JEE MAIN TEST – 5 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. If $0 < \theta < \pi$ , then minimum value of the expression $f(\theta) = 3 \sin \theta + \text{cosec}^3 \theta$ is:

(A) 4

(B) 3

(D) 6

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2. The value of $\sin \dfrac{\pi}{14} \sin \dfrac{3\pi}{14} \sin \dfrac{5\pi}{14} \sin \dfrac{7\pi}{14} \sin \dfrac{9\pi}{14} \sin \dfrac{11\pi}{14} \sin \dfrac{13\pi}{14}$ is equal to:

(A) $\dfrac{1}{8}$

(B) $\dfrac{1}{16}$

(D) $\dfrac{1}{64}$

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3. If $\sum_{i=1}^{5} (x_i - 10) = 5$ and $\sum_{i=1}^{5} (x_i - 10)^2 = 25$ , then standard deviation of observations $2x_1 + 7, 2x_2 + 7, 2x_3 + 7, 2x_4 + 7$ and $2x_5 + 7$ is equal to:

(A) 8

(B) 16

(D) 2

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4. If $z$ is a complex number satisfying $|z - 3| < 5$ , then range of $|z + 3i|$ is (where $i = \sqrt{-1}$ ):

(A) $[5 - 3\sqrt{2}, 5 + 3\sqrt{2}]$

(B) $[3\sqrt{2} - 5, 3\sqrt{2} + 5]$

(D) $[0, 5 - 3\sqrt{2}]$

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5. If matrix $A = [a_{ij}]_{3 \times 3}$ and $a_{ij} + a_{ji} \neq 0$ and element $a_{ij} \in \{0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7\}$ , then number of matrix $A$ is equal to:

(A) 3370

(B) 3300

(D) None of these

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6. A box contains 10 tickets numbered from 1 to 10. Two tickets are drawn one by one with replacement. The probability that the “difference between the first drawn ticket number and second is not less than 4” is:

(A) $\dfrac{7}{30}$

(B) $\dfrac{14}{30}$

(D) $\dfrac{10}{30}$

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7. $x_1, x_2$ & $x_3$ when divided by 4 leaves a remainder of 0, 1 & 2 respectively. Find number of non-negative integral solution of the equation $x_1 + x_2 + x_3 = 35$ :

(A) 45

(B) 55

(D) 190

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8. Two data sets each of size 10 has the variance as 4 and $k$ and the corresponding means as 2 and 4 respectively. If the variance of the combined data set is 5.5, then the value of $k$ is equal to:

(A) 5

(B) 6

(D) 3

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9. If $\phi(x) = f(x) + f(2a - x)$ and $f''(x) > 0$ , $a > 0$ , $0 \leq x \leq 2a$ , then:

(A) $\phi(x)$ increases in $(a, 2a)$

(B) $\phi(x)$ increases in $(0, a)$

(D) None

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10. The global maximum value of $f(x) = \log_{10} (4x^3 - 12x^2 + 11x - 3)$ , $x \in [2, 3]$ , is:

(A) $-\dfrac{3}{2} \log_{10} 3$

(B) $1 + \log_{10} 3$

(D) $\dfrac{3}{2} \log_{10} 3$

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11. Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f \left( \dfrac{x + y}{3} \right) = \dfrac{f(x) + f(y)}{3}$ , $f(0) = 3$ and $f'(0) = 3$ , then which of the following is correct?

(A) $\dfrac{f(x)}{x}$ is differentiable in $\mathbb{R}$

(B) $f(x)$ is continuous but not differentiable in $\mathbb{R}$

(D) None

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12. If $f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x 0 \end{cases}$ , then at $x = 0$ :

(A) $f(x)$ is continuous, when $a = 0$

(B) $f(x)$ is continuous, when $a = 8$

(D) None of these

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13. Area bounded by $y^2 = 8x$ and $x^2 = 12y$ , is:

(A) 32

(B) 16

(D) 8

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14. A curve passes through the point $\left( 1, \dfrac{\pi}{6} \right)$ . Let the slope of the curve at each point $(x, y)$ be $\dfrac{y}{x} + \sec \left( \dfrac{y}{x} \right)$ , $x > 0$ . Then the equation of the curve is:

(A) $\sin \left( \dfrac{y}{x} \right) = \ln x + \dfrac{1}{2}$

(B) $\text{cosec} \left( \dfrac{y}{x} \right) = \ln x + 2$

(D) $\cos \left( \dfrac{2y}{x} \right) = \ln x + \dfrac{1}{2}$

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15. If $\int \dfrac{dx}{x^2 (x^{2015} + 1)^{2014}} = -(1 + x^{-b})^{1/a} + C$ , then the value of $\dfrac{a}{b}$ is:

(A) -1

(B) 1

(D) $\dfrac{2016}{2015}$

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16. $\int \dfrac{dx}{(1 + \sqrt{x})^{2010}} = \dfrac{2}{(1 + \sqrt{x})^{\alpha}} - \dfrac{2}{(1 + \sqrt{x})^{\beta}} + C$ where $C$ is constant of integration and $\alpha > 0$ , $\beta > 0$ , then:

(A) $|\alpha - \beta| = 1$

(B) $\alpha, \beta, 2010$ (in order) are in arithmetic progression

(D) $\alpha + 1 = \beta + 1 = 2010$

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17. Area of the triangle $ABC$ where $A(9, -9)$ , $B(1, 3)$ and $C$ lies on $3x + 2y + 4 = 0$ is equal to:

(A) $\sqrt{13}$

(B) 26

(D) $\sqrt{208}$

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18. If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$ , then the locus of the foot of perpendicular from $O$ on $AB$ is:

(A) $(x^2 + y^2)^2 = 4Rx^2 y^2$

(B) $(x^2 + y^2)(x + y) = R^2 xy$

(D) $(x^2 + y^2)^2 = 4R^2 x^2 y^2$

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19. The slope of the line which belongs to family of lines $(1 + \lambda)x + (\lambda - 1)y + 2(1 - \lambda) = 0$ and makes shortest intercept on $x^2 = 4y - 4$ is:

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

(B) 0

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

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20. The equation of the line of shortest distance of the lines $\dfrac{x - 6}{3} = \dfrac{y - 7}{-1} = \dfrac{z - 4}{1}$ and $\dfrac{x - 3}{-3} = \dfrac{y + 9}{2} = \dfrac{z - 2}{4}$

(A) $\dfrac{x - 6}{2} = \dfrac{y - 7}{5} = \dfrac{z - 4}{-1}$

(B) $\dfrac{x}{2} = \dfrac{y + 9}{5} = \dfrac{z - 2}{-1}$

(D) $\dfrac{x - 3}{2} = \dfrac{y - 8}{5} = \dfrac{z - 3}{-1}$

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SECTION B: Numerical Value Type Questions

21. Mean and median of four numbers $a, b, c$ and $d$ ( $b < a < d < c$ ) is 35 and 25 respectively then the value of $b + c - a - d$ will be:

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

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23. Let $f(x)$ be twice differentiable function and $f''(0) = 5$ , then $\lim_{x \to 0} \dfrac{3f(x) - 4f(3x) + f(9x)}{x^2}$ is equal to:

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24. Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ x - [x + 2], & \text{if } [x] \text{ is even} \end{cases}$ , if $\int_{-1}^{2} f(x) \, dx$ is $K$ then find the value $[K]$ ?
[Note: $[K]$ denotes greatest integer less than or equal to $K$ ]

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25. If $\vec{a} \times \vec{b} = \vec{c}, \vec{b} \times \vec{c} = \vec{a}$ , then the value of $2\vec{a} + 3\vec{b} + 6\vec{c}$ equals (where $\vec{a}$ , $\vec{b}$ and $\vec{c}$ are unit vectors):

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