JEE MAIN TEST – 4 – JEE MATH APEX

SECTION A: Multiple Choice Questions

1. If all interior angles of a quadrilateral are in A.P. with a common difference of $10^\circ$ , then find the smallest angle.

(A) $60^\circ$

(B) $70^\circ$

(D) $75^\circ$

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2. If $\alpha, \beta, \gamma, \delta$ are the roots of $x^4 - 100x^3 + 2x^2 + 4x + 10 = 0$ then $\dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma} + \dfrac{1}{\delta}$ is equal to:

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

(B) $-\dfrac{1}{10}$

(D) $-\dfrac{2}{5}$

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3. If $\alpha, \beta, \gamma$ are the angles of triangle and the system of equations has non-trivial solutions:
$\cos (\alpha + \beta)x + \cos (\beta + \gamma)y + \cos (\gamma + \alpha)z = 0$
$\cos (\alpha - \beta)x + \cos (\beta - \gamma)y + \cos (\gamma - \alpha)z = 0$
$\sin (\alpha + \beta)x + \sin (\beta + \gamma)y + \sin (\gamma + \alpha)z = 0$
then the triangle is necessarily:

(A) Equilateral $\Delta$

(B) Acute angled $\Delta$

(D) Right angled $\Delta$

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4. The probability that in a family of 5 members, exactly two members have birthday on Sunday is:

(A) $\dfrac{(12 \times 5^3)}{7^5}$

(B) $\dfrac{(10 \times 6^2)}{7^5}$

(D) $\dfrac{(10 \times 6^3)}{7^5}$

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5. If $z = \dfrac{3}{2 + \cos \theta + i \sin \theta}$ , then locus of $z$ is:

(A) a straight line

(B) a circle having centre on $x$ -axis

(D) a parabola

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6. $16 \sin(20^\circ) \sin(40^\circ) \sin(80^\circ)$ is equal to:

(A) $\sqrt{3}$

(B) $2\sqrt{3}$

(D) $4\sqrt{3}$

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7. Let the mean and the variance of 5 observations $x_1, x_2, x_3, x_4, x_5$ be $\dfrac{24}{5}$ and $\dfrac{194}{25}$ respectively. If the mean and variance of the first 4 observations are $\dfrac{7}{2}$ and $a$ respectively, then $(4a + x_5)$ is equal to:

(A) 13

(B) 15

(D) 18

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8. The mean and variance of the data 4, 5, 6, 6, 7, 8, $x, y$ where $x < y$ are 6, and $\dfrac{9}{4}$ respectively. Then $x^4 + y^2$ is equal to:

(A) 162

(B) 320

(D) 420

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9. If $A_1$ and $A_2$ are areas bounded by:
$A_1 : y = \sin x, y = \cos x$ and $y$ -axis in $1^{\text{st}}$ quadrant.
$A_2 : y = \sin x, y = \cos x$ and $x$ -axis and $x = \frac{\pi}{2}$ , then:

(A) $A_1 : A_2 = 1 : \sqrt{2}; A_1 + A_2 = 1$

(B) $A_1 : A_2 = \sqrt{2} : 1; A_1 + A_2 = \sqrt{2} + 1$

(D) $A_1 : A_2 = 1 : 2; A_1 + A_2 = 1$

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10. Value of $\dfrac{29 \int_{0}^{1} (1 - x^4)^7 \, dx}{4 \int_{0}^{1} (1 - x^4)^6 \, dx}$ is equal to:

(A) 7

(B) -7

(D) $-\dfrac{1}{7}$

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11. Solution of differential equation $y (2x^4 + y) \dfrac{dy}{dx} = (1 - 4xy^2) x^2$ is:

(A) $3x^2 y + y^3 + x^3 = c$

(B) $3(x^2 y)^2 + y^3 + x^3 = c$

(D) $3x^2 y - y^3 - x^3 = c$

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12. If $\int x \log \left(1 + \dfrac{1}{x}\right) dx = f(x) \cdot \log(x + 1) + g(x) \cdot x^2 + Ax + C$ , then:

(A) $f(x) = x^2$

(B) $g(x) = \log x$

(D) None of these

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13. Range of $y = \cos^{-1} \sqrt{\log_{\lfloor x \rfloor} \left( \dfrac{\lfloor x \rfloor}{x} \right)}$ (where $\lfloor x \rfloor$ denotes greatest integer less than or equal to $x$ ).

(A) $[0, \pi]$

(B) $[-1, 1]$

(D) $\{1, -1\}$

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14. If $f(x) = \begin{cases} \dfrac{8^x - 4^x - 2^x + 1}{x^2} & \text{; When } x > 0 \\ e^x \sin x + 4x + k \ln 4 & \text{; When } x \leq 0 \end{cases}$ is continuous at $x = 0$ then find $k$ :

(A) 2

(B) $\ln 4$

(D) $-\ln 2$

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15. Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function having $f(2) = 6, f'(2) = \left( \dfrac{1}{48} \right)$ . Then $\lim_{x \to 2} \int_{x}^{6} \dfrac{4t^3}{x - 2} \, dt$ equals:

(A) 24

(B) 18

(D) None of these

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16. 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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17. The straight line $x + 2y = 1$ meets the coordinate axes at $A$ and $B$ . A circle is drawn through $A$ , $B$ and the origin. Then the sum of perpendicular distances from $A$ and $B$ on the tangent to the circle at the origin is:

(A) $\frac{\sqrt{5}}{4}$

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

(D) $4\sqrt{5}$

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18. From the point $(-1, 2)$ tangents are drawn to parabola $y^2 = 4x$ , then the area of the triangle formed by the chord of contact and the tangent is:

(A) $2\sqrt{2}$

(B) $3\sqrt{2}$

(D) $8\sqrt{2}$

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19. The eccentricity of the ellipse with centre at the origin which meets the straight line $\frac{x}{7} + \frac{y}{2} = 1$ on the axis of $x$ and the straight line $\frac{x}{3} - \frac{y}{5} = 1$ on the axis of $y$ and whose axes lie along the axes of coordinates is:

(A) $\frac{2\sqrt{6}}{7}$

(B) $\frac{3\sqrt{6}}{7}$

(D) None of these

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20. Vector $\vec{a} + 3\vec{b}$ is perpendicular to $7\vec{a} - 5\vec{b}$ and $\vec{a} - 5\vec{b}$ is perpendicular to $7\vec{a} + 3\vec{b}$ . The angle between non zero vectors $\vec{a}$ & $\vec{b}$ is:

(A) $\frac{\pi}{2}$

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

(D) data Insufficient

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SECTION B: Numerical Value Type Questions

21. Number of ways in which AAABBB can be placed in the squares of figure as shown so that no row remains empty is ?

[Insert Your Image Here]

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22. If $\sin^2(10^\circ) \sin(20^\circ) \sin(40^\circ) \sin(50^\circ) \sin(70^\circ) = a - \frac{1}{16} \sin(10^\circ)$ , then $16 + a^{-1}$ is equal to…

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23. $\int \frac{x + \cos 2x + 1}{x \cos^2 x} \, dx = f(x) + K (\ln |x|) + C$ where $f\left(\frac{\pi}{4}\right) = 1$ , then $f(0) + 10K$ is equal to…
(where $C$ is constant of integration)

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24. The integer $n$ for which $\lim_{x \to 0} \frac{(\cos x - 1)(\cos x - e^x)}{x^n}$ is a finite non-zero number, is…

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25. The projection of the line segment joining the points $(1, -1, 3)$ and $(2, -4, 11)$ on the line joining the points $(-1, 2, 3)$ and $(3, -2, 10)$ is…

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