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) |x^{2}| 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 3x^{2}.

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 x_{0} then f + g is not derivable at x_{0}.

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 = t^{2} and v= sin t^{2} then \(\frac{d v}{d u}\) = _______. (cos^{2} t, \(\frac{\sin }{t}\), sec t^{2}, cos t^{2})

Solution:

cos t^{2}

(iv) The tangent to the curve y = (1 + x^{2})^{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) = |x^{2}| 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) sin^{2} x

Solution:

Let y = sin^{2} x

(iii) cos x^{2}

Solution:

Let y = cos x^{2}

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

Solution:

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

Solution:

(vi) x^{2} sin x

Solution:

Let y = x^{2} 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) =[x^{2} + 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} (sec^{2} x + cosec^{2} 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) x^{2} e^{x} cosec x

Solution:

y = x^{2} e^{x} cosec x

\(\frac{d y}{d x}\) = \(\frac{d}{d x}\) (x^{2}) e^{x} . cosec x + x^{2} \(\frac{d}{d x}\) (e^{x}) . cosec x + x^{2} e^{x} \(\frac{d}{d x}\)

= 2x e^{x} cosec x + x^{2} e^{x} cosec x – x^{2} e^{x} . cosec x . cot x

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

Solution:

(viii) (x^{3} – 1)^{9} sec^{2} x

Solution:

(ix) sin^{2} (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 (x^{3} + 1) – x^{4}]^{2/3}

Solution:

(xxvi) log_{10} 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 x^{2} = cos θ)

Solution:

(ix) x^{2} cos \(\frac{\sqrt{x}-1}{\sqrt{x}+1}\) = x^{2} 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) x^{3} + y^{3} = 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) y^{2} + x^{2} = In (xy) + 1

Solution:

(vii) (cos x)^{y} = sin y

Solution:

(viii) y^{2} =a^{√x}

Solution:

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

Solution:

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

Solution:

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

Solution:

(xii) y^{2} = 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 \(\left(\frac{d y}{d x}\right)^2\) = n^{2}\(\left(\frac{y^2+4}{x^2+4}\right)\)

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 (x^{2} + 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) |x^{2} – 4|

Solution:

|x^{2} – 4| is not differentiable at the points where x^{2} – 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.