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

Q 1 :

Let $A = {x \in ℝ : [x + 3] + [x + 4] \leq 3},$

$B = {x \in ℝ : 3^{x} {(\sum_{r = 1}^{\infty} \frac{3}{10^{r}})}^{x - 3} < 3^{- 3 x}},$

where [t] denotes greatest integer function. Then, [2023]

$A \cap B = ϕ$
$B \subset C, A \neq B$
$A \subset B, A \neq B$
$A = B$

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(4)

We have, $[x + 3] + [x + 4] \leq 3$

$\Rightarrow [x] + 3 + [x] + 4 \leq 3 \Rightarrow 2 [x] \leq - 4 \Rightarrow [x] \leq - 2$

$\Rightarrow x \in (- \infty, - 1)$

Now,
$3^{x} (\sum_{r = 1}^{\infty} {(\frac{3}{10^{r}}))}^{x - 3} < 3^{- 3 x} \Rightarrow 3^{x} {(\frac{1}{3})}^{x - 3} < 3^{- 3 x}$

$\Rightarrow 3^{x} \cdot 3^{3 - x} < 3^{- 3 x} \Rightarrow 3^{3} < 3^{- 3 x} \Rightarrow 3 < - 3 x \Rightarrow x < - 1$

$\Rightarrow x \in (- \infty, - 1)$ $∴ Both sets A and B are equal.$

Q 2 :

Let the sets A and B denote the domain and range respectively of the function $f (x) = \frac{1}{\sqrt{⌈ x ⌉ - x}}$ , where $⌈ x ⌉$ denotes the smallest integer greater than or equal to x. Then among the statements

$(S 1) : A \cap B = (1, \infty) - N$ and

$(S 2) : A \cup B = (1, \infty)$ [2023]

only (S1) is true
both (S1) and (S2) are true
only (S2) is true
neither (S1) nor (S2) is true

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(1)

$f (x) = \frac{1}{\sqrt{⌈ x ⌉ - x}}$

If $x \in I$ , $⌈ x ⌉ = [x]$ (greatest integer function)

If $x \notin I$ , $⌈ x ⌉ = [x] + 1$

$\Rightarrow f (x) =$ ${\begin{matrix} \frac{1}{\sqrt{[x] - x}}, & x \in I \\ \frac{1}{\sqrt{[x] + 1 - x}}, & x \notin I \end{matrix}$

$\Rightarrow f (x) =$ ${\begin{matrix} \frac{1}{\sqrt{- {x}}}, & x \in I & (does not exist) \\ - \frac{1}{\sqrt{1 - {x}}}, & x \notin I \end{matrix}$

$\Rightarrow Domain of f (x) = R - I$

Now, $f (x) = \frac{1}{\sqrt{1 - {x}}}, x \notin I$

$\Rightarrow 0 < {x} < 1 \Rightarrow 0 < \sqrt{1 - {x}} < 1 \Rightarrow \frac{1}{\sqrt{1 - {x}}} > 1$

$\Rightarrow Range = (1, \infty) \Rightarrow A = R - I and B = (1, \infty)$

$So, A \cap B = (1, \infty) - N \Rightarrow A \cup B \neq (1, \infty)$

$\Rightarrow S 1 is true.$

Q 3 :

If $f (x) = \frac{(\tan 1^{o}) x + \log_{e} (123)}{x \log_{e} (1234) - (\tan 1^{o})}, x > 0,$ then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is [2023]

4
2
0
8

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(1)

Let $f (x) = \frac{A x + B}{C x - A}$

$f (f (x)) = \frac{A (\frac{A x + B}{C x - A}) + B}{C (\frac{A x + B}{C x - A}) - A} = \frac{A^{2} x + B C x}{B C + A^{2}} = x \dots (i)$

$f (f (\frac{4}{x})) = \frac{4}{x} [From (i)]$

$∴ f (f (x)) + f (f (\frac{4}{x})) = x + \frac{4}{x} \geq 4 [Using A.M. \geq G.M.]$

Q 4 :

The number of integral solutions x of $\log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2} \geq 0$ is [2023]

6
8
5
7

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(1)

Q 5 :

The domain of the function $f (x) = \frac{1}{\sqrt{{[x]}^{2} - 3 [x] - 10}}$ is (where [x] denotes the greatest integer less than or equal to x). [2023]

$(- \infty, - 3] \cup (5, \infty)$
$(- \infty, - 2) \cup [6, \infty)$
$(- \infty, - 3] \cup [6, \infty)$
$(- \infty, - 2) \cup (5, \infty)$

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(2)

${[x]}^{2} - 3 [x] - 10 > 0$

$\Rightarrow {[x]}^{2} - 5 [x] + 2 [x] - 10 > 0 \Rightarrow ([x] - 5) ([x] + 2) > 0$

$\Rightarrow [x] < - 2 or [x] > 5 \Rightarrow x < - 2 or x \geq 6$

$\Rightarrow x \in (- \infty, - 2) \cup [6, \infty)$

Q 6 :

For $x \in ℝ$ , two real valued functions $f (x)$ and $g (x)$ are such that, $g (x) = \sqrt{x} + 1$ and $f o g (x) = x + 3 - \sqrt{x} .$ Then $f (0)$ is equal to [2023]

5
1
0
- 3

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(*)

For $x \in R$ , $f (x)$ and $g (x)$ are real-valued functions.

