JEE Mathematics Monotonocity MCQs Set 01

Question. The function \(f(x) = \tan^{-1} (\sin x + \cos x)\) is an increasing function in
(a) \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\)
(b) \(\left(-\frac{\pi}{4}, \frac{\pi}{2}\right)\)
(c) \(\left(0, \frac{\pi}{2}\right)\)
(d) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
Answer: (b) \(\left(-\frac{\pi}{4}, \frac{\pi}{2}\right)\)

Question. A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched ?
(a) \((-\infty, -4] \quad x^3 + 6x^2 + 6\)
(b) \(\left(-\infty, \frac{1}{3}\right] \quad 3x^3 - 2x + 1\)
(c) \([2, \infty) \quad 2x^3 - 3x^2 - 12x + 6\)
(d) \((-\infty, \infty) \quad x^3 - 3x^2 + 3x + 3\)
Answer: (b) \(\left(-\infty, \frac{1}{3}\right] \quad 3x^3 - 2x + 1\)

Question. A function \(y = f(x)\) has a second order derivative \(f'' = 6(x - 1)\). If its graph passes through the point \((2, 1)\) and at that point the tangent of the graph is \(y = 3x - 5\), then the function is
(a) \((x-1)^2\)
(b) \((x-1)^3\)
(c) \((x+1)^3\)
(d) \((x+1)^2\)
Answer: (b) \((x-1)^3\)

Question. If \(f(x) = \frac{a \sin x + b \cos x}{c \sin x + d \cos x}\) is monotonically increasing, then
(a) \(ad \geq bc\)
(b) \(ad < bc\)
(c) \(ad \leq bc\)
(d) \(ad > bc\)
Answer: (d) \(ad > bc\)

Question. \(x^3 - 3x^2 - 9x + 20\) is
(a) -ve for \(x < 4\)
(b) +ve for \(x > 4\)
(c) -ve for \(x \in (0, 1)\)
(d) -ve for \(x \in (-1, 0)\)
Answer: (b) +ve for \(x > 4\)

Question. \(f(x) = x^2 - x \sin x\) is
(a) \(\uparrow\) for \(0 \leq x \leq \pi/2\)
(b) \(\downarrow\) for \(0 \leq x \leq \pi/2\)
(c) \(\downarrow\) for \([\pi/4, \pi/2]\)
(d) None of the options
Answer: (a) \(\uparrow\) for \(0 \leq x \leq \pi/2\)

Question. The number of values of 'c' of Lagrange's mean value theorem for the function, \(f(x) = (x - 1) (x - 2) (x - 3)\), \(x \in (0, 4)\) is
(a) 1
(b) 2
(c) 3
(d) None of the options
Answer: (b) 2

Question. The equation \(xe^x = 2\) has
(a) one root of \(x < 0\)
(b) two roots for \(x > 1\)
(c) no root in \((0, 1)\)
(d) one root in \((0, 1)\)
Answer: (d) one root in \((0, 1)\)

Question. If \(f(x) = 1 + x \ln \left[x + \sqrt{x^2 + 1}\right]\) and \(g(x) = \sqrt{x^2 + 1}\) then for \(x \geq 0\)
(a) \(f(x) < g(x)\)
(b) \(f(x) > g(x)\)
(c) \(f(x) \leq g(x)\)
(d) \(f(x) \geq g(x)\)
Answer: (d) \(f(x) \geq g(x)\)

Question. The set of values of the parameter 'a' for which the function; \(f(x) = 8ax - a \sin 6x - 7x - \sin 5x\) increases & has no critical points for all \(x \in R\), is
(a) \([-1, 1]\)
(b) \((-\infty, -6)\)
(c) \((6, +\infty)\)
(d) \([6, +\infty)\)
Answer: (c) \((6, +\infty)\)

Question. If \(f(x)\) and \(g(x)\) are differentiable in \([0, 1]\) such that \(f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2\), then Rolle's theorem is applicable for which of the following
(a) \(f(x) - g(x)\)
(b) \(f(x) - 2g(x)\)
(c) \(f(x) + 3g(x)\)
(d) None of the options
Answer: (b) \(f(x) - 2g(x)\)

