Sets, Relations & Functions PYQs Set 24 (PDF)

Set 24 of 25 · Questions 691-720 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 24 (Q691-720)
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691. $Let f (x) = \frac{x - 1}{x + 1}, x \in R - {0, - 1, 1} . If f_{(}^{n + 1} x_{)} = f_{(} f_{(}^{n} x_{))} for all n \in N, then f^{6} (6) + f^{7} (7) is equal to:$ (1) $\frac{7}{6}$ (2) $- \frac{3}{2}$ (3) $\frac{7}{12}$ (4) $- \frac{11}{12}$ 692. $Let f: R \to [0, \infty) be such that lim_{x \to 5} f(x) exists and lim_{x \to 5} \frac{( f ( x ) ) ^{2} - 9}{∣ x - 5∣} = 0$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 693. $Considering only the principal values of the inverse trigonometric functions, the domain of the function f (x) = cos^{- 1} (\frac{x ^{2} - 4 x + 2}{x ^{2} + 3}) is :$ (1) $(- \infty, \frac{1}{4}]$ (2) $[- \frac{1}{4}, \infty)$ (3) $(- \frac{1}{3}, \infty)$ (4) $(- \infty, \frac{1}{3}]$ 694. $If the number of elements in the sets G and A are 3 and 4 respectively, then match the items of List-I with those of List-II. List-I (A) The number of non-bijective functions from G \times G to G (B) The number of bijective functions from A to A (C) The number of functions from G to G \times A (D) The number of surjective functions from A to A \times A . List-II (I) 24 (II) 0 (III) 1728 (IV) 12 (V) 19683. The correct match is (A, B, C, D)$ (1) $V I III II$ (2) $V III IV II$ (3) $III IV V II$ (4) $I II III IV$ 695. $The domain of the function f(x) = \frac{1}{∣ x ∣ - x} is :$ (1) $(-\infty , \infty)$ (2) $(0, \infty)$ (3) $(-\infty, 0)$ (4) $(-\infty, \infty) - {0}$ 696. $The length of the segment of the tangent line to the curve x = a cos^{3} t, y = a sin^{3} t, at any point on the curve cut off by the coordinate axes is$ (1) $4 a$ (2) $a$ (3) $a^{2}$ (4) $2 a$ 697. $If f (x) is an even function, then f^{’} (x) is$ (1) $nothing can be said$ (2) $an odd function$ (3) $an even function$ (4) $may be even or may be odd$ 698. $If y = \frac{2}{a ^{2} - b ^{2}} tan^{- 1} (\frac{a - b}{a + b} tan \frac{x}{2}), then (\frac{d ^{2} y}{d x ^{2}})_{x = \frac{π}{2}} =$ (1) $\frac{b}{2 a ^{2}}$ (2) $\frac{b}{a ^{2}}$ (3) $\frac{2 b}{a}$ (4) $\frac{b ^{2}}{2 a}$ 699. $If f : [1, \infty) \to [0, \infty) is given by f (x) = x - \frac{1}{x}, then f^{- 1} (x) =$ (1) $x + x^{2} + 4$ (2) $\frac{x}{x ^{2} - 1}$ (3) $\frac{1}{2} (x + x^{2} + 4)$ (4) $\frac{1}{2} (x - x^{2} + 4)$ 700. $Which of the following function is not continuous at x = 0 ?$ (1) $f (x) = {(1 + 2 x)^{1/ x}, e^{2}, x \neq = 0 x = 0$ (2) $f (x) = {sin x - cos x, - 1, x \neq = 0 x = 0$ (3) $f (x) = ⎩ ⎨ ⎧ \frac{e ^{1/ x} - 1}{e ^{1/ x} + 1}, - 1, x \neq = 0 x = 0$ (4) $f (x) = ⎩ ⎨ ⎧ \frac{e ^{5 x} - e ^{2 x}}{sin 3 x}, 1, x \neq = 0 x = 0$ 701. $If A and B have n elements in common, then the number of elements common to A \times B and B \times A is$ (1) $0$ (2) $n$ (3) $2 n$ (4) $n^{2}$ 702. $Let f : [0, 10] \to [1, 20] be a function defined as f (x) = ⎩ ⎨ ⎧ \frac{60 - 5 x}{3}, 10, 31 - 3 x, 0 \leq x \leq 6 6 \leq x \leq 7 7 \leq x \leq 10, then f is$ (1) $bijective function$ (2) $one-one but not onto function$ (3) $onto but not one-one function$ (4) $neither one-one nor onto function$ 703. $Which one of the following is not correct for the features of exponential function given by f (x) = b^{x} where b > 1 ?$ (1) $For very large negative values of x, the function is very close to 0 .$ (2) $The domain of the function is R, the set of real numbers.$ (3) $The point (1, 0) is always on the graph of the function.