Sets, Relations & Functions PYQs Set 1 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 1 (Q1-30)
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1. $Let a, b, c be real numbers. A set formed with a . b, c whose order of occurrence is pre-assigned is called .......$ (1) $An ordered triad$ (2) $An ordered pair$ (3) $Both ordered triad and ordered pair$ (4) $None of these$ 2. $lim x \to 2 \frac{x ^{3} - x ^{2} - x - 2}{2 x ^{3} - 3 x ^{2} - 3 x + 2} =$ (1) $0$ (2) $\infty$ (3) $\frac{5}{7}$ (4) $\frac{7}{9}$ 3. $lim y \to 1 (\frac{1}{y ^{2} - 1} - \frac{2}{y ^{4} - 1}) =$ (1) $\frac{1}{2}$ (2) $\frac{1}{3}$ (3) $\frac{1}{4}$ (4) $0$ 4. $lim x \to - 2^{+} ([x]^{2} - [x] - 2) + lim x \to - 3^{-} ([x]^{2} - 4 [x] + 3) =$ (1) $39$ (2) $33$ (3) $28$ (4) $44$ 5. $For x \in (0, \frac{3}{2}), let f (x) = x, g (x) = tan x and h (x) = \frac{1 - x ^{2}}{1 + x ^{2}} . If φ (x) = ((hof) og) (x), then φ (\frac{π}{3}) is equal to :$ (1) $tan \frac{π}{12}$ (2) $tan \frac{11 π}{12}$ (3) $tan \frac{7 π}{12}$ (4) $tan \frac{5 π}{12}$ 6. $If x^{2} + y^{2} = t - \frac{1}{t}, x^{4} + y^{4} = t^{2} + \frac{1}{t ^{2}} then \frac{dy}{dx} =$ (1) $\frac{x}{y}$ (2) $\frac{- x}{y}$ (3) $\frac{y}{x}$ (4) $\frac{- y}{x}$ 7. $If the function f (x) = \frac{log ( 1 + ax ) - log ( 1 - bx )}{x}, x \neq = 0 is continuous at x = 0, then f (0) =$ (1) $log a - log b$ (2) $a + b$ (3) $log a + log b$ (4) $a - b$ 8. $If a function defined by f (x) = \frac{( 3 ^{x} - 1 ) ^{2}}{sin xlog ( 1 + x )}, x \neq = 0, is continuous at x = 0, then f (0) =$ (1) $2 log 3$ (2) $log 3^{2}$ (3) $2 + log 3$ (4) $(log 3)^{2}$ 9. $The range of the function f (x) = 9 - x^{2} is$ (1) $[0, 3]$ (2) $(0, 3]$ (3) $(0, 3)$ (4) $[0, 3)$ 10. $Let R_{1} ={(a,b) \in NxN:\|a-b\| \leq 13}and R_{2} ={(a,b) \in NxN: \|a-b\| \neq = 13}. Then on N:$ (1) $Both R_{1} and R_{2} are equivalence relations$ (2) $Neither R_{1} nor R_{2} is an equivalence relation$ (3) $R_{1} is an equivalence relation but R_{2} is not$ (4) $R_{2} is an equivalence relation but R_{1} is not$ 11. $On the set of all non-zero reals, an operation * is defined as a * b = \frac{3 ab}{2} . In this group, a solution of (2 * x) * 3^{- 1} = 4^{- 1} is$ (1) $6$ (2) $1$ (3) $1/6$ (4) $3/2$ 12. $lim_{x \to a} \frac{a + 2 x - 3 x}{3 a + x - 2 x} =$ (1) $\frac{2}{3}$ (2) $\frac{2}{3}$ (3) $\frac{3 3}{2}$ (4) $\frac{2}{3 3}$ 13. $The derivative of y = (sin x)^{x^{2}} with respect to x is$ (1) $(sin x)^{x^{2}} log (sin x)$ (2) $x^{2} (sin x)^{x^{2} - 1}$ (3) $2 x (sin x)^{x^{2}} cos x + 2 x (sin x)^{x^{2}} log (sin x)$ (4) $x^{2} (sin x)^{x^{2} - 1} cos x + 2 x (sin x)^{x^{2}} log (sin x)$ 14. $Lt_{x \to 0} \frac{x 2 ^{x} - x}{1 - cos x} =$ (1) $\frac{1}{2}$ (2) $2 log 2$ (3) $log 2$ (4) $\frac{1}{2} log 2$ 15. $Among the relations S ((a, b) : a, b ϵ R - (0), 2 + \frac{a}{b} > 0) and T = ((a, b) : a, b ϵ R, a^{2}, b^{2} ϵ Z),$ (1) $S is transitive but T is not$ (2) $T is symmetric but S is not$ (3) $Neither S nor T is transitive$ (4) $Both S and T are symmetric$ 16. $If