Question. \(f : [0, 4] \rightarrow R\) is a differentiable function then for some \(a, b \in (0, 4)\), \(f^2(4) - f^2(0)\) equals
(a) \(8f'(a) \cdot f(b)\)
(b) \(4f'(a) f(b)\)
(c) \(2f'(a) f(b)\)
(d) \(f'(a) f(b)\)
Answer: (a) \(8f'(a) \cdot f(b)\)

Question. Equation \(3x^2 + 4ax + b = 0\) has at least one root in \((0, 1)\) if
(a) \(4a + b + 3 = 0\)
(b) \(2a + b + 1 = 0\)
(c) \(b = 0, a = -\frac{3}{4}\)
(d) None of the options
Answer: (b) \(2a + b + 1 = 0\)

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) \(f(a, b) \geq 1\)
(c) \(f(a, b) \leq 1\)
(d) None of the options
Answer: (b) \(f(a, b) \geq 1\)

Question. Let \(f(x) = ax^4 + bx^3 + x^2 + x - 1\). If \(9b^2 < 24a\), then number of real roots of \(f(x) = 0\) are
(a) 4
(b) 2
(c) 0
(d) can't say
Answer: (b) 2

Question. Function for which LMVT is applicable but Rolle's theorem is not
(a) \(f(x) = x^3 - x, x \in [0, 1]\)
(b) \(f(x) = \begin{cases} x^2, & 0 \leq x < 1 \\ x, & 1 < x \leq 2 \end{cases}\)
(c) \(f(x) = e^x, x \in [-3, 3]\)
(d) \(f(x) = 1 - \sqrt[3]{x^2}, x \in [-1, 1]\)
Answer: (c) \(f(x) = e^x, x \in [-3, 3]\)

Question. LMVT is not applicable for which of the following ?
(a) \(f(x) = x^2, x \in [3, 4]\)
(b) \(f(x) = \ln x, x \in [1, 3]\)
(c) \(f(x) = 4x^2 - 5x^2 + x - 2, x \in [0, 1]\)
(d) \(f(x) = \{x^4 (x - 1)\}^{1/5}, x \in \left[-\frac{1}{2}, \frac{1}{2}\right]\)
Answer: (d) \(f(x) = \{x^4 (x - 1)\}^{1/5}, x \in \left[-\frac{1}{2}, \frac{1}{2}\right]\)

Question. If \(f(x) = (x - 1) (x - 2) (x - 3) (x - 4)\), then roots of \(f'(x) = 0\) not lying in the interval
(a) \([1, 2]\)
(b) \((2, 3)\)
(c) \((3, 4)\)
(d) \((4, \infty)\)
Answer: (d) \((4, \infty)\)

Question. If \(f(x) = 1 + x^m (x - 1)^n\), \(m, n \in N\), then \(f'(x) = 0\) has atleast one root in the interval
(a) \((0, 1)\)
(b) \((2, 3)\)
(c) \((-1, 0)\)
(d) None of the options
Answer: (a) \((0, 1)\)

Question. Which of the following statements is/are correct
(a) \(x + \sin x\) is increasing function
(b) \(\sec x\) is neither increasing nor decreasing function
(c) \(x + \sin x\) is decreasing function
(d) \(\sec x\) is an increasing function
Answer: (a) \(x + \sin x\) is increasing function, (b) \(\sec x\) is neither increasing nor decreasing function

Question. The function \(f(x) = 2 \ln (x - 2) - x^2 + 4x + 1\) increases in the intervals
(a) \((1, 2)\)
(b) \((2, 3)\)
(c) \(\left[\frac{5}{2}, 3\right]\)
(d) \((2, 4)\)
Answer: (b) \((2, 3)\), (c) \(\left[\frac{5}{2}, 3\right]\)

Question. If \(f(x) = 2x + \cot^{-1} x + \log \left(\sqrt{1 + x^2} - x\right)\), then \(f(x)\)
(a) increases in \([0, \infty)\)
(b) decreases in \([0, \infty)\)
(c) neither increases nor decreases in \([0, \infty)\)
(d) increases in \((-\infty, \infty)\)
Answer: (a) increases in \([0, \infty)\), (d) increases in \((-\infty, \infty)\)

