Let \(P Q R\) be a triangle of area \(\Delta\) with \(a=2, b=\frac{7}{2}\) and \(c=\frac{5}{2} ;\) where \(a, b\), and \(c\) are the lengths of the sides of the triangle opposite to the angles at \(P, Q\) and \(R\) respectively. Then \(\frac{2 \sin P-\sin 2 P}{2 \sin P+\sin 2 P}\) equals.

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\) be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that \(y_1>0\), and \(y_2>0\). Let C denote the circle \(x^2+y^2=9\), and \(M\) be the point \((3,0)\).
Suppose the line \(x=x_1\) intersects \(C\) at \(R\), and the line \(x=x_2\) intersects \(C\) at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\).
Then which of the following statements is (are) TRUE?

Consider the region \(R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(\mathcal{F}\) be the family of all circles that are contained in \(R\) and have centers on the \(x\)-axis. Let \(C\) be the circle that has largest radius among the circles in \(\mathcal{F}\). Let \((\alpha, \beta)\) be a point where the circle \(C\) meets the curve \(y^{2}=4-x\).
The value of $\alpha$ is _____.

Consider a pyramid$OPQRS$ located in the first octant $\left(x \ge 0, y \ge 0, z \ge 0\right)$ with$O$, as origin, and$OP$ and,$OR$ along the $x‐axis$ and the $y‐axis$, respectively. The base$OPQR$ of the pyramid is a square with$OP = 3$. The point S is directly above the mid-point T of diagonal, $OQ$,  such that, $TS = 3$. Then 

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

Let \(\omega=e^{i \pi / 3}\) and \(a, b, c, x, y, z\) be non-zero complex numbers such that \(a+b+c=x, a+b \omega+c \omega^2=y, a+b \omega^2+c \omega=z\).
Then, the value of \(\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}\) is

A plane passes through \((1,-2,1)\) and is perpendicular to two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of the plane from the point \((1,2,2)\) is

Match each of the reactions given in Column I with the corresponding products(s) given in Column II.

A table tennis ball has radius \((3 / 2) \times 10^{-2} \mathrm{~m}\) and mass \((22 / 7) \times 10^{-3} \mathrm{~kg}\). It is slowly pushed down into a swimming pool to a depth of \(d=0.7 \mathrm{~m}\) below the water surface and then released from rest. It emerges from the water surface at speed \(v\), without getting wet, and rises up to a height \(H\). Which of the following option(s) is(are) correct?
[Given: \(\pi=22 / 7, g=10 \mathrm{~m} \mathrm{~s}^{-2}\), density of water \(=1 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), viscosity of water \(=1 \times 10^{-3} \mathrm{~Pa}\)-s.]

If \(f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, & x \leq-\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0 < x \leq 1 \\ \log x, & x>1\end{array}\right.\), then

A circular coil of radius $R$ and $N$ turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge $Q$ through a switch. The coil is in a horizontal uniform magnetic field $B_o$ parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

Paragraph:

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

The value of \(r\) is

With respect to graphite and diamond, which of the statement(s) given below is (are) correct?