Let $\alpha$ and $\beta$ be the roots of the equation $px^2 + qx - r = 0$, where $p \neq 0$. If $p, q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}$, then the value of $\alpha - \beta^2$ is: (1) $\frac{80}{9}$ (2) 9 (3) $\frac{20}{3}$ (4) 8
If $z$ is a complex number such that $|z| \leq 1$, then the minimum value of $\left|z + \frac{1}{2}(3 + 4i)\right|$ is: (1) 2 (2) $\frac{5}{2}$ (3) $\frac{3}{2}$ (4) 3
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15} - S_5$ is equal to: (1) 800 (2) 890 (3) 790 (4) 690
Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is: (1) $\frac{4}{9}$ (2) $\frac{1}{9}$ (3) $\frac{1}{4}$ (4) $\frac{1}{4}$
Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is: (1) $\frac{1}{\sqrt{2}}$ (2) $\sqrt{2}$ (3) $\frac{1}{2}$ (4) 1
Let $P$ be a point on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2 + y^2 = 9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ = 4:3$ as $P$ moves on the ellipse, is: (1) $\frac{11}{19}$ (2) $\frac{13}{21}$ (3) $\frac{\sqrt{139}}{23}$ (4) $\frac{\sqrt{13}}{7}$
Let $f(x) = \begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases}$ $x \in \mathbb{N}$. If for some $a \in \mathbb{N}$, $f(f(f(a))) = 21$, then $\lim_{x \to a^-} \left(\frac{x^3}{a} - \left\lfloor\frac{x}{a}\right\rfloor\right)$, where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$, is equal to: (1) 121 (2) 144 (3) 169 (4) 225
Consider 10 observations $x_1, x_2, \ldots, x_{10}$, such that $\sum_{i=1}^{10} (x_i - \alpha) = 2$ and $\sum_{i=1}^{10} (x_i - \beta)^2 = 40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to: (1) 2 (2) $\frac{3}{2}$ (3) $\frac{5}{2}$ (4) 1
Let the system of equations $x + 2y + 3z = 5$, $2x + 3y + z = 9$, $4x + 3y + \lambda z = \mu$ have infinite number of solutions. Then $\lambda + 2\mu$ is equal to: (1) 28 (2) 17 (3) 22 (4) 15
If the domain of the function $f(x) = \frac{\sqrt{x^2 - 25}}{4 - x^2} + \log_{10}(x^2 + 2x - 15)$ is $(-\infty, \alpha) \cup (\beta, \infty)$, then $\alpha^2 + \beta^3$ is equal to: (1) 140 (2) 175 (3) 150 (4) 125
Let $f(x) = |2x^2 + 5x - 3|$, $x \in \mathbb{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to: (1) 5 (2) 2 (3) 0 (4) 3
If $\int_0^{\frac{\pi}{3}} \cos^4 x \, dx = a\pi + b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a + 8b$ is equal to: (1) 2 (2) 1 (3) 3 (4) $\frac{3}{2}$
Let $\alpha$ be a non-zero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$ and $\lim_{x \to -\infty} f(x) = 1$. If $f'(x) = \alpha f(x) + 3$, for all $x \in \mathbb{R}$, then $f(-\log_e 2)$ is equal to: (1) 1 (2) 5 (3) 9 (4) 7
Consider a $\triangle ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is: (1) $\frac{37}{2\sqrt{38}}$ (2) $\frac{\sqrt{38}}{2}$ (3) $\frac{39}{2\sqrt{38}}$ (4) $\sqrt{19}$
If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14\alpha + \beta + \gamma$ is: (1) 102 (2) 138 (3) 108 (4) 132
Let $P$ and $Q$ be the points on the line $\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$ which are at a distance of 6 units from the point $R(1,2,3)$. If the centroid of the triangle $PQR$ is $(\alpha, \beta, \gamma)$, then $\alpha^2 + \beta^2 + \gamma^2$ is: (1) 26 (2) 36 (3) 18 (4) 24
Let Ajay will not appear in JEE exam with probability $p = \frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q = \frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is: (1) $\frac{9}{35}$ (2) $\frac{18}{35}$ (3) $\frac{24}{35}$ (4) $\frac{3}{35}$
The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n = 1, 2, 3, \ldots, 10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:
If three successive terms of a G.P. with common ratio $r$ ($r > 1$) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r] + [-r]$ is equal to: