Application of Derivatives PYQs Set 3 (PDF)

Set 3 of 10 · Questions 61-90 of 274

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Application of Derivatives · Set 3 (Q61-90)
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61. $Let f : R \to R be a polynomial function of degree four having extreme values at x = 4 and x = 5 . If lim x \to 0 \frac{f ( x )}{x ^{2}} = 5, then f (2) is equal to$ (1) $8$ (2) $10$ (3) $12$ (4) $14$ 62. $The function f (x) = x^{2} - 2 x is strictly decreasing in the interval$ (1) $(- \infty, 1)$ (2) $(1, \infty)$ (3) $R$ (4) $(- \infty, \infty)$ 63. $Tangents are drawn to the curve y = sin x from the origin. The locus of the points of contact is$ (1) $xy = x + y$ (2) $x^{2} y^{2} = x^{2} - y^{2}$ (3) $xy = x - y$ (4) $x^{2} y^{2} = x^{2} + y^{2}$ 64. $Assertion (A) : The curves y^{2} = 4 x and x^{2} = - 2 y intersect at (1, 2) orthogonally. Reason (R) : If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each other orthogonally.The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation for (A)$ (2) $(A) is true, (R) is true but (R) is not the correct explanation for (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 65. $If the straight line xcos α + ysin α = p touches the curve (\frac{x}{a})^{n} + (\frac{y}{b})^{n} = 2 at the point (a, b) on it and \frac{1}{a ^{2}} + \frac{1}{b ^{2}} = \frac{k}{p ^{2}} then k =$ (1) $4$ (2) $5$ (3) $6$ (4) $7$ 66. $The sum of the. lengths of the subtangent and the subnormal drawn at θ = \frac{π}{3} on the cycloid x = a (θ - sin θ) : y = a (1 - cos θ) is$ (1) $2 a$ (2) $(23) a$ (3) $\frac{2 a}{3}$ (4) $\frac{a}{3}$ 67. $If the locus of the points on the curve x^{3} y^{2} + \frac{x ^{2}}{y} = 5 at which the tangent is parallel to X -axis is f (x, y) = 0, then the point that lies on this curve f (x, y) = 0 is$ (1) $(2, 33)$ (2) $(32, 3)$ (3) $(- 2, \frac{1}{3 3})$ (4) $(- 32, \frac{1}{3 3})$ 68. $If the area of a circle increases at the rate of \frac{1}{π} sq. unit / sec, then the rate (in units / sec) at which the perimeter of the circle changes. when perimeter is π units, is$ (1) $2$ (2) $4$ (3) $\frac{1}{π}$ (4) $π$ 69. $A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm / sec . At that instant, when the radius of circular wave is 8 cm, how fast is the enclosed area increasing?$ (1) $6 π cm^{2} / s$ (2) $8 π cm^{2} / s$ (3) $\frac{8}{3} cm^{2} / s$ (4) $80 π cm^{2} / s$ 70. $If the tangent drawn to the curve y = x^{3} - a x^{2} + x + 1 at each point x \in R, is inclined at an acute angle with the positive direction of X - axis, then the set of all possiblevalues of 'a' is$ (1) $R - (- 3, 3)$ (2) $[- 3, 3]$ (3) $R$ (4) $(- 3, 3)$ 71. $If the tangent and normal drawn to the curve x = a (θ + sin θ), y = a (1 - cos θ) at p (θ = \frac{π}{2}) cuts the X -axis at A and B respectively, then the area (in sq. units) of △ PAB is$ (1) $\frac{a ^{2}}{2}$ (2) $\frac{2}{a ^{2}}$ (3) $a^{2}$ (4) $2 a^{2}$ 72. $If c is a point at which Rolle's theorem holds for the function, f (x) = log_{e} (\frac{x ^{2} + α}{7 x}) in the interval [3, 4], where α \in R then f^{’ ’} (c) is equal to:$ (1) $\frac{1}{12}$ (2) $\frac{- 1}{12}$ (3) $\frac{1}{6}$ (4) $\frac{- 1}{6}$ 73. $Let f (x) = 3 x - 2 + 4 - x be a real valued function. If α and β are respectively the minimum and the maximum values of f, then α^{2} + 2 β^{2} is equal to$ (1) $42$ (2) $38$ (3) $24$ (4) $44$ 74. $If the parabola y = α x^{2} - 6 x + β passes through the point (0, 2) and has its tangent at x = \frac{3}{2} parallel