Sets, Relations & Functions PYQs Set 23 (PDF)

Set 23 of 25 · Questions 661-690 of 734

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Sets, Relations & Functions · Set 23 (Q661-690)
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661. $If the function f (x) defined by f (x) = {x sin \frac{1}{x}, k, x \neq = 0 x = 0 is continuous at x = 0, then k =$ (1) $0$ (2) $1$ (3) $-1$ (4) $1/2$ 662. $Let the relations R_{1} and R_{2} on the set X = 1, 2, 3, \dots, 20 be given by R_{1} = (x, y) : 2x-3y=2 and R_{2} = (x, y) : -5x+4y=0 . If M and N be the minimum number of elements required to be added in R_{1} and R_{2}, respectively, in order to make the relations symmetric, then M + N equals$ (1) $12$ (2) $16$ (3) $8$ (4) $10$ 663. $Which of the following is an empty set?$ (1) $(x : x^{2} - 9 = 0, x \in R)$ (2) $(x : x^{2} - 1 = 0, x \in R)$ (3) $(x : x^{2} = x + 2, x \in R)$ (4) $(x : x^{2} + 1 = 0, x \in R)$ 664. $Let f and g be real-valued functions. If lim x \to 0 \frac{2 f ( x ) - g ( x )}{[ f ( x ) + 7 ] ^{2/3}} = \frac{7}{4}, lim x \to 0 f (x) = 1 and lim_{x \to 0} g (x) = α, then h (x) = (sin (αx), 0 \leq x \leq \frac{π}{10} cos (2 αx), \frac{π}{10} < x \leq \frac{π}{5}) is$ (1) $continuous at x = \frac{π}{10} only$ (2) $discontinuous on (0, \frac{π}{5})$ (3) $discontinuous at x = \frac{π}{10}$ (4) $continuous on (0, \frac{π}{5})$ 665. $Which of the following is false?$ (1) $If a * b = a^{b} for all a, b \in N then * is commutative in N.$ (2) $Addition is associative in N .$ (3) $Addition is commutative in N .$ (4) $Multiplication is associative in N .$ 666. $The equation x^{3} + x - 1 = 0 has$ (1) $no real root.$ (2) $exactly two real roots.$ (3) $exactly one real roots.$ (4) $more than two real roots.$ 667. $f (x) and g (x) are differentiable functions such that \frac{f ( x )}{g ( x )} = a non zero constant. If \frac{f ^{’} ( x )}{g ^{’} ( x )} = α (x) and (\frac{f ( x )}{g ( x )})^{’} = β (x), then \frac{α ( x ) - β ( x )}{α ( x ) + β ( x )} =$ (1) $0$ (2) $f (x) + g (x)$ (3) $1$ (4) $f^{’} (x) + g^{’} (x)$ 668. $Let A = 1, 2, 3, \dots, 100 and R be a relation on A such that R = (a, b) : a=2b+1 . Let (a_{1}, (a_{2}), (a_{2}, a_{3}), (a_{3}, a_{4}), \dots, (a_{k}, a_{k + 1}) be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to:$ (1) $6$ (2) $8$ (3) $7$ (4) $5$ 669. $If the three functions f (x), g (x) and h (x) are such that h (x) = f (x) \cdot g (x) and f^{’} (x) \cdot g^{’} (x) = c, where c is a constant, then \frac{f ^{’ ’} ( x )}{f ( x )} + \frac{g ^{’ ’} ( x )}{g ( x )} + \frac{2 c}{f ( x ) \cdot g ( x )} is equal to$ (1) $\frac{h ( x )}{h ^{’ ’} ( x )}$ (2) $h^{’} (x) \cdot h^{’ ’} (x)$ (3) $\frac{h ( x )}{h ^{’} ( x )}$ (4) $\frac{h ^{’ ’} ( x )}{h ( x )}$ 670. $For any quadratic polynomial f (x), it is true that f (x) = f (a) + f^{’} (a) (x - a) + \frac{f ^{’ ’} ( a )}{2 !