Sets, Relations & Functions PYQs Set 6 (PDF)

Set 6 of 25 · Questions 151-180 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 6 (Q151-180)
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.
151. $Let f : R \to R be a function defined f (x) = \frac{x}{( 1 + x ^{4} ) ^{1/4}} and g (x) = f (f (f (f (x)))) then 18 \int_{0}^{25} x^{2} g (x) dx$ (1) $33$ (2) $36$ (3) $42$ (4) $39$ 152. $If f (x) is an even function and f^{'} (x) exists, then f^{'} (c) + f^{'} (- c) is$ (1) $\geq0$ (2) $< 0$ (3) $> 0$ (4) $0$ 153. $The domain of the function f (x) = log_{0.5} x! is$ (1) ${0, 1, 2, 3, \dots}$ (2) ${1, 2, 3, \dots}$ (3) $(0, \infty)$ (4) ${0, 1}$ 154. $If f : [- 3, 2] \to [0, 3 x] is an onto function defined by f (n) = (2 + 3 n, - 3 \leq n \leq - 1 n^{2/3}, - 1 \leq n \leq 2), then x =$ (1) $1$ (2) $2$ (3) $4$ (4) $6$ 155. $If f (x) = (1 + 6 x - 3 x^{2}, x \leq 1 x + log_{2} (b^{2} + 7), x > 1) is continuous at all real x, then b =$ (1) $\pm 1$ (2) $0$ (3) $\pm 5$ (4) $\pm 2$ 156. $Let D = R - 0, 1 and f : D \to D, g : D \to D and h : D \to D be three functions defined by f (x) = \frac{1}{x}; g (x) = 1 - x and h (x) = \frac{1}{1 - x} . If j : D \to D is such that (gojof) (x) = f (x) for all x \in D, then which one of the following is j (x) ?$ (1) $(f o g) (x)$ (2) $f (x)$ (3) $g (x)$ (4) $(g o h) (x)$ 157. $If y^{cos x} = x^{sin y}, then \frac{dy}{dx} =$ (1) $\frac{y ( xsin xlog y + sin y )}{x ( cos x - ylog xcos y )}$ (2) $\frac{y ( xsin xlog x - sin y )}{x ( cos x + ylog xcos y )}$ (3) $\frac{y ( sin y - xlog y )}{x ( x - ycos y ( log x ) )}$ (4) $\frac{y ( sin y + xlog y )}{x ( x + ycos y ( log x ) )}$ 158. $If \frac{y}{x} + 4 \frac{x}{y} = 4, then \frac{dy}{dx}$ (1) $xy$ (2) $x / y$ (3) $-4$ (4) $4$ 159. $Let a relation R on N \times N be defined as: (x_{1}, y_{1}) R (x_{2}, y_{2}) if and only if x_{1} \leq x_{2} or y_{1} \leq y_{2} . Consider the two statements: (I) R is reflexive but not symmetric. (II) R is transitive Then which one of the following is true?$ (1) $Both (I) and (II) are correct.$ (2) $Only (II) is correct.$ (3) $Neither (I) nor (II) is correct.$ (4) $Only (I) is correct.$ 160. $Let f (x) = \frac{2 ^{x + 2} + 16}{2 ^{2 x + 1} + 2 ^{x + 4} + 32} Then the value of 8 (f (\frac{1}{15}) + f (\frac{2}{15}) + \dots + f (\frac{59}{15})) is equal to$ (1) $92$ (2) $118$ (3) $102$ (4) $108$ 161. $If [a, b] is the range of the function \frac{x + 2}{2 x ^{2} + 3 x + 6} for x \in R, then$ (1) $a<0,b<0$ (2) $a < 0, b > 0$ (3) $a > 0, b > 0$ (4) $a>0,b<0$ 162. $The set A has 4 elements and the set B has 5 elements then the number of injective mappings that can be defined from A to B is$ (1) $144$ (2) $72$ (3) $60$ (4) $120$ 163. $Let f be a function such that f (x) + 3 f (\frac{24}{x}) = 4 x, x \neq = 0 . Then f (3) + f (8) is equal to$ (1) $13$ (2) $11$ (3) $10$ (4) $12$ 164. $The number of elements in the set S = {θ \in [0, 2 π] : 3 co s^{4} θ -5 co s^{2} θ -2 s i n^{6} θ + 2 = 0} is$ (1) $10$ (2) $8$ (3) $12$ (4) $9$ 165. $If the set x contains 7 elements and set y contains 8 elements, then the number of bijections from x to y is$ (1) $0$ (2) $7 !