Class 11 Mathematics Limits And Derivatives MCQs Set 12

Question. \( f(x) \) is differentiable function and \( (f(x) \cdot g(x)) \) is differentiable at \( x = a \), then
(a) \( g(x) \) must be differentiable at \( x = a \)
(b) if \( g(x) \) is discontinuous, then \( f(a) = 0 \)
(c) \( f(a) \neq 0 \), then \( g(x) \) must be differentiable
(d) nothing can be said
Answer: (c) \( f(a) \neq 0 \), then \( g(x) \) must be differentiable

Question. If \( f(x) = \begin{cases} |x| - 3, & x < 1 \\ |x - 2| + a, & x \geq 1 \end{cases} \) and \( g(x) = \begin{cases} 2 - |x|, & x < 2 \\ \text{sgn}(x - b), & x \geq 2 \end{cases} \) and \( h(x) = f(x) + g(x) \) is discontinuous at exactly one point, then which of the following values of \( a \) and \( b \) are possible:
(a) \( a = -3, b = 0 \)
(b) \( a = 2, b = 1 \)
(c) \( a = 2, b = 0 \)
(d) \( a = -3, b = 1 \)
Answer: (c) \( a = 2, b = 0 \)

Question. Let \([x]\) denote the greatest integer less than or equal to \( x \). If \( f(x) = [x \sin \pi x] \), then \( f(x) \) is
(a) continuous at \( x = 0 \)
(b) continuous in (-1, 0)
(c) differentiable at \( x = 1 \)
(d) differentiable in (-1, 1)
Answer: (a) continuous at \( x = 0 \)

Question. If \( f(x) = \min \{1, x^2, x^3\} \), then
(a) \( f(x) \) is continuous everywhere
(b) \( f(x) \) is continuous and differentiable everywhere
(c) \( f(x) \) is not differentiable at two points
(d) \( f(x) \) is not differentiable at one point
Answer: (a) \( f(x) \) is continuous everywhere

Question. For a function \( f(x) = \frac{\ln(\{ \sin x \}\{ \cos x \} + 1)}{\{ \sin x \}\{ \cos x \}} \), where \( \{.\} \) denotes fractional part function, then
(a) \( f(0^-) = f\left(\frac{\pi}{2}^+\right) \)
(b) \( f(0^+) = f\left(\frac{\pi}{2}^-\right) \)
(c) \( \lim_{x \to 0} f(x) = 1 \)
(d) \( \lim_{x \to \pi/2} f(x) = 1 \)
Answer: (c) \( \lim_{x \to 0} f(x) = 1 \)

Question. If \( f(x) = \begin{cases} \frac{x \log \cos x}{\log(1 + x^2)} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0, \end{cases} \) then
(a) \( f(x) \) is continuous at \( x = 0 \)
(b) \( f(x) \) is continuous but not differentiable at \( x = 0 \)
(c) \( f(x) \) is differentiable at \( x = 0 \)
(d) \( f(x) \) is not continuous at \( x = 0 \)
Answer: (a) \( f(x) \) is continuous at \( x = 0 \)

Question. If \( \lim_{x \to a} f(x) = \lim_{x \to a} [f(x)] \) (\([.]\) denotes the greatest integer function) and \( f(x) \) is non-constant continuous function, then
(a) \( \lim_{x \to a} f(x) \) is an integer
(b) \( \lim_{x \to a} f(x) \) is non-integer
(c) \( f(x) \) has local maximum at \( x = a \)
(d) \( f(x) \) has local minimum at \( x = a \)
Answer: (c) \( f(x) \) has local maximum at \( x = a \)

