CBSE Class 12 Mathematics Continuity and Differentiability MCQs Set 05

Question. Check whether the function \( f(x) = 3x - 5 \) is continuous at \( x = 0, x = - 5 \) and at \( x = 3 \).
(a) \( f(x) \) is continuous at \( x = 0 \)
(b) \( f(x) \) is not continuous at \( x = 0 \)
(c) \( f(x) \) is not continuous at \( x = - 5 \)
(d) \( f(x) \) is not continuous at \( x = 3 \)
Answer: (a) \( f(x) \) is continuous at \( x = 0 \)

Question. Find the continuity of \( f(x) = x \) at \( x = k \), \( k \) be any positive value:
(a) \( f(x) \) is continuous at \( x = k \)
(b) \( f(x) \) is not continuous at \( x = k \)
(c) \( f(x) \) is continuous at \( x = 0 \)
(d) None of the options
Answer: (a) \( f(x) \) is continuous at \( x = k \)

Question. Find the continuity of \( f(x) = \frac{x^2 - 16}{x + 4} , x \neq - 4 \) at \( x = k \), \( k \) be any positive value:
(a) \( f(x) \) is not continuous at \( x = k \)
(b) \( f(x) \) is continuous at \( x = k \)
(c) \( f(x) \) is continuous at \( x = - 4 \)
(d) \( f(x) \) is not continuous at \( x = - 4 \)
Answer: (b) \( f(x) \) is continuous at \( x = k \)

Question. Find all points of discontinuous of \( f \), where \( f \) is defined by \( f(x) = \begin{cases} x + 3, & \text{if } x \le 3 \\ x - 3, & \text{if } x > 3 \end{cases} \)
(a) \( f \) is discontinuous at \( x = 3 \)
(b) \( f \) is continuous at \( x = 3 \)
(c) \( f \) is continuous at \( x = - 3 \)
(d) \( f \) is discontinuous at \( x = - 3 \)
Answer: (a) \( f \) is discontinuous at \( x = 3 \)

Question. Find all points of discontinuity of \( f \), where \( f \) is defined by \( f(x) = \begin{cases} x^2 + 3, & \text{if } x \le 3 \\ x^2 - 3, & \text{if } x > 3 \end{cases} \)
(a) \( f \) is discontinuous at \( x = - 3 \)
(b) \( f \) is discontinuous at \( x = 3 \)
(c) \( f \) is continuous at \( x = - 3 \)
(d) \( f \) is continuous at \( x = 3 \)
Answer: (b) \( f \) is discontinuous at \( x = 3 \)

Question. Find all points of discontinuity of \( f \), where \( f \) is defined by \( f(x) = \begin{cases} x^2 + x + 3, & \text{if } x < 3 \\ 0, & \text{if } 0 \le x \le 1 \\ x^2 - x + 3, & \text{if } x > 1 \end{cases} \)
(a) \( f \) is discontinuous at all points of \( c \)
(b) \( f \) is continuous at all points of \( c \)
(c) \( f \) is discontinuous at \( x = 0 \)
(d) \( f \) is continuous at \( x = 0 \)
Answer: (a) \( f \) is discontinuous at all points of \( c \)

Question. Find all points of discontinuous of \( f \), where \( f \) is defined by \( f(x) = \begin{cases} 3, & \text{if } x \le -1 \\ 3x, & \text{if } -1 < x \le 1 \\ 3, & \text{if } x > 1 \end{cases} \)
(a) \( f \) is discontinuous at all points of \( c \)
(b) \( f \) is continuous at all points of \( c \)
(c) \( f \) is discontinuous at \( x = 0 \)
(d) \( f \) is continuous at \( x = 0 \)
Answer: (a) \( f \) is discontinuous at all points of \( c \)

Question. The function \( f(x) = 2 - 3x \) is:
(a) Increasing
(b) Decreasing
(c) Neither increasing nor decreasing
(d) None of the options
Answer: (b) Decreasing

Question. The function \( f(x) = e^{|x|} \) is :
(a) Continuous everywhere but not differentiable at \( x = 0 \)
(b) Continuous and differentiable everywhere
(c) Not continuous at \( x = 0 \)
(d) None of the options
Answer: (a) Continuous everywhere but not differentiable at \( x = 0 \)

Question. The function \( f(x) = \begin{cases} 1, & \text{if } x \neq 0 \\ 2, & \text{if } x \neq 0 \end{cases} \) is not continuous at :
(a) \( x = 0 \)
(b) \( x = 1 \)
(c) \( x = - 1 \)
(d) None of the options
Answer: (a) \( x = 0 \)

Question. The point of discontinuity of the function \( f(x) = \begin{cases} 2x + 3, & \text{if } x \le 2 \\ 2x - 3, & \text{if } x > 2 \end{cases} \) is :
(a) \( x = 0 \)
(b) \( x = 1 \)
(c) \( x = 2 \)
(d) None of the options
Answer: (c) \( x = 2 \)

Question. If \( f(x) = \begin{cases} \lambda(x^2 - 2x), & \text{if } x \le 0 \\ 4x + 1, & \text{if } x > 0 \end{cases} \), then which one of the following is correct:
(a) \( f(x) \) is continuous at \( x = 0 \) for any value of \( \lambda \)
(b) \( f(x) \) is discontinuous at \( x = 0 \) for any value of \( \lambda \)
(c) \( f(x) \) is discontinuous at \( x = 1 \) for any value of \( \lambda \)
(d) None of the options
Answer: (b) \( f(x) \) is discontinuous at \( x = 0 \) for any value of \( \lambda \)

Question. The function \( f(x) = \cot x \) is discontinuous on the set :
(a) \( \{x = n\pi : n \in Z\} \)
(b) \( \{x = 2n\pi : n \in Z\} \)
(c) \( \{x = (2n + 1)\frac{\pi}{2} : n \in Z\} \)
(d) \( \{x = \frac{n\pi}{2} : n \in Z\} \)
Answer: (a) \( \{x = n\pi : n \in Z\} \)

Question. The function defined by \( g(x) = x - [x] \) is discontinuous at:
(a) all rational points
(b) all irrational points
(c) all integral points
(d) None of the options
Answer: (c) all integral points

Question. The function \( f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 3, & \text{if } x = \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \), when \( k \) equals:
(a) – 6
(b) 6
(c) 5
(d) – 5
Answer: (b) 6

Question. The number of points at which the function \( f(x) = \frac{1}{x - [x]} \). [.] denotes the greatest integer function is not continuous is: [NCERT Exemplar]
(a) 1
(b) 2
(c) 3
(d) None of the options
Answer: (d) None of the options

Question. If \( f(x) = \begin{cases} \frac{\sqrt{1 + kx} - \sqrt{1 - kx}}{x}, & \text{for } - 1 \le x < 0 \\ 2x^2 + 3x - 2, & \text{for } 0 \le x \le 1 \end{cases} \) is continuous at \( x = 0 \), then \( k \) is equal to:
(a) – 4
(b) – 3
(c) – 2
(d) – 1
Answer: (c) – 2

Question. If \( f(x) = 2x \) and \( g(x) = \frac{x^2}{2} + 1 \), then which of the following can be discontinuous functions?
(a) \( f(x) + g(x) \)
(b) \( f(x) \cdot g(x) \)
(c) \( f(x) / g(x) \)
(d) \( g(x) / f(x) \)
Answer: (d) \( g(x) / f(x) \)

Question. The set of points, where the function \( f \) given by \( f(x) = | 2x - 1 | \sin x \) is differentiable is: [NCERT Exemplar]
(a) \( R \)
(b) \( R - \{ \frac{1}{2} \} \)
(c) \( (0, \infty) \)
(d) None of the options
Answer: (b) \( R - \{ \frac{1}{2} \} \)

Question. The differential coefficient of \( \sin (\cos (x^2)) \) with respect to \( x \) is:
(a) \( - 2x \sin x^2 \cos (\cos x^2) \)
(b) \( 2x \sin (x^2) \cos (x^2) \)
(c) \( 2x \sin (x^2) \cos (x^2) \cos x \)
(d) None of the options
Answer: (a) \( - 2x \sin x^2 \cos (\cos x^2) \)