$g (x) = \sqrt{x} + 1 and f o g (x) = x + 3 - \sqrt{x}$

$f o g (x) = x + 3 - \sqrt{x} \Rightarrow f (g (x)) = x + 3 - \sqrt{x}$

$= {(\sqrt{x} + 1)}^{2} - 3 (\sqrt{x} + 1) + 5$

$= {[g (x)]}^{2} - 3 [g (x)] + 5 (∵ g (x) = \sqrt{x} + 1)$

So, $f (x) = x^{2} - 3 x + 5$

$∴ f (0) = 0 - 3 \times 0 + 5 = 5$

Note: But if we consider the domain of the composite function $f o g (x)$ , then $f (0)$ will not be defined as $g (x)$ can’t be equal to zero.

Q 7 :

Let $f : R - {0, 1} \to ℝ$ be a function such that $f (x) + f (\frac{1}{1 - x}) = 1 + x .$ Then $f (2)$ is equal to [2023]

$\frac{7}{3}$
$\frac{9}{2}$
$\frac{9}{4}$
$\frac{7}{4}$

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(3)

$f (x) + f (\frac{1}{1 - x}) = 1 + x$

$x = 2 \Rightarrow f (2) + f (- 1) = 3$ $\dots (i)$

$x = - 1 \Rightarrow f (- 1) + f (\frac{1}{2}) = 0$ $\dots (i i)$

$x = \frac{1}{2} \Rightarrow f (\frac{1}{2}) + f (2) = \frac{3}{2}$ $\dots (i i i)$

$(i) + (i i i) - (i i) \Rightarrow 2 f (2) = \frac{9}{2}$ $∴$ $f (2) = \frac{9}{4}$

Q 8 :

The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

no solution
exactly two solutions in $(- \infty, \infty)$
a unique solution in $(- \infty, 1)$
a unique solution in $(- \infty, \infty)$

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(4)

$x^{2} - 4 x + [x] + 3 = x [x]$

$\Rightarrow x^{2} - x [x] - (x - [x]) + 3 (x - 1) = 0$

$\Rightarrow x (x - [x]) - 1 (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x] - 3) = 0$

$\Rightarrow (x - 1) ({x} - 3) = 0 \Rightarrow x = 1 as {x} \neq 3$

Q 9 :

If $f (x) = \frac{2^{2 x}}{2^{2 x} + 2}, x \in R,$ then

$f (\frac{1}{2023}) + f (\frac{2}{2023}) + . . . . + f (\frac{2022}{2023})$ is equal to [2023]

1011
2010
1010
2011

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(1)

Given, $f (x) = \frac{2^{2 x}}{2^{2 x} + 2} = \frac{4^{x}}{4^{x} + 2}$

$f (1 - x) = \frac{4^{1 - x}}{4^{1 - x} + 2}$

Now, $f (x) + f (1 - x) = \frac{4^{x}}{4^{x} + 2} + \frac{4^{(1 - x)}}{4^{(1 - x)} + 2}$

$f (x) + f (1 - x) = \frac{4^{x}}{4^{x} + 2} + \frac{4^{1 - x}}{4^{1 - x} + 2} = \frac{4^{x}}{4^{x} + 2} + \frac{4}{4 + 2 (4^{x})}$

$= \frac{4^{x}}{4^{x} + 2} + \frac{2}{2 + 4^{x}} = 1$

$f (\frac{1}{2023}) + f (\frac{2}{2023}) + \dots + f (\frac{2022}{2023})$

$= f (\frac{1}{2023}) + f (\frac{2}{2023}) + f (\frac{3}{2023}) + \dots + f (1 - \frac{3}{2023}) + f (1 - \frac{2}{2023}) + f (1 - \frac{1}{2023})$

$= 1 + 1 + 1 + \dots + 1 (1011 times) = 1011$

Q 10 :

Let $f (x)$ be a function such that $f (x + y) = f (x) \cdot f (y)$ for all $x, y \in N .$ If $f (1) = 3$ and $\sum_{k = 1}^{n} f (k) = 3279,$ then the value of n is [2023]

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(3)

$f (x + y) = f (x) \cdot f (y) \forall x, y \in N$

Put $x = y = 1, f (2) = 3^{2}; Put x = 2, y = 1$

$f (3) = 3^{3}, f (4) = 3^{4}$ and so on.

Now, $\sum_{k = 1}^{n} f (k) = 3279 \Rightarrow 3 + 3^{2} + 3^{3} + \dots + 3^{n} = 3279$

$\Rightarrow \frac{3 (3^{n} - 1)}{3 - 1} = 3279 \Rightarrow 3^{n} = 2187$ $∴$ $n = 7$

Q 11 :

The number of functions $f : {1, 2, 3, 4} \to {a \in : Z | a | \leq 8}$ satisfying $f (n) + \frac{1}{n} f (n + 1) = 1, \forall n \in {1, 2, 3}$ is [2023]

2
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4
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(1)

$f : {1, 2, 3, 4} \to {a \in Z : | a | \leq 8}$

$f (n) + \frac{1}{n} f (n + 1) = 1, \forall n \in {1, 2, 3}$

$f (n + 1)$ must be divisible by $n$

$f (4) \Rightarrow$ -6, -3, 0, 3, 6

$f (3) \Rightarrow$ -8, -6, -4, -2, 0, 2, 4, 6, 8

$f (2) \Rightarrow$ -8, ................, 8

$f (1) \Rightarrow$ -8, .............. , 8

$\frac{f (4)}{3} must be odd since f (3) is even.$

$∴$ Only two solutions possible.

