JEE MAIN TEST – 3 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. If $x \sin \left( \dfrac{y}{x} \right) dy = \left[ y \sin \left( \dfrac{y}{x} \right) - x \right] dx$ and $y(1) = \dfrac{\pi}{2}$ , then $\cos \left( \dfrac{y}{x} \right)$ is equal to:

(A) $x$

(B) $\dfrac{1}{x}$

(D) $e^x$

Click to View Answer

Correct Answer: [Insert Correct Option]

2. If the integral $\int_{0}^{10} \dfrac{[\sin 2\pi x]}{e^{x - [x]}} \, dx = \alpha e^{-1} + \beta e^{\dfrac{1}{2}} + \gamma$ , where $\alpha, \beta, \gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x$ , then the value of $\alpha + \beta + \gamma$ is equal to:

(A) 0

(B) 20

(D) 10

Click to View Answer

Correct Answer: [Insert Correct Option]

3. $\int \dfrac{dx}{\sqrt[4]{(x-1)^3 (x+2)^5}}$ is equal to:

(A) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\dfrac{1}{4}} + C$

(B) $\dfrac{3}{4} \left( \dfrac{x+2}{x-1} \right)^{\dfrac{5}{4}} + C$

(D) $\dfrac{4}{3} \left( \dfrac{x-1}{x+2} \right)^{\dfrac{1}{4}} + C$

Click to View Answer

Correct Answer: [Insert Correct Option]

4. The area of the shorter region bounded by $|y| = 4 - x^2$ and $|y| = 3x$ is given by $\left( 3K + \dfrac{1}{3} \right)$ sq-unit where $K$ is equal to:

(A) 1

(B) 2

(D) $3\dfrac{1}{3}$

Click to View Answer

Correct Answer: [Insert Correct Option]

5. Let $g(x) = f(\log x) + f(2 - \log x)$ and $f''(x) < 0 \, \forall \, x \in (0, 3)$ . Then find the interval in which $g(x)$ increases:

(A) $(0, 1)$

(B) $(1, 2)$

(D) $(0, e)$

Click to View Answer

Correct Answer: [Insert Correct Option]

6. If $f(x) = \begin{cases} 3(1 + |\tan x|)^{\dfrac{|\tan x|}{1 - |\tan x|}}, & -\dfrac{1}{2} < x < 0 \\ b, & x = 0 \\ 3 \left( 1 + \dfrac{|\sin x|}{3} \right)^{\dfrac{6}{|\sin x|}}, & 0 < x < \dfrac{2}{3} \end{cases}$ is continuous function at $x = 0$ , then:

(A) $a + \ln \left( \dfrac{b}{3} \right) = 4$

(B) $a \cdot \ln \left( \dfrac{b}{3} \right) = -4$

(D) $a - \ln \left( \dfrac{b}{3} \right) = 4$

Click to View Answer

Correct Answer: [Insert Correct Option]

7. Let $f(x) = 3^{\alpha x} + 3^{\beta x}$ , where $\alpha \neq \beta$ and $3 f(x) \log_3 e = 2 f(x) + f''(x) (\log_3 e)^2$ for all $x$ . Then the value of $\alpha + \beta$ is:

(A) 3

(B) 2

(D) 6

Click to View Answer

Correct Answer: [Insert Correct Option]

8. Range of $y = \cos^{-1} \sqrt{\log_{[x]} \dfrac{[x]}{x}}$ , (where $[x]$ denotes greatest integer less than or equal to $x$ )

(A) $[0, \pi]$

(B) $[-1, 1]$

(D) $\{1, -1\}$

Click to View Answer

Correct Answer: [Insert Correct Option]

9. Let $y = \dfrac{17 + 5 \sin x + 12 \cos x}{17 - 5 \sin x - 12 \cos x}$ . If $M$ and $m$ are the greatest and least value of $y$ for all $x \in \mathbb{R}$ , then $(Mm)$ is equal to:

(A) $\dfrac{4}{225}$

(B) $\dfrac{4}{15}$

(D) None

Click to View Answer

Correct Answer: [Insert Correct Option]

10. If $\sum_{i=1}^{5} (x_i - 10) = 5$ and $\sum_{i=1}^{5} (x_i - 10)^2 = 25$ , then standard deviation of observations $2x_1 + 7, 2x_2 + 7, 2x_3 + 7, 2x_4 + 7$ and $2x_5 + 7$ is equal to:

(A) 8

(B) 16

(D) 2

Click to View Answer

Correct Answer: [Insert Correct Option]

11. In a triangle $ABC$ , $A : B : C = 3 : 5 : 4$ . Then $[a + b + c\sqrt{2}]$ is equal to:

(A) $2b$

(B) $2c$

(D) $3a$

Click to View Answer

Correct Answer: [Insert Correct Option]

12. $\sqrt{\underbrace{111\ldots 1}_{200 \text{ digits}} - \underbrace{222\ldots 2}_{100 \text{ digits}}}$ equals:

(A) $\sqrt{\underbrace{1313\ldots 13}_{100 \text{ digits}}}$

(B) $\sqrt{\underbrace{33\ldots 3}_{100 \text{ digits}}}$

(D) $\sqrt{\underbrace{333\ldots 3}_{100 \text{ digits}}}$

Click to View Answer

Correct Answer: [Insert Correct Option]

13. If $b < 0$ and roots $x_1$ and $x_2$ of equation $2x^2 + 6x + b = 0$ satisfy the condition $\dfrac{x_1}{x_2} + \dfrac{x_2}{x_1} < k$ then $k =$ :

(A) -3

(B) -5

(D) -2

Click to View Answer

Correct Answer: [Insert Correct Option]

14. If the system of equation $2x + 3y = -1$ , $3x + y = 2$ , $\lambda x + 2y = \mu$ is consistent then:

(A) $\lambda - \mu = -2$

(B) $\lambda + \mu = -1$

(D) $\lambda - \mu + 8 = 0$

Click to View Answer

Correct Answer: [Insert Correct Option]

15. Let bag A contains 2 Red, 3 Green balls and bag B contains 3 Red, 2 Green balls. If two balls are drawn randomly from each of bag A & B and then put in to empty bag C. It is found that bag C contains equal number of red and green balls, then probability that 2 Red balls were drawn from bag B is:

(A) $\dfrac{9}{46}$

(B) $\dfrac{1}{15}$

(D) $\dfrac{3}{20}$

Click to View Answer

Correct Answer: [Insert Correct Option]

16. Let $z_1 = 6 + i$ and $z_2 = 4 - 3i$ . Let $z$ be a complex number such that $\arg \left( \dfrac{z - z_1}{z_2 - z} \right) = \dfrac{\pi}{2}$ , then $z$ satisfies:

(A) $|z - (5 - i)| = 5$

(B) $|z - (5 - i)| = \sqrt{5}$

(D) $|z - (5 + i)| = \sqrt{5}$

Click to View Answer

Correct Answer: [Insert Correct Option]

17. Consider a circle $C$ which touches the $y$ -axis at $(0, 6)$ and cuts off an intercept $6\sqrt{5}$ on the $x$ -axis. Then the radius of the circle $C$ is equal to:

(A) $\sqrt{53}$

(B) 9

(D) $\sqrt{82}$

Click to View Answer

Correct Answer: [Insert Correct Option]

18. A hyperbola passes through the foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities in one, then the equation of the hyperbola is:

(A) $\dfrac{x^2}{9} - \dfrac{y^2}{25} = 1$

(B) $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$

(D) $\dfrac{x^2}{9} - \dfrac{y^2}{4} = 1$

Click to View Answer

Correct Answer: [Insert Correct Option]

19. Let $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = 2\hat{i} - 3\hat{j} + 5\hat{k}$ . If $\vec{r} \times \vec{a} = \vec{b} \times \vec{r}$ , $\vec{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = 3$ and $\vec{r} \cdot (2\hat{i} + 5\hat{j} - \alpha \hat{k}) = -1$ , $\alpha \in \mathbb{R}$ , then the value of $\alpha + |\vec{r}|^2$ is equal to:

(A) 9

(B) 15

(D) 11

Click to View Answer

Correct Answer: [Insert Correct Option]

20. If the shortest distance between the lines $3(x - 1) = 6(y - 2) = 2(z - 1)$ and $4(x - 2) = 2(y - \lambda) = (z - 3)$ , $\lambda \in \mathbb{R}$ is $\dfrac{1}{\sqrt{38}}$ , then the integral value of $\lambda$ is equal to:

(A) 3

(B) 2

(D) -1

Click to View Answer

Correct Answer: [Insert Correct Option]

SECTION B: Numerical Value Type Questions

21. If the value of $\int_{\pi/2}^{5\pi/2} \dfrac{e^{\tan^{-1}(\sin x)}}{e^{\tan^{-1}(\sin x)} + e^{\tan^{-1}(\cos x)}} \, dx$ is $k\pi$ ( $k \in \mathbb{N}$ ) Find $k$ .

Click to View Answer

Correct Answer: [Insert Integer Value]

22. If $\lim_{x \to 2} \dfrac{x^{5/2} - x^{3/2} + x^2 - x - 2x^{1/2} - 2}{x - 2} = 3 \left( \sqrt{P} + 1 \right)$ , then value of $P (P \in \mathbb{N})$ is:

Click to View Answer

Correct Answer: [Insert Integer Value]

23. The minimum value of the expression $\dfrac{9x^2 \sin^2 x + 4}{x \sin x}$ for $x \in (0, \pi)$ is:

Click to View Answer

Correct Answer: [Insert Integer Value]

24. Number of five digit natural numbers, with sum of digits equal to 43 are:

Click to View Answer

Correct Answer: [Insert Integer Value]

25. Let the points of intersections of the lines $x - y + 1 = 0$ , $x - 2y + 3 = 0$ and $2x - 5y + 11 = 0$ are the mid points of the sides of a triangle $ABC$ . Then the area of the triangle $ABC$ is:

Click to View Answer

Correct Answer: [Insert Integer Value]