sets relations and functions jee mains questions | JEE Main Mathematics Sets Relations And Functions

Q 41 :
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3}$ and $(f o f) (x) = g (x)$ , where $g : R - {\frac{2}{3}} \to R - {\frac{2}{3}},$ then $(g o g o g) (4)$ is equal to [2024]

$\frac{19}{20}$
$- \frac{19}{20}$
$- 4$
$4$

1

2

3

4

(D)

Here, $f (x) = \frac{4 x + 3}{6 x - 4}$

$f [f (x)] = \frac{4 f (x) + 3}{6 f (x) - 4} = \frac{4 (\frac{4 x + 3}{6 x - 4}) + 3}{6 (\frac{4 x + 3}{6 x - 4}) - 4}$

$= \frac{16 x + 12 + 18 x - 12}{6 x - 4} = \frac{34 x}{34} = x$

$∴$ $f [f (x)] = x \Rightarrow g (x) = x$

Now, $(g o g o g) (4) = g o g [g (4)] = g o g (4) = g (4) = 4$

Q 42 :
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}} .$ If the composition of $\underset{10 times}{f, \underset{⏟}{(f o f o f o . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}},$ then the value of $\sqrt{3 α + 1}$ is equal to _________ . [2024]

0%

(1024)

We have, $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}}$

$∴$ $(f o f) (x) = \frac{2 f (x)}{\sqrt{1 + 9 (f {(x))}^{2}}} = \frac{\frac{4 x}{\sqrt{1 + 9 x^{2}}}}{\sqrt{1 + 9 \times \frac{4 x^{2}}{1 + 9 x^{2}}}} = \frac{4 x}{\sqrt{1 + 45 x^{2}}} = \frac{2^{2} x}{\sqrt{1 + 5 \times 9 x^{2}}}$

$(f o f o f) (x) = \frac{4 \times \frac{2 x}{\sqrt{1 + 9 x^{2}}}}{\sqrt{1 + 45 \times \frac{4 x^{2}}{1 + 9 x^{2}}}} = \frac{2^{3} x}{\sqrt{1 + 21 \times 9 x^{2}}}$

$(f o f o f o f) (x) = \frac{2^{4} x}{\sqrt{1 \times 85 \times 9 x^{2}}}$

$∴$ $\underset{n times}{\underset{⏟}{(f o f o f o . . . . . o f)}} (x)$ $= \frac{2^{n} x}{\sqrt{1 + 9 (\frac{4^{n} - 1}{3}) x^{2}}}$

$∴$ $\underset{10 times}{\underset{⏟}{(f o f o f o . . . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 (\frac{4^{10} - 1}{3}) x^{2}}} = \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}}$

( $∵$ Given)

On comparing, we get

$α = \frac{4^{10} - 1}{3} \Rightarrow 3 α + 1 = 4^{10}$

$\Rightarrow \sqrt{3 α + 1} = \sqrt{4^{10}} = 4^{5} = 1024 .$

Q 43 :
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m,$ $n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq {(2022)}^{2}$ is equal to ___________ . [2024]

0%

(1010)

We have, $f (m + n) = f (m) + f (n)$

So, $f (x) = k x$

$∵$ $f (1) = 1 \Rightarrow k = 1$

Hence, $f (x) = x$

Now, $\sum_{k = 1}^{2022} f (λ + k) = \sum_{k = 1}^{2022} (λ + k)$

$= \underset{2022}{\underset{⏟}{λ + λ + . . . + λ}}$ $+ (1 + 2 + . . . . + 2022)$

$= 2022 λ + \frac{2022 \times 2023}{2} \leq {(2022)}^{2}$ (Given)

$\Rightarrow λ \leq \frac{2021}{2}$

So, largest $λ = 1010$

Q 44 :
Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from A to B is equal to ________ . [2024]

0%

(24)

