JEE Mathematics Monotonocity MCQs Set 02

Question. The interval in which the function \(x^3\) increases less rapidly than \(6x^2 + 15x + 5\) is
(a) \((-\infty, -1)\)
(b) \((-5, 1)\)
(c) \((-1, 5)\)
(d) \((5, \infty)\)
Answer: (c) \((-1, 5)\)

Question. The function \(\frac{|x - 1|}{x^2}\) is monotonically decreasing in
(a) \((2, \infty)\)
(b) \((0, 1)\)
(c) \((0, 1)\) and \((2, \infty)\)
(d) \((-\infty, \infty)\)
Answer: (c) \((0, 1)\) and \((2, \infty)\)

Question. If \(y = (a + 2) x^3 - 3ax^2 + 9ax - 1\) decreases monotonically \(\forall x \in R\) then 'a' lies in the interval
(a) \((-\infty, -3]\)
(b) \((-\infty, -2) \cup (-2, 3)\)
(c) \((-3, \infty)\)
(d) None of the options
Answer: (a) \((-\infty, -3]\)

Question. The values of \(p\) for which the function \(f(x) = \left(\frac{\sqrt{p + 4}}{1 - p} - 1\right) x^5 - 3x + \ln 5\) decreases for all real x is
(a) \((-\infty, \infty)\)
(b) \(\left[-4, \frac{3 - \sqrt{21}}{2}\right] \cup (1, \infty)\)
(c) \(\left[-3, \frac{5 - \sqrt{27}}{2}\right] \cup (2, \infty)\)
(d) \((1, \infty)\)
Answer: (b) \(\left[-4, \frac{3 - \sqrt{21}}{2}\right] \cup (1, \infty)\)

Question. The true set of real values of x for which the function, \(f(x) = x \ln x - x + 1\) is positive is
(a) \((1, \infty)\)
(b) \((1/e, \infty)\)
(c) \([e, \infty)\)
(d) \((0, 1)\) and \((1, \infty)\)
Answer: (d) \((0, 1)\) and \((1, \infty)\)

Question. The set of all x for which \(\ln (1 + x) \leq x\) is equal to
(a) \(x > 0\)
(b) \(x > -1\)
(c) \(-1 < x < 0\)
(d) null set
Answer: (b) \(x > -1\)

Question. For which values of 'a' will the function \(f(x) = x^4 + ax^3 + \frac{3x^2}{2} + 1\) will be concave upward along the entire real line
(a) \(a \in [0, \infty)\)
(b) \(a \in (-2, 2)\)
(c) \(a \in [-2, 2]\)
(d) \(a \in (0, \infty)\)
Answer: (c) \(a \in [-2, 2]\)

Question. If the point \((1, 3)\) serves as the point of inflection of the curve \(y = ax^3 + bx^2\) then the value of 'a' and 'b' are
(a) \(a = 3/2\) & \(b = -9/2\)
(b) \(a = 3/2\) & \(b = 9/2\)
(c) \(a = -3/2\) & \(b = -9/2\)
(d) \(a = -3/2\) & \(b = 9/2\)
Answer: (d) \(a = -3/2\) & \(b = 9/2\)

Question. The function \(f(x) = x^3 - 6x^2 + ax + b\) satisfy the conditions of Rolle's theorem in \([1, 3]\). The value of a and b are
(a) 11, -6
(b) -6, 11
(c) -11, 6
(d) 6, -11
Answer: (a) 11, -6

Question. The function \(f(x) = x(x + 3) e^{-x/2}\) satisfies all the conditions of Rolle's theorem in \([-3, 0]\). The value of c which verifies Rolle's theorem, is
(a) 0
(b) -1
(c) -2
(d) 3
Answer: (c) -2

Question. If \(f(x) = \{a^{|x| \operatorname{sgn} x}\}\); \(g(x) = [a^{|x| \operatorname{sgn} x}]\) for \(a > 1, a \neq 1\) and \(x \in R\), where \(\{ * \}\) & \([ * ]\) denote the fractional part and integral part functions respectively, then which of the following statements holds good for the function \(h(x)\), where \((\ln a) h(x) = (\ln f(x) + \ln g(x))\).
(a) 'h' is even and increasing
(b) 'h' is odd and decreasing
(c) 'h' is even and decreasing
(d) 'h' is odd and increasing
Answer: (d) 'h' is odd and increasing

