Continuity & Differentiability PYQs Set 7 (PDF)

Set 7 of 13 · Questions 181-210 of 371

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 7 (Q181-210)
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181. $If f (x) = 3 x^{3} - 9 x^{2} - 27 x + 15, then the maximum value of f (x) is$ (1) $-66$ (2) $30$ (3) $-30$ (4) $66$ 182. $Suppose f (x) = x (x + 3) (x - 2), x \in [- 1, 4] . Then, a value of c in (- 1, 4) satisfying f^{’} (c) = 10 is$ (1) $2$ (2) $\frac{5}{2}$ (3) $3$ (4) $\frac{7}{2}$ 183. $Let f (x) = (x + 1)^{2} - 1, x \geq - 1 . Then (x : f (x) = f^{- 1} (x)) =$ (1) ${0, 1, - 1}$ (2) $(- 1, \frac{- 3 + i 3}{2}, \frac{- 3 - i 3}{2})$ (3) ${0, - 1}$ (4) $ϕ$ 184. $f (x) = (\frac{x - 4}{\| x - 4 \|} + a, x < 4 a + b, x = 4 \frac{x - 4}{\| x - 4 \|} + b, x > 4) If f (x) given above is continuous at x = 4, then find the values of ' a' and ' b'.$ (1) $a = 1, b = - 1$ (2) $a = - 1, b = 1$ (3) $a = 1, b = 1$ (4) $a = - 1, b = - 1$ 185. $Let f : R \to R be a function defined by f (x) = \frac{2 x + 1}{3} . If α is an element in the domain of f whose image is \frac{1}{α}, then the sum of all possible values of such α is$ (1) $\frac{- 1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{5}{2}$ (4) $0$ 186. $If the angle between the curves y = e^{2 (1 + x) - 4} and x^{2} y = 1 at the point (1, 1) is θ, then \| sin θ \| + \| cos θ \| =$ (1) $7/5$ (2) $3/5$ (3) $8/7$ (4) $6/5$ 187. $If (a + 2 bc os x) (a - 2 bc os y) = a^{2} - b^{2}, where a > b > 0, then \frac{dx}{dy} at (\frac{π}{4}, \frac{π}{4}) is:$ (1) $\frac{2 a + b}{2 a - b}$ (2) $\frac{a - b}{a + b}$ (3) $\frac{a + b}{a - b}$ (4) $\frac{a - 2 b}{a + 2 b}$ 188. $The maximum slope of the curve y = - x^{3} + 3 x^{2} + 2 x - 27 is$ (1) $1$ (2) $23$ (3) $5$ (4) $-23$ 189. $If log 10 (\frac{x ^{3} - y ^{3}}{x ^{3} + y ^{3}}) = 2, then \frac{dy}{dx} =$ (1) $\frac{x}{y}$ (2) $- \frac{y}{x}$ (3) $- \frac{x}{y}$ (4) $\frac{y}{x}$ 190. $If the minimum value of cos (sinh (log x) + cosh (log x)) is k, then cosh (k + 1) =$ (1) $\frac{e + e ^{- 1}}{2}$ (2) $\frac{e ^{2} + e ^{- 2}}{2}$ (3) $e$ (4) $1$ 191. $Let f be a differentiable function such that f (1) = 2 and f^{’} (x) = f (x) for all x \in R . If h (x) = f (f (x)), then h^{’} (1) is equal to :$ (1) $2 e^{2}$ (2) $4e$ (3) $2 e$ (4) $4 e^{2}$ 192. $If the curves, \frac{x ^{2}}{a} + \frac{y ^{2}}{b} = 1 and \frac{x ^{2}}{c} + \frac{y ^{2}}{d} = 1 intersect each other at an angle of 90^{\circ}, then which of the following relations is true?$ (1) $a + b = c + d$ (2) $a - b = c - d$ (3) $ab = \frac{c + d}{a + b}$ (4) $a - c = b + d$ 193. $If the error committed in measuring the radius of a circle is 0.05 %, then the corresponding error in calculating its area would be$ (1) $0.05 %$ (2) $0.0025 %$ (3) $0.25 %$ (4) $0.1 %$ 194. $Let f (x) = x^{3} + x^{2} f^{’} (1) + 2 x f^{’ ’} (2) + f^{’ ’ ’} (3), x \in R . Then the value of f^{’} (5) is:$ (1) $\frac{117}{5}$ (2) $\frac{657}{5}$ (3) $\frac{2}{5}$ (4) $\frac{62}{5}$ 195. $If x = 3 tan t and y = 3 sec t, then the value of \frac{d ^{2} y}{d x ^{2}} at t = \frac{π}{4}, is :$ (1) $\frac{1}{3 2}$ (2) $\frac{1}{6 2}$ (3) $\frac{3}{2 2}$ (4) $\frac{1}{6}$ 196. $If log (x + y) = log xy + 3, then \frac{dy}{dx} =$ (1) $(\frac{y}{x})^{2}$ (2) $- (\frac{x}{y})^{2}$ (3) $- (\frac{y}{x})^{2}$ (4) $(\frac{x}{y})^{2}$ 197. $The distance ' s' in meters travelled by a particle in ' t' seconds is given by s = \frac{2 t ^{3}}{3} - 18 t + \frac{5}{3} . The acceleration when the particle comes to rest is$ (1) $12 m^{2} / sec .$ (2) $3 m^{2} / sec .$ (3) $18 m^{2} / sec .$ (4) $10 m^{2} / sec .$ 198. $If x = a (t - \frac{1}{t}), y = a (t + \frac{1}{t}) where t be the parameter then \frac{dy}{dx} = ?$ (1) $\frac{y}{x}$ (2) $\frac{- x}{y}$ (3) $\frac{x}{y}$ (4) $\frac{- y}{x}$ 199. $The integral value of n for which lim_{x \to 0} \frac{( cos x - 1 ) ( cos x - e ^{x} )}{x ^{n}} is a finite non zero real number is$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 200. $Derivative of lo g_{e^{2}} (lo g x) with respect to x is$ (1) $\frac{2}{x log x}$ (2) $\frac{1}{x log x}$ (3) $\frac{1}{x log x ^{2}}$ (4) $\frac{2}{log x}$ 201. $If x log_{e} (log_{e} x) - x^{2} + y^{2} = 4 (y > 0), then \frac{dy}{dx} at x = e is equal to :$ (1) $\frac{( 1 + 2 e )}{2 4 + e ^{2}}$ (2) $\frac{( 2 e - 1 )}{2 4 + e ^{2}}$ (3) $\frac{( 1 + 2 e )}{4 + e ^{2}}$ (4) $\frac{e}{4 + e ^{2}}$ 202. $If y^{2} + log_{e} (cos^{2} x) = y, x \in (- \frac{π}{2}, \frac{π}{2}), then :$ (1) $(y^{’} (0)) + \| y " (0) \| = 3$ (2) $(y^{’} (0)) + \| y " (0) \| = 1$ (3) $y " (0) = 0$ (4) $\| y " (0) \| = 2$ 203. $If f (x) = (\frac{1 + kx - 1 - kx}{x} if - 1 \leq x < 0 \frac{2 x + 1}{x - 1} if 0 \leq x \leq 1) is continuous at x = 0, then the value of k is$ (1) $1$ (2) $-1$ (3) $0$ (4) $2$ 204. $lim_{x \to 0^{-}} \frac{\frac{1}{2} ( 1 - cos ^{2} x )}{x} is equal to$ (1) $\frac{1}{2}$ (2) $\frac{- 1}{2}$ (3) $-1$ (4) $does not exist$ 205. $If f (x) is a real valued function defined by f (x) = \frac{a x ^{10} + b x ^{8} + c x ^{6} + d x ^{4} + e x ^{2} + 12 x + 15}{x} (x \neq = 0) and f (4) = - 4, then f (- 4) =$ (1) $28$ (2) $39$ (3) $4$ (4) $24$ 206. $The value of c in the Lagrange’s mean value theorem for the function f (x) = x^{3} - 4 x^{2} + 8 x + 11, when x \in (0, 1) is:$ (1) $\frac{4 - 7}{3}$ (2) $\frac{4 - 5}{3}$ (3) $\frac{4 + 7}{3}$ (4) $\frac{4 + 5}{3}$ 207. $The derivative of f (sec x) with respect to g (tan x) at x = \frac{π}{4}, where f^{’} (2) = 4 and g^{’} (1) = 2, is$ (1) $2$ (2) $\frac{1}{2}$ (3) $2$ (4) $\frac{1}{2 2}$ 208. $The function f (x) = \frac{4 x ^{3} - 3 x ^{2}}{6} - 2 sin x + (2 x - 1) cos x :$ (1) $increases in (\frac{1}{2}, \infty)$ (2) $decreases (- \infty, \frac{1}{2})$ (3) $increases in (- \infty, \frac{1}{2})$ (4) $decreases (\frac{1}{2}, \infty)$ 209. $If the function f (x) = \frac{tan ( tan x ) - sin ( sin x )}{tan x - sin x} is continuous at x = 0, then f (0) is equal to_______.$ 210. $If the function f defined on (- \frac{1}{3}, \frac{1}{3}) by f (x) = (\frac{1}{x} log_{e} (\frac{1 + 3 x}{1 - 2 x}), when x \neq = 0 k, when x = 0) is continuous, then k is equal to \dots.$