Continuity & Differentiability PYQs Set 5 (PDF)

Set 5 of 13 · Questions 121-150 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 5 (Q121-150)
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121. $If the side of a cube is increased by 5 %, then the surface area of a cube is increased by$ (1) $60 %$ (2) $6 %$ (3) $20 %$ (4) $10 %$ 122. $If θ is the acute angle between the curves x^{2} + y^{2} = 4 and y^{2} = 3 x then tan θ =$ (1) $\frac{5}{3}$ (2) $\frac{3}{4}$ (3) $\frac{4}{3}$ (4) $\frac{3}{5}$ 123. $Let P (x) be polynomial of degree 2 , with P (2) = - 1, P^{’} (2) = 0, P^{’ ’} (2) = 2, then P (1.001) is$ (1) $0.002$ (2) $- 0.002$ (3) $0.004$ (4) $-0.004$ 124. $The range of the function f (x) = ⎩ ⎨ ⎧ 4 x - 1, x^{2} - 2, 3 x + 4, x > 3 - 2 \leq x \leq 3 x < - 2$ (1) $(- \infty, \infty)$ (2) $R - (- 3, 3)$ (3) $R - (7, 11]$ (4) $(7, 11]$ 125. $If θ is the angle between the curves x^{2} - y^{2} = 4 and y^{2} = 3 x then tan θ =$ (1) $\frac{5}{3 3}$ (2) $\frac{5}{6 3}$ (3) $\frac{5}{18}$ (4) $\frac{5}{6}$ 126. $If f (x) = 2 x^{2} + 3 x - 5, then the value of f^{’} (0) + 3 f^{’} (- 1) is equal to$ (1) $1$ (2) $0$ (3) $3$ (4) $2$ 127. $If the function f (x) = \frac{sin 3 x + α sin x - β cos 3 x}{x ^{3}}, x \in R, is continuous at x = 0, then f (0) is equal to :$ (1) $2$ (2) $-2$ (3) $4$ (4) $-4$ 128. $The maximum value of (\frac{1}{x})^{x} is$ (1) $e$ (2) $e^{e}$ (3) $e^{\frac{1}{e}}$ (4) $1 e^{e}$ 129. $For x > 1, if (2 x)^{2 y} = 4 e^{2 x - 2 y}, then (1 + log_{e} 2 x)^{2} \frac{dy}{dx} is equal to :$ (1) $x log_{e} 2 x -loge2x$ (2) $log_{e} 2 x$ (3) $x log_{e} 2 x +loge2x$ (4) $x log_{e} 2 x$ 130. $The domain of the real valued function f (x) = \frac{6 x ^{2} + 5 x - 6}{4 - x - x + 4} is$ (1) $(- 4, - \frac{3}{2}) \cup (\frac{2}{3}, 4)$ (2) $(- \infty, \frac{- 3}{2}) \cup (\frac{2}{3}, \infty)$ (3) $[- 4, 4]$ (4) $(- \frac{3}{2}, \frac{2}{3})$ 131. $For the function f (x) = (x - 1) (x - 2) defined on (0, \frac{1}{2}), the value of c satisfying Lagrange's mean value theorem is$ (1) $\frac{1}{5}$ (2) $\frac{1}{3}$ (3) $\frac{1}{7}$ (4) $\frac{1}{4}$ 132. $The angle of intersection between the curves y^{2} + x^{2} = a^{2} 2 and x^{2} - y^{2} = a^{2} is$ (1) $\frac{π}{3}$ (2) $\frac{π}{4}$ (3) $\frac{π}{6}$ (4) $\frac{π}{12}$ 133. $Derivative of log (sec θ + tan θ) with respect to sec θ at θ = π /4 is$ (1) $0$ (2) $1$ (3) $\frac{1}{2}$ (4) $2$ 134. $A stone is thrown vertically upwards and the height x ft . reached by the stone in t seconds is given by x = 80 t - 16 t^{2} . The stone reaches the maximum height in$ (1) $3 seconds$ (2) $1.5 seconds$ (3) $2 seconds$ (4) $2.5 seconds$ 135. $Let [t] denote the greatest integer less than or equal to t . If the function f (x) = ⎩ ⎨ ⎧ b^{2} sin (\frac{π}{2} [\frac{π}{2} (cos x + sin x) cos x]), \frac{s i n x - \frac{1}{2} s i n 2 x}{x ^{3}}, a, x < 0 x > 0 x = 0 is continuous at x = 0, then a^{2} + b^{2} is equal to$ (1) $\frac{5}{8}$ (2) $\frac{1}{2}$ (3) $\frac{9}{16}$ (4) $\frac{3}{4}$ 136. $The domain of the real valued function f (x) = \frac{\| x \| - x}{x - [ x ]} is$ (1) $Z$ (2) $ϕ$ (3) $R - Z$ (4) $R$ 137. $The absolute maximum value of the function f (x) = 2 x^{3} - 3 x^{2} - 36 x + 9 defined on [- 3, 3] is$ (1) $36$ (2) $53$ (3) $63$ (4) $72$ 138. $If x = ct and y = \frac{c}{t}, find \frac{dy}{dx} at t = 2 .$ (1) $4$ (2) $0$ (3) $\frac{1}{4}$ (4) $- \frac{1}{4}$ 139. $If x = e^{θ} (sin θ - cos θ), y = e^{θ} (sin θ + cos θ), then \frac{dy}{dx} at θ = \frac{π}{4} is$ (1) $1$ (2) $0$ (3) $\frac{1}{2}$ (4) $2$ 140. $In each of the choices given below a function and an interval are given, The correct choice having a function and the associated interval for which the Lagrange's mean value theorem is not valid is$ (1) $\| x \| : [1, 5]$ (2) $log x : [1, e]$ (3) $\frac{2 x - 1}{3 x - 4} : [1, 2]$ (4) $(x - 2)^{2} (x - 4)^{2} : [2, 4]$ 141. $If f^{’} (x) = sin (log x) and y = f (\frac{2 x + 3}{3 - 2 x}), then \frac{dy}{dx} at x=1 is.$ (1) $6 sin (log 5)$ (2) $5 sin (log 6)$ (3) $12 sin (log 5)$ (4) $5 sin (log 12)$ 142. $The value of lim_{x \to 0} \frac{\| x \|}{x} is$ (1) $1$ (2) $-1$ (3) $0$ (4) $Does not exist$ 143. $If the function f (x) = (k_{1} (x - π)^{2} - 1, x \leq π k_{2} cos x, x > π) is twice differentiable, then the ordered pair (k_{1}, k_{2}) is equal to:$ (1) $(1, 0)$ (2) $(\frac{1}{2}, 1)$ (3) $(1, 1)$ (4) $(\frac{1}{2}, - 1)$ 144. $The minimum value of the function f (x) = x log x is$ (1) $- \frac{1}{e}$ (2) $- e$ (3) $\frac{1}{e}$ (4) $e$ 145. $Using differentiation, approximate value of f (x) = x^{2} - 2 x + 1 at x = 2.99 is$ (1) $3.96$ (2) $9.96$ (3) $4.98$ (4) $5.98$ 146. $The two curves x^{3} - 3 x y^{2} + 2 = 0 and 3 x^{2} y - y^{3} = 2$ (1) $cut at right angle$ (2) $cut at angle \frac{π}{4}$ (3) $touch each other$ (4) $cut at angle \frac{π}{3}$ 147. $If f (x) = sin x \cdot sin 2 x \cdot sin 3 x and f^{’ ’} (x) = a (sin bx) + c (sin dx) + e (sin kx), then the value of (a + c + e) - (b + d + k) equals$ (1) $8$ (2) $-8$ (3) $16$ (4) $12$ 148. $Let f (x) = x [\frac{x}{2}], for - 10 < x < 10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to:$ 149. $Let f (x) = lim_{r \to x} (\frac{2 r ^{2} ( ( f ( r ) ) ^{2} - f ( x ) f ( r ) )}{r ^{2} - x ^{2}} - r^{3} e^{\frac{f ( r )}{r}}) be differentiable in (- \infty, 0) \cup (0, \infty) and f (1) = 1 . Then the value of ea, such that f (a) = 0, is equal to________$ 150. $Let f (x) = x^{3} + x^{2} f^{’} (1) + x f^{’ ’} (2) + f^{’ ’ ’} (3), x \in R . Then f^{’} (10) is equal to______$