Class 11 Mathematics Limits And Derivatives MCQs Set 09

Question. A point is moving along the curve \( y^3 = 27x \). Then the interval of values of x in which the ordinate changes faster than abscissa is
(a) \( x \in (-1, 1) \)
(b) \( x \in (-1, 1) - \{0\} \)
(c) \( x \in [-1, 1] - \{0\} \)
(d) \( x \in (-1, 1] \)
Answer: (b) \( x \in (-1, 1) - \{0\} \)

Question. An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long ?
(a) \( 100 \text{ cm}^3/\text{sec} \)
(b) \( 900 \text{ cm}^3/\text{sec} \)
(c) \( 800 \text{ cm}^3/\text{sec} \)
(d) \( 1200 \text{ cm}^3/\text{sec} \)
Answer: (b) \( 900 \text{ cm}^3/\text{sec} \)

Question. Let \( f(x) = \begin{cases} -x^2, & \text{for } x < 0 \\ x^2 + 8, & \text{for } x \ge 0 \end{cases} \). Then x intercept of the line that is tangent to the graph of \( f(x) \) is
(a) zero
(b) – 1
(c) –2
(d) – 4
Answer: (c) –2

Question. There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per additional tree drops by 10 apples. Number of trees that should be added to the existing orchard for maximising the output of the trees, is
(a) 5
(b) 10
(c) 15
(d) 20
Answer: (c) 15

Question. The ordinate of all points on the curve \( y = \frac{1}{2\sin^2 x + 3\cos^2 x} \) where the tangent is horizontal, is
(a) always equal to 1/2
(b) always equal to 1/3
(c) 1/2 or 1/3 according as n is an even or an odd integer.
(d) 1/2 or 1/3 according as n is an odd or an even integer.
Answer: (c) 1/2 or 1/3 according as n is an even or an odd integer.

Multi Answer Questions

Question. Which of the following statement(s) is/are TRUE?
Statement A: If function \( y = f(x) \) is continuous at \( x = c \) such that \( f(c) \neq 0 \) then \( f(x) f(c) > 0 \, \forall \, x \in (c-h, c+h) \) where \( h \) is sufficiently small positive quantity.
Statement B: \( \lim_{n \to \infty} \frac{1}{n} \ln \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \dots \left(1+\frac{n}{n}\right) \right] = 1 + 2\ln 2 \)
Statement C: Let \( f \) be a continuous and non-negative function defined on \( [a, b] \). If \( \int_a^b f(x) \, dx = 0 \) then \( f(x) = 0 \, \forall \, x \in [a, b] \).
Statement D: Let \( f \) be a continuous function defined on \( [a, b] \) such that \( \int_a^b f(x) \, dx = 0 \), then there exists at least one \( c \in (a, b) \) for which \( f(c) = 0 \).
(a) Statement A
(b) Statement B
(c) Statement C
(d) Statement D
Answer: (a) Statement A
(c) Statement C
(d) Statement D

Question. If \( a, b, c \in R^+ \) then \( \lim_{n \to \infty} \sum_{k=1}^n \frac{n}{(k + an)(k + bn)} \) is equal to
(a) \( \frac{1}{a-b} \ln \frac{b(b+1)}{a(a+1)} \) if \( a \neq b \)
(b) \( \frac{1}{a-b} \ln \frac{a(b+1)}{b(a+1)} \) if \( a \neq b \)
(c) non existent if \( a = b \)
(d) equals \( \frac{1}{a(a+1)} \) if \( a = b \)
Answer: (b) \( \frac{1}{a-b} \ln \frac{a(b+1)}{b(a+1)} \) if \( a \neq b \)
(d) equals \( \frac{1}{a(a+1)} \) if \( a = b \)

Question. Let \( f(x) = \begin{cases} \int_0^x (1 + |1-t|) \, dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases} \), then
(a) \( f(x) \) is not continuous at \( x = 2 \)
(b) \( f(x) \) is continuous but not differentiable at \( x = 2 \)
(c) \( f(x) \) is differentiable everywhere
(d) the right derivative of \( f(x) \) at \( x = 2 \) does not exist
Answer: (a) \( f(x) \) is not continuous at \( x = 2 \)
(d) the right derivative of \( f(x) \) at \( x = 2 \) does not exist

Question. If \( f'(x) = f(x) + \int_{-10}^{10} f(x) \, dx \), then
(a) \( f'(x) = f(x) \) if \( \lim_{x \to \infty} f(x) = 0 \)
(b) \( f'(x) - f(x) = c \) (any constant)
(c) \( f''(x) - f'(x) = c \) (any constant)
(d) None of the options
Answer: (a) \( f'(x) = f(x) \) if \( \lim_{x \to \infty} f(x) = 0 \)
(b) \( f'(x) - f(x) = c \) (any constant)
(c) \( f''(x) - f'(x) = c \) (any constant)

Question. Let \( f(x) \) be a non–constant twice differentiable function defined on \( (-\infty, \infty) \) such that \( f(x) = f(1-x) \) and \( f'\left(\frac{1}{4}\right) = 0 \). Then, 
(a) \( f''(x) \) vanishes at least twice on \( [0, 1] \)
(b) \( f'\left(\frac{1}{2}\right) = 0 \)
(c) \( \int_{-1/2}^{1/2} f\left(x + \frac{1}{2}\right) \sin x \, dx = 0 \)
(d) \( \int_0^{1/2} f(t) e^{\sin \pi t} \, dt = \int_{1/2}^1 f(1-t) e^{\sin \pi t} \, dt \)
Answer: (a) \( f''(x) \) vanishes at least twice on \( [0, 1] \)
(b) \( f'\left(\frac{1}{2}\right) = 0 \)
(c) \( \int_{-1/2}^{1/2} f\left(x + \frac{1}{2}\right) \sin x \, dx = 0 \)
(d) \( \int_0^{1/2} f(t) e^{\sin \pi t} \, dt = \int_{1/2}^1 f(1-t) e^{\sin \pi t} \, dt \)

Question. If the line \( ax + by + c = 0 \) is a normal to the rectangular hyperbola \( xy = 1 \), then
(a) \( a > 0, b > 0 \)
(b) \( a > 0, b < 0 \)
(c) \( a < 0, b > 0 \)
(d) \( a < 0, b < 0 \)
Answer: (b) \( a > 0, b < 0 \) and (c) \( a < 0, b > 0 \)

Question. Let \( f(x) = 2\sin^3 x - 3\sin^2 x + 12\sin x + 5, 0 \le x \le \pi/2 \). Then f(x) is
(a) decreasing in \( [0, \pi / 2] \)
(b) increasing in \( [0, \pi / 2] \)
(c) increasing in \( [0, \pi / 4] \) and decreasing in \( [\pi / 4, \pi / 2] \)
(d) none of the options
Answer: (b) increasing in \( [0, \pi / 2] \)

Question. Let \( f(x) = \frac{x^2 + 1}{[x]}, 1 \le x \le 3.9 \). [.] denotes the greatest integer function. Then
(a) f(x) is monotonically decreasing in [1, 3.9]
(b) f(x) is monotonically increasing in [1, 3.9]
(c) the greatest value of f(x) is \( \frac{1}{3} \times 16.21 \)
(d) the least value of f(x) is 2.
Answer: (c) the greatest value of f(x) is \( \frac{1}{3} \times 16.21 \)

Question. Let \( h(x) = f(x) - (f(x))^2 + (f(x))^3 \) for every real number x. Then
(a) h is increasing whenever f is increasing
(b) h is increasing whenever f is decreasing
(c) h is decreasing whenever f is decreasing
(d) nothing can be said in general
Answer: (a) h is increasing whenever f is increasing and (c) h is decreasing whenever f is decreasing

Question. Let the parabolas \( y = x^2 + ax + b \) and \( y = x(c - x) \) touch each other at a point (1, 0). Then
(a) \( a = - 3 \)
(b) \( b = 1 \)
(c) \( c = 2 \)
(d) \( b + c = 3 \)
Answer: (a) \( a = - 3 \) and (d) \( b + c = 3 \)

Question. A point on the ellipse \( 4x^2 + 9y^2 = 36 \) where the tangent is equally inclined to the axes is
(a) \( \left( \frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right) \)
(b) \( \left( \frac{9}{\sqrt{13}}, -\frac{4}{\sqrt{13}} \right) \)
(c) \( \left( -\frac{9}{\sqrt{13}}, \frac{4}{\sqrt{13}} \right) \)
(d) \( \left( -\frac{4}{\sqrt{13}}, -\frac{9}{\sqrt{13}} \right) \)
Answer: (a), (b), (c)

