Class 12

Odisha State Board CHSE Odisha Class 12 Foundations of Education Solutions Unit 1 Contribution of Educators Objective Questions and Answers.

CHSE Odisha 12th Class Foundations of Education Unit 1 Contribution of Educators Objective Questions

Question 1.
What is the other name of Basic Education?
(a) Vocational Education
(b) Professional Education
(c) New Train
(d) New Policy of Education
Answer:
(c) New Train

Question 2.
Which committee designed the curriculum of Basic Education?
(a) Hartog Committee
(b) Jakir Hussain Committee
(c) Basic Education Committee
(d) Mudaliar Committee
Answer:
(b) Jakir Hussain Committee

Question 3.
When did Gandhiji take birth?
(a) 2nd. October 1863
(b) 2nd. October 1869
(c) 9th. October 1937
(d) None of the above
Answer:
(b) 2nd. October 1869

Question 4.
In which year Wardha Education Conference was held?
(a) August 15,1947.
(b) October 22, 23, 1937
(c) January 30,1869
(d) 9, October, 1877
Answer:
(b) October 22, 23,1937

Question 5.
Who is the propounder of Class less Society in Odisha?
(a) Pandit Nilakantha Das
(b) RadhanathRay
(c) Madhusudan Das
(d) Gopabandhu Das
Answer:
(a) Pandit Nilakantha Das

Question 6.
Who established Satyabadi Vana Vidyalaya?
(a) Gandhi
(b) Gopabandhu
(c) Tagore
(d) Aurovindo
Answer:
(b) Gopabandhu

Question 7.
Where was Van Vidyalaya situated?
(a) Puri
(b) Satyabadi
(c) Dhenkanal
(d) Pipili
Answer:
(a) Puri

Question 8.
Who was the propounder of Basic Education?
(a) R. N. Tagore
(b) Mahatma Gandhi
(c) Sri Aurovindo
(d) Gopabandhu
Answer:
(b) Mahatma Gandhi

Question 9.
Where did the Basic Education Training Centre opened in Odisha?
(a) Cuttack
(b) Angul
(c) Puri
(d) Dhenkanal
Answer:
(b) Angul

Question 10.
Satyabadi School how many students were enrolled first?
(a) 25
(b) 19
(c) 23
(d) 24
Answer:
(b) 19

Question 11.
Education is the ‘Reconstruction of Experience’ whose definition is this?
(a) James Ross
(b) Mahatma Gandhi
(c) John Dewey
(d) Kingsley
Answer:
(c) John Dewey

Question 12.
Who is the propounder of‘Tri-polar Process of Education?
(a) John Adams
(b) John Rousseau
(c) John Dewey
(d) Gurukala Ashram
Answer:
(c) John Dewey

Question 13.
Who is the writer of the book ‘Emile’?
(a) John Dewey
(b) John Rousseau
(c) Gandhi
(d) Gopabandhu
Answer:
(b) John Rousseau

Question 14.
Who is the Editor of the Newspaper ‘The Samaja’?
(a) Gopabandhu
(b) Gandhi
(c) Radhanath Ratha
(d) None of the above
Answer:
(a) Gopabandhu

Question 15.
‘Harijan Patrika’ is written by?
(a) Gandhi
(b) Gopabandhu
(c) John Dewey
(d) None
Answer:
(a) Gandhi

Question 16.
In which year Basic Education Schools were opened in Rural areas?
(a) 1945
(b) 1937
(c) 1936
(d) 1947
Answer:
(a) 1945

Question 17.
When Basic Education was created?
(a) 1945
(b) 1937
(c) 1947
(d) 1950
Answer:
(b) 1937

Question 18.
The age range of Basic Education is?
(a) 6 – 14
(b) 7 – 14
(c) 6 – 15
(d) 5 – 14
Answer:
(b) 7 – 14

Question 19.
Who is the propounder of Craft Education?
(a) M. K. Gandhi
(b) R. N. Tagore
(c) Gopabandhu
(d) Rousseau
Answer:
(a) M. K. Gandhi

Question 20.
Which of the following leaders first attempted for legislation of Compulsory Primary Education in India?
(a) Gandhi
(b) G. K. Gokhale
(c) Aurovindo
(d) Gopbandhu
Answer:
(b) G K. Gokhale

Question 21.
Who is the propounder of ‘Integral Education’?
(a) Aurovinda
(b) R. N. Taogre
(c) John Rousseau
(d) Gandhi
Answer:
(a) Aurovindo

Question 22.
Education is the harmonious development of individual’s personality- man and child, with mind, body and spirit – Who told this?
(a) Gopbandhu
(b) Gandhi
(c) Aurovindo
(d) Rousseau
Answer:
(b) Gandhi

Question 23.
Who is the champion of Nationalism?
(a) Gandhi
(b) Rousseau
(c) Tagore
(d) John Dewey
Answer:
(b) Rousseau

Question 24.
In which year Satyabadi Vana Vidyalaya became a National School?
(a) 1909
(b) 1921
(c) 1915
(d) 1929
Answer:
(b) 1921

Question 25.
Who is the pioneer of Nationalism?
(a) Gandhi
(b) Rousseau
(c) Gopabandhu
(d) Aurovindo
Answer:
(b) Rousseau

Question 26.
Which of the following two educators believed in open air schooling?
(a) Gandhi and Rabindranath
(b) Gopabandhu and Gandhi
(c) Aurovindo and Gopabandhu
(d) Rabindranath and Gopabandhu
Answer:
(b) Gopabandhu and Gandhi

Question 27.
Who emphasized mother tongue as the medium of instruction?
(a) Gandhi
(b) Gopbandhu
(c) Both Gandhi and Gopabandhu
(d) None of the above
Answer:
(c) Both Gandhi and Gopabandhu

Question 28.
Where did Gopabandhu take birth?
(a) Suando
(b) Satyabadi
(c) Sakhigopal
(d) Saptasati
Answer:
(a) Suando

Question 29.
Who is the propounder of‘Craft Education’?
(a) M. K. Gandhi
(b) Rousseau
(c) Gopabandhu
(d) R. N. Tagore
Answer:
(a) M. K. Gandhi

Question 30.
What is taught in the class is real education – Who told this?
(a) Gandhi
(b) Gopabandhu
(c) Tagore
(d) Rousseau
Answer:
(b) Gopabandhu

Question 31.
In which year did Gopabandhu start the experiment on open air schooling?
(a) 2nd October 1909
(b) 12th August 1909
(c) 15th August 1909
(d) 26th August 1909
Answer:
(b) 12th August 1909

Question 32.
In which magazine Basic Education Curriculum was published?
(a) TheSamaja
(b) TheHarijana
(c) TheDharitii
(d) None of the above
Answer:
(b) The Harijana

Question 33.
Which aspect was neglected in Basic Education?
(a) Craftwork
(b) Creativity
(c) Aesthetic
(d) Writing
Answer:
(b) Creativity

Question 34.
Which is not normally a mass media of Education?
(a) Magazine
(b) Newspaper
(c) Computer
(d) Television
Answer:
(c) Computer

Question 35.
Who employed activity centered curriculum?
(a) Gandhi
(b) Gopabandhu
(c) John Dewey
(d) None
Answer:
(a) Gandhi

Question 36.
Who introduced ‘Correlation Teaching Methods’ in his curriculum?
(a) John Dewey
(b) John Rousseau
(c) Gandhiji
(d) Gopabandhu
Answer:
(c) Gandhiji

Question 37.
Who prepared the curriculum for Basic Education?
(a) G. K. Gokhale
(b) Dr. Jakir Hussain
(c) S. Radhakrishnan
(d) Utkalmani
Answer:
(b) Dr. Jakir Hussain

Question 38.
Whose philosophical thought is related to Naturalism and Negative Education?
(a) Gandhi
(b) John Dewey
(c) Rousseau
(d) Tagore
Answer:
(c) Rousseau

Question 39.
Gopbandhu died on?
(a) 17 June 1928
(b) 27 June 1928
(c) 7 June 1928
(d) None of the above
Answer:
(a) 17 June 1928

Question 40.
Inexpensive education is introduced by?
(a) John Rousseau
(b) Gopbandhu
(c) John Dewey
(d) Gandhi
Answer:
(b) Gopbandhu

Question 41.
What is the present structure of education in India?
(a) 10+3+2
(b) 10+2+3
(c) 11+2+2
(d) 10+3+3
Answer:
(b) 10+2+3

Question 42.
On which aspect Gandhi, Gopabandhu and Tagore emphasized?
(a) SUPW
(b) Craft Education
(c) Mother tongue as instruction
(d) None of the above
Answer:
(c) Mother tongue as instruction

Question 43.
Who considered school as ‘Man made industry’?
(a) John Dewey
(b) Gopabandhu
(c) Gandhi
(d) Tagore
Answer:
(b) Gopabandhu

Question 44.
Who introduced both activity method and play method?
(a) Aurovindo
(b) Rousseau
(c) John Dewey
(d) Gandhi
Answer:
(b) Rousseau

Question 45.
Who Introduced ‘Self Education?
(a) Rousseau
(b) Gopabandhu
(c) Aurovindo
(d) Gandhi
Answer:
(a) Rousseau

Very Short-Answer Type Questions

Question 1.
What did Gopabandhu consider the school?
Answer:
‘Man made industry’.

Question 2.
Who is the propounder of ‘Nature Endowment Theory’?
Answer:
Jean Jacues Rousseau.

Question 3.
What was the main aim of Satyabadi School?
Answer:
Nationalism and against social evils.

Question 4.
When Basic Education Conference was held at Wardha?
Answer:
In 1937, October 22nd and 23rd.

Question 5.
What was the philosophical foundation of Gandhi?
Answer:
Truth and Non-violence.

Question 6.
Which education system was in-expensive education?
Answer:
Open Air Schooling.

Question 7.
What was Rousseau’s method of teaching?
Answer:
Activity method and play method.

Question 8.
What is Education to Gandhi?
Answer:
To Gandhi, ‘Education is an all round drawing out of the best in child and man with mind body and spirit ’.

Question 9.
Name the propounder of Religious and Negative Education.
Answer:
RousseaRousseakuk.

Question 10.
What is ‘Nai Talim’ ?
Answer:
Nai Talim is the another name of Basic Education.

Question 11.
What is Auro Ville?
Answer:
Auro ville mean “Aurovindran Learning Centre”.

Question 12.
Mention aims of Satyabadi System of Education.
Answer:
To inculcate nationalism, patriotism and eradicate social evils.

Question 13.
Who is contemporary to Tagore in Odisha?
Answer:
Gopbandhu.

Question 14.
Wardha scheme, what it means?
Answer:
The All IndiaNational Education Conference convened at Wardha on 22nd and 23rd October, 1937, and the scheme is known as “Wardha Scheme”.

Question 15.
‘Education is the reconstruction of experience’ who told this?
Answer:
John Dewey.

Question 16.
Give the educational philosophy of Gopabandhu.
Answer:
“Education is the building of the hearts of the people”, is the educational philosophy of Gopabandhu.

Question 17.
What do you mean by “Open Air Schooling”?
Answer:
Open Air Schooling means teaching activity done under the sky in the surroundings of natural environment.

Question 18.
Give one similarity of Basic Education and Open Air Schooling.
Answer:
“Renaissance” is the common idealism of both Basic Education and Open Air Schooling.

Question 19.
In Satyabadi School how many students enrolled first?
Answer:
19 students

Question 20.
Where in which year Aurobindo was born?
Answer:
Sri Aurobindo was born on August 15, 1872, in Calcutta.

Question 21.
What was the name of Education of Sri Aurobindo?
Answer:
Integral Education.

Question 22.
What was the idea of Aurovindo regarding the Teacher?
Answer:
According to Sri Aurovindo the teacher must be a friend, philosopher and guide to pupils.

Question 23.
What is the other name of “Auroville”?
Answer:
‘The City of Dawn”.

Question 24.
What was the first name of‘The Mother”?
Answer:
The first name of the Mother” was Meera, the daughter of Roul Richard, France.

Question 25.
What is Satyabadi System of Education?
Answer:
Satyabadi System of Education is a serious experiment in Open Air Teaching of Gopabandhu.

Question 26.
What do you mean by ‘Basic Education?
Answer:
Education which linked with the Basic needs of life like food, clothing and shelter is known as Basic Education.

Fill in the Blanks with Appropriate Words

1. The other name of Basic Education Is ________
Answer: Nai-Talim

2. _______ propounded craft centered education in India.
Answer: M. K. Gandhi

3. ______ number of students enrolled first in Satyabadi School?
Answer: 19

4. _______ propounded religious and negative education.
Answer: Rousseau.

5. _______ is the contemporary educator to Tagore?
Answer: Gopabandhu.

6. Nature Experiment Theory was propounded by ________.
Answer: Jean Jacques Rousseau.

7. _______ is the propounder of Naturalism.
Answer: Rousseau.

8. The propounder of Craft Centred Education is __________.
Answer: M. K. Gandhi.

9. “Classless Society Education” in Odisha is introduced by _________?
Answer: Pandit Nilakantha Das

10. The write of “emile” is ________
Answer: Rousseau.

11. Basic Education curriculum published in the magazine ________.
Answer: The Harijan.

12. In ________ year Satyabadi School become a National School.
Answer: 1921.

13. “Education is the reconstruction of Experience” is the definition of ______ .
Answer: John Dewey.

14. “Auroville” is so named by ______ in ______ year.
Answer: “The Mother” in 1950.

15. ________ was the philosophical foundation of Gandhi?
Answer: Truth and Non-violence.

16. ________ Education system was in-expensive.
Answer: Satyabadi.

Question 17. The teaching in School is real education told by________.
Answer: Gopabandhu.

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Odisha State Board CHSE Odisha Class 12 Foundations of Education Solutions Unit 4 Educational Statistics Questions and Answers.

CHSE Odisha 12th Class Foundations of Education Unit 4 Educational Statistics Questions and Answers

Objective Type Question and Answers

Question 1.
What is the formula to find out the mean?
(a) \(\frac{\sum F x}{N}\)
(b) \(\frac{F x}{N}\)
(c) L+\(\frac{N_2 F}{M}\)
(d) H.S.-L.S.
Answer:
(a) \(\frac{\sum F x}{N}\)

Question 2.
Estimate the mode of the scores:
8, 9, 8, 9,10,11,8,9,10
(a) 9
(b) 10
(c) 11
(d) 8
Answer:
(d) 8

Question 3.
What is a formula to find median?
(a) H.S. – L.S.
(b) \(\frac{Q_3-Q_1}{Q}\)
(c) 3 median – 2 mean
(d) None of the above
Answer:
(c) 3 median – 2 mean

Question 4.
4. Estimate the mean of the scores: 8,9,10,11,12,13
(a) 8.5
(b) 10.5
(c) 11.5
(d) 12.5
Answer:
(b) 10.5

Question 5.
5. Estimate the class interval of25-29.
(a) 4
(b) 3
(c) 5
(d) None
Answer:
(c) 5

Question 6.
6. Estimate the mean of the scores:
2, 8, 10, 25,45
(a) 10
(b) 20
(c) 30
(d) 40
Answer:
(a) 10

Question 7.
7. Select the formula to find out the median?
(a) A.M.+\(\frac{\sum F d}{N}\) Xi
(b) U\(\frac{\frac{N}{2}-F}{f m}\) xi
(c) 3 median – 2 mean
(d) None
Answer:
(b) U\(\frac{\frac{N}{2}-F}{f m}\) xi

Answer in single word / single sentence

Question 1.
How can you find out Mode?
Answer:
The mode can be calculated as 3 median – 2 mean.

Question 2.
What is Mean?
Answer:
The mean is the sum of the group of scores divided by the number of scores.
M = \(\frac{\sum x}{N}\)

Question 3.
What do you mean by Median?
Answer:
The median is the midpoint or mid-value of the distribution of scores.

Question 4.
Give one use of Mean?
Answer:
Mean is used when a quick and easily computerized measure of central tendency is needed.

Question 5.
Give one use of Mode?
Answer:
Mode is used when the quickest of the central tendency is needed.

Question 6.
What is the limitation of the Median?
Answer:
A median is an algebraic measure and hence not suitable for the function of algebraic treatment.

Question 7.
How can you calculate the Mean?
Answer:
mean = L +\(\left|\frac{\frac{N}{2}-F}{F m}\right|\) Xi

Question 8.
What is the limitation of Mode?
Answer:
Mode is not based on all the observations in a series.

Question 9.
What is the formula for finding out the mean of the ungrouped data?
Answer:
Mean = \(\frac{\sum X}{N}\)

Question 10.
What is the mode of an ungrouped set score?
Answer:
The score which occurs frequently or number of times is known as a mode.

Question 11.
How can you measure the median in a simple method
Answer:
Median = \(\left(\frac{N+1}{2}\right)^{t h}\) Term

Question 12.
Find out the median of the following scores:
5.4, 3, 7, 8, 10, 4,6
Answer:
Arrange is chronological order :
3, 4, 5, 6, 7, 8, 9, 10
Median = \(\left(\frac{N+1}{2}\right)^{t h}\) Score
= \(\frac{8+1}{2}\)
= \(9 / 2\)
= 4.5^th
= \(\frac{7+8}{2}\)
= \(\frac{15}{2}\)
=7.5

Question 13.
what is the midpoint of class interval: 1.3 -1.7?
Answer: The midpoint of C, I.
= 1.3- 1.7= 1.5

Question 14.
What is Mode? Give example.
Answer:
The mode is the score that occurs frequently in a group.
Example – 5, 6, 7, 8,4, 8, 9 Here mode is = 5.

Question 15.
Find out H.S.&L.S. of 14.5 -19.5
Answer:
14- 19 & 15-20

Question 16.
find out the mean: 40, 30,20, 80,60
Answer:
Mean = \(\frac{\sum X}{N}\)
= \(\frac{160}{5}\)
= 32

Question 17.
Find out H.S. & L.S. of the three class intervals 100 – 109,10 – 14, 80 – 84
Answer:

C.I	L.S.	H.S.
100 – 109	99.5	109.5
10-14	9.5	14.5
80-84	79.5	84.5

Very Short Type Questions With Answer

Question 1:
Explain how the knowledge of statistics is helpful to a teacher.
Answer:
1. Statistics helps the teacher to draw general conclusions.
2. It enables the teacher to predict the future performance of the pupils.

Question 2:
What is measures of central tendency?
Answer:
Methods of finding out the central values or average value of a statistical series of quantitative information mean, median and mode are the measures of central tendency, to find out the arithmetic averages.

Question 3:
Explain the different steps for preparing a frequency distribution table.
Answer:
The following are the steps of frequency distribution:

• Find out the range,
• Deciding the length of class interval.
• Making tallies to find out the frequency of scores with each class.
• Making the total number of tallies in each class.

Short Type Questions With Answers

Question 1:
1. What is statistics? What is measure of central tendency?
Answer:
Statistics is a branch of mathematics. It is a science dealing with numerical facts collected systematically with the purpose of action and study. So statistics is a science dealing with the collection, analysis and interpretation of data. It always deals with numerical facts and figures. The data are collected from various sources, then it is analyzed, and tabulated for interpretation. Now, statistics is used to measure population growth in education, agriculture and industry in finding out the numerical facts and figures statistics is used.

Measures of Central Tendency

Measures of central tendency is the method of finding out the central values or average value of statistical series of quantitative information mean, median and mode are the measures of central tendencies. Mean is the arithmetical average of the scores of a group. Median is the score point in a distribution below and above which half of the cases live. Mode means the score which occurs most frequently.

Question 2:
Write the function of statistics in education.
Answer:

• Statistics helps in the collection, presentation of data in a systematic manner.
• It helps in classification.
• It provides a technique for making comparison of examination results among the students.
• It helps to understand complex data by simplifying it.
• It helps to study the relationship between different results of examinations.

Question 3:
Explain the use of mean and Median.
Answer:
Use of Mean :

• When the distribution is symmetrical score and are uniformly distributed mean is the centre of gravity have the same value and other scores distribute it to its distribution
• Mean is the most stable measure of central tendencies and is often in demand at the greatest statistical calculations.
• The mean has the greatest stability so when a measure of central tendency with the greatest stability is needed the mean is used.

Use of Median :

The Median is used when the exact midpoint the 50 % of the distribution is desired.

• It is scored that would markedly affect the mean.
• When it is desired that certain scores should influence the central tendency, the median is used.
• It is used when the distribution has no upper or lower class interval of complicated length.

Long Type Questions With Answers

Question 1.
What is statistics? What is measures of E.T. ? Explain.
Answer:
Statistics is a branch of mathematics. It is a science dealing with numerical facts collected systematically with the purpose of action and study.

So statistics is a science dealing with the collection, analysis and presentation of data and interpretation. Statistics always deals with numerical facts and figures. The data are collected systematically from various sources then it is analysed, and tabulated for interpretation. How statistics is used to measure population, growth in education, agriculture and industry.

Helpful to a Teacher:

•  It helps the teacher to provide the most exact type of description.
• It makes the teaching definite and exact is procedures and thinking.
• It enables the teacher to draw general conclusions.
• It enables the teacher to predict to future performances of the pupils.

Functions of statistics:

• Statistics help into the collection and presentation the data in a systematic manner.
• It helps to understand complex data by simplifying it.
• It helps to classify data.
• It provides a technique for making comparisons of exams and results among students.
• It helps to study the relationship between different results of examination.

Needs :

• Statistical methods are used for standardisation of various tests and measures like – achievement tests in various subjects, like intelligence tests, interest, aptitude scales and various other measures of personality measurement.
• The scores obtained from various tests and measures are always relative. Statistical methods help their proper presentation, comparison, analysis and interpretation, and statistics help better.
• To make predictions regarding to further progress.
• Statistics is used for the ID of our students to study their interest and I.Q.
• To compare the variability some Mean, Median and Mode and its central tendency.

What is the Measures of C.T :
Central value or central tendency means the value which lies in the centre or middle of the distribution. Thus it is located somewhere in between two extreme values in the distribution. In simple words, it is the value of a variable around which other values are distributed
A measure of C.T. of the measure of a single typical value which is the best representative of the whole group.

Measures of CT is the methods finding at the central values on the average value of a statistical series of quantitative information. Mean median and mode are the measures of central tendencies.

Mean is the score point in a distribution in between and above which half of the cases like a sum total of the score divided by its number is mean.
So μ = \(\frac{\sum X}{N}\)

where ∑ = sum total
X = individual scores
N = total no. of frequencies

Mean = \(\frac{\text { the sum of all the values }}{\text { the number of values }}\)

Suppose, 5 boys of a class are selected for the cricket team and their ages are 15 16 16 17 17 18.
We can find out the arithmetic mean of the ages of students by adding the ages of 5 students and then dividing that sum by the number of boys 5.

Arithmetic mean \(\frac{75}{5}\) = 15

Calculation of mean by the short method:

Marks	No of students	Midpoints	Deviations	Fd
C l	F	(m)	(d)
0-10	6	5	-30	-180
10-20	14	15	-20	-280
30-40	16	25	-10	-160
40-50	27	35	o	0
50-60	22	45	10	220
	15	55	20	300
	N=100			F d = -100

Arithmetic mean = \(\overline{\mathbf{X}}\) = A = \(\frac{\sum \mathrm{fd}}{\mathbf{N}}\)
Where A = Arithmetic means 35
∑Fd = Total deviations = -100
N=∑f=100
\(\overline{\mathbf{X}}\)
= 35+\(\frac{-100}{100}\)
=35-1
=34

Median: Median is the mid value of the middle item of the series when the series are arranged in ascending or descending order of magnitude, smallest to longest or largest to smallest.

Median = value of \(\left(\frac{N+1}{2}\right)^{\text {th }}\) item

For example – if the ages of five boys are 7, 8, 9, 10, 11

Median age will be the age of \(\left(\frac{N+1}{2}\right)^{\text {th }}\)
Boy = \(\left(\frac{S+1}{2}\right)\) boy

i.e. the 3rd boy which is equal to 9 years.
Example: Find out the median of the grouped data.

C.T.	F	Count
45-50	1	30
41-50	2	29
35-40	2	27
31-34	2	25
26-30	4	23
21-25	5	19
16-20	1	14
11-15	6	13
6-10	4	7
1-05	3	3
	N=31

Here L= 25 \(\frac{N}{2}\)
= \(\frac{30}{2}\)
= 15

F= I4 Fm=5 i=5
Median = 20.5+\(\frac{15-14}{5}\) x 5
= 20.5+\(\frac{1}{5}\) x5
= 20.5+\(5 / 5\)
= 20.5 + 1
= 21.5

Mode = mode is the frequently occurring value in a distribution.
Example = Calculate the mode from the following data of the marks obtained by 20 students 15, 10,25, 30, 25,40
Mode = 25
Formula = 3 median – 2 mean = mode
(1) Use of Mean: Mean is used:

• When the distribution is systematic, scores are distributed. Mean is the centre of gravity
  of the scores of distribution.
• Mean is the most stable measure of C.T. and greater statistical calculation.
• When the measure of C.T. with its greatest stability is needed the mean is used.

(2)The use of Median :

• The Median is used when the exact midpoint of the distribution is desired.
• It is scores which would markable affect the mean.
• When it is desired that certain scores should be with CT median is used.
• It is used when the distribution has no upper or lower interval of complicated length. Use of Mode: Mode is used when the quickest of the central tendency is needed.

Question 2.
Find out mean and median of the distribution?

C.L	F
10 – 14	4
15 – 19	5
20 – 24	9
25 – 29	18
30 – 34	11
35 – 39	5
40 – 44	6
45 – 49	3
50 – 54	1
	N=60

Answer:

C.I.		F	d1	Fd1
15-19	7	-3	-21	-39
20-24	5	-2	-10
25-29	8	-1	-8
30-34	15	0	0
35-39	10	+1	10	+49
40-44	6	+2	12
45-49	9	+3	27
	N=60	+3	∑fd1 = 10

Mean = A.m+ \(\frac{\sum f d}{N}\) xi
= 32+\(\frac{10}{60}\) x 5
= 32+ \(\frac{10}{12}\)
= 3282

Median = L + \(\left(\frac{\frac{N}{2}-F}{F m}\right)\) x i

Where L = 29.5
N/2=30 ,F=20 ,Fm=15 ,¡=5
= 29.5+\(\left(\frac{30-20}{15}\right)\) x5
= 29.5+\(\frac{10}{15}\) x 5
= 29.5+\(\frac{50}{15}\)
= 29.5 + 3.3
= 32.8

Question 3.
Calculate the mean, median and mode of the following frequency distribution?

CL	F
20-21	1
18-19	2
16-17	3
14-15	5
12-13	3
10-11	3
8-9	2
6-7	1
	N= 20

Answer:

C.I.	Midpoint	F	X1	Fx1
20-21	20.5	1	3	3
18-19	18.5	2	2	4
16-17	16.5	3	1	3
14-15	14.5	5	0	0
12-13	12.5	3	-1	-3
10-II	10.5	3	-2	-6
8-9	8.5	2	-3	-6
8-7	6.5	1	-4	-4
	N=20		Fx’ = -9

Mean = A.M. + \(\frac{\sum F x^1}{N}\) x i

Where,   A.M. = Assumed Mean
∑ = Sum of
Fx¹ = Frequency x deviates of the midpoints of the class intervals from the assumed mean in terms of class interval unit.
¡ = size of the class interval
Here,  A.M. = 14.5
∑FX¹ = -9
i=2

Mean = A.M. + \(\frac{\sum F x^1}{N}\) x i
= 14.5+\(\frac{-9}{20}\) x 2
= 14.5 + -9
= 13.6

Median = L + \(\left(\frac{\frac{N}{2}-F}{F m}\right)\) x i
Where L = Lower limit of the class interval containing the Median i.e. \(\frac{n}{2}\)th score.
F = Cumulative frequency up to ‘L’
Fm = Frequency of the C.I. containing the Medium
i = size of the class interval

When counting from the top or bottom we found that \(\frac{n}{2}\)th score i.e. the 10th. score lies in the class interval 14-15.
∴ L = 13.5
F=9 Fm=5 ¡=2

Mdn = 13.5+\(\left(\frac{\frac{20}{2}-9}{5}\right)\) x2
=13.5+4=13.9
Mode = 3Mdn – 2 Mean
=3 x 13.9 – 2 X 13.6
=41.7 – 27.2
=14.5
∴ Calculated Mean = (3.6), Median = (13.9) and Mode = 14.5

Question 4.
Calculate the mean, and median of the following frequency distribution?

C.I.	F
90-94	1
85-89	3
80-84	5
75-79	7
70-74	9
65-69	6
60-64	4
55-59	3
50-54	2
	N = 40

Answer:

C.I.		Midpoint	F	X1	Fx1
90-94	92	1	4	4
85-89	87	3	3	9
80-84	82	5	2	10
75-79	77	7	1	7
70-74	72	9	0	0
65-69	67	6	-1	-6
60-64	62	4	-2	-8
55-59	57	3	-3	-9
50-54	52	2	-4	-8
		N=40		∑Fx’ = -l

Mean = A.M. + \(\frac{\sum F x^1}{N}\) x i
= 72+\(\frac{-1}{10}\) x 5
=72-125
=71.89

Median = L=69.5 L=69.5
N/20 =2O  N/20 =20
F= 15
Fm = 9
i = 5
= L+\(\left(\frac{\frac{\mathrm{N}}{2}-\mathrm{F}}{\mathrm{Fm}}\right)\) xi
= 65.5+\(\frac{(20-15) \times 5}{9}\)
= 65.9+\(\frac{25}{9}\)
= 69.5 + 2.77
= 72.2

Questions 5.
Find Mean in a short method?

Class interval	Frequencies
90-94	2
85-89	2
80-84	4
75-79	8
70-74	6
65-69	11
60-64	9
	N = 42

Answer:
Calculation means by the short method :

C.I.	(M)	F	(d1)	(Fd1)
90-94	92	2	+3	+6
85-89	87	2	+2	+4
80-84	82	4	+1	+4
75-79	77	8	0	0
70-74	77	6	-1	-6
65-69	67	11	-2	-22
60-64	62	9	-3	-27
		N=42		∑fd1= -41

Mean = A.M. + \(\frac{\sum F x^1}{N}\) x i
Where, A.M. = 77¹
∑fd¹ = -41
\(\frac{n}{2}\) = 21
i = 5

So, mean  77+\(\frac{-41}{42}\) x5
= 77+\(\left(\frac{-41}{42} \times 5\right)\)
= 77+\(\frac{205}{42}\)
= 77 + -4.8
= 72.7

questions 6.
Find out the Median of the scores?

C.I.	10-19	20-29	30-39	40-49	50-59	60-69
F	10	05	15	20	10	10

Answer:

Class interval	Frequency	Cum F.
10-19	10	10
20-29	05	15
30-39	15	30
40-49	20	50
50-59	10	60
60-69	10	70

Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
L = Lower limit of the CI. in \(\frac{n}{2}\) F = The frequency below the CI. of \(\frac{n}{2}\)
Fm = The F at N/2 scores
i = class Interval
\(\frac{n}{2}\)= \(\frac{70}{2}\) = 35

L= 39.5   F = 30    F = 20  i =10
Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
= 39.5+\(\left(\frac{35-30}{20}\right)\)x10
= 39.5+\(\left(\frac{35-30}{20}\right)\)x10
= 39.5+\(\frac{5}{20}\)x10
= 39.5+\(\frac{5}{2}\)x10
= 39.5 + 2.5
= 42.6

questions 7.

CL	F
30-34	2
25-29	3
20-24	3
15-19	10
10-14	4
5-9	5
0-4	2
	N =30

Answer:

CL	F	Cum F
30-34	2	30
25-29	3	28
20-24	3	25
15-19	10	19
10-14	4	9
5-9	5	5
0-4	2	2
	N =30

Here \(\frac{n}{2}\) =15
F=15(15-19) in the CI.
Fm=1O i =15
Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
= 14.5+\(\frac{(15-19)}{10}\) x 5
= 14.5+\(\frac{4}{10}\) x5
= 14.5+2
= 16.5

Questions 8.
Calculate the Mean, Median and Mode from the class interval given?

CL	F
35-39	2
30-34	3
25-29	5
20-24	7
15-19	5
10-14	3
5-9	3
	N =28

Answer:

CL	F	D1	Fd1
35-39	2	+3	6	6
30-34	3	+2	5	+17
25-29	5	+1	5
20-24	7	0	0
15-19	5	-1	-5
10-14	3	-2	-6	-20
5-9	3	-3	-9
	N =28		∑fd1= -3

Mean=A.M. +C.I.
= 17+\(\frac{-3}{8}\) x5
= 17+\(\frac{-15}{28}\) x5
= 17-0.18
= 16.2

Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
= 14.5+\(\frac{14-11}{7}\) x5
= 14.5+\(\frac{15}{7}\)
= 14.5+2.1
= 16.7

Mode = 3 Median -2 Mean
=3X 16.7- 2X 16.2
=50.1 -32.4
= 17.6
∴ Calculated Mean, Median and Mode are 16.2, 17.7 and 17.6 respectively

Questions 9.
Find out the Mean, Median and Mode by the long method?

C.I.	F
55-59	1
50-54	1
45-49	3
40-44	4
35-39	6
30-34	7
25-29	12
20-24	6
15-19	8
10-14	2
	N = 50

Answer:

C.I.	F	X	Fx
55-59	1	57	57
50-54	1	52	52
45-49	3	47	141
40-44	4	42	168
35-39	6	37	222
30-34	7	32	224
25-29	12	27	324
20-24	6	22	132
15-19	8	17
	N = 50		∑Fx1 = 1472

mean = \(\frac{\sum F x}{N}\)
= \(\frac{1472}{50}\)
= 29.4
Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi

Where L = 34.5
N/2 = 25
F = 35
Fm = 6
i = 5

So, Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
= 34.5+\(\left(\frac{25-35}{6}\right)\) x5
= 34.5+\(\frac{-10}{6}\) x5
= 34.5+\(\frac{-50}{6}\)
= 34.5-8.3
= 26.2

Mode =3. Median -2 Mean
=3X 26.2 – 2X 29.4
= 76.6 – 58.8
= 19.5

Questions 10.
Find out the Mean, Median of the frequency distribution?

C.I.	F
20-21	1
18-19	2
16-17	3
14-15	5
12-13	3
10-11	3
8-9	2
6-7	1

Answer:

C.I.	Midpoint	F	X1	Fx1
20-21	20-5	1	3	3
18-19	18-5	2	2	4
16-17	16-5	3	1	3
14-15	14-5	5	0	0
12-13	12-5	3	-1	-3
10-11	10-5	3	-2	-6
8-9	8-5	2	-3	-6
6-7	6-5	1	-4	-4
		N = 20		∑fx1= -9

Mean = A.M. + \(\frac{\sum F x^1}{N}\) x i
where A.M, = Assumed mean
Σ=Sumtotal
Fx¹ = Frequency x deviations of the points of the class intervals from the assumed mean X¹ Terms of the class intervals.
i = size of the class intervals
Here, A.M. = 14.5
∑Fd =-9

Mean = 14.5+\(\frac{-9}{20}\) x2
= 14.5-9
= 13.6

Where L = Lower limit of the class interval containing the Median \(\frac{n}{2}\) score,
F = cumulative frequency up to L.

Fm = Frequency of the class interval containing the median.
i = size of the class interval
L= 13.5, f=9, Fm5, i=2
Median = 13.5+ \(\frac{10.9}{5}\) x2
= 13.5+\(\frac{1}{5}\)x2
= 13.5+4
= 13.9
Mode = 3 Median – 2 Mean
= 3 x 13.9 – 2 x 13.8
= 41.7 – 27.2
= 14.5

Question 11.
Find out the Mean and Median by the long method.

C.I.	F
10-14	4
15-19	5
20-24	9
25-29	18
30-34	11
35-39	5
40-44	6
45-49	3
50-54	1
	N = 60

Answer:

C.I.	X	F	FX
10-14	12	4	48
15-19	17	5	85
20-24	22	9	198
25-29	27	18	486
30-34	32	11	352
35-39	37	5	185
40-44	42	6	252
45-49	47	3	94
50-54	5	1	52
		N = 60	∑Fx = 1760

Mean = \(\frac{\sum F x}{N}\)
= \(\frac{1760}{60}\)
= 29.0

Calculation of Median;
Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
Where
L = 24.5\(\frac{N}{2}\)
= \(\frac{60}{2}\)
= 30
Fm = 18,F = 17 ,i=5
So, Median = L+\(\left(\frac{\frac{N}{2}-F}{F m}\right)\) xi
= 24.5+\(\left(\frac{30-17}{18}\right)\) x5
= 24.5+3.6
= 28.0

Mode = 3 Median – 2 Mean
=3 X28-2X29=26.3

Question 12.
Find out the Mean of the following distribution?

C.I.	X
15-19	6
20-24	5
25-29	8
30-34	15
35-39	10
40-44	9
45-49	7
	N = 60

Answer:

C.I.	X	D1	FD1
15-19	6	-3	-18
20-24	5	-2	-10
25-29	8	-1	-8
30-34	15	0	0
35-39	10	1	10
40-44	9	2	18
45-49	7	3	21
	N = 60		∑Fd1= -13

Mean = A.M.+\(\frac{\sum F d^{\prime}}{N}\) x i
Suppose, the mean is within the 30-34 class interval
∴ A.M. = 32
In C.I. 25-29,
d = \(\frac{27-32}{5}\)
= \(\frac{-5}{6}\)
= -1

∴ Mean = A.M.+\(\frac{\sum F d^{\prime}}{N}\) x i
= 32+\(\frac{13}{60}\)x5
= 32+\(\frac{13}{12}\)
= 32+1.09
= 33.09

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Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Direct and Reported Speech Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Direct and Reported Speech

Section -1

Revision
We use direct or indirect speech when we want to report what someone says.
In ‘Direct Speech’ we repeat the exact words of the speaker, and put them within quotation marks/inverted commas.

Example
Somesh said, “I get bored very easily.”
But in ‘Reported Speech’/’Indirect speech’, we do not repeat the exact words. We give only the meaning of what was said. Quotation marks are not used in such sentences.

Example
Somesh said (that) he got bored very easily.
Here we have changed I to he. We have also changed the tense of the verb used by the speaker. (from get to got)
In the above example, the verb said is known as the ‘reporting verb’ and ‘got’ as the verb of the reported speech

Section -2

Changes in the form of reported speech
Reported statements

A. Change in tense
When the reporting verb is in present tense and we want to do a sentence from direct to indirect, there will be no change in the tense form of the vebr of the sentence being reported (reported speech).

Example
He says, “I can do the sums.” (Direct)
He says that he can do the sums. (Indirect)

When the repoting verb is in the past tense i.e. said, replied etc., the tense of the verb in the (reported speech) sentence being reported is normally ‘pushed back’.

1. Present tense verb becomes past tense
(a) Present simple becomes past simple.
(b) Present progressive becomes past simple.
(c) Present perfect becomes past perfect.
(d) Present perfect progressive becomes past perfect progressive.

2. Past tense verbs
(a) Simple past becomes past perfect/simple past
(b) Past progressive becomes past perfect continuous progressive.
(c) Past perfect becomes past perfect (unchanged).
(d) Past perfect continuous becomes unchanged.

Present relevance
If the words (speaker’s) spoken by the speaker remain valid (true) at the time of reporting, it is possible to leave the tense of the verbs unchanged.
Example
“Sheela is coming this evening ”, may be reported as He said (that) Sheela is coming this evening – in case the reporting is done the same day, after a very short gap/period. It is also permissible or grammatical to change the tense, following the usual rules of reported speech, to
He said that Sheela was coming this evening.

Future time verbs

‘will’	changes to	‘would’
‘shall’	changes to	‘should’
‘may’	changes to	‘might’
‘can’	changes to	‘could’
‘must have’	changes to	‘can’t have’
‘must’	changes to	‘had to’

B. Change in adverbials of time and place
The following changes would take place in reported speech.

Speaker’s words	Reported speech
here	there
this	that/the
now	then
today	that day
tonight	that night
tomorrow	the next day/the following day
yesterday	the day before/the previous day
next Monday	the following Monday
last Monday	the previous Monday

But if the reporting takes place on the same day, at the same place, the adverbials of time and place remain unchanged.

Example
“I have asked Ramesh to come here today”, changes to He said that he has asked Ramesh to come here today. It can be also used ‘he said that he had asked Ramesh to come there that day, if the report takes place the same day, at the same place, so the ‘change’ in adverbirals of time and place depends on the context/situation.

Reported questions
(a) When in a direct speech (sentence) we find an interrogative (question) sentence, we use verbs such as asked, inquired, wanted to know etc. while making/changing it to ‘repoted speech’.

We know very well that there are two types interrogatives such as Wh- word and yes/no type interrogatives.
Do mark how we change “Wh- questions”
1. (direct) Teacher (to Bimal) : “When did you meet Shobha ?”
Indirect: The teacher asked Bimal when he has met Sobha.
2. (direct) Prem (to Sudhir) : “Where can I get the book ?”
Indirect: Prem asked Sudhir where he could get the book.
3. (direct) Smita : “How can I leave the place ?”
Indirect: Smita inquired how she could leave the place.

(b) Reporting yes/no questions.
Here are some examples of Yes or No questions and the same put in direct speech. Note the use of the conjunction if or whether in the indirect form.

Direct	Indirect
The doctor asked the patient, “Do you smoke ?”	The doctor asked the patient whether or if he smoked.
Bhupen: ‘May I go out, Teacher ?”	Bhupen asked the teacher if/whether he could go out.

(c) Reporting requests, offers, etc.
Note how the command or direction is expressed with ‘to + verb’ in the indirect form

Direct	Indirect
The captain to the men said, “Get ready to march.”	The captain order the men to get ready to march.
	Sita requested Mina to move the table.
Sita said to Mina, “Please move the table.”	He wished me a good morning./He greeted me.
He said, “Good morning.”
She said to me, “Congratulations.”	She congratulated me.

Activity – 1
Change the statements below into reported speech.

1. “We are visiting Konark on Sunday,” Harihar said.
2. “I will see you later,” Suresh told Sobha.
3. “Mr Mohanty has written the report,” the Chairman told me.
4. “I can’t speak Assamese,” I told her.
5. “We will have dinner at the Oberoi,” Umesh said.
6. “I don’t want to go shopping,” said Rabi.
7. “I lived in Cuttack when I was a boy,” he said.
8. “I don’t like balck coffee,” she said.
9. “We are going to be late,” Varsha said.

Answers
1. Harihar said that they were visiting Konark on Sunday.
2. Suresh told Sobha that he would see her later.
3. The Chairman told me that Mr. Mohanty had written the report.
4. I told her that I couldn’t speak Assamese.
5. Umesh said that they would have dinner at the Oberoi.
6. Rabi said that he didn’t want to go shopping.
7. He said that he (had) lived in Cuttack when he was a boy.
8. She said that she didn’t like black coffee.
9. Varsha said that they were going to be late.

Activity – 2
Complete the sentences in reported speech, making the necessary changes in person, place and time.
1. Nisha (to Seema): “Nina never speaks to me !”
Seema (to Nina): Nisha-said that ______________.
2. Suresh, “The match was played last week.”
You (a week later): Suresh said that _____________.
3. Ram: “I arrived in Delhi only yesterday.”
You (two days later): Ram said that ____________.
4. Renu (to Lipi): “I don’t have any money for the ticket.”
Lipi (to Pushpa) Renu told me that ______________.
5. Amiya (yesterday): “Tina and I are going on a picnic tomorrow.”
You (today): Amiya said that _____________.
6. Jina (in the library): “I’ll meet you and Mohan in the canteen.”
You (to Mohan, in the canteen): Jina told me that ____________.

Answers
1. Nisha (to Seema): “Nina never speaks to me !”
Seema (to Nina): Nisha said that you (Nina) never spoke to her.
2. Suresh, “The match was played last week.”
Suresh said that the match had- been played last week.
3. Ram: “I arrived in Delhi only yesterday.”
You (two days later): Ram said that he had arrived in Delhi only the day before/ the previous day.
4. Renu (to Lipi): “I don’t have any money for the ticket.”
Lipi (to Pushpa) Renu told me that she didn’t have any money for the ticket.
5. Amiya (yesterday): “Tina and I are going on a picnic tomorrow.”
You (today): Amiya said that Tina and I are going on a picnic (the next day) tomorrow.
6. Jina (in the library): “I’ll meet you and Mohan in the canteen.”
You (to Mohan, in the canteen): Jina told me that she would meet me and Mohan in the canteen.

Activity- 3
Respond to the statements in quotation marks. The speakers are all saying something different from what they had told you before. The first one has been done for you.
1. “I prefer chicken to mutton.”
But you had told me you prefer mutton.
2. “I like sweets more than I like fish.”
But you had said _____________
3. “I love classical music.”
But I remember you saying _______________
4. “I haven’t finished my homework.”
But I thought you said ___________
5. “I don’t have a computer at home.”
But ________________

Answers
1. “I prefer chicken to mutton.”
But you had told me you prefer mutton.
2. “I like sweets more than I like fish.”
But you had said to me that you like(d) fish.
3. “I love classical music.”
But I remember you saying That you love pop-music.
4. “I haven’t finished my homework.”
But I thought you said to me that you had finished your homework.
5. “I don’t have a computer at home.”
But you had told me that you had a computer at home.

Activity- 4
Kavita has been teaching in a school for five years. She can still remember the interview
she had to face and the questions that she was asked. This is what she tells her friend
Roma about the interview.
1. “Where are you from ?”
They asked me where I was from.
2. “Have you taught anywhere before ?”
They asked me _______________________
3. “Why are you interested in this job ?”
__________________________________
4. “Will you be able to come to the school at 7.00 every morning?
__________________________________
5. “Can you teach English in the higher classes ?”
__________________________________
6. “How much do you expect to be paid ?”
__________________________________

Answers
1. “Where are you from ?”
They asked me where I was from.
2. “Have you taught anywhere before ?”
They asked me if/whether I had taught anywhere.
3. “Why are you interested in this job ?”
They asked me why I was interested in that job
4. “Will you be able to come to the school at 7.00 every morning?
They asked me if/whether I would be able to come to school at 7.00 every morning.
5. “Can you teach English in the higher classes ?”
They asked me if/whether I could teach English in the higher classes.
6. “How much do you expect to be paid ?”
They asked me how much I expected to be paid.

Activity- 5
A famous magician has arrived in your city and just presented his first performance. The show has proved to be very popular. This is what the different newspapers had to say about it.

“The show is brilliant,” (the Samaj).
“It made me laugh and cry,” (the Sambad).
“We can’t remember a better show,” (the Dharitri).
“You’ll love it,” (the Prajatantra).
“You shouldn’t miss it.” (the Dinalipi)

Put their comments in reported speech.
Answers
(a) The Samaj reported that the show was brilliant.
(b) The Sambad said that had made him laugh and cry.
(c) The Dharitri said that they couldn’t remember a better show.
(d) The Prajatantra said we would love that.
(e) The Dinalipi said that we shouldn’t miss that.

Activity – 6
Change the sentences below into reported speech. Replace the word said, used in the sentences, by one of the verbs given below. Use each verb only once.

protested	suggested	admitted
shouted	boasted	exclaimed
muttered	agreed	objected
warned	insisted	claimed

1. “You can’t threaten me like that. I know my rights,” the old man said.
__________________________________
2. “This man doesn’t know what he’s talking about,” said someone in the audience.
__________________________________
3. “No, let’s finish this work first”, she said, “We can go to the movies some other day.”
__________________________________
4. “I can write a novel in a month,” said the man.
__________________________________
5. “Stand up now or I’ll beat you,” said the teacher.
__________________________________
6. “The book you are reading is my property,” the man said.
__________________________________
7. “If they can’t hand in the assignment today, they simply must hand it in tomorrow,” said the Principal.
__________________________________
8. “We’ll, the car has broken down and we can’t repair it now,” said the driver.
__________________________________
9. “Yes, I hit the boy with the school bag,” the girl said.
__________________________________
10. “We shall always be loyal to our country,” said the children.
__________________________________
11. “What a surprise to find him here today !” they said.
__________________________________
12. “I don’t have enough resources to build the house”, said the man.
__________________________________

Answers
1. “You can’t threaten me like that. I know my rights,” the old man said.
Answer:
The old man shouted that I couldn’t threaten him like that and he knew his rights.

2. “This man doesn’t know what he’s talking about,” said someone in the audience.
Answer:
Someone objected that the man didn’t know what he was talking about.

3. “No, let’s finish this work first”, she said, “We can go to the movies some other day.”
Answer:
She suggested that they should finish that work first and they could go to the movies some other day.

4. “I can write a novel in a month,” said the man.
Answer:
The man agreed that he could write a novel in a month.

5. “Stand up now or I’ll beat you,” said the teacher.
Answer:
The teacher warned me to stand up then or he would beat me.

6. “The book you are reading is my property,” the man said.
Answer:
The man claimed that the book he was reading was his property.

7. “If they can’t hand in the assignment today, they simply must hand it in tomorrow,” said the Principal.
Answer:
The Principal insisted that if they couldn’t hand in the assignment today, they simply had to hand it in the next day.

8. “We’ll, the car has broken down and we can’t repair it now,” said the driver.
Answer:
The driver muttered that the car had broken down and they couldn’t repair that then.

9. “Yes, I hit the boy with the school bag,” the girl said.
Answer:
The girl admitted that she had hit the boy with the school bag.

10. “We shall always be loyal to our country,” said the children.
Answer:
The children boasted that they should be loyal to their country.

11.“What a surprise to find him here today !” they said.
Answer:
They exclaimed that was a surprise to find him there that day.

12. “I don’t have enough resources to build the house”, said the man.
Answer:
The man protested that he didn’t have enough resources to build the house.

Activity – 7
Choose suitable adverbs from the list below and use while reporting the following statements. The first one has been done for you.

passionately agreeably sharply
accusingly sarcastically angrily
defiantly patiently

1. “I shall always love you,” the young boy said.
Ans. The young boy passionately said that he would always love her.
2. “I thought you said you were ill yesterday,” she said.
3. “You can go to the devil !” the boy said.
4. “I think we were having a wonderful party,” he said.
5. “You think you’re god, don’t you ?” she said.
6. “Let’s try again, shall we ? It is rather difficult,” the teacher said.
7. “Don’t keep asking such silly questions,” the husband said.
8. “Mind your own business,” the stranger said.

Answers
2. She defiantly said that she thought I said I was ill the previous day.
3. The’ boy angrily said that I could go to the devil.
4. He agreeably said that (he thought) they were having a wonderful party.
5. She sarcastically said that I thought whether I was sad or not.
6. The teacher patiently said that it was rather difficult and they should try again.
7. The husband accusingly said not to keep asking silly questions like that.
8. The stranger sharply told me to mind my own business.

Activity- 8
Last Sunday you met your cousin Harish. Here are some of the things Harish said to you.
1. “I’d like to go to a good college but my parents can’t afford the fees”
2. “I’m living with a friend now and working part-time.”
3. “I’m enjoying my work.”
4.“My bicycle was stolen a few days ago.”
5. “My friend has got a scholarship to study abroad.”
6. “I’ll tell my Mother that I met you.”

You want to tell your mother, the same day, what Harish said to you. (Use reported speech)
Answers
1. Harish said to me that he would like to go to a college but his parents can’t afford the fees.
2. He said that he is living with a friend now and working part-time.
3. He also said that he is enjoying his work.
4. He said that his bicycle was stolen a few days ago/had been stolen a few days before.
5. He said that his friend has got a scholarship to study abroad.
6. He said that he will/would tell his Mother that he met me.

Activity- 9
Imagine that you are a journalist and that you have to report a school debate. The topic for the debate was “Children should be encouraged to watch television.” Some speakers supported the topic while others opposed it. The main arguments are summarised below.
Topic: Children should be encouraged to watch television.

For	Against
Television stimulates thinking. It keeps the family together. It keeps one well-informed. It brings the outside world into your home. It provides a lot of information. It is a good babysitter.	It kills the art of conversation. It ruins family interaction. It makes people uncritical. It shuts out the real world. It ruins the reading habit. It makes children passive.

Begin as follows.
Some speakers said- that _________________________
but some other speakers were of the opinion that (continue) __________________
___________________________________________
___________________________________________
___________________________________________
___________________________________________

Answers
Some speakers said that television stimulates/stimulated thinking but some other speakers were of the opinion that it killed that art of conversation.
Some said that it keeps the family together but some other speakers were of the opinion that it ruined the family interaction.
Some said that it keeps one well informed but some others were of the opinion that it made people uncritical.
Some speakers said that it brought/brings the outside world into the home but some others were of the opinion that it shuts out the real world.
Some said that it provides a lot of information but some others were of the opinion that it ruined/ ruins the reading habit.
Some said that it is a good baby-sitter but some others were of the opinion that it made the children passive.

Activity – 10
Rewrite the following paragraph, using the sentences below the paragraph to fill in the gaps in the paragraph. The sentences have to be changed into reported speech.
Sometime ago Manoj applied for the post of clerk in a bank. Last Friday he was called for an
interview. When he went into the Manager’s office, the Manager told him 1 __________ and asked him 2 _________ Majoj told him 3 ___________. Then the Manager asked him 4 ____________and Manoj told him 5 ___________.He added 6 _____________. The Manager wanted to know 7 ___________ and Manoj replied 8 __________. Then the Manager asked him 9 __________ and Manoj replied 10 ___________. The Manager gave him a test and then said 11 _________. Manoj thanked him and asked 12 ____________ The Manager replied 13 _____________and Manoj promised 14 _________________.
1. “Please sit down.”
2. “What is your name ?”
3. “My name is Manoj Kumar.”
4. “Do you smoke ?”
5. “I don’t usually smoke.”
6. “I never smoke in the office.”
7. “Where have you worked before ?”
8. “I worked for two years in a private company.”
9. “Can you take shorthand ?”
10. “Yes sir, I can.”
11. “I can appoint you on probation for a month.”
12. “When can I start work ?”
13. “Tomorrow, if you wish.”
14. “I will be here on Monday.”

Answers
Sometime ago Manoj applied for the post of clerk in a bank. Last Friday he was called for an interview. When he went into the Manager’s office, the Manager told him to sit down and asked him what his name was. Manoj told him that his name was Manoj Kumar. Then the Manager asked him If/whether he smoked and Manoj told him that he didn’t usually smoke/doesn’t usually smoke. He added that he never smoke/smoked in the office. The Manager wanted to know where he had worked before and Manoj replied that he had worked for two years in a private company. Then the Manager asked him if/whether he could take shorthand and Manoj replied affirmatively that he could. The Manager gave him a test and then said that he could appoint him on probation for a month. Manoj thanked him and asked when he could start work. The Manager replied that if he wished, you could work tomorrow and Manoj promised that he could be there on Monday.

Activity- 11
Look at the following ‘tips’ given to candidates preparing for an interview.
1. Be sure to reach the place where the interview is to take place a few minutes before the time given to you.
2. Look relaxed when you are sitting, but sit erect. Don’t lounge.
3. Anticipate questions and be ready with your facts.
4. Do not talk more than is necessary.
5. Look into the interviewer’s eyes when you answer.
6. Be confident and do not become nervous.
Now tell a friend about these ‘tips’, using reported speech.
Begin
I have been advised to ______________________.I have also been asked to (continue) __________________________________
___________________________________________________________________________________________________
__________________________________________________________________.

Answers
I have been advised to be sure to reach the place where the interview is to take place a few minutes before the time given to me. I have also been asked to look relaxed when I am sitting, and to sit erect and not to lounge. I have also been asked to anticipate questions and be ready with my facts. I have also advised not to talk more than is necessary. I have been asked to look into the interviewer’s eyes when I answer. I have been advised to be confident and not become nervous.

Activity – 12
Rewrite the passage below using the reported speech.
“Have you been here long ?” the newcomer asked another person in the hotel lounge.
“No, just over a week; that’s all.”
“You. don’t know the place very well, then ?”
“I’m a stranger in these parts. But of course even a stranger learns a few things in a week.”
“Would you mind telling me if somebody called Sujit has been here ?”
“Sujit ?” said the middle-aged man. “It’s odd you should mention that name. I remember the
man, well; a big tall fellow, with a dark mustache. He left this morning.”
“Do you know where he went ?” asked the newcomer.
“Well, he said he was going to Delhi.”

Answers
The newcomer asked another person in the hostel lounge if/whether he had been there long. Another person replied negatively saying that he came just over a week and that was all. The newcomer asked another person if he did not know the place very well then. Another person
replied that he was a stranger in these parts and added saying that even a stranger learns a few things in a week. The newcomer asked another person if/whether he would mind telling him if somebody called Sujit had been there. The middle-aged man asked if it was Sujit and added that it was odd you should mention that name and he remembered the man well, a big tall fellow, with a dark mustache and he left this morning. The newcomer asked if/whether he did know where he went. He replied affirmatively that he had been going to Delhi.

Multiple-Choice Questions (MCQs) with Answers
Question 1.
He said, “I am busy now.”
He said that _________.
(A) he is busy now
(B) he had been busy now
(C) he was busy then
(D) he would be busy then
Answer:
(C) he was busy then

Question 2.
He said, “It may rain tomorrow.”
(A) He said that it might rain tomorrow.
(B) He said that it might rain the next day.
(C) He said that it might have been raining the next day.
(D) He said that it may rain tomorrow.
Answer:
(B) He said that it might rain the next day.

Question 3.
He said to us, “Are you going away today.”
(A) He asked us if we were going away that day.
(B) He asked us if we were going away today.
(C) He asked us that we were going away the next day.
(D) none of these
Answer:
(A) He asked us if we were going away that day.

Question 4.
He said to me, “Why did you insult my brother ?”
(A) He told me why I would insult his brother.
(B) He asked me why I had insulted his brother.
(C) He asked me if I insulted his brother.
(D) He asked me why I insult his brother.
Answer:
(B) He asked me why I had insulted his brother.

Question 5.
“Shoot the prisoner”, said the tyrant.
(A) The tyrant told them to shoot the prisoner.
(B) The tyrant advised them to shoot the prisoner.
(C) The tyrant wanted them to shoot the prisoner.
(D) The tyrant commanded them to shoot the prisoner.
Answer:
(D) The tyrant commanded them to shoot the prisoner.

Question 6.
The rebel said to the king, “Pardon my fault sir.”
(A) The rebel asked the king to pardon his fault.
(B) The rebel begged the king to pardon his fault.
(C) The rebel insisted the king on pardoning him.
(D) none of these
Answer:
(B) The rebel begged the king to pardon his fault.

Question 7.
He said, “Bravo! You have done well.”
(A) He said that he had done well.
(B) He said that he did well.
(C) He applauded him saying that he had done well.
(D) He remarked that he had done well.
Answer:
(C) He applauded him saying that he had done well.

Question 8.
I said to my students, “When are you meeting me ?”
(A) I said to my students when are you meeting him.
(B) I asked my students when you are meeting me.
(C) I asked my students when they were meeting me.
(D) I said to my students when they were meeting me.
Answer:
(C) I asked my students when they were meeting me.

Question 9.
The teacher said, “Life is full of sorrow.”
(A) The teacher said that life was full of sorrow.
(B) The teacher said that life is full of sorrow.
(C) The teacher explained that life was full of sorrow.
(D) all of these
Answer:
(B) The teacher said that life is full of sorrow.

Question 10.
He said, “Good-bye, friends.”
(A) He bade good-bye to his friends.
(B) He expressed good-bye to his friends.
(C) He wanted good-bye to his friends.
(D) none of these
Answer:
(A) He bade good-bye to his friends.

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Modal Verbs Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Modal Verbs

SECTION -1

Making requests (using can, could; may, will, would)
Read the sentences below carefully.
1. Suresh: Can you lend me your bicycle?
Mahesh: Of course, I’m not using it today.
2. Student: Could I borrow your dictionary, sir?
Teacher: Certainly. But it’s meant only for very advanced learners.
3. Shyam: Will you open the door, please?
Stranger: Sure.
4. Jatin: Would you mind repeating the last sentence, sir?
Speaker: Not at all. I said that…
5. Jina: May I ask a question, please?
Speaker: By all means.
6. Landlady: Will you please clean the kitchen?
Maid. Yes, ma’am. Right away.

The modal auxiliaries can, could, may, will, and would are used to make requests.
But we can well mark a slightly different way from the others.
1. The use of ‘can’ is informal, while could is more formal and polite. It suggests a more tentative attitude on the part of the speaker.
2. ‘May’ is used rather in a formal way to express a request.
3. ‘Will’ is also used to make requests when we don’t need to be particularly formal or polite.
4. ‘Would’ or would you mind is used for more polite requests or formal (with the 2nd person).

Activity – 1
Make a request to suit each of the situations suggested below, using the word supplied in brackets. In case no word has been suggested, you are free to choose your own words to make the request.
(a) You need some money. Ask a friend to lend you some. (can)
_________________________________
(b) You want a speaker in a seminar to explain something. Ask him to do so. (could/ would)
_________________________________
(c) You want your room key from the receptionist in a hotel. Ask the receptionist to give you the key. (can/may)
_________________________________
(d) You want a visitor to wait a minute while you get ready. (would)
_________________________________
(e) You are carrying a heavy bag. Ask someone to open the door for you. (could)
_________________________________
(f) You are on the phone. You want to speak to the chairman. (may)
_________________________________
(g) You want someone to tell you how to reach the railway station.
_________________________________
(h) You are buying a pair of shoes and want a receipt.
_________________________________
(i) You are working and want everybody to be quiet for some time.
_________________________________
(j) You need something to eat. Ask your mother.
_________________________________
(k) You are on a train and you would like to borrow a newspaper from another passenger, who has just finished reading it.
_________________________________
Answer:
(a) Can you lend me some money?
(b) Could you please explain something?
Would you explain something, please?
(c) Can you please give me the room key?
May I get the room key, please?
(d) Would you wait for a minute, please?
(e) Could you please open the door for me?
(f) May I talk to the chairman, please?
(g) Could you please tell me how to reach the railway station?
(h) Would you give me the receipt, please?
(Or) Could you please give me the receipt?
(i) Could you be silent for some time, please?
Would you mind keeping silent for some time?
(j) Can you give me something to eat?
(k) Would you mind giving me the newspaper for some time?

Activity – 2
Monisha is ill in bed and has to request her friend to do things for her. Use can, will, could or would for the requests that she must make. (More than one answer may be possible.)
(a) Monisha wants some books to read. She would like her friend to get some.
_________________________________
(b) Monisha doesn’t want to be disturbed by visitors.
_________________________________
(c) Monisha wants her friend to write a letter for her.
_________________________________
(d) Monisha wants some vegetable soup for lunch.
_________________________________
(e) Monisha wants her friend to fetch the mobile phone, which is in the next room.
_________________________________
(f) Monisha wants her friend not to go out and leave her alone.
_________________________________
(g) Monisha wants her friend to wash her clothes.
_________________________________
Answer:
(a) Can you lend me some books, please?
(b) Would you mind stop talking?
(c) Can you please write a letter for me?
(d) Could you/Can you provide me some vegetable soup for lunch, please?
(e) Can you get me the mobile phone from the next room, please?
(f) Would you stay for some time here, please?
(g) Can you please wash my clothes?

Activity – 3
Seema is staying in Leena’s home as a guest, and Leena has to request Seema to do a few things for her, which she is not able to do herself, for some reason or the other. Here is an example.
Seema, could you please clean the dishes after dinner? I’ve got to go to the market to buy a few things.
Can you make a few more requests to Seema on behalf of Leena, using the clues below? You also have to provide a reason for each request. (You should be specially polite while making more unusual requests.)
(a) switch off the lights
_________________________________
(b) make some tea
_________________________________
(c) water the plants in the garden
_________________________________
(d) feed the dog
_________________________________
(e) take the rubbish out of the dustbin
_________________________________
(f) take care of the neighbor’s children
_________________________________
(g) cook dinner
_________________________________
Answer:
(a) Could you please switch off the lights? I have to sweep the room.
(b) Could you please make me some tea?
I have got to wash my dresses.
(c) Would you please water the plants in the garden? I’ve to nurse my grandfather.
(d) Would you mind feeding the dog? I have got to do shopping.
I have to wash the utensils.
(e) Could you take the rubbish out of the dustbin, please?
I have to wash the utensils.
(f) Could you take care of the neighbor’s children, please? I have to have my bath.
(g) Could you please cook dinner?
I’ve, some sums to work out.

SECTION – 2

Asking for and granting permission
(using can, could, may, would)
Look at the examples below.
(a) Tarun: Can I use your typewriter to type out an application?
Friend: Yes, of course, you can.
(b) Tapas: Could I use the telephone?
Receptionist: Yes, sir. But this one is only for local calls.
(c) Journalist: Mr. Prime Minister, may I ask what you plan to do for unemployed engineers?
Prime Minister: New jobs are being created for them.
(d) Shefali: Would you mind if I sat here?
Stranger: No, not at all.
Can is used for permission usually in an informal style. When we talk to our intimate/ private men and women, we use ‘can’ for permission, and ‘could is used for permission when we are not sure whether the permission will be granted. It is usually used in ‘formal’ situations (i.e., for unknown people). Do remember this ‘could’ modal verb’ is never used to give permission to someone.
‘May and Would you mind’ are usually used for permission in a formal style. The speaker is even less sure of getting permission or feels that permission is likely to be refused.