We have, $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$

$B = {x : (x, y) \in A}$

$∴ A = {(1, 7), (4, 5), (7, 3), (10, 1)}$

and $B = {1, 4, 7, 10}$

Total number of one-one functions from A to B = 4! = 24

Q 45 :
Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions $f : A \to P (A)$ such that $a \in f (a)$ , $\forall$ $a \in A$ is $m^{n}$ , $m$ and $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]

0%

(44)

Given, $f : A \to P (A) \Rightarrow a \in f (a)$

It means $' a'$ will connect with subset which contain element $a .$

Total options for 1 will be $2^{6}$ ( $∵$ $2^{6}$ subsets contains 1)

Similarly, for every other element

Now, number of functions from A to P(A) = ${(2^{6})}^{7} = 2^{42}$

i.e., $m + n = 2 + 42 = 44$

Topic Question Set

Q 41 :
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3}$ and $(f o f) (x) = g (x)$ , where $g : R - {\frac{2}{3}} \to R - {\frac{2}{3}},$ then $(g o g o g) (4)$ is equal to [2024]

Q 42 :
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}} .$ If the composition of $\underset{10 times}{f, \underset{⏟}{(f o f o f o . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}},$ then the value of $\sqrt{3 α + 1}$ is equal to _________ . [2024]

0%

Q 43 :
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m,$ $n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq {(2022)}^{2}$ is equal to ___________ . [2024]

0%

Q 44 :
Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from A to B is equal to ________ . [2024]

0%

Q 45 :
Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions $f : A \to P (A)$ such that $a \in f (a)$ , $\forall$ $a \in A$ is $m^{n}$ , $m$ and $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]

0%

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 41 : If f(x)=4x+36x-4, x≠23 and (fof)(x)=g(x), where g:R-{23}→R-{23}, then (gogog)(4) is equal to [2024]

Q 42 : Consider the function f:R→R defined by f(x)=2x1+9x2. If the composition of f,(fofofo...of)⏟10 times (x) =210x1+9αx2, then the value of 3α+1 is equal to _________ . [2024]

0%

Q 43 : If a function f satisfies f(m+n)=f(m)+f(n) for all m, n∈N and f(1)=1, then the largest natural number λ such that ∑k=12022f(λ+k)≤(2022)2 is equal to ___________ . [2024]

0%

Q 44 : Let A={(x,y):2x+3y=23,x,y∈N} and B={x:(x,y)∈A}. Then the number of one-one functions from A to B is equal to ________ . [2024]

0%

Q 45 : Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions f:A→P(A) such that a∈f(a), ∀ a∈A is mn, m and n∈N and m is least, then m+n is equal to _____. [2024]

0%

Want Us to Email you About Special Offers & Updates?

Q 41 :
If $f (x) = \frac{4 x + 3}{6 x - 4}, x \neq \frac{2}{3}$ and $(f o f) (x) = g (x)$ , where $g : R - {\frac{2}{3}} \to R - {\frac{2}{3}},$ then $(g o g o g) (4)$ is equal to [2024]

Q 42 :
Consider the function $f : R \to R$ defined by $f (x) = \frac{2 x}{\sqrt{1 + 9 x^{2}}} .$ If the composition of $\underset{10 times}{f, \underset{⏟}{(f o f o f o . . . o f)}} (x)$ $= \frac{2^{10} x}{\sqrt{1 + 9 α x^{2}}},$ then the value of $\sqrt{3 α + 1}$ is equal to _________ . [2024]

Q 43 :
If a function $f$ satisfies $f (m + n) = f (m) + f (n)$ for all $m,$ $n \in N$ and $f (1) = 1$ , then the largest natural number $λ$ such that $\sum_{k = 1}^{2022} f (λ + k) \leq {(2022)}^{2}$ is equal to ___________ . [2024]

Q 44 :
Let $A = {(x, y) : 2 x + 3 y = 23, x, y \in N}$ and $B = {x : (x, y) \in A}$ . Then the number of one-one functions from A to B is equal to ________ . [2024]

Q 45 :
Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions $f : A \to P (A)$ such that $a \in f (a)$ , $\forall$ $a \in A$ is $m^{n}$ , $m$ and $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]