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## Continuity and Differentiability Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ questions on Continuity and Differentiability Class 12 MCQs Questions with Answers so you can show your skills during the exam.

Question 1.

The function

f(x) =

is continuous at x = 0, then the value of ‘k’ is:

(a) 3

(b) 2

(c) 1

(d) 1.5.

## Answer

Answer: (b) 2

Question 2.

The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at:

(a) 4

(b)-2

(c) 1

(d) 1.5.

## Answer

Answer: (d) 1.5.

Question 3.

The value of ‘k’ which makes the function defined by

continuous at x = 0 is

(a) -8

(b) 1

(c) -1

(d) None of these.

## Answer

Answer: (d) None of these.

Question 4.

Differential coefficient of sec (tan^{-1} x) w.r.t. x is

(a) \(\frac { x }{\sqrt{1+x^2}}\)

(b) \(\frac { x}{1+x^2}\)

(c) x\(\sqrt { 1+x^2}\)

(d) \(\frac { 1 }{\sqrt{1+x^2}}\)

## Answer

Answer: (a) \(\frac { x }{\sqrt{1+x^2}}\)

Question 5.

If y = log (\(\frac { 1-x^2 }{1+x^2}\)) then \(\frac { dy }{dx}\) is equal to:

(a) \(\frac { 4x^3 }{1-x^4}\)

(b) \(\frac { -4x}{1-x^4}\)

(c) \(\frac {1}{ 4-x^4}\)

(d) \(\frac { -4x^3 }{1-x^4}\)

## Answer

Answer: (b) \(\frac { -4x}{1-x^4}\)

Question 6.

If y = \(\sqrt { sin x+ y}\), then \(\frac { dy }{dx}\) is equal to

(a) \(\frac { cos x }{2y-1}\)

(b) \(\frac { cos x}{1-2y}\)

(c) \(\frac {sin x}{1-2y}\)

(d) \(\frac { sin x }{2y-1}\)

## Answer

Answer: (a) \(\frac { cos x }{2y-1}\)

Question 7.

If u = sin^{-1} (\(\frac { 2x }{1+x^2}\)) and u = tan^{-1} (\(\frac { 2x }{1-x^2}\)) then \(\frac { dy }{dx}\) is

(a) \(\frac { 1 }{2}\)

(b) x

(c) \(\frac {1-x^2}{1+x^2}\)

(d) 1

## Answer

Answer: (d) 1

Question 8.

If x = t², y = t³, then \(\frac { d^2y }{dx^2}\) is

(a) \(\frac { 3 }{2}\)

(b) \(\frac { 3 }{4t}\)

(c) \(\frac {3}{2t}\)

(d) \(\frac { 3t }{2}\)

## Answer

Answer: (b) \(\frac { 3 }{4t}\)

Question 9.

The value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in the interval [0, √3] is

(a) 1

(b) -1

(c) \(\frac {3}{2}\)

(d) \(\frac {1}{3}\)

## Answer

Answer: (a) 1

Question 10.

The value of ‘c’ in Mean Value Theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is

(a) \(\frac {3}{2}\)

(b) \(\frac {2}{3}\)

(c) \(\frac {1}{2}\)

(d) \(\frac {3}{4}\)

## Answer

Answer: (a) \(\frac {3}{2}\)

Question 11.

Let f : (- 1, 1) → R be a differentiable function with f(0) = – 1 and f'(0) = 1.

Let g(x) = [f (2f(x) + 2)]². Then g'(0) =

(a) 4

(b) -4

(c) log 2

(d) -log 2.

## Answer

Answer: (b) -4

Hint:

Here g (x)= [f (2 f(x) + 2)]²

g'(x) = 2[f(2f(x) + 2)] \(\frac { d }{dx}\) [2f(x) + 2 ]

= 2f(2f(x) + 2) . [2 f'(x)]

∴ g'(0) = 2f(2f(0) + 2) . [2f'(0)]

= 2f(2 (-1) +2). 2f’/(0)

= 2f(0) . 2f'(0) = 4f(0) f'(0)

= 4 (-1) (1) = -4.

Question 12.

\(\frac { d^2x }{dy^2}\) equals

## Answer

Answer: d

Hint:

Question 13.

If function f(x) is differentiable at x = a, then

\(\lim _{x \rightarrow a}\) \(\frac { x^2 f(a) – a^2 f(x) }{x-a}\) is

(a) a² f(a)

(b) af(a) – a² f'(a)

(c) 2a f(a) – a² f'(a)

(d) 2a f (a) + a² f'(a).

## Answer

Answer: (c) 2a f(a) – a² f'(a)

Hint:

Question 14.

If f: R → R is a function defined by

f(x) = [x] cos (\(\frac { 2x-1 }{2}\))π, where [x] denotes the greatest integer function, then ‘f’ is

(a) continuous for every real x

(b) discontinuous only at x = 0

(c) discontinuous only at non-zero integral values of x

(d) continuous only at x = 0.

## Answer

Answer: (a) continuous for every real x

Hint:

Continuous for every real x.

Question 15.

If y = sec (tan^{-1} x), then \(\frac { dy }{dx}\) at x = 1 is equal to

(a) \(\frac {1}{2}\)

(b) 1

(c) √2

(d) \(\frac {1}{√2}\)

## Answer

Answer: (d) \(\frac {1}{√2}\)

Hint:

Here y = sec (tan^{-1} x).

∴ \(\frac { dy }{dx}\) = sec (tan^{-1} x) tan (tan^{-1} x). \(\frac { 1 }{1+x^2}\)

= sec (tan^{-1} x). x . (tan^{-1} x). \(\frac { 1 }{1+x}\)

\(\left.\frac{d y}{d x}\right]_{x=1}\) = sec (tan^{-1} 1). \(\frac { 1 }{1+1}\)

= sec (\(\frac { π }{4}\)) \(\frac { 1 }{2}\) = \(\frac { √2 }{2}\) = \(\frac { 1 }{√2}\)

Question 16.

If g is the inverse of a function f and f'(x) = \(\frac { 1 }{1+x^5}\), then g'(x) is equal to

(a) 5x^{4}

(b) \(\frac {1}{1+{g(x)}^5}\)

(c) 1 + {g(x)}^{5}

(d) 1 + x^{5}

## Answer

Answer: (b) \(\frac {1}{1+{g(x)}^5}\)

Hint:

Here f(g(x)) = x. [∵ g is the inverse of f]

f'(g(x)) g'(x) = 1

⇒ g'(x) = \(\frac { 1 }{f'{g(x)}}\) = \(\frac { 1 }{1+{g(x)}^5}\)

Question 17.

If the function

g(x) = \(\left\{\begin{array}{ll}

k \sqrt{x+1} & ; 0 \leq x \leq 3 \\

m x+2 & ; 3 \end{array}\right.\)

is differentiable, then the value of k + m is

(a) 2

(b) \(\frac {16}{5}\)

(c) \(\frac {10}{3}\)

(d) 4

## Answer

Answer: (a) 2

Hint:

We have

g(x) = \(\left\{\begin{array}{ll}

k \sqrt{x+1} & ; 0 \leq x \leq 3 \\

m x+2 & ; 3 \end{array}\right.\)

When this function is differentiable, then it is continuous

⇒ \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\lim _{x \rightarrow 3^{+}}\) g(x) = g(3)

⇒ 2k = 3m + 2 = 2k

⇒ 2k = 3m + 2 ………… (1)

Also, LHD = \(\lim _{x \rightarrow 3^{-}}\) g(x) = \(\frac {k}{4}\)

RHD = \(\lim _{x \rightarrow 3^{+}}\) g(x) = m

∴ LHD = RHD k

⇒ \(\frac {k}{4}\) = m

Solving (1) and (2),

k = \(\frac {8}{5}\) and m = \(\frac {2}{5}\)

Hence, k + m = \(\frac {8}{5}\) + \(\frac {2}{5}\) = \(\frac {10}{2}\) = 2.

Question 18.

For x ∈ R, f(x) = |log 2 – sin x| and g(x) =f(f(x)), then

(a) g is not differentiable at x = 0

(b) g'(0) = cos (log 2)

(c) g'(0) = -cos (log 2)

(d) g is differentiable at x = 0 and g'(0) = – sin (log 2).

## Answer

Answer: (b) g'(0) = cos (log 2)

Hint:

We have : f(x) = log 2 – sin x

and g(x) = f(f cos x)

= log 2 – sin (log 2 – sin x).

Since ‘g’ is the sum of two differentiable functions,

∴ g is differentiable.

g'(x) = 0 – cos (log 2 – sin x) (0 – cos x)

= cos (log 2 – sin x) cos x.

Hence, g’ (x) = cos (log 2).

Fill in the blanks

Question 1.

If f(x) = \(\left\{\begin{array}{c}

\frac{x^{2}-1}{x-1}, \text { when } x \neq 1 \\

k, \text { when } x=1

\end{array}\right.\) is continuous then the value of k = …………………

## Answer

Answer: 2.

Question 2.

If f(x) = x + 7, and g(x) = x – 7, x ∈R, then \(\frac { d }{dx}\) (fog) (x) = ……………….

## Answer

Answer: 1.

Question 3.

If 2x + 3y = sin x, then \(\frac { dy }{dx}\) = …………………..

## Answer

Answer: \(\frac { cos x-2 }{3}\)

Question 4.

\(\frac { d }{dx}\) (cosec^{-1} x) = …………………

## Answer

Answer: \(\frac { -1 }{|x|\sqrt{x^2-1}}\)

Question 5.

\(\frac { d }{dx}\) (\(\sqrt { e^{ \sqrt{x}} }\)) = …………………

## Answer

Answer: \(\frac { 1 }{4√x}\) \(\sqrt { e ^{\sqrt{x}} }\)

Question 6.

If x = at², y = 2at, then \(\frac { dy }{dx}\) = ……………….

## Answer

Answer: \(\frac { 1 }{t}\)

Question 7.

The derivative of x^{x} w.r.t. x is.

## Answer

Answer: x^{x} (1 + log x).

Question 8.

If y = x² + 3x + 2, then \(\frac { d^2y }{dx^2}\) = ………………

## Answer

Answer: 2.

Question 9.

Value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in [-√3, 0] is ……………..

## Answer

Answer: c = -1.

Question 10.

Value of ‘c’ in LMV Theorem for f(x) = x² in [2, 4] is …………………

## Answer

Answer: c = 3.

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