Activity – 4
Ask for permission using can, could, or may. (More than one answer is possible.)
(a) You are at a seminar where someone is speaking. You want to ask the speaker a question.
Excuse me: _________________________________
(b) You are late for your English class. You want the teacher’s permission to get in.
_________________________________
(c) You are interviewing an actor. You would like to ask him something about his personal life.
_________________________________
(d) Yon wants to stay in the hostel. Ask the hostel warden for permission.
_________________________________
(e) You want to join the cricket team of your college. Ask the teacher in charge of the games.
_________________________________
(f) You have been invited to dinner by someone you do not know very well. You would like to ask for another piece of chicken.
_________________________________
(g) Your friend has a book that you want to read. Ask him/her.
_________________________________
Answer:
(a) You are at a seminar where someone is speaking. You want to ask the speaker a question.
Excuse me, could I ask you a question?
(b) You are late for your English class. You want the teacher’s permission to get in. Could, I get into the class? Or, May I come in?
(c) You are interviewing an actor. You would like to ask him something about his personal life.
Would you mind telling me a few words on your personal life?
Or, May I ask a few questions on your personal life?
(d) You want to stay in the hostel. Ask the hostel warden for permission.
Could you allow/grant me to stay in the hostel?
(e) You want to join the cricket team of your college. Ask the teacher in charge of the games.
Could I join the cricket team this year?
(f) You have been invited to dinner by someone you do not know very well. You would like to ask for another piece of chicken.
Could you give me one more piece of chicken?
(g) Your friend has a book that you want to read. Ask him/her.
Can you lend me your book for a night?

Activity – 5
Imagine that you are a guest at the places mentioned below. Use can (informal), could (semi-formal) or may (formal) to ask for permission to do a different thing at each place. (One has been done for you.) At your grandparents home Can I use the phone, grandfather?
(a) At your teacher’s home.
_________________________________
(b) At your neighbor’s home.
_________________________________
(c) At the house of a favorite uncle.
_________________________________
(d) At the house of a close friend.
_________________________________
(e) At the house of someone you are visiting for the first time.
_________________________________
(f) At the house of a friend of your elder brother.
_________________________________
(g) At a distant relative’s home
_________________________________
Answer:
(a) Could I use your bicycle?
(b) Can I use your comb?
(c) Can I have your photograph?
(d) Can I ride your scooter?
(e) Could I use your dictionary?
(f) Could I use your pen?
(g) Could I / May I switch on the TV?

Activity – 6
Say whether the sentences below are used to ask for permission or to make a request or whether they are used in formal or informal situations.
(a) Mr. Mohanty, would you please tell all the teachers to come for a meeting in my office at 3 o’clock?
(b) Can you put this parcel on the table, please?
(c) May I ask you a question about this job, sir?
(d) Would you mind sitting on that chair?
(e) Could you pass the salt, please?
(f) Can my friend stay here too?
(g) May I wait in the corridor outside?
Answer:
(a) Formal request
(b) Informal request
(c) Formal request
(d) More formal request
(e) Formal request
(f) Informal permission
(g) Formal permission

SECTION – 3

Making an offer to help (using will, would, shall, can, could)
(a) Rina: I want to take this bag with me.
Chhabi: I will (can) carry it for you.
Rina: Oh, thanks.
(b) Tarak: 1 must leave now.
Puma: Shall (Can) I get a taxi for you?
Tarak: Yes, please.
(c) Host: Would you like some more chicken?
Guest: Yes, please. Thank you.
(d) Pinaki : (to his friend’s friend) Could I buy the tickets for you?
Friends friend: Thank you very much.

Will, shall, and can are used in informal English to offer help to someone. will and shall are commonly abbreviated to ‘il (ill, We’ll, etc.) Will is used only in statements (e.g., I’ll carry your bag) but shall and can are used in questions as well as statements (e.g., Shall I carry your bag ? Can I carry your bag ? or I shall carry your bag / I can carn your bag). Could and would are used in questions as well as statements to offer help, in more formal situations. (e.g., could j carry your bag? ¡ could carry your bag!
Would you like me to carry your bag? I would be happy to carry your bag, etc.).

Activity – 7
Make offers to suit the following situations.
(a) Your friend accidentally drops his pen on the floor.
_________________________________
(b) An old man wants you to make room for him to sit down.
_________________________________
(c) A patient is sweating profusely and the windows are closed.
_________________________________
(d) An old man is carrying a heavy suitcase and gasping for breath.
_________________________________
(e) Your principal wants to get some medicines from a pharmacy.
_________________________________
(f) A stranger wants to go to a particular place and you are on a motorbike.
_________________________________
(g) Your friend does not have enough money to buy a book he needs.
_________________________________
(h) Your mother is not able to cook as she is unwell.
_________________________________
(i) Your sister is not able to work out a sum.
_________________________________
(j) Your younger brother is trying to reach a book on the top shelf of a cupboard.
_________________________________
Answer:
(a) Shall I/Can I lift it from the floor?
(Or) I’ll lift it form the floor.
(b) Would/Could I give you a seat here?
(c) Could/Would I open the windows for you?
(d) Could I carry the suitcase for you?
(e) Would / Could I bring some medicines for you, sir?
(f) Could I give you a lift?
(g) Shall / Can I lend you some money?
(h) I’ll cook for you.
(Or) Shall / Can I cook for you?
(i) Shall/Can I work out the sum for you?
(j) I’ll (will) get the book for you.
(Or) Shall/Can I get the book for you?

Activity-8
Study the offers below and say which ones are made in a formal situation in which each one could have been made.
(a) Shall I call the doctor for you?
_________________________________
(b) Would you like me to call the police?
_________________________________
(c) Shall I switch off the light?
_________________________________
(d) I can lend you two hundred rupees if you like.
_________________________________
(e) I’ll help you make the bed.
_________________________________
(f) Could I wait for you at the gate?
_________________________________
Answer:
(a) Shall I call the doctor for you?
Yes, I’m unwell, (formal)
(b) Would you like me to call the police?
Yes, I have to see the man stabbed.
(c) Shall I switch off the light?
Yes, I am awful tired now. (formal)
(d) I can lend you two hundred rupees if you like.
I’m badly in need of some money now. Oh, thanks.
(e) I’ll help you make the bed.
It’s very kind of you. I’m running a temperature now. (formal)
(f) Could I wait for you at the gate?
Yes, you may come and receive my other friends. (formal)

SECTION – 4

Making a suggestion (using shall, can, could)
(a) Pramod: It’s a lovely day. Shall we go for a walk?
Prashant: Yes, let’s do that.
(b) Nilima: Sisir is becoming more and more difficult to handle.
Manisha: We can talk to him if you like.
Nilima : No, I’ll manage without him.
(c) Girija: We need Sima’s room for the guest.
But she has gone home.
Gauri: We could go to her village tomorrow and collect the key.
In the sentences above, different ways of making suggestions have been shown.
The commonly used words are: Shall we /Can we /Could we (in questions), or You can /We can / We could….. (in statements), used together with if you like/want.
Suggestions can also be made by using let us (let’s), why don’t you/ we…. ? or how / what about…. ? Let’s and why don’t you/we…. ? are followed by the infinitive without to (e.g. Let’s stop here; why don’t you ask him ?, while How/What about….. ? are followed by an ‘-ing’ form (e.g., How about stopping here and eating lunch ?)

Activity – 9
Make suggestions to suit the following situations. (More than one answer is possible.)
(a) You think, your friend and you should go for a walk.
_________________________________
(b) You would like to have dinner at a restaurant with your neighbor.
_________________________________
(c) You feel like going to the park with your mother.
_________________________________
(d) You would like your father to take you to an uncle’s house.
_________________________________
(e) You have met somebody for the first time at a friend’s house. He is watching a film on television, but you would like to watch a cricket match, on a different channel.
_________________________________
(f) You feel like going to the theatre with some friends.
_________________________________
(g) It is 11.00 p.m. Your roommate in the hostel is reading a book, but you would like to sleep.
_________________________________
Answer:
(a) Let’s go for a walk.
Or, Can/Shall we go for a walk?
(b) Shall/Can we have dinner at a restaurant?
(c) Why don’t we got to the park now?
(d) Why don’t you take me to my uncle’s house?
(e) Could you change this channel? Let us see the cricket match in D.D. Sports.
(f) Let’s go to the theatre now.
Or, Why don’t we got to the theatre as we have some free moments now?
(g) Could you switch off the light? Let’s go to bed.

Activity – 10
Your friends and you are putting up a cultural show in your college. Make suggestions about the arrangements for the show, using the hints below. Study the example below and write your sentences.
(i) the place where the show is to be put up
(ii) decorating the stage
(iii) fixing the lights
(iv) getting singers and musicians to perform
(v) setting up the sound system
(vi) the guests to be invited
(vii) the refreshments to be served
Answer:
Example: We can invite the Chief Minister to be the Chief Guest.
(i) Shall we select/choose the old auditorium for the show?
(ii) Why don’t we decorate the stage by ourselves?
(iii) We shall invite the electrician for fixing the lights.
(iv) Let’s invite Akshay Mohanty and party for the musical show.
(v) What about setting up the sound system?
(vi) Why don’t we invite the Chief Minister and the Minister for Mass Education to be the guests?
(vii) The volunteers shall serve refreshments for the guests.

Activity – 11
Which of the following offer and which are suggestions? Say whether they are formal or informal.
(a) I can pay for your ticket if you like.
(b) What shall I do in case I don’t find Ram at home?
(c) Would you like them to wait for you?
(d) We could leave it until later.
(e) I can lend you the book.
(f) Could we paint this wall for you?
Answer:
(a) offer informal
(b) offer informal
(c) offer formal
(d) suggestion informal
(e) offer informal
(f) offer informal

SECTION – 5

1. Expressing ability to do something (using can be able to)
Study the following dialogues.
(a) Interviewer: Can you type?
Candidate: Yes sir. I can also take dictation in shorthand.
Interviewer: Will you be able to talk to our foreign clients?
Candidate: Yes sir. I can speak German and Italian.
(b) Teacher: Who can solve this problem?
Student: I can.
‘Can’ the modal in the above conversation is used to mean ‘ability ’ in general. ‘Be + able to’ is also used to mean the same meaning ‘ability’ to do something. ‘Can’ is informal/usual and ‘be + able to’ is usually formal.

Activity – 12
Complete the sentences below using can or be able to. (In some of the sentences, it will not be possible to use can.)
(a) Ask Karim about your friend’s condition. He might — tell you something.
(b) Ravi was bom and educated in Hyderabad. He — speaks Telugu.
(c) My friend can’t understand my problem. She’s never — understand it.
(d) I can’t lend you money but I — get the book for you.
(e) Meera — read Tamil but she doesn’t have any Tamil books.
(f) I haven’t — buy any books lately.
(g) We — travel a lot but we can’t do it now.
(h) Meet your teacher tomorrow. He — explain the question to you.
Answer:
(a) Ask Karim about your friend’s condition. He might be able to tell you something.
(b) Ravi was born and educated in Hyderabad. He can speak Telugu.
(c) My friend can’t understand my problem. She has never been able to understand it.
(d) I can’t lend you money but I will be able to get the book for you.
(e) Meera can read Tamil but she doesn’t have any Tamil books.
(f) I haven’t been able to buy any books lately.
(g) We used to be able to travel a lot but we can’t do it now.
(h) Meet your teacher tomorrow. He will be able to explain the question to you.

Activity – 13
List five things that you can do.
1. _________________________________
2. _________________________________
3. _________________________________
4. _________________________________
5. _________________________________
List five things that you can’t do.
1. _________________________________
2. _________________________________
3. _________________________________
4. _________________________________
5. _________________________________
Answer:
List five things that you can do.
1. I can lift this heavy box.
2. I will be able to get the first division.
3. I can speak and write in three languages.
4. I can give the answer to that question.
5. I will be able to swim in the river.

List five things that you can’t do.
1. I can’t lift an elephant.
2. I won’t be able to swim in the river.
3. I can’t read Greek.
4. I won’t be able to run 5 km at a stretch.
5. I won’t be able to get a medal in swimming

Activity – 14
A new student has joined your class. As the Secretary of the College Union, you would like to know which activities your new classmate would be able to take part in. Frame at least five questions that you might want to ask him and provide the answers that might be given.
Q. Can you draw and paint?
A. Yes, I can / No, I can’t.
1. Q. _________________________________
A. _________________________________
2. Q. _________________________________
A. _________________________________
3. Q. _________________________________
A. _________________________________
4. Q. _________________________________
A. _________________________________
5. Q. _________________________________
A. _________________________________
Answer:
Question 1.
Will you be able to contact the guests?
Answer:
Yes, I can.

Question 2.
Can you speak Hindi fluently?
Answer:
Yes, I can.

Question 3.
Will you be able to do shopping?
Answer:
Yes, I can do the shopping.

Question 4.
Will you be able to take care of the guests?
Answer:
Yes, I can.

Question 5.
Will you be able to see the seat arrangements?
Answer:
Yes, I can.

2. General ability versus the ability to do something in a particular situation. Look at the following sentences.

• Mohan could speak Assamese when he was a boy.
• When I was in London I could visit the museum every day.
• I had only a brief meeting with the Principal, but I was able to explain my difficulties to him.

In Sentence 1, could is used to refer to the general ability to do something that somebody possessed (in the past), while in Sentence 2, could indicate that the opportunity to do something was available. In Sentence 3, was able to indicate the ability, or the opportunity, to perform an action in a particular situation. Could is not used in such a context. Sometimes could is the past of ‘can’. We use ‘could’ to say that someone had the general ability to do something. But ‘was/were able to’ is used to indicate the past ‘ability + performance’ (result) which means ‘managed to do something in one particular situation.

Activity – 15
Use could or was/were able to in the sentences below. Use a negative if necessary. More than one answer is possible.
(a) The author of the book learned to write very early in life. He — write short stories when he was at school.
(b) My scooter didn’t start. Luckily I — borrowed a bicycle and got there in time for the interview.
(c) It was dark. We — see a thing.
(d) The children were warned that a fire had started, and they — move their belongings in time.
(e) I — meet my friend as the bus had left.
(f) The people in the neighboring village were having a party last night and you — have joined them.
(g) My friend ran after the bus and — get on.
(h) The freedom fighters had to live in the jungles for months. They — survive without food for several days.
Answer:
(a) The author of this book learned to write very early in life. He was able to write short stories when he was at school.
(b) My scooter didn’t start. Luckily I was able to borrow a bicycle and got there in time for the interview.
(c) It was dark. We couldn’t see a thing.
(d) The children were warned that a fire had started, and they were able to move their belongings in time.
(e) I could not meet my friend as the bus had left.
(f) The people in the neighboring village were having a party last night and you could have joined them.
(g) My friend ran after the bus and was able to get on.
(h) The freedom fighters had to live in the jungles for months. They were able to survive without food for several days.

Activity – 16
Saurabh is 40. Sometimes he feels that he has wasted his life. Read about Saurabh and replace some of the words with could have …, as shown in the example below.
Example: When Saurabh was 15 he had a chance to go abroad, but he decided not to go.
Ans. When Saurabh was 15 he could have gone abroad.
(a) Saurabh got a scholarship to study in a reputed public school. But he didn’t go.
_________________________________
(b) He was a good speaker, but he decided not to participate in debates.
_________________________________
(c) He had the opportunity to learn music under a famous teacher, but he decided against it.
_________________________________
(d) He was intelligent enough to come out first in the examination, but he did not try.
_________________________________
(e) He had the opportunity to start a business, but he did not want to.
_________________________________
(f) Everyone thought he had the creativity to write novels, but he never tried.
_________________________________
Answer:
(a) Saurabh could have studied in a reputed public school when he was young.
(b) He could have participated in debates when he was in school.
(c) He could have learned music under a famous teacher when he was 20 years old.
(d) He could have come out first in the examination if he had studied sincerely.
(e) He could have started a business when he was 30 years old.
(f) When he was 25 years old, he could have written novels.

Activity – 17
Use can, can’t, could, couldn’t was/were able to, where required. (Alternative answers are possible.)
(a) She — even speak properly when she was a girl, but now she’s a famous singer.
(b) Did you see anybody in the park? No, I — find anybody.
(c) An eagle — fly for hours.
(d) When my sister was just 7, she — speak Hindi much more fluently than she — now.
(e) On the fifth day the rescuers saw the children and — reach them without too much trouble.
(f) After weeks of training, I — climb some of the coconut trees.
Answer:
(a) She could not even speak properly when she was a girl, but now she’s a famous singer.
(b) Did you see anybody in the park? No, I could not find anybody.
(c) An eagle can fly for hours.
(d) When my sister was just 7, she was able to speak Hindi much more fluently than she can now.
(e) On the fifth day the rescuers saw the children and were able to reach them without too much trouble.
(f) After weeks of training, I was able to climb some of the coconut trees.

SECTION – 6

Indicating possibility (using may, can)
(a) This house is too small. We may sell it and buy a bigger one.
(b) That box is too heavy for you. You may drop it.
(c) Camels can survive for weeks in the desert, without water.
(d) Smoking can cause cancer.
‘May’ is used to express a specific possibility in the present or future. It can refer to a single occasion. But ‘can ’ is used to express a general theoretical possibility.

Activity – 18
Use can or may, as required, in the following sentences.
(a) The government — hike the bus fares next month.
(b) The villagers — convert this building into a hospital if everyone agrees.
(c) This medicine — cure any disease.
(d) My friends — put on a play here again next year.
(e) We’re going to do something different this year. We — decide to hire a professional drama group.
(f) Your boss is impressed with your work. You — get a promotion soon.
(g) The presence of lead particles in the air — cause brain damage in children.
(h) The machine has broken down and — takes some time to repair.
Answer:
(a) The government may hike the bus fares next month.
(b) The villagers may convert this building into a hospital if everyone agrees.
(c) This medicine can cure any disease.
(d) My friends may put on a play here again next year.
(e) We’re going to do something different this year. We may decide to hire a professional drama group.
(f) Your boss is impressed with your work. You may get a promotion soon.
(g) The presence of lead particles in the air can cause brain damage in children.
(h) The machine has broken down and may take some time to repair.

Activity – 19
A group of engineers has come up with a number of unusual products, for use at home. These products are :
1. A battery-operated fan.
2. A portable water heater.
3. Writing paper that glows in the dark.
4. Shirts made of edible material.
5. Shoes with a built-in alarm clock.
6. A bicycle that can float.
Think of some situations in which these products could prove to be very useful and frame sentences suggesting (to prospective customers) some possible uses for the products. One example is given.
Example
1. You can use the battery-operated fan to cool yourself when you are sitting in your garden.
2. _________________________________
3. _________________________________
4. _________________________________
5. _________________________________
6. _________________________________
Answer:
2. You can use the portable water heater when you go outside during chilly winter.
3. You can use writing paper that glows in the dark when there is no light.
4. You can use shirts made of edible material if you need.
5. You can use shoes with a built-in alarm clock to make use of the time properly.
6. You can use a bicycle that can float when you want to cross a flooded river.

SECTION – 7

Indicating present or future possibility (using may, might or could)
Look at the following sentences.
(a) Where’s Madhuri? She may/might / could be in the garden.
(b) There’s someone at the gate. It may / might / could be the postman.
(c) They’re not cooking dinner. They may / might / could be planning to go out tonight.
May, might, and could are used here to talk about present or future possibilities. Might is normally a little less sure than may, and could is less sure than may or might. Note that we can’t use can in the above examples. Can you say why? (Hint: Refer to the difference between can and may.)
‘May, might could’ all three models are used in sentences for possibility. There is still a little difference between them. May is a bit stronger than might or could. ‘Might’ is a little less sure than ‘may’ and ‘could’ is less sure than ‘may or might’.

Activity – 20
Rewrite the following sentences using ‘might’ or ‘could’.
(a) It is possible that she is cheating you.
_________________________________
(b) It is possible that they’ll come in the evening.
_________________________________
(c) It is possible that he is planning to sell the car.
_________________________________
(d) It is possible that she is telling the truth.
_________________________________
(e) This statement possibly contradicts his earlier statement.
_________________________________
(f) It is possible that one of them is a spy.
_________________________________
Answer:
(a) She might cheat me.
(b) They could come in the evening.
(c) He might sell the car.
(d) She might tell the truth.
(e) He might contradict his earlier statement.
(f) He might be a spy.

Activity – 21
Use may, might or could, together with the appropriate forms of the verbs in brackets.
1. Who is the girl Madhu is talking to? I’m not sure. She — (be) her sister.
2. Why are the boys shouting? I don’t know. They — (play).
3. Where is Manisha? I have no idea. She — (study) in the library.
4. Who is that man with Vijay? I’m not sure. It — (be) his uncle.
5. Shall I tell the Minister about you? You’d better not. He — (think) I want to ask him for a favor.
Answer:
1. Who is the girl Madhu is talking to? I’m not sure. She might be her sister.
2. Why are the boys shouting? I don’t know. The might he plane.
3. Where is Manisha? I have no idea. She might be studying in the library.
4. Who is that man with Vijay? I’m not sure. It might be his uncle.
5. Shall I tell the Minister about you? You’d better not. He may think I want to ask him for a favor.

Activity – 22
You are going for a walk, and you suddenly see a mysterious flying object, high up in the sky. Make five guesses about what the object could be.
1. _________________________________
2. _________________________________
3. _________________________________
4. _________________________________
5. _________________________________
Answers
1. It might be a skylark.
2. It may be a balloon.
3. It might be a parachute.
4. It could be the wing of a damaged airplane.
5. It might be a soldier falling from an aircraft.

SECTION – 8

Referring to possibility in the past (using may, might, or could)
Look at the examples below.
(a) She was not in the hostel yesterday. She might / could have gone home.
(b) A: Do you think he saw you?
B: No, he was too far away. He couldn’t have seen me.
(c) You were stupid to throw stones at the dog. It might have attacked you.
We use may/might/could, together with some verbs in the present perfect form, to talk about the possibility in the past.

