CBSE Class 12 Mathematics Continuity and Differentiability MCQs Set 07

Question. The value of \( k \) for which \( f(x) = \begin{cases} 3x + 5, & x \ge 2 \\ kx^2, & x < 2 \end{cases} \) is a continuous function, is
(a) \( -\frac{11}{4} \)
(b) \( \frac{4}{11} \)
(c) \( 11 \)
(d) \( \frac{11}{4} \)
Answer: (d) \( \frac{11}{4} \)

Question. The value of \( k \) for which the function \( f(x) = \begin{cases} \frac{1 - \cos 4x}{8x^2}, & \text{if } x \ne 0 \\ k, & \text{if } x = 0 \end{cases} \) is continuous at \( x = 0 \) is
(a) \( 0 \)
(b) \( -1 \)
(c) \( 1 \)
(d) \( 2 \)
Answer: (c) \( 1 \)

Question. The function \( f(x) = [x] \), where \( [x] \) denotes the greatest integer less than or equal to \( x \), is continuous at
(a) \( x = 1 \)
(b) \( x = 1.5 \)
(c) \( x = -2 \)
(d) \( x = 4 \)
Answer: (b) \( x = 1.5 \)

Question. The points, at which the function \( f \) given by \( f(x) = \begin{cases} \frac{x}{|x|}, & x < 0 \\ -1, & x \ge 0 \end{cases} \) is continuous, is/are
(a) \( x \in \mathbb{R} \)
(b) \( x = 0 \)
(c) \( x \in \mathbb{R} - \{0\} \)
(d) \( x = -1 \text{ and } 1 \)
Answer: (a) \( x \in \mathbb{R} \)

Question. For what value of \( x \) will the function, \( f(x) = \begin{cases} \frac{x + 4}{|x + 4|}, & x \ne -4 \\ 1, & x = -4 \end{cases} \) is discontinuous?
(a) \( -4 \)
(b) \( 4 \)
(c) \( 1 \)
(d) No point of discontinuity
Answer: (d) No point of discontinuity

Question. If \( f(x) = 2|x| + 3|\sin x| + 6 \), then the right hand derivative of \( f(x) \) at \( x = 0 \) is
(a) \( 6 \)
(b) \( 5 \)
(c) \( 3 \)
(d) \( 2 \)
Answer: (b) \( 5 \)

Question. The function \( f(x) = x|x| \) is
(a) continuous and differentiable at \( x = 0 \).
(b) continuous but not differentiable at \( x = 0 \).
(c) differentiable but not continuous at \( x = 0 \).
(d) neither differentiable nor continuous at \( x = 0 \).
Answer: (a) continuous and differentiable at \( x = 0 \).

Question. The value of \( k \) for which function \( f(x) = \begin{cases} x^2, & x \ge 0 \\ kx, & x < 0 \end{cases} \) is differentiable at \( x = 0 \) is
(a) \( 1 \)
(b) \( 2 \)
(c) any real number
(d) \( 0 \)
Answer: (d) \( 0 \)

Question. The set of all points, where the function \( f(x) = x + |x| \) is differentiable, is
(a) \( (0, \infty) \)
(b) \( (-\infty, 0) \)
(c) \( (-\infty, 0) \cup (0, \infty) \)
(d) \( (-\infty, \infty) \)
Answer: (c) \( (-\infty, 0) \cup (0, \infty) \)

Question. The function \( f(x) = |x| \) is
(a) continuous and differentiable everywhere.
(b) continuous and differentiable nowhere.
(c) continuous everywhere but differentiable everywhere except at \( x = 0 \).
(d) continuous everywhere but differentiable nowhere.
Answer: (c) continuous everywhere but differentiable everywhere except at \( x = 0 \).

