Sets, Relations & Functions PYQs Set 10 (PDF)

Set 10 of 25 · Questions 271-300 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 10 (Q271-300)
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271. $If cos y = x cos (a + y) with cos a \neq = \pm 1, then \frac{dy}{dx} is equal to$ (1) $\frac{sin a}{cos ^{2} ( a + y )}$ (2) $\frac{cos ^{2} ( a + y )}{sin a}$ (3) $\frac{cos a}{sin ^{2} ( a + y )}$ (4) $\frac{cos ^{2} ( a + y )}{cos a}$ 272. $For α \in N, consider a relation R on N given by R = {(x, y) : 3 x + αy is a multiple of 7} . The relation R is an equivalence relation if and only if:$ (1) $α = 14$ (2) $α is a multiple of 4$ (3) $4 is the remainder when a is divided by 10$ (4) $4 is the remainder when a is divided by 7$ 273. $Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b \in R, g^{’} (a) = 5 and g (a) = b, then f^{’} (b) is equal to:$ (1) $\frac{1}{5}$ (2) $\frac{2}{5}$ (3) $1$ (4) $5$ 274. $If 2 f (sin x) + f (cos x) = x, then f^{'} (x) =$ (1) $\frac{1}{1 - x ^{2}}$ (2) $\frac{- 1}{1 - x ^{2}}$ (3) $\frac{x}{1 - x ^{2}}$ (4) $\frac{- x}{1 - x ^{2}}$ 275. $If f (x) = a x^{2} + bx + c satisfies f (1) + 2 f (2) = 0 and 2 f (1) + f (2) = 0, then 3 a + b =$ (1) $2$ (2) $-1$ (3) $0$ (4) $1$ 276. $Binary operation * on R - -1 defined by a * b = \frac{a}{b + 1} is$ (1) $* is associative and commutative$ (2) $* is neither associative nor commutative$ (3) $* is commutative but not associative$ (4) $* is associative but not commutative$ 277. $If r = a e^{θ cot α} where a and α are real numbers, then \frac{d ^{2} r}{d θ ^{2}} - 4 r cot^{2} α is$ (1) $r$ (2) $\frac{1}{r}$ (3) $1$ (4) $0$ 278. $If f : [2, \infty) \to R is defined by f (x) = x^{2} - 4 x + 5, then the range of f is$ (1) $R$ (2) $[1, \infty)$ (3) $[4, \infty)$ (4) $[5, \infty)$ 279. $If (xe)^{y} = e^{x}, then \frac{dy}{dx} is$ (1) $\frac{1}{( 1 + log x ) ^{2}}$ (2) $\frac{log x}{( 1 + log x )}$ (3) $\frac{e ^{x}}{x ( y - 1 )}$ (4) $\frac{log x}{( 1 + log x ) ^{2}}$ 280. $If a relation R on the set 1, 2, 3 be defined by R = (1, 1), then R is$ (1) $Reflexive and transitive$ (2) $Symmetric and transitive$ (3) $Only symmetric$ (4) $Reflexive and symmetric$ 281. $The set A = x : \| 2x+3 \| <7 is equal to the set$ (1) $D = (x : 0 < x + 5 < 7)$ (2) $B = (x : - 3 < x < 7)$ (3) $E = (x : - 7 < x < 7)$ (4) $C = {x : - 13 < 2 x < 4}$ 282. $lim x \to 0 (1 + 3 x)^{2/ x} =$ (1) $6$ (2) $e^{6}$ (3) $e^{- 6}$ (4) $e^{\frac{1}{6}}$ 283. $The domain of f (x) = sin^{- 1} (log_{2} (\frac{x}{2})) is$ (1) $0\leqx\leq1$ (2) $0\leqx\leq4$ (3) $1\leqx\leq4$ (4) $4\leqx\leq6$ 284. $Let f:g:N-{1} \to N be functions defined by f_{(} a_{)} = α, where α is the maximum of the powers of those primes p such that p^{α} divides a, and g(a) = a + l, for all a \in N -_{1} . Then, the function f+g is$ (1) $one-one but not onto$ (2) $onto but not one-one$ (3) $both one-one and onto$ (4) $neither one-one nor onto$ 285. $The minimum number of elements that must be added to the relation R = ((a, b), (b, c)) on the set (a, b, c) so that it becomes symmetric and transitive is$ (1) $4$ (2) $7$ (3) $5$ (4) $3$ 286. $Let Z be the set of integers. If A = (x \in Z : 2^{(x + 2) (x^{2} - 5 x + 6)} = 1) and B = (x \in Z : - 3 < 2 x - 1 < 9), then the number of subsets of the set A \times B, is:$ (1) $2^{15}$ (2) $2^{18}$ (3) $2^{12}$ (4) $2^{10}$ 287. $Write the set builder form of A = -1, 1 .