Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is \(2 \pi / 3\), is \(\sqrt{3}\), then area of triangle is

The total number of local maxima and local minima of the function \(f(x)=\left\{\begin{array}{lc}(2+x)^3 ; & -3 < x \leq-1 \\ x^{\frac{2}{3}} ; & -1 < x < 2\end{array}\right.\) is

Let $m$ be the minimum possible value of ${\log }_3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$ , where $y_1, y_2, y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let $M$ be the maximum possible value of $\left({\log }_3{x}_1+{\text{log}}_3 x_2+{\text{log}}_3 x_3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of ${\text{log}}_2 \left(m^3\right)+{\text{log}}_3\left(M^2\right)$ is

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

Paragraph:
Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).

Question:
If \(a_{4}=28\), then \(p+2 q=\)

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi }{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.

(There are two questions based on $\mathrm{PARAGRAPH} " 1 "$, the question given below is one of them)
Let $\alpha$ be the area of the triangle $ABC$. Then the value of $(64\alpha {)}^2$ is

A ship is fitted with three engines \(E_{1}, E_{2}\) and \(E_{3}\). The engines function independently of each other with respective probabilities \(\frac{1}{2}, \frac{1}{4}\) and \(\frac{1}{4}\). For the ship to be operational at least two of its engines must function. Let \(X\) denote the event that the ship is operational and let \(X_{1}, X_{2}\) and \(X_{3}\) denote respectively the events that the engines \(E_{1}, E_{2}\) and \(E_{3}\) are functioning. Which of the following is(are) true?

Match the chemical substance in Column I with type of polymers/type of bonds in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Column I	Column II
A. \(C _6 H _5 CHO\)	p. Gives precipitate with 2, 4-dinitrophenylhydrazine
B. \(CH _3 \equiv CH\)	q. Gives precipitate with \(AgNO _3\)
C. \(CN ^{-}\)	r. Is a nucleophile
D. \(I ^{-}\)	s. Is involved in cyanohydrin formation

Let \(\alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\ldots+\) \(\frac{1}{\sin 118^{\circ} \sin 119^{\circ}}\)
Then the value of \(\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2\) is __________

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, \(A\) is the point of contact. \(B\) is the centre of the sphere and \(C\) is its topmost point. Then,

The IUPAC name of \(\mathrm{C}_6 \mathrm{H}_5 \mathrm{COCl}\) is

For any $y\in \mathrm{ℝ}$, let ${\cot }^{-1}\left(y\right)\in \left(0,\pi \right)$ and ${\tan }^{-1}\left(y\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<\left|y\right|<3$, is equal to

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $\frac{a}{4}$ is pivoted on this rod with its centre at a distance $\frac{a}{4}$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary conserver finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point $O$ is $\left[\left(\frac{M{a}^2\Omega }{48}\right)n\right]$. The value of $n$ is _________ .