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

Q 61 :

The sum of all local minimum values of the function $f (x) = {\begin{matrix} 1 - 2 x, & x < - 1 \\ \frac{1}{3} (7 + 2 | x |), & - 1 \leq x \leq 2 \\ \frac{11}{18} (x - 4) (x - 5), & x > 2 \end{matrix}$ is [2025]

$\frac{131}{72}$
$\frac{171}{72}$
$\frac{167}{72}$
$\frac{157}{72}$

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Graph of the function f(x)

The sum of local minimum values at A and B

$= \frac{7}{3} - \frac{11}{72} = \frac{157}{72}$ .

Q 62 :

If $f (x) = \frac{2^{x}}{2^{x} + \sqrt{2}}, x \in R$ , then $\sum_{k = 1}^{81} f (\frac{k}{82})$ is equal to [2025]

$\frac{81}{2}$
82
41
$81 \sqrt{2}$

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

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

$= \frac{2^{x}}{2^{x} + \sqrt{2}} + \frac{2}{2 + \sqrt{2} \cdot 2^{x}} = 1$

Now, $S = \sum_{k = 1}^{81} f (\frac{k}{82}) = f (\frac{1}{82}) + f (\frac{2}{82}) + . . . + f (\frac{81}{82})$

$\Rightarrow S = f (\frac{1}{82}) + . . . + f (1 - \frac{2}{82}) + f (1 - \frac{1}{82})$

$= f (\frac{1}{82}) + f (1 - \frac{1}{82}) + f (\frac{2}{82}) + f (1 - \frac{2}{82}) + . . . + upto 40 + f (\frac{41}{82})$

$= \underset{40 times}{\underset{⏟}{(1 + 1 + 1 + . . . + 1)}} + \frac{2^{1 / 2}}{2^{1 / 2} + \sqrt{2}} = 40 + \frac{1}{2} = \frac{81}{2}$ .

Q 63 :

Let $f : [0, 3] \to A$ be defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 7$ and $g : [0, \infty) \to B$ be defined by $g (x) = \frac{x^{2025}}{x^{2025} + 1}$ . If both the functions are onto and $S = {x \in Z : x \in A or x \in B}$ , then n(S) is equal to: [2025]

36
29
30
31

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As f(x) is onto, hence A is range of f(x).

Now, $f' (x) = 6 x^{2} - 30 x + 36 = 6 (x - 2) (x - 3)$

for extremum, f(2) = 16 – 60 + 72 + 7 = 35; f(3) = 54 – 135 + 108 + 7 = 34; f(0) = 7; F(1) = 30.

Hence, range $\in$ [7, 35] = A

Also, for range of g(x), g(x) = $1 - \frac{1}{x^{2025} + 1} \in [0, 1) = B$

$∴$ S = {0, 7, 8, ..., 35}

Hence, n(S) = 30

Q 64 :

If the domain of the function $\log_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $\log_{(x - 1)} (\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to : [2025]

195
174
186
179

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Let $f_{1} (x) = \log_{5} (18 x - x^{2} - 77)$

$∴ 18 x - x^{2} - 77 > 0$

$\Rightarrow x^{2} - 18 x + 77 < 0$

$\Rightarrow (x - 11) (x - 7) < 0$

$\Rightarrow x \in (7, 11) ∴ α = 7, β = 11$

Also, let $f_{2} (x) = \log_{(x - 1)} (\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4})$

$∴ x - 1 > 0 \Rightarrow x > 1, x - 1 \neq 1 \Rightarrow x \neq 2$ ,

$\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4} > 0 \Rightarrow \frac{(2 x - 1) (x + 2)}{(x - 4) (x + 1)} > 0$

$\Rightarrow x \in (4, \infty) ∴ γ = 4$

Now, $α^{2} + β^{2} + γ^{2}$ = 49 + 121 + 16 = 186.

Q 65 :

If the domain of the function

$f (x) = \log_{(10 x^{2} - 17 x + 7)} (18 x^{2} - 11 x + 1)$

is $(- \infty, a) \cup (b, c) \cup (d, \infty) - {e},$ then $90 (a + b + c + d + e)$ equals: [2026]

316
177
170
307

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Q 66 :

If the domain of the function $f (x)$ $= \sin^{- 1} (\frac{5 - x}{3 + 2 x}) + \frac{1}{\log_{e} (10 - x)}$ is $(- \infty, α] \cup [β, γ) - {δ},$ then $6 (α + β + γ + δ)$ is equal to [2026]

68
67
66
70

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$- 1 \leq \frac{5 - x}{2 x + 3} \leq 1 & 10 - x > 0, 10 - x \neq 1$

$| \frac{5 - x}{2 x + 3} |$ $\geq 1 & x < 10 & x \neq 9$

${(5 - x)}^{2} - {(2 x + 3)}^{2} \leq 0 & x < 10 & 4 x \neq 9$

$(x + 8) (3 x - 2) \geq 0 & x < 10 & x \neq 9$

$\Rightarrow (- \infty, - 8] \cup [\frac{2}{3}, 10) - {9}$

$\Rightarrow (α + β + γ + δ) = 6 (- 8 + \frac{2}{3} + 10 + 9)$

$= 70$

Q 67 :

The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$ where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]

4
-2
2
0

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Q 68 :

Given below are two statements :

Statement I : The function $f : ℝ \to ℝ$ defined by $f (x) = \frac{x}{1 + | x |}$ is one-one.

Statement II : The function $f : ℝ \to ℝ$ defined by $f (x) = \frac{x^{2} + 4 x - 30}{x^{2} - 8 x + 18}$ is many-one.

In the light of the above statements, choose the correct answer from the options given below : [2026]

Both Statement I and Statement II are false
Statement I is true but Statement II is false
Statement I is false but Statement II is true
Both Statement I and Statement II are true

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Q 69 :

Let $f$ be a function such that $3 f (x) + 2 f (\frac{m}{19 x}) = 5 x, x \neq 0,$ where $m = \sum_{i = 1}^{9} {(i)}^{2} .$ Then $f (5) - f (2)$ is equal to [2026]

18
- 9
9
36

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Q 70 :

Consider two sets $A = {x \in ℤ : | (| x - 3 | - 3) | \leq 1$ and

$B = {x \in ℝ - {1, 2} : \frac{(x - 2) (x - 4)}{x - 1} \log_{e} (| x - 2 |) = 0} .$

Then the number of onto functions $f : A \to B$ is equal to. [2026]

62
32
79
81

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$A : | | x - 3 | - 3 | \leq 1$

$\Rightarrow - 1 \leq | x - 3 | - 3 \leq 1$

$\Rightarrow 2 \leq$ $| x - 3 |$ $\leq 4$

$\Rightarrow 2 \leq (x - 3) \leq 4 or - 4 \leq (x - 3) \leq - 2$

$\Rightarrow 5 \leq x \leq 7 or - 1 \leq x \leq 1$

$A = {- 1, 0, 1, 5, 6, 7}$

$B \Rightarrow x = 4, | x - 2 | = 1 \Rightarrow x = 3 or 1 (reject)$

$\Rightarrow B = {3, 4}$

$Number of onto functions from A to B = 2^{6} - 2 = 62$

Topic Question Set

Q 61 :

The sum of all local minimum values of the function $f (x) = {\begin{matrix} 1 - 2 x, & x < - 1 \\ \frac{1}{3} (7 + 2 | x |), & - 1 \leq x \leq 2 \\ \frac{11}{18} (x - 4) (x - 5), & x > 2 \end{matrix}$ is [2025]

Q 62 :

If $f (x) = \frac{2^{x}}{2^{x} + \sqrt{2}}, x \in R$ , then $\sum_{k = 1}^{81} f (\frac{k}{82})$ is equal to [2025]

Q 63 :

Let $f : [0, 3] \to A$ be defined by $f (x) = 2 x^{3} - 15 x^{2} + 36 x + 7$ and $g : [0, \infty) \to B$ be defined by $g (x) = \frac{x^{2025}}{x^{2025} + 1}$ . If both the functions are onto and $S = {x \in Z : x \in A or x \in B}$ , then n(S) is equal to: [2025]

Q 64 :

If the domain of the function $\log_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $\log_{(x - 1)} (\frac{2 x^{2} + 3 x - 2}{x^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to : [2025]

Q 65 :

If the domain of the function

$f (x) = \log_{(10 x^{2} - 17 x + 7)} (18 x^{2} - 11 x + 1)$

is $(- \infty, a) \cup (b, c) \cup (d, \infty) - {e},$ then $90 (a + b + c + d + e)$ equals: [2026]

Q 66 :

If the domain of the function $f (x)$ $= \sin^{- 1} (\frac{5 - x}{3 + 2 x}) + \frac{1}{\log_{e} (10 - x)}$ is $(- \infty, α] \cup [β, γ) - {δ},$ then $6 (α + β + γ + δ)$ is equal to [2026]

Q 67 :

The sum of all the elements in the range of $f (x) = S g n (\sin x) + S g n (\cos x) + S g n (\tan x) + S g n (c o t x),$ $x \neq \frac{n π}{2}, n \in ℤ$ where $s g n (t) = {\begin{cases} 1 & i f \\ - 1 & i f \end{cases} \begin{matrix} t > 0 \\ t < 0 \end{matrix}$ is [2026]

Q 69 :

Let $f$ be a function such that $3 f (x) + 2 f (\frac{m}{19 x}) = 5 x, x \neq 0,$ where $m = \sum_{i = 1}^{9} {(i)}^{2} .$ Then $f (5) - f (2)$ is equal to [2026]

Q 70 :

Consider two sets $A = {x \in ℤ : | (| x - 3 | - 3) | \leq 1$ and

$B = {x \in ℝ - {1, 2} : \frac{(x - 2) (x - 4)}{x - 1} \log_{e} (| x - 2 |) = 0} .$

Then the number of onto functions $f : A \to B$ is equal to. [2026]

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