For any square matrix \(A ;\left(A-A^{\prime}\right)\) is :

Let \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\) and \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\). If \(A^T+A=I\), then:

If \(z\)-coordinate of a point P on the line joining the points A (2, 2, 1) and B (5, 1, -2) is -1, than \(x\)-coordinate of point P is

If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is/are possible?
(A) A = B
(B) P(A) = P(B)
(C) \(A \subset B\) but \(A \neq B\)
(D) \(A \cap B=\phi\)
Choose the correct answer from the options given below:

Let the equations of lines be \(L_1: \vec{r}_1=\vec{a}_1+\lambda \vec{b}_1\) and \(L_2: \vec{r}_2=\vec{a}_2+\lambda \vec{b}_2\)
Then the shortest distance between \(L_1\) and \(L_2\) is

The number of vectors of unit length perpendicular to both the vectors \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} + 3\hat{k}\) is:

for \(|x|<1, \sin \left(\tan ^{-1} x\right)\) equal to:

What will be the number of genotypes and phenotypes obtained in \(F _2\) generation when male parent RRyy is crossed with female parent rrYY?

At room temperature (\(25.0^\circ C\)) the resistance of a heating element is \(100 \Omega\). The temperature of the element at which resistance becomes \(125.5 \Omega\) is : (Given that the temperature coefficient of the material of the resistor is \(1.70 \times 10^{-4} \text{ } ^\circ C^{-1}\))

A metallic wire of resistance \( R \) and resistivity \( \rho \) is cut into six equal parts. Now each part is stretched to six times its length. The new resistance and resistivity of each part will be :

Given relation \(R =\{(x, y): y=x+5, x<4, x, y \in N\}\). Where N is a set of natural numbers then :

Read the passage carefully and answer the questions
The coordination compounds are of great importance. These compounds are widely present in the mineral, plants and animal world and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine. For example, coordination compounds find use in many qualitative and quantitative chemical analysis. Some important metal extraction and purification make use of complex formation. Hardness of water can be estimated by stable complex formation of Ca\(^{2+}\) and Mg\(^{2+}\)
Wilkinson catalyst used for hydrogenation of alkenes is a complex of ___________.

The solution of LPP :
Min. \(z=100 x+400 y\)
s.t. \(x+2 y \geq 8\)
\(2 x+5 y \leq 20\)
\(x \geq 0, y \geq 0\)
occurs at :