JEE MAIN TEST – 13 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. Let $z_1 = 6 + i$ and $z_2 = 4 - 3i$ . Let $z$ be a complex number such that $\arg \left( \dfrac{z - z_1}{z_2 - z} \right) = \dfrac{\pi}{2}$ , then $z$ satisfies:

(A) $|z - (5 - i)| = 5$

(B) $|z - (5 - i)| = \sqrt{5}$

(D) $|z - (5 + i)| = \sqrt{5}$

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2. If $|z_1| = 2$ , $|z_2| = 3$ , $|z_3| = 4$ , and $|16 z_1 z_2 + 9 z_1 z_3 + 4 z_2 z_3| = 96$ , then $|z_1 + z_2 + z_3| =$

(A) 2

(B) 3

(D) 6

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3. If $Z = \dfrac{2}{3 + \cos \theta + i \sin \theta}$ , then the locus of $Z$ will be:

(A) Straight line

(B) Circle

(D) Ellipse

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4. Eccentricity of the given curve is: If $|Z + 4| + |Z - 4| = 10$ .

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

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

(D) $\dfrac{4}{5}$

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5. If $z_1, z_2, z_3$ are complex numbers such that $|z_1| = 2$ , $|z_2| = 3$ , $|z_3| = 4$ , then the maximum value of $|z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2$ is:

(A) 87

(B) 79

(D) None

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6. If $x_p = \cos \left( \dfrac{\pi}{2^p} \right) + i \sin \left( \dfrac{\pi}{2^p} \right)$ , then the value of $\left( x_1 \times x_3 \times x_5 \cdots \infty \right) + \dfrac{1}{\left( x_2 \cdot x_4 \cdot x_6 \cdots \infty \right)}$ is:

(A) 0

(B) 1

(D) 3

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7. If $Z_1, Z_2$ are complex numbers such that $\text{Re}(Z_1) = |Z_1 - 2|$ , $\text{Re}(Z_2) = |Z_2 - 2|$ , and $\arg (Z_1 - Z_2) = \dfrac{\pi}{3}$ , then $\text{Im}(Z_1 + Z_2) =$

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

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

(D) $\sqrt{3}$

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8. If $z = (\tan 1 - i)^2$ , then:

(A) $|z| = \sec 1$ , $\text{Arg}(z) = -\dfrac{\pi}{2}$

(B) $|z| = \sec^2 1$ , $\text{Arg}(z) = \pi - 2$

(D) $|z| = \sec^2 1$ , $\text{Arg}(z) = \dfrac{\pi}{2} - 1$

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9. If $\left| \dfrac{z_1 - 3 z_2}{3 - z_1 z_2} \right| = 1$ and $|z_2| \neq 1$ , then $|z_1|$ is:

(A) 3

(B) 1

(D) 4

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10. If $A + B + C = \pi$ , $e^{i\theta} = \cos \theta + i \sin \theta$ , and $z = \begin{vmatrix} e^{i 2A} & e^{-iC} \\ e^{-iC} & e^{i 2B} \\ e^{-iB} & e^{-iA} \\ e^{-iA} & e^{i 2C} \end{vmatrix}$ , then:

(A) $\text{Re}(z) = 4$

(B) $\text{Im}(z) = 1$

(D) $\text{Re}(z) = 0$

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11. If $(a, b, c)$ is the image of the point $(1, 2, -3)$ in the line $\dfrac{x + 1}{2} = \dfrac{y - 3}{-2} = \dfrac{z}{-1}$ , then $a + b + c$ is equal to:

(A) -1

(B) 2

(D) 4

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12. The lines $x = ay - 1 = z - 2$ and $x = 3y - 2 = bz - 2$ , $(ab \neq 0)$ , are coplanar if:

(A) $b = 1$ , $a \in \mathbb{R} - \{0\}$

(B) $a = 1$ , $b \in \mathbb{R} - \{0\}$

(D) $a = 2$ , $b = 3$

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13. If the shortest distance between the lines $\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{\lambda}$ and $\dfrac{x - 2}{1} = \dfrac{y - 4}{4} = \dfrac{z - 5}{5}$ is $\dfrac{1}{\sqrt{3}}$ , then the sum of all possible values of $\lambda$ is:

(A) 16

(B) 6

(D) 15

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14. The equation of the line through the point $(0, 1, 2)$ and perpendicular to the line $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \dfrac{z - 1}{-2}$ is:

