JEE MAIN TEST – 11 – JEE MATH APEX

Mathematics Questions

1. Two circles of radii ‘ $a$ ‘ and ‘ $b$ ‘ touching each other externally are inscribed in the area bounded by $y = \sqrt{1 - x^2}$ and the $x$ -axis. If $b = \dfrac{1}{2}$ , then $4a$ is equal to:

(A) 1

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

(D) $\dfrac{5}{3}$

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2. If the lines $3x - 4y - 7 = 0$ and $2x - 3y - 5 = 0$ are two normals of a circle passing through $(2, 0)$ , then the equation of the circle is:

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

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

(D) $x^2 + y^2 + 2x + 2y - 8 = 0$

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3. 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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4. Let $R$ be the region satisfying $y x - 1$ , $x 0$ . Then the area of $R$ is:

(A) $\dfrac{11}{6}$

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

(D) 2

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5. The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\dfrac{dp(t)}{dt} = 0.5 p(t) - 450$ . If $p(0) = 850$ , then the time at which the population becomes zero is:

(A) $\ln 18$

(B) $2 \ln 18$

(D) $\dfrac{1}{2} \ln 18$

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6. Let $y(x)$ be a solution of $\dfrac{(2 + \sin x)}{(1 + y)} \dfrac{dy}{dx} = \cos x$ . If $y(0) = 2$ , then $y\left( \dfrac{\pi}{2} \right)$ equals:

(A) 2

(B) $\dfrac{5}{2}$

(D) 3

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7. The equation of the auxiliary circle of the ellipse which passes through $(1, 1)$ and has foci $(4, 5)$ and $(2, 3)$ is:

(A) $(x - 3)^2 + (y - 4)^2 = \left( \dfrac{5 - \sqrt{5}}{2} \right)^2$

(B) $(x - 3)^2 + (y - 4)^2 = \left( \dfrac{5 + \sqrt{5}}{2} \right)^2$

(D) $(x - 3)^2 + (y - 4)^2 = \left( \dfrac{7 - \sqrt{5}}{2} \right)^2$

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8. $A(0, 2)$ , $B$ , and $C$ are points on the parabola $y^2 = x + 4$ such that $\angle CBA = \dfrac{\pi}{2}$ . Then the least positive value of the ordinate of $C$ is:

(A) 3

(B) 4

(D) 2

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9. 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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10. A square matrix $M$ of order 3 satisfies $M^2 = I - M$ , where $I$ is an identity matrix of order 3. If $M^n = 2I - 3M$ , then $n$ is equal to:

(A) 4

(B) 5

(D) 7

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11. Let $A = \begin{bmatrix} t - 3 & a & b \\ c & 6 & d \\ e & f & 9 - t \end{bmatrix}$ , $B = \text{adj}(A)$ , and $C = \text{adj}(B)$ . If $|A| = 5$ , then $\text{tr}(C)$ is (where $|X|$ , $\text{tr}(X)$ , and $\text{adj}(X)$ denote determinant value, trace, and adjoint of matrix $X$ respectively):

(A) 5

(B) 12

(D) 60

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12. If $z$ and $w$ are two complex numbers satisfying $|z - 1| = 2$ and $|w - 5| = 3$ , then the maximum value of $|z - 4w|$ is:

(A) 14

(B) 15

(D) 31

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13. If $5 + i x^3 y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and $\arg(x + iy) = \theta$ , then $\tan^2 \theta$ is equal to:

(A) 4

(B) 5

(D) 7

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14. The image of the line $\dfrac{x - 1}{9} = \dfrac{y - 2}{-1} = \dfrac{z + 3}{-3}$ in the plane $3x - 3y + 10z = 26$ is the line:

(A) $\dfrac{x + 2}{9} = \dfrac{y - 3}{-1} = \dfrac{z + 7}{-3}$

(B) $\dfrac{x - 4}{9} = \dfrac{y + 1}{-1} = \dfrac{z - 7}{-3}$

(D) $\dfrac{x - 4}{9} = \dfrac{y - 2}{-1} = \dfrac{z + 1}{-3}$

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15. Vector equation of the line passing through $(-1, 1, 3)$ and perpendicular to the lines $\dfrac{x - 1}{1} = \dfrac{y + 2}{-1} = \dfrac{z + 3}{2}$ and $\dfrac{x - 1}{1} = \dfrac{y + 1}{2} = \dfrac{z + 1}{2}$ is:

(A) $\vec{r} = (-\hat{i} + \hat{j} + 3\hat{k}) + \lambda (2\hat{i} - 3\hat{j} - 4\hat{k})$

(B) $\vec{r} = (-\hat{i} + \hat{j} + 3\hat{k}) + \lambda (3\hat{i} - 2\hat{j} - 4\hat{k})$

(D) $\vec{r} = (-\hat{i} + \hat{j} + 3\hat{k}) + \lambda (3\hat{i} - 4\hat{j} - 3\hat{k})$

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16. Let $R$ be a relation defined as $a \, R \, b$ if $1 + ab > 0$ . Then, the relation $R$ is ( $a, b \in \mathbb{R}$ ):

(A) Reflexive

(B) Not symmetric

(D) None of these

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17. If the mean of a set of $n$ numbers $a_1, a_2, \ldots, a_n$ is $\bar{a}$ , then the mean of the numbers $a_i + 2i$ , $1 \leq i \leq n$ , will be:

(A) $\bar{a} + 2$

(B) $\bar{a} + n$

(D) $\bar{a} + n + 1$

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18. The variance of the observations $8, 12, 13, 15, 22$ is:

(A) 21

(B) 21.2

(D) None of these

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19. A box contains 20 identical balls of which 5 are white and 15 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $3^{\text{rd}}$ time on the $6^{\text{th}}$ draw is:

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

(B) $\dfrac{135}{2048}$

(D) $\dfrac{27}{4096}$

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20. If $A$ and $B$ are two events such that $P(A) = \dfrac{3}{5}, \quad P(\overline{B}) = \dfrac{1}{2}, \quad P(A \cap \overline{B}) = \dfrac{1}{5}$ , then $P(\overline{A} / A \cup \overline{B})$ is equal to:

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

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

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

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21. If $m$ is the slope of the common tangent of the parabola $y^2 = 16x$ and the circle $x^2 + y^2 = 8$ , then $m^2$ is equal to: [Image showing the common tangent to the parabola y^2 = 16x and the circle x^2 + y^2 = 8]

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22. Given $A$ is a square matrix of order 3, such that $|A| = 3$ and $\det\{\text{adj}(A \text{adj}(2A))\}$ is $2^a \times 3^b$ , then $a + b$ is:

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23. $a, b, c$ are three complex numbers on the unit circle $|z| = 1$ such that $abc = a + b + c$ , then $|ab + bc + ca|$ is equal to:

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24. If $d$ is the minimum distance between the lines $L_1: \vec{r} = (1 + \lambda)\hat{i} + (1 - 2\lambda)\hat{j} + \lambda\hat{k}$ and $L_2: \vec{r} = 2\mu\hat{i} + (1 - \mu)\hat{j} + (1 + \mu)\hat{k}$ , then $\left( \dfrac{11}{4} d^2 \right)$ is equal to:

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25. There are 10 pairs of shoes in a cupboard, from which 4 shoes are picked at random. If $p$ is the probability that there is at least one pair, find $323p$ .

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