Let \(S=\{1,2,3,4\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to

Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,....\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,....\right\}$ of rectangles as follows:
${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1;$
${\mathrm{R}}_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_1;$
${\mathrm{E}}_{\mathrm{n}}:$ ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}_{\mathrm{n}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}_{\mathrm{n}}^2}=1$ of largest area inscribed in ${\mathrm{R}}_{\mathrm{n}-1},\mathrm{n}>1;$
${\mathrm{R}}_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_{\mathrm{n}},\mathrm{n}>1.$
Then which of the following options is/are correct?

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

For any two points \(M\) and \(N\) in the \(X Y\)-plane, let \(\overrightarrow{M N}\) denote the vector from \(M\) to \(N\), and \(\overrightarrow{0}\) denote the zero vector. Let \(P, Q\) and \(R\) be three distinct points in the \(X Y\)-plane. Let \(S\) be a point inside the triangle \(\triangle P Q R\) such that \(\overrightarrow{S P}+5 \overrightarrow{S Q}+6 \overrightarrow{S R}=\overrightarrow{0}\). Let \(E\) and \(F\) be the mid-points of the sides \(P R\) and \(Q R\), respectively. Then the value of \(\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}\) is ________

Let \(A_1, B_1, C_1\) be three points in the \(x y\)-plane. Suppose that the lines \(A_1 C_1\) and \(B_1 C_1\) are tangents to the curve \(y^2=8 x\) at \(A_1\) and \(B_1\), respectively. If \(O=(0,0)\) and \(C_1=(-4,0)\), then which of the following statements is (are) TRUE?

Four persons independently solve a certain problem correctly with probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4},\frac{1}{8}$. Then the probability that the problem is solved correctly by at least one of them is

If \(x^2-10 a x-11 b=0\) have roots \(c\) and \(d\) and \(x^2-10 c x-11 d=0\) have roots \(a\) and \(b\), then, find \(a+b+c+d\).

Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?

In the following carbocation, \(\mathrm{H} / \mathrm{CH}_3\) that is most likely to migrate to the positively charged carbon is

A metal rod of length \(L\) and mass \(m\) is pivoted at one end. A thin disc of mass \(M\) and radius \(R( < L)\) is attached at its centre to the free end of the rod. Consider two ways the disc is attached. Case A-the disc is not free to rotate about its centre and case \(B\)-the disc is free to rotate about its centre. The rod-disc system performs
SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is/are true?

All the compounds listed in Column I react with water. Match the result of the respective reactions with the appropriate options listed in Column II.

A series \(R-C\) combination is connected to an \(\mathrm{AC}\) voltage of angular frequency \(\omega=500 \mathrm{rad} / \mathrm{s}\). If the impedance of the \(R-C\) circuit is \(R \sqrt{1.25}\), the time constant (in millisecond) of the circuit is

Let \(\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}\) and \(\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}\) be three vectors, where \(\alpha, \beta \in \mathbb{R}-\{0\}\) and \(O\) denotes the origin. If \((\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0\) and the point \((\alpha, \beta, 2)\) lies on the plane \(3 x+3 y-z+l=0\), then the value of \(l\) is ____