to x -axis, then$ (1) $α = 2, β = - 2$ (2) $α = - 2, β = 2$ (3) $α = 2, β = 2$ (4) $α = - 2, β = - 2$ 75. $The number of those tangents to the curve y^{2} - 2 x^{3} - 4 y + 8 = 0 which pass through the point (1, 2) is$ (1) $0$ (2) $2$ (3) $1$ (4) $3$ 76. $Length of the subtangent at (x_{1}, y_{1}) on x^{n} y^{m} = a^{m + n}, m, n > 0, is$ (1) $\frac{n}{m} (x_{1})$ (2) $\frac{n}{m} x_{1}$ (3) $\frac{m}{n} (x_{1})$ (4) $\frac{n}{m} (y_{1})$ 77. $The function f (x) = x^{3} - 6 x^{2} + 9 x + 2 has maximum value when x is$ (1) $1$ (2) $2$ (3) $3$ (4) $6$ 78. $A stone is dropped in a quiet lake and it is observed that waves move in circles. If the radius of a circular wave increases at the rate 2 cm / sec, then the rate of increase in its area at the instant when its radius is 10 cm, is cm^{2} / sec .$ (1) $40 π$ (2) $20 π$ (3) $10 π$ (4) $80 π$ 79. $The equation of the normal to the curve y = cosh x drawn at the point nearest to the origin is$ (1) $y = 0$ (2) $x = 1$ (3) $x = 0$ (4) $y = 1$ 80. $Let f (x) = \int \frac{x ^{2} - 3 x + 2}{x ^{4} + 1} dx, then function decreases in the interval$ (1) $(- \infty, - 2)$ (2) $(- 2, - 1)$ (3) $(1, 2)$ (4) $(2, \infty)$ 81. $Slope of normal to the curve y = x^{2} - \frac{1}{x ^{2}} at (- 1, 0) is$ (1) $- \frac{1}{4}$ (2) $-4$ (3) $\frac{1}{4}$ (4) $4$ 82. $If the function f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1, a > 0 has a local maximum at x = α and a local minimum at x = α^{2}, then α and α^{2} are the roots of the equation :$ (1) $x^{2} - 6 x + 8 = 0$ (2) $x^{2} + 6 x + 8 = 0$ (3) $8 x^{2} + 6 x - 1 = 0$ (4) $8 x^{2} - 6 x + 1 = 0$ 83. $The number of tangent to the curve y^{2} (x - a) = x^{2} (x + a) (a > 0) that are parallel to the X -axis is$ (1) $infinitely many$ (2) $0$ (3) $1$ (4) $2$ 84. $Which statement among the following is true? (i) the function f (x) = x \| x \| is strictly increasing on R - 0 . (ii) the function f (x) = log_{(1/4)} x is strictly increasing on (0, \infty) (iii) a one-one function is always an increasing function. (iv) f (x) = x^{1/3} is strictly decreasing on R$ (1) $(i)$ (2) $(ii)$ (3) $(iii)$ (4) $(iv)$ 85. $P is the point of contact of the tangent from the origin to the curve y = log_{e} x . The length of the perpendicular drawn from the origin to the normal at P is$ (1) $1/ e$ (2) $1/2 e$ (3) $e^{2} + 1$ (4) $2 e^{2} + 1$ 86. $Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(a), then in the interval [-1, 1]$ (1) $P (-1) is the minimum and P(a) is the maximum of P$ (2) $P (-1) is not minimum but P(a) is the maximum of P$ (3) $P (-1) is the minimum and P(a) is not the maximum of P$ (4) $neither P (-1) is the minimum nor P(a) is the maximum of P$ 87. $If the curve y = 2 x^{3} + a x^{2} + bx + c passes through the origin and the tangents drawn to it at x = - 1 and x = 2 are parallel to the X -axis, then the values of a, b and c are respectively$ (1) $12, - 3 and 0$ (2) $- 3, - 12 and 0$ (3) $- 3, 12 and 0$ (4) $3, - 12 and 0$ 88. $f (x) = 7 x + 9 is increasing for$ (1) $x \in R$ (2) $only for positive values of x \in R$ (3) $only for x \in (- 1, 1)$ (4) $none of these$ 89. $The slope of the tangent to the curve y = \int_{0}^{x} \frac{1}{1 + t ^{3}} dt at the point, where x = 1 is$ (1) $\frac{1}{4}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $1$ 90. $Let the set of all values of p, for which f (x) = (p^{2} - 6 p + 8) (sin^{2} 2 x - cos^{2} 2 x) + 2 (2 - p) x + 7 does not have any critical point, be the interval (a, b) . Then 16 ab is equal to_______$