JEE MAIN TEST – 7 – JEE MATH APEX

Mathematics Questions

1. If all interior angle of quadrilateral are in A.P. If common difference is $10^{\circ}$ , then find smallest angle?

(A) $60^{\circ}$

(B) $70^{\circ}$

(D) $75^{\circ}$

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2. If $\alpha, \beta$ are the roots of $a x^2 + b x + c = 0$ , then the roots of equation $a x^2 - b x (x - 1) + c (x - 1)^2 = 0$ are :-

(A) $\frac{\alpha}{\alpha - 1}, \frac{\beta}{\beta - 1}$

(B) $\frac{\alpha}{\alpha + 1}, \frac{\beta}{\beta + 1}$

(D) $\alpha, \beta$

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3. Let $a, b, c \in \mathbb{R}$ such that $a + b + c \neq 0$ , if system of equations $\begin{aligned} a x + b y + c z = 0 \\ b x + c y + a z = 0 \\ c x + a y + b z = 0 \end{aligned}$ has a non-trivial solutions then –

(A) $a + c - b = 0$

(B) $a = b = c$

(D) None of these

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

(A) 1

(B) 3

(D) 8

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5. There are two bags, one of which contain 3 black and 4 white balls while the second contains 4 black and 3 white balls. A dice is cast, if the face 1 or 3 turns up, a ball is taken from the first bag, and if any other face turn up, a ball is taken from second bag, then find the probability of choosing a black ball :-

(A) $\frac{10}{21}$

(B) $\frac{2}{21}$

(D) None of these

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6. If $\tan^2\left(\frac{\pi}{16}\right) + \tan^2\left(\frac{2 \pi}{16}\right) + \tan^2\left(\frac{3 \pi}{16}\right) + \ldots + \tan^2\left(\frac{7 \pi}{16}\right) = \lambda$ \& if $x^y + y^x = \lambda$ , then the value of $(x + y)^2$ must be

(A) 35

(B) 1225

(D) 2

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

(A) 8

(B) 16

(D) 2

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8. The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$ , where $p \neq 0$ and $q \neq 0$ . If the new mean and new s.d. become half of their original values, then $q$ is equal to

(A) $-20$

(B) 10

(D) $-5$

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9. Let $f(x) = (\sin x)^n + {}^n C_1 (\sin x)^{n-1} \cos x + {}^n C_2 (\sin x)^{n-2} \cos^2 x + \ldots + (\cos x)^n$ , where $n$ is an even number, then for $x \in [0, 2\pi]$ , number of maxima and minima are $p$ and $q$ respectively, then-

(A) $p = 2, q = 2$

(B) $p = 1, q = 3$

(D) $p = 3, q = 3$

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10. Given $f(x) = \begin{cases} \frac{\ln \left(1 + sgn(x) + (x)^2\right)}{1 - \cos (x)} & \text{if } x \neq 0, \\ k & \text{if } x = 0 \end{cases}$ (where $[.], \{.\}$ and $sgn x$ denotes greatest integer function, fractional part function and signum function respectively)

(A) $f(x)$ is continuous at $x = 0$ if $k = 2$

(B) for $k = 1$ , $f(x)$ has removable discontinuity at $x = 0$

(D) $\lim_{x \to 0} f(x)$ exists

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11. If $f(x) = x^3 + 3x + 4$ and $g$ is the inverse function of $f$ , then the value of $\frac{d}{dx} \left( \frac{g(x)}{g(g(x))} \right)$ at $x = 4$ equals :

(A) $\frac{-1}{3}$

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

(D) 6

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12. If $f(x)$ is a polynomial such that :- $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right); x \neq 0$ and $f(D) = -63$ then value of $f\left(\sqrt[3]{\frac{1}{5^{\log 7}} + \frac{1}{\sqrt{-\log_{10}(0.1)}}}\right)$ :-

(A) $-511$

(B) $-26$

(D) $-7$

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13. $\int \sqrt{1 + x \sqrt{1 + (x + 1) \sqrt{1 + (x + 2)(x + 4)}}} \, dx =$

(A) $\frac{x^2}{2} + x + C$

(B) $\frac{x^2}{2} - x + C$

(D) $x + C$

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14. $\int_0^\pi \frac{dx}{1 - 2a \cos x + a^2}, a < 1$ is equal to :-

(A) $\frac{\pi a \log 2}{4}$

(B) $\frac{4 \pi}{2 - a^2}$

(D) None of these

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15. The area (in sq. units) of the region $\left\{ (x, y) : y^2 \geq 2x \text{ and } x^2 + y^2 \leq 4x, x \geq 0, y \geq 0 \right\}$ is :-

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

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

(D) $\pi - \frac{4 \sqrt{2}}{3}$

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16. The solution of the differential equation $y \, dx - x \, dy + x y^2 \, dx = 0$ is :

(A) $\frac{x}{y} + x^2 = \lambda$

(B) $\frac{x}{y} + \frac{x^2}{2} = \lambda$

(D) None of these

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17. A ray of light coming from the point $P(1, 2)$ gets reflected from the point $Q$ on the $x$ -axis and then passes through the point $R(4, 3)$ . If the point $S(h, k)$ is such that $PQRS$ is a parallelogram, then $h k^2$ is equal to :

(A) 80

(B) 90

(D) 70

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18. Let $\overrightarrow{OA} = 2 \overrightarrow{a}$ , $\overrightarrow{OB} = 6 \overrightarrow{a} + 5 \overrightarrow{b}$ and $\overrightarrow{OC} = 3 \overrightarrow{b}$ , where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $OABC$ is equal to :

(A) 38

(B) 40

(D) 35

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19. The shortest distance between the line $\frac{x - 3}{4} = \frac{y + 7}{-11} = \frac{z - 1}{5} \text{ and } \frac{x - 5}{3} = \frac{y - 9}{-6} = \frac{z + 2}{1}$ is :

(A) $\frac{187}{\sqrt{563}}$

(B) $\frac{178}{\sqrt{563}}$

(D) $\frac{179}{\sqrt{563}}$

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20. Let the circles $C_1 : (x - \alpha)^2 + (y - \beta)^2 = r_1^2$ and $C_2 : (x - 8)^2 + \left(y - \frac{15}{2}\right)^2 = r_2^2$ touch each other externally at the point $(6, 6)$ . If the point $(6, 6)$ divides the line segment joining the centres of the circles $C_1$ and $C_2$ internally in the ratio $2:1$ , then $(\alpha + \beta) + 4 (r_1^2 + r_2^2)$ equals

(A) 110

(B) 130

(D) 145

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21. The number of 5-digit numbers of the form $xyzyx$ in which $x < y$ is :-

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22. If $0 < A < \pi/2$ and $\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$ and $\sin A$ and $\cos A$ are roots of equation $4 x^2 - 3 x + a = 0$ . Then value of $25 a$ is :-

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23. If $\lim_{x \to 0} \frac{a x - (e^{4x} - 1)}{a x (e^{4x} - 1)}$ exists and is equal to $b$ , then the value of $a - 2b$ is

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24. If $\int \sqrt{1 + \sin \left(\frac{x}{4}\right)} \, dx = k \left( \sin \frac{x}{a} - \cos \frac{x}{b} \right) + C$ ; then value of $(k + a + b) = ?$

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25. Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E : \frac{(x - 1)^2}{100} + \frac{(y - 1)^2}{75} = 1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$ . If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$ , then $3 \alpha^2 + 2 \beta^2$ is equal to :

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