Question. Let \( f(x) = \left|\begin{array}{cc} x^2 & \sin x \\ p & -1 \end{array}\right| \), where \( p \) is a constant. Then, the value of \( p \) for which \( f'(0) = 1 \) is
(a) \( \mathbb{R} \)
(b) \( 1 \)
(c) \( 0 \)
(d) \( -1 \)
Answer: (d) \( -1 \)

Question. If \( y = \frac{\cos x - \sin x}{\cos x + \sin x} \), then \( \frac{dy}{dx} \) is
(a) \( -\sec^2\left(\frac{\pi}{4}-x\right) \)
(b) \( \sec^2\left(\frac{\pi}{4}-x\right) \)
(c) \( \log \left|\sec\left(\frac{\pi}{4}-x\right)\right| \)
(d) \( -\log \left|\sec\left(\frac{\pi}{4}-x\right)\right| \)
Answer: (a) \( -\sec^2\left(\frac{\pi}{4}-x\right) \)

Question. If \( y = \log\sqrt{\sec\sqrt{x}} \), then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi^2}{16} \) is
(a) \( \frac{1}{\pi} \)
(b) \( \pi \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{4} \)
Answer: (a) \( \frac{1}{\pi} \)

Question. If \( f(x) = |\cos x| \), then \( f'\left(\frac{3\pi}{4}\right) \) is
(a) \( 1 \)
(b) \( -1 \)
(c) \( -\frac{1}{\sqrt{2}} \)
(d) \( \frac{1}{\sqrt{2}} \)
Answer: (d) \( \frac{1}{\sqrt{2}} \)

Question. If \( xe^y = 1 \), then the value of \( \frac{dy}{dx} \) at \( x = 1 \) is
(a) \( -1 \)
(b) \( 1 \)
(c) \( -e \)
(d) \( -\frac{1}{e} \)
Answer: (a) \( -1 \)

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

Question. If \( x = 3\cos\theta \) and \( y = 5\sin\theta \), then \( \frac{dy}{dx} \) is equal to
(a) \( -\frac{3}{5}\tan\theta \)
(b) \( -\frac{5}{3}\cot\theta \)
(c) \( -\frac{5}{3}\tan\theta \)
(d) \( -\frac{3}{5}\cot\theta \)
Answer: (b) \( -\frac{5}{3}\cot\theta \)

Question. If \( y = 10^{10^x} \), then \( \frac{dy}{dx} \) is equal to
(a) \( 10^{10^x}\log 10 \)
(b) \( 10^{10^x}(\log 10)^2 \)
(c) \( 10^{10^x} \cdot 10^x(\log 10)^2 \)
(d) \( 10^{10^x} \cdot 10^x\log 10 \)
Answer: (c) \( 10^{10^x} \cdot 10^x(\log 10)^2 \)

Question. Derivative of \( e^{\sin^2 x} \) with respect to \( \cos x \) is
(a) \( \sin x e^{\sin^2 x} \)
(b) \( \cos x e^{\sin^2 x} \)
(c) \( -2\cos x e^{\sin^2 x} \)
(d) \( -2\sin^2 x \cos x e^{\sin^2 x} \)
Answer: (c) \( -2\cos x e^{\sin^2 x} \)

Question. If \( x = at^2 \), \( y = 2at \), then \( \frac{d^2 y}{dx^2} \) is equal to
(a) \( -\frac{1}{t^2} \)
(b) \( \frac{1}{2at^3} \)
(c) \( -\frac{1}{t^3} \)
(d) \( -\frac{1}{2at^3} \)
Answer: (d) \( -\frac{1}{2at^3} \)

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

Question. If \( y = \sin^{-1} x \), then \( (1 - x^2)y_2 \) is equal to
(a) \( xy_1 \)
(b) \( xy \)
(c) \( xy_2 \)
(d) \( x^2 \)
Answer: (a) \( xy_1 \)

Question. If \( y^{1/m} + y^{-1/m} = 2x \), then \( (x^2 - 1)y_2 + xy_1 \) is equal to
(a) \( \pm m^2 y \)
(b) \( m^2 y \)
(c) \( -m^2 y \)
(d) \( -my \)
Answer: (b) \( m^2 y \)

Assertion-Reason Based Questions

Directions: In the questions given below are two statements labelled as Assertion (A) and Reason (R). In the context of the two statements, which one of the following is correct?
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.