Activity – 23
Complete the sentences below, expressing possibility in the past.
(a) The other teams weren’t good, so our team — (win). „
(b) You know where the library is, don’t you? You — (pass) it on your way here.
(c) Why didn’t your friend answer the phone? He — (not/ hear) it ringing.
(d) The light’s on. They — (forget) to switch it off.
(e) I did not hear the child’s voice, and I thought he — (fall asleep).
(f) Our teacher is not at home. He — (go) to meet a friend.
(g) How did you get that stain on your saree? I’m not sure. The baby — (drop) food on it.
(h) A: I can’t find my umbrella. Have you seen it?
B: No. You — (leave) in the restaurant.
Answer:
(a) The other teams weren’t good, so our team might have won.
(b) You know where the library is, don’t you? You might have passed it on your way here.
(c) Why didn’t your friend answer the phone? He could/might not have heard it ringing.
(d) The light’s on. They might have forgotten to switch it off.
(e) I did not hear the child’s voice, and I thought he could/might have fallen asleep.
(f) Our teacher is not at home. He could have gone to meet a friend.
(g) How did you get that stain on your saree? I’m not sure. The baby might/could have dropped food on it.
(h) A: I can’t find my umbrella. Have you seen it?
B: No. You might/could have left in the restaurant.

SECTION – 9

Making a deduction (using must have/can’t have)
Look at the following example.
The old man has not had anything to eat for the last two days. He must be starving.
The speaker is deducing something (starving). She / He takes the help of logical analysis or the basis of evidence that is available. She / He is very much sure that what he/she says is correct. So in the above example, ‘must’ is used for deduction. Can’t: In negative sentences for logical analysis/conclusion ‘can’t’ is used. The speaker is very sure about some negative conclusion that she/he has reached. The use of ‘must have ’ and ‘can’t have ’ shows that the speaker has made a deduction about something that happened in the past.
Example
1. Ramesh has done very well in the examination. He must have worked really hard.
2. The library was closed all day yesterday. You can’t have been there yesterday.

Activity – 24
Answer the questions in Column A, using must or can’t, and choosing a suitable reason from Column B. (One answer has been provided.)

Example: (She can’t be ill. She has gone swimming.)
Answer:
2. They must be sleeping. Their bedroom lights are off.
3. He must be happy. He has just got a job.
4. He can’t be a Bengali. He speaks Telugu.
5. The shopkeeper can’t be honest. He charged more than the price.
6. These boys must be naughty. They play tricks on everyone.
7. She can’t be married. She is too young.

Activity – 25
Consider the situations below and use the words in brackets to frame sentences with ‘/must have’ and ‘can’t have
(a) Satish got the first division in the exam. (He / works hard)
_________________________________
(b) Someone saw Sudha in the college library at 8.00 a.m. yesterday. (She / stays / in bed / all morning)
_________________________________
(c) The traffic lights were red and the car stopped, (the driver/see / the red light)
_________________________________
(d) He knows a lot about teaching. ( He / a teacher when he was young)
_________________________________
(e) Girish knew exactly what to do. (He / learns / the skills / very carefully)
_________________________________
(f) My friend knew everything about our plan. (He / listens / to our conversation)
_________________________________
(g) Janaki walked past me without speaking. (She / see / me)
_________________________________
(h) There are five stitches on her head. (She / have / an accident)
_________________________________
Answer:
(a) He must have worked hard.
(b) She can’t have stayed in bed all morning.
(c) The driver must have seen the red light.
(d) He must have been a teacher when he was young.
(e) He must have learnt the skills very carefully.
(f) He must have listened to our conversation.
(g) She can’t have seen me.
(h) She must have had an accident.

SECTION – 10

Offering advice (using should, ought to, had better, be supposed to)
(a) If your toothache is not getting better, you should see a dentist.
(b) They really looked after you well. You ought to write a letter of thanks.
(c) Your roommates are complaining. You had better turn the music down.
Should and ought to are used to ask for as well as give advice. Ought to is preferred when the advice is given by someone in authority.
‘Had better’ (generally abbreviated to better) indicates a strong recommendation, made for a particular situation.
Should have … and ought to have … are used to show that someone did not do something (in the past) which was advisable.
Look at the examples below.
1. I should have bought that book yesterday, but I didn’t have the money.
2. Haven’t you been to see your doctor? You ought to have seen him yesterday

Activity – 26
Complete the sentences in the “Advice” column, using ‘should’ or ‘ought to’.

Answer:
i. I should have bought a new pen yesterday, but I didn’t have money.
ii. I should have looked for other books, but I didn’t have free moments.
iii. I should have given some money to the poor, but I didn’t get the key of my drawer.
iv. They ought to have asked their teacher as they did not solve it themselves.
v. I should have gone and collected the glasses from the shop.
vi. I should have taken some days off.

Activity – 27
Produce sentences indicating what the speaker should (not) / ought to have done in given situations.
(a) I’m in trouble because I didn’t wish my boss “Good morning”!
(b) The pen he used to write the examination was not good enough.
(c) Nirmal did not meet Seeta until it was too late.
(d) I’m sorry that he didn’t take your advice.
(e) Smita had to go to the bank opposite the college, but she hired a taxi.
(f) I sent a messenger three days ago, but he has not arrived yet.
(g) She left her books in the canteen. She didn’t find them when she went back.
Answer:
(a) I’m in trouble because I didn’t wish my boss “Good morning”!
I should have wished my boss “Good morning.”
(b) The pen he used to write the examination was not good enough.
He ought to have used a good pen to write the examination.
(c) Nirmal did not meet Seeta until it was too late.
Nirmal should have met Seeta in time.
(d) I’m sorry that he didn’t take your advice. He ought to have taken your advice.
(e) Smita had to go to the bank opposite the college, but she hired a taxi. Smita should not have hired a taxi.
(I) I sent a messenger three days ago, but he has not arrived yet. He should have arrived by this time.
(g) She left her books in the canteen. She didn’t find them when she went back. She should have been very careful.

Activity – 28
What advice would you offer someone in a difficult or dangerous situation, such as an Are, flood, cyclone, a roof falling in, or a wild animal running loose in the city?
Think of at least 7 such situations and form sentences to offer advice, using either had better or had better not.
Example (a fire) You’d better phone the fire brigade.
1. (a burglary) You’d better not _________________________________
2. _________________________________
3. _________________________________
4. _________________________________
5. _________________________________
6. _________________________________
7. _________________________________
Answer:
1. You’d better not fight against them.
2. (a flood) You’d better take shelter on a hill.
3. (cyclone) You’d better leave that old house.
4. (a wild animal running loose) You’d better not open the door.
5. You’d better telephone the police about the wild animal.
6. (a roof falling in) You’d better leave the house with your family members.
7. (afire) You’d better make lose your cattle.

SECTION – 11

Expressing obligation or necessity (using must have to / had to, got to)
(a) I’ve got terrible pain in my back. I must go and see the doctor.
(b) I have got an appointment. I have to go and see the doctor at 9.00 tomorrow morning.
(c) Our teacher had given us fifty sums to work out. So I had to work late yesterday.
(d) My friend must be waiting for me. I’ve got to hurry.
‘Must’ is used for an internal obligation or when authority comes from the speaker. ‘Have to ’ is normally used for outside direction/obligation or when the authority comes from someone other than the speaker.
We use ‘had to’ when talking about past obligations or necessity. ‘Have got’ is sometimes used instead or ‘have to’ to talk about obligation or necessity but it is more informal.

Activity – 29
Complete the sentences using must or some form of have to. (In some cases, more than one answer is possible.)
(a) I really — find something cheaper. This is too expensive.
(b) A. — you — get a visa to go to Nepal?
B. No, you don’t but you — get one for Pakistan.
(c) You — fill in this form first. You can’t do anything until it is completed.
(d) Sudhir: I’ll meet you at the airport at 9 o’clock then.
Kamal: What! We — check in by 7.30.
(e) Jiban: This medicine tastes horrible. I’m going to take it only once a day.
Jiten: Now, don’t be silly. The doctor says you — take it 3 times a day.
Jiban: All right, I’ll take the medicine after meals.
Jiten: No, you — take the medicine before meals.
(f) You — remember to brush your teeth. If you don’t, you’ll have to go to the dentist.
(g) I — stay in bed yesterday because I wasn’t very well.
(h) I couldn’t go to the party last night because I — look after my youngest sister, who was ill.
Answer:
(a) I really must find something cheaper. This is too expensive.
(b) A: Have you got to get a visa to go to Nepal?
B: No, you don’t, but you have to get one for Pakistan.
(c) You have to fill in this form first. You can’t do anything until it is completed.
(d) Sudhir: I’ll meet you at the airport at 9 o’clock then.
Kamal: What! We have to check in by 7.30.
(e) Jiban: I’ve never tasted such horrible medicine. I’m going to take it only once a day.
Jiten: Now, don’t be silly. The doctor says you have to take it 3 times a day.
Jiban: All right, I’ll take the medicine after meals.
Jiten: No, you have to take the medicine before meals.
(f) You must remember to brush your teeth. If you don’t, you’ll have to go to the dentist.
(g) I had to stay in bed yesterday because I wasn’t very well.
(h) I couldn’t go to the party last night because I had to look after my youngest sister.

SECTION – 12

Expressing prohibition (using mustn’t, needn’t, don’t have to)
(a) You mustn’t jump over that wall. You’ll break your legs.
(b) You mustn’t drive that car. The brakes don’t work.
(c) Tomorrow is a holiday. You needn’t get up early.
(d) They needn’t buy the book. They can get it in the library.
(e) We don’t have to / need to go to a hotel. My mother can cook for us. ‘Mustn’t’ is usually used for prohibition or it is forbidden to do something.
But needn ’t is used for absence or obligation/necessity. Doesn ’t/ don’t have to, doesn’t / don’t need to’ are used when it is not necessary to do something.

Activity – 30
Choose the correct modal.
(a) They’ve been late for class three times this week. They mustn ’t/ needn’t be late again.
(b He mustn’t / needn’t tell our teacher about the meeting. Sudha has already told him.
(c) You mustn’t / don’t have to make any noise going into the house.
It’s very late and everybody is asleep.
(d) I mustn’t / don’t have to hurry. I’ve got plenty of time.
(e) You mustn’t / needn’t open the door before the bus stops. You could fall out.
(f) You don’t have to / mustn’t wash that shirt. It isn’t dirty.
(g) Sima needn’t/ mustn’t finish her essay today. She’s got to hand it in by next Friday.
(h) You needn’t / mustn’t wear your best clothes. You can wear what you like.
Answer:
(a) The mustn’t be late again.
(b) He needn’t tell our teacher about the meeting. Sudha has already told him.
(c) You mustn’t make any noise going into the house. It’s very late and everybody is asleep.
(d) I don’t have to hurry. I’ve got plenty of time.
(e) You mustn’t open the door before the bus stops. You could fall out.
(f) You don’t have to wash that shirt. It isn’t dirty.
(g) Sima needn’t finish her essay today.
She’s got to hand it in by next Friday.
(h) You needn’t wear your best clothes. You can wear what you like.

Activity – 31
List five things which a visitor to a zoo must not do.
1. (A) Visitors to a zoo mustn’t…..
2. (S)he…
3. _________________________________
4. _________________________________
5. _________________________________
6. _________________________________
7. _________________________________
8. _________________________________
Answer:
1. He mustn’t tease the wild animals.
2. He mustn ’t row boats himself in the lake without a guide.
3. He mustn’t throw stones at the birds.
4. He mustn’t give food to any animals there.
5. He mustn’t put his fingers inside a cage.
6. He mustn’t disfigure the walls.
7. He mustn’t go near the crocodiles.
8. He mustn’t terrify the small animals.

Activity – 32
You went on a picnic to some far-away place with a group of friends. As you did not know what things would be available there, you made elaborate preparations. Now, after your return, you are narrating your experiences to another group of friends, who would also like to go on a picnic to the same spot.
Tell your friends which of the arrangements made by you were necessary, and which of them were either unnecessary or proved to be unnecessary. Use had to, needed to, didn’t need to, didn’t have to, and need not have to talk about the arrangements which were either necessary or unnecessary.
Here are some of the arrangements that were made.
1. hiring a bus
2. buying fire-wood for cooking
3. carrying water for drinking and cooking
4. carrying food materials: rice daal, vegetables, meat
5. finding a place to cook in
6. finding a place to serve the food
7. going to places of interest nearby
8. thinking of ways to keep yourself entertained
9. washing utensils after the meal
10. making arrangements for the return journey.
Answer:
1. We had to hire a bus for the picnic.
2. We needn’t have to / didn’t have to buy firewood for cooking as it as plentily available there.
3. We didn’t have to carry water for drinking and cooking.
4. We had to carry food materials: rice, daal, vegetables, and meat.
5. We needed to / had to find a place to cook in.
6. We needed to find a place to serve the food.
7. We needn’t have to go to places of interest nearby.
8. We had to think of ways to keep ourselves entertained.
9. We needed to wash the utensils after the meal.
10. We needn’t have to make arrangements for the return journey as we had already arranged before.

Multiple-Choice Questions (MCQs) with Answers

Question 1.
Rain has stopped; you ___________ carry an umbrella with you.
(A) mustn’t
(B) ought not
(C) needn’t
(D) won’t
Answer:
(C) needn’t

Question 2.
A judge ___________ be upright.
(A) shall
(B) will
(C) must
(D) could
Answer:
(C) must

Question 3.
He ___________be at least thirty years old.
(A) can
(B) should
(C) will
(D) must
Answer:
(D) must

Question 4.
Shut the window, ___________?
(A) will you
(B) would you
(C) could you
(D) none of these
Answer:
(B) would you

Question 5.
___________ you open the door, please?
(A) Shall
(B) Will
(C) Can
(D) May
Answer:
(B) Will

Question 6.
You do ___________as you are told.
(A) will
(B) can
(C) must
(D) shall
Answer:
(D) shall

Question 7.
I ___________ try if you wish.
(A) shall
(B) will
(C) should
(D) would
Answer:
(B) will

Question 8.
_____ you rather have tea or coffee?
(A) Should
(B) Will
(C) Would
(D) Can
Answer:
(C) Would

Question 9.
He _____ prefers to have coffee.
(A) will
(B) can
(C) would
(D) none of these
Answer:
(C) would

Question 10.
We _____ disobey our parents.
(A) needn’t
(B) mustn’t
(C) can’t
(D) wouldn’t
Answer:
(B) mustn’t

Question 11.
I ____ do it now.
(A) could
(B) should
(C) can
(D) might
Answer:
(C) can

Question 12.
They worked hard and they succeed.
(A) may
(B) could
(C) might
(D) should
Answer:
(C) might

Question 13.
Let’s go for cricket, _____ we?
(A) should
(B) could
(C) would
(D) shall
Answer:
(D) shall

Question 14.
It _____ rain tonight.
(A) can
(B) could
(C) should
(D) may
Answer:
(D) may

Question 15.
I ___________ do it long ago.
(A) could
(B) might
(C) should
(D) would
Answer:
(A) could

Question l6.
He is so weak that he _____ walks.
(A) will not
(B) would not
(C) cannot
(D) may not
Answer:
(C) cannot

Question 17.
You ____ go now.
(A) can
(B) could
(C) ought
(D) none of these
Answer:
(A) can

Question 18.
_____ I use your telephone directory?
(A) May
(B) Can
(C) Shall
(D) both (A) and (B)
Answer:
(D) both (A) and (B)

Question 19.
If you do this, you _____ be punished.
(A) will
(B) would
(C) shall
(D) can
Answer:
(A) will

Question 20.
I ____ do as you did.
(A) could
(B) should
(C) can
(D) would
Answer:
(D) would

Question 21.
Take heed less you ____ fall.
(A) would
(B) should
(C) might
(D) could
Answer:
(B) should

Question 22.
I _____ have my work here.
(A) must
(B) may
(C) can
(D) none of these
Answer:
(A) must

Question 23.
You _____ if the Court had instructed you to do so.
(A) shall have attended
(B) would have to attend
(C) would have had to attend
(D) would attend
Answer:
(C) would have had to attend

Question 24.
Tell my mother that she _____ telephone me at work for I will be home by seven o’clock this evening.
(A) needs not to
(B) need not to
(C) has no need
(D) need not
Answer:
(D) need not

Question 25.
The fire spread through the building quickly but everybody ___________.
(A) was able to escape
(B) managed to escape
(C) could escape
(D) both (A) and (B)
Answer:
(D) both (A) and (B)

Question 26.
The phone is ringing. It ___________be Tim.
(A) might
(B) can
(C) could
(D) both (A) and (C)
Answer:
(D) both (A) and (C)

Question 27.
Why did you stay at a hotel when you went to New York? You _____ with Barbara.
(A) can stay
(B) could stay
(C) could have stayed
(D) none of these
Answer:
(C) could have stayed

Question 28.
I’ve lost one of my gloves. I ___________ it somewhere.
(A) must drop
(B) must have dropped
(C) must be dropping
(D) must have been dropping
Answer:
(B) must have dropped

Question 29.
Take an umbrella with you when you go out, It _____ rain later.
(A) may
(B) might
(C) can
(D) could
Answer:
(A) may

Question 30.
What was wrong with you? Why ___________ go to the hospital?
(A) had you to
(B) did you have to
(C) must you
(D) none of these
Answer:
(B) did you have to

Question 31.
There’s plenty of time. You ___________ hurry.
(A) don’t have to
(B) mustn’t
(C) needn’t
(D) none of these
Answer:
(A) don’t have to

Question 32.
It was a great party last night. You _____ come. Why didn’t you?
(A) must have
(B) should have
(C) ought to have
(D) had to
Answer:
(B) should have

Question 33.
Jane ______ a car.
(A) suggested that I buy
(B) suggested that I should buy
(C) suggested me to buy
(D) none of these
Answer:
(A) suggested that I buy

Question 34.
I think all drivers _____ seat belts.
(A) should wear
(B) had better wear
(C) had better to wear
(D) none of these
Answer:
(A) should wear

Question 35.
It’s late. It’s time ___________ home.
(A) we go
(B) we must go
(C) we should go
(D) we went
Answer:
(D) we went

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(f) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(f)

Find the following limits:
Question 1.
\(\lim _{x \rightarrow 0} \frac{\tan a x}{x}\)
Solution:

Question 2.
\(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\)
Solution:

Question 3.
\(\lim _{x \rightarrow 1} \frac{\ln x}{x^2-1}\)
Solution:

Question 4.
\(\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}\)
Solution:

Question 5.
\(\lim _{x \rightarrow 0} \frac{x-\sin x}{\tan ^3 x}\)
Solution:

Question 6.
\(\lim _{x \rightarrow 2} \frac{x^3-12 x+16}{3 x^3-8 x^2-4 x+16}\)
Solution:

Question 7.
\(\lim _{x \rightarrow 1} \frac{{In}(2-x)}{1-x^2}\)
Solution:

Question 8.
\(\lim _{x \rightarrow 0+} \frac{\sqrt{1-x}-\sqrt{1+x}}{\sqrt{x}}\)
Solution:

Question 9.
\(\lim _{x \rightarrow 1} \frac{2 \sqrt{x}-3 \sqrt[3]{x}+1}{(x-1)^2}\)
Solution:

Question 10.
\(\lim _{x \rightarrow \infty} \frac{x^3-3 x+1}{2 x^3-7 x^2+5}\)
Solution:

Question 11.
\(\lim _{x \rightarrow 2} \frac{4^x-2^{3+x}+16}{(x-2)^2}\)
Solution:

Question 12.
\(\lim _{x \rightarrow 0+} \frac{\ln \tan x}{\ln \sin 2 x}\)
Solution:

Question 13.
\(\lim _{x \rightarrow 0} \frac{e^x-e^{-x}-2 \sin x}{x \sin x}\)
Solution:

Question 14.
\(\lim _{x \rightarrow \infty} \frac{{In}\left(\frac{1}{x}\right)}{x^x}\)
Solution:

Question 15.
\(\lim _{x \rightarrow 0+} \frac{e^{-\frac{1}{x}}}{x}\)
Solution:

Question 16.
\(\lim _{x \rightarrow \infty} \frac{x^n+x^{-1}}{(x+2)^n}\)
Solution:

Question 17.
\(\lim _{x \rightarrow 0} \frac{e^x-e^{-x}}{x \cos x}\)
Solution:

Question 18.
\(\lim _{x \rightarrow \infty} \frac{\sin ^{-1} x}{x}\)
Solution:

Question 19.
\(\lim _{x \rightarrow 0+}\) log tanx tan 2x
Solution:

Question 20.
\(\lim _{x \rightarrow \pi / 2}\)(tan x)^{cos x}
Solution:

Question 21.
\(\lim _{x \rightarrow 1} x^{\frac{1}{1-x}}\)
Solution:

Question 22.
\(\lim _{x \rightarrow 0+} x^{\sin x}\)
Solution:

Question 23.
\(\lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{1}{\ln x}\right)\)
Solution:

Question 24.
\(\lim _{x \rightarrow \infty}\left(x-\sqrt{x^2-1}\right)\)
Solution:

Question 25.
\(\lim _{x \rightarrow \infty}\left(x-\sqrt{x^2+2 x}\right)\)
Solution:

Question 26.
\(\lim _{x \rightarrow 0}\left(\frac{1}{\sin ^2 x}-\frac{1}{x^2}\right)\)
Solution:

Question 27.
\(\lim _{x \rightarrow \pi / 2}\) (tan 3x – tan x)
Solution:

Question 28.
\(\lim _{x \rightarrow 0+} \frac{\sqrt{a^x-b^x}}{\sqrt{x}}\), a > b
Solution:

Question 29.
\(\lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right)^x\)
Solution:

Question 30.
\(\lim _{x \rightarrow \infty}\left(e^{\frac{1}{x}}\right)^{\ln (1+x)}\)
Solution:

Question 31.
\(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{1 / x}\)
Solution:

Question 32.
\(\lim _{x \rightarrow 0}\left(1+x^2\right)^{1 / x}\)
Solution:

Question 33.
\(\lim _{x \rightarrow 0}\) (cot x – cosec x)
Solution:

Question 34.
\(\lim _{x \rightarrow 1}\) (2 – x)^{cosec πx}
Solution:

Question 35.
\(\lim _{x \rightarrow 0}\left(\frac{x^2+2 x-1}{x^2-1}\right)^{\frac{1}{x}}\)
Solution:

Question 36.
\(\lim _{x \rightarrow 0} \frac{(1+x)^{1 / x}-e}{x}\)
Solution:

Question 37.
\(\lim _{x \rightarrow 0}(\tan x)^{1 / \ln x}\)
Solution:

Question 38.
\(\lim _{x \rightarrow 0} \frac{x^3 \sin \frac{1}{x}}{\tan x}\)
Solution:

Question 39.
\(\lim _{x \rightarrow 0+} \frac{2^x-1}{\sqrt{(1+x)}-1}\)
Solution:

Question 40.
\(\lim _{x \rightarrow 1}(1-x) \tan \frac{\pi x}{2}\)
Solution:

Question 41.
\(\lim _{x \rightarrow 0} \frac{x \cos x-\sin x}{x^2 \sin x}\)
Solution:

Question 42.
\(\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^2}}}{x^2}\)
Solution:

Question 43.
\(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{\frac{1}{x^2}}\)
Solution:

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 2 The Present Perfect Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 2 The Present Perfect

SECTION- 1:
Examine the use of the Present Perfect in the following sentences.
(a) A: Where’s your T.V. set? I don’t see it.
B: I have sold it.
(b) A: Why are you looking so happy?
B: I have just got a job.
In the above examples, the speaker ‘B’ talks about some events beginning in the past and lasting up to the present moment (or still continuing). Though the event started in the past, it is connected to the present moment of speaking. In this sense, we used the Present Perfect Tense.

