A group of 9 students, \(\mathrm{s}_1, \mathrm{~s}_2, \ldots \ldots, \mathrm{s}_9\), is to be divided to from three teams \(\mathrm{X}, \mathrm{Y}\), and \(\mathrm{Z}\) of sizes 2,3 , and 4 , respectively. Suppose that \(s_1\) cannot be selected for the team \(\mathrm{X}\), and \(\mathrm{s}_2\) cannot be selected for the team Y. Then the number of ways to from such teams, is

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

Let $\mathrm{y}\left(\mathrm{x}\right)$ be a solution of the differential equation $\left(1+{\mathrm{e}}^{\mathrm{x}}\right)\mathrm{ }{\mathrm{y}}^{\mathrm{\prime }}+\mathrm{y}{\mathrm{e}}^{\mathrm{x}}=1.$ If $\mathrm{y}\left(0\right)=2$ , then which of the following statements is (are) true?

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3}a\cos x+2b\sin x=c, x\in \left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of $\frac{b}{a}$ is

Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,....\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,....\right\}$ of rectangles as follows:
${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1;$
${\mathrm{R}}_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_1;$
${\mathrm{E}}_{\mathrm{n}}:$ ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}_{\mathrm{n}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}_{\mathrm{n}}^2}=1$ of largest area inscribed in ${\mathrm{R}}_{\mathrm{n}-1},\mathrm{n}>1;$
${\mathrm{R}}_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_{\mathrm{n}},\mathrm{n}>1.$
Then which of the following options is/are correct?

Let $f\left(x\right)=x+{\log }_{e}x-x{\log }_{\mathrm{e}}x,x\in \left(0, \infty \right).$
$•$   Column 1 contains information about zeros of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right).$
$•$   Column 2 contains information about the limiting behaviour of $f\left(x\right), f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.
$•$   Column 3 contains information about increasing-decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right).$
 

Column 1	Column 2	Column 3
(I) $f\left(x\right)=0$ for some $x\in \left(1, e^2\right)$	(i) $\underset{x\to \infty }{\lim }f\left(x\right)=0$	(P) $f$ is increasing in (0, 1)
(II) $f^{\prime }\left(x\right)=0$ for some $x in \left(1, e\right)$	(ii) $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$	(Q) $f$ is decreasing in $\left(e, e^2\right)$
(III) $f^{\prime }\left(x\right)=0$ for some $x\in \left(0, 1\right)$	(iii) $\underset{x\to \infty }{\lim }{f}^{\prime }\left(x\right)=-\infty$	(R) $f^{\prime }$ is increasing in (0, 1)
(IV) $f^{\prime \prime }\left(x\right)=0$ for some $x\in \left(1, e\right)$	(iv) $\underset{x\to \infty }{\lim }{f}^{\prime \prime }\left(x\right)=0$	(S) $f^{\prime }$ is decreasing in $\left(e, e^2\right)$

Which of the following options is the only Correct combination?

Statement I The curve \(y=-\frac{x^2}{2}+x+1\) is symmetric with respect to the line \(x=1\)
Statement II A parabola is symmetric about its axis.

Paragraph I: An organic compound \(\mathbf{P}\) with molecular formula \(\mathrm{C}_9 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathbf{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathbf{Q}\) and \(\mathbf{R}\). While compound \(\mathbf{Q}\) shows positive iodoform test, compound \(\mathbf{R}\) does not give positive iodoform test. \(\mathbf{Q}\) and \(\mathbf{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathbf{S}\) and \(\mathbf{T}\), respectively. Both \(\mathbf{S}\) and \(\mathbf{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathbf{Q}\) and 500 moles of \(\mathbf{R}\) gives one mole of a single acyclic copolymer \(\mathbf{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: Sum of number of oxygen atoms in \(\mathbf{S}\) and \(\mathbf{T}\) is

In a \(\triangle A B C\) with fixed base \(B C\), the vertex \(A\) moves such that \(\cos B+\cos C=4 \sin ^2 \frac{A}{2}\). If \(a, b\) and \(c\) denote the lengths of the sides of the triangle opposite to the angles \(A, B\) and \(C\) respectively, then

A wide slab consisting of two media of refractive indices $n_1$ and $n_2$ is placed in air as shown in the figure. A ray of light is incident from medium $n_1$ to $n_2$ at an angle $\theta$, where $\sin \theta$ is slightly larger than $\frac{1}{n_1}$. Take refractive index of air as $1$. Which of the following statement(s) is (are) correct?

Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are \(\beta_{G}, \beta_{R}\) and \(\beta_{B}\), respectively. Then,

If the function \(f(x)=x^3+e^{\frac{x}{2}}\) and \(g(x)=f^{-1}(x)\), then the value of \(g^{\prime}(1)\) is

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 ____.