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

Question. Let \( f(x) = x^3 + ax^2 + bx + 5 \sin^2 x \) be an increasing function in the set of real numbers R. Then a and b satisfy the condition
(a) \( a^2 - 3b - 15 \le 0 \)
(b) \( a^2 - 3b + 15 \ge 0 \)
(c) \( a^2 - 3b + 15 \le 0 \)
(d) \( a > 0 \) and \( b > 0 \)
Answer: (c) \( a^2 - 3b + 15 \le 0 \)

Question. Let \( f(x) = (x-1)^4 \cdot (x-2)^n, n \in N \). Then f(x) has
(a) a local maximum at x = 1 if n is odd
(b) a local maximum at x = 1 if n is even
(c) a local minimum at x = 2 if n is even
(d) a local maximum at x = 2 if n is odd
Answer: (a) and (c)

Question. Let \( f(x) = \begin{cases} x^3 + x^2 - 10x & -1 \le x < 0 \\ \sin x & 0 \le x < \frac{\pi}{2} \\ 1 + \cos x & \frac{\pi}{2} \le x \le \pi \end{cases} \) then f(x) has
(a) local maxima at \( x = \frac{\pi}{2} \)
(b) local minima at \( x = \frac{\pi}{2} \)
(c) absolute maxima at x = 0
(d) absolute maxima at x = –1
Answer: (b) local minima at \( x = \frac{\pi}{2} \)

Question. If \( f(x) = \sin x, -\pi / 2 \le x \le \pi / 2 \), then
(a) f(x) is increasing in the interval \( [-\pi / 2, \pi / 2] \)
(b) f{f(x)} is increasing in the interval \( [-\pi / 2, \pi / 2] \)
(c) f{f(x)} is decreasing in \( [-\pi / 2, 0] \) and increasing in \( [0, \pi / 2] \)
(d) f{f(x)} is invertible in \( [-\pi / 2, \pi / 2] \)
Answer: (a), (b) and (d)

Question. If \( f(x) = x^4 - 4x - 1 \), then
(a) f ( x ) has exactly one positive real root
(b) f ( x ) has exactly one negative real root
(c) f ( x ) has exactly two real roots
(d) minimum value of f ( x ) = -3
Answer: (a), (b) and (c)

Question. The extremum values of the function \( f(x) = \frac{1}{\sin x + 4} - \frac{1}{\cos x - 4} \), where \( x \in R \) is
(a) \( \frac{4}{8 - \sqrt{2}} \)
(b) \( \frac{2\sqrt{2}}{8 - \sqrt{2}} \)
(c) \( \frac{2\sqrt{2}}{4\sqrt{2} + 1} \)
(d) \( \frac{4\sqrt{2}}{8 + \sqrt{2}} \)
Answer: (b) \( \frac{2\sqrt{2}}{8 - \sqrt{2}} \)

Question. If \( f''(x) < 0 \forall x \in R \) and \( g(x) = f(x^2 - 2) + f(6 - x^2) \) then
(a) g(x) is an increasing in [0, 2]
(b) g(x) is an increasing in [2, \(\infty\))
(c) g(x) has a local minima at x = –2
(d) g(x) has a local maxima at x = 2
Answer: (a) and (d)

Question. The function \( f(x) = \int_{-1}^{x} t(e^t - 1)(t - 1)(t - 2)^3(t - 3)^5 dt \) has a local minimum at x =
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1 and (d) 3

Question. The function \( y = \frac{2x - 1}{x - 2} \) (x ≠ 2)
(a) is its own inverse
(b) decreases for all values of x
(c) has a graph entirely above x-axis
(d) is bound for all x.
Answer: (a) is its own inverse and (b) decreases for all values of x

Question. The function \( \frac{\sin(x + a)}{\sin(x + b)} \) has no maxima or minima if
(a) \( b - a = n\pi, n \in I \)
(b) \( b - a = (2n + 1)\pi, n \in I \)
(c) \( b - a = 2n\pi, n \in I \)
(d) \( b - a = (2n + 1)\frac{\pi}{2}, n \in I \)
Answer: (a) \( b - a = n\pi, n \in I \)

