Sets, Relations & Functions PYQs Set 25 (PDF)

Set 25 of 25 · Questions 721-734 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 25 (Q721-734)
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.
721. $If x = a is a root of multiplicity two of a polynomial equation f (x) = 0, then$ (1) $f^{’} (a) = f^{’ ’} (a) = 0$ (2) $f^{’ ’} (a) = f (a) = 0$ (3) $f^{’} (a) \neq = 0 \neq = f^{’ ’} (a)$ (4) $f (a) = f^{’} (a) = 0; f^{’ ’} (a) \neq = 0$ 722. $If x = a cos^{3} θ, y = a sin^{3} θ, then 1 + (\frac{dy}{dx})^{2} is$ (1) $tan^{2} θ$ (2) $1$ (3) $tan θ$ (4) $sec^{2} θ$ 723. $Let A = {x \in R : [x + 3] + [x + 4] \leq 3}, B = {x \in R : 3^{x} (\sum_{r = 1}^{\infty} \frac{3}{1 0 ^{r}}) < 3^{- 3 x}}, where [t] denotes greatest integer function. Then,$ (1) $B \subset C, A \neq = B$ (2) $A \cap B = φ$ (3) $A \subset B, A \neq = B$ (4) $A = B$ 724. $For each t\inR let [t] be the greatest integer less than or equal to t. Then lim_{x \to 0^{+}} x ([\frac{1}{x}] + [\frac{2}{x}] + ...... + [\frac{15}{x}])$ (1) $is equal to 120$ (2) $does not exist (in R)$ (3) $is equal to 0$ (4) $is equal to 15$ 725. $Let f : N \to R be a function such that f(x+y)=2f(x)f(y) for natural numbers x and y. If f(1) = 2, then the value of a for which \sum_{k = 1}^{10} f (α + k) = \frac{512}{3} (2^{20} - 1) holds, is$ (1) $2$ (2) $3$ (3) $4$ (4) $6$ 726. $For each t\inR, let [t] be the greatest integer less than or equal to t. Then lim_{x \to 1 +} \frac{( 1 - ∣ x ∣ + s i n ∣1 - x ∣ ) s i n ( \frac{π}{2} [ 1 - x ] )}{∣1 - x ∣ [ 1 - x ]},$ (1) $does not exist$ (2) $equals 1$ (3) $equals -1$ (4) $equals 0$ 727. $For every integer n, let an and bn be real numbers. Let function f : ΙR \to ΙR be given by f(x) = {a_{n} + sin π x, b_{n} + cos π x, for x \in [2 n, 2 n + 1] for x \in (2 n - 1, 2 n), for all integers n. If f is continuous, then which of the following hold(s) for all n ?$ (1) $a_{n - 1} - b_{n - 1} = 0$ (2) $a_{n} - b_{n} = 1$ (3) $a_{n} - b_{n + 1} = 1$ (4) $a_{n - 1} - b_{n} = - 1$ 728. $If y = f (x) l a g (x) m b h (x) n c, then \frac{d y}{d x} is equal to$ (1) $f^{'} (x) l a g^{'} (x) m b h^{'} (x) n c$ (2) $l f^{'} (x) a m g^{'} (x) b n h^{'} (x) c$ (3) $f^{'} (x) g^{'} (x) h^{'} (x) l m n a b c$ (4) $none of the above$ 729. $lim n \to \infty ((1 + \frac{1}{n ^{2}}) (1 + \frac{2 ^{2}}{n ^{2}}) \dots (1 + \frac{n ^{2}}{n ^{2}}))^{1/ n} =$ (1) $e$ (2) $2 e$ (3) $2 e^{\frac{π - 2}{2}}$ (4) $2 e^{\frac{π - 4}{2}}$ 730. $Consider the following statements Statement I : If a_{0} + \frac{a _{1}}{2} + \frac{a _{2}}{3} + \dots . + \frac{a _{n}}{n + 1} = 0, where a_{0}, a_{1}, \dots ., a_{n} . are real numbers, then the polynomial a_{0} + a_{1} x + a_{2} x^{2} + \dots . + a_{n} x^{n} has a zero in the interval (0, 1) . Statement II: If f : [a, b] \to R is continuous on [a, b] and f is differentiable in (a, b), where a > 0 and if \frac{f ( a )}{a} = \frac{f ( b )}{b}, then there exists c \in (a, b), such that c f^{’} (c) = f (c) Which one of the following options is true?$ (1) $Only I is true$ (2) $Only II is true$ (3) $Neither (I) nor (II) is true$ (4) $Both (I) and (II) are true$ 731. $Let f (x) = {- a, x + a, - a \leq x \leq 0 0 < x \leq a where a > 0 and g (x) = (f (∣ x ∣) - ∣ f (x) ∣) /2 . Then the function g : [- a, a] \to [- a, a] is$ (1) $neither one-one nor onto.$ (2) $onto.$ (3) $both one-one and onto.$ (4) $one-one.$ 732. $The number of one-one function f: {a, b, c, d} {0, 1,2,... .,10} such that 2f(a) - f(b) + 3f(c) + f (d) = 0 is_____.$ 733. $Number of functions f : 1, 2, \dots, 100 \to 0, 1, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to -$ 734. $Let [t] denote the greatest integer \leq 1 . The number of points where the function f (x) = [x] [x^{2} - 1] + sin (\frac{π}{[ x ] + 3}) - [x + 1], x \in (- 2, 2) is not continuous is .$