Maths

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

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is

For $k\in R,$ let the solutions of the equation $cos\left(si{n}^{-1}\left(xcot\left(ta{n}^{-1}\left(cos\left(si{n}^{-1}x\right)\right)\right)\right)\right)$ =, 0 < $\left|x\right|<\frac{1}{\sqrt[]{2}}$ be and , where the inverse trigonometric functions take only principal values. If the solutions of the equation x2 – bx – 5 = 0 are $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}and\frac{\alpha }{\beta }then\frac{b}{k^2}$ is equal to

Let x = $sin\left(2ta{n}^{-1}\alpha \right)$ and y = $sin\left(\frac{1}{2}ta{n}^{-1}\frac{4}{3}\right).$ If S = $\left\{\alpha \in R:y^2=1-x\right\},$ then ${\displaystyle \sum _{a\in S}^{}16{\alpha }^3}$ is equal to