Consider the functions $f,g:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=x^2+\frac{5}{12}$ and $g\left(x\right)=\left\{\begin{array}{cc}2\left(1-\frac{4\left|x\right|}{3}\right),& \left|x\right|\le \frac{3}{4},\\ 0,& \left|x\right|>\frac{3}{4}\end{array}\right.$
If $\alpha$ is the area of the region $\left\{\left(x,y\right)\in \mathrm{ℝ}\times \mathrm{ℝ}:\left|x\right|\le \frac{3}{4},0\le y\le \min \left\{f\left(x\right),g\left(x\right)\right\}\right\}$, then the value of $9\alpha$ is _____ .

In a triangle $PQR$ , let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$ . If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}.\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}.\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$, then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is _______

Paragraph:
Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The function \(f^{\prime}(x)\) is

Let the function $f:\left[0,1\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)=\frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$  is_________  

Let \(f\) and \(g\) be real valued functions defined on interval \((-1,1)\) such that \(g^{\prime \prime}(x)\) is continuous, \(g(0) \neq 0, g^{\prime}(0)=0, g^{\prime \prime}(0) \neq 0\) and \(f(x)=g(x) \sin x\).
Statement \(1 \lim _{x \rightarrow 0}[g(x) \cos x-g(0) \operatorname{cosec} x]=f^{\prime \prime}(0)\).
Statement \(2 f^{\prime}(0)=g(0)\).

\(\frac{d^2 x}{d y^2}\) equals

Equation of the plane containing the straight line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the straight lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.

For a solution formed by mixing liquids $\text{L}$ and $\text{M}$, the vapour pressure of $\text{L}$ plotted against the mole fraction of $\text{M}$ in solution is shown in the following figure, Here, ${\mathrm{X}}_{\mathrm{L}}$ and ${\mathrm{X}}_{\mathrm{M}}$ represent mole fractions of $\text{L}$ and $\text{M}$, respectively, in the solution. The correct statement(s) applicable to this system is(are) -

Statement 1 The plot of atomic number ( \(Y\)-axis) versus number of neutrons ( \(x\)-axis) for stable nuclei shows a curvature towards \(X\)-axis from the line of \(45^{\circ}\) slope as the atomic number is increased.
Statement 2 Proton-proton electrostatic repulsions begin to overcome attractive forces involving protons and neutrons in heavier nuclides.

Paragraph I: In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of \(0.8 \mathrm{~mm}\). The distance between the slits at time \(t\) is given by \(d=(0.8+0.04 \sin \omega t) \mathrm{mm}\), where \(\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}\). The distance of the screen from the slits is \(1 \mathrm{~m}\) and the wavelength of the light used to illuminate the slits is \(6000 Ã…\). The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point \(O\).

The \(8^{\text {th }}\) bright fringe above the point \(\mathrm{O}\) oscillates with time between two extreme positions. The separation between these two extreme positions, in micrometer \((\mu \mathrm{m})\), is ___________ .

Paragraph:
Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The real number \(s\) lies in the interval

Let $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be three non - coplanar unit vectors such that the angle between every pair of them is $\frac{\pi }{3}.$ If $\overrightarrow{a} \times \overrightarrow{b}+ \overrightarrow{b} \times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c}$ , where p, q and r are scalars, then the value of $\frac{p^2+2q^2+r^2}{q^2}$ is _______