JEE MAIN TEST – 16 – JEE MATH APEX

Mathematics Questions

1. A square matrix $P$ satisfies $P^2 = I - P$ . If $P^n = 5I - 8P$ , then $n$ is:

(A) 4

(B) 5

(D) 7

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2. Let $A = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix}$ and $(A + I)^{50} - 50A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ . Then the value of $a + b + c + d$ is:

(A) 2

(B) 1

(D) None of these

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3. Let $A$ and $B$ be $3 \times 3$ matrices such that $AB + A + B = 0$ , then:

(A) $(A + B)^2 = A^2 + 2AB + B^2$

(B) $|A| = |B|$

(D) None of these

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4. If $\alpha, \beta$ are the roots of the equation $x^2 - 2x + 4 = 0$ and $A = \begin{bmatrix} \alpha + \beta & \alpha^2 + \beta^2 & \alpha^3 + \beta^3 \\ \alpha^2 + \beta^2 & \alpha^3 + \beta^3 & \alpha^4 + \beta^4 \\ \alpha^3 + \beta^3 & \alpha^4 + \beta^4 & \alpha^5 + \beta^5 \end{bmatrix}$ , then:

(A) $A$ is an involutory matrix

(B) $A$ is a singular matrix

(D) None of these

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5. Consider the parabola $y^2 + 6x - 2y + 13 = 0$ . Then:

Column-I	Column-II
(A) Vertex	(P) $\left( -\dfrac{1}{2}, 1 \right)$
(B) Focus	(Q) $\left( -\dfrac{7}{2}, 4 \right)$
(C) End of Latus Rectum	(R) $\left( -\dfrac{7}{2}, 1 \right)$
(D) Point of intersection directrix and axis	(S) $(-2, 1)$

(A) A-P, B-Q, C-R, D-S

(B) A-S, B-R, C-Q, D-P

(D) A-S, B-P, C-Q, D-R

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6. If the eccentricity of the hyperbola $x^2 - y^2 \sec^2 \theta = 4$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2 \sec^2 \theta + y^2 = 16$ , then the value of $\theta$ equals:

(A) $\dfrac{\pi}{6}$

(B) $\dfrac{3\pi}{4}$

(D) $\dfrac{\pi}{2}$

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7. The coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is 4 are:

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

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

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

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8. The eccentricity of the hyperbola with asymptotes $3x + 4y = 2$ and $4x - 3y = 2$ is:

(A) 2

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

(D) $\sqrt{2}$

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9. Minimum distance between the parabola $y^2 = 8x$ and its image with respect to the line $x + y + 4 = 0$ is:

(A) $2\sqrt{2}$

(B) $3\sqrt{2}$

(D) $5\sqrt{2}$

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10. If the line $4x - 5y + 8 = 0$ is a chord of the parabola $y^2 = 8x$ , then its length is:

(A) $\dfrac{1}{2} \sqrt{41}$

(B) $\sqrt{41}$

(D) $2\sqrt{41}$

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11. The axis of a parabola lies along the $x$ -axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive $x$ -axis, then which of the following points does not lie on it?

(A) $(4, -4)$

(B) $(5, 2\sqrt{6})$

(D) $(6, 4\sqrt{2})$

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12. If the area of the triangle whose one vertex is at the vertex of the parabola $y^2 + 4(x - a^2) = 0$ and the other two vertices are the points of intersection of the parabola and the $y$ -axis is 250 sq. units, then a value of ‘ $a$ ‘ is:

(A) $5\sqrt{5}$

(B) $(10)^{2/3}$

(D) 5

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13. A ray of light moving parallel to the $x$ -axis gets reflected from a parabolic mirror whose equation is $(y - 2)^2 = 4(x + 1)$ . After reflection, the ray must pass through the point:

(A) $(-2, 0)$

(B) $(-1, 2)$

(D) $(2, 0)$

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14. Value of ‘ $t$ ‘ for which the intercept cut off by the chord $x + 2y + t = 0$ of the parabola $x^2 = 6y$ subtends a right angle at its vertex is:

(A) $t = -6$

(B) $t = -12$

(D) $t \in \emptyset$

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15. The ellipse $x^2 + 4y^2 = 4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point $(4, 0)$ . Then the equation of the ellipse is:

(A) $x^2 + 12y^2 = 16$

(B) $4x^2 + 48y^2 = 48$

(D) $x^2 + 16y^2 = 16$

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16. Planet $M$ orbits around its sun, $S$ , in an elliptical orbit with the sun at one of the foci. When $M$ is closest to $S$ , it is 2 units away. When $M$ is farthest from $S$ , it is 18 units away. Then the equation of motion of planet $M$ around its sun $S$ , assuming $S$ is at the center of the coordinate plane and the other focus lies on the negative $y$ -axis, is:

(A) $\dfrac{x^2}{36} + \dfrac{(y - 8)^2}{100} = 1$

(B) $\dfrac{x^2}{36} + \dfrac{(y + 8)^2}{100} = 1$

(D) $\dfrac{x^2}{64} + \dfrac{(y + 8)^2}{100} = 1$

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17. A man running around a racecourse notes that the sum of the distances of two flag-posts from him is always 10 meters and the distance between the flag-posts is 8 meters. The area of the path he encloses in square meters is:

(A) $15\pi$

(B) $12\pi$

(D) $8\pi$

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18. If the area of the auxiliary circle of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ( $a > b$ ) is twice the area of the ellipse, then the eccentricity of the ellipse is:

(A) $\dfrac{1}{\sqrt{2}}$

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

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

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19. The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is:

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

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

(D) 2

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20. The foci of the ellipse $ax^2 + 16y^2 = 16a$ and those of the hyperbola $\left( \dfrac{x}{12} \right)^2 - \left( \dfrac{y}{9} \right)^2 = \left( \dfrac{1}{5} \right)^2$ coincide, then $a =$

(A) 3

(B) 7

(D) $\sqrt{7}$

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21. If matrix $A = [a_{ij}]_{3 \times 3}$ and $a_{ij} + a_{ji} = 0$ and element $a_{ij} \in \{0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7\}$ , then the number of matrices $A$ is equal to:

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22. Let $A$ and $B$ be two non-singular matrices of order 3 such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$ , then $|\text{adj}(4AB)|$ is (where $\text{adj}(A)$ is the adjoint of matrix $A$ ):

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23. If the line $2x + y + \lambda = 0$ is a focal chord of the parabola $y^2 = -8x$ , then the value of $\lambda$ is:

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24. The radius of the largest circle that passes through the focus of the parabola $y^2 = 4x$ and is contained in it is:

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25. The radius of the circle passing through the foci of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$ and having its center at $(0, 3)$ is:

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26. A point $P$ on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ has the eccentric angle $\dfrac{\pi}{8}$ . The sum of the distances of $P$ from the two foci is:

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27. The focal length of the hyperbola $x^2 - 3y^2 - 4x - 6y - 11 = 0$ is:

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28. A hyperbola is such that its length of latus rectum equals 6 and the length of its conjugate axis is equal to one third of its distance between foci. If its eccentricity is $\dfrac{a}{b\sqrt{b}}$ (where $a$ and $b$ are prime), then the value of $ab$ is:

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29. If the equations of the directrices of the hyperbola $9x^2 - 16y^2 - 18x + 32y - 151 = 0$ are $x = a$ and $x = b$ , then the value of $a + b$ is:

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30. Let $S_1$ and $S_2$ be the foci of the hyperbola $\dfrac{x^2}{\left( \alpha + \dfrac{1}{\alpha} \right)^2} - \dfrac{y^2}{5} = 1$ , $\alpha > 0$ . If $P$ is a variable point on the hyperbola, then the minimum value of $|S_1 P - S_2 P|$ is:

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