A ball is projected from the ground with a speed $15 \mathrm{~m}\mathrm{~s}^{-1}$ at an angle $\theta$ with horizontal so that its range and maximum height are equal, then $\tan\theta$ will be equal to (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) 2 (4) 4
A particle is moving in a straight line such that its velocity is increasing at $5 \mathrm{~m}\mathrm{~s}^{-1}$ per meter. The acceleration of the particle is $\_\_\_\_$ $\mathrm{m}\mathrm{~s}^{-2}$ at a point where its velocity is $20 \mathrm{~m}\mathrm{~s}^{-1}$.
Three identical spheres each of mass $M$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3 m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt{x}$ m. The value of $x$ is
For $z \in \mathbb{C}$ if the minimum value of $(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is $\_\_\_\_$. (1) 3 (2) $\frac{7}{2}$ (3) 4 (4) $\frac{9}{2}$
The value of $2\sin\frac{\pi}{22}\sin\frac{3\pi}{22}\sin\frac{5\pi}{22}\sin\frac{7\pi}{22}\sin\frac{9\pi}{22}$ is equal to: (1) $\frac{1}{16}$ (2) $\frac{5}{16}$ (3) $\frac{7}{16}$ (4) $\frac{3}{16}$
Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L_1: 3x - 4y + 12 = 0$, and $L_2: 8x + 6y + 11 = 0$. If $P$ lies below $L_1$ and above $L_2$, then $100(\alpha + \beta)$ is equal to (1) $-14$ (2) 42 (3) $-22$ (4) 14
The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2 - 2x - 2y = 1$ meet at the point $P$. Then the area (in unit$^2$) of the triangle $PAB$ is: (1) 4 (2) 6 (3) 7 (4) 8
If the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ meets the line $\frac{x}{7} + \frac{y}{2\sqrt{6}} = 1$ on the $x$-axis and the line $\frac{x}{7} - \frac{y}{2\sqrt{6}} = 1$ on the $y$-axis, then the eccentricity of the ellipse is (1) $\frac{5}{7}$ (2) $\frac{2\sqrt{6}}{7}$ (3) $\frac{3}{7}$ (4) $\frac{2\sqrt{5}}{7}$
Let the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{7} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{\alpha} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is: (1) $\frac{32}{9}$ (2) $\frac{18}{5}$ (3) $\frac{27}{4}$ (4) $\frac{27}{10}$
If the mean deviation about median for the number $3, 5, 7, 2k, 12, 16, 21, 24$ arranged in the ascending order, is 6 then the median is (1) 11.5 (2) 10.5 (3) 12 (4) 11
The number of real values of $\lambda$, such that the system of linear equations $2x - 3y + 5z = 9$ $x + 3y - z = -18$ $3x - y + (\lambda^2 - |\lambda|)z = 16$ has no solutions, is (1) 0 (2) 1 (3) 2 (4) 4
Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101} \left([\sin(\pi x)] + e^{[\cos(2\pi x)]}\right) dx$ is equal to (1) $\frac{52(1-e)}{e}$ (2) $\frac{52}{e}$ (3) $\frac{52(2+e)}{e}$ (4) $\frac{104}{e}$
Let a smooth curve $y = f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$. If the curve passes through the points $(1, 2)$ and $(8, 1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to (1) $2\log_e 2$ (2) 4 (3) 1 (4) $4\log_e 2$
Let $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$ and $\vec{a} \cdot \vec{b} = 3$. Then the projection of $\vec{b}$ on the vector $\vec{a} - \vec{b}$ is: (1) $\frac{2}{\sqrt{21}}$ (2) $2\sqrt{\frac{3}{7}}$ (3) $\frac{2}{3}\sqrt{\frac{7}{3}}$ (4) $\frac{2}{3}$
A plane $E$ is perpendicular to the two planes $2x - 2y + z = 0$ and $x - y + 2z = 4$, and passes through the point $P(1, -1, 1)$. If the distance of the plane $E$ from the point $Q(a, a, 2)$ is $3\sqrt{2}$, then $(PQ)^2$ is equal to (1) 9 (2) 12 (3) 21 (4) 33