CBSE Class 12 Mathematics Continuity and Differentiability MCQs Set 06

Assertion and Reason Questions

Choose the correct option :
(a) Both (A) and (R) are true and R is the correct explanation A.
(b) Both (A) and (R) are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : The function \( f(x) = | x | \) is discontinuous at \( x = 0 \).
Reason (R) : The function \( f(x) = | x | \) is non differentiable at \( x = 0 \).
Answer: (d) A is false but R is true.

Question. Assertion (A) : \( f(x) = \frac{1}{\{x\}} \) is discontinuous for integral values of \( x \), where \( \{ \} \) denotes the fractional part function.
Reason (R) : For integral values of \( x \), \( f(x) \) is not defined.
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : \( f(x) = \sqrt{x - 2} + \sqrt{2 - x} \) is continuous at \( x = 2 \)
Reason (R) : \( f(x) \) is a point function.
Answer: (d) A is false but R is true.

Question. Assertion (A) : The function \( f(x) = \begin{cases} [x] + \sqrt{x - [x]}, & x \ge 0 \\ \sin x, & x < 0 \end{cases} \) where \( [ ] \) denotes the greatest integer function, is continuous everywhere.
Reason (R) : \( f(x) \) is a periodic function.
Answer: (c) A is true but R is false.

Question. Assertion (A) : The function \( f(x) = \{x\} \), where \( \{ \} \) denotes the fractional part function, is discontinuous at \( x = 1 \).
Reason (R) : \( \lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x) \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : The function \( f(x) = \frac{(27 - 2x)^{1/3} - 3}{9 - 3(243 + 5x)^{1/5}} \) is continuous eveywhere is \( f(0) = 2 \).
Reason (R) : For continuous function \( f(0) = \lim_{x \to 0} f(x) \).
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Question. Assertion (A) : If \( f(x) \) is continuous, then \( | f ( | x | )| \) is also continuous.
Reason (R) : If \( | f(x) | \le | x | \forall x \in R \), then \( | f(x) | \) is continuous at \( x = 0 \).
Answer: (a) Both (A) and (B) are true and R is the correct explanation A.

Question. Assertion (A) : \( f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) is continuous at \( x = 0 \).
Reason (R) : Both \( h(x) = x^2, g(x) = \begin{cases} \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) are continuous at \( x = 0 \).
Answer: (c) A is true but R is false.

Question. Assertion (A) : \( f(x) = x \left( \frac{1 + e^{1/x}}{1 - e^{1/x}} \right) (x \neq 0), f(0) = 0 \) is continuous at \( x = 0 \).
Reason (R) : A function is said to be are continuous at \( a \) if both limits are exists and equal to \( f(a) \).
Answer: (a) Both (A) and (R) are true and R is the correct explanation A.

Case Study 1

Consider the following values. \( x = a \cos^3 \theta, y = a \sin^3 \theta \). On the basis of above information, answer the following questions:

Question. \( \frac{dx}{d\theta} |_{\theta = \frac{\pi}{4}} \)
(a) \( \frac{3a}{2\sqrt{2}} \)
(b) \( - \frac{3a}{2\sqrt{2}} \)
(c) \( \frac{4\sqrt{2}}{3a} \)
(d) \( - \frac{1}{\sqrt{3}} \)
Answer: (b) \( - \frac{3a}{2\sqrt{2}} \)

Question. \( \frac{dy}{d\theta} |_{\theta = \frac{\pi}{4}} \)
(a) \( \frac{3a}{2\sqrt{2}} \)
(b) \( - \frac{3a}{2\sqrt{2}} \)
(c) \( \frac{4\sqrt{2}}{3a} \)
(d) \( - \frac{1}{\sqrt{3}} \)
Answer: (a) \( \frac{3a}{2\sqrt{2}} \)

Question. \( \frac{dy}{dx} = \)
(a) \( \tan \theta \)
(b) \( - \tan \theta \)
(c) \( \cot \theta \)
(d) \( - \cot \theta \)
Answer: (b) \( - \tan \theta \)

