Let $n\ge 2$ be a natural number and $f:\left[0, 1\right]\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)=\left\{\begin{array}{ccc}n\left(1-2nx\right)& \mathrm{if}& 0\le x\le \frac{1}{2n}\\ 2n\left(2nx-1\right)& \mathrm{if}& \frac{1}{2n}\le x\le \frac{3}{4n}\\ 4n\left(1-nx\right)& \mathrm{if}& \frac{3}{4n}\le x\le \frac{1}{n}\\ \frac{n}{n-1}\left(nx-1\right)& \mathrm{if}& \frac{1}{n}\le x\le 1\end{array}\right.$

If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f\left(x\right)$ is $4$, then the maximum value of the function $f$ is

Let \(a, b, c, p, q\) be real numbers.
Suppose, \(\alpha, \beta\) are the roots of the equation \(x^2+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^2+2 b x+c=0\), where \(\beta^2 \notin\{-1,0,1\}\).
Statement \(1\left(p^2-q\right)\left(b^2-a c\right) \geq 0\).
Statement \(2 b \neq p a\) or \(c \neq q a\).

Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\cdots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
\(P:\left|z_2-z_1\right|+\left|z_3-z_2\right|+\cdots+\left|z_{10}-z_9\right|\) \(+\left|z_1-z_{10}\right| \leq 2 \pi \)
\( Q:\left|z_2^2-z_1^2\right|+\left|z_3^2-z_2^2\right|+\cdots+\left|z_{10}^2-z_9^2\right|\) \(+\left|z_1^2-z_{10}^2\right| \leq 4 \pi\)

For a non-zero complex number $z$, let arg$\left(z\right)$ denote the principal argument with $-\pi <\mathrm{a}\mathrm{r}\mathrm{g}\left(z\right)\le \pi$ then, which of the following statement(s) is (are) FALSE?

Let $\mathrm{\Delta }\mathrm{P}\mathrm{Q}\mathrm{R}$ be a triangle. Let $\overrightarrow{\mathrm{a}}=\mathrm{ }\overrightarrow{\mathrm{Q}\mathrm{R}},\mathrm{ }\mathrm{ }\overrightarrow{\mathrm{b}}=\mathrm{ }\overrightarrow{\mathrm{R}\mathrm{P}}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\overrightarrow{\mathrm{c}}=\mathrm{ }\overrightarrow{\mathrm{P}\mathrm{Q}}.$ If $\left|\overrightarrow{\mathrm{a}}\right|=12,\mathrm{ }\left|\overrightarrow{\mathrm{b}}\right|=4\sqrt{3}$ and $\overrightarrow{\mathrm{b}}\mathrm{ }\cdot \mathrm{ }\overrightarrow{\mathrm{c}}=24,$ then which of the following is (are) true ?

The smallest value of \(k\), for which both the roots of the equation \(x^2-8 k x+16\left(k^2-k+1\right)=0\) are real, distinct and have values atleast 4 , is

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be functions defined by
\(f(x)=\left\{\begin{array}{ll}x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0,\end{array} \quad\right.\) and \(g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2} \\ 0, & \text { otherwise }\end{cases}\)
Let \(a, b, c, d \in \mathbb{R}\). Define the function \(h: \mathbb{R} \rightarrow \mathbb{R}\) by
\(h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R}\)
Match each entry in List-I to the correct entry in List-II.

The correct option is :

Consider the planes \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\).
Statement I The parametric equations of the line of intersection of the given planes are \(x=3+14 t, y=1+2 t\) and \(z=15 t\).
Statement II The vectors \(14 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+15 \hat{\mathbf{k}}\) is parallel to the line of intersection of the given planes.

A solid sphere is in pure rolling motion on an inclined surface having inclination \(\theta\).

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency \(v_1\) and ejects the electron with a kinetic energy of 10 eV . The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency \(v_2\). The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV . It is given that positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV ) is \(\qquad\)

Let two non-collinear unit vectors \(\mathbf{a}\) and \(\hat{\mathbf{b}}\) form an acute angle.
A point \(P\) moves so that at any time \(t\) the position vector \(\mathbf{O P}\) (where, \(O\) is the origin) is given by \(\mathbf{a} \cos t+\hat{\mathbf{b}} \sin t\). When \(P\) is farthest from origin \(O\), let \(M\) be the length of \(\mathbf{O P}\) and \(\hat{\mathbf{u}}\) be the unit vector along \(\mathbf{O P}\). Then,

Match the following.
 

	List – I		List – II
(A)	The number of polynomials f(x) with non – negative integer coefficients of degree $\le 2,$ satisfying f(0) =0 and $\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1, is$	(P)	8
(B)	The number of point in the interval $\left[- \sqrt{13}, \sqrt{13}\right]$ at which $f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is	(Q)	2
(C)	$\underset{-2}{\overset{2}{\int }}\frac{3x^2}{\left(1+e^x\right)}dx$ equals	(R)	4
(D)	$\frac{\left(\underset{-\frac{1}{2}}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{\frac{1}{2}}{\int }}\cos 2x .\log \left(\frac{1+x}{1-x}\right)dx\right)}$	(S)	0

Assume that the nuclear binding energy per nucleon \((B / A)\) versus mass number \((A)\) is as shown in the figure. Use this plot to choose the correct choice (s) given below.