} (x - a)^{2}, where a is any real number. If \frac{3 x ^{2} + 4 x + 7}{( x - 2 ) ^{3}} = \frac{A}{( x - 2 ) ^{3}} + \frac{B}{( x - 2 ) ^{2}} + \frac{C}{( x - 2 )} and g (x) = 3 x^{2} + 4 x + 7, then A + B + C is equal to$ (1) $g (2) + g^{’} (2) + g^{’ ’} (2)$ (2) $g^{’ ’} (2) + 2 g (2) + \frac{g ^{’} ( 1 )}{2 !}$ (3) $g (2) + g^{’} (2) + g^{’ ’} (2) 2!$ (4) $2 g (2) + 2 g^{’} (2) + \frac{g ^{’ ’} ( 2 )}{2 !}$ 671. $Let E_{1} = (x \in R : x \neq = 1) and (\frac{x}{x - 1} > 0) and E_{2} = (x \in E_{1} : sin^{- 1} (log_{e} (\frac{x}{x - 1}))) is a real number } (Here, the inverse trigonometric function sin^{- 1} x assumes values in (- \frac{π}{2}, \frac{π}{2}) Let f : E_{1} \to R be the function defined by f(x)=logexx-1and g : E_{2} \to R be the function defined byg(x)=sin-1logexx-1. (JEE Advanced 2018) List I (A) The range of f is (B) The range of g contains (C) The domain of f contains (D) the domain of g is (T) (- \infty, \frac{e}{e - 1}) List II (P) (- \infty, \frac{1}{1 - e}) \cup (\frac{e}{e - 1}, \infty) (Q) (0, 1) (R) [- 1/2, 1/2] (S) (- \infty, 0) \cup (0, \infty) (U) (- \infty, 0) \cup (\frac{1}{2}, \frac{e}{e - 1}) The correct option is$ (1) $(A) \to (S); (B) \to (Q); (C) \to (P); (D) \to (P)$ (2) $(A) \to (R); (B) \to (R); (C) \to (U); (D) \to (T)$ (3) $(A) \to (S); (B) \to (Q); (C) \to (P); (D) \to (U)$ (4) $(A) \to (S); (B) \to (R); (C) \to (U); (D) \to (T)$ 672. $Let g (x) \neq = 0, g^{’} (x) \neq = 0, f (x) \neq = 0, f^{’} (x) \neq = 0 . If F (x) = f (x) g (x), G (x) = f^{’} (x) g^{’} (x) and F^{’} (x) = G (x) H (x) = F (x) K (x), then H (x) + K (x) =$ (1) $\frac{f ^{’}}{f} + \frac{f}{f ^{’}} + \frac{g}{g ^{’}}$ (2) $\frac{f ^{’}}{f} + \frac{g}{g ^{’}} + \frac{g ^{’}}{g}$ (3) $\frac{f ^{’} g ^{’} + fg}{f f ^{’} g g ^{’}}$ (4) $\frac{f ^{’}}{f} + \frac{g}{g ^{’}} + \frac{f}{f ^{’}} + \frac{g ^{’}}{g}$ 673. $Let X = R \times R . Define a relation R on X as: (a_{1}, b_{1}) R (a_{2}, b_{2}) \Leftrightarrow b_{1} = b_{2} Statement I: R is an equivalence relation. Statement II: For some (a, b) \in X, the set S = (x, y) \in X : (x, y) R (a, b) represents a line parallel to y = x . choose the correct answer from the options given below :$ (1) $Both Statement I and Statement II are false$ (2) $Statement I is true but Statement II is false$ (3) $Both Statement I and Statement II are true$ (4) $Statement I is false but Statement II is true$ 674. $Survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be:$ (1) $37$ (2) $55$ (3) $29$ (4) $65$ 675. $Let S be the set of all real roots of the equation, 3^{x} (3^{x} - 1) + 2 = ∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣ . Then S :$ (1) $singleton set$ (2) $two elements$ (3) $at least four elements$ (4) $infinite elements$ 676. $Let f, g : R \to R be defined as : f (x) = ∣ x - 1∣ and g (x) = {e^{x}, x + 1, x \geq 0 x \leq 0 Then the function f (g (x)) is$ (1) $neither one-one nor onto.