Question. Let \(g(x) = 2f(x/2) + f(1 - x)\) and \(f''(x) < 0\) in \(0 \le x \le 1\) then \(g(x)\)
(a) decreases in \(\left[0, \frac{2}{3}\right]\)
(b) decreases in \(\left[\frac{2}{3}, 1\right]\)
(c) increases in \(\left[0, \frac{2}{3}\right]\)
(d) increases in \(\left[\frac{2}{3}, 1\right]\)
Answer: (b) decreases in \(\left[\frac{2}{3}, 1\right]\), (c) increases in \(\left[0, \frac{2}{3}\right]\)

Question. Let the function \(f(x) = \sin x + \cos x\), be defined in \([0, 2\pi]\), then \(f(x)\)
(a) increases in \((\pi/4, \pi/2)\)
(b) decreases in \([\pi/4, 5\pi/4]\)
(c) increases in \([0, \pi/4] \cup [5\pi/4, 2\pi]\)
(d) decreases in \([0, \pi/4) \cup (\pi/2, 2\pi]\)
Answer: (b) decreases in \([\pi/4, 5\pi/4]\), (c) increases in \([0, \pi/4] \cup [5\pi/4, 2\pi]\)

Question. If \(f(x) = \tan^{-1}x - (1/2) \ln x\) then
(a) the greatest value of \(f(x)\) on \([1/\sqrt{3}, \sqrt{3}]\) is \(\pi/6 + (1/4) \ln 3\)
(b) the least value of \(f(x)\) on \([1/\sqrt{3}, \sqrt{3}]\) is \(\pi/3 - (1/4) \ln 3\)
(c) \(f(x)\) decreases on \((0, \infty)\)
(d) \(f(x)\) increases on \((-\infty, 0)\)
Answer: (a) the greatest value of \(f(x)\) on \([1/\sqrt{3}, \sqrt{3}]\) is \(\pi/6 + (1/4) \ln 3\), (b) the least value of \(f(x)\) on \([1/\sqrt{3}, \sqrt{3}]\) is \(\pi/3 - (1/4) \ln 3\), (c) \(f(x)\) decreases on \((0, \infty)\)

Question. If \(f(x) = \log(x - 2) - 1/x\), then
(a) \(f(x)\) is M.I. for \(x \in (2, \infty)\)
(b) \(f(x)\) is M.I. for \(x \in [-1, 2]\)
(c) \(f(x)\) is always concave downwards
(d) \(f^{-1}(x)\) is M.I. wherever defined
Answer: (a) \(f(x)\) is M.I. for \(x \in (2, \infty)\), (c) \(f(x)\) is always concave downwards, (d) \(f^{-1}(x)\) is M.I. wherever defined

Question. Which of the following functions do not satisfy conditions of Rolle's Theorem?
(a) \(e^x \sin x, x \in \left[0, \frac{\pi}{2}\right]\)
(b) \((x + 1)^2 (2x - 3)^5, x \in \left[-1, \frac{3}{2}\right]\)
(c) \(\sin |x|, x \in [\pi, 2\pi]\)
(d) \(\sin \frac{1}{x}, x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
Answer: (a) \(e^x \sin x, x \in \left[0, \frac{\pi}{2}\right]\), (d) \(\sin \frac{1}{x}, x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)

Question. Let \(f(x) = x^{m/n}\) for \(x \in \mathbb{R}\) where \(m\) and \(n\) are integers, \(m\) even and \(n\) odd and \(0 < m < n\). Then
(a) \(f(x)\) decreases on \((-\infty, 0]\)
(b) \(f(x)\) increases on \([0, \infty)\)
(c) \(f(x)\) increases on \((-\infty, 0]\)
(d) \(f(x)\) decreases on \([0, \infty)\)
Answer: (a) \(f(x)\) decreases on \((-\infty, 0]\), (b) \(f(x)\) increases on \([0, \infty)\)