Question. Let \( f(x) = \frac{1}{[\sin x]} \), (where \([.]\) denotes the greatest integer function), then
(a) domain of \( f(x) \) is \( (2n\pi + \pi, 2n\pi + 2\pi) \cup \{2n\pi + \pi/2\} \), where \( n \in I \)
(b) \( f(x) \) is continuous, when \( x \in (2n\pi + \pi, 2n\pi + 2\pi) \), where \( n \in I \)
(c) \( f(x) \) is differentiable at \( x = \pi/2 \)
(d) None of the options
Answer: (a) domain of \( f(x) \) is \( (2n\pi + \pi, 2n\pi + 2\pi) \cup \{2n\pi + \pi/2\} \), where \( n \in I \)

Question. Let \( f(x) = (x - \lambda)^m |x - \lambda| \), where \( m \) is a non-negative integer, then at \( x = \lambda \)
(a) \( f(x) \) is not differentiable, if \( m = 0 \)
(b) \( f(x) \) is differentiable, if \( m \geq 1 \)
(c) \( f'(x) \) is not differentiable, if \( m = 1 \)
(d) \( f'(x) \) is differentiable, if \( m \geq 2 \).
Answer: (a) \( f(x) \) is not differentiable, if \( m = 0 \)

Question. If \( f(x) = \begin{cases} x^2 (\text{sgn}[x] + \{x\}), & 0 \leq x < 2 \\ \sin x + |x - 3|, & 2 \leq x < 4 \end{cases} \) where \([ ]\) and \(\{ \}\) represents the greatest integer and the fractional part function, respectively
(a) \( f(x) \) is differentiable at \( x = 1 \)
(b) \( f(x) \) is continuous but non differentiable at \( x = 1 \)
(c) \( f(x) \) is non differentiable at \( x = 2 \)
(d) \( f(x) \) is discontinuous at \( x = 2 \)
Answer: (d) \( f(x) \) is discontinuous at \( x = 2 \)

Question. Consider the function \( f(x) = \left( \frac{ax + 1}{bx + 2} \right)^x \), where \( a^2 + b^2 \neq 0 \) then \( \lim_{x \to \infty} f(x) \):
(a) exists for all values of \( a \) and \( b \)
(b) is zero for \( 0 < a < b \)
(c) is non-existent for \( a > b > 0 \)
(d) is \( e^{-(1/a)} \) or \( e^{-(1/b)} \) for \( a = b \)
Answer: (b) is zero for \( 0 < a < b \)

Question. A function \( f(x) \) satisfies the relation \( f(x + y) = f(x) + f(y) + xy(x + y), \forall x, y \in R \). If \( f'(0) = 1 \) then
(a) \( f(x) \) is a polynomial function
(b) \( f(x) \) is an exponential function
(c) \( f(x) \) is twice differentiable for all \( x \in R \)
(d) \( f'(3) = 8 \)
Answer: (a) \( f(x) \) is a polynomial function

Question. If \( f(x) = \begin{cases} 3x^2 + 12x - 1, & -1 \leq x \leq 2 \\ 37 - x, & 2 < x \leq 3 \end{cases} \), then
(a) \( f(x) \) is increasing on [-1, 2]
(b) \( f(x) \) is continuous on [-1, 3]
(c) \( f'(2) \) does not exist
(d) \( f(x) \) has the maximum value at \( x = 2 \)
Answer: (b) \( f(x) \) is continuous on [-1, 3]

Question. Let \( f(x + y) = f(x) + f(y) + 2xy - 1, \forall x, y \in R \). If \( f(x) \) is differentiable and \( f'(0) = \sin \phi \), then
(a) \( f(x) < 0, \forall x \in R \)
(b) \( f(x) > 0, \forall x \in R \)
(c) \( f(x) \geq \frac{3}{4}, \forall x \in R \)
(d) \( -1 \leq f(x) \leq 1, \forall x \in R \)
Answer: (c) \( f(x) \geq \frac{3}{4}, \forall x \in R \)

