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

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$ If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

Consider the hyperbola $H :x^2-y^2=1$ and a circle S with center $N\left(x_2, 0\right).$ Suppose that H and S touch each other at point $P\left(x_1, y_1\right)$ with $x_1>1 and y_1>0.$ The common tangent to H and S at P intersects the x - axis at point M. If $(l, m)$ is the centroid of the triangle $\mathrm{\Delta }PMN,$ then the correct expression(s) is(are)

Consider the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$. Let $H\left(\alpha ,0\right),0<\alpha <2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\varphi$ with the positive $x$-axis.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	If $\varphi =\frac{\pi }{4}$, then the area of the triangle $FGH$ is	$\left(\mathrm{P}\right)$	$\frac{{\left(\sqrt{3}-1\right)}^4}{8}$
$\left(\mathrm{II}\right)$	If $\varphi =\frac{\pi }{3}$, then the area of the triangle $FGH$ is	$\left(\mathrm{Q}\right)$	$1$
$\left(\mathrm{III}\right)$	If $\varphi =\frac{\pi }{6}$, then the area of the triangle $FGH$ is	$\left(\mathrm{R}\right)$	$\frac{3}{4}$
$\left(\mathrm{IV}\right)$	If $\varphi =\frac{\pi }{12}$, then the area of the triangle $FGH$ is	$\left(\mathrm{S}\right)$	$\frac{1}{2\sqrt{3}}$
		$\left(\mathrm{T}\right)$	$\frac{3\sqrt{3}}{2}$

The correct option is:

Let $E_1$ and $E_2$ be two ellipse whose centers are at the origin. The major axes of $E_{1}and E_2$ lie along the x - axis and the y - axis, respectively. Let S be the circle $x^2+{\left(y-1\right)}^2=2.$ The straight line $x+y=3$ touches the curves $S, E_1$ and $E_2$ at $P, Q$ and $R,$ respectively. Suppose that $PQ=PR= \frac{2\sqrt{2}}{3}.$ If$e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2,$respectively, then the correct expression(s) is(are)

If $3^{\text{x}}=4^{\text{x}-1}$, then x =

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The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

The amount of water produced ($\mathrm{in} \mathrm{g}$) in the oxidation of $1$ $\mathrm{mole}$of rhombic sulphur by conc. ${\mathrm{H}\mathrm{N}\mathrm{O}}_3$ to a compound with the highest oxidation state of sulphur is
(Given data: Molar mass of water$=18\mathrm{ }\mathrm{g}\mathrm{ }{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ )

A block \((B)\) is attached to two unstretched \(S_1\) and \(S_2\) with spring constants \(k\) and \(4 k\), respectively (see figure I). The other ends are attached to identical supports \(M_1\) and \(M_2\) not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block \(B\) is displaced towards wall 1 by a small distance \(x\) (figure II) and released. The block returns and moves a maximum distance \(y\) towards wall 2 . Displacements \(x\) and \(y\) are measured with respect to the equilibrium position of the block \(B\). The ratio \(\frac{y}{-1}\) is

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In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The empty space in this \(\mathrm{HCP}\) unit cell is

Let \(f(x)=x^4+a x^3+b x^2+c\) be a polynomial with real coefficients such that \(f(1)=-9\). Suppose that \(i \sqrt{3}\) is a root of the equation \(4 x^3+3 a x^2+2 b x=0\), where \(i=\sqrt{-1}\). If \(\alpha_1, \alpha_2, \alpha_3\), and \(\alpha_4\) are all the roots of the equation \(f(x)=0\), then \(\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2\) is equal to ________.

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that the trace of \(A\) is not divisible by \(p\) but det \((A)\) is divisible by \(p\) is
[Note : The trace of a matrix is the sum of its diagonal entries.]

Regarding the molecular orbital (MO) energy levels for homonuclear diatomic molecules, the INCORRECT statement(s) is(are)