$\begin{matrix} f (4) & f (3) & f (2) & f (1) \\ - 3 & 2 & 0 & 1 \\ 3 & 0 & 1 & 0 \end{matrix}$

Q 12 :

Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N, a n d f (4) = 133, f (5) = 255 .$ Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is [2023]

59
60
61
58

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(2)

$f (x) = 2 x^{n} + λ, λ \in R, n \in N$

$f (4) = 2 {(4)}^{n} + λ = 133$ ... $(i)$

$f (5) = 2 {(5)}^{n} + λ = 255$ ... $(ii)$

(ii) - (i), we get

$\Rightarrow f (5) - f (4) = 2 {(5)}^{n} - 2 {(4)}^{n} = 122; 5^{n} - 4^{n} = 61$

which gives $n = 3$ .

Then $f (3) = 2 {(3)}^{3} + λ, f (2) = 2 {(2)}^{3} + λ$

$f (3) = 54 + λ,$ $f (2) = 16 + λ .$ So, $f (3) - f (2) = 38$

Positive divisions of 38 are 1, 2, 19, 38 and whose sum = 1 + 2 + 19 + 38 = 60

Q 13 :

Let $f : R \to R$ be a function defined by $f (x) = \log_{\sqrt{m}} {\sqrt{2} (\sin x - \cos x) + m - 2},$ for some $m$ , such that the range of $f$ is [0, 2]. Then the value of $m$ is [2023]

3
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(2)

Since, $- \sqrt{2} \leq \sin x - \cos x \leq \sqrt{2}$

$∴$ $- 2 \leq \sqrt{2} (\sin x - \cos x) \leq 2$

Let $\sqrt{2} (\sin x - \cos x) = t$

$- 2 \leq t \leq 2$ ...(i)

$f (x) = \log_{\sqrt{m}} (t + m - 2)$

We have, $0 \leq f (x) \leq 2$

$\Rightarrow 0 \leq \log_{\sqrt{m}} (t + m - 2) \leq 2$

$1 \leq t + m - 2 \leq m$

$- m + 3 \leq t \leq 2$ ...(ii)

From (i) and (ii), we get $- m + 3 = - 2 \Rightarrow m = 5$

Q 14 :

Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

$f (x)$ is many-one in $(- \infty, - 1)$
$f (x)$ is one-one in $(- \infty, \infty)$
$f (x)$ is many-one in $(1, \infty)$
$f (x)$ is one-one in $[1, \infty)$ but not in $(- \infty, \infty)$

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(4)

Given, $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1}$

So, $f (x) = 1 + \frac{2 x}{x^{2} + 1}$

$f' (x) = \frac{(x^{2} + 1) \cdot 2 - 2 x \times 2 x}{{(x^{2} + 1)}^{2}}$

$= \frac{2 - 2 x^{2}}{{(x^{2} + 1)}^{2}} = \frac{2 (1 - x^{2})}{{(x^{2} + 1)}^{2}}$

The graph of the function is, By horizontal line test, we can say that

$f (x) is one-one in [1, \infty) but not in (- \infty, \infty) .$

Q 15 :

The domain of $f (x) = \frac{\log_{(x + 1)} (x - 2)}{e^{2 \log_{e} x} - (2 x + 3)}, x \in ℝ$ is [2023]

$(2, \infty) - {3}$
$ℝ - {3}$
$ℝ - {- 1, 3}$
$(- 1, \infty) - {3}$

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(1)

Given, $f (x) = \frac{\log_{(x + 1)} (x - 2)}{e^{2 \log_{e} x} - (2 x + 3)} = \frac{\log_{(x + 1)} (x - 2)}{x^{2} - 2 x - 3}$

Logarithmic function will be defined

$x - 2 > 0 \Rightarrow x > 2 and x + 1 > 0 \Rightarrow x > - 1$

$\Rightarrow x + 1 \neq 1 \Rightarrow x \neq 0 and x > 0$

In denominator, $x^{2} - 2 x - 3 \neq 0$

$\Rightarrow (x - 3) (x + 1) \neq 0 \Rightarrow x \neq - 1, 3 .$

So, $x \in (2, \infty) - {3}$

Q 16 :

Consider a function $f : ℕ \to ℝ,$ satisfying $f (1) + 2 f (2) + 3 f (3) + . . . + x f (x) = x (x + 1) f (x); x \geq 2$ with $f (1) = 1 .$ Then $\frac{1}{f (2022)} + \frac{1}{f (2028)}$ is equal to [2023]

8400
8200
8100
8000

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(3)

Let us consider a function $f : N \to R$ satisfying

$f (1) + 2 f (2) + 3 f (3) \dots + x f (x) = x (x + 1) f (x)$

where $x \geq 2$ with $f (1) = 1$ . We have for $x \geq 2$

$f (1) + 2 f (2) + 3 f (3) + \dots + x f (x) = x (x + 1) f (x)$

Now we will replace $x$ by $(x + 1)$ , we get

$x (x + 1) f (x) + (x + 1) f (x + 1) = (x + 1) (x + 2) f (x + 1)$

$\Rightarrow \frac{x}{f (x + 1)} + \frac{1}{f (x)} = \frac{x + 2}{f (x)} \Rightarrow x f (x) = (x + 1) f (x + 1) x \geq 2$

$f (2) = \frac{1}{4}, f (3) = \frac{1}{6}$

Now, $f (2022) = \frac{1}{4044}$

Similarly, $f (2028) = \frac{1}{4056}$

So, $\frac{1}{f (2022)} + \frac{1}{f (2028)} = 4044 + 4056 = 8100$

Q 17 :

The range of the function $f (x) = \sqrt{3 - x} + \sqrt{2 + x}$ is [2023]

