Sets, Relations & Functions PYQs Set 8 (PDF)

Set 8 of 25 · Questions 211-240 of 734

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 · Sets, Relations & Functions · 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. $On the set of integers Z, define f : Z \to Z as f (n) = (\frac{n}{2}, n is even 0, n is odd) then f is$ (1) $surjective but not injective$ (2) $bijective$ (3) $injective but not surjective$ (4) $neither injective nor surjective$ 212. $At x = 1, the function f (x) = {x^{3} - 1, x - 1, 1 < x < \infty - \infty < x \leq 1 is$ (1) $continuous and differentiable$ (2) $continuous and non-differentiable$ (3) $discontinuous and differentiable$ (4) $discontinuous and non-differentiable$ 213. $Let R be a relation on Z \times Z defined by (a, b) R (c, d) if and only if ad - bc is divisible by 5. Then R is$ (1) $Reflexive and symmetric but not transitive$ (2) $Reflexive but neither symmetric not transitive$ (3) $Reflexive, symmetric and transitive$ (4) $Reflexive and transitive but not symmetric$ 214. $Let the range of the function f (x) = 6 + 16 co s x \cdot co s (\frac{π}{3} - x) \cdot co s (\frac{π}{3} + x) \cdot si n 3 x c os 6 x, x \in R be [α, β] . Then the distance of the point (α, β) from the line 3 x + 4 y + 12 = 0 is:$ (1) $11$ (2) $8$ (3) $10$ (4) $9$ 215. $lim_{x \to 0} \frac{log _{e} ( 1 + x )}{3 ^{x} - 1} =$ (1) $log_{e} 3$ (2) $0$ (3) $log_{3} e$ (4) $1$ 216. $lim_{x \to 0} (tan (x + \frac{π}{4}))^{1/ x} is equal to$ (1) $e^{2}$ (2) $e$ (3) $e^{3/2}$ (4) $e^{- 1}$ 217. $lim x \to \frac{π}{6} \frac{3 sin x - 3 cos x}{6 x - π} =$ (1) $\frac{- 1}{3}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $\frac{- 1}{2}$ 218. $Consider the following statements: Statement 1: If y = log_{10} x + log_{e} x, then \frac{dy}{dx} = \frac{log _{10} e}{x} + \frac{1}{x} Statement 2: If \frac{d}{dx} (log_{10} x) = \frac{log x}{log 10} and \frac{d}{dx} (log_{e} x) = \frac{log x}{log e}$ (1) $Statement 1 is true; Statement 2 is false$ (2) $Statement 1 is false; Statement 2 is true$ (3) $Both statements 1 and 2 are true$ (4) $Both statements 1 and 2 are false$ 219. $If y = e^{sin^{- 1} (t^{2} - 1)} & x = e^{sec^{- 1} (\frac{1}{t ^{2} - 1})} then \frac{dy}{dx} is equal to$ (1) $\frac{x}{y}$ (2) $\frac{- y}{x}$ (3) $\frac{y}{x}$ (4) $\frac{- x}{y}$ 220. $If y = log (\frac{1 + x}{1 - x})^{1/4} - \frac{1}{2} tan^{- 1} (x), then \frac{dy}{dx} at x = \frac{1}{2} equals$ (1) $\frac{- 4}{3}$ (2) $\frac{4}{3}$ (3) $\frac{- 2}{3}$ (4) $\frac{2}{3}$ 221. $R is a relation on N given by R = {(x, y) ∣ 4 x + 3 y = 20} . Which of the following belongs to R ?$ (1) $(3, 4)$ (2) $(2, 4)$ (3) $(- 4, 12)$ (4) $(5, 0)$ 222. $Let A = -3, -2, -1, 0, 1, 2, 3 . Let R be a relation on A defined by xRy if and only if 0 \leq x^{2} + 2 y \leq 4 . Let l be the number of elements in R and m be the minimum number of elements required to be added in R to make it a reflexive relation. Then l + m is equal to$ (1) $18$ (2) $20$ (3) $17$ (4) $19$ 223. $lim_{n \to \infty} (n sin \frac{2 π}{3 n} \cdot cos \frac{2 π}{3 n}) =$ (1) $\frac{2 π}{3}$ (2) $\frac{π}{6}$ (3) $\frac{π}{3}$ (4) $1$ 224. $The domain of the function f (x) = sin^{- 1} (\frac{\| x \| + 5}{x ^{2} + 1}) is (- \infty, - a] \cup [a, \infty) . Then a is equal to:$ (1) $\frac{1 + 17}{2}$ (2) $\frac{17}{2} + 1$ (3) $\frac{17 - 1}{2}$ (4) $\frac{17}{2}$ 225. $If A = 1, 2, 3, 4, 5, 6, then the number of subsets of A which contains atleast two elements is$ (1) $63$ (2) $57$ (3) $58$ (4) $64$ 226. $If y = f (x) is twice differentiable function such that at a point P, \frac{dy}{dx} = 4, \frac{d ^{2} y}{d x ^{2}} = - 3 then (\frac{d ^{2} x}{d y ^{2}})_{P} =$ (1) $\frac{64}{3}$ (2) $\frac{16}{3}$ (3) $\frac{3}{16}$ (4) $\frac{3}{64}$ 227. $If f (x) = \frac{2 ^{2 x}}{2 ^{2 x} + 1}, x ϵ R then f (\frac{1}{2023}) + f (\frac{2}{2023}) + \dots + f (\frac{2022}{2023}) is equal to$ (1) $2011$ (2) $1010$ (3) $2010$ (4) $1011$ 228. $lim x \to 0 \frac{tan 2 x - 2 tan x}{( 1 - cos x ) ( 2 ^{x} - 1 )} =$ (1) $\frac{2}{log 2}$ (2) $\frac{1}{log 4}$ (3) $4 log 2$ (4) $\frac{4}{log 2}$ 229. $If f (x) = \frac{( x + 1 ) sinh x}{e ^{2 x} tan x} and \frac{f ^{’} ( x )}{f ( x )} = \frac{1}{x + 1} + coth x + g (x) then g (x) =$ (1) $- 2 + \frac{1}{sin xcos x}$ (2) $2 - 2 cosec 2 x$ (3) $- 2 (1 + cosec 2 x)$ (4) $2 - \frac{1}{sin xcos x}$ 230. $If A = a, b, c, then the number of binary operations on A is$ (1) $3^{6}$ (2) $3^{9}$ (3) $3^{3}$ (4) $3$ 231. $If y = (x - 1)^{2} (x - 2)^{3} (x - 3)^{5}, then \frac{dy}{dx} at x = 4 is equal to$ (1) $108$ (2) $54$ (3) $36$ (4) $516$ 232. $Let f, g : N \to N such that f (n + 1) = f (n) + f (1) \forall n \in N and g be any arbitrary function. Which of the following statements is NOT true?$ (1) $f is one-one$ (2) $If fog is one-one, then g is one-one$ (3) $If g is onto, then fog is one-one$ (4) $If f is onto, then f (n) = n \forall n \in N$ 233. $If a function f is defined by f (x) = \frac{cot ^{3} x - tan x}{cos ( x + π /4 )} (x \neq = π /4), then lim_{x \to \frac{π}{4}} f (x) =$ (1) $4$ (2) $8$ (3) $\frac{8}{3}$ (4) $16$ 234. $If y = \frac{ax + b}{cx + d} and \frac{dx}{dy} = \frac{ad - bc}{P y ^{2} + Qy + R} then P + Q + R =$ (1) $(a + c)^{2}$ (2) $(a - c)^{2}$ (3) $\frac{ad - bc}{a ^{2} - c ^{2} - 2 ac}$ (4) $1 (a - c)^{2}$ 235. $If f (x) = sin^{- 1} (\frac{2 x}{1 + x ^{2}}), then f^{’} (3) is$ (1) $\frac{1}{2}$ (2) $\frac{1}{3}$ (3) $- \frac{1}{3}$ (4) $- \frac{1}{2}$ 236. $If y = \frac{x sin ^{- 1} x}{1 - x ^{2}} + log 1 - x^{2}, then \frac{dy}{dx} =$ (1) $\frac{sin ^{- 1} x}{1 - x ^{2}}$ (2) $\frac{sin ^{- 1} x}{( 1 - x ^{2} ) ^{3/2}}$ (3) $\frac{x}{1 - x ^{2}}$ (4) $\frac{x sin ^{- 1} x}{1 - x ^{2}} - \frac{2 x}{1 - x ^{2}}$ 237. $Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K % of them are suffering from both ailments, then K can not belong to the set:$ (1) ${80, 83, 86, 89}$ (2) ${84, 86, 88, 90}$ (3) ${79, 81, 83, 85}$ (4) ${84, 87, 90, 93}$ 238. $The domain and range for the function f (x) = e^{\| x \| sin x} are$ (1) $domain =R, range (0, \infty)$ (2) $domain = R, range = R$ (3) $domain =R, range = [0, \infty)$ (4) $domain =R, range = [1, \infty)$ 239. $\frac{d}{dx} (cos^{2} (cot^{- 1} \frac{2 + x}{2 - x})) is$ (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) $\frac{- 1}{2}$ (4) $\frac{- 3}{4}$ 240. $The derivative of y = tan^{- 1} (\frac{1 + sin x + 1 - sin x}{1 + sin x - 1 - sin x}) with respect to x is equal to$ (1) $-1$ (2) $0$ (3) $\pm 2$ (4) $\pm \frac{1}{2}$