CBSE Class 12 Mathematics Continuity and Differentiability MCQs Set 03

Question. The function \( f(x) = \begin{cases} \frac{\sin x}{x} + \cos x, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( k \) is
(a) 3
(b) 2
(c) 1
(d) 1.5
Answer: (b) 2

Question. 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: (d) 1.5

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

Question. The value of \( k \) which makes the function defined by \( f(x) = \begin{cases} \sin \frac{1}{x}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases} \), continuous at \( x = 0 \) is
(a) 8
(b) 1
(c) -1
(d) None of the options
Answer: (d) None of the options

Question. The value of \( c \) in Rolle's Theorem for the function \( f(x) = e^x \sin x \), \( \in [0, \pi] \) is 
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{3\pi}{4} \)
Answer: (d) \( \frac{3\pi}{4} \)

Question. The value of \( c \) in Mean Value Theorem for the function \( f(x) = x(x - 2) \), \( x \in [1, 2] \) is
(a) \( \frac{3}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{7}{4} \)
Answer: (a) \( \frac{3}{2} \)

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

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

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) \( \left\{ x = (2n + 1)\frac{\pi}{2} ; n \in Z \right\} \)
(d) \( \left\{ x = \frac{n\pi}{2} ; n \in Z \right\} \)
Answer: (a) \( \{x = n\pi : n \in Z\} \)

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. 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. 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 \( x = n\pi : n \in Z \)

Question. The value of \( c \) in Rolle's theorem for the function \( f(x) = x^3 - 3x \) in the interval \( [0, \sqrt{3}] \) is
(a) 1
(b) -1
(c) \( \frac{3}{2} \)
(d) \( \frac{1}{3} \)
Answer: (a) 1

Question. The set of points where the functions \( f \) given by \( f(x) = |x - 3| \cos x \) is differentiable is
(a) \( R \)
(b) \( R - \{3\} \)
(c) \( (0, \infty) \)
(d) None of the options
Answer: (b) \( R - \{3\} \)

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

Question. The function \( f(x) = |x| + |x - 1| \) is
(a) continuous at \( x = 0 \) as well as at \( x = 1 \).
(b) continuous at \( x = 1 \) but not at \( x = 0 \).
(c) discontinuous at \( x = 0 \) as well as at \( x = 1 \).
(d) continuous at \( x = 0 \) but not at \( x = 1 \).
Answer: (a) continuous at \( x = 0 \) as well as at \( x = 1 \).

Question. 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. If \( u = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \) and \( v = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \), then \( \frac{du}{dv} \) is
(a) \( \frac{1}{2} \)
(b) \( x \)
(c) \( \frac{1 - x^2}{1 + x^2} \{4, -4\}, \phi \)
(d) 1
Answer: (d) 1

Question. The function \( f(x) = \frac{4 - x^2}{4x - x^3} \) is
(a) discontinuous at only one point
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of the options
Answer: (c) discontinuous at exactly three points

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) \( \frac{\cos x}{2y - 1} \)
 

Assertion-Reason Questions

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : If \( f(x).g(x) \) is continuous at \( x = a \). then \( f(x) \) and \( g(x) \) are separately continuous at \( x = a \).
Reason (R) : Any function \( f(x) \) is said to be continuous at \( x = a \), if \( \lim_{h \to 0} f(a + h) = f(a) \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.
Solution: Let \( f(x) = x \) and \( g(x) = \sin \left(\frac{1}{x}\right) \) then \( F(x) = f(x)g(x) = x \sin \left(\frac{1}{x}\right) \).
Then \( \lim_{h \to 0} F(0 + h) = F(0) \). Hence \( F(x) = f(x) g(x) \) is continuous at \( x = 0 \), But \( g(x) = \sin \left(\frac{1}{x}\right) \) is not continuous at \( x = 0 \). Assertion (A) is false and Reason (R) is true.
Hence, (d) is the correct option.

Question. Assertion (A) : If \( f(x) \) and \( g(x) \) are two continuous functions such that \( f(0) = 3 \), \( g(0) = 2 \), then \( \lim_{x \to 0} \{f(x) + g(x)\} = 5 \).
Reason (R) : If \( f(x) \) and \( g(x) \) are two continuous functions at \( x = a \) then \( \lim_{x \to a} \{f(x) + g(x)\} = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: \( \lim_{x \to 0} \{f(x) + g(x)\} = f(0) + g(0) = 3 + 2 = 5 \)
Hence, (a) is the correct option.

Question. Assertion (A) : \( |\sin x| \) is a continuous function.
Reason (R) : If \( f(x) \) and \( g(x) \) both are continuous functions, then \( gof(x) \) is also a continuous function.
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) Both A and R are true and R is the correct explanation for A.
Solution: We have \( f(x) = \sin x \) and \( g(x) = |x| \)
\( \therefore gof(x) = g(f(x)) = g(\sin x) = |\sin x| \)
Here, \( \sin x \) and \( |x| \) both are continuous functions.
\( \therefore |\sin x| \) is also continuous function.
Clearly, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
Hence, (a) is the correct option.

Question. Assertion (A) : If \( y = \sin x \), then \( \frac{d^3 y}{dx^3} = -1 \) at \( x = 0 \).
Reason (R) : If \( y = f(x) \cdot g(x) \), then \( \frac{dy}{dx} = f(x) \cdot \frac{d}{dx}g(x) + g(x) \frac{d}{dx}f(x) \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: We have, \( y = \sin x \)
\( \implies \) \( \frac{dy}{dx} = \cos x \)
\( \implies \) \( \frac{d^2 y}{dx^2} = -\sin x \)
\( \therefore \quad \frac{d^3 y}{dx^3} = -\cos x = -1 \), at \( x = 0 \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.

Question. Assertion (A) : If \( f(x) = \sin^{-1} x + \cos^{-1} x + 2 \) then \( f'(1) = 0 \).
Reason (R) : \( \frac{d}{dx} \sin x = \cos x \)
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.
Solution: We have, \( f(x) = \sin^{-1} x + \cos^{-1} x + 2 \)
\( f(x) = \frac{\pi}{2} + 2 \quad \left[ \because \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \right] \)
\( \therefore \quad f'(x) = 0 \)
\( \implies \) \( f'(1) = 0 \)
Clearly, both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, (b) is the correct option.