JEE MAIN TEST – 9 – JEE MATH APEX

Mathematics Questions

1. The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to (where $C$ is a constant of integration):

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

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

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

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2. If $\int_0^{100} \dfrac{\sin^2 x}{e^{\left( \frac{x}{\pi} \left[ \frac{x}{\pi} \right] \right)}} \, dx = \dfrac{\alpha \pi^3}{1 + 4\pi^2}$ , $\alpha \in \mathbb{R}$ , where $[x]$ is the greatest integer less than or equal to $x$ , then the value of $\alpha$ is:

(A) $200 (1 - e^{-1})$

(B) $100 (1 - e)$

(D) $150 (e^{-1} - 1)$

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3. Let $y = y(x)$ be the solution of the differential equation $x \, dy = (y + x^3 \cos x) \, dx$ with $y(\pi) = 0$ . Then $y\left( \dfrac{\pi}{2} \right)$ is equal to:

(A) $\dfrac{\pi^2}{4} + \dfrac{\pi}{2}$

(B) $\dfrac{\pi^2}{2} + \dfrac{\pi}{4}$

(D) $\dfrac{\pi^2}{4} - \dfrac{\pi}{2}$

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4. The area (in sq. units) of the region given by the set $\left\{ (x, y) \in \mathbb{R} \times \mathbb{R} \mid x \geq 0, 2x^2 \leq y \leq 4 - 2x \right\}$ is:

(A) $\dfrac{8}{3}$

(B) $\dfrac{17}{3}$

(D) $\dfrac{7}{3}$

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5. The last common term to the sequences $1, 11, 21, 31$ (100 terms) and $31, 36, 41, 46$ (100 terms) is:

(A) 381

(B) 521

(D) None

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6. If $A = \begin{bmatrix} 1 + a^2 + a^4 & 1 + ab + a^2 b^2 & 1 + ac + a^2 c^2 \\ 1 + ab + a^2 b^2 & 1 + b^2 + b^4 & 1 + bc + b^2 c^2 \\ 1 + ac + a^2 c^2 & 1 + bc + b^2 c^2 & 1 + c^2 + c^4 \end{bmatrix}$ and $\det(A) = \det(4I)$ , where $I$ is the $3 \times 3$ identity matrix, then $\left| (a - b)^3 + (b - c)^3 + (c - a)^3 \right|$ is equal to:

(A) 24

(B) 34

(D) 32

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7. From a pack of 52 well-shuffled cards, cards are drawn one by one without replacement. If the $4^{\text{th}}$ drawn card is found to be an ace, then the probability that there are no more aces left in the pack is:

(A) $\dfrac{1}{{}^{48}C_3 + 3 {}^{49}C_2 + 1}$

(B) $\dfrac{1}{{}^{48}C_3 + {}^{49}C_2 + 1}$

(D) $\dfrac{1}{{}^{52}C_4 + 1}$

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8. If $|z_1| = 2$ , $|z_2| = 3$ , $|z_3| = 4$ , and $|2z_1 + 3z_2 + 4z_3| = 9$ , then the value of $|8z_2 z_3 + 27z_3 z_1 + 64z_1 z_2|$ is equal to:

(A) 216

(B) 18

(D) None

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9. Let $\lambda \in \mathbb{R}$ . The system of linear equations $\begin{aligned} 2x_1 - 4x_2 + \lambda x_3 &= 1 \\ x_1 - 6x_2 + x_3 &= 2 \\ \lambda x_1 - 10x_2 + 4x_3 &= 3 \end{aligned}$ is inconsistent for:

(A) Exactly one negative value of $\lambda$

(B) Exactly one positive value of $\lambda$

(D) Exactly two values of $\lambda$

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10. If the value of $\dfrac{3 \cos 36^\circ + 5 \sin 18^\circ}{5 \cos 36^\circ - 3 \sin 18^\circ}$ is $\dfrac{a \sqrt{5} - b}{c}$ , where $a, b, c$ are natural numbers and $\gcd(a, c) = 1$ , then $a + b + c$ is equal to:

(A) 50

(B) 40

(D) 54

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11. If the variance of the frequency distribution is 160, then the value of $c \in \mathbb{N}$ is:

x	c	2c	3c	4c	5c	6c
f	2	1	1	1	1	1

(A) 5

(B) 8

(D) 6

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12. Let $\alpha, \beta \in \mathbb{R}$ . Let the mean and the variance of 6 observations $-3, 4, 7, -6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

(A) $\dfrac{13}{3}$

(B) $\dfrac{16}{3}$

(D) $\dfrac{14}{3}$

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13. $f(x) = \begin{cases} 2 - |x^2 + 5x + 6|, & x \neq -2 \\ a^2 + 1, & x = -2 \end{cases}$ . Then the range of $a$ , so that $f(x)$ has a maximum at $x = -2$ , is:

(A) $|a| \geq 1$

(B) $|a| < 1$

(D) $a < 1$

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14. Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f\left( \dfrac{x + y}{3} \right) = \dfrac{f(x) + f(y)}{3}$ , $f(0) = 3$ , and $f'(0) = 3$ . Then which of the following is correct?