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

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

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

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15. Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $\ell + m - n = 0$ and $\ell^2 + m^2 - n^2 = 0$ . Then the value of $\sin^4 \alpha + \cos^4 \alpha$ is:

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

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

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

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16. If the foot of the perpendicular from point $(4, 3, 8)$ on the line $L_1: \dfrac{x - a}{\ell} = \dfrac{y - 2}{3} = \dfrac{z - b}{4}$ , $\ell \neq 0$ , is $(3, 5, 7)$ , then the shortest distance between the line $L_1$ and line $L_2: \dfrac{x - 2}{3} = \dfrac{y - 4}{4} = \dfrac{z - 5}{5}$ is equal to:

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

(B) $\dfrac{1}{\sqrt{6}}$

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

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17. Let the position vectors of two points $P$ and $Q$ be $3\hat{i} - \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} - 4\hat{k}$ , respectively. Let $R$ and $S$ be two points such that the direction ratios of lines $PR$ and $QS$ are $(4, -1, 2)$ and $(-2, 1, -2)$ , respectively. Let lines $PR$ and $QS$ intersect at $T$ . If the vector $\overrightarrow{TA}$ is perpendicular to both $\overrightarrow{PR}$ and $\overrightarrow{QS}$ and the length of vector $\overrightarrow{TA}$ is $\sqrt{5}$ units, then the modulus of a position vector of $A$ is:

(A) $\sqrt{482}$

(B) $\sqrt{171}$

(D) $\sqrt{227}$

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18. If the two lines $\ell_1: \dfrac{x - 2}{3} = \dfrac{y + 1}{-2}$ , $z = 2$ and $\ell_2: \dfrac{x - 1}{1} = \dfrac{2y + 3}{\alpha} = \dfrac{z + 5}{2}$ are perpendicular, then an angle between the lines $\ell_2$ and $\ell_3: \dfrac{x - 1}{3} = \dfrac{2y - 1}{-4} = \dfrac{z}{4}$ is:

(A) $\cos^{-1} \left( \dfrac{29}{4} \right)$

(B) $\sec^{-1} \left( \dfrac{29}{4} \right)$

(D) $\cos^{-1} \left( \dfrac{2}{\sqrt{29}} \right)$

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19. If two straight lines whose direction cosines are given by the relations $l + m - n = 0$ , $3l^2 + m^2 + cnl = 0$ are parallel, then the positive value of $c$ is:

(A) 6

(B) 4

(D) 2

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20. If the length of the perpendicular drawn from the point $P(a, 4, 2)$ , $a > 0$ , on the line $\dfrac{x + 1}{2} = \dfrac{y - 3}{3} = \dfrac{z - 1}{-1}$ is $2\sqrt{6}$ units and $Q(\alpha_1, \alpha_2, \alpha_3)$ is the image of the point $P$ in this line, then $a + \sum_{i=1}^3 \alpha_i$ is equal to:

(A) 7

(B) 8

(D) 14

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21. Let $z = \dfrac{1 - i\sqrt{3}}{2}$ , $i = \sqrt{-1}$ , then the least value of $21 + \left( z + \dfrac{1}{z} \right)^3 + \left( z^2 + \dfrac{1}{z^2} \right)^3 + \left( z^3 + \dfrac{1}{z^3} \right)^3 + \cdots + \left( z^{21} + \dfrac{1}{z^{21}} \right)^3$ is:

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22. If $|z| = 1$ , then $\left| 3 + \dfrac{1}{z} \right|^2 + |3 - z|^2$ is equal to?

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23. Value of $\left| \sum_{i=1}^4 z_i^4 \right| + \sum_{i=1}^4 z_i^5$ is equal to:

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24. If the shortest distance between the lines $\vec{r}_1 = \alpha \hat{i} + 2\hat{j} + 2\hat{k} + \lambda (\hat{i} - 2\hat{j} + 2\hat{k}), \quad \lambda \in \mathbb{R}, \quad \alpha > 0$ and $\vec{r}_2 = -4\hat{i} - \hat{k} + \mu (3\hat{i} - 2\hat{j} - 2\hat{k}), \quad \mu \in \mathbb{R}$ is 9, then $\alpha$ is equal to:

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25. Let $a, b \in \mathbb{R}$ . If the mirror image of the point $P(a, 6, 9)$ with respect to the line $\dfrac{x - 3}{7} = \dfrac{y - 2}{5} = \dfrac{z - 1}{-9}$ is $(20, b, -a - 9)$ , then $|a + b|$ is equal to:

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