# MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers

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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.

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.

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.

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: (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: (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: (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

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: (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) $$\frac {1}{3}$$

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: (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.

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  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: (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: (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: (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) 5x4
(b) $$\frac {1}{1+{g(x)}^5}$$
(c) 1 + {g(x)}5
(d) 1 + x5

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

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: (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 = …………………

Question 2.
If f(x) = x + 7, and g(x) = x – 7, x ∈R, then $$\frac { d }{dx}$$ (fog) (x) = ……………….

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

Answer: $$\frac { cos x-2 }{3}$$

Question 4.
$$\frac { d }{dx}$$ (cosec-1 x) = …………………

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

Question 5.
$$\frac { d }{dx}$$ ($$\sqrt { e^{ \sqrt{x}} }$$) = …………………

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

Question 6.
If x = at², y = 2at, then $$\frac { dy }{dx}$$ = ……………….

Answer: $$\frac { 1 }{t}$$

Question 7.
The derivative of xx w.r.t. x is.

Answer: xx (1 + log x).

Question 8.
If y = x² + 3x + 2, then $$\frac { d^2y }{dx^2}$$ = ………………

Question 9.
Value of ‘c’ in Rolle’s Theorem for the function f(x) = x³ – 3x in [-√3, 0] is ……………..

Question 10.
Value of ‘c’ in LMV Theorem for f(x) = x² in [2, 4] is …………………