Question. Assertion (A) \( \sin^{-1} x \) is continuous on \( [-1, 1] \).
Reason (R) \( \cos x^2 \) is discontinuous on \( x \in \mathbb{R} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (c) A is correct; R is incorrect.

Question. Assertion (A) \( |\sin x| \) is not 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 correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A) If \( y = \cos x \), then \( \left(\frac{dy}{dx}\right)_{x=\pi/2} \) is 0.
Reason (R) \( \frac{d}{dx}(\cos x) = -\sin x \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A) If \( y = \cos^2 x^2 \), then \( \frac{dy}{dx} = -4x\cos x^2 \sin x^2 \).
Reason (R) For the curve \( \sqrt{x} + \sqrt{y} = 1 \), \( \frac{dy}{dx} \) at \( \left(\frac{1}{4}, \frac{1}{4}\right) \) is \( -1 \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A) If \( f(x) = \sqrt{1 + \cos^2(x^2)} \), then \( f'\left(\frac{\sqrt{\pi}}{2}\right) \) is \( \frac{\pi}{3} \).
Reason (R) If \( f(x) = e^x \), \( g(x) = \sin^{-1} x \) and \( h(x) = f(g(x)) \), then \( \frac{h'(x)}{h(x)} \) is \( \frac{1}{\sqrt{1 - x^2}} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A) If \( e^{xy} + \log(xy) + \cos(xy) + 4 = 0 \), then \( \frac{dy}{dx} = -\frac{y}{x} \).
Reason (R) \( \frac{d}{dx}(xy) = 0 \Rightarrow \frac{dy}{dx} = -\frac{y}{x} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) If \( f(x) = \sin^{-1} x + \cos^{-1} x + 4 \), then \( f'(4) = 0 \).
Reason (R) \( \frac{d}{dx}(\cos x) = -\sin x \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A) If \( f(x) \) is an odd function, then \( f'(x) \) is an even function.
Reason (R) If \( u = f(x) \), \( v = g(x) \), then the derivative of \( f \) with respect to \( g \) is \( \frac{du}{dv} = \frac{du/dx}{dv/dx} \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A) If \( y = \sin x + \cos x \), then \( \left(\frac{dy}{dx}\right)_{x=\pi/4} = 0 \).
Reason (R) \( \frac{d^2 y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A) If \( y = \tan^{-1} x \), then \( \frac{dy}{dx} = \frac{1}{1 + x^2} \).
Reason (R) \( \frac{d^2 y}{dx^2} = \left(\frac{dy}{dx}\right)^2 \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (c) A is correct; R is incorrect.

Case Study Based Questions - I

Ambika, a mathematics teacher is conducting a practice session on calculus where she is discussing continuity and differentiability of several functions in their domains. Shown below are four cards that contain a function each along with their domains and graphs.
Card I: \( f_1(x) = x^2 + 2x \)
Card II: \( f_2(x) = \frac{1}{2x} \), where \( x \ne 0 \)
Card III: \( f_3(x) = \begin{cases} -x, & x \le 0 \\ 0, & x \ge 0 \end{cases} \)
Card IV: \( f_4(x) = \begin{cases} 1, & x > 0 \\ 0, & x \le 0 \end{cases} \)
While discussing in the group, four students claimed as follows:
Leela: 'As the function on card I is both continuous and differentiable, we can say that every continuous function is differentiable.'
Irfan: 'As the graph is not in one piece, the function on card II is discontinuous.'
Deepak: 'The function on card III is continuous.'
Kiran: 'The function on card IV is discontinuous.'