Activity — 9
Complete the sentences marked B. Use the verb in brackets, together with ‘just/already/yet’.
A: What does your wife think of your plan?
B : I __________(not tell) her yet.
A: Would you like something to eat?
B : No, thanks I __________(just/eat).
A: Is your brother here yet?
B : Yes, he __________(just/arrive).
A: What’s on TV today?
B: I don’t know. I __________(not see) the programme yet.
A: Do you know where Bidhu lives?
B : Yes, he __________(just/move) to Satyanagar.
A: Are your friends coming to the circus with us?
B : No, they __________(already/see) it.
A: When is Prakash leaving?
B : He __________(already/leave).
Answer:
1. A: What does your wife think of your plan?
B: I have not told her yet.
A: Would you like something to eat?
B: No, thanks I have just eaten.
A: Is your brother here yet?
B: Yes, he has just arrived.
A: What’s on TV today?
B: I don’t know. I have not seen the program yet.
A: Do you know where Bidhu lives?
B: Yes, he has just moved to Satyanagar.
A: Are your friends coming to the circus with us?
B: No, they have already seen it.
A: When is Prakash leaving?
B: He has already left.

Activity – 10
Study the situations suggested below and makeup sentences with “yet”, “already” or “just”.
1. You are going to Koraput next Sunday. You phone your travel agent to buy a ticket for you.
Later your father says. ‘Shall I get the ticket for you ?”
You: No, (buy) ___________________________________.
2. Alok goes to the Post Office but returns after a while. His friend asks you if he is still at the Post Office.
You: No, (come back) ___________________________________.
3. You know that one of your classmates is looking for a house. When you meet him, you want to know if he has been successful.
You: ___________________________________.
4. You visit a friend’s house after lunch. He asks if you would like something to eat.
You: ___________________________________.
5. You are doing your homework. Your brother thinks that you have finished and turns the light off. What would you tell him?
You: ___________________________________.
6. Amar goes out. Ten minutes later his friend comes and asks you if he can meet Amar.
You: ___________________________________.
Answer:
1. No, I have already bought it from a travel agent.
2. No, he has already come back.
3. Have you got a house yet?
4. Have you had your lunch yet?
5. I have not finished my homework yet.
6. Amar has iust gone out

Activity – 11
Below is a list of things that your parents have asked you to do today. You have checked the things you’ve done so far. Talk about the things you’ve already done and the things you haven’t done yet. (Two have been done for you as examples.)
1. do the washing up
2. do your homework ✓
3. wash the scooter
4. write to the brother
5. read today’s newspapers ✓
6. defrost the fridge
7. buy some fruits ✓
8. watch the news on TV
9. clean the windows ✓
10. water the plants
11. empty the dustbins ✓
12. phone uncle
Answer:
1. I haven’t done the washing up yet.
2. I have already done my homework.
3. I have not washed the scooter yet.
4. I have not written to my brother yet.
5. I have already read today’s newspaper.
6. I have not de-frosted the fridge yet.
7. I have already bought some fruit.
8. I have not watched the news on TV.
9. I have already cleaned the windows.
10. I haven’t watered the plants yet.
11. I have not emptied the dustbin yet.
12. I have not phoned my uncle yet.

Activity – 12
Anil, Anima, Mohan and Anand are talking about the places they have visited. Fill in the spaces using the information in the chart below.

	Kolkata	Koraput	Puri	Sambalpur	Shillong
Anil	Yes	No	No	Yes	No
Anima	Yes	No	Yes	Yes	No
Mohan	No	No	Yes	No	No
Anand	Yes	No	Yes	Yes	No

1. Anil __________been to Kolkata, but Mohan ___________.
2. Three people __________ been to Puri.
3. Only one person __________been to Sambalpur.
4. Mohan is the only one who __________ visited only one place.
5. Nobody __________ been to Shillong.
6. Two people __________ been to three places.
7. Anima and Mohan __________ both been to Puri, but neither __________ been to Koraput.
Answer:
1. Anil has been to Calcutta, but Mohan hasn ’t.
2. Three people have been to Puri.
3. Only one person hasn’t been to Sambalpur.
4. Mohan is the only one who has visited only one place.
5. Nobody has been to Shillong.
6. Two people have been to three places.
7. Anima and Mohan have both been to Puri, but neither haven’t been to Koraput.

Note:
Look at the difference between ‘have /has been ’ and ‘have/has gone ‘Has /have been’ means went and returned. But ‘has gone/have gone means went and not returned. The Present Perfect form of ‘go’ (has/have gone) is not used when the subject of the sentence is 7, we or you.’ The Present Perfect is used to refer to some happening in the past, for which the time of action is not given.

SECTION – 2
Look at the sentences below.
1. Hari didn’t have a beard six months ago. He has a beard now. He has grown a beard.
2. Malati was very shy. She is smart now. She has become smart.
3. She was a little baby when I last saw her. She is a young girl now. She has grown up.
When a change has taken place between now and sometime before now, we use the Present Perfect Tense.

Activity – 13
Study the situations below and make up appropriate sentences using the verbs suggested.
1. Yesterday my sister bought a pen. She can’t find it now. (lose)
______________________________________________.
2. The children were playing here some time ago. Now they are not seen, (leave)
______________________________________________.
3. My friend weighed 50 kilograms. Now he weighs 70. (gain weight)
______________________________________________.
4. The man met with an accident. Now he is not able to speak, (loses voice)
______________________________________________.
5. It was raining in the morning. Now the sky is clear. (stop)
______________________________________________.
6. The tiger attacked the man. He is dead now. (kill)
______________________________________________.
7. He had some paper with him. Now he does not have any to write on. (run out)
______________________________________________.
8. My teacher got a job in a bank. He is not coming to school anymore, (resign)
______________________________________________.
Answer:
1. She has lost it somewhere.
2. They have already left.
3. He has gained weight.
4. He has lost voice in the accident.
5. The rain has already stopped.
6. The tiger has already killed the man.
7. He has already run out of paper.
8. He has already resigned from school.

SECTION – 3
Present Perfect is often used with the following time expressions.

lately	until now	ever	for five years
not yet	never	always	over the last eight years
recently	just	so far	in the past Iwo years
in recent years	already	since 1990

Do remember that the expressions like “last year, ago, yesterday” etc. cannot be used in the Present Perfect.

Activity – 14
Rewrite the following sentences putting the words in brackets in the right place. The first one has been done for you.
1. My teacher has wanted to be a writer. (never)
Ans. My teacher has never wanted to be a writer.
2. I’ve found him helpful, (always)
______________________________________________.
3. People have misunderstood him. (often)
______________________________________________.
4. I’ve had lunch. (just)
______________________________________________.
5. Has he been to Puri? (ever)
______________________________________________.
6. Don’t panic. The police have arrested the culprit, (already)
______________________________________________.
Answer:
2. I’ve always found him helpful.
3. People have often misunderstood him.
4. I’ve Just had lunch.
5. Has he ever been to Puri?
6. Don’t panic. The police have already arrested the culprit.

Activity – 15
Imagine that you suddenly run into an old friend whom you have not met for the last five years. But he has changed so much that you can hardly recognize him. Describe the changes that have taken place in your friend.

a) _______________________________
b) _______________________________
c) _______________________________
d) _______________________________
e) _______________________________
Answer:
1. He has grown into a young man.
2. He has become strong and healthy.
3. He has grown fair and tall.
4. He has made himself smart and confident.
5. He has grown a thick beard.

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Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(e) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(e)

Question 1.
Determine the differentials in each of the following cases.
(i) y = x³ – 1
Solution:
y= x³ – 1
Then dy = 3x² dx

(ii) y = sin² x
Solution:
y = sin² x
Then dy = 2 sin x cos x dx = sin 2x dx

(iii) y = \(\frac{1+\sqrt{x}}{1-\sqrt{x}}\)
Solution:

(iv) z = cos 2t – 2 cot t
Solution:
z = cos 2t – 2 cot t
dz = (-2 sin 2t + 2 cosec² t) dt

(v) r = \(\frac{4}{1+\sin \theta}\)
Solution:
r = \(\frac{4}{1+\sin \theta}\)
dz = \(\frac{-4 \cos \theta}{(1+\sin \theta)^2}\) . dθ
= \(-\frac{4 \cos \theta}{(1+\sin \theta)^2}\) dθ

(vi) x²y = 2
Solution:
⇒ y = \(\frac{2}{x^2}\)
dy = \(\frac{4}{x^3}\)

(vii) xy² + yx² = 1
Solution:
⇒ dx . y² + x . 2y dy + dy . x² + y . 2x dx = 0
⇒ (2 xy + x²) dy = – (y² + 2xy) dx
⇒ dy = –\(\frac{y^2+2 x y}{x^2+2 x y}\)dx

Question 2.
Find δf and df when
(i) f(x) = 2x² – 1, x = 1, δx = 0.02
Solution:
f(x) = 2x² – 1, x = 1, δx = 0.02
Then δf = f(x + δx) – f(x)
= f(1.02) – f(1)
= 2 (1.02)² – 1 – (2 – 1)
= 2.0808 – 2 = 0.0808
Again df = 4x dx
= 4 × 1 × 0.02 = 0.08

(ii) f(x) = √x, x = 16, δx = 0.3
Solution:
f(x) = √x, x = 16, δx = 0.3

(iii) f(x) = (x + 1)³, x = 8, δx = 0.04
Solution:
f(x) = (x + 1)³, x = 8, δx = 0.04
δf = f(x + δx) – f(x)
= (9.04)³ – (8 +1)³
= (9.04)³ – 9³ = 9.7632
Again df = 3(x + 1)² dx = 3 × 9² × 0.04
= 243 × 0.04 = 9.72

(iv) f(x) = In (1 + x), x = 1, δx = 0.04
Solution:
f(x) = In (1 + x), x = 1, δx = 0.04
δf = f(x + 8x) – f(x)
= In (1 + x + 5x) – ln (1 + x)
= In (2.04) – In (2) = 0.0198
Again df = \(\frac{1}{1+x}\)dx = \(\frac{1}{2}\) × 0.04 = 0.02

Question 3.
Find approximate values of the following:
(i) \(\sqrt[3]{28}\)
Solution:

(ii) \(\sqrt[6]{63}\)
Solution:

(iii) \(\sqrt{48 \cdot 96}\)
Solution:

(iv) (1.99)⁷
Solution:
Let y = x⁷
Then dy = 7x⁶ dx
⇒ δy = 7x⁶ δx
⇒ (x + δx)⁷ – x⁷ = 7x⁶ δx
(x + δx)⁷ = x⁷ + 7x⁶ δx
Put x = 2 and δx = -0.01
Then (1.99)⁷ = 2⁷ – 7 × 2⁶ × 0.01
= 128 – 7 × 64 × 0.01
= 128 – 4.48 = 123.52

(v) 2^3.02
Solution:
Let y = 2^x
Then dy = 2^x In 2 dx
⇒ δy = 2^x In 2 . δx
⇒ 2^{x + δx} – 2^x = 2^x In 2 . δx
⇒ 2^{x + δx} = 2^x + 2^x In 2 δx
Then 2^3.02 = 2³ + 2³ In 2 × 0.02
= 8 + 8 In 2 × 0.02 = 8.1109

(vi) sin 59°
Solution:
Let y = sin x
Then dy = cos x dx ⇒ δy = cos x . δx
⇒ sin (x + δx) – sin x = cos x × δx
⇒ sin (x + δx) = sin x + cos x × δx
Put x = 60°, δx = -1°
Then sin 59° = sin 60° + cos 60° x – \( \frac{\pi}{180} \)
\(\frac{\sqrt{3}}{2}\) – \(\frac{1}{2}\) × \(\frac{\pi}{180}\)
[ ∵ 1° = \(\frac{\pi}{180}\) radian = 0.85729

Question 4.
find the percentage of error in calculation of the surface area of a spherical balloon of diameter 14.02 m. if the true diameter is 14m.
Solution:
Let S be the surface area of a spherical balloon of radius V.
Then S = 4πr²
Then dS = 8πr dr

Question 5.
Find approximately the difference between the volumes of two cubes of sides 3 cm and 3.04 cm.
Solution:
Let V be the volume of a cube of side x cm.
Then V = x³ ⇒ dV = 3x² dx
Here x = 3 cm and dx = (3.04 – 3) cm = 0.04 cm
Thus dV = 3 × 9 × 0.04 = 1.08 cm³
∴ Difference of two volumes is 1.08 cm³.

Question 6.
The height of a regular cone is 3 times the radius of its base. The radius of the base was wrongly measured to be 5 cm. whereas its true radius is 4.88 cm. Find the relative error in measuring the curved surface area of the cone.
Ans.
Consider a regular cone of height h and radius of its base x.
Then h = 3x.
If S is the area of the curved surface of the cone
then S = πxl
Where l is the slant height of the cone.

= 0.049

Must Refer:

Bank Nifty Point Calculator

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(h) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(h)

Find \(\frac{d y}{d x}\)
Question 1.
x = a cos θ, y = a sin θ
Solution:
x = a cos θ, y = a sin θ

Question 2.
x = at², y = 2 at at t = \( \frac{1}{2}\)
Solution:
x = at², y = 2 at at t = \( \frac{1}{2}\)

Question 3.
x = a cos³ t, y = a sin³ t at t = \( \frac{\pi}{4} \)
Solution:
x = a cos³ t, y = a sin³ t

Question 4.
sin x = \(\frac{2 t}{1+t^2}\), tan y = \(\frac{2 t}{1-t^2}\)
Solution:

Question 5.
x = 3 cos t – 2 cos³ t,
y = 3 sin t – 2 sin³ t.
Solution:
x = 3 cos t – 2 cos³ t

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 7 Future Time Reference Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 7 Future Time Reference

SECTION – 1

The Present Progressive is used for future reference
A: What are you doing tomorrow evening?
B: I’m going to the town hall. My friends are coming. We are putting up a show for the handicapped. We are also meeting the Minister for some funds. Are you going somewhere tomorrow evening?
A: Yes, I’m going to the library. I thought I could ask you to come along.
In the dialogue above you can only see the use of more number of Present Progressive constructions/sentences.

Here are a few questions for you to answer.
1. Which period of time does the use of the Present Progressive refer to: time which is past, time which is present, or future time?
2. Are ‘A’ and ‘B’ talking about actions that they have already planned and arranged to do?
3. Is the arrangement personal (made by either A or B) or is official (made by someone else, who is in a position of authority)?
Answer:
1. The time in all the Present Progressive sentences refers to a future time (futurity).
2. Yes. In the dialogue between A and B, we see the actions have already been planned/arranged.
3. The arrangements are personal. No official planning/arrangements.
The above sentences (Present Progressive form) are based on a personal plan or decision or need or requirement. So now we can say these sentences are based on the ‘internal evidence’ of the speaker or writer.

Activity – 26
Surabhi, who is 16, wants to go on an excursion with her friends and teachers. . Her mother is worried and has a lot of questions to ask about the arrangements. Look at the hints supplied and complete the dialogue between them, using the appropriate forms of the verbs.
Mother: Who / you / go / with?
______________________
Surabhi: friends/teachers
______________________
Mother: Where / you / go?
______________________
Surabhi: Darjeeling
______________________
Mother: When / go / there?
______________________
Surabhi: next Monday
______________________
Mother: How / you / get there?
______________________
Surabhi: by bus?
______________________
Mother: Where / you / stay?
______________________
Mother: a hotel.
______________________
Answer:
Mother: Who are you going with?
Surabhi: I’m going with my friends and teachers.
Mother: Where are you going?
Surabhi: I’m (We are) going to Darjeeling.
Mother: When are you going there?
Surabhi: I’m (We are) going there next Monday.
Mother: How are you getting there?
Surabhi: I’m (We are) getting there by bus.
Mother: Where are you staying?
Surabhi: I’m (We are) staying in a hotel.
Mother: What are you doing the next Tuesday there?
Surabhi: I’m visiting a cinema with my cousin.
Mother: When are you coming back?
Surabhi: I’m (We are) coming back next Sunday.

Activity – 27
Bindu maintains a diary in which she writes down a list of the things that she plans to do during the week ahead. Here is an outline of the entries in her diary for the next week. Complete the entries, using the hints given below. Use the Present Progressive.
Monday: meet music teacher.
______________________
Tuesday: go to the cinema with cousin
______________________
Wednesday: play badminton
______________________
Thursday: see off Maya at the station.
______________________
Friday: throw a party for friends.
______________________
Saturday: visit a dentist in the evening.
______________________
Sunday: rest.
______________________
Answer:
Monday: Bindu is meeting her music teacher on Monday.
Tuesday: She is going to the cinema with her cousin.
Wednesday: She is playing badminton in the evening.
Thursday: She is seeing off Maya at the station.
Friday: She is throwing a party for her friends.
Saturday: She is visiting the dentist in the evening.
Sunday: She is having a rest on Sunday.
Or, She is resting on Sunday.

Activity – 28
You plan to visit Koraput after your examination. Mention five things that you have arranged to do there.
(a) ……………………………………………………………………………………
(b) ……………………………………………………………………………………
(c) ……………………………………………………………………………………
(d) ……………………………………………………………………………………
(e) ……………………………………………………………………………………
Answer:
(a) I’m meeting one of my school friends there.
(b) I’m attending her birthday ceremony.
(c) I’m visiting the Damanjodi project there.
(d) I’m helping her with the party.
(e) I’m visiting new places there with my friend.

SECTION – 2

The Present Simple for future time reference
Minister: Have you drawn up my tour program for the next week?
P. A.: Yes sir. You leave for Sambalpur at 7.00 a.m. on Monday morning.
Minister: When do I get there?
P. A. : You reach Sambalpur at 1.00 p.m. You halt at Angul for a few minutes, on the way Then, after lunch, you meet the Commissioner at 3.00 for a discussion.
Minister: Where do I stay in Sambalpur?
P.A.: The Guest House at the Hirakud Dam has been reserved for you, sir. Then, the next morning at 8.00, you proceed to Rourkela. The Commissioner accompanies you to Rourkela.
Minister: And when do I return to Bhubaneswar?
P. A. . On Thursday, sir.
Look at the use of the present simple in the dialogue. What is the time reference here: past time, present time, or future time?
The Present Simple Tense’ can be used for future time reference. We use this tense for future events, i.e. fixed by the calendar or an official or departmental timetable. It is used to talk about the activity which has already been planned and fixed.

Activity — 29
The following is the list of official engagements of the Chief Minister for next Monday. Write one sentence to describe each item. Use the Present Simple form of the verb given in brackets :
8.30 a.m. inaugural address, Conference on Preservation of Human Rights (deliver)
9.30 a.m. Speech on Syllabus Reform, Utkal University. (give)
1.00 p.m. Cabinet Committee meeting (preside over)
4.30 p.m. National Book Fair (inaugurate)
5.50 p.m. Parliamentary delegation from Turkey (welcome)
8.00 p.m. Dinner party in honor of the Prime Minister of Bangladesh (host)
____________________________________________
____________________________________________
____________________________________________
____________________________________________
____________________________________________
Answers
8.30 a.m. The Chief Minister delivers his inaugural address at the Conference on Preservation of Human Rights.
9.30 a.m. The Chief Minister gives his speech on Syllabus Reform, at Utkal University.
1.00 p.m. The Chíef Minister presides over the Cabinet Committee meeting.
4.30 p.m. The Chief Minister inaugurates the National Book Fair at Unit — III, Exhibition field, Bhubaneswar.
5.50 p.m. The Chief Minister welcomes the Parliamentary delegation from Turkey.
8.00 p.m. The Chief Minister hosts the Dinner party in honor of the Prime Minister of Bangladesh.

