JEE MAIN TEST – 6 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. Number of rational terms in the expansion of $(7^{1/7} + 11^{1/11})^{711}$ is:

(A) 7

(B) 8

(D) 10

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2. The least value of $|Z - 3 - 4i|^2 + |Z + 2 - 7i|^2 + |Z - 5 + 2i|^2$ occurs when $Z =$ (Where $Z$ is complex no.)

(A) $1 + 3i$

(B) $3 + 3i$

(D) None of these

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3. Let $U_1$ and $U_2$ be two bags such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed if head appears, then 1 ball is drawn at random from $U_1$ and put into $U_2$ . However if tail appears, then 2 balls are drawn at random from $U_1$ and put into $U_2$ . Now, 1 ball is drawn at random from $U_2$ then. The probability of the drawn ball from $U_2$ being white is :-

(A) $\frac{13}{30}$

(B) $\frac{23}{30}$

(D) $\frac{11}{30}$

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4. For $a, b \in \mathbb{R} - \{0\}$ , let $f(x) = ax^2 + bx + a$ satisfies $f\left(x + \frac{7}{4}\right) = f\left(\frac{7}{4} - x\right) \ \forall \ x \in \mathbb{R}$ . Also the $ax^2 + bx + a = 7x + a$ has only one real solution then $a + b$ is equal to :-

(A) 4

(B) 5

(D) 7

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5. If $A$ is a square matrix of order 4 and $B = \text{adj}A$ , where $|B| = 27$ , then value of $|A^{-1} \text{adj}(3AB)|$ is :

(A) $3^{20}$

(B) $3^{21}$

(D) $3^{23}$

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6. The function $f(x) = \int_{1}^{x} \left\{ 2(t - 1)(t - 2)^3 + 3(t - 1)^2 (t - 2)^2 \right\} \, dt$ attains its maximum at $x$ is equal to :-

(A) 1

(B) 2

(D) 4

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7. A spherical balloon is expanding. If at any instant rate of increase of its volume is 16 times of rate of increase of its radius, then its radius at that instant, is :

(A) $\frac{1}{\sqrt{\pi}}$

(B) $\frac{2}{\sqrt{\pi}}$

(D) $\frac{4}{3\sqrt{\pi}}$

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8. If $f(x) = 1 + 2x^2 + 4x^4 + 6x^6 + \ldots + 100x^{100}$ is a polynomial in a real variable $x$ , then $f(x)$ has

(A) neither a maximum nor a minimum

(B) only one maximum

(D) None of these

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9. If $f(x) = \begin{cases} \frac{\sqrt{1 + px} - \sqrt{1 - px}}{2x + 1}, & -1 < x < 0 \\ \frac{2x + 1}{x - 2}, & 0 \leq x \leq 1 \end{cases}$ is continuous in the interval $[-1, 1]$ then $p$ equals

(A) $-1$

(B) $1$

(D) $-1/2$

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10. The area of the bounded region enclosed by the curve $y = 3 - \left| x - \frac{1}{2} \right| - |x + 1|$ and the $x$ -axis is :

(A) $\frac{9}{4}$

(B) $\frac{45}{16}$

(D) $\frac{63}{16}$

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11. $\int_{0}^{5} \cos \left( \pi \left( x - \left\lfloor \frac{x}{2} \right\rfloor \right) \right) \, dx$ Where $\lfloor t \rfloor$ denotes greatest integer less than or equal to $t$ , is equal to :

(A) $-3$

(B) $-2$

(D) $0$

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12. Let the solution curve $y = y(x)$ of the differential equation, $\left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] \frac{dy}{dx} = x + \left[ \frac{x}{\sqrt{x^2 - y^2}} + e^{\frac{y}{x}} \right] y$ pass through the points $(1, 0)$ and $(2\alpha, \alpha)$ . Then $\alpha$ is $\alpha > 0$ equal to

(A) $\frac{1}{2} \exp \left( \frac{\pi}{6} + \sqrt{e} - 1 \right)$

(B) $\frac{1}{2} \exp \left( \frac{\pi}{3} + \sqrt{e} - 1 \right)$

(D) $2 \exp \left( \frac{\pi}{3} + \sqrt{e} - 1 \right)$

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13. $\int \frac{(x^2 + 1) e^x}{(x + 1)^2} \, dx = f(x) e^x + C$ , Where $C$ is a constant, then $\frac{d^3 f}{dx^3}$ at $x = 1$ is equal to :

(A) $-\frac{3}{4}$

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

(D) $\frac{3}{2}$

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14. If $\cot \alpha = 1$ and $\sec \beta = -\frac{5}{3}$ , where $\pi < \alpha < \frac{3\pi}{2}$ and $\frac{\pi}{2} < \beta < \pi$ , then the value of $\tan (\alpha + \beta)$ and the quadrant in which $\alpha + \beta$ lies, respectively are :

(A) $-\frac{1}{7}$ and IV $^{\text{th}}$ quadrant

(B) $7$ and I $^{\text{st}}$ quadrant

(D) $\frac{1}{7}$ and I $^{\text{st}}$ quadrant

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15. The value of $\cos \left( \frac{2\pi}{7} \right) + \cos \left( \frac{4\pi}{7} \right) + \cos \left( \frac{6\pi}{7} \right)$ is equal to :

(A) $-1$

(B) $-\frac{1}{2}$

(D) $-\frac{1}{4}$

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16. If the mean deviation about median for the number 3, 5, 7, 2k, 12, 16, 21, 24 arranged in the ascending order, is 6 then the median is

(A) 11.5

(B) 10.5

(D) 11

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17. Maximum distance of any point on the curve $x^2 + 2y^2 + 2xy = 1$ from the origin is :-

(A) $\frac{2}{3 - \sqrt{5}}$

(B) $\frac{2}{2 + \sqrt{5}}$

(D) $\sqrt{\frac{2}{3 + \sqrt{5}}}$

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18. In the right angle triangle as shown, an altitude is drawn from the right to the hypotenuse. Circles are inscribed within each of the smaller triangles. What is the distance between the centres of these circles ?

[Diagram: Right-angled triangle with sides 15, 20 and inscribed circles in sub-triangles]

(A) 5

(B) 7

(D) $\sqrt{50}$

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

(A) 3

(B) 7

(D) $\sqrt{7}$

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20. The distance of the point having position vector $-\hat{i} + 2\hat{j} + 6\hat{k}$ from the straight line passing through the point $(2, 3, -4)$ and parallel to the vector, $6\hat{i} + 3\hat{j} - 4\hat{k}$ is :

(A) 7

(B) $4\sqrt{3}$

(D) 6

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21. If $x, y, z \in \mathbb{R}^+$ and $xyz = 32$ , then the minimum value of $x^2 + 4xy + 4y^2 + z^2$ is equal to :

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22. Let $f(x)$ be a twice-differentiable function and $f''(0) = 2$ . Then evaluate. $\lim_{x \to 0} \frac{2f(x) - 3f(2x) + f(4x)}{x^2}$

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23. For real numbers $a, b \ (a > b > 0)$ , let $\text{Area} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi$ and $\text{Area} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi$ . Then the value of $(a - b)^2$ is equal to.

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24. Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in the worst case, the number of students can fail is :

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

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