Continuity & Differentiability PYQs Set 8 (PDF)

Set 8 of 13 · Questions 211-240 of 371

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Continuity & Differentiability · Set 8 (Q211-240)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
211. $If f (x) = [x], where [x] is the greatest integer not greater than x, then f^{’} (1^{+}) =$ (1) $1$ (2) $2$ (3) $0$ (4) $-1$ 212. $Consider the function f : R \to R defined by f (x) = ((2 - sin (\frac{1}{x})) \| x \|, x \neq = 00, x = 0) . Then f is$ (1) $monotonic on (- \infty, 0) \cup (0, \infty)$ (2) $not monotonic on (- \infty, 0) and (0, \infty)$ (3) $monotonic on (0, \infty) only$ (4) $monotonic on (- \infty, 0) only$ 213. $Let f : R - 0 \to R be a function satisfying f (\frac{x}{y}) = \frac{f ( x )}{f ( y )} for all x, y, f (y) \neq = 0 . If f^{’} (1) = 2024, then$ (1) $x f^{’} (x) - 2024 f (x) = 0$ (2) $xf f^{’} (x) + 2024 f (x) = 0$ (3) $x f^{’} (x) + f (x) = 2024$ (4) $x f^{’} (x) - 2023 f (x) = 0$ 214. $Let f : R \to R be a positive increasing function with lim_{x \to \infty} \frac{f ( 3 x )}{f ( x )} = 1. Then lim_{x \to \infty} \frac{f ( 2 x )}{f ( x )} .$ (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $3$ (4) $1$ 215. $Let f : R \to R be defined as f (x) = ⎩ ⎨ ⎧ \frac{a - b c o s 2 x}{x ^{2}}; x^{2} + c x + 2; 2 x + 1; x < 0 0 \leq x \leq 1 x > 1 If f is continuous everywhere in R and m is the number of points where f is NOT differentiable then m + a + b + c equals :$ (1) $1$ (2) $4$ (3) $3$ (4) $2$ 216. $Let f (x) = \frac{x}{1 - x} + \frac{1 - x}{x} . If lim_{x \to m} f (x) = 5/2, then set of all possible finite values of 1 and m is$ (1) ${0, 1}$ (2) $(0, \frac{1}{3}, \frac{2}{3})$ (3) $(0, \frac{2}{5}, \frac{3}{5})$ (4) $(\frac{1}{5}, \frac{4}{5})$ 217. $If xy = tan^{- 1} (xy) + cot^{- 1} (xy), then (\frac{dy}{dx})_{(4, 2)} = (where x, y \in R)$ (1) $\frac{- 1}{2}$ (2) $\frac{1}{2}$ (3) $2$ (4) $-2$ 218. $If f (x) = sin (π^{2}) x + cos (- π^{2}) x then f^{’} (x) is, here (π^{2}) and (- π^{2}) are greatest integer function not greater than its value$ (1) $-1$ (2) $sin 9 x + cos 9 x$ (3) $9 cos 9 x - 10 sin 10 x$ (4) $0$ 219. $An equilateral triangle is of side 10 units. In measuring the side, if an error of 0.05 unit is made. Then, the percentage error in the area of the triangle is$ (1) $5$ (2) $4$ (3) $1$ (4) $0.5$ 220. $Let f (x) = {x^{3} - x^{2} + 10 x - 7, - 2 x + lo g_{2} (b^{2} - 4), x \leq 1 x > 1 Then the set of all values of b, for which f (x) has maximum value at x = 1, is$ (1) $(- 6, - 2)$ (2) $(2, 6)$ (3) $[- 6, - 2) \cup (2, 6]$ (4) $[- 6, - 2) \cup (2, 6]$ 221. $If the function f (x) = (\frac{1 - cos x}{x ^{2}} for x \neq = 0 k for x = 0) is continuous at x = 0, then the value of k is$ (1) $1$ (2) $0$ (3) $\frac{1}{2}$ (4) $-1$ 222. $If y (x) = sin x 271 cos x 281 sin x + cos x + 1 271, x \in R, then \frac{d ^{2} y}{d x ^{2}} + y is equal to$ (1) $28$ (2) $27$ (3) $-1$ (4) $1$ 223. $Let f : R \to R be a function such that f (x) = x^{3} + x^{2} f^{’} (1) + x f^{’ ’} (2) + 6, x \in R, then f (2) equals$ (1) $30$ (2) $-4$ (3) $-2$ (4) $8$ 224. $If f (x) = (x for x \leq 00 for x > 0) then f (x) at x = 0 is$ (1) $Continuous but not differentiable$ (2) $Not continuous but differentiable$ (3) $Continuous and differentiable$ (4) $Not continuous and not differentiable$ 225. $The value(s) of x for which the function f (x) = ⎩ ⎨ ⎧ 1 - x, (1 - x) (2 - x), 3 - x, x < 1 1 \leq x \leq 2 x > 2 fails to be continuous is(are)$ (1) $1$ (2) $2$ (3) $3$ (4) $all real numbers$ 226. $If x = a^{sin^{- 1} t}, y = a^{cos^{- 1} t}, then \frac{dy}{dx} =$ (1) $\frac{- y}{x}$ (2) $\frac{x}{y}$ (3) $\frac{y}{x}$ (4) $\frac{- x}{y}$ 227. $The value of C in mean value theorem for the function f (x) = x^{2} in [2, 4] is$ (1) $2$ (2) $4$ (3) $\frac{7}{2}$ (4) $3$ 228. $The number of discontinuities of the greatest integer function f (x) = [x], x \in (- \frac{7}{2}, 100) is$ (1) $104$ (2) $100$ (3) $102$ (4) $103$ 229. $The derivative of tan^{- 1} (\frac{1 + x ^{2} - 1}{x}) with respect to tan^{- 1} (\frac{2 x 1 - x ^{2}}{1 - 2 x ^{2}}) at x = \frac{1}{2} is:$ (1) $\frac{2 3}{3}$ (2) $\frac{2 3}{5}$ (3) $\frac{3}{10}$ (4) $\frac{3}{12}$ 230. $The function f (x) = x^{3} - 4 x^{2} + 4 x + 3 defined on [- 1, 3] has$ (1) $minimum value -6 at x = - 1$ (2) $minimum value 6 at x = 3$ (3) $minimum value 3 at x = 2$ (4) $minimum value 9 at x = 3$ 231. $For a > 0, if the function f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1 attains its maximum value at p and minimum value at q such that p^{2} = q, then a =$ (1) $\frac{1}{2}$ (2) $1$ (3) $2$ (4) $4$ 232. $The range of the real valued function f (x) = \frac{x ^{2} + x + 1}{x} is$ (1) $(- \infty, 1) \cup (1, \infty)$ (2) $(- \infty, - 1] \cup [1, \infty)$ (3) $(- \infty, - 2] \cup [3, \infty)$ (4) $(- \infty, - 1] \cup [3, \infty)$ 233. $If f (x) is twice differentiable and continuous function in x \in (a, b) also f^{'} (x) > 0 and f^{''} (x) < 0 and c \in (a, b) then \frac{f ( c ) - f ( a )}{f ( b ) - f ( c )} is greater than$ (1) $\frac{b - c}{c - a}$ (2) $1$ (3) $\frac{a + b}{b - c}$ (4) $\frac{c - a}{b - c}$ 234. $The stationary points of the curve y = 8 x^{2} - x^{4} - 4 are$ (1) $(0, - 4), (2, 12), (- 2, 12)$ (2) $(0, 4), (- 2, 12), (1, 2)$ (3) $(0, - 4), (- 1, 2), (2, 12)$ (4) $(0, 4), (- 1, 2), (1, 2)$ 235. $If g (x) is the inverse function of f (x) and f^{’} (x) = \frac{1}{1 + x ^{4}}, then g^{’} (x) is$ (1) $1 + [g (x)]^{4}$ (2) $1 - [g (x)]^{4}$ (3) $1 + [f (x)]^{4}$ (4) $1 1 + [g (x)]^{4}$ 236. $The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is$ (1) $8 π sq. units$ (2) $4 sq. units$ (3) $5 sq. units$ (4) $8 sq. units$ 237. $The perimeter of a sector is a constant. If its area is to be maximum, then the sectorial angle is$ (1) $2^{c}$ (2) $\frac{π ^{c}}{6}$ (3) $\frac{π ^{c}}{4}$ (4) $4^{c}$ 238. $If the curves y^{2} = 6x, 9 x^{2} + b y^{2} = 16 intersect each other at right angels , then the value of b is :$ (1) $4$ (2) $\frac{9}{2}$ (3) $6$ (4) $\frac{7}{2}$ 239. $The number of points of discontinuity of the function f (x) = [\frac{x ^{2}}{2}] - [x], x \in [0, 4], where [•] denotes the greatest integer function, is ______.$ 240. $Let for a differentiable function f : (0, \infty) \to R, f (x) - f (y) \geq log_{e} (\frac{x}{y}) + x - y, \forall x, y \in (0, \infty) . Then \sum_{n = 1}^{20} f^{’} (\frac{1}{n ^{2}}) is equal to_____$