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

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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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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2

3

4

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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4

(A)

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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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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4

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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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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(D)

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

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

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

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

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

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

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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4

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]

8
6
7
9

1

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3

4

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 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]

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]

(A)

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]

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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]

(A)

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]

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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 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]

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]