Question. The function, \( f(x) = \max \{(1 - x), (1 + x), 2\}, x \in (-\infty, \infty) \) is
(a) continuous at all points
(b) differentiable at all points
(c) differentiable at all points except at \( x = 1 \) and \( x = -1 \)
(d) continuous at all points except at \( x = 1 \) and \( x = -1 \), where it is discontinuous
Answer: (a) continuous at all points

Question. Evaluate \( \lim_{x \to 0} \left( \frac{(1 + \{x\})^{1/\{x\}}}{e} \right)^{1/x} \) if it exists, where \( \{x\} \) denotes the fractional part of \( x \).
(a) \( 2/e \)
(b) \( e^{-1/2} \)
(c) 2
(d) does not exist
Answer: (d) does not exist

Question. Let \( L = \lim_{x \to 0} \frac{a - \sqrt{a^2 - x^2} - \frac{x^2}{4}}{x^4} \), \( a > 0 \). If \( L \) is finite, then
(a) \( a = 2 \)
(b) \( a = 1 \)
(c) \( L = 1/64 \)
(d) \( L = 1/34 \).
Answer: (a) \( a = 2 \)

Question. Let \( f : R \to R \) be a function such that \( f(x + y) = f(x) + f(y), \forall x, y \in R \). If \( f(x) \) is differentiable at \( x = 0 \), then
(a) \( f(x) \) is differentiable only in a finite interval containing zero
(b) \( f(x) \) is continuous \( \forall x \in R \)
(c) \( f'(x) \) is continuous \( \forall x \in R \)
(d) \( f(x) \) is differentiable except at finitely many points
Answer: (b) \( f(x) \) is continuous \( \forall x \in R \)

Question. If \( f(x) = \begin{cases} -x - \frac{\pi}{2}, & x \leq -\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x - 1, & 0 < x \leq 1 \\ \ln x, & x > 1 \end{cases} \), then
(a) \( f(x) \) is continuous at \( x = -\frac{\pi}{2} \)
(b) \( f(x) \) is not differentiable at \( x = 0 \)
(c) \( f(x) \) is differentiable at \( x = 1 \)
(d) \( f(x) \) is differentiable at \( x = -3/2 \)
Answer: (b) \( f(x) \) is not differentiable at \( x = 0 \)

Question. For every integer \( n \), let \( a_n \) and \( b_n \) be real numbers. Let function \( f : R \to R \) be given by \( f(x) = \begin{cases} a_n + \sin \pi x, & \text{for } x \in [2n, 2n + 1] \\ b_n + \cos \pi x, & \text{for } x \in (2n - 1, 2n) \end{cases} \), for all integers \( n \). If \( f \) is continuous, then which of the following hold(s) for all \( n \)
(a) \( a_{n-1} - b_{n-1} = 0 \)
(b) \( a_n - b_n = 1 \)
(c) \( a_n - b_{n+1} = 1 \)
(d) \( a_{n-1} - b_n = -1 \)
Answer: (b) \( a_n - b_n = 1 \)

COMPREHENSIONS QUESTIONS

Passage - 1:
Let \( f(x) = \lim_{n \to \infty} \left( \cos \frac{x}{\sqrt{n}} \right)^n \), \( g(x) = \lim_{n \to \infty} \left( 1 + \frac{x + x\sqrt{e}}{n} \right)^n \). Now, consider the function \( y = h(x) \), where \( h(x) = \tan^{-1}(g^{-1}(f^{-1}(x))) \).
 

