Sets, Relations & Functions PYQs Set 9 (PDF)

Set 9 of 25 · Questions 241-270 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 9 (Q241-270)
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241. $The set (x \in R : 16 (2^{x}) > 16^{\frac{- 1}{x}}) =$ (1) ${x \in R : x > 0}$ (2) ${x \in R : x < 0}$ (3) $R$ (4) ${x \in R : x > 2}$ 242. $The derivative of tan^{- 1} (\frac{sin x}{1 + cos x}) with respect to tan^{- 1} (\frac{cos x}{1 + sin x}) is$ (1) $2$ (2) $-1$ (3) $0$ (4) $-2$ 243. $f (x) = \frac{sin ^{2} x}{1 + cot x} + \frac{cos ^{2} x}{1 + tan x}, then f^{’} (\frac{π}{4}) is$ (1) $3$ (2) $\frac{1}{3}$ (3) $0$ (4) $- 3$ 244. $Let N be the set of natural numbers and a relation R on B be defined by R = {(x, y) \in N\timesN : x^{3} - 3x2y - x y^{2} + 3 y^{3} = 0} Then the relation R is:$ (1) $symmetric but neither reflexive nor transitive$ (2) $reflexive but neither symmetric nor transitive$ (3) $reflexive and symmetric, but not transitive$ (4) $an equivalence relation$ 245. $Let A, B and C be sets such that φ \neq = A \cap B \subseteq C . Then which of the following statements is not true?$ (1) $B\capC\neqϕ$ (2) $If A-B\subseteqC, then A \subseteq C$ (3) $(C \cup A) \cap (C \cup B) = C$ (4) $If A-C\subseteqB, then A \subseteq B$ 246. $Let f (x) = x^{5} + 2 x^{3} + 3 x + 1, x \in R, and g (x) be a function such that g (f (x)) = x for all x \in R . Then \frac{g ( 7 )}{g ^{’} ( 7 )} is equal to :$ (1) $14$ (2) $42$ (3) $7$ (4) $1$ 247. $Domain of the function f (x) = \frac{1}{[ x ] ^{2} - [ x ] - 6}, where [x] is greatest integer \leq x, is$ (1) $(- \infty, - 2) \cup [4, \infty)$ (2) $(- \infty, - 2) \cup [3, \infty)$ (3) $(- \infty, - 2] \cup [4, \infty)$ (4) $(- \infty, 2] \cup [3, \infty)$ 248. $If y = x^{sin x} + (sin x)^{x} then \frac{dy}{dx} at x = \frac{π}{2} is$ (1) $\frac{4}{π}$ (2) $1$ (3) $π log \frac{π}{2}$ (4) $\frac{π ^{2}}{2}$ 249. $If f is a relation from set of positive real numbers to the set of positive real numbers defined by f (x) = 3 x^{2} - 2 then f is$ (1) $one-one but not onto$ (2) $onto but not one-one$ (3) $a bijection$ (4) $not a function$ 250. $If the function f (x) defined by f (x) = \frac{x ^{100}}{100} + \frac{x ^{99}}{99} + \dots .. + \frac{x ^{2}}{2} + x + 1, then f^{’} (0) =$ (1) $100 f^{’} (0)$ (2) $100$ (3) $1$ (4) $-1$ 251. $The coordinates of a point on the curve x = a (θ + sin θ), y = a (1 - cos θ) where the tangent is inclined at an angle \frac{π}{4} to the positive X -axis, are$ (1) $(a (\frac{π}{2} - 1), a)$ (2) $(a (\frac{π}{2} + 1), a)$ (3) $(a \frac{π}{2}, a)$ (4) $(a, a)$ 252. $The limit lim_{x \to 1} \frac{1 - cos 2 ( x - 1 )}{x - 1}$ (1) $exists and is equal to 2$ (2) $exists and is equal to - 2$ (3) $does not exist$ (4) $exists and is equal to (\frac{1}{2})$ 253. $lim_{x \to - \infty} \frac{3 \| x \| ^{3} - x ^{2} + 2 \| x \| - 5}{- 5 \| x \| ^{3} + 3 x ^{2} - 2 \| x \| + 7} is equal to$ (1) $\frac{3}{5}$ (2) $\frac{- 5}{7}$ (3) $\frac{5}{7}$ (4) $\frac{- 3}{5}$ 254. $lim x \to 0 \frac{x ^{2} log ( cos x )}{log ( 1 + x ^{2} )} =$ (1) $0$ (2) $1$ (3) $-1$ (4) $\infty$ 255. $A survey shows that 73% of persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be:$ (1) $63$ (2) $36$ (3) $38$ (4) $54$ 256. $The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to:$ (1) $5/16$ (2) $9/16$ (3) $11/16$ (4) $13/16$ 257. $Which one of the following is not true always?