Application of Derivatives PYQs Set 4 (PDF)

Set 4 of 10 · Questions 91-120 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 4 (Q91-120)
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91. $The tangent to the curve y = x^{3} + 1 at (1, 2) makes an angle θ with y -axis, then the value of tan θ is$ (1) $- \frac{1}{3}$ (2) $3$ (3) $-3$ (4) $\frac{1}{3}$ 92. $The rate of change of x^{2} + 16 with respect to \frac{x}{x - 1} at x = 5 is$ (1) $\frac{- 80}{41}$ (2) $\frac{80}{41}$ (3) $\frac{12}{5}$ (4) $\frac{- 12}{5}$ 93. $Let (2, 3) be the largest open interval in which the function f (x) = 2 lo g_{e} (x - 2) - x^{2} + a x + 1 is strictly increasing and (b, c) be the largest open interval, in which the function g (x) = (x - 1)^{3} (x + 2 - a)^{2} is strictly decreasing. Then 100 (a + b - c) is equal to:$ (1) $420$ (2) $360$ (3) $160$ (4) $280$ 94. $A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the rate of 0.05 cm / sec . The rate at which its area is increasing when the radius is 5.2 cm is$ (1) $5.05 π cm^{2} / sec$ (2) $5.2 π cm^{2} / sec$ (3) $0.52 π cm^{2} / sec$ (4) $27.4 π cm^{2} / sec$ 95. $Electric current (I) is measured by galvanometer, the current being proportional to the tangent of the angle (θ) of deflection. If the deflection is read as 45^{\circ} and an error of 1 % is made in reading it, the percentage error in the current is$ (1) $π$ (2) $\frac{π}{2}$ (3) $\frac{π}{3}$ (4) $\frac{π}{4}$ 96. $The maximum value of the function f (x) = 3 x^{3} - 18 x^{2} + 27 x - 40 on the set S = (x \in R / x^{2} + 30 \leq 11 x) is$ (1) $122$ (2) $-122$ (3) $-222$ (4) $222$ 97. $The sides of an equilateral triangle are increasing at the rate of 4 cm / sec . The rate at which its area is increasing, when the side is 14 cm is$ (1) $103 cm^{2} / sec$ (2) $42 cm^{2} / sec$ (3) $143 cm^{2} / sec$ (4) $283 cm^{2} / se c$ 98. $For all twice differentiable functions f : R \to R with f (0) = f (1) = f^{’} (0) = 0$ (1) $f^{’ ’} (0) = 0$ (2) $f^{’ ’} (x) = 0 for some x\in(0,1)$ (3) $f^{’ ’} (x) = 0 at every point x\in(0,1)$ (4) $f^{’ ’} (x) \neq = 0 at every point x\in(0,1)$ 99. $The maximum value of \frac{log x}{x} in (2, \infty) is$ (1) $e$ (2) $1/ e$ (3) $1$ (4) $2/ e$ 100. $The length of the subtangent to the curve x^{2} y^{2} = a^{4} at (- a, a) is$ (1) $a /2$ (2) $2 a$ (3) $a$ (4) $a /3$ 101. $The range in which y = - x^{2} + 6 x - 3 is increasing is$ (1) $x < 3$ (2) $x > 3$ (3) $7 < x < 8$ (4) $5 < x < 6$ 102. $Suppose f (x) is a polynomial of degree four, having critical points at - 1, 0, 1 . If T = x \in Rf (x) =f (0) then the sum of squares of all the elements of T is :$ (1) $4$ (2) $2$ (3) $6$ (4) $8$ 103. $Let f be a real valued function, defined on R - -1, 1 and given by f (x) = 3 log_{e} (\frac{x - 1}{x + 1}) - \frac{2}{x - 1} Then in which of the following intervals, function f (x) is increasing?$ (1) $(- \infty, - 1) \cup ((\frac{1}{2}, \infty) - {1})$ (2) $(- \infty, \infty) - {- 1, 1}$ (3) $(- 1, \frac{1}{2})$ (4) $(- \infty, \frac{1}{2}) - {- 1}$ 104. $A balloon which always remains spherical is being inflated by pumping in 10 cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm .$ (1) $\frac{1}{9 π} cm / sec$ (2) $\frac{1}{π} cm / sec$ (3) $\frac{1}{90 π} cm / sec$ (4) $\frac{1}{30 π} cm / sec$ 105. $The sum of the intercepts made by a tangent drawn to the curve (\frac{x}{a})^{n} + (\frac{y}{b})^{n} = 2 at (a, b) on the coordinate axes is$ (1) $a + b$ (2) $a^{2} + b^{2}$ (3) $2 (a - b)$ (4) $2 (a + b)$ 106. $If the slope of the line through (0, 0) which is tangent to the curve y = x^{3} + x + 16 is m then the value of m - 4 is$ (1) $9$ (2) $10$ (3) $12$ (4) $13$ 107. $If the volume of spherical ball is increasing at the rate of 4 π cc / sec then the rate of change of its surface area when the volume is 288 π cc is$ (1) $\frac{4}{3} π cm^{2} / sec$ (2) $\frac{2}{3} π cm^{2} / sec$ (3) $4 π cm^{2} / sec$ (4) $2 π cm^{2} / sec$ 108. $Maximum area of the rectangle that can be formed with the fixed perimeter ' p' cm ...........$ (1) $\frac{p ^{2}}{8} cm^{2}$ (2) $\frac{p ^{2}}{16} cm^{2}$ (3) $\frac{p ^{2}}{64} cm^{2}$ (4) $\frac{p ^{2}}{32} cm^{2}$ 109. $For the function f (x) = sin x + 3 x - \frac{2}{π} (x^{2} + x), where x \in (0, \frac{π}{2}), consider the following two statements: (I) f is increasing in (0, \frac{π}{2}) . (II) f^{’} is decreasing in (0, \frac{π}{2}) . Between the above two statements,$ (1) $only (II) is true.$ (2) $only (I) is true.$ (3) $neither (I) nor (II) is true.$ (4) $both (I) and (II) are true$ 110. $The distance between the point (1, 1) and the tangent to the curve y = e^{2 x} + x^{2} drawn at the point x = 0 is$ (1) $\frac{1}{5}$ (2) $\frac{- 1}{5}$ (3) $\frac{2}{5}$ (4) $\frac{- 2}{5}$ 111. $The side of a square sheet of metal is increasing at the rate of 3 cm / min . At what rate is the area increasing when the length of the side is 6 cm long?$ (1) $12 cm^{2} / min$ (2) $18 cm^{2} / min$ (3) $9 cm^{2} / min$ (4) $36 cm^{2} / min$ 112. $f : R^{+} \to R be an increasing function such that f (x) > 0 for all x . If lim_{x \to \infty} \frac{f ( 9 x )}{f ( 3 x )} = 1, then lim_{x \to \infty} \frac{f ( 6 x )}{f ( 3 x )} =$ (1) $1$ (2) $2$ (3) $\frac{3}{2}$ (4) $\frac{2}{3}$ 113. $The points on the curve y^{2} = x + sin x at which the normal is parallel to the Y-axis lie on$ (1) $a line parallel to Y-axis$ (2) $a circle with centre at origin$ (3) $a parabola$ (4) $a pair of lines bisecting the angle between the coordinate axes$ 114. $The position of a moving car at time t is given by f (t) = a t^{2} + bt + c, t > 0 where a, b and c are real numbers greater than 1 . Then the average speed of the car over the time interval [t_{1}, t_{2}] is attained at the point:$ (1) $2 a (t_{1} + t_{2}) + b$ (2) $(t_{2} - t_{1}) /2$ (3) $a (t_{2} - t_{1}) + b$ (4) $(t_{1} + t_{2}) /2$ 115. $In the interval (7, \infty) f (x) = \| x - 5 \| + 2 \| x - 7 \| is$ (1) $increasing function$ (2) $decreasing function$ (3) $constant function$ (4) $attains maximum value$ 116. $The set of real values of x for which f (x) = \frac{x}{log x} is increasing, is$ (1) $empty$ (2) ${x:x\geqe}$ (3) ${1}$ (4) ${x : x < e}$ 117. $If the tangent to the curve y = x^{3} at the point P (t, t^{3}) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is:$ (1) $- 2 t^{3}$ (2) $- t^{3}$ (3) $0$ (4) $2 t^{3}$ 118. $If A > 0, B > 0 and A + B = \frac{π}{3}, then the maximum value of tan A tan B is$ (1) $\frac{1}{3}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $3$ 119. $The equation of tangent of the curve y = 9 - 2 x^{2} at the point where the ordinate and abscissa are equal is$ (1) $2 x + y - 33 = 0$ (2) $2 x + y + 33 = 0$ (3) $2 x - y - 33 = 0$ (4) $2 x - y + 33 = 0$ 120. $Let a be an integer such that all the real roots of the polynomial 2 x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 10 x + 10 lie in the interval (a, a + 1) Then, \| a \| is equal to$