Question. \( \lim_{x \to 0} \frac{\ln(f(x))}{\ln(g(x))} \) is equal to
(a) \( 1/2 \)
(b) \( -1/2 \)
(c) 0
(d) 1
Answer: (b) \( -1/2 \)

Question. Domain of the function \( y = h(x) \) is
(a) \( (0, \infty) \)
(b) R
(c) (0, 1)
(d) [0, 1]
Answer: (b) R

Question. Range of the function \( y = h(x) \) is
(a) \( \left( 0, \frac{\pi}{2} \right) \)
(b) \( \left( -\frac{\pi}{2}, 0 \right) \)
(c) R
(d) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
Answer: (d) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)

Passage - 2:
Let \( f(x) = \begin{cases} x + 2, & 0 \leq x < 2 \\ 6 - x, & x \geq 2 \end{cases} \), \( g(x) = \begin{cases} 1 + \tan x, & 0 \leq x < \pi/4 \\ 3 - \cot x, & \pi/4 \leq x < \pi \end{cases} \)

Question. \( f(g(x)) \) is
(a) discontinuous at \( x = \frac{\pi}{4} \)
(b) differentiable at \( x = \frac{\pi}{4} \)
(c) continuous but non differentiable at \( x = \frac{\pi}{4} \)
(d) differentiable at \( x = \frac{\pi}{4} \), but derivative is not continuous.
Answer: (c) continuous but non differentiable at \( x = \frac{\pi}{4} \)

Question. The number of points of non differentiability of \( h(x) = |f(g(x))| \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The range of \( h(x) = f(g(x)) \) is
(a) \( (-\infty, \infty) \)
(b) \( (4, \infty) \)
(c) \( (-\infty, 4] \)
(d) \( [4, \infty) \)
Answer: (c) \( (-\infty, 4] \)

Passage - 3:
Let \( f(x) = \frac{\sin^{-1}(1 - \{x\}) \cdot \cos^{-1}(1 - \{x\})}{\sqrt{2 \{x\}}(1 - \{x\})} \), where \( \{.\} \) denotes the fractional part function.

Question. If \( R = \lim_{x \to 0^+} f(x) \), then the value of \( \cos(100 R) \) is :
(a) -1
(b) 0
(c) 1/2
(d) 1
Answer: (d) 1

Question. If \( L = \lim_{x \to 0^-} f(x) \), then the value of \( \sin(99 \sqrt{2} L) \) is :
(a) -1
(b) 0
(c) 1/2
(d) 1
Answer: (b) 0

Question. The value of \( [2R^2 + 4L^2] \) is [where \([.]\) denotes the greatest integer function] :
(a) 3
(b) 6
(c) 9
(d) 12
Answer: (b) 6

Passage - 4:
Let \( f(x) = \begin{cases} [x], & -2 \leq x \leq -1/2 \\ 2x^2 - 1, & -1/2 < x \leq 2 \end{cases} \) and \( g(x) = f([x]) + [f(x)] \), where \( [ ] \) represents greatest integer function.

Question. The number of points where \( |f(x)| \) is non-differentiable is
(a) 3
(b) 4
(c) 2
(d) 5
Answer: (a) 3

Question. The number of points where \( g(x) \) is non-differentiable is
(a) 4
(b) 5
(c) 2
(d) 3
Answer: (b) 5

Question. The number of points where \( g(x) \) is discontinuous is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (b) 2
 

FILL IN BLANKS QUESTIONS

Question. Let \( f(x) = x |x| \). The set of points where \( f(x) \) is twice differentiable is ....... 
Answer: \( R - \{0\} \)

Question. Let \( f(x) = (x - 1)(x - 2)(x - 3).....(x - n), n \in N \) and \( f'(n) = 5040 \) then ‘n’ is
Answer: 8

Question. Let \( f(xy) = xf(y) + yf(x) \) for all \( x, y \in R_{+} \) and \( f(x) \) be differentiable in \( (0, \infty) \), then determine \( f(x) \).
Answer: \( f(x) = cx \ln x \)

ASSERTION-REASONING QUESTIONS

Question. Statement-1: The set of all points where the function \( f(x) = \begin{cases} 0, & x = 0 \\ \frac{x}{1+e^{1/x}}, & x \neq 0 \end{cases} \) is differentiable is \( (-\infty, \infty) \).
Statement-2: \( Lf'(0) = 1, Rf'(0) = 0 \) and \( f'(x) = \frac{1+e^{1/x} + e^{1/x} \cdot \frac{1}{x}}{(1+e^{1/x})^2} \), which exists when \( x \neq 0 \).
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (d) Statement –1 is False, statement–2 is True.

