If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is: (1) $\frac{8}{5}$ (2) $\frac{4}{3}$ (3) $1$ (4) $\frac{7}{4}$
If the sum of the first ten terms of the series $\left(1\frac{3}{5}\right)^{2}+\left(2\frac{2}{5}\right)^{2}+\left(3\frac{1}{5}\right)^{2}+4^{2}+\left(4\frac{4}{5}\right)^{2}+\ldots$, is $\frac{16}{5} m$, then $m$ is equal to: (1) 102 (2) 101 (3) 100 (4) 99
The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for: (1) infinitely many values of $\lambda$ (2) exactly one value of $\lambda$ (3) exactly two values of $\lambda$ (4) exactly three values of $\lambda$
The number of distinct real roots of the equation $\tan^{2}x - \sec^{10}x + 1 = 0$ in the interval $\left(0, \frac{\pi}{3}\right)$ is: (1) 0 (2) 1 (3) 2 (4) 3
A value of $\theta$ for which $\frac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary, is: (1) $\frac{\pi}{3}$ (2) $\frac{\pi}{6}$ (3) $\sin^{-1}\left(\frac{\sqrt{3}}{4}\right)$ (4) $\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$
A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $= x$ units and a circle of radius $= r$ units. If the sum of the areas of the square and the circle so formed is minimum, then: (1) $2x = (\pi + 4)r$ (2) $(4-\pi)x = \pi r$ (3) $x = 2r$ (4) $2x = r$
A straight line through origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio: (1) $2:3$ (2) $1:2$ (3) $4:1$ (4) $3:4$
Let $P$ be the point on the parabola, $y^2 = 8x$ which is at a minimum distance from the centre $C$ of the circle, $x^2 + (y+6)^2 = 1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is: (1) $x^2 + y^2 - 4x + 8y + 12 = 0$ (2) $x^2 + y^2 - x + 4y - 12 = 0$ (3) $x^2 + y^2 - \frac{x}{4} + 2y - 24 = 0$ (4) $x^2 + y^2 - 4x + 9y + 18 = 0$
The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is: (1) $\frac{4}{3}$ (2) $\frac{4}{\sqrt{3}}$ (3) $\frac{2}{\sqrt{3}}$ (4) $\sqrt{3}$
The area (in sq. units) of the region $\{(x,y): y^2 \geq 2x$ and $x^2 + y^2 \leq 4x, x \geq 0, y \geq 0\}$ is: (1) $\pi - \frac{4\sqrt{2}}{3}$ (2) $\pi - \frac{8}{3}$ (3) $\pi - \frac{4}{3}$ (4) $\frac{\pi}{2} - \frac{2\sqrt{2}}{3}$
If a curve $y = f(x)$ passes through the point $(1,-1)$ and satisfies the differential equation, $y(1+xy)dx = x\,dy$, then $f\left(-\frac{1}{2}\right)$ is equal to: (1) $-\frac{2}{5}$ (2) $-\frac{4}{5}$ (3) $\frac{2}{5}$ (4) $\frac{4}{5}$
If the standard deviation of the numbers $2, 3, a$ and $11$ is $3.5$, then which of the following is true? (1) $3a^2 - 26a + 55 = 0$ (2) $3a^2 - 32a + 84 = 0$ (3) $3a^2 - 34a + 91 = 0$ (4) $3a^2 - 23a + 44 = 0$
Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of dice $A$ and $B$ is odd, then which of the following statements is NOT true? (1) $E_1$ and $E_3$ are independent. (2) $E_1$, $E_2$ and $E_3$ are independent. (3) $E_1$ and $E_2$ are independent. (4) $E_2$ and $E_3$ are independent.
If $0 \leq x < 2\pi$, then the number of real values of $x$, which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is: (1) 3 (2) 5 (3) 7 (4) 9
Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b}+\vec{c})$. If $\vec{b}$ is not parallel to $\vec{c}$, then the angle between $\vec{a}$ and $\vec{b}$ is: (1) $\frac{3\pi}{4}$ (2) $\frac{\pi}{2}$ (3) $\frac{2\pi}{3}$ (4) $\frac{5\pi}{6}$
If the line $\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$ lies in the plane $lx + my - z = 9$, then $l^2 + m^2$ is equal to: (1) 18 (2) 5 (3) 2 (4) 26
The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is: (1) $3\sqrt{10}$ (2) $10\sqrt{3}$ (3) $\frac{10}{\sqrt{3}}$ (4) $\frac{20}{3}$
If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is: (1) 46th (2) 59th (3) 52nd (4) 58th