Maths

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $ta{n}^{-1}\frac{3}{4}.$  Water is poured in it a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is……….

  $\frac{2^3-1^3}{1*7}+\frac{4^3-3+2^3-1^3}{2*11}+\frac{6^3-5^3+4^3+2^3-1^3}{3*15}+.......+$

$\frac{30^3-29^3+28^3-27^3+....+2^3-1^3}{15*63}isequalto.............$

Let for the 9^th term in the binomial expansion of ${\left(3+6x\right)}^n$ , in the increasing powers of 6x, to be the greatest for x $=\frac{3}{2}$ , the least value of n is n₀. If k is the ratio of the coefficient of x⁶ to the coefficient of x³, then k + n₀ is equal to:

The number of functions f, from the set A = $\left\{x\in N:x^2-10x+9\le 0\right\}$  to the set B = $\left\{n^2:n\in N\right\}$  such that f(x) $\le$ (x – 3)² + 1, for every x $\in$  A, is…………

Consider a matrix A = $\left[\begin{array}{ccc}\hfill \alpha \hfill & \hfill \beta \hfill & \hfill \gamma \hfill \\ \hfill {\alpha }^2\hfill & \hfill {\beta }^2\hfill & \hfill {\gamma }^2\hfill \\ \hfill \beta +\gamma \hfill & \hfill \gamma +\alpha \hfill & \hfill \alpha +\beta \hfill \end{array}\right]$ $\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill {}^{}\hfill & \hfill {}^{}\hfill & \hfill {}^{}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}$ , where , , $\gamma$  $$ are three distinct natural numbers.

If $\frac{det\left(adj\left(adj\left(adj\left(adjA\right)\right)\right)\right)}{{\left(\alpha -\beta \right)}^{16}{\left(\beta -\gamma \right)}^{16}{\left(\gamma -\alpha \right)}^{16}}=2^{32}*3^{16},$ $\frac{}{{}^{}{}^{}{}^{}}{}^{}{}^{}$ then the number of such 3 – tuples (, , $\gamma$ $$ ) is…………

Let S $=\left\{z\in C:z^2+\overline{Z}=0\right\}.$ Then ${\displaystyle \sum _{z\in R}^{}\left(Re\left(z\right)+Im\left(z\right)\right)}$ is equal to

If the length of the latus rectum of the ellipse x² + 4y² + 2x + 3y $-\lambda =0$ is 4, and $\mathcal{l}$ is the length of the its major axis, then $\lambda +\mathcal{l}$ is equal to

Let S be the set containing all 3 * 3 matrices with entries from $\left\{-1,0,1\right\}$ . The total number of matrices $A\in S$ such that the sum of all the diagonal elements of A^TA is 6 is

Let f(x) = 2x² – x – 1 and $S=\left\{n\in Z:\left|f\left(n\right)\right|<800\right\}.$ Then, the value of ${\displaystyle \sum _{n\in S}^{}f\left(n\right)}$ is equal to

Let M and N be the number of points on the curve y⁵ – 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals