Continuity & Differentiability PYQs Set 6 (PDF)

Set 6 of 13 · Questions 151-180 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 6 (Q151-180)
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151. $If f (x) = ⎩ ⎨ ⎧ \frac{1}{∣ x ∣}, a x^{2} + b, ∣ x ∣ \geq 1 ∣ x ∣ < 1 is differentiable at every point of the domain, then the values of a and b are respectively:$ (1) $\frac{1}{2}, \frac{1}{2}$ (2) $\frac{1}{2}, - \frac{3}{2}$ (3) $\frac{5}{2}, - \frac{3}{2}$ (4) $- \frac{1}{2}, \frac{3}{2}$ 152. $If y = sec (tan^{- 1} x), then \frac{dy}{dx} at x = 1 is equal to$ (1) $\frac{1}{2}$ (2) $1$ (3) $2$ (4) $\frac{1}{2}$ 153. $If lim_{x \to \infty} (\frac{x ^{3} + 1}{x ^{2} + 1} - (αx + β)) exists and equal to 2 , then the ordered pair (α, β) of real numbers is$ (1) $(1, - 1)$ (2) $(- 2, 1)$ (3) $(- 1, 1)$ (4) $(1, - 2)$ 154. $If the function f (x) = \frac{e ^{x} ( e ^{tanx - x} - 1 ) + log _{e} ( secx + tanx ) - x}{tanx - x} is continuous at x = 0, then the value of f (0) is equal to$ (1) $\frac{2}{3}$ (2) $2$ (3) $\frac{3}{2}$ (4) $\frac{1}{2}$ 155. $If the two curves y = a^{x} and y = b^{x} intersect at angle α, then tan α =$ (1) $\frac{log a - log b}{1 + log a log b}$ (2) $\frac{log a + log b}{1 - log a log b}$ (3) $\frac{π}{4}$ (4) $\frac{π}{2}$ 156. $If a function f (x) defined by f (x) = (a e^{x} + b e^{- x}, - 1 \leq x < 1 c x^{2}, 1 \leq x \leq 3 a x^{2} + 2 cx, 3 < x \leq 4) be continuous for some a, b, c \in R and f^{’} (0) + f^{’} (2) = e, then the value of a is:$ (1) $\frac{e}{e ^{2} + 3 e + 13}$ (2) $\frac{e}{e ^{2} - 3 e + 13}$ (3) $\frac{e}{e ^{2} - 3 e - 13}$ (4) $\frac{1}{e ^{2} - 3 e + 13}$ 157. $The sum of two positive numbers is given. If the sum of their cubes is minimum, then$ (1) $one is thrice the other$ (2) $they are equal$ (3) $one is twice the other$ (4) $they are unequal$ 158. $The maximum area of a right angled triangle with hypotenuse h is$ (1) $h^{2} /2 2$ (2) $h^{2} /2 .$ (3) $h^{2} /2$ (4) $h^{2} /4$ 159. $Let [x] denote the largest integer \leq x . If the number of solutions of sin x 4 cos^{2} x = \frac{2 + x - [ x ]}{1 - x + [ x ]} is k, then for x \in (\frac{π}{4}, \frac{π}{3}), the value of k^{tan^{2} x}$ (1) $is equal to 1$ (2) $lies in between 2^{1} and 2^{3}$ (3) $is equal to zero$ (4) $lies in between 1 2^{3} and \frac{1}{2}$ 160. $Let y = y (x) be a function of x satisfying y 1 - x^{2} = k - x 1 - y^{2} where k is a constant and y (\frac{1}{2}) = - \frac{1}{4} . Then \frac{dy}{dx} at x = \frac{1}{2}, is equal to:$ (1) $\frac{5}{7}$ (2) $- \frac{5}{4}$ (3) $- \frac{5}{2}$ (4) $\frac{5}{2}$ 161. $If f (x) = log_{e} (e^{2 x} (\frac{3 x + 5}{5 - 3 x})^{\frac{2}{3}}), x \neq = \frac{- 5}{3}, \frac{5}{3}, then the value of \frac{df}{dx} at x = 1, is$ (1) $\frac{5}{4}$ (2) $\frac{7}{4}$ (3) $\frac{11}{4}$ (4) $\frac{13}{4}$ 162. $Let y = log_{e} (\frac{1 - x ^{2}}{1 + x ^{2}}), - 1 < x < 1 . Then at x = \frac{1}{2}, the value of 225 (y^{’} - y^{’ ’}) is equal to$ (1) $732$ (2) $746$ (3) $742$ (4) $736$ 163. $Consider two families of curves y^{2} = 4 ax (a is a parameter) and x^{2} + \frac{y ^{2}}{2} = c^{2} (c is parameter). If one