JEE MAIN TEST – 14 – JEE MATH APEX

Mathematics Questions

1. The solution of the differential equation $\left( x^3 - xy \right) dx = \left( 1 + x^2 \right) dy$ , where $y(0) = -\dfrac{2}{3}$ , is:

(A) $x^2 + 3y + 2 = 0$

(B) $x + 3y + 2 = 0$

(D) $3y + 2 = x^2$

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2. Let $P$ and $Q$ be two square and invertible matrices such that $Q = -P^{-1} Q P$ , then $(P + Q)^2$ is equal to:

(A) Null matrix

(B) $P^2 + 2PQ + Q^2$

(D) Identity matrix

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3. A point moves such that the sum of the squares of its distances from four sides of a square of unit length is equal to 3. If the point moves on a circle, then its radius is:

(A) 1

(B) 2

(D) 9

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4. The area bounded by the parabola $x^2 = 8y$ and the line $x - 2y + 8 = 0$ is:

(A) 36

(B) 72

(D) 9

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5. Common area bounded by regions $0 \geq y \geq x^2 + 2x$ and $x - y \geq 0$ is:

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

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

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

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6. Solution of the differential equation $x \, dy + y \, dx = e^{xy - \ln x^2} (x \, dy - y \, dx)$ is given by (where $C$ is an arbitrary constant):

(A) $\dfrac{y}{x} + e^{-xy} = C$

(B) $\dfrac{x}{y} + e^{-xy} = C$

(D) $-\dfrac{x}{y} + e^{-xy} = C$

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7. If the curve $y(x)$ satisfying $x \, dx = \left( \dfrac{x^2}{y} - y^3 \right) dy$ passes through $(0, 2)$ , then the value of $(y(4))^2 \left( 4 - (y(4))^2 \right)$ is:

(A) 8

(B) 16

(D) 12

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8. $F_1, F_2$ are two foci of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ . Let $P$ be a point on the ellipse such that $PF_1 = 2 PF_2$ , then the area of $\triangle PF_1 F_2$ is:

(A) 3

(B) 4

(D) $\dfrac{\sqrt{13}}{2}$

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9. If a hyperbola centered at the origin has one of its directrices as $x = 2$ and the ordinate of one of the endpoints of a latus rectum is 12, then find its eccentricity:

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

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

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

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10. Let $A = \begin{bmatrix} \lambda^2 - 3\lambda + 2 & 3 & -6 \\ -3 & \lambda^3 - 6\lambda^2 + 11\lambda - 6 & 4 \\ 6 & -4 & \tan \dfrac{\lambda x}{4} - 1 \end{bmatrix}$ . If $A$ is a skew-symmetric matrix, then $\lambda$ is:

(A) 1

(B) 2

(D) 0

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11. $3 \times 3$ matrices are formed using the elements from the set $\{-3, -2, -1, 0, 1, 2, 3\}$ . Then the number of matrices each of whose trace is at least 7 is:

(A) $10 \times (7^7)$

(B) $10 \times (7^6)$

(D) $7^6$

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12. The image of the circle $x^2 + y^2 - 2x + 4y - 4 = 0$ with respect to the line $x - y + 5 = 0$ is:

(A) $x^2 + y^2 + 14x - 12y + 76 = 0$

(B) $x^2 + y^2 - 14x - 12y + 76 = 0$

(D) $x^2 + y^2 - 14x - 12y - 76 = 0$

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13. If $\ell$ is the length of the latus rectum of the parabola $3x^2 - 10x - 5y - 20 = 0$ , then $\ell$ equals:

(A) 5

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

(D) $\dfrac{10}{3}$

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14. The focus and directrix of a parabola are $(1, 0)$ and $3x - 4y + 2 = 0$ , respectively. If the length of the latus rectum of the parabola is $M$ , then the value of $M$ is equal to:

(A) 4

(B) 2

(D) 8

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15. Eccentricity of the hyperbola conjugate to the hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{12} = 1$ is:

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

(B) 2

(D) $\dfrac{4}{3}$

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16. The eccentric angle of a point on the ellipse $9x^2 + 4y^2 = 36$ at a distance $\dfrac{\sqrt{31}}{2}$ units from the center of the ellipse is ‘ $\theta$ ‘, then the value of $|2 \cos \theta|$ is:

(A) 1

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

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

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17. Circle $K$ is inscribed in the first quadrant touching the circle $x^2 + y^2 = 36$ internally. The length of the radius of the circle $K$ is:

(A) $\dfrac{6 - \sqrt{2}}{2}$

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

(D) $6(\sqrt{2} - 1)$

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18. Solution of the differential equation $\sin y \dfrac{dy}{dx} + \dfrac{1}{x} \cos y = x^4 \cos^2 y$ is (where $C$ is the constant of integration):

(A) $x \sec y = x^6 + C$

(B) $6x \sec y = x + C$

(D) $6x \sec y = 6x^6 + C$

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19. The focal chord of $y^2 = 16x$ is a tangent to $(x - 6)^2 + y^2 = 2$ , then the possible values of the slope of this chord are:

(A) $1, -1$

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

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

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20. Tangents are drawn from every point on the line $x + 5y = 4$ to the ellipse $\dfrac{x^2}{5} + \dfrac{y^2}{3} = 1$ , then the corresponding chord of contact always passes through the point $(\alpha, \beta)$ , then $(\alpha + \beta)$ equals:

(A) 5

(B) 3

(D) 8

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21. If $A \cdot \text{adj}(A^2) = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$ , then the absolute value of the sum of elements of $\text{adj } A$ is (where $\text{adj}(X)$ denotes the adjoint of matrix $X$ ):

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22. Sum of all possible slopes of tangents to the circle $(x - 2)^2 + (y - 3)^2 = 1$ that pass through the origin is:

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23. Area bounded by the curve $12x - y^2 \geq 0$ and $y \geq 2x$ is equal to:

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24. Let $y = f(x)$ be a real-valued differentiable function on $\mathbb{R}$ (the set of all real numbers) such that $f(1) = 1$ . If $f(x)$ satisfies $x f'(x) = x^2 + f(x) - 2$ , then the area bounded by $f(x)$ with the $x$ -axis between ordinates $x = 0$ and $x = 3$ is equal to:

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25. Length of the normal chord of the parabola $y^2 = 4x$ , which makes an angle of $\dfrac{\pi}{4}$ with the axis of $x$ , is $K\sqrt{2}$ , then $K$ is equal to:

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