	0	3

Step – 2: Let us consider the point (0, 0) which is not on the line. Putting x = 0, y = 0 in the inequality we have 0 ≥ 12 (false).
∴ (0, 0) does not satisfy the inequality.
⇒ The half-plane that does not contain (0, 0) is the solution region.
Step – 3: The shaded region is the solution region.

Question 3.
x – y > 0
Solution:
x – y > 0
Step – 1: Let us draw the dotted line x = y.

X	0	1
y	0	1

Step – 2: Let us consider (1, 0) which is not on the line.
Putting x = 1, y = 0 in the inequation we get 1 > 0 (True)
⇒ (1, 0) satisfies the inequation.
⇒ The half-plane containing (1, 0) is the solution region.

Question 4.
x + 2y – 5 ≤ 0
Solution:
x + 2y – 5 ≤ 0
Step – 1: Let us draw the line x + 2y – 5 = 0
⇒ y = \(\frac{5-x}{2}\)

X	5	1
y	0	2

Step – 2: Let us consider the point (0, 0) which does not lie on the line putting x = 0, y = 0 in the inequation we get – 5 < 0 (True).
⇒ The point satisfies the inequation.
⇒ The half-plane containing (0, 0) is the solution region.
Step – 3: The shaded region is the solution region.

Question 5.
7x – 4y < 14
Solution:
7x – 4y < 14
Step – 1: Let us draw the dotted line 7x – 4y = 14

X	2	6
y	0	7

Step – 2: Let us consider the point (0, 0) which does not lie on the line. Putting x = 0, y = 0 in the inequality we get 0 < 14 (True).
⇒ (0, 0) satisfies the inequation.
⇒ The half-plane including (0, 0) is the solution region.
Step – 3: The solution region is the shaded region.

Question 6.
x + 8y + 10 > 0
Solution:
x + 8y + 10 > 0
Step – 1: Let us draw the dotted line x + 8y + 10 = 0

X	-10	-2
y	0	-1

Step – 2: Let us consider the point (0, 0) which does not lie on the line. Putting x = 0, y = 0 in the inequation we get 10 > 0 (True)
⇒ (0, 0) satisfies the inequality.
⇒ The half-plane containing origin is the solution region.
Step – 3: The solution region’ is the shaded region.

Question 7.
5x + 6y < 12
Solution:
5x + 6y < 12
Step – 1: Let us draw the dotted line 5x + 6y = 12

x	-6	0	6
y	7	2	-3

Step – 2: Putting x = 0, y = 0 in the equation we get, 0 < 12 (True)
⇒ The (0, 0) satisfies the equation.
Step – 3: The shaded region is the required solution.

Question 8.
– 3x + y > 0
Solution:
Step – 1: Let us draw the dotted line – 3x + y = 0

x	0	1	-1
y	0	3	-3

Step – 2: Putting x – 1, y = 0 in the equation we get, – 3 > 0 (false)
∴ Point (1, 0) does not satisfy the in the equation.
Step – 3: The shaded half-plane is the solution.

Question 9.
3x + 8y > 24
Solution:
Step- 1: Let us draw the dotted graph of 3x + 8y = 24

x	8	0
y	0	3

Step- 2: Putting x = 0, y- 0 we get, 0 > 24 (false)
∴ 0, (0, 0) does not satisfy the in equality.

Question 10.
x + y > 1
Solution:
Step – 1: Let us draw the graph x + y = 1

x	1	0
y	0	1

Step – 2: Putting x = 0, y = 0 in the equation we get, 0 > 1 (false)
∴ 0 (0, 0) does not satisfy the in equation
Step – 3: The shaded region is the solution.

Question 11.
x ≤ 0
Solution:
Step – 1: Let us draw the graph x = 0

Step – 2: Putting x = – 1 we get, – 1 ≤ 0 (True)
Thus, the shaded region is the solution.

Question 12.
y > 5
Solution:
Step – 1: Let us draw the dotted graph of y = 5

x	0	1	-1
y	5	5	5

Step- 2: Putting x = 0, y = 0 we have 0 > 5 (false)
we 0(0, 0) does not satisfy. the inequality.
Step – 3: The shaded region is the solution.

Odisha State Board Elements of Mathematics Class 11 CHSE Odisha Solutions Chapter 7 Linear Inequalities Ex 7(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 7 Linear Inequalities Exercise 7(a)

Question 1.
Determine whether the solution set is finite or infinite or empty:
(i) x < 1000, x ∈ N
Solution:
Finite

(ii) x < 1, x ∈ Z (set of integers)
Solution:
Infinite

(iii) x < 2, x is a positive integer.
Solution:
Finite

(iv) x < 1, x is a positive integer.
Solution:
Empty

Question 2.
Solve as directed:
(i) 5x ≤ 20 in positive integers, in integers.
Solution:
5x ≤ 20
⇒ \(\frac{5 x}{5} \leq \frac{20}{5}\)
⇒ x ≤ 4
If x is a positive integer, then the solution set is {1, 2, 3, 4}
If x is an integer, then the solution set is:
S = {x : x ∈ Z and x ≤ 4}
= { ….. -3, -2, -1, 0, 1, 2, 3, 4}

(ii) 2x + 3 > 15 in integers, in natural numbers.
Do you mark any difference in the solution sets?
Solution:
2x + 3 > 15
⇒ 2x + 3 – 3 > 15 – 3
⇒ 2x > 12
⇒ \(\frac{2 x}{2}>\frac{12}{2}\)
⇒ x > 6
If x ∈ Z, then the solution set is S = (x : x ∈ Z and x > 6}
= {7, 8, 9…… }
If x ∈ N. then the solution set is S = {x : x ∈ N and x > 6}
= {7, 8, 9…… }
Two solution sets are the same.

