In a $∆PQR$, $P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $PQ, QR$ and $RP$ at $N, L$ and $M$ respectively, such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

Consider all rectangles lying in the region $\left\{\left(x, y\right)\in R\times R:0\le x\le \frac{\pi }{2}and0\le y\le 2\sin \left(2x\right)\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

Let $E$ denote the parabola $y^2=8x.$ Let $P=\left(-2,4\right)$, and let $Q$ and $Q^{\text{'}}$ be two distinct points on $E$ such that the lines $PQ$ and $P{Q}^{\text{'}}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE?

Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $MN$ Then, the mid-point of the line segment must lie on the curve

Let \(\theta, \varphi \in[0,2 \pi]\) be such that \(2 \cos \theta(1-\sin \varphi)\) \(=\sin ^{2} \theta ~\times\) \(\left(\tan \frac{\theta}{2}+\cot \frac{\theta}{2}\right) \cos \varphi-1, \tan\) \((2 \pi-\theta)>0\) and \(-1 < \sin \theta < -\frac{\sqrt{3}}{2}\), then \(\varphi\) cannot satisfy

If \(\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5}\), then

Paragraph:
The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
Question:
\(\mathrm{XeF}_4\) and \(\mathrm{XeF}_6\) are expected to be

Consider the lines $L_1\text{:}\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1}\text{,}{L}_2\text{:}\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1\text{:}7x+y+2z=3\text{,}{P}_2\text{:}3x+5y-6z=4\text{.}$ Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of L₁ and L₂, and perpendicular to planes P₁ and P₂.
Match List- I with List - II and select the correct answer using the code given below the lists :
 

	List I		List II
A.	a =	P.	13
B.	b =	Q.	-3
C.	c =	R.	1
D.	d =	S.	-2

An ideal gas in a thermally insulated vessel at internal pressure = ${\mathrm{P}}_1$ , volume = ${\mathrm{V}}_1$ and absolute temperature = ${\mathrm{T}}_1$ expands irreversibly against zero external pressure, as shown in the diagram. The final internal pressure, volume and absolute temperature of the gas are ${\mathrm{P}}_2$ , ${\mathrm{V}}_2$ and ${\mathrm{T}}_2$ , respectively. For this expansion,

A small block of mass of \(0.1 \mathrm{~kg}\) lies on a fixed inclined plane PQ which makes an angle \(\theta\) with the horizontal. A horizontal force of \(1 \mathrm{~N}\) acts on the block through its centre of mass as shown in the figure.
The block remains stationary if (take \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) )

A spring - block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0N{m}^{-1}$ and the mass of the block is $2.0kg$ . Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0kg moving with a speed of $2.0m{s}^{-1}$ collides elastically with the first block. The collision is such that the 2.0kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.