$[2 \sqrt{2}, \sqrt{11}]$
$[\sqrt{5}, \sqrt{10}]$
$[\sqrt{5}, \sqrt{13}]$
$[\sqrt{2}, \sqrt{7}]$

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(2)

Given, $f (x) = \sqrt{3 - x} + \sqrt{2 + x}$

Let $y = \sqrt{3 - x} + \sqrt{2 + x}$

$\Rightarrow y^{2} = 3 - x + 2 + x + 2 \sqrt{3 - x} \sqrt{2 + x}$

$= 5 + 2 \sqrt{6 + x - x^{2}}$

$= 5 + 2 \sqrt{\frac{25}{4} - {(x - \frac{1}{2})}^{2}}$

$y_{m a x}^{2}$ when ${(x - \frac{1}{2})}^{2} = 0$

$∴$ $y_{\max}^{2} = 5 + 2 \times \frac{5}{2} = 10 \Rightarrow y_{\max} = \sqrt{10}$

$y_{m i n}^{2}$ when $\sqrt{\frac{25}{4} - {(x - \frac{1}{2})}^{2}} = 0 \Rightarrow y_{\min} = \sqrt{5}$

$Range of f (x) = [\sqrt{5}, \sqrt{10}]$

Q 18 :

If the domain of the function $f (x) = \frac{[x]}{1 + x^{2}},$ where [x] is greatest integer $\leq x,$ is [2,6), then its range is [2023]

$(\frac{5}{37}, \frac{2}{5}]$
$(\frac{5}{26}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$
$(\frac{5}{26}, \frac{2}{5}]$
$(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$

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(1)

We have, $f (x) = \frac{[x]}{1 + x^{2}} and domain = [2, 6)$

$∴$ $f (x) =$ ${\begin{matrix} \frac{2}{1 + x^{2}}; & [2, 3) \\ \frac{3}{1 + x^{2}}; & [3, 4) \\ \frac{4}{1 + x^{2}}; & [4, 5) \\ \frac{5}{1 + x^{2}}; & [5, 6) \end{matrix}$

For $x \in [2, 6)$ , $f (x) > 0$ and it is a decreasing function.

At $x = 2, f (x) = \frac{2}{5}$ and at $x = 6, f (x) = \frac{5}{37}$

$Hence, Range = (\frac{5}{37}, \frac{2}{5}]$

Q 19 :

Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} - 8 x + 12} .$ Then range of $f$ is [2023]

$(- \infty, - \frac{21}{4}] \cup [\frac{21}{4}, \infty)$
$(- \infty, - \frac{21}{4}] \cup [0, \infty)$
$(- \infty, - \frac{21}{4}) \cup (0, \infty)$
$(- \infty, - \frac{21}{4}] \cup [1, \infty)$

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(2)

Let $y = \frac{x^{2} + 2 x + 1}{x^{2} - 8 x + 12}$

By cross multiplying, we get

$y x^{2} - 8 x y + 12 y - x^{2} - 2 x - 1 = 0$

$\Rightarrow x^{2} (y - 1) - x (8 y + 2) + (12 y - 1) = 0$

Case I: When $y \neq 1$ ; $D \geq 0$

$\Rightarrow {(8 y + 2)}^{2} - 4 (y - 1) (12 y - 1) \geq 0$

$\Rightarrow y (4 y + 21) > 0$

$\Rightarrow y \in (- \infty, - \frac{21}{4}] \cup [0, \infty) - {1}$

Case II: When $y = 1$

$x^{2} + 2 x + 1 = x^{2} - 8 x + 12$

$\Rightarrow 10 x = 11$

$\Rightarrow x = \frac{11}{10}$

So, $y$ can be 1. Hence, $y \in (- \infty, - \frac{21}{4}] \cup [0, \infty)$

Q 20 :

If domain of the function

$\log_{e} (\frac{6 x^{2} + 5 x + 1}{2 x - 1}) + \cos^{- 1} (\frac{2 x^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ],$ then 18 $(α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to ____________. [2023]

(20)

For domain, $\frac{6 x^{2} + 5 x + 1}{2 x - 1} > 0$

$\Rightarrow \frac{(3 x + 1) (2 x + 1)}{2 x - 1} > 0 \Rightarrow x \in (\frac{- 1}{2}, \frac{- 1}{3}) \cup (\frac{1}{2}, \infty)$ $...(A)$

and

$\Rightarrow \frac{2 x^{2} - 1}{3 x - 5} \geq 0 and \frac{2 x^{2} - 6 x + 9}{3 x - 5} \leq 0 \Rightarrow 3 x - 5 < 0$

$\Rightarrow x \in [\frac{- 1}{\sqrt{2}}, \frac{1}{\sqrt{2}}] \cup (\frac{5}{3}, \infty) ...(B)$

$x < \frac{5}{3}$ $...(C)$

$∴$ $A \cap B \cap C \equiv (\frac{- 1}{2}, \frac{- 1}{3}) \cup (\frac{1}{2}, \frac{1}{\sqrt{2}}]$

$So, 18 (α^{2} + β^{2} + γ^{2} + δ^{2}) = 18 (\frac{1}{4} + \frac{1}{9} + \frac{1}{4} + \frac{1}{2})$

$= 18 + 2 = 20$

Q 21 :

Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

(180)

Total onto functions

$= 4^{5} - {}^{4}{C_{1}} \times 3^{5} + {}^{4}{C_{2}} \times 2^{5} - {}^{4}{C_{3}} \times 1^{5} = 240$

