Question

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

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let \(E_1\) be the event that the ball chosen belonged to Box I and let \(E_2\) be the event that the ball chosen belonged to Box II. Let \(F_1\) be the event that the ball chosen is red and let \(F_2\) be the event that the ball chosen is green. Then which of the following statements is (are) TRUE?

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?