Let $f:\left[0, 2\right]\to ℝ$ be the function defined by $f\left(x\right)=\left(3-\sin \left(2\pi x\right)\right)\sin \left(\pi x-\frac{\pi }{4}\right)-\sin \left(3\pi x+\frac{\pi }{4}\right)$. If $\alpha , \beta \in \left[0, 2\right]$ are such that $\left\{x\in \left[0, 2\right] :f\left(x\right)\ge 0\right\}=\left[\alpha , \beta \right]$, then the value of $\beta -\alpha$ is _____

Match the statements of Column I with these in Column II.
[Note : Here \(z\) takes values in the complex plane and \(\operatorname{Im}(z)\) and \(\operatorname{Re}(z)\) denote respectively, the imaginary part and the real part of \(z\) ]

Column I	Column II
(A)The set of points \(z\) satisfying \(|z-i| z||=|z+i| z||\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(|z+4|+|z-4|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(|w|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(|\operatorname{Im} z| \leq 1\)
(D) If \(|w|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(|\operatorname{Re} z| \leq 2\)
	(t) the set of points \(z\) satisfying \(|z| \leq 3\)

Let \(I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x\).
Then, for an arbitrary constant \(C\), the value of \(J-I\) equals

For the function \(f(x)=x \cos \frac{1}{x}, x \geq 1\),

Let $X$ and $Y$ be two events such that $P\left(X\right)=\frac{1}{3}$, $P\left(X|Y\right)=\frac{1}{2}$ and $P\left(Y|X\right)=\frac{2}{5}.$ Then

Paragraph:
Read the following passage and answer the questions.
Let \(A B C D\) be a square of side length 2 units. \(C_2\) is the circle through vertices \(A, B, C, D\) and \(C_1\) is the circle touching all the sides of square \(A B C D\). \(L\) is the line through \(A\).Question:
If \(P\) is a point on \(C_1\) and \(Q\) is a point on \(C_2\), then \(\frac{P A^2+P B^2+P C^2+P D^2}{Q A^2+Q B^2+Q C^2+Q D^2}\) is equal to

Paragraph:
Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Let \(g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t\). Which of the following is true ?

Consider a triangle $PQR$ having sides of lengths $p,q$ and $r$ opposite to the angles $P,Q$ and $R$, respectively. Then which of the following statements is (are) TRUE?

An electron in a hydrogen atom undergoes a transition from an orbit with a quantum number $n_i$ to another quantum number $n_f$ . $V_i \text{and} V_f$ are the initial and final potential energies of the electron. If $\frac{V_i}{V_f}=6.25$ then the smallest $n_f$ is

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

A particle of mass \(m\) is under the influence of the gravitational field of a body of mass \(M(\gg m)\). The particle is moving in a circular orbit of radius \(r_0\) with time period \(T_0\) around the mass \(M\). Then, the particle is subjected to an additional central force, corresponding to the potential energy \(V_{\mathrm{c}}(r)=m \alpha / r^3\), where \(\alpha\) is a positive constant of suitable dimensions and \(r\) is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius \(r_0\) in the combined gravitational potential due to \(M\) and \(V_{\mathrm{c}}(r)\), but with a new time period \(T_1\), then \(\left(T_1^2-T_0^2\right) / T_1^2\) is given by
[ \(G\) is the gravitational constant.]

Paragraph : If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
The above derivation makes the assumption that \(\Delta r / x \ll 1, \Delta y / y < 1\). Therefore, the higher powers of these quantities are neglected.
Question : Consider the ratio \(r=\frac{(1-a)}{(1+a)}\) to be determined by measuring a dimensionless quantity a If the error in the measurement of a is \(\Delta a(\frac{\Delta a}{a}<<1)\), then what is the error \(\Delta r\) in determining \(r\) ?

A particle of mass $1 \mathrm{kg}$ is subjected to a force which depends on the position as $\overrightarrow{F}=-k\left(x\hat{i}+y\hat{j}\right) \mathrm{kg} \mathrm{m} {\mathrm{s}}^{-2}$ with $k=1 \mathrm{kg} {\mathrm{s}}^{-2}$. At time $t=0$, the particle's position $\overrightarrow{r}=\left(\frac{1}{\sqrt{2}}\hat{i}+\sqrt{2}\hat{j}\right) \mathrm{m}$ and its velocity $\overrightarrow{v}=\left(-\sqrt{2}\hat{i}+\sqrt{2}\hat{j}+\frac{2}{\pi }\hat{k}\right) \mathrm{m} {\mathrm{s}}^{-1}$. Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z=0.5 \mathrm{m}$, the value of $\left(x{v}_y-y{v}_x\right)$ is ______ ${\mathrm{m}}^2 {\mathrm{s}}^{-1}$.