Now, when $f (a) = 1$

$⌊ 4 + \frac{⌊ 4}{⌊ 2 ⌊ 2} \times ⌊ 3 = 24 + 36 = 60$

So, required number of onto functions $= 240 - 60 = 180$

Q 22 :

Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

(8)

${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}, where 2 a > 0 and b c > 0$

$\Rightarrow \log_{e} a (\log_{e} 2 + \log_{e} a) = \log_{e} b (\log_{e} b + \log_{e} c)$ ...(i)

Also, $b^{\log_{e} 2} = a^{\log_{e} c}$

$\Rightarrow \log_{e} 2 \log_{e} b = \log_{e} c \log_{e} a$ ...(ii)

Let $\log_{e} 2 = α, \log_{e} a = x, \log_{e} b = y, and \log_{e} c = z$

$∴ From (i) and (ii), we get x (α + x) = y (y + z) and α y = x z$

$\Rightarrow x (\frac{x z}{y} + x) = y (y + z)$ $[∵ α = \frac{x z}{y}]$

$\Rightarrow x^{2} (z + y) = y^{2} (y + z)$

$\Rightarrow y + z = 0 or x^{2} = y^{2} \Rightarrow x = - y$ $[∵ x = y is not possible]$

$\Rightarrow b c = 1 or a b = 1$

Now, $b c = 1 \Rightarrow 2 a \log_{e} a = 1 \Rightarrow a = 1 or \frac{1}{2}$

$Case - 1: Distinct possible values of a, b, c can be (\frac{1}{2}, p, \frac{1}{p}), where p \neq 1, 2, \frac{1}{2}$

$∴ 6 a + 5 b c = 6 (\frac{1}{2}) + 5 (p) (\frac{1}{p}) = 3 + 5 = 8$

$Case - 2: (a, b, c) can be (p, \frac{1}{p}, \frac{1}{2}), where p \neq (1, 2, \frac{1}{2})$

So, in this case infinite answers are possible.

Q 23 :

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

(360)

We have, $A = {1, 2, 3, 4, 5}; B = {1, 2, 3, 4, 5, 6}$

Also, $f (1) + f (2) = f (4) - 1 \Rightarrow f (1) + f (2) + 1 = f (4)$

Here, $f (4) \leq 6$

$f (1) + f (2) + 1 \leq 6$

$f (1) + f (2) \leq 5$

At $f (1) = 1, then, f (2) = 1, 2, 3, 4 \Rightarrow 4 mappings,$

$f (1) = 2, then, f (2) = 1, 2, 3 \Rightarrow 3 mappings,$

$f (1) = 3, then, f (2) = 1, 2 \Rightarrow 2 mappings$

$f (1) = 4, then, f (2) = 1 \Rightarrow 1 mapping$

$f (3) and f (5) have 6 mappings each.$

$∴ Total number of functions = 10 \times 6 \times 6 = 360$

Q 24 :

Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

(825)

$\sum_{r = 1}^{120} [\sqrt{r}] = \sum_{r = 1}^{3} 1 + \sum_{r = 4}^{8} 2 + \sum_{r = 9}^{15} 3 + \sum_{r = 16}^{25} 4 + \dots + \sum_{r = 81}^{99} 9 + \sum_{r = 100}^{120} 10$

$= 3 (1) + 5 (2) + 7 (3) + 9 (4) + \dots + 21 (10)$

$= \sum_{r = 1}^{10} (2 r + 1) r = 2 \times \frac{10 \times 11 \times 21}{6} + \frac{10 \times 11}{2}$

$= \frac{10 \times 11}{2} \times 15 = 825$

Q 25 :

For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

(2039)

Given, $f (x) = a x - 3, g (x) = x^{b} + c$ and ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{\frac{1}{3}}$

Let $f o g (x) = h^{- 1} (x) .$ So, $h^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3}$

Let $y = {(\frac{x - 7}{2})}^{1 / 3}$ $\Rightarrow$ $y^{3} = \frac{x - 7}{2} \Rightarrow x = 2 y^{3} + 7$

$∴$ $h (x) = f o g (x) = 2 x^{3} + 7$

Now, $f o g (x) = a (x^{b} + c) - 3 \Rightarrow 2 x^{3} + 7 = a (x^{b} + c) - 3$

Comparing the coefficients of like powers, we get

$a = 2, b = 3$ and $c = 5$

So, $f o g (a c) = f o g (10) = 2 {(10)}^{3} + 7 = 2007$

$g [f (x)] = {(a x - 3)}^{b} + c = {(2 x - 3)}^{3} + 5$

$g o f (b) = (2 {(3) - 3)}^{3} + 5 = 27 + 5 = 32$

$∴ f o g (a c) + g o f (b) = 2007 + 32 = 2039$

Q 26 :

Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

(10)

Given $f (x + y) = f (x) + f (y) and f (1) = \frac{1}{5}$

So, $f (2) = f (1) + f (1) = \frac{2}{5}; f (3) = f (2) + f (1) = \frac{3}{5}$

$∴ \sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{5} \sum_{n = 1}^{m} (\frac{1}{n + 1} - \frac{1}{n + 2})$

$= \frac{1}{5} (\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \dots + \frac{1}{m + 1} - \frac{1}{m + 2})$

$= \frac{1}{5} (\frac{1}{2} - \frac{1}{m + 2}) = \frac{m}{10 (m + 2)} = \frac{1}{12}$

$∴ m = 10$

Q 27 :

Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

(3240)

Q 28 :

Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

(3125)

We have, $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in ℝ - {\frac{- 2}{3}}$

$f^{2} (x) = \frac{13 x + 12}{12 x + 13}; f^{3} (x) = \frac{63 x + 62}{62 x + 63}$

