Let $n \ge 2$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is ________

A circle S passes through the point (0, 1) and is orthogonal to the circles ${\left(x-1\right)}^2+y^2=16$ and $x^2+y^2=1$. Then

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is

Match the statements/expressions given in Column I with the values given in Column II.

Let S be the set of all twice differentiable functions f from $\mathrm{ℝ}$ to $\mathrm{ℝ}$ such that $\frac{d^{2}f}{d{x}^2}\left(x\right)>0$ for all $x\in \left(-1,1\right)$. For $f\in S$, let $X_f$ be the number of points $x\in \left(-1,1\right)$ for which $f\left(x\right)=x$. Then which of the following statements is(are) true?

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Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The real number \(s\) lies in the interval

Consider the lines given by
\[
L_1: x+3 y-5=0, L_2: 3 x-k y-1=0, L_3: 5 x+2 y-12=0
\]
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Vessel-1 contains \(\mathbf{w}_2 \mathrm{~g}\) of a non-volatile solute \(\mathbf{X}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Vessel- 2 contains \(\mathbf{w}_2 \mathrm{~g}\) of another non-volatile solute \(\mathbf{Y}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Both the vessels are at the same temperature and pressure. The molar mass of \(\mathbf{X}\) is \(80 \%\) of that of \(\mathbf{Y}\). The van't Hoff factor for \(\mathbf{X}\) is 1.2 times of that of \(\mathbf{Y}\) for their respective concentrations.
The elevation of boiling point for solution in Vessel-1 is ________. \(\%\) of the solution in Vessel-2.

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$is released on the slide from rest at a height $h$from the top of the terrace. The ball leaves the slide with a velocity ${\overrightarrow{u}}_0=u_0\stackrel{\wedge }{x}$and falls on the ground at a distance $d$from the building making an angle $\theta$with the horizontal. It bounces off with a velocity $\overrightarrow{v}$and reaches a maximum height $h_1$. The acceleration due to gravity is $g$and the coefficient of restitution of the ground is $\frac{1}{\sqrt{3}}$. Which of the following statement(s) is(are) correct?

Suppose that $\overrightarrow{p}, \overrightarrow{q} and \overrightarrow{r}$ are three non - coplanar vectors in ${\mathbb{R}}^3$ . Let the components of a vector $\overrightarrow{s}$ along \(p, q\) and \(r\) be 4,3 and 5 respectively. If the components of $\left(-\overrightarrow{p}+ \overrightarrow{q}+ \overrightarrow{r}\right), \left(\overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right) and \left(- \overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right)$ are x, y and z, respectively, then the value of $2x+y+z$ is

On a frictionless horizontal plane, a bob of mass $m=0.1 \mathrm{kg}$ is attached to a spring with natural length $l_0=0.1 \mathrm{m}$. The spring constant is $k_1=0.009 \mathrm{N} {\mathrm{m}}^{-1}$ when the length of the spring $l>l_0$ and is $k_2=0.016 \mathrm{N} {\mathrm{m}}^{-1}$ when $l<l_0$. Initially the bob is released from $l=0.15 \mathrm{m}$. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is $T=\left(n\pi \right)\mathrm{s}$, then the integer closest to $n$ is _______.

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The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of the circle with \(A B\) as its diameter is

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?