JEE MAIN TEST – 15 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. Consider a sequence of 101 terms as $\dfrac{{}^{100}C_0}{1 \cdot 2 \cdot 3 \cdot 4}, \dfrac{{}^{100}C_1}{2 \cdot 3 \cdot 4 \cdot 5}, \dfrac{{}^{100}C_2}{3 \cdot 4 \cdot 5 \cdot 6}, \ldots, \dfrac{{}^{100}C_{100}}{101 \cdot 102 \cdot 103 \cdot 104}$ If the $n^{\text{th}}$ term is the greatest term of the sequence, then $n$ is equal to:

(A) 48

(B) 49

(D) 51

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2. Number of rational terms in the binomial expansion of $\left( 7^{\frac{1}{7}} + 11^{\frac{1}{11}} \right)^{711}$ is:

(A) 7

(B) 8

(D) 10

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3. If $\left( 1 + x + x^2 \right)^5 = a_0 + a_1 x + a_2 x^2 + \cdots + a_{10} x^{10}$ , then $\left( a_0 - a_2 + a_4 - a_6 + a_8 - a_{10} \right)^2 + \left( a_1 - a_3 + a_5 - a_7 + a_9 \right)^2$ is equal to:

(A) 0

(B) $2^{10}$

(D) -1

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4. The coefficient of $x^5$ in the expansion of $(1 + 5x)^5 + (1 + 5x)^6 + \cdots + (1 + 5x)^{19}$ is:

(A) ${}^{20}C_4 \cdot 5^5$

(B) ${}^{20}C_6 \cdot 5^5$

(D) ${}^{20}C_5 \cdot 5^6$

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5. Sum of all the rational terms in the expansion $\left( 3^{1/4} + 4^{1/3} \right)^{12}$ is:

(A) 27

(B) 256

(D) None of these

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6. Find the remainder when $x = 5^{5^{5 \cdots 5}}$ (24 times 5) is divided by 24:

(A) 5

(B) 21

(D) 0

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7. $x_1, x_2$ , and $x_3$ , when divided by 4, leave remainders of 0, 1, and 2 respectively. Find the number of nonnegative integer solutions of the equation $x_1 + x_2 + x_3 = 35$ :

(A) 45

(B) 55

(D) 190

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8. 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:

(A) 3375

(B) 2744

(D) 5488

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9. How many words can be made by using all letters of the word ‘BAHUBALI’ if all words start and end with vowels?

(A) 2160

(B) 900

(D) 780

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10. If the coefficient of $a^7 b^8$ in the expansion of $(a + ab + 4ab)^{10}$ is $K \cdot 2^{16}$ , then $K$ is equal to:

(A) 215

(B) 315

(D) None

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11. In a circle, 20 persons are seated. Then the number of ways of selecting 5 persons such that no two persons are consecutive is:

(A) ${}^{16}C_5$

(B) ${}^{16}C_5 - {}^{14}C_3$

(D) None

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12. How many six-digit numbers are there in which no digit is repeated, even digits appear at even places, odd digits appear at odd places, and the number is divisible by 4?

(A) 3600

(B) 2700

(D) 1440

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13. Number of integer solutions of $xyzw = 360$ :

(A) 800

(B) 12800

(D) 3200

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14. If a vector $\vec{r}$ of magnitude $3\sqrt{6}$ is directed along the bisector of the angle between the vectors $\vec{a} = 7\hat{i} - 4\hat{j} - 4\hat{k}$ and $\vec{b} = -2\hat{i} - \hat{j} + 2\hat{k}$ , then $\vec{r}$ is:

(A) $\hat{i} - 7\hat{j} + 2\hat{k}$

(B) $\hat{i} + 7\hat{j} - 2\hat{k}$

(D) $\hat{i} - 7\hat{j} - 2\hat{k}$

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15. The set of values of $c$ for which the angle between the vectors $c x \hat{i} - \hat{j} + 3\hat{k}$ and $x \hat{i} - 2\hat{j} + 2cx \hat{k}$ is acute for every $x \in \mathbb{R}$ is:

