Class 11 Mathematics Limits And Derivatives MCQs Set 06

Question. Let \( f : [2,7] \to [0, \infty) \) be a continuous and differentiable function. Then, the value of \( \frac{(f(7) - f(2)) \left( \frac{(f(7))^2 + (f(2))^2 + f(2) \cdot f(7)}{3} \right)}{?} \) is (where \( c \in (2,7) \))
(a) \( 3f^2(c) f'(c) \)
(b) \( 5f^2(c) \cdot f(c) \)
(c) \( 5f^2(c) \cdot f'(c) \)
(d) None of the options
Answer: (c) \( 5f^2(c) \cdot f'(c) \)
 

Question. The tangent to the curve \( y = e^x \) drawn at the point \( (c, e^c) \) intersects the line joining the points \( (c - 1, e^{c-1}) \) and \( (c + 1, e^{c+1}) \)
(a) on the left of \( x = c \)
(b) on the right of \( x = c \)
(c) at no point
(d) at all points
Answer: (c) at no point

Question. If \( f \) is continuous function in \( [1, 2] \) such that \( |f(1) + 3| < |f(1)| + 3 \) and \( |f(2) + 10| = |f(2)| + 10 \), (\( f(2) \neq 0 \)), then the function \( f \) in \( (1, 2) \) has
(a) at least one root
(b) no root
(c) exactly one root
(d) none of the options
Answer: (a) at least one root

Question. The function ‘g’ defined by \( g(x) = f(x^2 - 2x + 8) + f(14 + 2x - x^2) \), where \( f(x) \) is twice differentiable function, \( f''(x) \geq 0 \) for all real numbers \( x \). The function \( g(x) \) is increasing in the interval
(a) \( [-1, 1] \cup [2, \infty) \)
(b) \( (-\infty, -1] \cup [1, 3] \)
(c) \( [-1, 1] \cup [3, \infty) \)
(d) \( (-\infty, -2] \cup [1, \infty) \)
Answer: (b) \( (-\infty, -1] \cup [1, 3] \)

Question. The length of a longest interval in which the function \( 3 \sin x - 4 \sin^3 x \) is increasing, is
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{3\pi}{2} \)
(d) \( \pi \)
Answer: (a) \( \frac{\pi}{3} \)

Question. In \( [0, 1] \) Lagrange's Mean Value theorem is not applicable to
(a) \( f(x) = \begin{cases} \frac{1}{2} - x & x < \frac{1}{2} \\ \left( \frac{1}{2} - x \right)^2 & x \geq \frac{1}{2} \end{cases} \)
(b) \( f(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0 \\ 1 & x = 0 \end{cases} \)
(c) \( f(x) = x|x| \)
(d) \( f(x) = |x| \)
Answer: (a) \( f(x) = \begin{cases} \frac{1}{2} - x & x < \frac{1}{2} \\ \left( \frac{1}{2} - x \right)^2 & x \geq \frac{1}{2} \end{cases} \)

Question. Tangent is drawn to ellipse \( \frac{x^2}{27} + y^2 = 1 \) at \( (3\sqrt{3} \cos \theta, \sin \theta) \) (where \( \theta \in (0, \pi/2) \)). Then the value of \( \theta \) such that sum of intercepts on axes made by this tangent is minimum, is 
(a) \( \frac{\pi}{3} \)
(b) \( \frac{\pi}{6} \)
(c) \( \frac{\pi}{8} \)
(d) \( \frac{\pi}{4} \)
Answer: (d) \( \frac{\pi}{4} \)

Question. If the tangent at \( (x_1, y_1) \) to the curve \( x^3 + y^3 = a^3 \) meets the curve again at \( (x_2, y_2) \) then
(a) \( \frac{x_2}{x_1} + \frac{y_2}{y_1} = -1 \)
(b) \( \frac{x_2}{y_1} + \frac{x_1}{y_2} = -1 \)
(c) \( \frac{x_1}{x_2} + \frac{y_1}{y_2} = -1 \)
(d) \( \frac{x_2}{x_1} + \frac{y_2}{y_1} = 1 \)
Answer: (a) \( \frac{x_2}{x_1} + \frac{y_2}{y_1} = -1 \)

Question. If \( 0 < a < b < \frac{\pi}{2} \) and \( f(a,b) = \frac{\tan b - \tan a}{b - a} \), Then
(a) \( f(a,b) \geq 2 \)
(b) \( f(a,b) > 1 \)
(c) \( f(a,b) \leq 1 \)
(d) None of the options
Answer: (b) \( f(a,b) > 1 \)

Question. \( f(x) \) is a polynomial of degree 4 with real coefficients such that \( f(x) = 0 \) is satisfied by \( x = 1, 2, 3 \) only, then \( f'(1) \cdot f'(2) \cdot f'(3) \) is equal to
(a) 0
(b) 2
(c) -1
(d) None of the options
Answer: (a) 0

Question. If \( f(x) \) is a polynomial of degree 5 with real coefficients such that \( f(|x|) = 0 \) has 8 real roots, then \( f(x) = 0 \) has
(a) 4 real roots
(b) 5 real roots
(c) 3 real roots
(d) nothing can be said
Answer: (b) 5 real roots

Question. If the function \( f(x) = |x^2 + a|x| + b| \) has exactly three points of non-differentiability, then which of the following can be true?
(a) \( b = 0, a < 0 \)
(b) \( b < 0, a \in \mathbb{R} \)
(c) \( b > 0, a \in \mathbb{R} \)
(d) All of the options
Answer: (a) \( b = 0, a < 0 \)

Question. Let \( f \) be continuous and differentiable function such that \( f(x) \) and \( f'(x) \) have opposite signs everywhere. Then
(a) \( f \) is increasing
(b) \( f \) is decreasing
(c) \( |f| \) is non-monotonic
(d) \( |f| \) is decreasing
Answer: (d) \( |f| \) is decreasing

Question. If \( f(x) = 4x^3 - x^2 - 2x + 1 \) and \( g(x) = \begin{cases} \min\{f(t) : 0 \leq t \leq x\}, & 0 \leq x \leq 1 \\ 3 - x, & 1 < x \leq 2 \end{cases} \) then \( g\left(\frac{1}{4}\right) + g\left(\frac{3}{4}\right) + g\left(\frac{5}{4}\right) \) has the value equal to
(a) 7/4
(b) 9/4
(c) 13/4
(d) 5/2
Answer: (d) 5/2

Question. The largest term in the sequence \( a_n = \frac{n^2}{n^3 + 200} \) is given by
(a) 529/49
(b) 8/89
(c) 49/543
(d) 89/8
Answer: (c) 49/543

Question. The number of values of \( k \) for which the equation \( x^3 - 3x + k = 0 \) has two distinct roots lying in the interval \( (0, 1) \) is
(a) three
(b) two
(c) infinitely many
(d) zero
Answer: (d) zero

Question. The maximum value of \( \left( \sqrt{-3 + 4x - x^2} + 4 \right)^2 + (x - 5)^2 \), (where \( 1 \leq x \leq 3 \)) is
(a) 34
(b) 36
(c) 32
(d) 20
Answer: (b) 36

Question. On the interval \( \left[ \frac{5\pi}{4}, \frac{4\pi}{3} \right] \) the least value of the function \( f(x) = \int_{5\pi/4}^x (3 \sin t + 4 \cos t) dt \) is
(a) \( \frac{3}{2} + \frac{1}{\sqrt{2}} - 2\sqrt{3} \)
(b) \( \frac{3}{2} - \frac{1}{\sqrt{2}} + 2\sqrt{3} \)
(c) \( \frac{3}{2} - \frac{1}{\sqrt{2}} - 2\sqrt{3} \)
(d) \( \frac{3}{2} - 2\sqrt{3} \)
Answer: (c) \( \frac{3}{2} - \frac{1}{\sqrt{2}} - 2\sqrt{3} \)

Question. The function \( f(x) = \frac{\ln(\pi + x)}{\ln(e + x)} \)
(a) increasing on \( (0, \infty) \)
(b) decreasing on \( (0, \infty) \)
(c) increasing on \( (0, \pi/e) \), decreasing on \( (\pi/e, \infty) \)
(d) decreasing on \( (0, \pi/e) \), increasing on \( (\pi/e, \infty) \)
Answer: (b) decreasing on \( (0, \infty) \)

Question. One corner of a long rectangular sheet of paper of width 1 unit is folded over so as to reach the opposite edge of the sheet. The minimum length of the crease is
(a) \( \frac{3\sqrt{3}}{4} \)
(b) \( \frac{3\sqrt{3}}{2} \)
(c) \( 4\sqrt{3} \)
(d) \( 3\sqrt{3} \)
Answer: (b) \( \frac{3\sqrt{3}}{2} \)

Question. Let \( S \) be a square of unit area. Consider any quadrilateral which has one vertex on each side of \( S \). If \( a, b, c \) and \( d \) denote the length of the sides of the quadrilateral, then \( a^2 + b^2 + c^2 + d^2 \) lies in
(a) \( [3, 5] \)
(b) \( [2, 4] \)
(c) \( [1, 3] \)
(d) \( [0, 2] \)
Answer: (b) \( [2, 4] \)