$∴$ $f^{5} (x) = \frac{1563 x + 1562}{1562 x + 1563} = \frac{a x + b}{b x + a}$

$∴$ $a + b = 1563 + 1562 = 3125$

Q 29 :

Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

(432)

Let $A = {1, 2, 3, 5, 8, 9}$

$f (m \cdot n) = f (m) \cdot f (n) for every m, n \in A and m \cdot n \in A$

Clearly, $f (1) = 1$ , i.e. $f (3) = 1$ or $3$

Total functions = $1 \times 6 \times 2 \times 6 \times 6 \times 1 = 432$

Q 30 :

Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

4
16
8
2

1

2

3

4

(2)

We have, $(g o f) (x) = x \Rightarrow g (f (x)) = x$

Differentiating w.r.t. $x$ , we get

$g' (f (x)) \cdot f' (x) = 1 \Rightarrow g' (f (x)) = \frac{1}{f' (x)}$

Now, $f (x) = x^{5} + 2 e^{x / 4} \Rightarrow f' (x) = 5 x^{4} + \frac{1}{2} e^{x / 4}$

$∴$ $f' (0) = \frac{1}{2}, f (0) = 2 \Rightarrow g' (2) = 2$

Hence, $8 g' (2) = 16$

Topic Question Set

Q 1 :

Let $A = {x \in ℝ : [x + 3] + [x + 4] \leq 3},$

$B = {x \in ℝ : 3^{x} {(\sum_{r = 1}^{\infty} \frac{3}{10^{r}})}^{x - 3} < 3^{- 3 x}},$

where [t] denotes greatest integer function. Then, [2023]

Q 3 :

If $f (x) = \frac{(\tan 1^{o}) x + \log_{e} (123)}{x \log_{e} (1234) - (\tan 1^{o})}, x > 0,$ then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is [2023]

Q 4 :

The number of integral solutions x of $\log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2} \geq 0$ is [2023]

(1)

Q 5 :

The domain of the function $f (x) = \frac{1}{\sqrt{{[x]}^{2} - 3 [x] - 10}}$ is (where [x] denotes the greatest integer less than or equal to x). [2023]

Q 6 :

For $x \in ℝ$ , two real valued functions $f (x)$ and $g (x)$ are such that, $g (x) = \sqrt{x} + 1$ and $f o g (x) = x + 3 - \sqrt{x} .$ Then $f (0)$ is equal to [2023]

Q 7 :

Let $f : R - {0, 1} \to ℝ$ be a function such that $f (x) + f (\frac{1}{1 - x}) = 1 + x .$ Then $f (2)$ is equal to [2023]

Q 8 :

The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

Q 9 :

If $f (x) = \frac{2^{2 x}}{2^{2 x} + 2}, x \in R,$ then

$f (\frac{1}{2023}) + f (\frac{2}{2023}) + . . . . + f (\frac{2022}{2023})$ is equal to [2023]

Q 10 :

Let $f (x)$ be a function such that $f (x + y) = f (x) \cdot f (y)$ for all $x, y \in N .$ If $f (1) = 3$ and $\sum_{k = 1}^{n} f (k) = 3279,$ then the value of n is [2023]

Q 11 :

The number of functions $f : {1, 2, 3, 4} \to {a \in : Z | a | \leq 8}$ satisfying $f (n) + \frac{1}{n} f (n + 1) = 1, \forall n \in {1, 2, 3}$ is [2023]

Q 12 :

Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N, a n d f (4) = 133, f (5) = 255 .$ Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is [2023]

Q 13 :

Let $f : R \to R$ be a function defined by $f (x) = \log_{\sqrt{m}} {\sqrt{2} (\sin x - \cos x) + m - 2},$ for some $m$ , such that the range of $f$ is [0, 2]. Then the value of $m$ is [2023]

Q 14 :

Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

Q 15 :

The domain of $f (x) = \frac{\log_{(x + 1)} (x - 2)}{e^{2 \log_{e} x} - (2 x + 3)}, x \in ℝ$ is [2023]

Q 16 :

Consider a function $f : ℕ \to ℝ,$ satisfying $f (1) + 2 f (2) + 3 f (3) + . . . + x f (x) = x (x + 1) f (x); x \geq 2$ with $f (1) = 1 .$ Then $\frac{1}{f (2022)} + \frac{1}{f (2028)}$ is equal to [2023]

Q 17 :

The range of the function $f (x) = \sqrt{3 - x} + \sqrt{2 + x}$ is [2023]

Q 18 :

If the domain of the function $f (x) = \frac{[x]}{1 + x^{2}},$ where [x] is greatest integer $\leq x,$ is [2,6), then its range is [2023]

Q 19 :

Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} - 8 x + 12} .$ Then range of $f$ is [2023]

Q 20 :

If domain of the function

$\log_{e} (\frac{6 x^{2} + 5 x + 1}{2 x - 1}) + \cos^{- 1} (\frac{2 x^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ],$ then 18 $(α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to ____________. [2023]

Q 21 :

Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

Q 22 :

Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

Q 23 :

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

Q 24 :

Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

Q 25 :

For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

Q 26 :

Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

Q 27 :

Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

(3240)

Q 28 :

Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

Q 29 :

Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

Q 30 :

Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]

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Topic Question Set

Q 1 : Let A={x∈ℝ:[x+3]+[x+4]≤3}, B={x∈ℝ:3x(∑r=1∞310r)x-3<3-3x}, where [t] denotes greatest integer function. Then, [2023]