$ (1) $A={x:x is an integer }$ (2) $A={x:x is a root of the equation (x^{2} + 1 = 0)$ (3) $A={x:x is a real number }$ (4) $A={x:x is a root of the equation (x^{2} = 1)$ 288. $Let f : R - 0 \to R be a function such that f (x) - 6 f (\frac{1}{x}) = \frac{35}{3 x} - \frac{5}{2} . If the l im_{x \to 0} (\frac{1}{αx} + f (x)) = β; α, β \in R, then α + 2 β is equal to$ (1) $5$ (2) $3$ (3) $4$ (4) $6$ 289. $Let f : R - (0, 1) \to R be a function such that f (x) + f (\frac{1}{1 - x}) = 1 + x . Then f (2) is equal to$ (1) $\frac{9}{2}$ (2) $\frac{9}{4}$ (3) $\frac{7}{4}$ (4) $\frac{7}{3}$ 290. $If f (f (0)) = 0, where f (x) = x^{2} + ax + b, b \neq = 0 then a + b =$ (1) $2$ (2) $1$ (3) $-1$ (4) $-2$ 291. $If the fraction part of the number is , then k is equal to :$ (1) $6$ (2) $8$ (3) $4$ (4) $14$ 292. $Consider two sets A = {x \in Z : \| (\| x - 3 \| - 3) \| \leq 1} and B = (x \in R - 1, 2 : \frac{( x - 2 ) ( x - 4 )}{x - 1} log_{e} (\| x - 2 \|) = 0) . Then the number of onto functions f : A \to B is equal to$ (1) $32$ (2) $79$ (3) $62$ (4) $81$ 293. $Let S be the set of all real numbers. A relation R has been defined on S by aRb \Leftrightarrow \| a - b \| \leq 1, then R is$ (1) $symmetric and transitive but not reflexive$ (2) $reflexive and transitive but not symmetric$ (3) $reflexive and symmetric but not transitive$ (4) $an equivalence relation$ 294. $If y = tan^{- 1} (\frac{1}{1 + x + x ^{2}}) + tan^{- 1} (\frac{1}{x ^{2} + 3 x + 3}) + tan^{- 1} (\frac{1}{x ^{2} + 5 x + 7}) + \dots . n terms then y^{’} (0) is$ (1) $- \frac{n ^{2}}{1 + n ^{2}}$ (2) $\frac{π}{2}$ (3) $\frac{2 n}{1 + n ^{2}}$ (4) $\frac{n ^{2}}{1 + n ^{2}}$ 295. $On the set Q of all rational numbers the operation * which is both associative and commutative is given by a * b =$ (1) $a + b + ab$ (2) $a^{2} + b^{2}$ (3) $ab + 1$ (4) $2 a + 3 b$ 296. $The total number of functions, f:{1,2,3,4}.(1,2,3,4,5,6} such that f(1) + f(2) = f(3), is equal to :$ (1) $60$ (2) $90$ (3) $108$ (4) $126$ 297. $If f (x) = - sin x cos x 0 2 sin 2 x 4 sin^{2} x - cos x 4 cos^{2} x 2 sin 2 x sin x, then f (\frac{5 π}{4}) + f^{'} (\frac{5 π}{4}) =$ (1) $0$ (2) $-1$ (3) $-2$ (4) $-4$ 298. $If x \neq = 0 and f (x) satisfies 8 f (x) + 6 f (\frac{1}{x}) = x + 5, then \frac{d}{dx} (x^{2} f (x)) at x = 1 is$ (1) $- 1/14$ (2) $25/14$ (3) $9/14$ (4) $19/14$ 299. $Let A = {x : ∣ x^{2} - 10∣ \leq 6} and B = {x : ∣ x - 2∣ > 1} . Then$ (1) $A \cup B = (- \infty, 1] \cup (2, \infty)$ (2) $B - A = (- \infty, - 4) \cup (- 2, 1) \cup (4, \infty)$ (3) $A - B = [2, 3)$ (4) $A \cap B = [- 4, - 2] \cup [3, 4]$ 300. $Let A = {n N : H.C.F. (n, 45) = 1} and Let B = {2k : k {1, 2,..., 100}}. Then the sum of all the elements of AB is__________.$