Question. Check whether Deepak and Kiran's claims are correct. Justify your answer.
Answer: Deepak's claim is correct. For the function \( f_3(x) = \begin{cases} -x, & x \le 0 \\ 0, & x \ge 0 \end{cases} \):
LHL \( = \lim_{x \to 0^-} f(x) = \lim_{h \to 0} (-(-h)) = 0 \)
RHL \( = \lim_{x \to 0^+} f(x) = \lim_{h \to 0} 0 = 0 \)
At \( x = 0 \), \( f_3(0) = 0 \).
Since LHL = RHL = \( f_3(0) \), the function on Card III is continuous. Hence, Deepak is right.
Kiran's claim is also correct. For the function \( f_4(x) = \begin{cases} 1, & x > 0 \\ 0, & x \le 0 \end{cases} \):
LHL \( = \lim_{x \to 0^-} f(x) = 0 \)
RHL \( = \lim_{x \to 0^+} f(x) = 1 \)
Since LHL \( \ne \) RHL, \( f_4(x) \) is discontinuous. Hence, Kiran is right.

Question. Is Irfan's claim correct? Does the reason given by him support his claim? Justify.
Answer: Irfan's claim is incorrect. The function is \( f_2(x) = \frac{1}{2x} \), where \( x \ne 0 \). This function is continuous everywhere in its domain (for all \( x \ne 0 \)). The point \( x = 0 \) is not in the domain of the function. Therefore, the function on card II is continuous. His reason 'As the graph is not in one piece...' does not support his claim, and his conclusion is wrong.

Question. Is Leela's claim true for all continuous functions? Justify with a valid reason or provide a counter example.
Answer: Leela's claim is not true for all continuous functions. Every continuous function need not be differentiable. For example, the modulus function \( f(x) = |x| \) is continuous everywhere but not differentiable at \( x = 0 \).

Case Study Based Questions - II

Let \( f(x) \) be a real valued function. Then, its Left Hand Derivative (LHD) is defined as:
\[ Lf'(a) = \lim_{h \to 0} \frac{f(a-h) - f(a)}{-h} \]
Right Hand Derivative (RHD) is defined as:
\[ Rf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \delta \]
Also, a function \( f(x) \) is said to be differentiable at \( x = a \), if its LHD and RHD at \( x = a \) exist and both are equal.
For the function:
\[ f(x) = \begin{cases} |x-3|, & x \ge 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & x < 1 \end{cases} \]

Question. What is RHD of \( f(x) \) at \( x = 1 \)?
Answer: For \( x \ge 1 \), \( f(x) = |x - 3| \). Near \( x = 1 \), \( x - 3 < 0 \), so \( f(x) = -(x-3) = 3-x \).
Therefore, \( Rf'(1) = \frac{d}{dx}(3-x)_{x=1} = -1 \).

Question. What is LHD of \( f(x) \) at \( x = 1 \)?
Answer: For \( x < 1 \), \( f(x) = \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4} \).
\( Lf'(1) = \frac{d}{dx}\left(\frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}\right)_{x=1} = \left(\frac{2x}{4} - \frac{3}{2}\right)_{x=1} = \frac{1}{2} - \frac{3}{2} = -1 \).

Question. Check if the function \( f(x) \) is differentiable at \( x = 1 \).
Answer: Since LHD \( = \) RHD \( = -1 \) at \( x = 1 \), the function \( f(x) \) is differentiable at \( x = 1 \).

Question. Find \( f'(2) \) and \( f'(-1) \).
Answer: For \( x = 2 \), which lies in the interval \( 1 < x < 3 \), \( f(x) = -(x - 3) = 3 - x \), hence \( f'(2) = -1 \).
For \( x = -1 \), which lies in the interval \( x < 1 \), \( f'(x) = \frac{x}{2} - \frac{3}{2} \), hence \( f'(-1) = \frac{-1}{2} - \frac{3}{2} = -2 \).