Question. \( \frac{dy}{dx} |_{\theta = \frac{\pi}{6}} \)
(a) \( \frac{3a}{2\sqrt{2}} \)
(b) \( - \frac{3a}{2\sqrt{2}} \)
(c) \( \frac{4\sqrt{2}}{3a} \)
(d) \( - \frac{1}{\sqrt{3}} \)
Answer: (d) \( - \frac{1}{\sqrt{3}} \)

Question. \( \frac{d^2y}{dx^2} = \)
(a) \( \frac{3a}{2\sqrt{2}} \)
(b) \( - \frac{3a}{2\sqrt{2}} \)
(c) \( \frac{4\sqrt{2}}{3a} \)
(d) \( - \frac{1}{\sqrt{3}} \)
Answer: (c) \( \frac{4\sqrt{2}}{3a} \)

Case Study 2

The derivative of \( f \) at \( x = c \) is defined by \( f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \). A function is said to be differentiable at a point \( c \) if left hand derivative at \( x = c \) is equal to the right hand derivative at \( x = c \). Similarly, a function is said to be differentiable in an interval \( (a, b) \) if it is differentiable at every point of \( (a, b) \). Based on the above information, answer the following questions :

Question. Derivative of \( f(x) = \cos(\sqrt{x}) \) is :
(a) \( - \sin(\sqrt{x}) \)
(b) \( - \frac{\sin(\sqrt{x})}{2\sqrt{x}} \)
(c) \( \sin(\sqrt{x}) \)
(d) \( \frac{1}{2} \sin(\sqrt{x}) \)
Answer: (b) \( - \frac{\sin(\sqrt{x})}{2\sqrt{x}} \)

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

Question. \( f(x) = | x | \) is :
(a) Differentiable at all points \( x \in R \)
(b) Differentiable at all points \( x \in R - \{0\} \)
(c) Not Differentiable at \( x = 1 \)
(d) None of the options
Answer: (b) Differentiable at all points \( x \in R - \{0\} \)

Question. Derivative of function \( f(x) = \sin (x^2) \) is :
(a) \( 2 \cos (x^2) \)
(b) \( 2x \cos (x^2) \)
(c) \( 2x^2 \sin (x) \)
(d) \( 2 \cos (x) \)
Answer: (b) \( 2x \cos (x^2) \)

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

Case Study 3

A function is continuous at \( x = c \), if the function is defined at \( x = c \) and if the value of the function at \( x = c \) equals the limit of the function at \( x = c \), i.e., \( \lim_{x \to c} f(x) = f(c) \). Based on the above information answer the following questions :

Question. The relationship between \( a \) and \( b \) so that the below function is continuous at \( x = 3 \), \( f(x) = \begin{cases} ax + 1; & x \le 3 \\ bx + 3; & x > 3 \end{cases} \)
(a) \( a = b + 2 \)
(b) \( a + b = \frac{2}{3} \)
(c) \( a = b + \frac{2}{3} \)
(d) None of the options
Answer: (b) \( a = b + \frac{2}{3} \)

Question. \( f(x) = \begin{cases} kx^2; & x \le 2 \\ 3; & x > 2 \end{cases} \) is continuous at \( x = 2 \) then \( k \) is :
(a) \( k = 0.25 \)
(b) \( k = 3 \)
(c) \( k = 0.75 \)
(d) \( k = 1 \)
Answer: (c) \( k = 0.75 \)

Question. If \( f(x) \) and \( g(x) \) is continuous at \( x = c \) then :
(a) \( f \pm g \) is continuous at \( x = c \)
(b) \( f \cdot g \) is continuous at \( x = c \)
(c) \( f \pm g \) may or may not continuous
(d) None of the options
Answer: (a) \( f \pm g \) is continuous at \( x = c \)

Question. \( f(x) = \begin{cases} 2x; & x < 6 \\ x - 1; & x \ge 6 \end{cases} \) at \( x = 6 \) :
(a) Continuous
(b) Discontinuous
(c) May or may not continuous
(d) None of the options
Answer: (b) Discontinuous

Question. \( f(x) = \begin{cases} \frac{|x|}{x}; & x \neq 0 \\ 0; & x = 0 \end{cases} \) at \( x = 0 \) :
(a) Continuous
(b) Discontinuous
(c) May or may not continuous
(d) None of the options
Answer: (b) Discontinuous