$ (4) $The range of the function is the set of all positive real numbers$ 704. $If x = 3 cos t and y = 4 sin t, then \frac{d ^{2} y}{d x ^{2}} at the point (x_{0}, y_{0}) = (\frac{3}{2} 2, 22), is$ (1) $\frac{4 2}{9}$ (2) $- \frac{4 2}{9}$ (3) $\frac{8 2}{9}$ (4) $- \frac{8 2}{9}$ 705. $f : (- \infty, 0] \to [0, \infty) is defined as f (x) = x^{2} . The domain and range of its inverse is$ (1) $Domain f^{- 1} =[0,\infty). Range of f^{- 1} =(-\infty,0]$ (2) $Domain of f^{- 1} =[0,\infty). Range of f^{- 1} =(-\infty,\infty]$ (3) $Domain of f^{- 1} =[0,\infty). Range of f^{- 1} =(0,\infty)$ (4) $f^{- 1} does not exist$ 706. $lim x \to \infty (\frac{2 x ^{2} + 3 x + 4}{x ^{2} - 3 x + 5})^{\frac{3 ( x ) + 1}{2 ( x ) - 1}} =$ (1) $\frac{3}{2}$ (2) $22$ (3) $3$ (4) $2$ 707. $Suppose f (x) = (x + 1)^{2} for x \geq - 1 . If g (x) is a function whose graph is the reflection of the graph of f (x) in the line y = x, then g (x) =$ (1) $\frac{1}{( x + 1 ) ^{2}}; x > - 1$ (2) $- x - 1$ (3) $x + 1$ (4) $x - 1$ 708. $Let f (x) = 1 - x, g (x) = \frac{1}{1 - x}, h (x) = \frac{1}{x} be three functions, for x \neq = 0, 1 . If a function F (x) satisfies f (F (h (x))) = g (x), then$ (1) $F (2022) = f (2022)$ (2) $F (2022) = g (2022)$ (3) $F (2022) = h (2022)$ (4) $F (2022) = \frac{1}{2022} f (2022)$ 709. $Let the relation R is defined in N by aRb, if 3 a + 2 b = 27 then R is$ (1) ${(1, 12), (3, 9), (5, 6), (7, 3)}$ (2) ${(1, 12), (3, 9), (5, 6), (7, 3), (9, 0)}$ (3) $((0, \frac{27}{2}), (1, 12), (3, 9), (5, 6), (7, 3))$ (4) ${(2, 1), (9, 3), (6, 5), (3, 7)}$ 710. $f : [- 2, 2] \to [- 2, 2] and g : [- 2, 2] \to [0, 4] are two functions defined as f (x) = (- 2, - 2 \leq x \leq 0 x^{2} - 2, 0 \leq x \leq 2) and g (x) = \| f (x) \| + f (\| x \|), then$ (1) $f and g are injective mappings$ (2) $f and g are surjective mappings$ (3) $f is bijective mapping and g is injective mapping$ (4) $f is not bijective mapping and g is surjective mapping$ 711. $lim x \to 0 \frac{tan ^{2} ( π sec ^{4} x )}{π ^{2} x ^{4}} =$ (1) $0$ (2) $4$ (3) $1$ (4) $16$ 712. $If y = log_{2} (log_{2} x), then \frac{dy}{dx} is equal to$ (1) $\frac{log _{e} 2}{x log _{e} x}$ (2) $\frac{1}{log _{e} ( 2 x ) ^{x}}$ (3) $\frac{1}{( x log _{e} x ) log _{e} 2}$ (4) $\frac{1}{x ( log _{2} x ) ^{2}}$ 713. $If the function f (x), defined below is continuous in the interval [0, π], then f (x) = ⎩ ⎨ ⎧ x + a 2 sin x, 2 x cot x + b, a cos 2 x - b sin x, 0 \leq x < \frac{π}{4} \frac{π}{4} \leq x \leq \frac{π}{2} \frac{π}{2} < x \leq π$ (1) $a = \frac{π}{6}, b = \frac{π}{12}$ (2) $a = \frac{- π}{6}, b = \frac{π}{12}$ (3) $a = \frac{- π}{6}, b = \frac{- π}{12}$ (4) $a = \frac{π}{6}, b = \frac{- π}{12}$ 714. $lim x \to \frac{π}{2} \frac{1 - tan \frac{x}{2}}{1 + tan \frac{x}{2}} \cdot \frac{1 - sin x}{( π - 2 x ) ^{3}} =$ (1) $\frac{1}{32}$ (2) $0$ (3) $\frac{1}{16}$ (4) $\frac{1}{8}$ 715. $A function f : R - 0 \to R is defined as f (x) = (x^{2} + 3 x - 7, x > 0 h (x), x < 0) If f (x) is an odd function, then h (x) =$ (1) $x^{2} + 3 x + 7$ (2) $x^{2} + 3 x - 7$ (3) $- x^{2} + 3 x + 7$ (4) $- x^{2} - 3 x + 7$ 716. $If a \neq = 0, x = a (t + sin t) and y = a (1 - cos t), then \frac{d ^{2} y}{d x ^{2}} at t = \frac{2 π}{3} is$ (1) $\frac{4}{a}$ (2) $\frac{1}{4 a}$ (3) $4 a$ (4) $\frac{a}{4}$ 717. $If y = tan^{- 1} (\frac{x}{1 + 1 - x ^{2}}) + sin (2 tan^{- 1} \frac{1 - x}{1 + x}), then \frac{dy}{dx} =$ (1) $\frac{1 - 2 x}{2 1 - x ^{2}}$ (2) $\frac{1 - 2 x}{x 1 - x ^{2}}$ (3) $\frac{2 x + 1}{x 1 - x}$ (4) $\frac{2 - x}{2 1 - x ^{2}}$ 718. $Let N be the set of all natural numbers, Z be the set of all integers and σ : N \to Z be defined by σ (n) = (\frac{n}{2}, if n is even - \frac{n - 1}{2}, if n is odd .) Then,$ (1) $σ is onto but not one-one$ (2) $σ is one-one but not onto$ (3) $σ is neither one-one nor onto$ (4) $σ is one-one and onto$ 719. $Let f : R \to R be a function defined by f (x) = (2 (1 - \frac{x ^{25}}{2}) (2 + x^{25}))^{\frac{1}{50}} . If the function g(x) = f(f(f(x))) + f(f(x)), the the greatest integer less than or equal to g (1) is______$ 720. $Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p, p \neq = 0 and f (1) = \frac{1}{3} . If the equation f(x) = 0 and fofofof(x) = 0 have a common real root, then f(-3) is equal to .......................$