Question. Statements-1: \( f(x) = \begin{cases} 3 - x^2, & x > 1 \\ x^3 + 1, & x \leq 1 \end{cases} \) then \( f(x) \) is differentiable at \( x = 1 \)
Statements-2: A function \( y = f(x) \) is said to have a derivative if \( \lim_{h \to 0^{+}} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0^{-}} \frac{f(x+h) - f(x)}{h} \)
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (d) Statement –1 is False, statement–2 is True.

Question. Statement – 1: \( \lim_{x \to 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} \) does not exist
Statement – 2: L.H.L. = 1 and R.H.L. = –1.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (c) Statement–1 is True, Statement–2 is False

Question. Consider the function \( f(x) = (|x| - |x - 1|)^2 \)
Statement – 1: \( f(x) \) is continuous everywhere but not differentiable at \( x = 0 \) and 1.
Statement – 2: \( f'(0^{-}) = 0, f'(0^{+}) = -4, f'(1^{-}) = 4, f'(1^{+}) = 0 \).
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Question. Let \( f \) and \( g \) be real valued functions defined on interval \( (-1, 1) \) such that \( g''(x) \) is continuous \( g(0) \neq 0, g'(0) = 0, g''(0) \neq 0 \), and \( f(x) = g(x) \sin x \).
Statement–1: \( \lim_{x \to 0} [g(x) \cot x - g(x) \csc x] = f''(0) \).
Statement–2: \( f'(0) = g(0) \)
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (d) Statement –1 is False, statement–2 is True.

Question. Statement 1 : If \( f(x) \) and \( g(x) \) are continuous and differentiable function, then there exist \( c, c \in [a, b] \) such that \( \frac{f'(c)}{f(a) - f(c)} + \frac{g'(c)}{g(b) - g(c)} = 1 \)
Statement 2: If \( f(x) \) and \( g(x) \) are continuous and differentiable function, then \( h(x) = (f(a) - f(x))(g(b) - g(x)) e^x \) is continuous and differentiable.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Question. Statement 1 : The function \( f(x) = (3x - 1) |4x^2 - 12x + 5| \cos \pi x \) is differentiable at \( x = \frac{1}{2}, \frac{5}{2} \).
Statement 2: \( \cos (2n + 1) \frac{\pi}{2} = 0, \forall n \in I \)
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Question. Statement 1 : \( \lim_{x \to a} f(x) \) exists = k, but \( \lim_{x \to k} g(x) \) does not exist. If \( \lim_{x \to a} (g(f(x))) \) exists, then \( x = a \) is a point of extremum for \( y = f(x) \), If \( f(x) \) is non linear
Statement 2: \( \lim_{x \to k} g(x) \) does not exist but \( \lim_{x \to a} (g(f(x))) \) exists, \( f(x) \) will approach k when \( x \to a \) through only one side.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

Question. Let \( f(0) = 0, f\left(\frac{\pi}{2}\right) = 1, f\left(\frac{3\pi}{2}\right) = -1 \) be a continuous and twice differentiable function.
Statement 1 : \( |f''(x)| \leq 1 \) for at least one \( x \in \left(0, \frac{3\pi}{2}\right) \)
Statement 2: According to Rolles theorem if \( y = g(x) \) is continuous and differentiable \( \forall x \in [a, b] \) and \( g(a) = g(b) \), then there exist at least one \( c \) such that \( g'(c) = 0 \).
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(b) Statement-1 is True, Statement-2 is True; Statement–2 NOT a correct explanation for Statement-1.
(c) Statement–1 is True, Statement–2 is False
(d) Statement –1 is False, statement–2 is True.
Answer: (d) Statement –1 is False, statement–2 is True.