Class 11 Mathematics Limits And Derivatives MCQs Set 08

Question. The line which is parallel to x-axis and crosses the curve \( y = \sqrt{x} \) at an angle of \( \frac{\pi}{4} \) is
(a) \( y = -\frac{1}{2} \)
(b) \( x = \frac{1}{2} \)
(c) \( y = \frac{1}{4} \)
(d) \( y = \frac{1}{2} \)
Answer: (d) \( y = \frac{1}{2} \)

Question. A function \( y = f(x) \) is given by \( x = \frac{1}{1 + t^2} \) & \( y = \frac{1}{t(1 + t^2)} \) for all \( t > 0 \) then f is
(a) increasing in \( (0, 3/2) \) & decreasing in \( (3/2, \infty) \)
(b) increasing in \( (0, 1) \)
(c) increasing in \( (0, \infty) \)
(d) decreasing in \( (0, 1) \)
Answer: (b) increasing in \( (0, 1) \)

Question. If the normal to the curve \( y = f(x) \) at the point (3, 4) makes an angle \( \frac{3\pi}{4} \) with the positive x-axis, then \( f'(3) = ... \)
(a) –1
(b) \( -\frac{3}{4} \)
(c) \( \frac{4}{3} \)
(d) 1
Answer: (d) 1

Question. Let \( f(x) = \begin{cases} |x|, & \text{for } 0 < |x| \le 2 \\ 1, & \text{for } x = 0 \end{cases} \) then at \( x = 0, f \) has 
(a) a local maximum
(b) no local maximum
(c) a local minimum
(d) no extremum
Answer: (a) a local maximum

Question. For all \( x \in (0, 1) \)
(a) \( e^x < 1 + x \)
(b) \( \log_e(1 + x) < x \)
(c) \( \sin x > x \)
(d) \( \log_e x > x \)
Answer: (b) \( \log_e(1 + x) < x \)

Question. If \( f(x) = x e^{x(1 - x)} \), then \( f(x) \) is
(a) increasing on \( [-1/2, 1] \)
(b) decreasing on R
(c) increasing on R
(d) decreasing on \( [-1/2, 1] \)
Answer: (a) increasing on \( [-1/2, 1] \)

Question. \( f(x) = 2 \cdot e^{x^2 - 4x} \) decreases in
(a) \( (2, \infty) \)
(b) \( (2, -\infty) \)
(c) \( (-\infty, 2) \)
(d) R
Answer: (c) \( (-\infty, 2) \)

Question. The values ‘a’ for which the function \( f(x) = (a + 2)x^3 - 3ax^2 + 9ax - 1 \) decreases for all real values of x, is
(a) \( a < -2 \)
(b) \( a > -2 \)
(c) \( a < -3 \)
(d) \( -3 < a < -2 \)
Answer: (c) \( a < -3 \)

Question. Maximum value of \( f(x) = x + \sin 2x, x \in [0, 2\pi] \) is
(a) \( \pi \)
(b) \( 2\pi \)
(c) \( 3\pi \)
(d) \( \pi / 2 \)
Answer: (b) \( 2\pi \)

Question. Minimum value of \( f(x) = x^2 \log x, x \in [1, e] \) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0

Question. Consider the curve \( y = c e^{x/a} \). The equation of normal to the curve where the curve cut y-axis is
(a) \( cx + ay = c^2 \)
(b) \( cy + ax = c^2 \)
(c) \( x + y = 2 \)
(d) \( x + y = 4 \)
Answer: (b) \( cy + ax = c^2 \)

Question. Which of the following function does not satisfies the condition for requirement of Lagrange’s Mean Value Theorem (LMVT)
(a) \( f(x) = \begin{cases} \frac{\sin 3x}{x}, & 0 < x \le 5 \\ 3, & x = 0 \end{cases}, x \in [0, 5] \)
(b) \( f(x) = \tan \frac{\pi x}{2}, x \in [-\frac{1}{2}, \frac{1}{2}] \)
(c) \( f(x) = \begin{cases} x^3 \sin \frac{1}{x}, & 0 < x \le 1 \\ 0, & x = 0 \end{cases}, x \in [0, 1] \)
(d) \( f(x) = \begin{cases} x^2 + \sin x, & x < 0 \\ x^3 - 5x, & x \ge 0 \end{cases}, x \in [-2, 3] \)
Answer: (d) \( f(x) = \begin{cases} x^2 + \sin x, & x < 0 \\ x^3 - 5x, & x \ge 0 \end{cases}, x \in [-2, 3] \)

Question. Rolle’s theorem is not applicable for the function \( f(x) = |x| \) in the interval [–1, 1] because
(a) \( f'(1) \) does not exist
(b) \( f'(-1) \) does not exist
(c) \( f(x) \) is discontinuous at \( x = 0 \)
(d) \( f'(0) \) does not exist
Answer: (d) \( f'(0) \) does not exist

