JEE MAIN TEST – 2 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. The number of solutions of the equation $\cos\left(\pi \sqrt{x} - 4\right) \cos\left(\pi \sqrt{x}\right) = 1$ is:

(A) None

(B) One

(D) More than two

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2. The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$ , where $p \neq 0$ and $q \neq 0$ . If the new mean and new s.d. become half of their original values, then $q$ is equal to:

(A) -20

(B) 10

(D) -5

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3. Let $T_n$ be the $n^{\text{th}}$ term of an A.P. If $\sum_{m=1}^{5^{99}} T_{2m} = 5^{100}$ and $\sum_{m=1}^{5^{99}} T_{2m-1} = 5^{99}$ , then the common difference of A.P. is:

(A) 3

(B) 5

(D) 7

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4. If the equation $x^3 + 2x^2 - 4x + 5 = 0$ has roots $\alpha, \beta, \gamma$ , then the value of $\dfrac{(\alpha^3 + 5)(\beta^3 + 5)(\gamma^3 + 5)}{8\alpha\beta\gamma}$ is:

(A) 104

(B) 8

(D) -104

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5. Let $A$ be a non-singular square matrix of order 3 such that $\text{Tr}(A^{-1}) = 3$ and $\det(A^{-1}) = \dfrac{1}{5}$ . If $A^{-1}BA = 2(\text{adj} A)$ then:
[Note: $\text{Tr}(P)$ and $\text{adj} P$ denote trace of square matrix $P$ and adjoint matrix of square matrix $P$ respectively.]

(A) $\det(B) = 5000$

(B) $\det(B) = 200$

(D) $\text{Tr}(B) = 15$

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6. Match the column

Column-I	Column-II
(a) ${}^{24}C_{2}+{}^{23}C_{2}+{}^{22}C_{2}+{}^{21}C_{2}+{}^{20}C_{2}+{}^{20}C_{3}$ is equal to	(p) 102
(b) In the figure, number of progressive ways to reach from (0, 0) to (4, 4) passing through point (2, 2) are: [Insert Grid Image Here]	(q) 2300
(c) The number of 4 digit numbers that can be made with the digits 1, 2, 3, 4, 3, 2	(r) 82
(d) If $\left( \dfrac{500!}{14^k} \right) = 0$ , then the maximum natural value of $k$ is equal to (where $\{ \cdot \}$ is fractional part function)	(s) 36

(A) (a) → q; (b) → s; (c) → p; (d) → r

(B) (a) → p; (b) → s; (c) → r; (d) → q

(D) (a) → p; (b) → s; (c) → p; (d) → r

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7. Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\dfrac{2}{3}$ and $\dfrac{1}{3}$ , respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then probability that the roots of the quadratic equation $x^2 - \alpha x + \beta = 0$ are real and equal is:

(A) $\dfrac{40}{81}$

(B) $\dfrac{20}{81}$

(D) $\dfrac{1}{4}$

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8. If $z_1, z_2$ are complex numbers such that $\text{Re}(z_1) = |z_1 - 1|$ , $\text{Re}(z_2) = |z_2 - 1|$ and $\arg(z_1 - z_2) = \dfrac{\pi}{6}$ , then $\text{Im}(z_1 + z_2)$ is equal to:

(A) $\dfrac{\sqrt{3}}{2}$

(B) $\dfrac{2}{\sqrt{3}}$

(D) $2\sqrt{3}$

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9. If $\sum_{r=0}^{20} (2r + 3^r) \left(^{20}C_r\right) - \sum_{r=0}^{40} \left(^{40}C_r\right) = 2^a 5$ , then $a$ is:

(A) 20

(B) 32

(D) None

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10. There are 3 bags A, B & C. Bag A contains 1 Red & 2 Green balls, bag B contains 2 Red & 1 Green balls and bag C contains only one green ball. One ball is drawn from bag A & put into bag B then one ball is drawn from B & put into bag C & finally one ball is drawn from bag C & put into bag A. When this operation is completed, probability that bag A contains 2 Red & 1 Green balls, is:

(A) $\dfrac{1}{4}$

(B) $\dfrac{1}{2}$

(D) $\dfrac{1}{6}$

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11. If $f(x) = \begin{cases} \dfrac{1 - \sin^3 x}{3 \cos^2 x}, & x \dfrac{\pi}{2} \end{cases}$ is continuous at $x = \dfrac{\pi}{2}$ , then value of $a$ and $b$ are:

(A) $\dfrac{1}{2}, \dfrac{1}{4}$

(B) 2, 4

(D) $\dfrac{1}{4}, 2$

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12. $f(x)$ is a function such that $f''(x) = -f(x)$ and $f'(x) = g(x)$ . Also $h(x)$ is a function such that $h(x) = [f(x)]^2 + [g(x)]^2$ and $h(5) = 5$ , then the value of $h(10)$ will be:

(A) 0

(B) 5

(D) 15

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13. 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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14. If $f(x) + f(y) = f\left( \dfrac{x + y}{1 - xy} \right)$ for all $x, y \in \mathbb{R}$ and $xy \neq 1$ and $\lim_{x \to 0} \dfrac{f(x)}{x} = 2$ , then the value of $\dfrac{1500}{\pi} \dfrac{f\left( {\sqrt{3}} \right)}{f'(-2)}$ must be:

(A) 2000

(B) 2500

(D) 3500

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15. Area enclosed between the curves $|y| = 1 - x^2$ and $x^2 + y^2 = 1$ is:

(A) $\dfrac{3\pi - 8}{3}$ sq. units

(B) $\dfrac{\pi - 8}{3}$ sq. units

(D) None of these

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16. The solution of $\dfrac{dy}{dx} = \sin(x + y) + \cos(x + y)$ is:

(A) $\log \left[ 1 + \tan \left( \dfrac{x + y}{2} \right) \right] + c = 0$

(B) $\log \left[ 1 + \tan \left( \dfrac{x + y}{2} \right) \right] = x + c$

(D) None of these

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17. If $f(x) = \int_{0}^{x} \sin^4 t \, dt$ , then $f(x + \pi)$ is equal to:

(A) $f(\pi)$

(B) $f(x)$

(D) $f(x) \cdot f(\pi)$

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18. If the shortest distance between the line $\vec{r} = (-\hat{i} + 3\hat{k}) + \lambda (\hat{i} - a\hat{j})$ and $\vec{r} = (\hat{j} + 2\hat{k}) + \mu (\hat{i} - \hat{j} + \hat{k})$ is $\dfrac{\sqrt{2}}{3}$ , then the integral value of ‘ $a$ ‘ is equal to:

(A) 1

(B) -1

(D) 4

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19. Given $A(1,2)$ & $B(3,7)$ are two vertices of the $\triangle ABC$ . If the locus of centroid of $\triangle ABC$ is $2x - y = 0$ then the minimum distance between the locus of the vertex $C$ from the line $2x - y = 0$ is:

(A) 0

(B) 1

(D) $\dfrac{1}{\sqrt{5}}$

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20. The co-ordinates of a point on the parabola $y^2 = 8x$ whose focal distance is 4 is:

(A) $(2, \pm 4)$

(B) $(\pm 2, 4)$

(D) $(\pm 2, -4)$

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SECTION B: Numerical Value Type Questions

21. If $0 < A < \pi/2$ and $\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$ and $\sin A$ and $\cos A$ are roots of equation $4x^2 - 3x + a = 0$ . Then the value of $25a$ is:

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22. For real numbers $\alpha$ and $\beta$ , consider the following system of linear equations: $x + y - z = 2$ , $x + 2y + \alpha z = 1$ , $2x - y + z = \beta$ . If the system has infinite solutions, then $\alpha + \beta$ is equal to:

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23. Number of integral values of parameter $\lambda$ for which $f(x) = 2x^3 - 3(2 + \lambda)x^2 + 12\lambda x + \log(16 - \lambda^2)$ has exactly one local maxima & one local minima is:

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24. Let $I_1 = \int_{0}^{1} \cot^{-1} (1 - x + x^2) \, dx$ and $I_2 = \int_{0}^{1} \tan^{-1} x \, dx$ . Then $I_1/I_2$ is:

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25. If the foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{b^2} = 1$ and the hyperbola $\dfrac{x^2}{144} - \dfrac{y^2}{81} = \dfrac{1}{25}$ coincide, then the value of $b^2$ is:

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