Question

Which one of the following matrices can be obtained by performing elementary row transformations on the \(3\times3\) identity matrix?

Considering only the principal values of the inverse trigonometric functions, the value of $$\cot^{-1}(\cot(-11))+10\sin\left(2\cos^{-1}\left(\frac{1}{\sqrt2}\right)\right)+10\sin\left(2\tan^{-1}(2)\right)$$ is

Let \(P\) be the plane such that it contains the straight line $$\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$$ and is perpendicular to the plane $$x+2y+3z=4.$$ Let \(P_1\) be the plane which passes through the point \((4,2,2)\) and is parallel to \(P\). Then which of the following statements is (are) TRUE?

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f:\mathbb{R}\to\mathbb{R}\) be an arbitrary function and let \(g:\mathbb{R}\to\mathbb{R}\) be the function defined by $$g(x)=x f(x),\quad \forall x\in\mathbb{R}.$$ Then which of the following statements is (are) TRUE?

Consider the matrix $$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$ Let \(p,q,r,s,a,b,c,d\) be integers such that $$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}$$ and $$\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$ Then which of the following statements is (are) TRUE?