$ (2) $one-one but not onto.$ (3) $onto but not one-one.$ (4) $both one-one and onto.$ 677. $If f (x) = {2 + 2 x, 1 - \frac{x}{3}, - 1 \leq x < 0 0 \leq x \leq 3; g (x) = {- x, x, - 3 \leq x \leq 0 0 < x \leq 1, then range of (f \circ g (x)) is$ (1) $(0, 1]$ (2) $[0, 3)$ (3) $[0, 1]$ (4) $[0, 1)$ 678. $If l, m (l < m) are roots of a x^{2} + b x + c = 0, then lim_{x \to α} \frac{∣ a x ^{2} + b x + c ∣}{a x ^{2} + b x + c} =$ (1) $\frac{\| a \|}{a}, \forall α \in R$ (2) $\frac{- \| a \|}{a}, when α \in / (l, m)$ (3) $\frac{- \| a \|}{a}, when α\in(l,m)$ (4) $\frac{\| a \|}{a}, when α\in(l,m)$ 679. $If f : R \to R is defined by f (x) = 2 x + 3, then f^{- 1} (x)$ (1) $does not exist because ' f' is not surjective$ (2) $is given by \frac{x - 3}{2}$ (3) $is given by \frac{1}{2 x + 3}$ (4) $does not exist because ' f' is not injective$ 680. $The domain of the function f (x) = sin^{- 1} (\frac{x ^{2} - 3 x + 2}{x ^{2} + 2 x + 7}) is$ (1) $[1, \infty)$ (2) $(-1,2]$ (3) $[-1, \infty)$ (4) $(- \infty,2]$ 681. $If lim_{x \to \infty} (\frac{x ^{2} + x + 1}{x + 1} - a x - b) = 4, then$ (1) $a = 1, b = 4$ (2) $a = 1, b = -4$ (3) $a = 2, b = -3$ (4) $a = 2, b = 3$ 682. $The number of integral solution x of lo g_{(x + \frac{7}{2})} (\frac{x - 7}{2 x - 3})^{2} \geq 0 is$ (1) $7$ (2) $8$ (3) $6$ (4) $5$ 683. $The domain of the function cos^{- 1} (\frac{2 s i n ^{- 1} ( \frac{1}{4 x ^{2} - 1} )}{π}) is:$ (1) $R - {- \frac{1}{2}, \frac{1}{2}}$ (2) $(- \infty, - 1] \cup [1, \infty) \cup_{0}$ (3) $(- \infty, \frac{- 1}{2}) \cup (\frac{1}{2}, \infty) \cup {0}$ (4) $(- \infty, \frac{- 1}{2}] \cup [\frac{1}{2}, \infty) \cup {0}$ 684. $The number of real solutions of the equation, x^{2} - \|x\| - 12 = 0 is:$ (1) $2$ (2) $3$ (3) $1$ (4) $4$ 685. $The range of the function f (x) = 3 - x + 2 + x$ (1) $[5, 10]$ (2) $[22, 11]$ (3) $[5, 13]$ (4) $[2, 7]$ 686. $The number of elements in the set S = {x \in R : 2 cos (\frac{x ^{2} + x}{6}) = 4^{x} + 4^{- x}} is:$ (1) $1$ (2) $3$ (3) $0$ (4) $infinite$ 687. $In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?$ (1) $P and Q$ (2) $P and R$ (3) $None of these$ (4) $Q and R$ 688. $In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _______$ 689. $A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics \& Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is_____$ 690. $Let f : [0, 3] \to R be defined by f(x) = min{x - [x], 1 + [x] - x} where is the greatest integer less than or equal to x. Let P denote the set containing all x \in [0, 3] where is discontinuous, and Q denote the set containing all x \in (0, 3) where is not differentiable. Then the sum of number of elements in P and Q is equal to$