Question. The tangent lines for the curve \( y = \int_0^x |t| dt \) which are parallel to the bisector of the first coordinate angle, is given by
(a) \( y = x + \frac{3}{4}, y = x - \frac{1}{4} \)
(b) \( y = -x + \frac{1}{4}, y = -x + \frac{3}{4} \)
(c) \( x + y = 2, x - y = 1 \)
(d) \( y = x + \frac{1}{4}, y = x - \frac{1}{4} \)
Answer: (d) \( y = x + \frac{1}{4}, y = x - \frac{1}{4} \)

Question. If \( f(x) = x^3 + bx^2 + cx + d \) and \( 0 < b^2 < c \), then in \( (-\infty, \infty) \) 
(a) \( f(x) \) is a strictly increasing function
(b) \( f(x) \) has a local maxima
(c) \( f(x) \) is strictly decreasing function
(d) \( f(x) \) is bounded
Answer: (a) \( f(x) \) is a strictly increasing function

Question. If \( f(x) = x^\alpha \log x \) and \( f(0) = 0 \), then the value of \( \alpha \) for which Rolle’s theorem can be applied in \( [0, 1] \) is
(a) -2
(b) 0
(c) 1/2
(d) -1/2
Answer: (c) 1/2

Question. Consider the two curves: \( C_1 : y^2 = 4x \); \( C_2 : x^2 + y^2 - 6x + 1 = 0 \), Then
(a) \( C_1 \) and \( C_2 \) touch each other only at one point
(b) \( C_1 \) and \( C_2 \) touch each other exactly at two points
(c) \( C_1 \) and \( C_2 \) intersect (but do not touch) at exactly two points
(d) \( C_1 \) and \( C_2 \) neither intersect nor touch each other
Answer: (b) \( C_1 \) and \( C_2 \) touch each other exactly at two points

Question. The total number of local maxima and local minima of the function \( f(x) = \begin{cases} (2 + x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases} \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. Let the function \( g : (-\infty, \infty) \to \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) be given by \( g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} \). Then, \( g \) is
(a) even and is strictly increasing in \( (0, \infty) \)
(b) odd and is strictly decreasing in \( (-\infty, \infty) \)
(c) odd and is strictly increasing in \( (-\infty, \infty) \)
(d) neither even nor odd, but is strictly increasing in \( (-\infty, \infty) \)
Answer: (c) odd and is strictly increasing in \( (-\infty, \infty) \)

Question. Let \( f \) be a non-negative function on the interval \( [0, 1] \). If \( \int_0^x \sqrt{1 - (f'(t))^2} dt = \int_0^x f(t) dt \), \( 0 \leq x \leq 1 \), and \( f(0) = 0 \), then 
(a) \( f\left(\frac{1}{2}\right) < \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) > \frac{1}{3} \)
(b) \( f\left(\frac{1}{2}\right) > \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) > \frac{1}{3} \)
(c) \( f\left(\frac{1}{2}\right) < \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) < \frac{1}{3} \)
(d) \( f\left(\frac{1}{2}\right) > \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) < \frac{1}{3} \)
Answer: (c) \( f\left(\frac{1}{2}\right) < \frac{1}{2} \) and \( f\left(\frac{1}{3}\right) < \frac{1}{3} \)

Question. Interval in which \( \tan^{-1}(\sin x + \cos x) \) is increasing
(a) \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)
(b) \( \left( 0, \frac{\pi}{2} \right) \)
(c) \( \left( \frac{\pi}{8}, \frac{5\pi}{8} \right) \)
(d) \( \left( 0, \frac{\pi}{8} \right) \)
Answer: (d) \( \left( 0, \frac{\pi}{8} \right) \)

Question. The second degree polynomial \( f(x) \), satisfying \( f(0) = 0, f(1) = 1, f'(x) > 0 \) for all \( x \in (0, 1) \)
(a) \( f(x) = \phi \)
(b) \( f(x) = ax + (1 - a)x^2; \forall a \in (0, \infty) \)
(c) \( f(x) = ax + (1 - a)x^2, a \in (0, 2) \)
(d) no such polynomial
Answer: (c) \( f(x) = ax + (1 - a)x^2, a \in (0, 2) \)

Question. Let \( f, g, \) and \( h \) be real valued functions defined on the interval \( [0, 1] \) by \( f(x) = e^{x^2} + e^{-x^2}, g(x) = xe^{x^2} + e^{-x^2}, h(x) = x^2 e^{x^2} + e^{-x^2} \). If \( a, b \) and \( c \) denotes respectively, the absolute maximum of \( f, g \) and \( h \) on \( [0, 1] \) then
(a) \( a = b \) and \( c \neq b \)
(b) \( a = c \) and \( a \neq b \)
(c) \( a = b \) and \( c = b \)
(d) \( a = b = c \)
Answer: (d) \( a = b = c \)

