Class 11 Mathematics Limits And Derivatives MCQs Set 11

Question. The integral value of n for which \( \lim_{x \to 0} \frac{\cos^2 x - \cos x - e^x \cos x + e^x - \left(\frac{x^3}{2}\right)}{x^n} \) is finite and non zero is
(a) 2
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. The value of \( \lim_{x \to \infty} \frac{(2x)^{1/2} + (3x)^{1/3} + (4x)^{1/4} + ...... + (nx)^{1/n}}{(2x-3)^{1/2} + (2x-3)^{1/3} + ...... + (2x-3)^{1/n}} \) is
(a) \( \sqrt{2} \)
(b) 2
(c) \( \frac{1}{\sqrt{3}} \)
(d) 0
Answer: (b) 2

Question. ABC is an isosceles triangle inscribed in a circle of radius r. If AB = AC and h is the altitude from A to BC, then the triangle ABC has perimeter P = \( 2\left[\sqrt{(2hr - h^2)} + \sqrt{2hr}\right] \) and area A, then \( \lim_{h \to 0} \frac{A}{P^3} \)
(a) \( \frac{1}{r} \)
(b) \( \frac{1}{64r} \)
(c) \( \frac{1}{128r} \)
(d) \( \frac{1}{2r} \)
Answer: (c) \( \frac{1}{128r} \)

Question. The integer n for which \( \lim_{x \to 0} \frac{(\cos x - 1)(\cos x - e^x)}{x^n} \) is a finite non-zero number is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. If \( A_i = \frac{x - a_i}{|x - a_i|} \), i = 1, 2, 3,....,n and \( a_1 < a_2 < a_3 < ..... < a_n \), then \( \lim_{x \to a_m} (A_1 A_2 .... A_n) \), \( 1 \leq m \leq n \)
(a) is equal to \( (-1)^m \)
(b) is equal to \( (-1)^{m+1} \)
(c) is equal to \( (-1)^{m-1} \)
(d) Does not exist
Answer: (d) Does not exist

Question. The value of \( \lim_{x \to \infty} \left\{ \frac{x}{x + \frac{\sqrt[3]{x}}{x + \frac{\sqrt[3]{x}}{x + \sqrt[3]{x} \dots \infty}}} \right\} \) is
(a) 1
(b) 0
(c) 2
(d) \( \frac{1}{2} \)
Answer: (a) 1

Question. If \( \lim_{x \to 0} \left[ \frac{(1 - \sqrt{\cos 2x} \sqrt[3]{\cos 3x} \sqrt[4]{\cos 4x} ..... \sqrt[n]{\cos nx})}{x^2} \right] \) is equal to 10, then the value of n is
(a) 5
(b) 4
(c) 6
(d) 3
Answer: (b) 4

Question. Let \( a_1, a_2, \dots, a_n \) be sequence of real numbers with \( a_{n+1} = a_n + \sqrt{1 + a_n^2} \) and \( a_1 = 0 \). Then \( \lim_{n \to \infty} \left( \frac{a_n}{2^{n-1}} \right) = \)
(a) \( \frac{2}{\pi} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{2}{\pi} - 1 \)
(d) \( \frac{\pi}{2} - 1 \)
Answer: (b) \( \frac{\pi}{2} \)

Question. If \( \lim_{x \to 0} (x^{-3} \sin 3x + ax^{-2} + b) \) exists and is equal to 0, then
(a) a = -3 and \( b = \frac{9}{2} \)
(b) a = 3 and \( b = \frac{9}{2} \)
(c) a = -3 and \( b = -\frac{9}{2} \)
(d) a = 3 and \( b = -\frac{9}{2} \)
Answer: (a) a = -3 and \( b = \frac{9}{2} \)

Question. If \( \lim_{x \to 0} [1 + x \ln(1+b^2)]^{1/x} = 2b \sin^2 \theta, b > 0 \) and \( \theta \in (-\pi, \pi] \), then the value of \( \theta \) is 
(a) \( \pm \frac{\pi}{4} \)
(b) \( \pm \frac{\pi}{3} \)
(c) \( \pm \frac{\pi}{6} \)
(d) \( \pm \frac{\pi}{2} \)
Answer: (d) \( \pm \frac{\pi}{2} \)

Question. If \( \lim_{x \to \infty} \left( \frac{x^2+x+1}{x+1} - ax - b \right) = 4 \) then, [IIT-2012]
(a) a = 1, b = 4
(b) a = 1, b = -4
(c) a = 2, b = -3
(d) a = 2, b = 3
Answer: (b) a = 1, b = -4

Question. Let \( \alpha(a) \) and \( \beta(a) \) be the roots of the equation \( (\sqrt[3]{1+a}-1)x^2 + (\sqrt{1+a}-1)x + (\sqrt[6]{1+a}-1) = 0 \) where a > -1. The \( \lim_{a \to 0^+} \alpha(a) \) and \( \lim_{a \to 0^+} \beta(a) \) are
(a) \( -\frac{5}{2} \) and 1
(b) \( -\frac{1}{2} \) and -1
(c) \( -\frac{7}{2} \) and 2
(d) \( -\frac{9}{2} \) and 3
Answer: (b) \( -\frac{1}{2} \) and -1

Question. Which of the following is differentiable at x = 0 ? 
(a) \( \cos(|x|) + |x| \)
(b) \( \cos(|x|) - |x| \)
(c) \( \sin(|x|) + |x| \)
(d) \( \sin(|x|) - |x| \)
Answer: (b) \( \cos(|x|) - |x| \)

Question. For the function \( f(x) = \begin{cases} \frac{x}{1+e^{1/x}}, & x \neq 0 \\ 0, & x = 0 \end{cases} \), the derivative from the right, \( f'(0^+) \) .............. and the derivative from the left, \( f'(0^-) \) are
(a) {0, 1}
(b) {1, 0}
(c) {1, 1}
(d) {0, 0}
Answer: (a) {0, 1}

Question. Let \( f : [0, \pi/2] \to R \) be a function defined by \( f(x) = \max \{ \sin x, \cos x, 3/4 \} \), then number of points where f(x) is non differentiable is
(a) 1
(b) 2
(c) 3
(d) 0
Answer: (b) 2

Question. Let \( f(x) = \begin{cases} \frac{1}{|x|} & \text{for } |x| \geq 1 \\ ax^2 + b & \text{for } |x| < 1 \end{cases} \). If f(x) is continuous and differentiable everywhere, then
(a) \( a = \frac{1}{2}, b = -\frac{3}{2} \)
(b) \( a = -\frac{1}{2}, b = \frac{3}{2} \)
(c) a = 1, b = -1
(d) a = b = 1
Answer: (b) \( a = -\frac{1}{2}, b = \frac{3}{2} \)

Question. Let \( f(x) = [3 + 2 \cos x], x \in (-\pi/2, \pi/2) \), where [.] denotes the greatest integer function. Then number of points of discontinuity of f(x) is
(a) 3
(b) 2
(c) 5
(d) 6
Answer: (b) 2

Question. Let f(x) be a real function not identically zero in R, such that \( f(x+y^{2n+1}) = f(x) + \{f(y)\}^{2n+1}, n \in N \) and \( x, y \in R \). If \( f'(0) \geq 0 \), then f'(6) is equal to
(a) 0
(b) 1
(c) -1
(d) 2
Answer: (b) 1

Question. A function f : R → R satisfies the equation \( f(x) f(y) - f(xy) = x + y, \forall x, y \in R \) and f(1) > 0, then
(a) \( f(x) f^{-1}(x) = x^2 - 4 \)
(b) \( f(x) f^{-1}(x) = x^2 - 6 \)
(c) \( f(x) f^{-1}(x) = x^2 - 1 \)
(d) \( f(x) f^{-1}(x) = x^2 + 6 \)
Answer: (c) \( f(x) f^{-1}(x) = x^2 - 1 \)

Question. If \( f(x) = \frac{x^2-1}{x^2+1} \), for every real number x, then the minimum value of 
(a) does not exist because f is unbounded
(b) is not attained even though f is bounded
(c) is equal to 1
(d) is equal to -1
Answer: (d) is equal to -1

