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

Q 71 :

Let the domain of the function $f (x) = \log_{3} \log_{5} (7 - \log_{2} (x^{2} - 10 x + 85)) + \sin^{- 1} (| \frac{3 x - 7}{17 - x} |)$ be $(α, β] . Then α + β$ is equal to: [2026]

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12

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(3)

$Let x^{2} - 10 x + 85 = λ$

$∴$ $Domain for first term$

$λ > 0 . . . (1)$

$& 7 - \log_{2} λ > 0 \Rightarrow λ < 2^{7} . . . (2)$

$& \log_{5} (7 - \log_{2} λ) > 0 \Rightarrow λ < 2^{6} . . . (3)$

$∴$ $from (1), (2) & (3)$

$0 < λ < 2^{6}$

$0 < x^{2} - 10 x + 85 < 64$

$\Rightarrow x \in (3, 7) . . . (A)$

$& domain for second term - 1 \leq \frac{3 x - 7}{x - 17} \leq 1$

$\Rightarrow x \in [- 5, 6] . . . (B)$

$From (A) & (B), domain of function will be (3, 6]$

$\Rightarrow α = 3, β = 6$

$\Rightarrow α + β = 9$

Q 72 :

Let $f$ and $g$ be functions satisfying $f (x + y) = f (x) f (y)$ , $f (1) = 7$ and $g (x + y) = g (x y)$ , $g (1) = 1$ for all $x, y \in ℕ$ . If $\sum_{x = 1}^{n} ((\frac{f (x)}{g (x)})) = 19607,$ then n is equal to: [2026]

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(2)

$f (x + y) = f (x) f (y) \Rightarrow f (x) = a^{x}$

$(∵ f (1) = 7 \Rightarrow a^{1} = 7)$

$So f (x) = 7^{x}$

$Now$

$g (x + y) = g (x y) (put y = 1)$

$\Rightarrow g (x + 1) = g (x)$

$so g (1) = g (2) = g (3) = \dots = g (n) = 1$

$Given \sum_{x = 1}^{n} \frac{f (x)}{g (x)} = 19607$

$\sum_{x = 1}^{n} \frac{7^{x}}{1} = 19607$

$\Rightarrow 7 (\frac{7^{n} - 1}{7 - 1}) = 19607$

$7^{n} - 1 = \frac{6}{7} \times 19607$

$7^{n} = 16807$

$\Rightarrow n = 5$

Q 73 :

Let $f (x) = {[x]}^{2} - [x + 3] - 3, x \in ℝ,$ where $[\cdot]$ is the greatest integer function. Then: [2026]

$f (x) < 0 only for x \in [- 1, 3)$
$\int_{0}^{2} f (x) d x = - 6$
$f (x) > 0 only for x \in [4, \infty)$
$f (x) = 0 for finitely many values of x$

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(1)

$f (x) = {[x]}^{2} - [x] - 6 = ([x] + 2) ([x] - 3)$

$(1) f (x) > 0 \Rightarrow [x] \in (- \infty, - 2) \cup (3, \infty)$

$\Rightarrow x \in (- \infty, - 2) \cup [4, \infty)$

$(2) f (x) < 0 \Rightarrow [x] \in (- 2, 3)$

$\Rightarrow x \in [- 1, 3)$

$option (2) is correct$

$(3) \int_{0}^{2} f (x) d x = \int_{0}^{1} (0 - 0 - 6) d x + \int_{1}^{2} (1 - 1 - 6) d x$

$= - 6 - 6$

$= - 12$

$(4) f (x) = 0 \Rightarrow [x] = 3 or [x] = - 2$

$infinitely many solutions$

Q 74 :

If $g (x) = 3 x^{2} + 2 x - 3, f (0) = - 3 a n d 4 g (f (x)) = 3 x^{2} - 32 x + 72$ , then f(g(2)) is equal to: [2026]

$- \frac{25}{6}$
$\frac{7}{2}$
$- \frac{7}{2}$
$\frac{25}{6}$

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(2)

Topic Question Set

Q 71 :

Let the domain of the function $f (x) = \log_{3} \log_{5} (7 - \log_{2} (x^{2} - 10 x + 85)) + \sin^{- 1} (| \frac{3 x - 7}{17 - x} |)$ be $(α, β] . Then α + β$ is equal to: [2026]

Q 72 :

Let $f$ and $g$ be functions satisfying $f (x + y) = f (x) f (y)$ , $f (1) = 7$ and $g (x + y) = g (x y)$ , $g (1) = 1$ for all $x, y \in ℕ$ . If $\sum_{x = 1}^{n} ((\frac{f (x)}{g (x)})) = 19607,$ then n is equal to: [2026]

Q 73 :

Let $f (x) = {[x]}^{2} - [x + 3] - 3, x \in ℝ,$ where $[\cdot]$ is the greatest integer function. Then: [2026]

Q 74 :

If $g (x) = 3 x^{2} + 2 x - 3, f (0) = - 3 a n d 4 g (f (x)) = 3 x^{2} - 32 x + 72$ , then f(g(2)) is equal to: [2026]

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