Question. Let \(f(x) = (x - 4) (x - 5) (x - 6) (x - 7)\) then,
(a) \(f'(x) = 0\) has four roots
(b) Three roots of \(f'(x) = 0\) lie in \((4, 5) \cup (5, 6) \cup (6, 7)\)
(c) The equation \(f'(x) = 0\) has only one real root
(d) Three roots of \(f'(x) = 0\) lie in \((3, 4) \cup (4, 5) \cup (5, 6)\)
Answer: (b) Three roots of \(f'(x) = 0\) lie in \((4, 5) \cup (5, 6) \cup (6, 7)\)

Question. For what values of a does the curve \(f(x) = x(a^2 - 2a - 2) + \cos x\) is always strictly monotonic \(\forall x \in R\).
(a) \(a \in R\)
(b) \(|a| < \sqrt{2}\)
(c) \(1 - \sqrt{2} \leq a \leq 1 + \sqrt{2}\)
(d) \(|a| < \sqrt{2} - 1\)
Answer: (c) \(1 - \sqrt{2} \leq a \leq 1 + \sqrt{2}\)

Question. Given that f is a real valued differentiable function such that \(f(x) f'(x) < 0\) for all real x, it follows that
(a) \(f(x)\) is an increasing function
(b) \(f(x)\) is a decreasing function
(c) \(|f(x)|\) is an increasing function
(d) \(|f(x)|\) is a decreasing function
Answer: (d) \(|f(x)|\) is a decreasing function

Question. If \(f(x) = \frac{x^2}{2 - 2\cos x}\); \(g(x) = \frac{x^2}{6x - 6\sin x}\) where \(0 < x < 1\), then
(a) both 'f' and 'g' are increasing functions
(b) 'f' is decreasing & 'g' is increasing function
(c) 'f' is increasing & 'g' is decreasing function
(d) both 'f' & 'g' are decreasing function
Answer: (c) 'f' is increasing & 'g' is decreasing function

Question. If the function \(f(x) = x^3 - 6ax^2 + 5x\) satisfies the conditions of Lagrange's mean theorem for the interval \([1, 2]\) and the tangent to the curve \(y = f(x)\) at \(x = 7/4\) is parallel to the chord joining the points of intersection of the curve with the ordinates \(x = 1\) and \(x = 2\). Then the value of a is
(a) \(35/16\)
(b) \(35/48\)
(c) \(7/16\)
(d) \(5/16\)
Answer: (b) \(35/48\)

Question. \(f : R \rightarrow R\) be a differentiable function \(\forall x \in R\). If tangent drawn to the curve at any point \(x \in (a, b)\) always lie below the curve, then
(a) \(f'(x) > 0, f''(x) < 0 \ \forall x \in (a, b)\)
(b) \(f'(x) < 0, f''(x) < 0 \ \forall x \in (a, b)\)
(c) \(f'(x) > 0, f''(x) > 0 \ \forall x \in (a, b)\)
(d) None of the options
Answer: (c) \(f'(x) > 0, f''(x) > 0 \ \forall x \in (a, b)\)

Question. A value of C for which the conclusion of Mean Value Theorem holds for the function \(f(x) = \log_e x\) on the interval \([1, 3]\) is
(a) \(2 \log_3 e\)
(b) \(\frac{1}{2} \log_e 3\)
(c) \(\log_3 e\)
(d) \(\log_e 3\)
Answer: (a) \(2 \log_3 e\)

Question. Let \(f\) and \(g\) be two functions defined on an interval \(I\) such that \(f(x) \ge 0\) and \(f(x) \le 0\) for all \(x \in I\) and \(f\) is strictly decreasing on \(I\) while \(g\) is strictly increasing on \(I\) then
(a) the product function \(fg\) is strictly increasing on \(I\)
(b) the product function \(fg\) is strictly decreasing on \(I\)
(c) \(fog(x)\) is monotonically increasing on \(I\)
(d) \(fog(x)\) is monotonically decreasing on \(I\)
Answer: (a) the product function \(fg\) is strictly increasing on \(I\), (d) \(fog(x)\) is monotonically decreasing on \(I\)

Question. The function \(y = 2x^2 - \ln |x|\) is monotonically increasing in the interval \(I_1\) and monotonically decreasing in the interval \(I_2\), \(x \neq 0\), then
(a) \(I_1 = \left(-\frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty\right)\)
(b) \(I_2 = \left(-\infty, -\frac{1}{2}\right) \cup \left(0, \frac{1}{2}\right)\)
(c) \(I_1 = \left(-\infty, -\frac{1}{2}\right) \cup \left(0, \frac{1}{2}\right)\)
(d) \(I_2 = \left(-\frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty\right)\)
Answer: (a) \(I_1 = \left(-\frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty\right)\), (b) \(I_2 = \left(-\infty, -\frac{1}{2}\right) \cup \left(0, \frac{1}{2}\right)\)

