Maths

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

Let y = y(x) be the solution curve of the differential equation $sin\left(2x^2\right)lo{g}_e\left(tan{x}^2\right)dy+\left(4xy-4\sqrt[]{2}xsin\left(x^2-\frac{\pi }{4}\right)\right)dx=0,$ $0<x<\sqrt[]{\frac{\pi }{2},}$ which passes through the point $\left(\sqrt[]{\frac{\pi }{6},1}\right)$ . Then $\left|y\left(\sqrt[]{\frac{\pi }{3}}\right)\right|$ is equal to

An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1.$ Let the major and minor axes of the ellipse E coincide with transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be $\frac{1}{2}.$ If $\mathcal{l}$ is the length of the latus rectum of the ellipse E, then the value of 113 $\mathcal{l}$ is equal to

Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and 4ax – y + 5z – 7a = 0 = 2x – 5y – z – 3, a $\in R$ at the point $P\left(\alpha ,\beta \gamma \right).$ Then the value of + + $\gamma$ equals