cos^{- 1} (\frac{y}{b}) = n log (\frac{x}{n}), then$ (1) $x y_{1} = n b^{2} - y^{2}$ (2) $x y_{1} + n b^{2} - y^{2} = 0$ (3) $y_{1} = x b^{2} - y^{2}$ (4) $x y_{1} - b^{2} - y^{2} = 0$ 17. $The number of points on the parabola y^{2} = x at which the slope of the normal drawn at the point is equal to the x -coordinate of that point is$ (1) $\infty$ (2) $1$ (3) $2$ (4) $0$ 18. $If f (x) = \frac{1 + sec x}{2 ( sec x - 1 )} for 0 < x < \frac{π}{2} and f^{’} (x) = f (x) \cdot g (x), then g (x) =$ (1) $cosec x$ (2) $- cosec x$ (3) $2 cosec x$ (4) $- 2 cosec x$ 19. $If D is the domain and G is the range of the real valued function f (x) = \frac{1 - x ^{2}}{1 + x ^{2}}, then D \cap G =$ (1) $[0, \infty)$ (2) $[0, 1]$ (3) $(0, \frac{1}{2})$ (4) $[- 1, 1]$ 20. $In a certain town 65 % families own cell phones, 15000 families own scooter and 15 % families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is$ (1) $20000$ (2) $30000$ (3) $40000$ (4) $50000$ 21. $If A = (x \in R / \frac{π}{4} \leq x \leq \frac{π}{3}) and f (x) = sin x - x, then f (A) is equal to$ (1) $(\frac{3}{2} - \frac{π}{3}, \frac{1}{2} - \frac{π}{4})$ (2) $(\frac{- 1}{2} - \frac{π}{4}, \frac{3}{2} - \frac{π}{3})$ (3) $(- \frac{π}{3}, - \frac{π}{4})$ (4) $(\frac{π}{4}, \frac{π}{3})$ 22. $If f (x) = cot^{- 1} (\frac{x ^{x} - x ^{- x}}{2}), then f^{’} (1) =$ (1) $- log 2$ (2) $log 2$ (3) $1$ (4) $-1$ 23. $If y = \frac{( x + 1 ) ^{2} x - 1}{( x + 4 ) ^{3} e ^{x}} then \frac{dy}{dx} =$ (1) $\frac{( x + 1 ) ^{3} x - 1}{( x + 4 ) ^{3} e ^{x}} (\frac{2}{x + 1} + \frac{1}{2 ( x - 1 )} - \frac{3}{x + 4} - 1)$ (2) $\frac{( x + 1 ) ^{2} x - 1}{( x + 4 ) ^{3} e ^{x}} (\frac{2}{x + 1} + \frac{1}{2 ( x - 1 )} + \frac{3}{x + 4} - 1)$ (3) $\frac{( x + 1 ) ^{2} x - 1}{( x + 4 ) ^{3} e ^{x}} (\frac{2}{x + 1} + \frac{1}{2 ( x - 1 )} - \frac{3}{x + 4} - 1)$ (4) $\frac{( x + 1 ) x - 1}{( x + 4 ) ^{2} e ^{x}} (\frac{2}{x + 1} + \frac{1}{x - 1} - \frac{3}{4} + x - 1)$ 24. $The range of the function f (x) = log_{0.5} (x^{4} - 2 x^{2} + 3) is$ (1) $(- \infty, \infty)$ (2) $(- \infty, - 1]$ (3) $[- 1, \infty]$ (4) $[- 1, 1]$ 25. $Let A = 1, 2, 3 . The number of relations on A, containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is -$ 26. $The number of elements in the set {n Z :\| - 10n + 19\|< 6} is_______.$ 27. $If the range of f (θ) = \frac{sin ^{4} θ + 3 cos ^{2} θ}{sin ^{4} θ + cos ^{2} θ}, θ \in R is [α, β], then the sum of the infinite G.P., whose first term is 64 and the common ratio is \frac{α}{β}, is equal to_______$ 28. $If a + α = 1, b + β = 2 and af (x) + αf (\frac{1}{x}) = bx + \frac{β}{x}, x \neq = 0, then the value of the expression \frac{f ( x ) + f ( \frac{1}{x} )}{x + \frac{1}{x}} is$ 29. $Let A = {-4, -3, -2, 0,1,3,4} and R = {(a, b) \in A x A : b = \|a\| or b^{2} = a + 1} be a relation on A. Then the minimum number of elements, that must be added to the relation R so that it becomes reflexive and symmetric, is$ 30. $The number of elements in the set {nN: 10 n 100 and - 3 is a multiple of 7} is_______$