If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

Let $f\mathbb{ }:\mathbb{R}\to \left(0, \infty \right)$ and $g\mathbb{ }:\mathbb{R}\to \mathbb{R}$ be twice differentiable functions such that $f^{″}$ and $g^{″}$ are continuous functions on $\mathbb{R}$ . Suppose $f^{\prime }\left(2\right)=g\left(2\right)=0, f^{″}\left(2\right)\ne 0$ and $g^{\prime }\left(2\right) \ne 0.\underset{x\to 2}{\lim }\frac{f\left(x\right) g\left(x\right)}{f^{\prime } \left(x\right) g^{\prime }\left(x\right)}=1,$ then

The largest value of the non - negative integer a for which $\underset{x\to 1}{\lim }{\left\{\frac{-ax+\sin \left(x-1\right)+a}{x+\sin \left(x-1\right)-1}\right\}}^{\frac{1-x}{1- \sqrt{x}}}=\frac{1}{4}$ is _________

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) has a unique solution, is

The number of distinct real roots of \(x^4-4 x^3+12 x^2+x-1=0\) is

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Let \(S\) be the focus of the parabola \(y^{2}=8 \mathrm{x}\) and let \(P Q\) be the common chord of the circle \(x^{2}+y^{2}-2 x-4 y=0\) and the given parabola. The area of the triangle \(P Q S\) is

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

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Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The vapour pressure of the solution \(M\) is

At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current $\mathrm{I}\left(\mathrm{t}\right)={\mathrm{I}}_0\cos \left(\mathrm{\omega }\mathrm{t}\right)$ , with ${\mathrm{I}}_0=1\mathrm{A}$ and $\mathrm{\omega }=500\mathrm{}$ rad/s starts flowing in it with the initial direction shown in the figure. At $\mathrm{t}=\frac{7\mathrm{\pi }}{6\mathrm{\omega }}$ , the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If $\mathrm{C}=20\mathrm{}\mathrm{\mu }\mathrm{F},\mathrm{}\mathrm{R}=10\mathrm{}\mathrm{\Omega }$ and the battery is ideal with of 50V, identify the correct statement(s).

The maximum number of electrons that can have principal quantum number, \(n=3\) and spin quantum number, \(m_s=-\frac{1}{2}\), is

Sunlight of intensity $1.3 kW m^{-2}$ is incident normally on a thin convex lens of focal length $20 cm$ . Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in $kW m^{-2}$ , at a distance $22 cm$ from the lens on the other side is __________.

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is