Question. Let \(\phi(x) = f(x)^3 - 3(f(x))^2 + 4f(x) + 5x + 3 \sin x + 4 \cos x \forall x \in \mathbb{R}\), then
(a) \(\phi\) is increasing whenever \(f\) is increasing
(b) \(\phi\) is increasing whenever \(f\) is decreasing
(c) \(\phi\) is decreasing whenever \(f\) is decreasing
(d) \(\phi\) is decreasing if \(f'(x) = -11\)
Answer: (a) \(\phi\) is increasing whenever \(f\) is increasing, (d) \(\phi\) is decreasing if \(f'(x) = -11\)

Question. If \(\phi(x) = f(x) + f(2a - x)\) and \(f''(x) > a\), \(a > 0\), \(0 \le x \le 2a\), then
(a) \(\phi(x)\) increases in \((a, 2a)\)
(b) \(\phi(x)\) increases in \((0, a)\)
(c) \(f(x)\) decreases in \((0, a)\)
(d) \(\phi(x)\) decreases in \((1, 2a)\)
Answer: (a) \(\phi(x)\) increases in \((a, 2a)\), (c) \(f(x)\) decreases in \((0, a)\)

Question. For the function \(f(x) = x^4 (12 \ln x - 7)\)
(a) the point \((1, -7)\) is the point of inflection
(b) \(x = e^{1/3}\) is the point of minima
(c) the graph is concave downwards in \((0, 1)\)
(d) the graph is concave upwards in \((1, \infty)\)
Answer: (a) the point \((1, -7)\) is the point of inflection, (b) \(x = e^{1/3}\) is the point of minima, (c) the graph is concave downwards in \((0, 1)\), (d) the graph is concave upwards in \((1, \infty)\)

Question. The function \(f(x) = 3x^4 + 4x^3 - 12x^2 - 7\) is
(a) \(\uparrow\) in \([-2, 0]\) & \([1, \infty)\)
(b) \(\downarrow\) in \((-\infty, -2]\) & \([0, 1]\)
(c) \(\downarrow\) in \([-2, 0]\) & \([1, \infty)\)
(d) \(\uparrow\) in \((-\infty, -2]\) & \([0, 1]\)
Answer: (a) \(\uparrow\) in \([-2, 0]\) & \([1, \infty)\), (b) \(\downarrow\) in \((-\infty, -2]\) & \([0, 1]\)

Question. The function \(f(x) = x^2/(x - 1)\), \(x \neq 1\) is
(a) \(\uparrow [0, 1) \cup (1, 2]\)
(b) \(\downarrow (-\infty, 0] \cup [2, \infty)\)
(c) \(\downarrow [0, 1) \cup (1, 2]\)
(d) \(\uparrow (-\infty, 0] \cup [2, \infty)\)
Answer: (c) \(\downarrow [0, 1) \cup (1, 2]\), (d) \(\uparrow (-\infty, 0] \cup [2, \infty)\)

Question. If \(p, q, r\) be real then the intervals in which,
\[f(x) = \begin{vmatrix} x + p^2 & pq & pr \\ pq & x + q^2 & qr \\ pr & qr & x + r^2 \end{vmatrix}\]
(a) increases is \(x < -\frac{2}{3}(p^2 + q^2 + r^2), x > 0\)
(b) decrease is \(\left(-\frac{2}{3}(p^2 + q^2 + r^2), 0\right)\)
(c) decrease is \(x < -\frac{2}{3}(p^2 + q^2 + r^2), x > 0\)
(d) increase is \(\left(-\frac{2}{3}(p^2 + q^2 + r^2), 0\right)\)
Answer: (a) increases is \(x < -\frac{2}{3}(p^2 + q^2 + r^2), x > 0\), (b) decrease is \(\left(-\frac{2}{3}(p^2 + q^2 + r^2), 0\right)\)

Question. Which of the following inequalities are valid
(a) \(|\tan^{-1} x - \tan^{-1} y| \le |x - y| \forall x, y \in \mathbb{R}\)
(b) \(|\tan^{-1} x - \tan^{-1} y| \ge |x - y|\)
(c) \(|\sin x - \sin y| \le |x - y|\)
(d) \(|\sin x - \sin y| \ge |x - y|\)
Answer: (a) \(|\tan^{-1} x - \tan^{-1} y| \le |x - y| \forall x, y \in \mathbb{R}\), (c) \(|\sin x - \sin y| \le |x - y|\)