Question. The greatest value of the function \( f(x) = \frac{\sin 2x}{\sin(x + \frac{\pi}{4})} \) on the interval \( [0, \frac{\pi}{2}] \) is
(a) \( \frac{1}{\sqrt{2}} \)
(b) \( \sqrt{2} \)
(c) 1
(d) \( -\sqrt{2} \)
Answer: (c) 1

Question. The range of \( y = (\text{arc cos} x) (\text{arc sin} x) \) is
(a) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
(b) \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)
(c) \( [-\frac{\pi^2}{2}, \frac{\pi^2}{16}] \)
(d) \( [0, \frac{\pi^2}{16}] \)
Answer: (c) \( [-\frac{\pi^2}{2}, \frac{\pi^2}{16}] \)

Question. Rolle’s theorem holds for the function \( x^3 + bx^2 + cx, 1 \le x \le 2 \) at the point \( x = 4/3 \), the values of b and c are
(a) \( b = 8, c = -5 \)
(b) \( b = -5, c = 8 \)
(c) \( b = 5, c = -8 \)
(d) \( b = -5, c = -8 \)
Answer: (b) \( b = -5, c = 8 \)

Question. If a, b, c, d are real numbers such that \( \frac{3a + 2b}{c + d} + \frac{3}{2} = 0 \), Then the equation \( ax^3 + bx^2 + cx + d = 0 \) has.
(a) at least one root in [–2, 0]
(b) at least one root in [0, 2]
(c) at least two roots in [–2, 2]
(d) No root in [–2, 2]
Answer: (b) at least one root in [0, 2]

Question. If \( y = a \ln |x| + bx^2 + x \) has its local extremum values at \( x = -1 \) and \( x = 2 \), then
(a) \( a = 2, b = -1 \)
(b) \( a = 2, b = -\frac{1}{2} \)
(c) \( a = -2, b = \frac{1}{2} \)
(d) \( a = 2, b = 1/2 \)
Answer: (b) \( a = 2, b = -\frac{1}{2} \)

Question. The equations of the tangents to the curve \( y = x^4 \) from the point (2, 0) other than x-axis, is
(a) \( y = 0 \)
(b) \( y - 1 = 5(x - 1) \)
(c) \( y - \frac{4096}{81} = \frac{2048}{27} (x - \frac{8}{3}) \)
(d) \( y - \frac{32}{243} = \frac{80}{81} (x - \frac{2}{3}) \)
Answer: (c) \( y - \frac{4096}{81} = \frac{2048}{27} (x - \frac{8}{3}) \)

Question. The greatest value of the expression \( P(x) = (1 - x)^5(1 + x)(1 + 2x)^2 \)
(a) 3
(b) 1
(c) 4
(d) 6
Answer: (b) 1

Question. Which one of the following curves cut the parabola \( y^2 = 4ax \) at right angles?
(a) \( x^2 + y^2 = a^2 \)
(b) \( y = e^{-x/2a} \)
(c) \( y = ax \)
(d) \( x^2 = 4ay \)
Answer: (b) \( y = e^{-x/2a} \)

Question. The function defined by \( f(x) = (x + 2) e^{-x} \) is
(a) decreasing for all x
(b) decreasing in \( (-\infty, -1) \) and increasing in \( (-1, \infty) \)
(c) increasing for all x
(d) decreasing in \( (-1, \infty) \) and increasing in \( (-\infty, -1) \)
Answer: (d) decreasing in \( (-1, \infty) \) and increasing in \( (-\infty, -1) \)

Question. On the interval [0, 1] the function \( x^{25}(1 - x)^{75} \) takes its maximum value at the point
(a) 0
(b) \( 1/4 \)
(c) \( 1/2 \)
(d) \( 1/3 \)
Answer: (b) \( 1/4 \)

Question. If \( f(x) = \frac{x}{\sin x} \) and \( g(x) = \frac{x}{\tan x} \), where \( 0 < x \le 1 \), then in this interval
(a) both \( f(x) \) and \( g(x) \) are increasing functions
(b) both \( f(x) \) and \( g(x) \) are decreasing functions
(c) \( f(x) \) is an increasing function
(d) \( g(x) \) is an increasing function
Answer: (c) \( f(x) \) is an increasing function

Question. The function \( f(x) = \sin^4 x + \cos^4 x \) increases if
(a) \( 0 < x < \frac{\pi}{8} \)
(b) \( \frac{\pi}{4} < x < \frac{3\pi}{8} \)
(c) \( \frac{3\pi}{8} < x < \frac{5\pi}{8} \)
(d) \( \frac{5\pi}{8} < x < \frac{3\pi}{4} \)
Answer: (b) \( \frac{\pi}{4} < x < \frac{3\pi}{8} \)

Question. Let \( f(x) = \int e^x (x - 1)(x - 2) dx \). Then f decreases in the interval
(a) \( (-\infty, 2) \)
(b) \( (1, 2) \)
(c) \( (2, \infty) \)
(d) \( (-\infty, 1) \)
Answer: (b) \( (1, 2) \)

Question. If \( 2a + 3b + 6c = 0 \), then the equation \( ax^2 + bx + c = 0 \) has at least one real root in
(a) (0, 1)
(b) (0, 1/2)
(c) (1/4, 1/2)
(d) (-1, 1)
Answer: (a) (0, 1)

Question. Let f be differentiable for all x. If \( f(1) = -2 \) and \( f'(x) \ge 2 \) for all \( x \in [1, 6] \). Then
(a) \( f(6) = 8 \)
(b) \( f(6) \ge 8 \)
(c) \( f(6) \le 8 \)
(d) \( f(6) < 8 \)
Answer: (b) \( f(6) \ge 8 \)

Question. If an interval (a, b) contains k roots of a real polynomial P(x) then it, contains
(a) at least (k - 1) roots of \( P'(x) = 0 \)
(b) at most (k - 1) roots of \( P'(x) = 0 \)
(c) at most k roots of \( P'(x) = 0 \)
(d) atleast k roots of \( P'(x) = 0 \)
Answer: (a) at least (k - 1) roots of \( P'(x) = 0 \)

Question. Let \( \alpha, \beta (\alpha < \beta) \), be two real roots of the equation \( ax^2 + bx + c = 0 \). Then \( -\frac{b}{2a} \) lies in
(a) \( (\beta, \alpha) \)
(b) \( (-\alpha, \alpha) \)
(c) \( (-\beta, \beta) \)
(d) \( (\alpha, \beta) \)
Answer: (d) \( (\alpha, \beta) \)

Question. The triangle formed by the tangent to the curve \( f(x) = x^2 + bx - b \) at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is
(a) –1
(b) 3
(c) –3
(d) 1
Answer: (c) –3

Question. Let \( f(x) = (1 + b^2)x^2 + 2bx + 1 \) and let m(b) be the minimum value of \( f(x) \). As b varies, the range of m(b) is
(a) [0, 1]
(b) [0, 1/2]
(c) [1/2, 1]
(d) (0, 1]
Answer: (d) (0, 1]

Question. If a variable tangent to the curve \( x^2y = c^3 \) makes intercepts a, b on x and y axis respectively, then the value of \( a^2b \) is
(a) \( 27 c^3 \)
(b) \( \frac{27}{4} c^3 \)
(c) \( \frac{4}{27} c^3 \)
(d) \( \frac{4}{9} c^3 \)
Answer: (b) \( \frac{27}{4} c^3 \)

Question. If \( f(x) = x^3 + 7x - 1 \) then \( f(x) \) has a zero between \( x = 0 \) and \( x = 1 \). The theorem which best describes this, is
(a) Squeeze play theorem
(b) Mean value theorem
(c) Maximum-Minimum value theorem
(d) Intermediate value theorem
Answer: (d) Intermediate value theorem

Question. \( f(x) = \begin{cases} x \sin \frac{\pi}{x} & \text{for } x > 0 \\ 0 & \text{for } x = 0 \end{cases} \) then the number of points in (0, 1) where the derivative \( f'(x) \) vanishes, is
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (d) infinite

Question. Suppose that \( f(0) = -3 \) and \( f'(x) \le 5 \) for all values of x. Then the largest value which \( f(2) \) can attain is
(a) 7
(b) – 7
(c) 13
(d) 8
Answer: (a) 7

Question. Equation of the line through the point (1/2, 2) and tangent to the parabola \( y = -\frac{x^2}{2} + 2 \) and secant to the curve \( y = \sqrt{4 - x^2} \) is
(a) \( 2x + 2y - 5 = 0 \)
(b) \( 2x + 2y - 3 = 0 \)
(c) \( y - 2 = 0 \)
(d) \( 2x - 1 = 0 \)
Answer: (a) \( 2x + 2y - 5 = 0 \)

Question. A curve is represented by the equations, \( x = \sec^2 t \) and \( y = \cot t \) where t is a parameter. If the tangent at the point P on the curve where \( t = \pi/4 \) meets the curve again at the point Q then \( |PQ| \) is equal to
(a) \( \frac{5\sqrt{3}}{2} \)
(b) \( \frac{5\sqrt{5}}{2} \)
(c) \( \frac{2\sqrt{5}}{3} \)
(d) \( \frac{3\sqrt{5}}{2} \)
Answer: (b) \( \frac{5\sqrt{5}}{2} \)

Question. For all a, b \( \in \) R the function \( f(x) = 3x^4 - 4x^3 + 6x^2 + ax + b \) has
(a) no extremum
(b) exactly one extremum
(c) exactly two extremum
(d) three extremum
Answer: (b) exactly one extremum

Question. The set of values of p for which the equation \( |\ln x| - px = 0 \) possess three distinct roots is
(a) \( (0, \frac{1}{e}) \)
(b) (0, 1)
(c) (1, e)
(d) (0, e)
Answer: (a) \( (0, \frac{1}{e}) \)