Q 2 : Let the sets A and B denote the domain and range respectively of the function f(x)=1⌈x⌉-x, where ⌈x⌉ denotes the smallest integer greater than or equal to x. Then among the statements (S1):A∩B=(1,∞)-N and (S2): A∪B=(1,∞) [2023]

Q 3 : If f(x)=(tan1o)x+loge(123)xloge(1234)-(tan1o),x>0, then the least value of f(f(x))+f(f(4x)) is [2023]

Q 4 : The number of integral solutions x of log(x+72)(x-72x-3)2≥0 is [2023]

(1)

Q 5 : The domain of the function f(x)=1[x]2-3[x]-10 is (where [x] denotes the greatest integer less than or equal to x). [2023]

Q 6 : For x∈ℝ, two real valued functions f(x) and g(x) are such that, g(x)=x+1 and fog(x)=x+3-x. Then f(0) is equal to [2023]

Q 7 : Let f:R-{0,1}→ℝ be a function such that f(x)+f(11-x)=1+x. Then f(2) is equal to [2023]

Q 8 : The equation x2-4x+[x]+3=x[x], where [x] denotes the greatest integer function, has [2023]

Q 9 : If f(x)=22x22x+2,x∈R, then f(12023)+f(22023)+....+f(20222023) is equal to [2023]

Q 10 : Let f(x) be a function such that f(x+y)=f(x)·f(y) for all x,y∈N. If f(1)=3 and ∑k=1nf(k)=3279, then the value of n is [2023]

Q 11 : The number of functions f:{1,2,3,4}→{a∈:Z|a|≤8} satisfying f(n)+1nf(n+1)=1, ∀n∈{1,2,3} is [2023]

Q 12 : Let f(x)=2xn+λ,λ∈R,n∈N, and f(4)=133,f(5)=255. Then the sum of all the positive integer divisors of (f(3)-f(2)) is [2023]

Q 13 : Let f:R→R be a function defined by f(x)=logm{2(sinx-cosx)+m-2}, for some m, such that the range of f is [0, 2]. Then the value of m is [2023]

Q 14 : Let f:R→R be a function such that f(x)=x2+2x+1x2+1. Then [2023]

Q 15 : The domain of f(x)=log(x+1)(x-2)e2logex-(2x+3),x∈ℝ is [2023]

Q 16 : Consider a function f:ℕ→ℝ, satisfying f(1)+2f(2)+3f(3)+...+xf(x)=x(x+1)f(x);x≥2 with f(1)=1. Then 1f(2022)+1f(2028) is equal to [2023]

Q 17 : The range of the function f(x)=3-x+2+x is [2023]

Q 18 : If the domain of the function f(x)=[x]1+x2, where [x] is greatest integer ≤x, is [2,6), then its range is [2023]

Q 19 : Let f:R-{2,6}→R be real valued function defined as f(x)=x2+2x+1x2-8x+12. Then range of f is [2023]

Q 20 : If domain of the function loge(6x2+5x+12x-1)+cos-1(2x2-3x+43x-5) is (α,β)∪(γ,δ], then 18(α2+β2+γ2+δ2) is equal to ____________. [2023]

Q 21 : Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions f:R→S such that f(a)≠1, is equal to ________ . [2023]

Q 22 : Let a, b, c be three distinct positive real numbers such that (2a)logea=(bc)logeb and bloge2=alogec. Then 6a + 5bc is equal to _____________ . [2023]

Q 23 : Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions f:A→B satisfying f(1)+f(2)=f(4)-1 is equal to __________ . [2023]

Q 24 : Let [α] denote the greatest integer ≤α. Then [1]+[2]+[3]+...+[120] is equal to __________ . [2023]

Q 25 : For some a, b, c∈N, let f(x)=ax-3 and g(x)=xb+c,x∈R. If (fog)-1(x)=(x-72)1/3, then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

Q 26 : Suppose f is a function satisfying f(x+y)=f(x)+f(y) for all x,y∈ℕ and f(1)=15. If ∑n=1mf(n)n(n+1)(n+2)=112, then m is equal to ___________ . [2023]

Q 27 : Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions f:S→P(S), where P(S) denote the power set of S, such that f(n)⊂f(m) where n<m is __________ . [2023]

(3240)

Q 28 : Let f1(x)=3x+22x+3,x∈R-{-32} For n≥2, define fn(x)=f1ofn-1(x). If f5(x)=ax+bbx+a,gcd(a,b)=1, then a + b is equal to _________ . [2023]

Q 29 : Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions f:A→A such that f(m·n)=f(m)·f(n) for every m,n∈A with m·n∈A is equal to __________________ . [2023]

Q 30 : Let f(x)=x5+2ex/4 for all x∈R. Consider a function g(x) such that (gof) (x)=x for all x∈R. Then the value of 8g'(2) is [2024]

Want Us to Email you About Special Offers & Updates?