$ (3) $^{8} P_{7}$ (4) $8 !$ 166. $The number of distinct real solutions of the equation x \| x + 4 \| + 3 \| x + 2 \| + 10 = 0 is$ (1) $2$ (2) $1$ (3) $3$ (4) $0$ 167. $For x \neq = - 1, y \neq = - 1, if x = \frac{1 - 3 y}{1 + 3 y}, then \frac{dx}{dy} =$ (1) $\frac{- 6 ( 1 - x ) ^{2}}{( 1 + x ) ^{4}}$ (2) $\frac{- ( 1 + x ) ^{4}}{6 ( 1 - x ) ^{2}}$ (3) $\frac{4 ( 1 - x ) ^{4}}{( 1 + x ) ^{6}}$ (4) $\frac{- 6 ( 1 + x ) ^{2}}{( 1 - x ) ^{4}}$ 168. $If g is the inverse of a function f and f'(x) = \frac{1}{1 + x ^{5}}, then g'(x) equal to$ (1) $\frac{1}{1 + g ( x ) ^{5}}$ (2) $1 + {g(x)}5$ (3) $1 + x5$ (4) $5x4$ 169. $If x = e^{y + e^{y + e^{y + \dots}}}, then \frac{dy}{dx} =$ (1) $\frac{1 - x}{x}$ (2) $\frac{1}{x}$ (3) $\frac{x}{1 + x}$ (4) $\frac{1 + x}{x}$ 170. $Let f : R - (\frac{- 1}{2}) \to R and g : R - (\frac{- 5}{2}) \to R be defined as f (x) = \frac{2 x + 3}{2 x + 1} and g (x) = \frac{\| x \| + 1}{2 x + 5} . Then the domain of the function fog is :$ (1) $R - {- \frac{5}{2}}$ (2) $R$ (3) $R - {- \frac{7}{4}}$ (4) $R - {- \frac{5}{2}, - \frac{7}{4}}$ 171. $lim_{x \to 0} \frac{( 1- cos 2 x ) ( 3 + cos x )}{x t an 4 x} is equal to$ (1) $- \frac{1}{4}$ (2) $\frac{1}{2}$ (3) $1$ (4) $2$ 172. $If f is a function defined on (0, 1) by f (x) = min x- [x], -x- [x], then (fofofof) (x) is equal to ⟶ ([\cdot] greatest integer function)$ (1) $x$ (2) $- x$ (3) $4 x$ (4) $2 x$ 173. $On the set Z, of all integers * is defined by a * b = a + b - 5 . If 2 * (x * 3) = 5 then x =$ (1) $5$ (2) $10$ (3) $0$ (4) $3$ 174. $\sum_{r = 1}^{n} (2 r - 1) = x then lim_{n \to \infty} (\frac{1 ^{3}}{x ^{2}} + \frac{2 ^{3}}{x ^{2}} + \frac{3 ^{3}}{x ^{2}} + \dots .. + \frac{n ^{3}}{x ^{2}}) =$ (1) $\frac{1}{2}$ (2) $1$ (3) $\frac{1}{4}$ (4) $4$ 175. $If the function f (x) = ((tan (\frac{π}{4} + x))^{1/ x} for x \neq = 0 K for x = 0) is continuous at x = 0, then K = ?$ (1) $e$ (2) $e^{- 1}$ (3) $e^{2}$ (4) $e^{- 2}$ 176. $Let f : R \to R be defined by f (x) = \frac{1}{x} \forall x \in R, then f is$ (1) $onto$ (2) $not defined$ (3) $one-one$ (4) $bijective$ 177. $If f (x) and g (x) are two real valued functions such that f (x) = 3 x - 2 and g (x) = x^{2} + 2 then [(gof) + (fog)] (x) =$ (1) $2 g (x) + 2 f (x)$ (2) $12 g (x) - 4 f (x) - 22$ (3) $3 g (x) + f (x) - 2$ (4) $2 f (x) + 4 g (x) - 32$ 178. $If 3 sin xy + 4 cos xy = 5, then \frac{dy}{dx} is equal to$ (1) $\frac{3 sin xy + 4 cos xy}{3 cos xy - 4 sin xy}$ (2) $\frac{3 cos xy + 4 sin xy}{4 cos xy - 3 sin xy}$ (3) $\frac{- y}{x}$ (4) $\frac{x}{y}$ 179. $Let A = (n \in N ∣ n^{2} \leq n + 10, 000), B = 3k+1 ∣ k \in N and C = 2k ∣ k \in N, then the sum of all the elements of the set A \cap (B - C) is equal to____.$ 180. $Let A = {1, 2, 3, 4} and R be a relation on the set A x A defined by R = {((a, b), (c, d)): 2a + 3b = 4c + 5d}. Then the number of elements in R is$