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

Let \(\alpha, \beta\) be the roots of the equation \(x^2-p x+r=0\) and \(\frac{\alpha}{2}, 2 \beta\) be the roots of the equation \(x^2-q x+r=0\). Then, the value of \(r\) is

Let $n_1 and n_2$ be the number of red and black balls, respectively, in box I. Let $n_{3}and n_4$ be the number of red and black balls, respectively, in box II.
One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The balls was found to be red. If the probability that this red ball was drawn from box II is $\frac{1}{3}$ , then the correct option(s) with the possible values of $n_1, n_2, n_{3}and n_4$ is(are)

Match List-I with List - II and select the correct answer using the code given below the lists 
 

	List-I		List-II
A.	Volume of parallelepiped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2\left(\overrightarrow{a}\times \overrightarrow{b}\right),3\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ and $\left(\overrightarrow{c}\times \overrightarrow{a}\right)$ is	P.	$100$
B.	Volume of parallel piped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3\left(\overrightarrow{a}+\overrightarrow{b}\right),\left(\overrightarrow{b}+\overrightarrow{c}\right)$ and $2\left(\overrightarrow{c}+\overrightarrow{a}\right)$ is	Q.	$30$
C	Area of a triangle with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $20$.Then the area of the triangle with adjacent sides determined by vectors $\left(2\overrightarrow{a}+3\overrightarrow{b}\right)$ and $\left(\overrightarrow{a}-\overrightarrow{b}\right)$  is	R.	$24$
D	Area of a parallelogram with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\overrightarrow{a}$ is	S.	$60$

Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \(\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17 <0\right\}\) is

Let $F\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F\left(1\right)=0, F\left(3\right)= -4$ and $F\prime \left(x\right)<0$ all $x\in \left(\frac{1}{2}, 3\right).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in \mathbb{ }\mathbb{R}$ .
If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)dx= -12$ and $\underset{1}{\overset{3}{\int }}{x}^3 F^{\prime \prime }\left(x\right)dx=40,$ then the correct expression(s) is(are)

Let $\mathrm{\alpha }$ and $\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2-\mathrm{x}-1=0,$ with $\mathrm{\alpha }>\mathrm{\beta }.$ For all positive integers $\mathrm{n},$ define
${\mathrm{a}}_{\mathrm{n}}=\frac{{\mathrm{\alpha }}^{\mathrm{n}}-{\mathrm{\beta }}^{\mathrm{n}}}{\mathrm{\alpha }-\mathrm{\beta }},\mathrm{n}\ge 1$
${\mathrm{b}}_1=1$ and ${\mathrm{b}}_{\mathrm{n}}={\mathrm{a}}_{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}+1},\mathrm{n}\ge 2.$
Then which of the following options is/are correct?

A train with cross-sectional area $S_t$ is moving with speed $v_t$ inside a long tunnel of cross-sectional area $S_0\left(S_0=4S_t\right)$. Assume that almost all the air (density $\rho$) in front of the train flows back between its sides and the walls of the tunnel. Also, the airflow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $P_0$. If the pressure in the region between the sides of the train and the tunnel walls is $P$, then $\left(P_0-P\right)=\frac{7}{2N}\rho v_t^2.$ The value of $N$ is _______.

Paragraph :
In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


Question :
Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

The boiling point of water in a \(0.1\) molal silver nitrate solution (solution \(\mathbf{A}\) ) is \(\mathbf{x}^{\circ} \mathrm{C}\). To this solution \(\mathbf{A}\), an equal volume of \(0.1\) molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}\).
(Assume: Densities of the solutions \(\mathbf{A}\) and \(\mathbf{B}\) are the same as that of water and the soluble salts dissociate completely.
Use: Molal elevation constant (Ebullioscopic Constant), \(K_{b}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\); Boiling point of pure water as \(100^{\circ} \mathrm{C}\).)
The value of $\left|\mathrm{y}\right|$ is ___.

A block of mass \(0.18 \mathrm{~kg}\) is attached to a spring of force constant \(2 \mathrm{~N} / \mathrm{m}\). The coefficient of friction between the block and the floor is 0.1. Initially, the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the figure. The block slides a distance of \(0.06 \mathrm{~m}\) and comes to rest for the first time. The initial velocity of the block in \(\mathrm{m} / \mathrm{s}\) is \(v=\frac{N}{10}\). Then, \(N\) is

There are two vernier calipers both of which have $1\mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is