Question. Let \( f(x) = \lim_{n \to \infty} \sum_{r=1}^n 3^{r-1} \sin^3 \left(\frac{x}{3^r}\right) \) and \( g(x) = x - 4f(x) \) then \( \lim_{x \to 0} (1 + g(x))^{\cot x} \) is
(a) \( \frac{1}{e} \)
(b) \( e^2 \)
(c) \( e \)
(d) 1
Answer: (c) \( e \)

Question. Let \( f(x) \) be a real valued function defined on the interval \( (0, \infty) \) by \( f(x) = \ln x + \int_0^x \sqrt{1 + \sin t} \, dt \). Then which of the following statements is (are) true? [IIT-2010]
(a) \( f''(x) \) exists for all \( x \in (0, \infty) \)
(b) \( f'(x) \) exists for all \( x \in (0, \infty) \) and \( f'(x) \) is continuous on \( (0, \infty) \) but not differentiable on \( (0, \infty) \)
(c) There exists \( \alpha > 1 \) such that \( |f'(x)| < |f(x)| \) for all \( x \in (\alpha, \infty) \)
(d) There exists \( \beta > 0 \) such that \( |f(x)| + |f'(x)| \leq \beta \)
Answer: (b) \( f'(x) \) exists for all \( x \in (0, \infty) \) and \( f'(x) \) is continuous on \( (0, \infty) \) but not differentiable on \( (0, \infty) \)

Question. Let \( f(x) = \text{degree of } (ux^2 + u^2 + 2u - 3) \) (where \( u \) is a real parameter, \( x \) is variable) then
(a) \( f(x) \) is continuous at \( x = \sqrt{2} \) but not differentiable
(b) \( f(x) \) is neither differentiable nor continuous at \( x = \sqrt{2} \)
(c) \( f(x) \) is discontinuous at \( x = \sqrt{2} \)
(d) \( f(x) \) is differentiable at \( x = \sqrt{2} \)
Answer: (a) \( f(x) \) is continuous at \( x = \sqrt{2} \) but not differentiable

Question. Given \( f(x) = \left\{ \log_a (a [x] + [-x])^x \left[ \frac{2 a^{\frac{1}{3 + a^{1/|x|}}}}{\frac{[x] + [-x]}{|x|}} \right]^{-5} \right\} \); \( x \neq 0, a > 1 \) and \( f(x) = 0 \) for \( x = 0 \) (where [.] is G.I.F) then
(a) \( f \) is continuous but not differentiable at \( x = 0 \)
(b) \( f \) is continuous and differentiable at \( x = 0 \)
(c) The differentiability of \( f \) at \( x = 0 \) depends on the value of \( a \)
(d) \( f \) is continuous and differentiable at \( x = 0 \) and for \( a = e \) only
Answer: (b) \( f \) is continuous and differentiable at \( x = 0 \)

Question. Let \( f(0) = 0 \) and \( f'(0) = 1 \). For a positive integer \( K \), \( \lim_{x \to 0} \frac{1}{x} \left( f(x) + f\left(\frac{x}{2}\right) + \dots + f\left(\frac{x}{k}\right) \right) \) is
(a) \( \frac{k(k+1)}{2} \)
(b) \( \frac{k+1}{2k^2} \)
(c) \( 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{k} \)
(d) \( \frac{1}{2} \)
Answer: (c) \( 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{k} \)

Question. Let \( f : [a, b] \to [1, \infty) \) be a continuous function and \( g : R \to R \) be defined as
\( g(x) = \begin{cases} 0, & \text{if } x < a \\ \int_a^x f(t) \, dt, & \text{if } a \leq x \leq b \\ \int_a^b f(t) \, dt, & \text{if } x > b \end{cases} \) Then,
(a) \( g(x) \) is continuous but not differentiable at \( a \)
(b) \( g(x) \) is differentiable on \( R \)
(c) \( g(x) \) is continuous but not differentiable at \( b \)
(d) \( g(x) \) is continuous and differentiable at either \( a \) or \( b \) but not both
Answer: (d) \( g(x) \) is continuous and differentiable at either \( a \) or \( b \) but not both

