If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, is

Tangent is drawn at any point \(P\) of a curve which passes through \((1,1)\) cutting \(X\)-axis and \(Y\)-axis at \(A\) and \(B\), respectively. If \(A P: B P=3: 1\), then

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $x+2y+z=7$ $x+\alpha z=11$ $2x-3y+\beta z=\gamma$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma =28$ then the system has	$\left(1\right)$	a unique solution
$\left(\mathrm{Q}\right)$	If $\beta =\frac{1}{2}\left(7\alpha -3\right)$ and $\gamma \ne 28$, then the system has	$\left(2\right)$	no solution
$\left(\mathrm{R}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma \ne 28$, then the system has	$\left(3\right)$	infinitely many solutions
$\left(\mathrm{S}\right)$	If $\beta \ne \frac{1}{2}\left(7\alpha -3\right)$ where $\alpha =1$ and $\gamma =28$, then the system has	$\left(4\right)$	$x=11, y=-2$ and $z=0$ as a solution
		$\left(5\right)$	$x=-15, y=4$ and $z=0$ as a solution

The correct option is:

Equation of common tangent of \(y=x^2, y=-x^2+4 x-4\) is

If \(f(x)\) is twice differentiable function such that \(f(a)=0, f(b)=2, f(c)=1, f(d)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeros of \(g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)\) in the interval \([a, e]\) is

There are three bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3.$ The bag ${\mathrm{B}}_1$ contains $5$ red and $5$ green balls, ${\mathrm{B}}_2$ contains $3$ red and $5$ green balls, and ${\mathrm{B}}_3$ contains $5$ red and $3$ green balls, Bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

The value of the integral $\underset{0}{\overset{\pi /2}{\int }}\frac{3\sqrt{\cos \theta }}{{\left(\sqrt{\cos \theta }+\sqrt{\sin \theta }\right)}^5}d\theta$ equals

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2y}{x+y+z}$, then which of the following statements is/are TRUE?

Paragraph :
Two plane harmonic sound waves are expressed by the equations.
\(\begin{aligned}
& y_1(x, t)=A \cos (0.5 \pi x-100 \pi t) \\
& y_2(x, t)=A \cos (0.46 \pi x-92 \pi t)
\end{aligned}\)
(All parameters are in MKS)
Question :
How many times does an observer hear maximum intensity in one second?

A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $X$is ____.

Statement I A block of mass \(m\) starts moving on a rough horizontal surface with a velocity \(v\). It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of \(30^{\circ}\) with the horizontal and the same block is made to go up on the surface with the same initial velocity \(v\). The decrease in the mechanical energy in the second situation is smaller than that in the first situation.
Statement II The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.

If $y=y\left(x\right)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}dx$ and $x>0, y\left(0\right)=\sqrt{7},$ then $y\left(256\right)=$