Activity — 30
A travel agency offers a number of packages for tourists. Here is an outline of such a travel package.
Imagine that you are a salesman in a travel agency and that you are describing the program to a group of tourists, who want to know the details. Use complete sentences, with the Present Simple, to describe the program.
Monday → Leave Bhubaneswar for Hyderabad by Konark Express
March 2 →2 days’ sight-seeing in Hyderabad
Thursday → Board Kaveri Express for Chennai
March 5 → 2 nights in Chennai, Visit to Mahabalipuram
Saturday → Board Vrindaban Express for Bangalore
March 7 → 2 nights in Bangalore
Monday → By Deluxe bus to Mysore
March 9 → 1 night in Mysore
Tuesday → Mysore to Goa by taxi
March 10 → 2 nights in Goa
Thursdaý → Board Flight IC 765 for Bhubaneswar
March 12
Answer:
Monday → The tourists leave Bhubaneswar for Hyderabad by Konark Express.
March 2 → They go for 2 days sightseeing in Hyderabad.
Thursday → They board the Kaveri Express for Chennai.
March 5 → They stay 2 nights in Chennai and visit Mahabalipuram.
Saturday → They board the Vrindaban Express for Bangalore.
March 7 → They halt 2 nights in Bangalore.
Monday → They go by Deluxe bus to Mysore.
March 9 → They halt for 1 night in Mysore.
Tuesday → They go from Mysöre to Goa by taxi.
March 10 → They put up 2 nights in Goa.
Thursday → They board flight IC 765 for Bhubaneswar.
March 12

SECTION-3

Future time reference with ‘be going to’
1. I’m sweating already. I think it’s going to be a very hot day.
2. Be careful of that dog; it’s going to bite.
3. Mohan is eating too much; he’s going to be sick.
In the above sentences ‘be going to’ structure is used. In ‘be’ verb family we have eight verbs such as: am, is, are, was, were, be, being, and been. But in this structure for futurity (future time reference) am, is, are only these three are used.
be going to = is / am / are going to.
We can use this ‘be going to ’ structure for ‘external evidence ’ at the moment of speaking. We can better mark in the above three sentences some evidence or reason at the moment of speaking which helps the speaker to predict/believe something that is likely to happen in the near future.

Activity – 31
Complete the sentences using ‘be going to’ and the verbs in brackets.
(a) Look at those dark clouds. It’s _____________(rain).
(b) The lady is gasping for breath. I think she’s _____________(faint).
(c) My neighbor has packed up all his belongings. I think he _____________ (leave) the house.
(d) Rakesh _____________(fail) the exam. I don’t see him studying at all.
(e) Bijoy Babu _____________(lose) in the election. The voters are very unhappy with him.
Answer:
(a) Look at those dark clouds. It’s going to rain.
(b) The lady is gasping for breath. I think she is going to faint.
(c) My neighbor has packed up all his belongings. I think he is going to leave the house.
(d) Rakesh is going to fail the exam. I don’t see him studying at all.
(e) Bijoy Babu is going to lose in the election. The voters are very unhappy with

Activity – 32
Study the situation and guess what is going to happen. (One example has been given)
(a) The old man has been ill for a long time. He stopped taking food 5 days ago.
Ans. He is going to die.
(b) My friend has been reading the “matrimonial” column in the newspaper and collecting photographs of girls.
_____________________________________________________.
(c) A man is getting into the house opposite through the window. The people who live there are away on holiday.
_____________________________________________________.
(d) Do you see that man is trying to walk on the ice? His feet are sleeping.
____________________________________________________.
(e) The policeman is running after the thief and pointing his gun at him.
_____________________________________________________.
(f) The boy has taken the book from the shelf and put it on the table. He is drawing up a chair now.
_____________________________________________________.
(g) India meets Pakistan in the final match today. Sachin Tendulkar is injured and will not be able to play.
_____________________________________________________.
(h) Water has got into the boat.
_____________________________________________________.
Answer:
(b) My friend is going to marry soon.
(c) The man is going to steal from the house.
(d) The man is going to fall.
(e) The policeman is going to fire.
(f) He is going to read the book.
(g) India is going to lose the match.
(h) The boat is going to sink.

SECTION – 4

Here are some more sentences with ‘be going to’.
(a) A. Why have you bought so many books?
What are you going to do?
B. I‘m going to read all of them for my project.
(b) She has decided not to leave the house this year. She is going to stay on for another year.
(c) A. I’m going to buy a scooter
B. How‘re you going to pay for it?
A. I’m going to ask my brother to lend me some.
Look at these sentences with ‘be going to’. How are they different in meaning from the sentences in Section 3 (above)?
In this case, be going to is used to express an ‘intention’ to do something in the future. We can say: a decision or plan is based on a personal need or requirement.

Activity-33

There are a number of things that you haven’t done yet but intend to do. Answer
the questions below, using be going to as well as the words in brackets.
Friend: Have you had lunch?
You: No, but _____________. (after my friend arrives)
Friend: Have you written a letter to your father?
You: Not yet, _____________. (tomorrow)
Friend: Have you read the new novel by Vikram Seth?
You: No, _____________. (next week),
Friend: Have you watered the plants?
You: Not yet, _____________. (this afternoon)
Answer:
You: No, but I am going to have my lunch after my friend’s arrival.
Friend: Have you written a letter to your father?
You: Not yet, I am going to write a letter to my father tomorrow.
Friend: Have you read the new novel by Vikram Seth?
You: No, I am going to read the new novel by Vikram Seth next week.
Friend: Have you watered the plants?
You: Not yet, I am going to water the plants this afternoon.

Activity – 34
The members of a Youth Club have taken a vow on Gandhi Jayanti. They have promised that each of them will do at least one good deed, in memory of Mahatma Gandhi.
Can you draw up a list of the good things that the boys intend to do? Here is an example.
I’m going to plant 100 trees inside the school compound.
(a) ____________________________________________
(b) ____________________________________________
(c) ____________________________________________
(d) ____________________________________________
(e) ____________________________________________
(f) ____________________________________________
(g) ____________________________________________
Answer:
(a) I’m going to clean the rubbish in our lane.
(b) I’m going to help the poor and sick.
(c) I’m going to be friends with people of other religions.
(d) I’m going to be truthful.
(e) I’m going to fight against injustice.
(f) I’m going to be punctual.
(g) I’m going to preach equality among people.

Multiple-Choice Questions (MCQs) with Answers

Question 1.
London ___________the capital of the United Kingdom.
(A) were
(B) are
(C) is
(D) is being
Answer:
(C) is

Question 2.
It ___________since yesterday evening.
(A) was snowing
(B) had been snowing
(C) snowed
(D) has been snowing
Answer:
(D) has been snowing

Question 3.
Just as I was entering the room, the family was ___________for a party.
(A) leaving
(B) left
(C) had left
(D) going
Answer:
(A) leaving

Question 4.
The arm was so badly injured that he ___________it amputated.
(A) has to have
(B) had had
(C) had to have
(D) must have
Answer:
(C) had to have

Question 5.
I ___________ very busy lately.
(A) have been
(B) will be
(C) would be
(D) having being
Answer:
(A) have been

Question 6.
He was not thirsty because he ___________too much water.
(A) had drank
(B) had drunk
(C) had been drunk
(D) had drunken
Answer:
(B) had drunk

Question 7.
Where ___________since we last played together?
(A) were you playing
(B) had you been playing
(C) have you been playing
(D) do you play
Answer:
(C) have you been playing

Question 8.
As soon as she noticed the workmen, she asked them what they ___________.
(A) have been doing
(B) are doing
(C) have done
(D) had been done
Answer:
(C) have done

Question 9.
The bus has been traveling for days and it times it ___________.
(A) stops
(B) will stop
(C) stopped
(D) would stop
Answer:
(C) stopped

Question 10.
You need not ___________when the barrister asked you where you were when the crime was committed.
(A) be lying
(B) to tell a lie
(C) tell a lie
(D) have tell a lie
Answer:
(C) tell a lie

Question 11.
The news about the disaster ___________six weeks ago.
(A) broadcasted
(B) was broadcasting
(C) was broadcasted
(D) was broadcast
Answer:
(D) was broadcast

Question 12.
___________daily keeps people healthy and vigorous.
(A) In eating
(B) Eating
(C) Having eaten
(D) To eat
Answer:
(D) To eat

Question 13.
I cannot imagine myself ___________ car repairs in the garage.
(A) to do
(B) doing
(C) to be doing
(D) in doing
Answer:
(B) doing

Question 14.
The pianist ___________his performance to be well accepted by the audience willingly practiced his art.
(A) expect
(B) expecting
(C) expected
(D) expects
Answer:
(B) expecting

Question 15.
After our football team lost the cup final, the manager told them ___________ and not winning is the most important.
(A) playing the game
(B) the game
(C) the play
(D) the gaming
Answer:
(B) the game

Question 16.
I noticed the culprit ___________ away from the house.
(A) run
(B) would be running
(C) ran
(D) had running
Answer:
(B) would be running

Question 17.
I am convinced he ___________to school after his insolence.
(A) dare not return
(B) dare not returning
(C) dare not to return
(D) does not dare returning
Answer:
(A) dare not return

Question 18.
‘___________this week?’ ‘No, she’s on holiday.’
(A) Is Susan working
(B) Does Susan work
(C) Does work Susan
(D) none of these
Answer:
(A) Is Susan working

Question 19.
I don’t understand this sentence. What ___________?
(A) does mean this word
(B) does this word mean
(C) means this word
(D) none of these
Answer:
(B) does this word mean

Question 20.
John ___________tennis once or twice a week.
(A) is playing usually
(B) is usually playing
(C) usually plays
(D) none of these
Answer:
(C) usually plays

Question 21.
How ___________now ? Better than before?
(A) you are feeling
(B) do you feel
(C) are you feeling
(D) none of these
Answer:
(B) do you feel

Question 22.
It was a boring weekend ___________anything.
(A) I didn’t
(B) I don’t do
(C) I didn’t do
(D) none of these
Answer:
(C) I didn’t do

Question 23.
Tom ___________his hand when he was cooking the dinner.
(A) burnt
(B) was burning
(C) has burnt
(D) none of these
Answer:
(A) burnt

Question 24.
Jim is away on holiday. He ___________to Spain.
(A) is gone
(B) has gone
(C) has been
(D) none of these
Answer:
(B) has gone

Question 25.
Everything is going well. We ___________any problems so far.
(A) didn’t have
(B) don’t have
(C) haven’t had
(D) none of these
Answer:
(C) haven’t had

Question 26.
Linda has lost her passport again. It’s the second time this ___________.
(A) has happened
(B) happens
(C) happened
(D) none of these
Answer:
(A) has happened

Question 27.
You’re out of breath ___________?
(A) Are you running
(B) Have you run
(C) Have you been running
(D) none of these
Answer:
C) Have you been running

Question 28.
Where’s the book I gave you? What ___________with it?
(A) have you, done
(B) have you been doing
(C) are you doing
(D) none of these
Answer:
(A) have you, done

Question 29.
We’re good friends. We __________ each other for a long time.
(A) know
(B) have known
(C) have been knowing
(D) knew
Answer:
(B) have known

Question 30.
Sally has been working here __________.
(A) for six months
(B) since six months
(C) six months ago
(D)none of these
Answer:
(A) for six months

Question 31.
It’s two years __________ Joe.
(A) that I don’t see
(B) that I haven’t seen
(C) since I didn’t see
(D) since I saw
Answer:
(D) since I saw

Question 32.
They __________out after lunch and they’ve just come back.
(A) went
(B) have gone
(C) are gone
(D) none of these
Answer:
(A) went

Question 33.
The Chinese __________printing.
(A) invented
(B) have invented
(C) had invented
(D) none of these
Answer:
(A) invented

Question 34.
I am __________in Scotland for ten years. Now he lives in London.
(A) lived
(B) has lived
(C) has been living
(D) none of these
Answer:
(A) lived

Question 35.
The man sitting next to me on the plane was nervous because he __________before.
(A) hasn’t flown
(B) didn’t fly
(C) hadn’t flown
(D) none of these
Answer:
(C) hadn’t flown

Question 36.
__________a car when they were living in London?
(A) Had they
(B) Did they have
(C) Were they having
(D) Have they had
Answer:
(B) Did they have

Question 37.
I __________television a lot but I don’t anymore.
(A) was watching
(B) was used to watch
(C) used to watch
(D) none of these
Answer:
(C) used to watch

Question 38.
__________ tomorrow, so we can go out somewhere.
(A) I’m not working
(B) I don’t work
(C) I won’t work
(D) none of these
Answer:
(A) I’m not working

Question 39.
That hag looks heavy, __________ you with it.
(A) I’m helping
(B) I help
(C) I’ll help
(D) none of these
Answer:
(C) I’ll help

Question 40.
I think the weather __________be nice later.
(A) will
(B) shall
(C) is going to
(D) none of these
Answer:
(A) will

Question 41.
‘Ann is in hospital.’ ‘Yes, I know. __________her tomorrow.’
(A) I visit
(B) I’m going to visit
(C) I’ll visit
(D) none of these
Answer:
(B) I’m going to visit

Question 42.
We’re late. The film __________by the time we get to the cinema.
(A) will already start
(B) will be already started
(C) will already have started
(D) none of these
Answer:
(C) will already have started

Question 43.
Don’t worry __________late tonight.
(A) if I am
(B) when I am
(C) when I’ll be
(D) if I’ll be
Answer:
(A) if I am

Question 44.
The film __________by the time we get to the cinema.
(A) will start
(B) will have started
(C) will be starting
(D) will have been starting
Answer:
(B) will have started

Question 45.
Next year they __________for 25 years.
(A) will have been marrying
(B) will have been married
(C) will be marrying
(D) none of these
Answer:
(B) will have been married

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Conditionals Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Conditionals

Section – 1

The ‘if-sentences’ are generally known as the ‘conditional’ sentences. In this ‘if- sentences’ one can well mark a condition along with a result/effect. A certain action will be fulfilled only if a certain/particular ‘condition’ is performed.

Example
1. I will pardon you if you admit your mistake.
2. If you touch that plate, you’ll bum your fingers.
In the above sentences, we can well see two parts in each sentence, i.e. (i) if-part and (ii) the main part.

The ‘if-part’ is known as the if-clause or conditional clause or the subordinate clause. But the ‘main part’ is usually known as the stem of the long sentence i.e. the principal/main clause, the word ‘if’ is the ‘conjunction’ that joins both the parts of the long sentence. This if part/clause states the condition that must be satisfied before the main part/clause may be true.

In the example in Sentence 1 above, ‘admit’ is the condition and ‘pardon’ is the result. Likewise in Sentence 2, the clause/condition is the ‘touching action ’ and result is the ‘burning action’.

A question comes to mind: “Which action happens first?”
The answer is very simple that the action in if-clause/conditional clause happens first and the action of the main clause happens next/second/later.
In Sentence 1 the main clause is written first and then the if- clause. Sentence 2 is written in reverse order i.e. the if-clause first and the main clause next.
Both the sentences in the examples are grammatical and acceptable. The conditional/if-clause is written/spoken first only when the condition tends to become more emphatic.
Do mark that when the conditional clause is written first, there should be a ‘comma ’ just after the conditional clause. No ‘comma’ is used if the conditional clause is written later or the main clause is written first. A condition can also be signaled without the use of ‘if’.

Example
3. Admit your mistake and I will pardon you.

Types of conditionals
O-Type Conditionals
1. If you heat ice, it melts.
2. If you drop a glass, it breaks.
3. If I make a promise, I keep it.

Let us divide the above sentences into two parts. Look at Sentence 1.
If you heat ice, = X (If-clause)
it melts = Y (Main clause)

X = (Simple present)- heat, heating action- the cause/condition- happens first.
Y = (Simple present)- melts, melting action- the result/- happens later.
The condition as well as the effect (result) remains in a O(zero) state. They are in a balancing state. So the grammarians have rightly named it ‘0- conditionals’.
We may say it a scientific truth/ universal truth. But Sentence 3 above tends the meaning of general happenings or what usually happens. In such cases, ‘Simple Present’ verb is used in both the parts. Here ‘if’ can be substituted by ‘when (ever)’.

(A) Open Condition
If you send the money today, he will get it tomorrow.
In If-part ‘sendign money is the condition and ‘getting it’ is the result in ‘main part’. ‘Your sending money’ is more possible i.e. the condition is open to the speaker and ‘his getting money’ is also more possible in the future. The time reference is futurity.

Notice both the parts of the sentence.
In ‘if-part’ a simple present verb ‘send’ is used and in ‘main part’ ‘will + get’ is used. Will/shall modals are not used in the ‘if-part’ in such conditions. We use this type of conditionals (real condition) sentences to talk about what is probable or possible or more like to happen in the future. This condition is, therefore, called the real condition.
Look at the following sentence.
If you wake up before me, give me a call.
Here ‘if-part, takes a simple present verb and the ‘main part’ is an imperative sentence. This condition is also an open or a real condition for future possibility/probability. Some variations/deviations

Study the following sentences, which are also examples of ‘open’ conditionals, but with a difference.
1. If it’s a holiday tomorrow, we may go for a swim.
2. If I should see him, I will ask him to write.
3. Stop borrowing money or you will be in trouble.
When the imperative is used in place of an ‘if- clause’ the condition changes into a threat, request etc.

Example
If we should miss the 9 o’ clock train, we shouldn’t get there till after lunch. The introduction of ‘should’ (sometimes stressed) in the conditional clause has the effect of making it less likely that the condition will be fulfilled. Here ‘should’ means ‘by any chance’. We may call this a condition of ‘remote possibility’. Note that only ‘should’ (never would) is used in this way.

Now look at the following.
If you will reserve seats, we shall be sure of a comfortable journey. We saw earlier that ‘will’ is not used in a predictive sense in the conditional clause, though the sentence has a future time reference. Here the verb ‘will’ is not an auxiliary indicating future; it is a modal verb which means agreement/cooperation/willingness.
“If you will reserve” = if you are willing/agreeing. The verb ‘will’ when means ‘insistence’, it is used in a conditional clause of this type.
Example
If my brother will mix with the bad boys, what shall I do?
“If my brother will mix” = If my brother is insisting on mixing (not in a predictive sense)

(B) ‘Hypothetical’ or ‘Unreal’ Condition
Example
If I became the Chief Minister, I would put things right at once. Notice that both the parts i.e. the ‘if-part’ as well as ‘the main-part’ contain verbs in the past tense form i.e. became and would. We use this conditional when we talk about unreal or imaginary events/actions.
The use of the past tense has the effect of making the condition seem ‘remote’ or unlikely/distant. That is why the condition is said to be ‘hypothetical’ or unreal. The time reference is futurity.
If I knew his address. I could tell you.
The above sentence represents us with a totally imaginary (or unreal) situation with reference to the time of speaking. It implies that I really don’t know his address and I can’t tell you now. Note that the past tense is used here to indicate present unreality.

The Subjunctive
Compare the two sentences below.
1. If I were you, I would take my work more seriously.
2. If I was you. I would take my work more seriously.
One might think that the first sentence is wrong sicne the verb were being in the plural form does not agree with the singular subject I. This plural verb is called ‘the subjunctive’. It is used to express the unreal or hypothetical condition. In modem English the subjunctive is not used
commonly. Most educated native speakers would prefer now to use “IfI was you” (instead of If I were you …)

Variations
If we were to miss the 9 o’ clock train, we would not get there till after lunch. The use of ‘were to’ in the conditional clause sometimes has the effect of emphasizing the suppositional nature of the condition and some ways analogous to the use of ‘should’ in conditional clause A. ‘were to’ can be substituted ‘by any chance’ without changing the meaning.

(C) Unfulfilled Condition.
Example
If you had asked me for the money, I would have given it to you. Notice that ‘if-part’ contains ‘past perfect form ’ and the main clause ‘would + have with past participle form.
The use of these past- verbs in both the parts tells us that the action (giving of money) was not performed because the condition was not fulfilled. The sentence actually means ‘you really did not ask me for the money and I didn’t give it to you.’ This conditional sentence represents what is contrary to the fact. The past perfect tense is ued to indicate past unreality.
The same meaning is expressed by
Had you asked me for the money, I would have given it to you. (without the use of if) The unfulfilled condition is often used to comment on, or to express regret for, something that did not happen.

Activity – 1
Complete the sentences below, using appropriate words of your own.

1. If we finish early today,________________________.
2. If I lose all my money,___________________.
3. If you go away on a holiday this summer. ________________________.
4. If our friend gets into trouble. _________________________.
5. Will you help me if _____________________________
6. I would be rather disappointed if ________________________.
7. This school will have to be closed if ___________________________
8. We will all be very happy if _____________________________.
9. They will not listen to anybody if _________________________.

Answers
1. If we finish early today, we’ll put some free time to see the match on T. V.
2. If I lose all my money, I’ll be in trouble.
3. If you go away on a holiday this summer. I’ll be very happy.
4. If our friend gets into trouble, we’ll help him.
5. Will you help me if I am in trouble? ‘
6. I would be rather disappointed if von didn’t help me.
7. This school will have to be closed if the lunar eclipse falls on tomorrow.
8. We will all be happy if our team wins the trophy.
9. They will not listen to anybody if von don’t behave them properly

Activity -2
The following are examples of some common superstitions.
1. If you hear an owl hooting at night, a friend will die.
2. If your left hand begins to itch, you will give money away.

Can you think of five other common superstitions? (Use the word if, as shown above.)
Answers
1 . If a cat passes in front of somebody, she/he will face some danger.
2. If three persons start a journey together, they will reap a negative result.
3. If an owl sits on the roof of a person’s house, one of the members of the family will die soon.
4. The man will die if a lizard falls on his shoulders.
5. Man will be crowned if a lizard falls on his forehead.

Activity – 3
Fill in the blank spaces to make meaningful conditions.
If I work hard, I’ll get good marks, If I ____________good marks, I ____________go to a good college and __________a degree. If I _____________ go abroad, I ___________ a highly-paid job. If __________name, fame and money, I _____________ marry a pretty girl. If ______________ only one or two children, I ____________ educate them properly. If _____________ come back to my country, I ______________ some useful work.

Answers
If I work hard, I’ll get good marks, If I get good marks, I will go to a good college and do a degree. If I get go abroad, I will set a highly-paid job. If gel name, fame and money, I will marry a pretty girl. If I only one or two children, I can educate them properly. If1 come back to my country, I will do some useful work.

