A body of mass $m$ is projected with a speed $u$ making an angle of $45^\circ$ with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as $\dfrac{\sqrt{Z}\, m u^3}{X g}$. The value of $X$ is $\_\_\_\_$.
The time period of simple harmonic motion of mass $M$ in the given figure is $\pi\sqrt{\dfrac{\alpha M}{5K}}$, where the value of $\alpha$ is $\_\_\_\_$.
Let $z_1$ and $z_2$ be two complex number such that $z_1 + z_2 = 5$ and $z_1^3 + z_2^3 = 20 + 15i$. Then $z_1^4 + z_2^4$ equals- (1) $30\sqrt{3}$ (2) 75 (3) $15\sqrt{15}$ (4) $25\sqrt{3}$
The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is (1) 406 (2) 130 (3) 142 (4) 136
Let $2^{\text{nd}}$, $8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{st}}$, $2^{\text{nd}}$ and $3^{\text{rd}}$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to- (1) 980 (2) 960 (3) 990 (4) 970
If for some $m, n$; ${}^{6}C_m + 2\,{}^{6}C_{m+1} + {}^{6}C_{m+2} > {}^{8}C_3$ and ${}^{n-1}P_3 : {}^{n}P_4 = 1 : 8$, then ${}^{n}P_{m+1} + {}^{n+1}C_m$ is equal to (1) 380 (2) 376 (3) 384 (4) 372
Let $A(a, b)$, $B(3, 4)$ and $(-6, -8)$ respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point $P(2a+3, 7b+5)$ from the line $2x + 3y - 4 = 0$ measured parallel to the line $x - 2y - 1 = 0$ is (1) $\dfrac{15\sqrt{5}}{7}$ (2) $\dfrac{17\sqrt{5}}{6}$ (3) $\dfrac{17\sqrt{5}}{7}$ (4) $\dfrac{\sqrt{5}}{17}$
Let a variable line passing through the centre of the circle $x^2 + y^2 - 16x - 4y = 0$, meet the positive coordinate axes at the point $A$ and $B$. Then the minimum value of $OA + OB$, where $O$ is the origin, is equal to (1) 12 (2) 18 (3) 20 (4) 24
Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2x + y = 6$. Let an ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, $a > b$ of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then the square of the length of the latus rectum of $E$, is (1) $\dfrac{385}{8}$ (2) $\dfrac{347}{8}$ (3) $\dfrac{512}{25}$ (4) $\dfrac{656}{25}$
Let $f : \mathbb{R} \rightarrow (0, \infty)$ be strictly increasing function such that $\lim_{x \rightarrow \infty} \dfrac{f(7x)}{f(x)} = 1$. Then, the value of $\lim_{x \rightarrow \infty} \left(\dfrac{f(5x)}{f(x)} - 1\right)$ is equal to (1) 4 (2) 0 (3) $\dfrac{7}{5}$ (4) 1
Let the mean and the variance of 6 observations $a, b, 68, 44, 48, 60$ be 55 and 194, respectively. If $a > b$, then $a + 3b$ is (1) 200 (2) 190 (3) 180 (4) 210
Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}2\\0\\0\end{pmatrix}$, $A\begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}4\\0\\0\end{pmatrix}$, $A\begin{pmatrix}1\\1\\1\end{pmatrix} = \begin{pmatrix}2\\1\\1\end{pmatrix}$. Then, the system $(A - 3I)\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$ has (1) unique solution (2) exactly two solutions (3) no solution (4) infinitely many solutions
If the function $f : (-\infty, -1] \rightarrow [a, b]$ defined by $f(x) = e^{x^3 - 3x + 1}$ is one-one and onto, then the distance of the point $P(2b+4, a+2)$ from the line $x + e^{-3}y = 4$ is: