Continuity & Differentiability PYQs Set 4 (PDF)

Set 4 of 13 · Questions 91-120 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 4 (Q91-120)
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91. $The function f (x) = \frac{x}{3} + \frac{3}{x} decreases in the interval$ (1) $(- 3, 3)$ (2) $(- \infty, 3)$ (3) $(3, \infty)$ (4) $(- 9, 9)$ 92. $Let f (x) be a differentiable function at x = a with f^{’} (a) = 2 and f (a) = 4 . Then lim x \to a \frac{xf ( a ) - af ( x )}{x - a} equals:$ (1) $2 a + 4$ (2) $2 a - 4$ (3) $4 - 2 a$ (4) $a + 4$ 93. $The angle between the curve 2 y = e^{- x /2} and the y -axis is tan^{- 1} (k) then k =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 94. $The derivative of e^{ax} cos bx with respect to x is r e^{ax} cos (bx + tan^{- 1} \frac{b}{a}) When a > 0, b > 0, the value of r is$ (1) $1 ab$ (2) $a^{2} + b^{2}$ (3) $a + b$ (4) $ab$ 95. $If an error k % is made in measuring the radius of a sphere, then percentage error in its volume is$ (1) $k %$ (2) $3 k %$ (3) $2 k %$ (4) $\frac{k}{3} %$ 96. $f (x) = x^{2} + 1 : g (x) = \frac{x + 1}{x ^{2} + 1} : h (x) = 2 x - 3, then the value of f^{’} (h^{’} (g^{’} (x))) is equal to$ (1) $5$ (2) $\frac{2}{5}$ (3) $\frac{5}{2}$ (4) $\frac{1}{5}$ 97. $Find the domain of the real valued function f (x) = ([x]^{2} - [x] - 2)^{- 1/2}, where [\cdot] is the greatest integer function.$ (1) $R - (- 1, 3]$ (2) $R - [- 1, 3)$ (3) $R - (- 1, 3)$ (4) $R - [- 1, 3]$ 98. $If an open cylinder of given surface area has maximum volume then its radius is$ (1) $Height of the cylinder$ (2) $Height of the cylinder /2$ (3) $2 times Height of the cylinder$ (4) $3 times Height of the cylinder$ 99. $If f (x) = (\frac{log _{e} x}{x - 1}; x \neq = 1 k; x = 1) is continuous at x = 1, then the value of k is$ (1) $e$ (2) $1$ (3) $-1$ (4) $0$ 100. $Let f (x) = (\frac{a x ^{2} + 2 ax + 3}{4 x ^{2} + 4 x - 3}, x \neq = - \frac{3}{2}, \frac{1}{2} b, x = - \frac{3}{2}, \frac{1}{2}) be continuous at x = - \frac{3}{2} . If f \circ f (x) = \frac{7}{5}, then x is equal to:$ (1) $4$ (2) $0$ (3) $2$ (4) $1$ 101. $If f (1) = 1, f^{'} (1) = 3, then the derivative of f (f (f (x))) + (f (x))^{2} at x = 1 is$ (1) $12$ (2) $19$ (3) $23$ (4) $33$ 102. $Let the function, f : (- 7, 0) \to R be continuous on (- 7, 0) and differentiable on (- 7, 0) . If f (- 7) = - 3 and f ’ (x) \leq 2, for x \in (- 7, 0), then for all such functions f, f (- 1) + f (0) lies in the interval:$ (1) $(- 5, - 7)$ (2) $(- \infty, 6)$ (3) $(- \infty, 20)$ (4) $(- 5, 3)$ 103. $If [x] represent greatest integer \leq x and [α, β] is the set of all real values of x for which the real function f (x) = \frac{3 + x + 3 - x}{[ x ] + 2} is defined, then f^{2} (α + 1) + 5 f^{2} (β) =$ (1) $0$ (2) $\frac{36}{5}$ (3) $12$ (4) $1$ 104. $Suppose for a differentiable function h, h (0) = 0, h (1) = 1 and h^{’} (0) = h^{’} (1) = 2 . If g (x) = h (e^{x}) e^{h (x)}, then g^{’} (0) is equal to:$ (1) $5$ (2) $4$ (3) $8$ (4) $3$ 105. $The point on the curve 6 y = x^{3} + 2 at which y -co-ordinate is changing 8 times as fast as x -co-ordinate is$ (1) $(4, 11)$ (2) $(4, - 11)$ (3) $(-4,-11)$ (4) $(-4,11)$ 106. $For all real values of x, the minimum value of \frac{1 - x + x ^{2}}{1 + x + x ^{2}} is$ (1) $0$ (2) $\frac{1}{3}$ (3) $1$ (4) $3$ 107. $f (x) = (\frac{sin ( a + 2 ) x + sin x}{x}; x < 0 b; x = 0 \frac{( x + 3 x ^{2} ) ^{\frac{1}{3}} - x ^{\frac{1}{3}}}{x ^{\frac{4}{3}}}; x > 0) Function is continuous at x = 0, find a + 2 b .$ (1) $0$ (2) $4$ (3) $6$ (4) $8$ 108. $If y (α) = 2 (\frac{tan α + cot α}{1 + tan ^{2} α}) + \frac{1}{sin ^{2} α} α \in (\frac{3 π}{4}, π) then \frac{dy}{dα} at α = \frac{5 π}{6} is$ (1) $4$ (2) $2$ (3) $3$ (4) $- 4$ 109. $The maximum value of 'a' such that the second derivative of x^{4} + a x^{3} + \frac{3 x ^{2}}{2} + 1 is positive for all real x is .$ (1) $3$ (2) $-3$ (3) $2$ (4) $-2$ 110. $The approximate value of f (x) = x^{3} + 5 x^{2} - 7 x + 9 at x = 1.1 is$ (1) $8.6$ (2) $8.5$ (3) $8.4$ (4) $8.3$ 111. $If 2 x^{2} + 3 xy - y^{2} + 4 x - 5 y + 6 = 0, then the value of \frac{dy}{dx} at (x, y) = (1, - 2) is$ (1) $1$ (2) $-1$ (3) $\frac{7}{2}$ (4) $0$ 112. $If y = [(x + 1) (2 x + 1) (3 x + 1) \dots \dots .. (nx + 1)]^{n}, then \frac{dy}{dx} at x = 0 is$ (1) $\frac{n ( n + 1 )}{2}$ (2) $\frac{n ^{2} ( n + 1 )}{2}$ (3) $\frac{n ( n + 1 )}{4}$ (4) $\frac{n ^{2} ( n - 1 )}{2}$ 113. $Let x^{k} + y^{k} = a^{k} where a, k > 0 and \frac{dy}{dx} + (\frac{y}{x})^{\frac{1}{3}} = 0 then find k$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $\frac{4}{3}$ (4) $2$ 114. $If Rolle's theorem holds for the function f (x) = x^{3} - a x^{2} + bx - 4, x \in [1, 2] with f^{’} (\frac{4}{3}) = 0, then ordered pair (a, b) is equal to :$ (1) $(- 5, 8)$ (2) $(5, 8)$ (3) $(5, - 8)$ (4) $(- 5, - 8)$ 115. $If f (x) = x + 2 2 x - 4 + x - 2 2 x - 4, then the value of 10 \times f^{’} (102) =$ (1) $1$ (2) $2$ (3) $102$ (4) $-1$ 116. $Let f : R \to R be a twice differentiable function such that (sin xcos y) (f (2 x + 2 y) - f (2 x - 2 y)) = (coxsin y) (f (2 x + 2 y) + f (2 x - 2 y)), for all x, y \in R . If f^{’} (0) = \frac{1}{2}, then the value of 24 f^{’ ’} (\frac{5 π}{3}) is:$ (1) $2$ (2) $3$ (3) $-3$ (4) $-2$ 117. $The value of lim_{θ \to 0} \frac{1 - cos 4 θ}{1 - cos 6 θ} is$ (1) $\frac{9}{4}$ (2) $\frac{4}{9}$ (3) $\frac{9}{3}$ (4) $\frac{3}{4}$ 118. $For every pair of continuous functions f, g:[0, 1] \to R such that max {f(x) : x \in [0,1]} = max {g(x) : x \in [0, 1]},the correct statement(s) is (are) :$ (1) $(f(c))2 + 3f(c) = (g(c))2 + 3g(c) for some c \in [0, 1]$ (2) $(f(c))2 + f(c) = (g(c))2 + 3g(c) for some c \in [0, 1]$ (3) $(f(c))2 + 3f(c) = (g(c))2 + g(c) for some c \in [0, 1]$ (4) $(f(c))2 = (g(c))2 for some c \in [0, 1]$ 119. $If f (x) = (x^{3} sin (\frac{1}{x}), x \neq = 00, x = 0) then$ (1) $f^{’ ’} (\frac{2}{π}) = \frac{24 - π ^{2}}{2 π}$ (2) $f^{’ ’} (\frac{2}{π}) = \frac{12 - π ^{2}}{2 π}$ (3) $f^{’ ’} (0) = 1$ (4) $f^{’ ’} (0) = 0$ 120. $If the function f (x) = {\frac{1}{∣ x ∣}, a x^{2} + 2 b, ∣ x ∣ \geq 2 ∣ x ∣ < 2 is differentiable on R, then 48 (a + b) is equal to_________$