$ (1) $if f (x) is not continuous at x = a, then it is not differentiable at x = a$ (2) $if f (x) is continuous at x = a, then it is differentiable at x = a$ (3) $if f(x) and g(x) are differentiable at x=a, then f (x) + g (x) is also differentiable at x=a$ (4) $if a function f(x) is continuous at x=a, then lim_{x \to a} f (x) exists$ 258. $Let f : N \to N defined by f (n) = (\frac{n + 1}{2} if n is odd \frac{n}{2} if n is even) then f is$ (1) $onto but not one-one$ (2) $one-one and onto$ (3) $neither one-one nor onto$ (4) $one-one but not onto$ 259. $Let R be an equivalence relation defined on a set containing 6 elements. The minimum number of ordered pairs that R should contain is$ (1) $6$ (2) $12$ (3) $36$ (4) $64$ 260. $Let f : A \to B be defined as f (x) = \frac{1}{2} - tan (\frac{πx}{2}) and g : B \to C be defined g (x) = 3 + 4 x - 4 x^{2} . If A, B, C are subsets of R and f is an onto function then the range of the function f (x) is$ (1) $(- \infty, \infty)$ (2) $[0, \infty)$ (3) $(- \frac{1}{2}, \frac{3}{2})$ (4) $[- 1, 1]$ 261. $In a class of 60 students, 25 students play cricket and 20 students play tennis and 10 students play both the games, then the number of students who play neither is$ (1) $45$ (2) $0$ (3) $25$ (4) $35$ 262. $If A = {1, 2, 3, \dots ., 10} then number of subsets of A containing only odd numbers is$ (1) $31$ (2) $32$ (3) $27$ (4) $30$ 263. $If A = {x ∣ x \in N, x \leq 5}, B = (x ∣ x \in Z, x^{2} - 5 x + 6 = 0), then the number of onto functions from A to B is$ (1) $2$ (2) $30$ (3) $23$ (4) $32$ 264. $If the function f (x), defined below, is continuous on the interval [0, 8], then f (x) = ⎩ ⎨ ⎧ x^{2} + a x + b, 3 x + 2, 2 a x + 5 b, 0 \leq x < 2 2 \leq x \leq 4 4 < x \leq 8$ (1) $a = 3, b = - 2$ (2) $a = - 3, b = 2$ (3) $a = - 3, b = - 2$ (4) $a = 3, b = 2$ 265. $If f (x) = \frac{e ^{- x} sin x}{log _{e} x} and f^{’} (x) = f (x) \cdot g (x), then g^{’} (e) =$ (1) $e^{- 2} - cosec^{2} (e)$ (2) $3e-2-cosec2(e)$ (3) $2 e^{- 2} - cosec^{2} (e)$ (4) $2 e^{- 2} + cosec^{2} (e)$ 266. $If f (x) = lim_{n \to \infty} (\frac{log ( 2 + x ) - x ^{2 n} sin x}{1 + x ^{2 n}}), 0 \leq x \leq \frac{π}{2} x \in R, then at x = 1, f (x) is$ (1) $differentiable$ (2) $discontinuous$ (3) $continuous$ (4) $continuous but not differentiable$ 267. $Let A = 1, 2, 3, \dots, 10 and B = (\frac{m}{n} : m, n \in A, m < n) and (gcd (m, n) = 1) . Then n (B) is equal to:$ (1) $36$ (2) $31$ (3) $37$ (4) $29$ 268. $A is a set having 6 distinct elements. The number of distinct functions from A to A which are not bijections is$ (1) $6! - 6$ (2) $6^{6} - 6$ (3) $6^{6} - 6!$ (4) $6 !$ 269. $Let the functions f : R \to R and g : R \to R be defined as : f (x) = (x + 2, x < 0 x^{2}, x \geq 0) and g (x) = (x^{3}, x < 13 x - 2, x \geq 1) Then, the number of points in R where (fog) (x) is NOT differentiable is equal to$ (1) $3$ (2) $1$ (3) $0$ (4) $2$ 270. $Let and be relations on the set {1, 2, ..., 50} such that = {(p, ): p is a prime and n0 is an integer} and = {(p, ): p is a prime and n = 0 or 1}. Then, the number of elements in - is_____.$