Question. The function \( f(x) = [x]^2 - [x^2] \) (where [x] is the greatest integer less than or equal to x), is discontinuous at : 
(a) all integers
(b) all integers except 0 and 1
(c) all integers except 0
(d) all integers except 1
Answer: (b) all integers except 0 and 1

Question. The left hand derivative of \( f(x) = [x] \sin(\pi x) \) at x = k, k an integer ([.] denotes G.I.F) is 
(a) \( (-1)^k(k-1)\pi \)
(b) \( (-1)^{k-1}(k-1)\pi \)
(c) \( (-1)^k k\pi \)
(d) \( (-1)^{k-1} k\pi \)
Answer: (a) \( (-1)^k(k-1)\pi \)

Question. Let f : R → R be any function. Define g : R → R by g(x) = |f(x)| for all x. Then g is
(a) onto if f is onto
(b) one - one if f is one - one
(c) continuous if f is continuous
(d) differentiable if f is differentiable
Answer: (c) continuous if f is continuous

Question. If \( f(x) = \frac{x}{1+x} + \frac{x}{(x+1)(2x+1)} + \frac{x}{(2x+1)(3x+1)} + \dots \infty \), then at x = 0, f(x)
(a) has no limit
(b) is discontinuous
(c) is continuous but not differentiable
(d) is differentiable
Answer: (b) is discontinuous

Question. The domain of the derivative of the function \( f(x) = \begin{cases} \tan^{-1} x & \text{if } |x| \leq 1 \\ \frac{1}{2}(|x|-1) & \text{if } |x| > 1 \end{cases} \) is 
(a) R - {0}
(b) R - {1}
(c) R - {-1}
(d) R - {-1, 1}
Answer: (d) R - {-1, 1}

Question. Let f(x) be a strictly increasing and differentiable function, then \( \lim_{x \to 0} \frac{f(x^2) - f(x)}{f(x) - f(0)} \), equals
(a) 2
(b) 1
(c) -1/2
(d) -1
Answer: (d) -1

Question. Let f(x) = | |x| - 1 | , then points where f(x) is differentiable is (are)
(a) \( 0, \pm 1 \)
(b) \( \pm 1 \)
(c) 0
(d) 1
Answer: (a) \( 0, \pm 1 \)

Question. If f is differentiable function satisfying \( f(1/n) = 0 \) for all \( n \geq 1, n \in I \), then 
(a) \( f(x) = f'(x) = 0, x \in (0, 1] \)
(b) \( f'(0) = 0 = f(0) \)
(c) \( f(0) = 0 \) but \( f'(0) \) not necessarily zero
(d) \( |f(x)| \leq 1, x \in (0, 1] \)
Answer: (b) \( f'(0) = 0 = f(0) \)

Question. If y = f(x) and \( y \cos x + x \cos y = \pi \) for \( x \in (-r, r) \), where r > 0, then the value of f''(0) is 
(a) \( \pi \)
(b) \( -\pi \)
(c) 0
(d) \( 2\pi \)
Answer: (a) \( \pi \)

Question. f''(x) = - f(x) where f(x) is a continuous double differentiable function & \( g(x) = f'(x) \). If \( F(x) = \left[f\left(\frac{x}{2}\right)\right]^2 + \left[g\left(\frac{x}{2}\right)\right]^2 \) and F(5) = 5, then F(10) is
(a) 0
(b) 5
(c) 10
(d) 25
Answer: (b) 5

Question. If f is a periodic function with period T & [.] denotes GIF, then, \( \lim_{n \to \infty} \frac{\left[f(x+T)\right]^{f(x+T)} + \left[2^2 f(x+2T)\right]^{f(x+2T)} + \left[3^2 f(x+3T)\right]^{f(x+3T)} + \dots + \left[n^2 f(x+nT)\right]^{f(x+nT)}}{n^3} \)
(a) \( \frac{(f(x))^{f(x)}}{2} \)
(b) \( \frac{(f(x))^{f(x)}}{4} \)
(c) \( \frac{2(f(x))^{f(x)}}{3} \)
(d) \( \frac{(f(x))^{f(x)}}{3} \)
Answer: (d) \( \frac{(f(x))^{f(x)}}{3} \)

Question. Let \( g(x) = \frac{(x-1)^n}{\log \cos^m(x-1)} \); 0 < x < 2, m and n are integers, \( m \neq 0, n > 0 \), and let p be the left hand derivative of |x - 1| at x = 1. If \( \lim_{x \to 1^+} g(x) = p \), then 
(a) n = 1, m = 1
(b) n = 1, m = -1
(c) n = 2, m = 2
(d) n > 2, m = n
Answer: (c) n = 2, m = 2

Question. Number of points, where the function \( f(x) = \text{Max} \{ \text{sgn}(x), -\sqrt{(9-x^2)}, x^3 \} \) is continuous but not differentiable is:
(a) 6
(b) 5
(c) 4
(d) 3
Answer: (b) 5

Question. The value of K such that the function \( f(x) = |x^2 + (k-1)|x| - k| \) is non-differentiable at exactly five points is
(a) 3
(b) 4
(c) -2
(d) 5
Answer: (a) 3

Question. A function f : R → R is defined as \( f(x) = \lim_{n \to \infty} \frac{ax^2 + bx + c + e^{nx}}{1 + c \cdot e^{nx}} \), where f is continuous on R. the values of a, b and c are
(a) c = 1, a, b ∈ R
(b) c = 0, a, b ∈ R
(c) a = 0, b, c ∈ R
(d) c = 3, a, b ∈ R
Answer: (a) c = 1, a, b ∈ R

Question. Let \( f(x) = \frac{\sin^{-1}(1-\{x\}) \cdot \cos^{-1}(1-\{x\})}{\sqrt{2\{x\}} \cdot (1-\{x\})} \), where {x} denotes the fractional part of x. Then
(a) \( \lim_{x \to 0^+} f(x) = \infty \)
(b) \( \lim_{x \to 0^-} f(x) = \frac{\pi}{2\sqrt{2}} \)
(c) \( \lim_{x \to 0^+} f(x) = -\frac{\pi}{2} \)
(d) \( \lim_{x \to 0^-} f(x) = 0 \)
Answer: (b) \( \lim_{x \to 0^-} f(x) = \frac{\pi}{2\sqrt{2}} \)

Question. If \( \lim_{x \to 0} \frac{a \sin x - bx + cx^2 + x^3}{2x^2 \log_e(1+x) - 2x^3 + x^4} \) exists and is finite then
(a) a = 6
(b) b = 6
(c) a(b + c) = 36
(d) c = 2
Answer: (a), (b), (c)

Question. Let f(x) = [x] and \( g(x) = \begin{cases} 0, & x \in I \\ x^2, & x \notin I \end{cases} \), then
(a) \( \lim_{x \to 1} g(x) \) exists, but g(x) is not continuous at x = 1
(b) \( \lim_{x \to 1} f(x) \) does not exist and f(x) is not continuos at x = 1
(c) gof is continuous for all x
(d) fog is continuous for all x
Answer: (a), (b), (c)

Question. Which of the following function(s) has/have removable discontinuuity at x = 1?
(a) \( f(x) = \frac{1}{\ln |x|} \)
(b) \( f(x) = \frac{x^2 - 1}{x^3 - 1} \)
(c) \( f(x) = 2^{-2^{1/(1-x)}} \)
(d) \( f(x) = \frac{\sqrt{x+1}-\sqrt{2x}}{x^2 - x} \)
Answer: (a), (b), (d)

Question. The function \( f(x) = ||2x - 3| - 10| \) is non differentiable at
(a) \( x \in \left\{ \frac{-7}{2}, \frac{13}{2} \right\} \)
(b) \( x \in \left\{ \frac{-7}{2}, \frac{13}{2}, \frac{3}{2} \right\} \)
(c) \( x \in \left\{ \frac{3}{2} \right\} \)
(d) \( x \in \left\{ -\frac{3}{2} \right\} \)
Answer: (b) \( x \in \left\{ \frac{-7}{2}, \frac{13}{2}, \frac{3}{2} \right\} \)