Continuity & Differentiability PYQs Set 3 (PDF)

Set 3 of 13 · Questions 61-90 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 3 (Q61-90)
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61. $The value of \frac{d}{dx} (log (sin \frac{x ^{2} + 1}{x ^{2} + 2})) when x = 2, is$ (1) $\frac{2 cot ( \frac{3}{2} )}{6 3}$ (2) $\frac{2 tan ( \frac{3}{2} )}{6 3}$ (3) $\frac{2 cot ( \frac{3}{2} )}{8 3}$ (4) $\frac{2 tan ( \frac{3}{2} )}{8 3}$ 62. $If x^{y} = e^{x - y}, then \frac{dy}{dx} is$ (1) $\frac{1 + x}{1 + log x}$ (2) $\frac{1 - log x}{1 + log x}$ (3) $not defined$ (4) $\frac{log x}{( 1 + log x ) ^{2}}$ 63. $Let P (x) be a polynomial of degree 3 having extreme value at x = 1 . If lim_{x \to 0} (\frac{P ( x ) + 4}{x ^{2}} + 2) = 6, then (\frac{dP}{dx})_{x = \frac{1}{2}} =$ (1) $2$ (2) $0$ (3) $-2$ (4) $4$ 64. $For x^{2} - 4 \neq = 0, the value of \frac{d}{dx} (log (e^{x} (\frac{x - 2}{x + 2})^{\frac{3}{4}})) at x = 3 is$ (1) $\frac{8}{5}$ (2) $2$ (3) $1$ (4) $\frac{8 e ^{3}}{5}$ 65. $If x is real, then the minimum value of x^{2} - 8 x + 17 is$ (1) $2$ (2) $4$ (3) $1$ (4) $3$ 66. $If f (x) = \sum_{p = 1}^{7} p^{2} sin^{- 1} (\frac{4}{5} sin (px) - \frac{3}{5} cos (px)) then the value of \frac{df}{dx} at x = 1 is (Given that sin^{- 1} (sin x) = x)$ (1) $0$ (2) $628$ (3) $1140$ (4) $784$ 67. $Let f (x) = x^{2} + 2 x + 2, g (x) = - x^{2} + 2 x - 1 and a, b be the extreme values of f (x), g (x) respectively. If c is the extreme value of \frac{f}{g} (x) (for x \neq = 1), then a + 2 b + 5 c + 4 =$ (1) $2$ (2) $1$ (3) $4$ (4) $3$ 68. $The sum of the global minimum and global maximum values of the function f (x) = \frac{4}{3} x^{3} - 4 x in [0, 2] is$ (1) $0$ (2) $8/3$ (3) $- 8/3$ (4) $1$ 69. $The domain of the real valued function f (x) = sin (log (\frac{4 - x ^{2}}{1 - x})) is$ (1) $(1, 4)$ (2) $(- 1, 1)$ (3) $(- 2, 1)$ (4) $(- 2, 4)$ 70. $If y = cos^{2} (\frac{5 x}{2}) - sin^{2} (\frac{5 x}{2}), then \frac{d ^{2} y}{d x ^{2}} is equal to$ (1) $- 5 1 - y^{2}$ (2) $25 y$ (3) $- 25 y$ (4) $25 1 - y^{2}$ 71. $The diameter of a sphere is measured as 42 cm . If there is an error of 1/77 cm in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is$ (1) $33$ (2) $\frac{24}{7}$ (3) $36 .$ (4) $\frac{36}{7}$ 72. $If Rolle's theorem for f (x) = e^{x} (sin x - cos x) is verified on [π /4, 5 π /4], then the value of c is$ (1) $π /3$ (2) $π /2$ (3) $3 π /4$ (4) $π$ 73. $The value that should be assigned to f (0) so that the function f (x) = (x + 1)^{cot x} is continuous at x = 0, is$ (1) $e$ (2) $1$ (3) $2$ (4) $e^{- 1}$ 74. $Let f (x) = \frac{2 - x + 4}{sin 2 x}, x \neq = 0 . In order that f (x) is continuous at x = 0, f (0) is to be defined as$ (1) $\frac{- 1}{8}$ (2) $\frac{1}{2}$ (3) $1$ (4) $\frac{1}{8}$ 75. $A stone is thrown vertically upwards from the top of a tower 64 metres high according to the law s = 48 t - 16 t^{2} . The greatest height attained by the stone above the ground is$ (1) $100 metres$ (2) $64 metres$ (3) $36 metres$ (4) $32 metres$ 76. $If the function f (x) = (\frac{72 ^{x} - 9 ^{x} - 8 ^{x} + 1}{2 - 1 + cos x}, x \neq = 0 a log_{e} 2 log_{e} 3, x = 0) is continuous at x = 0, then the value of a^{2} is equal to$ (1) $968$ (2) $1152$ (3) $746$ (4) $1250$ 77. $Let f (x) be continuous on [0, 4], differentiable on (0, 4), f (0) = 4 and f (4) = - 2 . If g (x) = \frac{f ( x )}{x + 2} then the value of g^{’} (c) for some Lagrange's constant c \in (0, 4) is$ (1) $\frac{1}{2}$ (2) $\frac{5}{12}$ (3) $- \frac{5}{12}$ (4) $- \frac{7}{12}$ 78. $If θ is the angle between the curves y^{2} = 4 x and x^{2} + y^{2} = 5 then \| tan θ \| =$ (1) $5$ (2) $4$ (3) $3$ (4) $2$ 79. $If x = sin θ, y = sin^{3} θ then \frac{d ^{2} y}{d x ^{2}} at θ = \frac{π}{2} is$ (1) $3$ (2) $6$ (3) $1/6$ (4) $1/3$ 80. $The minimum value of f (x) = \frac{x ^{2} - 2 x + 3}{x ^{2} - 4 x + 7} is$ (1) $1 + \frac{1}{3}$ (2) $\frac{3 - 3}{3}$ (3) $2 - \frac{1}{3}$ (4) $3 - \frac{1}{3}$ 81. $If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area,$ (1) $2.16 π cm^{2}$ (2) $21.6 π cm^{2}$ (3) $216 π cm^{2}$ (4) $0.216 π cm^{2}$ 82. $Assertion (A): lim_{x \to 0} \frac{1}{x} = \infty . Reason (R): As the value of x decreases, the value of \frac{1}{x} increases$ (1) $Both A, R are true, and R is correct explanation of A$ (2) $A, R are both true and R is not the correct explanation of A$ (3) $A is true and R is false$ (4) $A is false and R is true$ 83. $If f (x) defined as given below, is continuous on R, then the value of a + b is equal to f (x) = ⎩ ⎨ ⎧ sin x, x^{2} + a, b x + 3, - 3, x \leq 0 0 < x < 1 1 \leq x \leq 3 x > 3$ (1) $0$ (2) $2$ (3) $-2$ (4) $3$ 84. $Consider the function. f (x) = (\frac{a ( 7 x - 12 - x ^{2} )}{b ( x ^{2} - 7 x + 12 )}, x < 3 2^{\frac{sin ( x - 3 )}{x - [ x ]}}, x > 3 b, x = 3) Where [x] denotes the greatest integer less than or equal to x . If S denotes the set of all ordered pairs (a, b) such that f (x) is continuous at x = 3, then the number of elements in S is :$ (1) $2$ (2) $Infinitely many$ (3) $4$ (4) $1$ 85. $The angle between the curves 2 x^{2} + y^{2} = 20 and 4 y^{2} - x^{2} = 8 at a point where they intersect in the IV th quadrant is$ (1) $0$ (2) $\frac{π}{2}$ (3) $\frac{π}{4}$ (4) $\frac{π}{6}$ 86. $If the function f (x) = \frac{x}{5} + \frac{5}{x}, (x \neq = 0) attains its relative maximum value at x = a then a^{2} + 2 a - 6 =$ (1) $10$ (2) $6$ (3) $5$ (4) $3$ 87. $Let α \in R be such that the function f (x) = ⎩ ⎨ ⎧ \frac{cos ^{- 1} ( 1 - { x } ^{2} ) sin ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}}, α, x \neq = 0 x = 0 is continuous at x = 0, where {x} = x - [x], [x] is the greatest integer less than or equal to x . Then :$ (1) $α = \frac{π}{2}$ (2) $α = 0$ (3) $no such α exists$ (4) $α = \frac{π}{4}$ 88. $If the function f (x) = ⎩ ⎨ ⎧ (1 + ∣ cos x ∣)^{λ /∣ c o s x ∣}, μ, e^{c o t 6 x / c o t 4 x}, 0 < x < \frac{π}{2} x = \frac{π}{2} \frac{π}{2} < x < π is continuous at x = \frac{π}{2}, then 9 λ + 6 lo g_{e} μ + μ^{6} - e^{6 λ} is equal to$ (1) $11$ (2) $8$ (3) $2 e^{4} + 8$ (4) $10$ 89. $If lim_{x \to 0} \frac{sin ( 2 + x ) - sin ( 2 - x )}{x} = A cos B, then the values of A and B respectively are$ (1) $2,1$ (2) $2,2$ (3) $1,1$ (4) $1,2$ 90. $Let [t] denote the greatest integer less than or equal to t . Let f : [0, \infty) \to R be a function defined by f (x) = [\frac{x}{2} + 3] - [x] . Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then \sum_{a \in S} a is equal to ______$