Q 1 :

Let $A = {x \in ℝ : [x + 3] + [x + 4] \leq 3},$

$B = {x \in ℝ : 3^{x} {(\sum_{r = 1}^{\infty} \frac{3}{10^{r}})}^{x - 3} < 3^{- 3 x}},$

where [t] denotes greatest integer function. Then, [2023]

Q 3 :

If $f (x) = \frac{(\tan 1^{o}) x + \log_{e} (123)}{x \log_{e} (1234) - (\tan 1^{o})}, x > 0,$ then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is [2023]

Q 4 :

The number of integral solutions x of $\log_{(x + \frac{7}{2})} {(\frac{x - 7}{2 x - 3})}^{2} \geq 0$ is [2023]

Q 5 :

The domain of the function $f (x) = \frac{1}{\sqrt{{[x]}^{2} - 3 [x] - 10}}$ is (where [x] denotes the greatest integer less than or equal to x). [2023]

Q 6 :

For $x \in ℝ$ , two real valued functions $f (x)$ and $g (x)$ are such that, $g (x) = \sqrt{x} + 1$ and $f o g (x) = x + 3 - \sqrt{x} .$ Then $f (0)$ is equal to [2023]

Q 7 :

Let $f : R - {0, 1} \to ℝ$ be a function such that $f (x) + f (\frac{1}{1 - x}) = 1 + x .$ Then $f (2)$ is equal to [2023]

Q 8 :

The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

Q 9 :

If $f (x) = \frac{2^{2 x}}{2^{2 x} + 2}, x \in R,$ then

$f (\frac{1}{2023}) + f (\frac{2}{2023}) + . . . . + f (\frac{2022}{2023})$ is equal to [2023]

Q 10 :

Let $f (x)$ be a function such that $f (x + y) = f (x) \cdot f (y)$ for all $x, y \in N .$ If $f (1) = 3$ and $\sum_{k = 1}^{n} f (k) = 3279,$ then the value of n is [2023]

Q 11 :

The number of functions $f : {1, 2, 3, 4} \to {a \in : Z | a | \leq 8}$ satisfying $f (n) + \frac{1}{n} f (n + 1) = 1, \forall n \in {1, 2, 3}$ is [2023]

Q 12 :

Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N, a n d f (4) = 133, f (5) = 255 .$ Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is [2023]

Q 13 :

Let $f : R \to R$ be a function defined by $f (x) = \log_{\sqrt{m}} {\sqrt{2} (\sin x - \cos x) + m - 2},$ for some $m$ , such that the range of $f$ is [0, 2]. Then the value of $m$ is [2023]

Q 14 :

Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

Q 15 :

The domain of $f (x) = \frac{\log_{(x + 1)} (x - 2)}{e^{2 \log_{e} x} - (2 x + 3)}, x \in ℝ$ is [2023]

Q 16 :

Consider a function $f : ℕ \to ℝ,$ satisfying $f (1) + 2 f (2) + 3 f (3) + . . . + x f (x) = x (x + 1) f (x); x \geq 2$ with $f (1) = 1 .$ Then $\frac{1}{f (2022)} + \frac{1}{f (2028)}$ is equal to [2023]

Q 17 :

The range of the function $f (x) = \sqrt{3 - x} + \sqrt{2 + x}$ is [2023]

Q 18 :

If the domain of the function $f (x) = \frac{[x]}{1 + x^{2}},$ where [x] is greatest integer $\leq x,$ is [2,6), then its range is [2023]

Q 19 :

Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} - 8 x + 12} .$ Then range of $f$ is [2023]

Q 20 :

If domain of the function

$\log_{e} (\frac{6 x^{2} + 5 x + 1}{2 x - 1}) + \cos^{- 1} (\frac{2 x^{2} - 3 x + 4}{3 x - 5})$ is $(α, β) \cup (γ, δ],$ then 18 $(α^{2} + β^{2} + γ^{2} + δ^{2})$ is equal to ____________. [2023]

Q 21 :

Let R = {a, b, c, d, e} and S = {1, 2, 3, 4}. Total number of onto functions $f : R \to S$ such that $f (a) \neq 1,$ is equal to ________ . [2023]

Q 22 :

Let a, b, c be three distinct positive real numbers such that ${(2 a)}^{\log_{e} a} = {(b c)}^{\log_{e} b}$ and $b^{\log_{e} 2} = a^{\log_{e} c}$ .

Then 6a + 5bc is equal to _____________ . [2023]

Q 23 :

Let A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}. Then the number of functions $f : A \to B$ satisfying $f (1) + f (2) = f (4) - 1$ is equal to __________ . [2023]

Q 24 :

Let $[α]$ denote the greatest integer $\leq α$ . Then $[\sqrt{1}] + [\sqrt{2}] + [\sqrt{3}] + . . . + [\sqrt{120}]$ is equal to __________ . [2023]

Q 25 :

For some a, b, c $\in$ N, let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R .$ If ${(f o g)}^{- 1} (x) = {(\frac{x - 7}{2})}^{1 / 3},$ then (fog)(ac) + (gof)(b) is equal to ___________ . [2023]

Q 26 :

Suppose $f$ is a function satisfying $f (x + y) = f (x) + f (y)$ for all $x, y \in ℕ$ and $f (1) = \frac{1}{5} .$ If $\sum_{n = 1}^{m} \frac{f (n)}{n (n + 1) (n + 2)} = \frac{1}{12},$ then $m$ is equal to ___________ . [2023]

Q 27 :

Let S = {1, 2, 3, 4, 5, 6}. Then the number of one-one functions $f : S \to P (S),$ where $P (S)$ denote the power set of S, such that $f (n) \subset f (m)$ where $n < m$ is __________ . [2023]

Q 28 :

Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$

For $n \geq 2,$ define $f^{n} (x) = f^{1} o f^{n - 1} (x) .$

If $f^{5} (x) = \frac{a x + b}{b x + a}, g c d (a, b) = 1,$ then a + b is equal to _________ . [2023]

Q 29 :

Let A = {1, 2, 3, 5, 8, 9}. Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to __________________ . [2023]

Q 30 :

Let $f (x) = x^{5} + 2 e^{x / 4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g o f)$ $(x) = x$ for all $x \in R$ . Then the value of $8 g' (2)$ is [2024]