Question. Consider the polynomial \( f(x) = 1 + 2x + 3x^2 + 4x^3 \). Let \( s \) be the sum of all distinct real roots of \( f(x) \) and let \( t = |s| \). The function \( f'(x) \) is
(a) Increasing in \( \left( -t, -\frac{1}{4} \right) \) and decreasing in \( \left( -\frac{1}{4}, t \right) \)
(b) Decreasing in \( \left( -t, -\frac{1}{4} \right) \) and increasing in \( \left( -\frac{1}{4}, t \right) \)
(c) Increasing in \( (-t, t) \)
(d) Decreasing in \( (-t, t) \)
Answer: (b) Decreasing in \( \left( -t, -\frac{1}{4} \right) \) and increasing in \( \left( -\frac{1}{4}, t \right) \)

Question. The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. The rate of change of the volume of the cylinder, in \( \text{cm}^3 / \text{min} \), when the radius is 2 cm and the height is 3 cm, is
(a) \( -2\pi \)
(b) \( -\frac{8\pi}{5} \)
(c) \( -\frac{3\pi}{5} \)
(d) \( \frac{2\pi}{5} \)
Answer: (d) \( \frac{2\pi}{5} \)

Question. Let \( f(x) = |x^2 - 3x - 4|, -1 \leq x \leq 4 \). Then
(a) \( f(x) \) is monotonically increasing in \( [-1, 3/2] \)
(b) \( f(x) \) is monotonically decreasing in \( (3/2, 4] \)
(c) the maximum value of \( f(x) \) is 25/4
(d) the minimum value of \( f(x) \) is 0.
Answer: (c) the maximum value of \( f(x) \) is 25/4 and (d) the minimum value of \( f(x) \) is 0.

Question. The critical point(s) of \( f(x) = \frac{|2-x|}{x^2} \) is/are
(a) \( x = 0 \)
(b) \( x = 2 \)
(c) \( x = 4 \)
(d) \( x = 1 \)
Answer: (b) \( x = 2 \) and (c) \( x = 4 \)

Question. If the tangent at any point \( P(4m^2, 8m^3) \) of \( x^3 - y^2 = 0 \) is also a normal to the curve \( x^3 - y^2 = 0 \), then the value of \( m \) is
(a) \( m = \frac{\sqrt{2}}{3} \)
(b) \( m = -\frac{\sqrt{2}}{3} \)
(c) \( m = \frac{3}{\sqrt{2}} \)
(d) \( m = -\frac{3}{\sqrt{2}} \)
Answer: (a) \( m = \frac{\sqrt{2}}{3} \) and (b) \( m = -\frac{\sqrt{2}}{3} \)

Question. The angle between the tangents at any point \( P \) and the line joining \( P \) to the origin, where \( P \) is a point on the curve in \( (x^2 + y^2) = c \tan^{-1} \frac{y}{x} \), \( c \) is a constant, is
(a) independent of \( x \)
(b) independent of \( y \)
(c) independent of \( x \) but dependent on \( y \)
(d) independent of \( y \) but dependent on \( x \)
Answer: (a) independent of \( x \) and (b) independent of \( y \)

Question. Let \( f(x) = \frac{1}{1 + |x|} + \frac{1}{1 + |x - 1|} \) then
(a) \( f(x) \) has global maximum
(b) \( f(x) \) has local minimum
(c) \( f(x) \) has absolute minimum
(d) \( f(x) \) has local maximum
Answer: (a) \( f(x) \) has global maximum and (b) \( f(x) \) has local minimum

Question. If \( f(x) = (\sin^2 x - 1)^n \), then \( x = \frac{\pi}{2} \) is a point of
(a) local maximum, if \( n \) is odd
(b) local minimum, if \( n \) is odd
(c) local maximum, if \( n \) is even
(d) local minimum, if \( n \) is even
Answer: (b) local minimum, if \( n \) is odd and (d) local minimum, if \( n \) is even

Question. Let \( f(x) = \log(2x - x^2) + \sin \frac{\pi x}{2} \). Then which of the following is/are true?
(a) graph of \( f \) is symmetrical about the line \( x = 1 \)
(b) maximum value of \( f \) is 1
(c) absolute minimum value of \( f \) does not exist
(d) \( f(x) \) is a periodic function
Answer: (a) graph of \( f \) is symmetrical about the line \( x = 1 \), (b) maximum value of \( f \) is 1, and (c) absolute minimum value of \( f \) does not exist