Question. If \( y = \sqrt{3x + 2} + \frac{1}{\sqrt{2x^2 + 4}} \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{3}{2\sqrt{3x + 2}} - \frac{2x}{(2x^2 + 4)^{3/2}} \)
(b) \( \frac{3}{2\sqrt{3x + 2}} + \frac{2x}{(2x^2 + 4)^{3/2}} \)
(c) \( \frac{3}{2\sqrt{3x + 2}} \times \frac{2x}{(2x^2 + 4)^{3/2}} \)
(d) None of the options
Answer: (a) \( \frac{3}{2\sqrt{3x + 2}} - \frac{2x}{(2x^2 + 4)^{3/2}} \)

Question. Let \( f(x) = \begin{cases} (x - 1) \sin \frac{1}{(x - 1)}, & \text{if } x \neq 1 \\ 0, & \text{if } x = 1 \end{cases} \). Then, which of the following is true?
(a) \( f \) is differentiable at \( x = 1 \) but not at \( x = 0 \)
(b) \( f \) is neither differentiable at \( x = 0 \) nor at \( x = 1 \)
(c) \( f \) is differentiable at \( x = 0 \) and \( x = 1 \)
(d) \( f \) is differentiable at \( x = 0 \) but not at \( x = 1 \)
Answer: (d) \( f \) is differentiable at \( x = 0 \) but not at \( x = 1 \)

Question. The derivative of \( 2x + 3y = \sin y \) is:
(a) \( \frac{2}{\cos y} \)
(b) \( \frac{2}{\cos y + 3} \)
(c) \( \frac{2}{\cos y - 3} \)
(d) None of the options
Answer: (c) \( \frac{2}{\cos y - 3} \)

Question. If \( x + \sin y = \log x \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{1 - x}{x \sin y} \)
(b) \( \frac{1 - x}{x \cos y} \)
(c) \( \frac{1 + x}{x \cos y} \)
(d) None of the options
Answer: (b) \( \frac{1 - x}{x \cos y} \)

Question. If \( 2x + 3y = \sin x \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{\cos x + 2}{3} \)
(b) \( \frac{\cos x - 2}{3} \)
(c) \( \cos x + 2 \)
(d) None of the options
Answer: (b) \( \frac{\cos x - 2}{3} \)

Question. 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. If \( \cos y = x \cos (a + y) \) with \( \cos a = 1 \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{\sin^2(a + y)}{\sin a} \)
(b) \( \frac{\cos^2(a + y)}{\sin a} \)
(c) \( \sin^2 (a + y) \sin a \)
(d) None of the options
Answer: (b) \( \frac{\cos^2(a + y)}{\sin a} \)

Question. If \( y = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{1}{1 + x^2} \)
(b) \( \frac{2}{1 + x^2} \)
(c) \( \frac{2}{1 - x^2} \)
(d) \( \frac{2}{1 + x} \)
Answer: (b) \( \frac{2}{1 + x^2} \)

Question. If \( y = \tan^{-1} \left( \frac{3x - x^3}{1 - 3x^2} \right) , \frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}} \), then \( \frac{dy}{dx} \) is:
(a) \( \frac{3}{1 + x^2} \)
(b) \( \frac{1}{1 + x^2} \)
(c) \( \frac{3}{1 - x^2} \)
(d) \( \frac{3}{1 - x^2} \)
Answer: (a) \( \frac{3}{1 + x^2} \)

Question. If \( y = \sin^{-1} x + \sin^{-1} \sqrt{1 - x^2} \), \( -1 < x < 1 \), then \( \frac{dy}{dx} \) in equal to:
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0

Question. Derivative of \( \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right] , 0 < x < \frac{\pi}{2} \) is:
(a) \( \frac{1}{2} \)
(b) 1
(c) 2
(d) None of the options
Answer: (a) \( \frac{1}{2} \)

Question. If \( y^x = e^{y - x} \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{1 + \log y}{y \log y} \)
(b) \( \frac{(1 + \log y)^2}{y \log y} \)
(c) \( \frac{1 + \log y}{(\log y)^2} \)
(d) \( \frac{(1 + \log y)^2}{\log y} \)
Answer: (d) \( \frac{(1 + \log y)^2}{\log y} \)

Question. If \( x = e^{x - y} \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \frac{x - y}{x \log x} \)
(b) \( \frac{y - x}{\log x} \)
(c) \( \frac{y - x}{x \log x} \)
(d) \( \frac{x - y}{\log x} \)
Answer: (a) \( \frac{x - y}{x \log x} \)

Question. If \( x = at^2 \) and \( y = 2at \), then \( \frac{dy}{dx} \) is equal to:
(a) \( t \)
(b) \( \frac{1}{t} \)
(c) \( \frac{-1}{t^2} \)
(d) None of the options
Answer: (b) \( \frac{1}{t} \)

Question. If \( x = a(\cos \theta + \theta \sin \theta) \) and \( y = a(\sin \theta - \theta \cos \theta) \), then \( \frac{dy}{dx} \) is equal to:
(a) \( \tan \theta \)
(b) \( \cos \theta \)
(c) \( \sin \theta \)
(d) \( \cos \theta \)
Answer: (a) \( \tan \theta \)

Question. The derivative of \( \cos^{-1} (2x^2 - 1) \) w.r.t. \( \cos^{-1} x \) is:
(a) 2
(b) \( \frac{-1}{2\sqrt{1 - x^2}} \)
(c) \( \frac{2}{x} \)
(d) \( 1 - x^2 \)
Answer: (a) 2

Question. The derivative of \( \sin^2 x \) with respect to \( e^{\cos x} \) is:
(a) \( \frac{2 \cos x}{e^{\cos x}} \)
(b) \( - \frac{2 \cos x}{e^{\cos x}} \)
(c) \( \frac{2}{e^{\cos x}} \)
(d) None of the options
Answer: (b) \( - \frac{2 \cos x}{e^{\cos x}} \)

Question. If \( y = \cos^{-1} x \), then the value of \( \frac{d^2y}{dx^2} \) in terms of \( y \) alone is:
(a) \( - \cot y \text{ cosec}^2 y \)
(b) \( \text{cosec } y \cot^2 y \)
(c) \( - \cot y \text{ cosec } y \)
(d) None of the options
Answer: (a) \( - \cot y \text{ cosec}^2 y \)

Question. The function \( f(x) = \tan x \) is discontinuous on the set
(a) \( \{x = n\pi : n \in Z\} \)
(b) \( \{x = 2n\pi : n \in Z\} \)
(c) \( \{x = (2n + 1)\frac{\pi}{2} : n \in Z\} \)
(d) \( \{x = \frac{n\pi}{2} : n \in Z\} \)
Answer: (a) \( \{x = (2n + 1)\frac{\pi}{2} : n \in Z\} \)

Question. The function \( f(x) = e^{|x|} \) is:
(a) \( f'(0) = 1 \)
(b) \( f'(0) = -1 \)
(c) \( f'(0) = 0 \)
(d) \( f'(0) \) does not exist
Answer: (d) \( f'(0) \) does not exist

Question. If \( f(x) = x^2 \sin \frac{1}{x} \), where \( x \neq 0 \), then the value of the function \( f \) at \( x = 0 \), so that the function is continuous at \( x = 0 \), is:
(a) 0
(b) – 1
(c) 1
(d) None of the options
Answer: (a) 0

Question. If \( f(x) = \begin{cases} mx + 1, & \text{if } x < \frac{\pi}{2} \\ \sin x + n, & \text{if } x > \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \), then
(a) \( m = 1, n = 0 \)
(b) \( m = \frac{n\pi}{2} + 1 \)
(c) \( n = \frac{m\pi}{2} \)
(d) \( m = n = \frac{\pi}{2} \)
Answer: (c) \( n = \frac{m\pi}{2} \)

Question. Let \( f(x) = | \sin x | \). Then
(a) \( f \) is everywhere differentiable
(b) \( f \) is everywhere continuous but not differentiable at \( x = n\pi, n \in Z \)
(c) \( f \) is everywhere continuous but not differentiable at \( x = (2n + 1)\frac{\pi}{2}, n \in Z \)
(d) None of the options
Answer: (b) \( f \) is everywhere continuous but not differentiable at \( x = n\pi, n \in Z \)