(A) $\dfrac{f(x)}{x}$ is differentiable in $\mathbb{R}$

(B) $f(x)$ is continuous but not differentiable in $\mathbb{R}$

(D) None

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

(A) 3

(B) 2

(D) 6

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16. If $f: [-4, 4] \to \mathbb{R}$ , where $f(x) = \left[ \dfrac{x^4 + 1}{a} \right] \sin x + \cos x + \dfrac{e^x + e^{-x}}{2}$ (where $[.]$ is the greatest integer function) is an even function, then:

(A) $a \in (0, 257]$

(B) $a \in [257, \infty)$

(D) $a \in (0, 257)$

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17. The vertices of a variable triangle are $(3, 4)$ , $(5 \cos \theta, 5 \sin \theta)$ , and $(5 \sin \theta, -5 \cos \theta)$ , where $\theta \in \mathbb{R}$ . The locus of its orthocenter is:

(A) $(x + y - 1)^2 + (x - y - 7)^2 = 100$

(B) $(x + y - 7)^2 + (x - y - 1)^2 = 100$

(D) $(x + y - 7)^2 + (x - y + 1)^2 = 100$

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18. The point $(a^2, a + 1)$ is a point in the angle between the lines $3x - y + 1 = 0$ and $x + 2y - 5 = 0$ containing the origin, if:

(A) $a \geq 1$ or $a \leq -3$

(B) $a \in (0, 1)$

(D) None of these

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19. In the right angle triangle as shown, an altitude is drawn from the right angle to the hypotenuse. Circles are inscribed within each of the smaller triangles. What is the distance between the centres of these circles?

(A) 5

(B) 7

(D) $\sqrt{50}$

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20. Vector $\hat{a}$ in the plane of $\vec{b} = 2\hat{i} + \hat{j}$ and $\vec{c} = \hat{i} - \hat{j} + \hat{k}$ is such that it is equally inclined to $\vec{b}$ and $\vec{d}$ where $\vec{d} = \hat{j} + 2\hat{k}$ . The value of $\hat{a}$ is:

(A) $\dfrac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$

(B) $\dfrac{\hat{i} - \hat{j} + \hat{k}}{\sqrt{3}}$

(D) None of these

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21. For real numbers $\alpha, \beta, \gamma$ , and $\delta$ , if $\int \dfrac{(x^2 - 1) + \tan^{-1} \left( \dfrac{x^2 + 1}{x} \right)}{(x^4 + 3x^2 + 1) \tan^{-1} \left( \dfrac{x^2 + 1}{x} \right)} \, dx = \alpha \log_e \left( \tan^{-1} \left( \dfrac{x^2 + 1}{x} \right) \right) + \beta \tan^{-1} \left( \dfrac{\gamma (x^2 - 1)}{x} \right) + \delta \tan^{-1} \left( \dfrac{x^2 + 1}{x} \right) + C$ where $C$ is an arbitrary constant, then the value of $10(\alpha + \beta \gamma + \delta)$ is equal to:

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22. The number of ways 16 identical cubes, of which 11 are blue and the rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is:

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23. The number of solutions of $\sin^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0$ , where $-\pi \leq x \leq \pi$ , is:

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24. Value of $\lim_{x \to 0^-} \left[ \dfrac{\sin |x|}{x} \right] + \lim_{x \to 0^+} \left[ \dfrac{\sin^{-1} |x|}{|x|} \right] + \lim_{x \to 0^-} \left[ \dfrac{-2x}{\tan x} \right]$ (where $[.]$ denotes the greatest integer function):

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25. Let a line having direction ratios $1, -4, 2$ intersect the lines $\dfrac{x - 7}{3} = \dfrac{y - 1}{-1} = \dfrac{z + 2}{-1}$ and $\dfrac{x}{2} = \dfrac{y - 7}{3} = \dfrac{z}{1}$ at the points $A$ and $B$ . Then $(AB)^2$ is equal to:

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