No Author Solutions for Exercise 6: EXAM GALLERY

The value of ${\cos }^4\frac{\mathrm{\pi }}{8}+{\cos }^4\frac{3\mathrm{\pi }}{8}+{\cos }^4\frac{5\mathrm{\pi }}{8}+{\cos }^4\frac{7\mathrm{\pi }}{8}$ is

The value of $\sin \left(\frac{\mathrm{\pi }}{14}\right)\sin \left(\frac{3\mathrm{\pi }}{14}\right)\sin \left(\frac{5\mathrm{\pi }}{14}\right)$ is

Equation of a common tangent (if it exists) to $x^2+y^2=2$ and $x^3{y}^3=1,$ is

Let $f:\left(-\infty ,\infty \right)\to \left(-\infty ,\infty \right)$ be defined by $f\left(x\right)=\left\{\begin{array}{cc}{x}^3,& x>0\\ 0,& x\le 0\end{array}\right.$ and let $f^{\text{'}}\left(x\right)$ denotes the derivative of $f\left(x\right)$.
Statement $\mathrm{I}$ : $\underset{x\to 0}{\lim }{f}^{\text{'}}\left(x\right)=f^{\text{'}}\left(0\right)=0$
Statement $\mathrm{II}$ : $f^{\text{'}}\left(x\right)$ exists, for each $x\in \left(-\infty ,\infty \right)$ and $f^{\text{'}}\left(x\right)$ is continuous at $x=0$.

Let $f:[11,\infty )\to \left(0,\infty \right)$ be given by $f\left(x\right)=\coprod _{l=1}^{10}\frac{1}{x-l}$, where, for real numbers $a_1, a_2 ,\dots , a_n,$ $\underset{l=1}{\overset{n}{\cup }}$ denotes the product $a_1\times a_2\times \dots \times a_n$
Statement I: $\int f\left(x\right)dx=\sum _{l=1}^{10}\frac{{\left(-1\right)}^l\ln |x-l|}{\left(l-1\right)!\left(10-l\right)!}$
Statement II: For $x\in [11, \infty ),$ $f\left(x\right)=\sum _{l=1}^{10}\frac{A_l}{x-l}$, where, $A_l=\coprod _{\begin{array}{c}j=1\\ j\ne l\end{array}}^{10}\frac{1}{\left(l-j\right)}, l=1, 2, \dots , 10$

The directrix of a parabola is $x+y=2$ and the focus is $\left(\sqrt{2}+1, \sqrt{2}+1\right).$
Statement I: Length of the latusrectum is $4$ because
Statement II: The distance of directrix from focus is $3.$

An urn contains four balls bearing numbers $1, 2, 3$ and $123$, respectively. A ball is drawn at random from the urn. Let $E_i, i=1, 2, 3,$ denote the event that digit $i$ appears on the ball drawn.
Statement I : $P\left(E_1\cap E_2\right)=P\left(E_1\cap E_3\right)=P\left(E_2\cap E_3\right)=\frac{1}{4}$
Statement II : $P\left(E_1\right)=P\left(E_2\right)=P\left(E_3\right)=\frac{1}{2}$

Let $f:\left(-\infty ,\infty \right)\to \left(-\infty ,\infty \right)$ be such that $f\left(x\right)$ is continuous at $x=0$ and let $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}$, for every $x, y\in \left(-\infty ,\infty \right)$.

Statement $\mathrm{I}$ : $f\left(x\right)$ is continuous at every point on $\left(-\infty ,\infty \right)$.

Statement $\mathrm{II}$: $2 f\left(x+h\right)+f\left(0\right)=2 f\left(x\right)+f\left(2h\right)$ for every $x, h\in \left(-\infty ,\infty \right)$.