Activity- 4

How imaginative are you? Complete the following sentences.
(a) If Ibecame a cloud, I would ………………………….
(b) If I could go back to the past,………………………..
(c) If I were a rat,………………………
(d) If I were a tiger, ……………………….
(e) If I ……………………………. I’d write a great novel.
(f) If people ……………………………they would die of boredom.
(g) If the earth …………………… ……, no living creature would survive.
(h) I would stand on my head if …………………………
(i) Would you be well prepared if ……………………..?

Answers
(a) If I became a cloud, I would sail in the sky.
(b) If I could go back to the past, I would not make my hair-dye.
(c) If I were a rat, I would make holes.
(d) If I were a tiger, / would live in the jungle.
(e) If I were a Jane Austin. I’d write a great novel.
(f) If people were dumb and deaf, they would die of boredom.
(g) If the earth was/were a sun, no living creature would survive.
(h) I would stand on my head if I were not a man.
(i) Would you be well prepared if you were a literate?

Activity- 5
How would your life be different if you were :

older      more intelligent     stronger
taller       less intelligent        richer
shorter    more patient         poorer

Answers
1. If I were older, I could beg you for some help.
2. If I were taller, I would join the military service.
3. If I were shorter, I could play the role of a joker.
4. If I were more intelligent, I could/would tackle the world problem.
5. If I were less intelligent, I could/would be in difficulty.
6. If I were more patient, I would tolerate your insulting words.
7. If I were stronger, I could/would fight with a tiger.
8. If I were richer, I would make you my accountant.
9. If I were poorer, I would beg for food.

Activity- 6

Write 5 sentences on the things you would do if you won a lottery. (Use if + past simple in the ‘if – clause” and would in the main clause) If I won a lottery, I would get a lot of money, If I got a lot of money …

Answers
If I won a lottery, I would get a lot of money.
(a) If I got a lot of money, I would buy a car.
(2) If I bought a car, I would take my parents for an outing.
(3) If I went on an outing, I would drive the car.
(4) If I drove the car, I would be very cautious.
(5) If my parents saw my driving, they would be very happy.

Activity – 7

Write a comment on each of the following situations, using the unfulfilled condition. (One example is given)
(a) I didn’t study, so I didn’t do very well in the examination.
Comment: If you had studied, you would have done better.
(b) I didn’t like the food because it was too hot.
Comment: If ______________________________________________________________
(c) They waited at the station for four hours because they did not know that the train was delayed.
Comment: If ______________________________________________________________
(d) The boy met with an accident because he was careless.
Comment: If _______________________________________________________________
(e) She heard the news because she turned on the radio this morning.
Comment: If ______________________________________________________________
(f) They didn’t see us, so they could not laugh at us.
Comment: If ______________________________________________________________
(g) My father earned a lot of money, so life was easy for us.
Comment: If ______________________________________________________________

Answers
(b) I didn’t like the food because it was too hot.
Comment: I would/could have liked the food if it had not been hot.
(c) They waited at the station for four hours because they did not know that the train was delayed.
Comment: If they had known that the train was delayed, they would not have waited at the station.
(d) The boy met with an accident because he was careless.
Comment: The boy could not have met with an accident if he had been careful/if he had not been careless.
(e) She heard the news because she turned on the radio this morning.
Comment: If she had not turned on the radio this morning, she could not have heard the news.
(f) They didn’t see us, so they could not laugh at us.
Comment: If they had seen us, they could have laughed at us.
(g) My father earned a lot of money, so life was easy for us.
Comment: If my father had not earned a lot of money, life would/could not have been easy for us.

Activity – 8
Read the text below and then complete the sentences that follow.
Namita went to Cuttack yesterday. The sun was shining brightly, so she never thought of taking her umbrella. But what a terrible day she had! Everything that could possibly go wrong went wrong. The weather changed, as the weather report had predicted, and Namita got soaked. She slipped on the stairs of a shop and had a bad fall. Someone stole her purse when she was talking to a friend and had her back turned. The bus broke down on the way back and was delayed. Namita’s mother was angry because she returned so late.

(a) If Namita had taken her umbrella, ………………………………
(b) If she had listened to the weather forecast, ………………………
(c) If she had taken the lift in the store ………………………..
(d) If she hadn’t turned her back, …………………….
(e) If the bus hadn’t broken down …………………….
(f) If she’d returned home on time, …………………………
Answers
(a) If Namita had taken her umbrella, she couldn’t have got soaked.
(b) If she had listened to the weather forecast, she could not have drenched.
(c) If she had taken the lift in the store, she could not have fallen.
(d) If she hadn’t turned her back, someone couldn’t have stolen her purse.
(e) If the bus hadn’t broken down, Namita could have returned in time.
(f) If she’d returned home on time, her mother could not have been angry.

Section -2

Other ways of expressing a condition
Look at the following sentences
1. He can’t work unless he eats something.
2. I won’t go to the party unless they invite me.
These sentences with unless could be re-written as
3. He can’t work ifhe doe snot eat something.
4. I won’t go to the party if I am not invited.
Unless has the meaning “if …. not”. It is used to express a negative condition. The same meaning can be expressed by using ‘only if’. So the same sentences could be re-written as
5. He can work only if he eats something.
6. I will go to the party only if he invites me

Activity – 9
Rewrite each of the following sentences using ‘unless’.
(The first sentence has been worked out for you.)
1. We must leave now or we’ll miss the start of the film.
Unless we leave now we’ll miss the start of the film.
2. You should wear your coat or you’ll be cold.
_______________________________________
3. You must give me your address or I can’t write to you.
_______________________________________
4. You have to speak very loudly or we won’t be able to hear you.
_______________________________________
5. You must stop smoking or your cough won’t get better.
_______________________________________
6. You must say you’re sorry or he won’t forgive you
_______________________________________

Answers
1. We must leave now or we’ll miss the start of the film.
Unless we leave now, we’ll miss the start of the film.

2. You should wear your coat or you’ll be cold.
Unless you wear your coat, you’ll be cold.

3. You must give me your address or I can’t write to you.
Unless you give me your address, I can’t write to you.

4. You have to speak very loudly or we won’t be able to hear you.
Unless you speak very loudly, we won’t be able to hear you.

5. You must stop smoking or your cough won’t get better.
Unless you stop smoking, your cough won’t get better.

6. You must say you’re sorry or he won’t forgive you.
Unless you say you’re sorry, he won’t forgive you.

Multiple-Choice Questions (MCQs) with Answers

Question 1.
Unless you ___________the truth, you will suffer.
(A) spoke
(B) speak
(C) do not speak
(D) should
Answer:
(A) spoke

Question 2.
If you were a bird, you ____________ fly.
(A) could
(B) should
(C) would
(D) none of these
Answer:
(C) would

Question 3.
She would come, if she ____________ time.
(A) has
(B) had
(C) would have
(D) should have
Answer:
(B) had

Question 4.
If I had known you, I_ told you.
(A) would have
(B) should have
(C) will have
(D) shall have
Answer:
(A) would have

Question 5.
If I __________ rich, I would help the poor.
(A) was
(B) am
(C) were
(D) shall be
Answer:
(C) were

Question 6.
_________ I find the pen, I will let you have it.
(A) Could
(B) Should
(C) Would
(D) Can
Answer:
(B) Should

Question 7.
If I had had time, I ___________visited the zoo.
(A) would have
(B) should have
(C) could have
(D) shall have
Answer:
(B) should have

Question 8.
If he _____________ a taxi, she would have reached the station in time.
(A) has taken
(B) would take
(C) had taken
(D) should take
Answer:
(C) had taken

Question 9.
If he ___________, he will suffer from indigestion.
(A) ate
(B) had eaten
(C) eats
(D) will eat
Answer:
(C) eats

Question 10.
If they _________ here, I should speak to them.
(A) were
(B) had been
(C) will be
(D) would be
Answer:
(A) were

Question 11.
I’m not tired enough to go to bed yet. I wouldn’t sleep if I ____________ to bed now.
(A) go
(B) went
(C) had gone
(D) would go
Answer:
(B) went

Question 12.
If I were you, I _____________ that coat. It was much too expensive.
(A) won’t buy
(B) don’t buy
(C) am not going to buy
(D) wouldn’t buy
Answer:
(D) wouldn’t buy

Question 13.
I decided to stay at home last night. I would have gone out if I ___________so tired.
(A) wasn’t
(B) weren’t
(C) wouldn’t have been
(D) hadn’t been
Answer:
(D) hadn’t been

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(k) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(k)

Question 1.
State True (T) or False (F).
(i) There is no function whose derivative is log π.
Solution:
False

(ii) There is no function which is its own derivative.
Solution:
False

(iii) A function is not differentiable at x = c ⇒ f is not continuous at x = c.
Solution:
False

(iv) |x²| is differentiable on (- 1, 1).
Solution:
True

(v) |x + 2| is not differentiable at x = 2.
Solution:
False

(vi) Derivative of e^{3 log x} w.r.t. x is 3x².
Solution:
True

(vii) The derivative of a non constant even function is always an odd function.
Solution:
True

(viii) If f and g are not derivable at x₀ then f + g is not derivable at x₀.
Solution:
False

Question 2.
Fill up the gaps by using the correct answer.
(i) If a is a constant and v is a variable then \(\frac{d u^v}{d v}\) = _______. (u^v In v, vu^{v – 1}, u^v In u, uv^{v – 1})
Solution:
u^v In v

(ii) If t = e^a then \(\frac{d}{d x}\)x^t = _______. (tx^{t – 1}, x^t, x^t In a, tx^t)
Solution:
tx^{t – 1}

(iii) If u = t² and v= sin t² then \(\frac{d v}{d u}\) = _______. (cos² t, \(\frac{\sin }{t}\), sec t², cos t²)
Solution:
cos t²

(iv) The tangent to the curve y = (1 + x²)² at x = -1 has slope _______. (4, -4, 8, -8)
Solution:
-8

(v) If v = (gof) (x) then \( \frac{d y}{d x}\) = _______. (\(\frac{d g}{d x} \frac{d x}{d f}\), \(\frac{d g}{d f} \frac{d f}{d x}\), \(\frac{d f}{d x} \frac{d x}{d g}\), \(\frac{d f}{d g} \frac{d g}{d x}\))
Solution:
\(\frac{d g}{d f}\frac{d f}{d x}\)

(vi) If y = sec^-1 \(\frac{\sqrt{x}+1}{\sqrt{x}}\) + \(\frac{\sqrt{x}}{\sqrt{x}+1}\) then \(\frac{d y}{d x}\) = _______. (0, undefined, \(\frac{\pi}{2}\), 1)
Solution:
0

(vii) If (x) = \(\sqrt{x^2-2 x+1}\), x ∈ [0, 2] then at x = 1, f(x) = _______. (1, 0, -1, does not exist)
Solution:
does not exist

(viii) If f(x) = |x²| then f'(\(\frac{3}{2}\)) = _______. (0, 2, 3, does not exist)
Solution:
0

Question 3.
Differentiate from first principles.
(i) e^2x
Solution:
Let y = e^2x

(ii) sin² x
Solution:
Let y = sin² x

(iii) cos x²
Solution:
Let y = cos x²

(iv) \(\boldsymbol{e}^{x^2} \)
Solution:

(v) \(\sqrt{\tan x} \)
Solution:

(vi) x² sin x
Solution:
Let y = x² sin x

(vii) In sin x
Solution:
Let y = In sin x
Then e^y = sin x  … (1)
Let dx be an increment of x and δv be the corresponding increment of y.
Then e^{y + δy} = sin (x + δx)  … (2)
Subtracting (1) from (2) we get,

(viii) sin √x
Solution:
y = sin √x
Put u = √x
Then y = sin u
Let δx be an increment of x and δu, δy be the corresponding increment of u and y respectively.

(ix) cos In x
Solution:
Let y = cos (In x)
Let u = In x
Then y = cos u
Suppose that δx be an increment of x and δu, δy be corresponding increments of u and y respectively.
Then y + δy = cos (u + δu)  ….(3)
and u + δu = In (x + δx)  ….(4)
Subtracting (2) from (3) and (1) from (4) we get
δy = cos (u + δu) – cos u
and δu = In (x + δx) – In x

Question 4.
Test differentiability of the following functions at the indicated points.
(i) f(x) =[x² + 1] at x = –\(\frac{1}{2}\)
Solution:

(ii) f(x) = \(\begin{cases}1-2 x, & x \leq \frac{1}{2} \\ x-\frac{1}{2}, & x>\frac{1}{2}\end{cases}\) at x = \(\frac{1}{2}\)
Solution:

(iii) f(x) = x + |cos x| at x = \(\frac{\pi}{2}\)
Solution:
f(x) = x + |cos x| at c = \(\frac{\pi}{2}\)

Hence onwards domain of a function is to be understood to be its natural domain unless stated otherwise.

Question 5.
Differentiate.
(i) \(\frac{1}{\ln (x \sqrt{x+1})}\)
Solution:

(ii) \(\frac{\ln x}{e^x \sin x}\)
Solution:

(iii) e^x (tan x – cot x)
Solution:
y = e^x (tan x – cot x)
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\) (e^x) (tan x – cot x) + e^x \(\frac{d}{d x}\) (tan x – cot x)
= e^x (tan x – cot x) + e^x (sec² x + cosec² x)

(iv) \(\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)\)x tan x
Solution:

(v) \(\frac{\cos 3 x-\cos x}{\cos 5 x-\cos 3 x}\)
Solution:

(vi) x² e^x cosec x
Solution:
y = x² e^x cosec x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\) (x²) e^x . cosec x + x² \(\frac{d}{d x}\) (e^x) . cosec x + x² e^x \(\frac{d}{d x}\)
= 2x e^x cosec x + x² e^x cosec x – x² e^x . cosec x . cot x

(vii) \(\frac{(x+1) \ln x}{\sqrt{x+2}}\)
Solution:

(viii) (x³ – 1)⁹ sec² x
Solution:

(ix) sin² (cos^-1 x)
Solution:

(x) a^x \(\left(x+\frac{1}{x}\right)^{10}\)
Solution:

(xi) In \(\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\)
Solution:

(xii) In \(\frac{4 x^2(2 x-7)^3}{\left(3 x^2-7\right)^5}\)
Solution:

(xiii) 5^{ln sin x}
Solution:

(xiv) \(\sqrt{\sin \sqrt{x}}\)
Solution:

(xv) x^{sin x} + (tan x)^x
Solution:
Let y = x^{sin x} + (tan x)^x
Put u = x^{sin x} , v = (tan x)^x
Then y = u + v

(xvi) \(e^{e^x}\)
Solution:

(xvii) \(x^{\sqrt{x}}\)
Solution:

(xviii) sec^-1(e^x + x)
Solution:

(xix) ln cos e^x
Solution:

(xx) \(a^{\sin ^{-1} x^2}\)
Solution:

(xxi) cos^-1 \(\left(\frac{x^4-1}{x^4+1}\right)\)
Solution:

(xxii) \(\left(\mathbf{x}^{\mathbf{e}}\right)^{\mathbf{e}^{\mathrm{x}}}\) + \(\left(\mathrm{e}^{\mathrm{x}}\right)^{\mathrm{x}^e}\)
Solution:

(xxiii) \(\boldsymbol{x}^{\left(\boldsymbol{x}^x\right)}\)
Solution:

(xxiv) \(\frac{\left(x+1^2\right) \sqrt{x-1}}{\left(x^2+3\right)^3 3^x}\)
Solution:

(xxv) [5 In (x³ + 1) – x⁴]^2/3
Solution:

(xxvi) log₁₀ sin x + log_x 10, 0 < x > π.
Solution:

Question 6.
Differentiate
(i) sec^-1 \(\left(\frac{x^2+1}{x^2-1}\right)\)
Solution:

(ii) \(e^{\tan ^{-1} x^2}\)
Solution:

(iii) \(\frac{x \sin ^{-1} x}{\sqrt{1+x^2}}\)
Solution:

(iv) tan^-1 e^2x
Solution:

(v) tan^-1 \(\frac{\cos x}{1+\sin x}\)
Solution:

(vi) tan^-1 \(\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right) \)
Solution:

(vii) tan^-1 \(\frac{7 a x}{a^2-12 x^2}\)
Solution:

(viii) tan^-1 \(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\) (Put x² = cos θ)
Solution:

(ix) x² cos \(\frac{\sqrt{x}-1}{\sqrt{x}+1}\) = x² cosec^-1 \(\frac{\sqrt{x}+1}{\sqrt{x}-1}\)
Solution:

(x) tan^-1 \(\frac{x}{1+\sqrt{1-x^2}}\)
Solution:

(xi) tan^-1 \(\left(\frac{x \sin \alpha}{1-x \cos \alpha}\right)\)
Solution:

Question 7.
Find \(\frac{d y}{d x}\) if
(i) x³ + y³ = 12xy
Solution:

(ii) \(\left(\frac{x}{a}\right)^{2 / 3}\) + \( \left(\frac{y}{b}\right)^{2 / 3}\) = 1
Solution:

(iii) x^y = c
Solution:

(iv) y^x = c
Solution:
y^x = c ⇒ x In y = In c
⇒ in y + \(\frac{x}{y} \frac{d y}{d x}\) = 0
⇒ \(\frac{d y}{d x}\) = –\(\frac{y \ln y}{x}\)

(v) x cot y + cosec x = 0
Solution:

(vi) y² + x² = In (xy) + 1
Solution:

(vii) (cos x)^y = sin y
Solution:

(viii) y² =a^√x
Solution:

(ix) x^m yⁿ = \(\left(\frac{x}{y}\right)^{m+n}\)
Solution:

(x) y = x cot ^-1 \(\left(\frac{x}{y}\right)\)
Solution:

(xi) y = (sin y)^{sin 2x}
Solution:

(xii) y² = x^y
Solution:

(xiii) (x + y)^{cos x} = e ^{x + y}
Solution:

(xiv) x tan y + y tan x = 0
Solution:

(xv) \(\sqrt{x^2+y^2}\) k tan^-1 \(\left(\frac{y}{x}\right)\)
Solution:

Question 8.
Differentiate
(i) tan^-1 \(\frac{2}{1-x^2}\) w.r.t. sin^-1 \(\frac{2}{1+x^2}\)
Solution:

(ii) sec^-1 \(\left(\frac{1}{2 x^2-1}\right)\) w.r.t. \(\sqrt{1-x^2}\)
Solution:

(iii) tan^-1 \(\left(\frac{1+\sin x}{1-\sin x}\right)\) w.r.t. log \(\left(\frac{1+\cos x}{1-\cos x}\right)\)
Solution:

Question 9.
Find the \(\frac{d y}{d x}\) when
(i) x = a [cos t + log tan ( t/2)], y = a sin t
Solution:

(ii) sin x = \(\frac{2 t}{1+t^2}\), tan y = \(\frac{2 t}{1-t^2}\)
Solution:

(iii) cos x= \(\sqrt{\frac{1}{1+t^2}}\), siny = \(\frac{2 t}{1+t^2}\)
Solution:

(iv) cos x = \(\sqrt{\sin 2 u}\), y = \(\sqrt{\cos 2 u}\)
Solution:

(v) x = \(\frac{\cos ^3 t}{\sqrt{\cos 2 t}}\), y = \(\frac{\sin ^3 t}{\sqrt{\cos 2 t}}\)
Solution:

Question 10.
Assuming the validity of the operations on the r.h.s. find \(\frac{d y}{d x}\).
(i) y = [ sin x + { sin x + (sin x +….)}]
Solution:

(ii) y = 1 ÷ [ x + 1 ÷ (x + 1 ÷ (x + 1 ÷ …))]
Solution:

(iii) y = In [x + In (x + In (x + ….))]
Solution:

Question 11.
If cos y = x cos (a + y) then prove that
(i) \(\frac{d y}{d t}\) = \(\frac{\cos ^2(a+y)}{\sin a}\)
Solution:

(ii) If e^θΦ = c + 4 θΦ , show that Φ + θ \(\frac{d \phi}{d \theta}\) = 0.
Solution:

Question 12.
Can you differentiate log log |sin x|? Justify your answer.
Solution:
Clearly for all x ∈ R
sin x ∈ [- 1, 1]
⇒ |sin x| ∈ [ 0,1]
⇒ log |sin x| ∈ (-∞ , 0]
⇒ log log |sin x| is not well defined for all x ∈ R
∴ Log log |sin x| is not a differentiable function.

Question 13.

Solution:

Question 14.
If x = \(\frac{1-\cos ^2 \theta}{\cos \theta}\), y = \(\frac{1-\cos ^{2 n} \theta}{\cos ^n \theta}\) then show that \(\left(\frac{d y}{d x}\right)^2\) = n²\(\left(\frac{y^2+4}{x^2+4}\right)\)
Solution:

Question 15.
Show the \(\frac{d y}{d x}\) is independent of t if.
x = cos^-1\(\frac{1}{\sqrt{t^2+1}}\), y = sin^-1\(\frac{t}{\sqrt{t^2+1}}\)
Solution:

Question 16.
If y \(\sqrt{x^2+1}\) = {\(\sqrt{x^2+1}\) – x} then prove that (x² + 1) \(\frac{d y}{d x}\) + xy + 1 = 0
Solution:

Question 17.
If e^y/x = \(\frac{x}{a+b x}\), then show that x3 \(\frac{d}{d x}\) \(\left(\frac{d y}{d x}\right)\) = \(\left(x \frac{d y}{d x}-y\right)^2\)
Solution:

Question 18.
Find the points where the following functions are not differentiable.
(i) e^|x|
Solution:
e^|x| is not differentiable at x = 0 because |x| is not differentiable at x = 0

(ii) |x² – 4|
Solution:
|x² – 4| is not differentiable at the points where x² – 4 = 0 i.e, x =± 2.

(iii) |x – 1| + |x – 2|
Solution:
|x – 1| + |x – 2| is not differentiable at x = 1 and x = 2.

(iv) sin |x|
Solution:
sin |x| is not differentiable at x = 0.