COMPREHENSION TYPE

Passage : L’ Hospital’s rule has many versions one them is this suppose \( f, g : (a, b) \to R \) are differentiable on \( (a, b) \). Suppose further that
i) \( g'(x) \neq 0 \) for \( x \in (a, b) \)
ii) \( \lim_{x \to a^+} g(x) \to \infty (\text{or } -\infty) \)
iii) \( \lim_{x \to a^+} \frac{f'(x)}{g'(x)} = L \) then \( \lim_{x \to a^+} \frac{f(x)}{g(x)} = L \)
(This rule can be extended to cover the case where \( a \) and \( b \) tends to infinity or \( L \) tends to infinity)

Question. Let ‘f’ be a differentiable function on \( (0, \infty) \).
If \( \lim_{x \to \infty} \left( \sin\left(\frac{\pi}{10}\right) f(x) + f'(x) \right) = \sec\left(\frac{\pi}{5}\right) \)
then \( \lim_{x \to \infty} f(x) \) equals
(a) \( \frac{1}{4} \)
(b) 4
(c) \( 3 - \sqrt{5} \)
(d) \( \frac{3 + \sqrt{5}}{4} \)
Answer: (b) 4

Question. Let ‘f’ be a differentiable function on \( (0, \infty) \).
If \( \lim_{x \to \infty} \left( \tan\left(\frac{\pi}{8}\right) f(x) + 2\sqrt{x} f'(x) \right) = \cot\left(\frac{\pi}{12}\right) \)
then \( \lim_{x \to \infty} f(x) \) equals
(a) \( \sqrt{8} - \sqrt{6} + \sqrt{4} - \sqrt{3} \)
(b) \( \sqrt{8} + \sqrt{6} - \sqrt{4} - \sqrt{3} \)
(c) \( \sqrt{3} + \sqrt{4} + \sqrt{6} + \sqrt{8} \)
(d) \( \sqrt{8} - \sqrt{6} - \sqrt{4} + \sqrt{3} \)
Answer: (c) \( \sqrt{3} + \sqrt{4} + \sqrt{6} + \sqrt{8} \)

Passage : Let ‘f’ be a polynomial function satisfying
\( f(x).f(y) + 2 = f(x) + f(y) + f(xy) \quad \forall x, y \in R^+ \cup \{0\} \) and \( f(x) \) is one-one with \( f(0) = 1 \) and \( f'(1) = 2 \) then answer the following

Question. The function \( y = f(x) \) is given by
(a) \( x^2 - 1 \)
(b) \( x^{1/3} - 1 \)
(c) \( 1 + \frac{2x^3}{3} \)
(d) \( 1 + x^2 \)
Answer: (d) \( 1 + x^2 \)

Question. The number of points of non-differentiability of \( h(x) = \min \left\{ \frac{2}{f(x)}, x^2, |1 - |x|| \right\}, x \geq 0 \) is
(a) 6
(b) 5
(c) 3
(d) 4
Answer: (c) 3

Question. Match the following :
Column I
A) \( f(x) = \begin{cases} \frac{\left( \frac{\pi}{2} - \sin^{-1}(1 - \{x\}^2) \right) (\sin^{-1}(1 - \{x\}))}{\sqrt{2} (\{x\} - \{x\}^3)}, & x > 0 \\ k, & x = 0 \\ \frac{A \sin^{-1}(1 - \{x\}) \cos^{-1}(1 - \{x\})}{\sqrt{2 \{x\}} (1 - \{x\})}, & x < 0 \end{cases} \)
is continuous at \( x = 0 \) then the value of \( \sin^2 k + \cos^2 \left(\frac{A\pi}{\sqrt{2}}\right) \) is
B) \( f(x) = [2 + 5n \sin x] \), \( n \in Z \) has exactly 19 points of non differentiability in \( x \in (0, \pi) \) then possible values of \( n \) are
C) If \( f(x) = \begin{cases} x \cdot \frac{(3/4)^{1/x} - (3/4)^{-1/x}}{(3/4)^{1/x} + (3/4)^{-1/x}}, & x \neq 0 \\ 0, & x = 0 \end{cases} \)
if \( P = f'(0^-) - f'(0^+) \) then \( \lim_{x \to P^-} \frac{( \exp((x+2) \ln 4) - 16 ) 4^{\frac{1}{[x+1]}}}{4^x - 16} \) is \(\leq\)
Column II
P) 2
Q) –2
R) 3
S) –3
Answer: A-P, B- P, Q, C-Q, R