(A) $(0, 4/3)$

(B) $[0, 4/3]$

(D) $[0, 4/3)$

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16. A unit vector which is perpendicular to the vector $2\vec{i} - \vec{j} + 2\vec{k}$ and is coplanar with the vectors $\vec{i} + \vec{j} - \vec{k}$ and $2\vec{i} + \vec{j} - \vec{k}$ is:

(A) $\dfrac{3\vec{i} + 2\vec{j} - 2\vec{k}}{\sqrt{17}}$

(B) $\dfrac{2\hat{j} + \hat{k}}{\sqrt{5}}$

(D) $\dfrac{2\vec{i} + 2\vec{j} - \vec{k}}{3}$

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17. Let $ABCD$ be a parallelogram such that $\overrightarrow{AB} = \vec{q}$ , $\overrightarrow{AD} = \vec{p}$ , and $\angle BAD$ is an acute angle. If $\vec{r}$ is the vector that coincides with the altitude directed from the vertex $B$ to the side $AD$ , then $\vec{r}$ is given by:

(A) $\vec{r} = -3\vec{q} + \dfrac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}$

(B) $\vec{r} = 3\vec{q} - \dfrac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}$

(D) $\vec{r} = \vec{q} - \left( \dfrac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}} \right) \vec{p}$

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18. The vector $\hat{i} + x \hat{j} + 3 \hat{k}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4 \hat{i} + (4x - 2) \hat{j} + 2 \hat{k}$ . The values of $x$ are:

(A) $\left\{ -\dfrac{2}{3}, 2 \right\}$

(B) $\left\{ \dfrac{1}{3}, 2 \right\}$

(D) $\{2, 7\}$

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19. If $\vec{a}, \vec{b}$ , and $\vec{c}$ are unit vectors, then $|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2$ does not exceed:

(A) 4

(B) 9

(D) 6

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20. Let $x^2 + 3y^2 = 3$ be the equation of an ellipse in the $x$ – $y$ plane. $A$ and $B$ are two points whose position vectors are $-\sqrt{3} \hat{i}$ and $-\sqrt{3} \hat{i} + 2 \hat{k}$ . Then the position vector of a point $P$ on the ellipse such that $\angle APB = \pi/4$ is:

(A) $\pm \hat{j}$

(B) $\pm (\hat{i} + \hat{j})$

(D) None of these

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21. The number of values of ‘ $x$ ‘ for which the fourth term in the expansion $\left( \left( 4^x + 44 \right)^{1/5} + \left( 2^{x-1} + 7 \right)^{-1/3} \right)^8$ is 336 is:

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22. The $13^{\text{th}}$ term in the expansion of $\left( x^2 + \dfrac{2}{x} \right)^n$ is independent of $x$ , then the sum of the divisors of $n$ is:

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23. If $\left( 2 \cdot {}^{35} \cdot 3^{16} \right)$ is divided by 11, then the remainder is:

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24. The number of shortest paths from $A$ to $B$ , which are neither passing through $P$ nor $Q$ nor $R$ nor $S$ , is:

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25. The total number of two-digit numbers ‘ $n$ ‘ such that $3^n + 7^n$ is a multiple of 10 is:

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26. 20 persons are sitting in a particular arrangement around a circular table. Three persons are to be selected for leaders. The number of ways of selecting three persons such that no two were sitting adjacent to each other is:

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27. The maximum number of points of intersection of five lines and four circles is (if no two of them are similar): [Image of intersecting lines and circles]

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28. Consider the set of eight vectors $V = \{ a\hat{i} + b\hat{j} + c\hat{k} ; a, b, c \in \{-1, 1\} \}$ . Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is:

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29. If $\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}$ , $\vec{y} = \hat{i} + 4\hat{j} - 3\hat{k}$ , and $\vec{z} = 3\hat{i} - 4\hat{j} - 12\hat{k}$ , then the magnitude of the projection of $\vec{x} \times \vec{y}$ on $\vec{z}$ is:

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30. If vectors $\vec{a}, \vec{b}, \vec{c}$ satisfy the condition $|\vec{a} - \vec{c}| = |\vec{b} - \vec{c}|$ , then $(\vec{b} - \vec{a}) \cdot \left( \vec{c} - \dfrac{\vec{a} + \vec{b}}{2} \right)$ is equal to:

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