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Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The radius of the circumcircle of the \(\triangle P R S\) is

Let $f :\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to R$ be given by $f\left(x\right)={\left(\log \left(\sec x+\tan x\right)\right)}^3.$ Then

The positive integer value of \(n>3\) satisfying the equation
\(\frac{1}{\sin \left(\frac{\pi}{n}\right)}=\frac{1}{\sin \left(\frac{2 \pi}{n}\right)}+\frac{1}{\sin \left(\frac{3 \pi}{n}\right)} \text { is }\)

Let M and N be two $3 \times 3$ matrices such that MN = NM. Further, if $M \ne N^2$ and $M^2=N^4$ , then

The coefficients of three consecutive terms of ${\left(1+\text{x}\right)}^{\text{n}+5}$ are in the ratio $\text{5 : 10 : 14}$. Then $\text{n} =$

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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 \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) is

Let \(I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\) and \(P=\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right)\). Let \(Q=\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ \mathrm{z} & 4\end{array}\right)\) for some non-zero real numbers \(x, y\), and \(z\), for which there is \(2 \times 2\) matrix \(R\) with all entries being non-zero real numbers, such that \(Q R=R P\). Then which of the following statements is (are) TRUE?

Two concentric circular loops, one of radius $R$ and the other of radius $2R$, lie in the $xy$-plane with the origin as their common center, as shown in the figure. The smaller loop carries current $I_1$ in the anti-clockwise direction and the larger loop carries current $I_2$ in the clockwise direction, with $I_2>2I_1. \overrightarrow{B}\left(x,y\right)$ denotes the magnetic field at a point $\left(x,y\right)$ in the $xy$-plane. Which of the following statement(s) is (are) correct?

A disc of mass \(M\) and radius \(R\) is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass \(M\) and radius \(R / 2\) is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed \(\omega\). If the angular speed at which the large disc rotates is \(\omega / n\), then the value of \(n\) is ______ .

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $\left(\mathrm{V}-\mathrm{T}\right)$ diagram. The correct statement(s) is/are:
[ $\mathrm{R}$ is the gas constant]

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

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An acylic hydrocarbon \(P\), having molecular formula \(\mathrm{C}_6 \mathrm{H}_{10}\), gave acetone as the only organic product through the following sequence of reactions, in which \(Q\) is an intermediate organic compounds.

Question:
The structure of compound P is