curve from each family is chosen, then the angle between those two curves is$ (1) $π$ (2) $\frac{π}{4}$ (3) $\frac{3 π}{4}$ (4) $\frac{π}{2}$ 164. $(x \in R / \frac{\| x \| ^{2} - 2 \| x \| - 8}{log ( 2 - x - x ^{2} )} is a real number) =$ (1) $(- \infty, - 4] \cup [4, \infty)$ (2) $ϕ$ (3) $(- 1, 2)$ (4) $(- \infty, - 4] \cup (- 1, 2) \cup [4, \infty)$ 165. $The range of the real valued function f (x) = \| x - 2 \| + \| x - 3 \| is$ (1) $[3, \infty)$ (2) $[1, \infty)$ (3) $[2, \infty)$ (4) $(0, 2] \cup [3, \infty)$ 166. $If an error of 0.02 sq. cm is found in the surface area of a sphere when its radius is measured as 10 cm, then the approximate error that occurs in the volume of the sphere, in cubic centimetres, is$ (1) $0.2$ (2) $0.01$ (3) $0.3$ (4) $0.1$ 167. $The nearest approximate value of 2023 is$ (1) $(6.6)^{2}$ (2) $44.9778$ (3) $(6.8)^{2}$ (4) $44.7777$ 168. $The function f (x) = (\| x - 3 \|, x \geq 1 \frac{x ^{2}}{4} - \frac{3 x}{2} + \frac{13}{4}, x < 1) is$ (1) $continuous at x = 1$ (2) $differentiable at x = 1$ (3) $continuous at x = 3$ (4) $All of these$ 169. $The function f (x) = x 1 - x, where x \in (0, 1) has local maximum at x =$ (1) $\frac{2}{3}$ (2) $\frac{1}{4}$ (3) $\frac{3}{4}$ (4) $\frac{1}{3}$ 170. $The maximum value of f (x) = \frac{log x}{x} (x \neq = 0, x \neq = 1) is$ (1) $e$ (2) $\frac{1}{e}$ (3) $e^{2}$ (4) $1 e^{2}$ 171. $Derivative of tan^{- 1} (\frac{x}{1 - x ^{2}}) with respect to sin^{- 1} (3 x - 4 x^{3}) is$ (1) $\frac{1}{1 - x ^{2}}$ (2) $\frac{3}{1 - x ^{2}}$ (3) $3$ (4) $\frac{1}{3}$ 172. $The value of lim_{x \to 0} \frac{5 ^{x} - 5 ^{- x}}{2 x} =$ (1) $log 5$ (2) $0$ (3) $1$ (4) $2 log 5$ 173. $The values of x at the stationary points of f (x) = x^{3} + 3 x^{2} - 2 are$ (1) $0,2$ (2) $1,2$ (3) $0, - 2$ (4) $1,1$ 174. $If y (x) = x^{x}, x > 0, then y " (2) - 2 y ’ (2) is equal to$ (1) $8 log_{e} 2 - 2$ (2) $4 log_{e} 2 + 2$ (3) $4 (log_{e} 2)^{2} - 2$ (4) $4 (log_{e} 2)^{2} + 2$ 175. $If x = 2 sin θ - sin 2 θ and y = 2 cos θ - cos 2 θ, θ \in (0, 2 π), then \frac{d ^{2} y}{d x ^{2}} at θ = π is:$ (1) $\frac{3}{8}$ (2) $\frac{3}{2}$ (3) $\frac{3}{4}$ (4) $- \frac{3}{4}$ 176. $If the function f (x) = a x^{3} + b x^{2} + 11 x - 6 satisfies the conditions of Rolle's theorem in [1, 3] and f^{’} (2 + \frac{1}{3}) = 0 then a + b =$ (1) $-5$ (2) $-3$ (3) $4$ (4) $7$ 177. $If f (x) = x^{4} - x^{3} + 7 x^{2} + 14, then what is the value of f^{’} (5) ?$ (1) $594$ (2) $549$ (3) $954$ (4) $495$ 178. $If a function f (x) defined by f (x) = ⎩ ⎨ ⎧ a x + b, 2 x^{2} + 2 b x - \frac{a}{2}, 7, x \leq - 1 - 1 < x < 1 x \geq 1 is continuous on R, then (a, b) =$ (1) $(- 22, - 3)$ (2) $(22, - 3)$ (3) $(11, - 6)$ (4) $(- 22, - 6)$ 179. $For a, b > 0, let f (x) = ⎩ ⎨ ⎧ \frac{t a n (( a + 1 ) x ) + b t a n x}{x}, 3, \frac{a x + b ^{2} x ^{2} - a x}{b a x x}, x < 0 x = 0 x > 0 be a continous function at x = 0 . Then \frac{b}{a} is equal to :$ (1) $6$ (2) $4$ (3) $5$ (4) $8$ 180. $Let f : R \to R be a thrice differentiable odd function satisfying f^{'} (x) \geq 0, f^{''} (x) = f (x), f (0) = 0, f^{'} (0) = 3 . Then 9 f (lo g_{e} 3) is equal to ______.$