Class 12th

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

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

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

In the following reaction

The % yield for reaction I is 60% and that of reaction II is 50%. The overall yield of the complete reaction is___________% [Nearest integer]

Optical activity of an enantiomeric mixture is + 12.6° and the specific rotation of (+) isomer is +30°. The optical purity is_____________%

In the titration of KMnO₄ and oxalic acid in acidic medium, the change in oxidation number of carbon at the end point is____________.

The conductivity of a solution of complex with formula CoCl₃(NH₃)₄ corresponds to 1 : 1 electrolyte, then the primary valency of central metal ion is__________________.