(iii) 5x + 7 < 32 in integers, in non-negative integers.
Solution:
5x + 7 < 32
⇒ 5x + 7 – 7 < 32 – 7
⇒ 5x < 25
⇒ \(\frac{5 x}{5}<\frac{25}{5}\)
⇒ x < 5
If x ∈ Z, then the solution set is S = { x : x ∈ Z and x < 5 }
= {…..-3, -2, -1, 0, 1, 2, 3, 4}
If x is a non-negative solution then the solution set is S = {x : x is a non-negative integer < 5}
= (0, 1,2, 3,4}

(iv) -3x – 8 > 19, in integers, in real numbers.
Solution:
– 3x – 8 > 19
⇒ – 3 x – 8 + 8 > 19 + 8
⇒ – 3x > 27
⇒ \(\frac{-3 x}{-3}<\frac{27}{-3}\)
⇒ x < – 9
If x ∈ Z, then the solution set is S = (x : x ∈ Z and x < – 9}
= { ……..- 11, – 10}
If x ∈ R then the solution set is S = {x : x ∈ R and x < – 9}
= (∞, – 9)

(v) |x – 3| < 11, in N and in R.
Solution:
|x – 3| < 11
⇒ – 1 < x – 3 < 11
⇒ – 11 + 3 < x – 3 + 3 < 11+3
⇒ – 8 < x < 14
If x ∈ N the solution set is S = {1, 2, 3, 4, 5……..12, 13}
If x ∈ R then the solution set is: S = {x : x ∈ R and – 8 < x < 14}
= (- 8, 14)

Question 3.
Solve as directed:
(i) 2x + 3 > x – 7 in R
Solution:
2x + 3 > x – 7
⇒ 2x – x > – 7 – 3
⇒  x > – 10
x ∈ R, the solution set is S = (x : x ∈ R and x > – 10} = (-10, ∞)

(ii) \(\frac{x}{2}+\frac{7}{3}\) <  3x – 1 in R
Solution:
\(\frac{x}{2}+\frac{7}{3}\) <  3x – 1
\(\frac{3 x+14}{6}\) <  3x – 1
⇒ 3x + 14 < 18x – 6
⇒ 3x – 18x < – 6 – 14
⇒ – 15x < – 20
⇒ \(\frac{-15 x}{-15}>\frac{-20}{-15}\)
⇒ x > \(\frac{4}{3}\)
If x ∈ R, the solution set is S = \(\left(\frac{4}{3}, \infty\right)=\left\{x: x \in R \text { and } x>\frac{4}{3}\right\}\)

(iii) \(\frac{x}{2}-\frac{x}{3}+\frac{x}{5} \leq \frac{11}{3}\) for non-negative real numbers.
Solution:
\(\frac{x}{2}-\frac{x}{3}+\frac{x}{5} \leq \frac{11}{3}\)
⇒ \(\frac{15 x-10 x+6 x}{30}\) ≤ \(\frac{11}{3}\)
⇒ 11x ≤ \(\frac{11}{3}\) × 30
⇒ 11x ≤ 110
⇒ x ≤ 10
If x is a non-negative real number then the solution set is S = {x : x ∈ R and 0 ≤ x ≤ 10}
= {0, 10}

(iv) 2(3x – 1) < 7x + 1 < 3 (2x + 1) for real values.
Solution:
2(3x – 1) < 7x + 1 < 3(2x + 1)
⇒ 6x – 2 < 7x + 1< 6x + 3
⇒ – 2 < x + 1 < 3
⇒ – 3 < x < 2
If x ∈ R, the solution set is S = (x : x ∈ R and -3 < x < 2}
= {-3, 2}

(v) 7(x – 3) ≤ 4 (x + 6), for non-negative integral values.
Solution:
7(x – 3) ≤ 4(x + 6)
⇒ 7x – 21 ≤ 4x + 24
⇒ 7x – 4x ≤ 24 + 21
⇒ 3x ≤ 45
⇒ x ≤ 9
If x is a non-negative integer the solution set is S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

(vi) Convert to linear inequality and solve for natural numbers: (x – 2) (x – 3) < (x + 3) (x – 1)
Solution:
(x – 2) (x – 3) < (x + 3) (x – 1)
⇒ x² – 5x + 6  <  x² + 2x – 3
⇒ – 5x + 6 < 2x – 3
⇒ – 5x – 2x < – 3 – 6
⇒ – 7x < – 9
⇒ x > \(\frac{9}{7}\)
If x ∈ N, the solution set is S = {2, 3, 4 }

(vii) Solve in R, \(\frac{x}{2}\) + 1 ≤ 2x – 5 < x. Also, find its solution in N.
Solution:
\(\frac{x}{2}\) + 1 ≤ 2x – 5 < x
⇒ \(\frac{x}{2}\) +1 ≤ 2x – 5 and 2x – 5 < x
⇒ \(\frac{x}{2}\) – 2x ≤ – 5 – 1 and x < 5
⇒ \(\frac{-3x}{2}\) ≤ – 6 and x < 5
⇒ – 3x ≤ – 12 and x < 5
⇒ x ≥ 4 and x < 5
⇒ 4 ≤ x < 5
If x ∈ R, the solution set is S = {x : x ∈ R and 4 < x < 5}
= {4, 5}
If x ∈ N, the solution set is S = { 4 }

(viii) Solve in R and also in Z: \(\frac{3 x+1}{5} \geq \frac{x+2}{3}-\frac{5-3 x}{5}\)
Solution:
\(\frac{3 x+1}{5} \geq \frac{x+2}{3}-\frac{5-3 x}{5}\)
⇒ \(\frac{3 x+1}{5} \geq \frac{5 x+10-15+9 x}{15}\)
⇒ 3x + 1 ≥ \(\frac{14 x-5}{3}\)
⇒ 9x + 3 ≥ 14x – 5
⇒ 9x – 14x ≥ – 5 – 3
⇒ – 5x ≥ – 8
⇒ x ≤ \(\frac{8}{5}\)
If x ∈ R, then the solution set is S = (x : x ∈ R and x ≤ \(\frac{8}{5}\)}
= (- ∞, \(\frac{8}{5}\))
If x ∈ Z, then the solution set is S = { x : x ∈ Z and x ≤ \(\frac{8}{5}\)}
= {……. -3, -2, -1, 0, 1}

Question 4.
Solve |x – 1| >1 and represent the solution on the number line.
[Exhaustive hints: By definition of modulus function
For x – 1 ≥ 0 or x ≥ 1, |x – 1| > 1
⇔ x – 1 > 1 ⇔ x > 2 ⇔ x ∈ (2, ∞)
For x- 1 < 0 or x < 1, |x – 1| > 1
⇔ – (x – 1) > 1
⇔ x – 1 < -1 (multiplication by -1 reverses the inequality)
⇔ x < 0 ⇔ x ∈ ( -∞, 0)
∴ The solution set is the Union,
(-∞, 0) ∪ (2, ∞) Show this as two disjoint open intervals on the number line, i.e., real line.]
Solution:
|x – 1| > 1
⇒ – 1 > x – 1 > 1
⇒ 0 > x > 2
⇒ x < 0 and x > 2
∴ The solution set is S = {x : x ∈ R, x < 0 and x > 2}
= (-∞, 0) ∪ (2, ∞)
We can show this solution in the number line as

Question 5.
Solve in R and represent the solution on the number line.
(i) |x – 5| < 1
Solution:
|x – 5| < 1
⇒ – 1< x – 5 < 1
⇒ 4 < x < 6
If x ∈ R, then the solution set is S = (4, 6)
We can represent the solution on the number line as

(ii) \(\frac{x}{5}<\frac{2 x+1}{3}+\frac{1-3 x}{6}\)
Solution:
\(\frac{x}{5}<\frac{2 x+1}{3}+\frac{1-3 x}{6}\)
⇒ \(\frac{x}{5}<\frac{4 x+2+1-3 x}{6}\)
⇒ \(\frac{x}{5}<\frac{x+3}{6}\)
⇒ 6x < 5x + 15
⇒ x < 15
If x ∈ R, the solution set is S = (-∞, 5)
We can represent the solution on the number line as

(iii) 2x + 1 ≥ 0
Solution:
2x + 1 ≥ 0
⇒ 2x ≥ -1
⇒ x ≥ -1/2
If x ∈ R, then the solution set is S = [\(-\frac{1}{2}\), ∞]
We can represent the solution on the number line as

(iv) \(\frac{x-1}{2} \leq \frac{x+1}{3}<\frac{3 x-1}{6}\)
Solution:
\(\frac{x-1}{2} \leq \frac{x+1}{3}<\frac{3 x-1}{6}\)
⇒ 3x – 3 ≤ 2x + 2 < 3x – 1
⇒ 3x – 3 ≤ 2x + 2 and 2x + 2 < 3x – 1
⇒ x ≤ 5 and – x < – 3
⇒ x ≤ 5 and x > 3
⇒ 3 < x ≤ 5
If x ∈ R, the solution set is S = {3, 5}
We can represent the solution on the number line as

Question 6.
In a triangle, ABC; AB, BC, and CA are x, 3x + 2, and x + 4 units respectively where x ∈ N. Find the length of its sides. (Hint: Apply triangle-inequality).
Solution:
Given AB = x
BC = 3x + 2
and CA = x + 4
Now AB + AC > BC (Triangle inequality)
⇒ x + x + 4 > 3x + 2
⇒ 2x + 4 > 3x + 2
⇒ – x > – 2
⇒ x < 2
As x ∈ N we have x = 1
The sides of triangle ABC are
AB = 1 unit
BC = 5 units
and CA = 5 units

Question 7.
The length of one side of a parallelogram is 1 cm. shorter than that of its adjacent side. If its perimeter is at least 26 c.m., find the minimum possible lengths of its sides.
Solution:
Let the longer side = x cm
∴ The smaller side = (x – 1) cm
Perimeter = 2(x + (x – 1)) = 4x – 2 cm
According to the question
4x – 2 ≥ 26
⇒ 4x ≥ 28
⇒ x > 7
The minimum value of x = 7.
∴ The minimum length of the sides is 7cm and 6 cm.

Question 8.
The length of the largest side of a quadrilateral is three times that of its smallest side. Out of the other two sides, the length of one is twice that of the smallest and the other is 1 cm. longer than the smallest. If the perimeter of the quadrilateral is at most 36 c.m., then find the maximum possible lengths of its sides.
Solution:
Let the smallest side = x cm.
Largest side = 3 times x = 3x cm.
The other two sides are 2x cm and x + 1 cm.
⇒ The perimeter = x + 3x + 2x + x + 1
= 7x + 1 cm
According to the question:
7x + 1 ≤ 36
⇒ 7x ≤ 35
⇒ x ≤ 5
Maximum value of x = 5
∴ The maximum possible length of sides are x = 5 cm, 3x = 15 cm, 2x = 10 cm, and x + 1 = 6 cm.

Question 9.
Find all pairs of consecutive odd numbers each greater than 20, such that their sum is less than 60.
Solution:
Let two consecutive odd numbers are
2n – 1 and 2n + 1
Now 2n – 1 > 20 and 2n + 1 > 20
But their sum = 2n – 1 + 2n + 1
= 4n < 60
⇒ n < 15
for n = 14 two numbers are 27, 29
for n = 13 two numbers are 25, 27
for n = 12 two numbers are 23, 25
for n = 11 two numbers are 21, 23
∴ All pairs are 21, 23; 23, 25; 25, 27 and 27, 29

Question 10.
Find all pairs of even numbers each less than 35, such that their sum is at least 50.
Solution:
Let two even numbers be x and y.
According to the question
x < 35, y < 35 and x + y ≥ 50
⇒ x ≤ 34, y ≤ 34 and x + y ≥ 50
⇒ x + y ≤ 70, x + y ≥ 50
⇒ 50 ≤ x + y ≤ 70
If x + y = 50 the numbers are {34, 16}, {32, 18}, {30, 20}, {28, 22}, {26, 24}
If x + y = 52 the numbers are {34, 18}, {32, 20}, {30, 22}, {28, 24}, {26, 26}
If x + y = 34 the numbers are {34, 20}, {32, 22}, {30, 24}
If x + y = 56 the numbers are {34, 22}, {32, 24}, {30, 26}, {28, 28}
If x + y = 58 the numbers are {34, 24}, {32, 26}, {30, 28}
If x + y = 60 the numbers are {34, 26}, {32, 28}, {30, 30}
If x + y = 62 the numbers are {34, 28}, {32, 30}
If x + y = 64 the numbers are {34, 30}, {32, 32}
If x + y = 68 the numbers are {34, 34}

Odisha State Board CHSE Odisha Class 11 Math Solutions Chapter 3 Relations And Functions Ex 3(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 3 Relations And Functions Exercise 3(a)

Question 1.
Compute the product A × B when
(i) A = {0} = B
(ii) A = {a, b}, B = {a, b, c}
(iii) A = Z, B = Φ
Solution:
(i) A = {0} = B
∴ A × B = {(0, 0)}

(ii) A = {a, b}, B = {a, b, c}
∴ A × B = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c)}

(iii) A = Z, B = Φ ∴ AxB = Φ

Question 2.
If |A| = m, |B| = n, what can you say about
(i) |A × B| (ii) |P(A) × P(B)|
Solution:
If |A| = m. |B| = n then

(i) lA × B| = mn.

(ii) |P(A)| = 2^m . |P(B)| = 2ⁿ
∴  |P(A) × P(B)| =2^m × 2ⁿ = 2^m+n

Question 3.
Find x, y if
(i) (x, y) = (-3, 2)
(ii) {x + y, 1) = (1, x – y)
(iii) (2x + y, 1) = (x, 2x + 3y)
Solution:

(i) ∴ x = – 3, y = 2

(ii) ∴ x + y = 1, x – y = 1
∴ 2x = 2 or, x = 1
∴ y=0

(iii) ∴ 2x + y = x, 1 = 2x + 3y
∴ {x + y = 0} × 2
2x + 3y = 1
–      –      –
∴ – y = – 1 or, y = 1
∴ x = – 1

Question 4.
If, A × B = B × A then what can you
Solution:
If A × B = B × A then A = B

Question 5.
|A × B| = 6. If ( -1, y ), (1, x), (0, y) are in A × B. Write other elements in A × B, where x ≠ y.
Solution:
Let |A × B| = 6 and (-1, y) (1, .x) (0, y) ∈ A × B
⇒ -1, 1, 0 ∈ A and x, y ∈ B
As |A × B| = 6 and 3 × 2 = 6
We have A = {-1, 1, 0} and B = {x, y}
Thus other elements of A × B is (-1, x) , (1, y), (0, x)

Odisha State Board Elements of Mathematics Class 11 CHSE Odisha Solutions Chapter 4 Trigonometric Functions Ex 4(d) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 4 Trigonometric Functions Exercise 4(d)

Question 1.
Fill in the blanks choosing the correct answer from the brackets.

(i) In Δ ABC, b =____________. (b cos B + c cos C, a cos A + c cos C, c cos A + a cos C)
Solution:
c cos A + a cos C

(ii) If a cot A = b cot B then Δ ABC is__________. (isosceles, right-angled, equilateral)
Solution:
isosceles

(iii) In Δ ABC if b sin C = c sin B = 2 then b sin C = ___________. (0, 1, 2)
Solution:
1

(iv) In Δ ABC if \(\frac{\cos \mathrm{A}}{a}=\frac{\cos \mathrm{B}}{b}=\frac{\cos \mathrm{C}}{c}\) then the tringle is_________ (equilateral, isosceles, scalene)
Solution:
equilateral

(v) If sin A = sin B and b = 1/2, then a = _______________. (2, 1/2, 1)
Solution:
a = 1/2

(vi) In Δ ABC if A = 60°, B = 45° a : b = __________. ( √2 : √3, √6 : 2, √3 : 2)
Solution:
√6 : 2

(vii) In Δ ABC if b² + c² < a² , then _________ angle is obtuse. (A, B, C)
Solution:
A

(viii) If a cos B = b cos A. then cos B = _____________. \(\left(\frac{c}{a}, \frac{a}{2 c}, \frac{c}{2 a}\right)\)
Solution:
cos B = \(\frac{c}{2 a}\)

(ix) If a – b cos C, then __________ angle is a right angle. (A, B, C)
Solution:
∠B is a right angle

(x) If a = 12, b = 7, C = 30°, then Δ = ______________. (42, 84, 21)
Solution:
Δ = 21

Question 2.
Prove that
(i) a sin A – b sin B = c sin (A – B)
Solution:
R.H.S. = c sin (A – B)
= 2R sin C sin (A – B)
= 2R sin (A + B) sin (A – B)
[∴ A + B + C = π or, A + B = π – C
or sin (A + B) = sin (π – C) sin C]
= 2R (sin² A – sin²  B)
= 2R sin A sin A – 2R sin B sin B
= a sin A – b sin B = L.H.S.

(ii) b cos B + c cos C = a cos (B – C)
Solution:
R.H.S. = a cos (B – C)
= 2R sin A cos (B – C)
= 2R sin (B + C) cos (B – C)
= R sin (B + C + B – C) + sin (B + C – B + C)}
= R [(sin 2B + sin 2C)
= R(2sin B cos B + 2 sin C cos C)
= 2R sin B cos B + 2R sin C cos C
= b cos B + c cos C = L.H.S.

(iii) If (a + b + c)(b + c – a) = 3bc, then A = 60°
Solution:

(iv) If \(\frac{b+c}{5}=\frac{c+a}{6}=\frac{a+b}{7}\) then sin A : sin B : sin C = 4 : 3 : 2
Solution:

(v) If A: B: C = 1 : 2 : 3 then sin A: sin B: sin C = 1 : 2
Solution:

(vi) If b² + c² – a² = bc, then A = 60°
Solution:
If b² + c² – a² = bc, then A = 60°
or, \(\frac{b^2+c^2-a^2}{2 b c}\) = 1/2 or, cos A = 1/2
or, A = 60°

(vii) If A : B: C = 1 : 2 : 7, then c: a = (√5 + 1) : (√5 – 1)
Solution:

But we know that \(\frac{\sin C}{\sin A}=\frac{c}{a}\)
∴ \(\frac{c}{a}=\frac{\sqrt{5}+1}{\sqrt{5}-1}\)

Question 3.
(i) If cos A = \(\frac{12}{13}\), cos B = \(\frac{5}{13}\) then find a : b.
Solution:

(ii) If a = 7, b = 3, c = 5 then find A.
Solution:

(iii) If a = 8, b = 6, c = 4 find tan \(\frac{B}{2}\)
Solution:

(iv) If \(\frac{a}{\sec A}=\frac{b}{\sec B}\) and a ≠ b then find C.
Solution:

(v) If a = 48, b = 35, ∠C = 60° then find c.
Solution:

In Δ ABC prove that (Q. 4 to Q. 26)

Question 4.
a sin (B – C) + b sin (C – A) + c sin (A – B) = 0
Solution:
a sin (B – C) + b sin (C – A) + c sin (A – B)
= 2R sin A sin (B – C) + 2R sin B sin (C – A) + 2R sin C sin (A – B)
= 2R [sin (B + C) sin (B – C) + sin (C + A) sin (C – A) + sin (A + B) sin (A – B)]
= 2R[sin² B – sin² C + sin² C – sin² A + sin² A – sin² A]
= 2R x 0 = 0

Question 5.
\(\frac{\sin (B-C)}{\sin (B+C)}=\frac{b \cos C-c \cos B}{b \cos C+c \cos B}\)
Solution:

Question 6.
\(\sum \frac{a^2 \sin (B-C)}{\sin (B+C)}=0\)
Solution:

Question 7.
a²(cos² B – cos² C) + b²(cos² C – cos² A) + c²(cos² A – cos² B) = 0
Solution:

Question 8.
\(\frac{b^2-c^2}{a^2} \sin 2 A+\frac{c^2-a^2}{b^2} \sin 2 B\) \(+\frac{a^2-b^2}{c^2} \sin 2 \mathrm{C}=0\)
Solution:

Question 9.
\(\frac{a^2\left(b^2+c^2-a^2\right)}{\sin 2 \mathrm{~A}}=\frac{b^2\left(c^2+a^2-b^2\right)}{\sin 2 \mathrm{~B}}\) \(=\frac{c^2\left(a^2+b^2-c^2\right)}{\sin 2 C}\)
Solution:

Question 10.
\(\Sigma \frac{\cos A}{\sin B \cdot \sin C}=2\)
Solution:

Question 11.
(a² – b² + c²) tan B = (a² + b² – c²) tan C
Solution:

Question 12.
(b² – c²) cot A + (c² – a²) cot B + (a² – b²) cot C = 0
Solution:

Question 13.
\(\frac{b+c}{a}=\frac{\cos \mathbf{B}+\cos C}{1-\cos A}\)
Solution:

Question 14.
\(\sum a^3 \sin (B-C)=0\)
Solution:

Question 15.
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c
Solution:
L.H.S. = (b + c) cos A + (c + a) cos B + (a + b) cos C
= b cos A + c cos A + a cos B + c cos B + a cos C + b cos C
= (b cos C+ c cos B) +(c cos A + a cos C) + (a cos B + b cos A)
= a + b + c = R.H.S.

Question 16.
2 (bc cos A + ca cos B + ab cos C) = a² + b² + c²
Solution:
2 (bc cos A + ca cos B + ab cos C) = a² + b² + c²
\(=2\left(b c \times \frac{\left(b^2+c^2-a^2\right)}{2 b c}+c a \times \frac{c^2+a^2-b^2}{2 c a}\right.\) \(\left.+a b \times \frac{a^2+b^2-c^2}{2 a b}\right)\)
= b² + c² – a² + c² + a² – b² + a² + b²– c²
= a² + b² + c²

Question 17.
a (b² + c²) cos A + b (c² + a²) cos B + c(a² + b²) cos C = 3 abc.
Solution:
a (b² + c²) cos A + b (c² + a²) cos B + c(a² + b²) cos C
= ab² cos A + ac² cos A + bc² cos B + ba² cos B + ca² cos C + cb² cos C
= ab² cos A + ba² cos B + ac² cos A + ca² cos C + bc² cos B + cb² cos C
= ab (b cos A + a cos B) + ac (c cos A + a cos C) bc (c cos B + b cos C)
= abc = abc + abc = 3abc

Question 18.
a³ cos (B – C) + b³ cos (C – A) + c³ cos (A – B) = 3 abc
Solution:
1st term of L.H.S. = a³ cos (B – C)
= a² a cos (B – C)
= a² . 2R sin A cos (B – C)
= 2a²R sin (B + C) cos (B- C)
= a²R [sin (B + C + B – C) + sin (B + C – B + C)]
= a² R (sin 2B + sin 2C)
= a²R [2 sin B cos B + 2 sin C cos C]
= a² [2R sin B cos B + 2R sin C cos C]
= a² (b cos B + c cos C)
Similarly, 2nd term
= b² (c cos C + a cos A) and
3rd term = c² (a cos A + b cos B)
∴ L.H.S.= a²b cos B+a²c cos C+b²c cos C + b²a cos A + c²a cos A + c²b cos B
= ab (a cos B + b cos A) + bc (b cos C + c cos B) + ca (c cos A + a cos C)
= abc + bca + cab = 3abc = R.H.S.

Question 19.
a (cos B + cos C) = 2(b + c) sin² \(\frac{A}{2}\)
Solution:
Refer to Q. N. 13.

Question 20.
(b + c – a) tan \(\frac{A}{2}\) = (c + a – b) tan \(\frac{B}{2}\) = (a + b – c) tan \(\frac{C}{2}\)
Solution:

Question 21.
\((b+c-a)\left(\cot \frac{B}{2}+\cot \frac{C}{2}\right)\) \(=2 a \cot \frac{A}{2}\)
Solution:

Question 22.
(a – b)² cos² \(\frac{C}{2}\) + (a + b)² sin² \(\frac{C}{2}\) = c²
Solution:
L.H.S = (a – b)² cos² \(\frac{C}{2}\) + (a + b)² sin² \(\frac{C}{2}\)

Question 23.
1 – tan \(\frac{A}{2}\) tan \(\frac{B}{2}\) = \(\frac{c}{2}\)
Solution:

Question 24.
(b – c) cot \(\frac{A}{2}\) + (c – a) cot \(\frac{B}{2}\) + (a – b) cot \(\frac{C}{2}\) = 0
Solution:

Question 25.
cot A + cot B + cot C = \(\frac{a^2+b^2+c^2}{4 \Delta}\)
Solution:

Question 26.
a² cot A + b² cot B + c² cot C = 4Δ
Solution:

Question 27.
If \(\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}\) then prove C = 60°.
Solution:

or, cos C = 1/2 or, ∠C = 60°

Question 28.
If a = 2b and A = 3B, find the measures of the angles of the triangle.
Solution:
If a = 2b and A = 3B, we have \(\frac{a}{b}\) = 2
or, \(\frac{\sin A}{\sin B}\) = 2
or, sin A = 2 sin B         …..(1)
Also  sin A = sin 3B (as a = 3B)    …..(2)
∴ From (1) and (2), we have
2 sin B = sin 3B = 3 sin B – 4 sin³ B
or, 4 sin³ B – sin B = 0
or, sin B(4 sin² B – 1) = 0
or, sin B = 0, 4 sin² B = 1
Now sin B = 0 ⇒ B = 0 (Impossible)
∴ sin² B = \(\frac{1}{2}\) or, sin B = ± \(\frac{1}{2}\)
If sin B = \(\frac{1}{2}\) then ∠B = 30°
∴ A = 3B = 3 x 30° = 90°
∠C = 60°

Question 29.
If a⁴ + b⁴ + c⁴ = 2c² (A² + b²), prove that m∠ACB = 45° or 135°.
Solution:
a⁴ + b⁴ + c⁴ = 2c² (a² + b²)
or, a⁴ + b⁴ + c⁴ + 2a²b² – 2b²c² – 2c²a² = 2a²b²
or, (a² + b² – c²)² = 2a²b²
or, a² + b² – c² = ± √2 ab
or, \(\frac{a^2+b^2-c^2}{2 a b}=\pm \frac{1}{\sqrt{2}}\)
or, cos C = ± \(\frac{1}{\sqrt{2}}\)
∴ ∠C = 45° or 135°.

Question 30.
If x² + x + 1, 2x + 1, and x² – 1 are lengths of sides of a triangle, then prove that the measure of the greatest angle is 120°.
Solution:

Question 31.
if cos B = \(\frac{\sin A}{2 \sin C}\), prove that the triangle is isosceles.
Solution:
cos B = \(\frac{\sin A}{2 \sin C}\)
⇒ \(\frac{c^2+a^2-b^2}{2 c a}=\frac{a}{2 c}\) ⇒ c² + a² – b² = a²
or, c² = b² or, c = b
∴ The triangle is isosceles.

Question 32.
If a tan A + b tan B = (a + b)tan \(\frac{1}{2}\) (A + B) prove that the triangle is isosceles.
Solution:

Question 33.
If (cos A + 2 cos C) : (cos A + 2 cos B) = sin B : sin C prove that the triangles are either isosceles or right-angled.
Solution:
\(\frac{\cos A+2 \cos C}{\cos A+2 \cos B}=\frac{\sin B}{\sin C}\)
⇒ cos A sin C = cos A sin B + 2 cos B sin B
⇒ cos A (sin B – sin C) + (sin 2B – sin 2c) = 0
⇒ cos A (sin B – sin C) + 2 cos (B + C) sin (B – C) = 0
⇒ cos A (sin B – sin C) – 2 cos A sin (B – C) = 0
(∴ cos (B + C) = cos (π – A) = – cos A)
⇒ cos A = 0 or sin B – sin C – 2 sin (B – C) = 0
cos A = 0 ⇒ A = 90°
i.e. the triangle is right-angled. sin B sin C – 2 sin (B – C) = 0

Question 34.
If cos A = sin B – cos C, prove that the triangle is right-angled.
Solution:
cos A = sin B – cos C
or, cos C + cos A = sin B

or, 2C = π
or, C = \(\frac{\pi}{2}\)

Question 35.
If a², b², c²  be in A.P., prove that cot A, cot B, cot C are also in A.P.
Solution:
If a², b², c²  be in A.P.
then b² – a² = c² – b²
or, 2b² = c² + a²
or, \(b^2=\frac{c^2+a^2}{2}\)
or, 2b² = c² + a² …..(1)
We have to prove that cot A, cot B, cot C are in A.P.
i.e. to prove cot B – cot A = cot C – cot B
i.e. 2 Cot B = cot C + cot A
∴ R.H.S. = cot C + cot A

Question 36.
If sin A: sin C = sin (A – B) : sin (B – C) prove that a², b², c² are in A.P.
Solution:
\(\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}\)
or, sin A sin (B – C) = sin C sin (A – B)
or, sin (B + C) sin (B – C) = sin (A + B) sin (A – B)
or, sin² B – sin² C = sin² A – sin² B
or, 2 sin² B = sin² C + sin² A
or, \(2 \frac{b^2}{4 \mathrm{R}^2}=\frac{c^2}{4 \mathrm{R}^2}+\frac{a^2}{4 \mathrm{R}^2}\)
or, 2b² = c² + a²
or, b² – a² = c² – b²
∴ a², b², c² are in A.P

Question 37.
If the side lengths a, b, and c are in A.P., then prove that cos \(\frac{1}{2}\) (A – C) = 2 sin \(\frac{1}{2}\) B.
Solution:
If a,b, and c are in A.P. then b – a – c – b or, 2b = c + a
We have to prove that

Question 38.
If the side lengths a, b, and c are in A.P., prove that cot \(\frac{1}{2}\) A, cot \(\frac{1}{2}\) B, cot \(\frac{1}{2}\) C are in A.P.
Solution:

Odisha State Board Elements of Mathematics Class 11 Solutions CHSE Odisha Chapter 2 Sets Ex 2(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 2 Sets Exercise 2(b)

Question 1.
An examination was conducted in physics, chemistry, and mathematics. If P.C.M. denotes respectively the sets of students who passed in Physics, Chemistry, and Mathematics, express the following sets using union, intersection, and different symbols.
(a) Set of candidates who passed in Mathematics and Chemistry, but not in Physics.
(b) Set of candidates who passed in all three subjects.
(c) Set of candidates who passed in Mathematics only.
(d) Set of candidates who failed in Mathematics, but passed in at least one subject.
(e) Set of candidates who passed in at least two subjects.
(f) Set of candidates who failed in one subject only.
Solution:
An examination was conducted in Physics, Chemistry, and Mathematics. P, C, and M denoted the set of students who passed Physics, Chemistry, and Mathematics, respectively. Then.
(a) Set of candidates who passed in Mathematics and Chemistry, but not in Physics (M ∩ C) – P.
(b) Set of candidates who passed in all three subjects M ∩ C ∩ P.
(c) Set of candidates who passed in Mathematics only M – C – P.
(d) Set of candidates who failed in Mathematics, but passed in at least one subject (P ∪ C) – M.
(e) Set of candidates who passed in at least two subjects.
(f) Set of candidates who failed in one subject only.
(P ∩ C – M) ∪ (P ∩ M – C) ∪ (M ∩ C – P)

Question 2.
What can you say about sets A and B if
(i) A ∪ B= Φ
(ii) A Δ B = Φ
(iii) A \ B = Φ
(iv) A \ B = A
(v) A ∩ B= U, Where U is the Universal set, A \ B = U?
Solution:
(i) if A ∪ B = Φ then A = Φ =B
(ii) A Δ B = Φ ⇒ A = B
(iii) A – B = Φ ⇒ A ⊆ B
(iv) A – B = A ⇒ B = Φ
(v) A ∩ B = U ⇒ A = B = U
(vi) A – B = U ⇒ A = U and B = Φ

Question 3.
Are differences and symmetric commutative? Give reason.
Solution:
The difference of the two sets is not commutative but the symmetric of the two sets is commutative.
Reason:
Let x ∈ A – B ⇔ x ∈ A ∧ x ∉ B
≠ x ∈ B ∧ x ∉ A ⇔ x ∈ b – A
A- B ≠ B – A
But if y ∈ A Δ ⇔ y ∈ (A-B) ∪ (B – A)
⇒ y ∈ (B – A) ∪ (A – B) ⇔ y ∈ B Δ A
∴ A Δ B = B Δ A.

Question 4.
If B ⊂ C, prove that A/B = A/C. Is this result true when a difference is replaced by a symmetric difference? Give reason.
Solution:
If B ⊂ C, then x ∈ A ⇒ x ∈ C
Now x ∈ A – C ⇔ x: x ∈ A ∧ x ∉ C
⇔ {x: x ∈ A ∧ x ∉ B}
⇔ {x: x ∈ A – C}
∴ A – C = A – B
but, A Δ B ≠ A Δ C.

Question 5.
Prove the following :
(i) (A\B)\C = (A\C)\B = A\(B ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∪ B)
(iii) A Δ (B Δ C) = (A Δ B) Δ C
(iv) A ⊂ B ⇔ B’ ⊂ A’ A ⇔ A’ ∪ B = U
⇔ B’ ∩ A = Φ, where U is the universal set.
(v) A ∪ B = U and A ∩ B = Φ
⇒ B = A’
(iv) A ∪ B = A for all A ⇒ B = Φ
Solution:

(i) Let x ∈ (A – B) – C    ……(1)
⇔ x ∈ A- B ∧ x ∉ C
⇔ (x ∈ A ∧ x ∉ B) ∧ x ∉ C   ……(2)
⇔ (x ∈ A ∧ x ∉ C) ∧ x ∉ B
⇔ x ∈ A – C ∧ x ∉ B
⇔ x ∈ (A – C) – B   ……(3)
∴ from (2), we have
x ∈ A ∧ ∉ B ∧ x ∉ C
⇔ x ∈ A ∧ (x ∉ B ∧ x ∉ C)
⇔ x ∈ A ∧ x ∉ B ∪ C
[∴ ~ (p ∨ q) = ~ p ∧ ~ q]
⇔ x ∈ A – (B ∪ C)   …….(4)
∴ From (1), (3), and (4), we have
(A – B) – C = (A – C)-B = A – (B ∪ C)

(ii) Let x ∈ A ∩ (B ∪ C)
⇔ x ∈ A ∧ x ∈ B Δ C
⇔ x ∈ A ∧ (x ∈ B – C ∨ x ∉ C – B)
⇔ x ∈ A ∧ (x ∈ B ∧ x ∉ C ∨ x ∈ C ∧ x ∉ B)
⇔ x ∈ A ∧ (x ∈ B ∧ x ∈ A ∧ C) ∨ x ∈ A ∧ (x ∈ B ∧ x ∈ B ∧ x ∈ A ∧ x ∉ B)
⇔ (x ∈ A ∩ B ∧ x ∉ A ∩ C) ∨ (x ∈ A ∩ C  ∨ x ∉ A ∩ B)
⇔ x ∈ (A ∩ B) – (A ∩ C) ∨ x ∈ (A ∩ C) – (A ∩ B)
⇔ x ∈ (A ∩ B) Δ (A ∩ C)   ……(2)
∴ From (1) and (2), we have
A ∩ (B Δ C) = (A ∩ B) Δ (A ∩ C)

(iii) A Δ (B Δ C)
= A ∪ (B Δ C)- A ∩ (B Δ C)
= A Δ (B ∪ C)- A Δ (B ∩ C)
[ ∴ A ∪ (B Δ C) = A Δ (B ∪ C)]
and A n (B Δ C) = A Δ (B ∩ C)
= (A Δ B) ∪ C- (A Δ B) ∩ C
[∴ A Δ (B ∪ C) = (A Δ B) ∪ C and A Δ (B ∩ C) = (A Δ B) ∩ C
= (A Δ B) Δ C
∴ A Δ (B Δ C)= (A Δ B) Δ C
(Proved)

(iv) If A ⊂ B then x ∈ B’ or ⇒ x ∉ B
⇒ x ∉ B ⇒ x ∈ A’ (∴ A ⊂ B)
∴ B’ ⊂ A’
Again, let y ∈ A ⇒ y ∉ A’ ⇒ y ∈ B’
(∵ B’ ⊂ A’)
⇒ y ∈ B ∴ A ⊂ B
∴ A ⊂ B ⇔ B’ ⊂ A’
∴ Again as A ⊂ B, we have
U = A ∪ B = B = U, where U is the universal set of A and B.
∴ A’= B – A ⇒ A’ ∪ B
= (B – A) ∪ B = B = U
∴ A ⊂ B ⇒ A’ ∪ B = U
Again A’ ∪ B = U
⇒ A ∩ (A’ ∪ B) = A ∩ U = A
⇒ (A ∩ A’) ∪ (A ∩ B) = A
⇒ Φ ∪ (A ∩ B) = A
⇒ A ∩ B = A ⇒ A ⊂ B
Lastly, B’ = U’ = Φ
∴ B’ ∩ A = Φ

(v) Let A ∪ B = U and A ∩ B = Φ
∴ Let x ∈ B ⇔ x ∉ B’ ⇔ x ∉ U – B
⇔ x ∉ A ⇔ x ∉ A’

(vi) As A ∪ B = A for all A
we have B ⊂ A for all A
∴ B ⊂ A even for A = Φ Thus B = Φ

Question 6.
Prove all the results of sections 1.13 and 1.14 that are started without proof.
Solution:
(i) A ∪ B = B ∪ A
Let x ∈ A ∪ B ⇔ x ∈ A ∨ x ∈ B
⇔ x ∈ B ∨ x ∈ B ⇔ x ∈ B ∪ A

(ii) A ∩ B = B ∩ A
Let x ∈ A ∩ B ⇔ x ∈ A ∧ x ∈ B
∴ A ∩ B = B ∩ A

(iii) A ∩ (B ∪ C)
= (A ∩ B) ∪ (A ∩ C)
Let x ∈ A ∩ (B ∪ C)
⇔ x ∈ A ∧ x ∈ A ∪ C
⇔ x ∈ A ∧ (x ∈ A ∨ x ∈ C)
⇔ (x ∈ A ∩ B ∨ x ∈ A ∩ C)
⇔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈C)
⇔ x ∈ (A ∩ B) ∪ (A ∩ C)
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Question 7.
Prove that
(i) \(\mathbf{A}-\bigcup_{i=1}^n \mathbf{B}_i=\bigcap_{i=1}^n\left(\mathbf{A}-\mathbf{B}_i\right)\)
Solution:
Let x ∈ \(A-\bigcup_{i=1}^n B_i \Rightarrow x \in A \wedge x \notin \bigcup_{i=1}^n B_i\)
⇔ x ∈ A  ∧ x ∉(B₁ ∪ B₂ ∪….∪ B_n )
⇔ x ∈ A  ∧ (x ∉ B₁ ∧ x ∉ B₂ ∧…..∧ x ∉ B_n )
⇔ (x ∈ A  ∧ x ∉ B₁ ) ∧ (x ∈ A ∧ x ∉ B₂ ) ∧….∧ (x ∈ A  ∧ x ∉ B_n )
⇔ x ∈ A – B₁ ∧ x ∈ A – B₂ ∧……..∧ x ∈ A – B_n
⇔ x ∈ (A – B₁ ) ∩ (A – B_i ) ∩…..∩ (A – B_n )
⇔ \(x \in \bigcap_{i=1}^n\left(\mathrm{~A}-\mathrm{B}_{\mathrm{i}}\right)\)
∴ \(\mathrm{A}-\cup_{i=1}^n \mathrm{~B}_{\mathrm{i}}=\bigcap_{i=1}^n\left(\mathrm{~A}-\mathrm{B}_{\mathrm{i}}\right)\)

(ii) ∴ \(\mathbf{A}-\bigcap_{i=1}^n \mathbf{B}_i=\bigcup_{i=1}^n\left(\mathbf{A}-\mathbf{B}_i\right)\)
Solution:
Let x ∈ \(A-\bigcap_{i=1}^n B_i\)
⇔ x ∈ A ∧ x ∈ \(\bigcap_{i=1}^n \mathrm{~B}_{\mathrm{i}}\)
⇔ x ∈ A ∧ x ∉ (B₁ ∩ B₂ ∩….∩ B_n )
⇔ x ∈ A ∧ (x ∉ B₁ ∨ x ∉ B₂ ∨….∨ x ∉ B_n )
⇔ (x ∈ A ∧ x ∉ B₁ ) ∨ (x ∈ A ∧ x ∉ B₂ ) ∨….∨ (x ∈ A ∧ x ∉ B_n )
⇔ x ∈ A – B₁ ∨ x ∈ A – B₂ ∨……..∨ x ∈ A – B_n
⇔ x ∈ (A – B₁ ) ∪ (A – B₂ )……(A – B_n )
⇔ x ∈ \(\cup_{i=1}^n\left(\mathrm{~A}-\mathrm{B}_{\mathrm{i}}\right)\)
∴ \(\mathrm{A} \bigcap_{i=1}^n \mathrm{~B}_1=\bigcup_{u=1}^n\left(\mathrm{~A}-\mathrm{B}_i\right)\)

Question 8.
Prove that |A ∪ B ∪ C|
Solution:
= |A| + |B| + |C| + |A ∩ B ∩ C| – |A ∩ B| – |B ∩ C| – |C ∩ A|
L.H.S. =|A ∪ B ∪ C| = |A ∪ D|
where D = B ∪ C
= |A| + |D| – |A ∩ D|
( ∵ |A ∪ B| =|A| + |B| – |A ∩ B|)
= |A| + |B ∪ C| – | A ∩ (B ∪ C)|
= |A| + |B| + |C| – |B ∩ C| – |(A ∩ B) ∪ (A ∩ C)|
= |A| + |B| + | C |- |B ∩ C| – [|A ∩ B| + |A ∩ C| – |(A ∩ B) ∩ (A ∩ C)|]
= |A| + |B| + |C| – |B ∩ C| – |A∩ B| – |A ∩ C| + |A ∩ B ∩ Cl
= |A| + |B| + |C| – | A ∩ Bl – |B ∩ C| – |C ∩ A| + |A ∩ B  ∩ Cl = R.H.S.

Question 9.
If X and Y are two sets such that X ∪ Y has 20 objects, X has 10 objects and Y has 15 objects; how many objects does X ∩ Y have?
Solution:
Given |X ∪ Y| = 20
|X| = 10
|Y| = 15
We know that |X ∪ Y|
= |X| + |Y| – |X ∩ Y|
⇒ 20 = 10 + 15 – |X ∩ Y|
⇒ |X ∩ Y| = 25 -20 = 5
∴ X ∩ Y has 5 elements.

Question 10.
In a group of 450 people, 300 can speak Hindi and 250 can speak English. How many people can speak both Hindi and English?
Solution:
Let H = The set of people who can speak Hindi
E = The set of people who can speak English.
According to the question we have
|H ∪ E| = 450, |H| = 300,
|E| = 250
We want to find 1 H ∩ E
|H ∩ E| = |H| + |E|-  |H ∪ E|
= 300 + 250 – 450 = 100
∴ 100 people can speak both Hindi and English.

Question 11.
In a group of people,37 like coffee, 52 like tea and each person in the group likes at least one of the two drinks. 19 people like both tea and
coffee, how many people are in the group?
Solution:
Let T = The set of persons who like Tea. ,
C = The set of persons who like coffee According to the question
|C| = 37, |T| = 52 and |T ∩ C| = 19
Total number of persons in the group
= |T ∩ C| = |T| + |C| – |T ∩ C|
= 37 + 52 – 19 = 70

Question 12.
In a class of 35 students, each student likes to play either cricket or hockey. 24 students like to play cricket and 5 students like to play both games; how many students play hockey?
Solution:
Let C = Set of students like to play cricket
H = The set of students like to play Hockey.
According to the question
|C ∪ H| = 35
|C| = 24, |C ∩ H| = 5
Now |C ∪ H| = |C| + |H| – |C ∩ H|
⇒ 35 = 24 + |H| – 5
⇒ |H| = 16
16 students like to play Hockey.

Question 13.
In a class of 400 students, 100 were listed as taking apple juice, 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice or orange juice.
Solution:
Let A = The set of students take apple juice
O = The set of students take orange juice
According to the question
|A| = 100, |O| = 150 and
|A ∩ O| = 75
∴ Number of students take at least one of the juice = |A ∪ O|
= |A| + |O| – |A ∩ O|
= 100 + 150 – 75 = 175
Total number of students
= |U| = 400
Number of students taking neither of these juice
= |U| – |A ∪ O|
= 400 – 175 = 225

Question 14.
In a group of 65 people, 40 like cricket, and 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Solution:
Let C = The set of persons like cricket
T = The set of people who like tennis.
According to the question

|C| = 40 , |C ∩ T| = 10 and
|C ∪ T| = 65
A number of people like tennis only but not cricket = |C ∪ T| – |C|
= 65 – 40 = 15
Number of persons like tennis
= |C ∩ T| – |C| + |C ∩ T|
= 65 – 40 + 10 = 25

Question 15.
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12
people liked products C and A, 14 people liked products B and C and 8 liked all the three products, find how many liked products C only.
Solution:
Let E = Set of persons like product A
F = Set of persons like product B
G = Set of persons like product C

According to the question
a + b + d + e = 21
c + b + f + e = 26
g + f + d + e = 29
b + e = 14
f + e = 14
d + e = 12
e = 8
⇒ e = 8    g = 11
d = 4        c = 6
f = 6          a = 3
b = 6
Number of persons like product C only
O = g = 11