Consider a block kept on an inclined plane (inclined at \(45^{\circ}\) ) as shown in the figure. If the force required to just push it up the incline is \(2\) times the force required to just prevent it from sliding down, the coefficient of friction between the block and inclined plane \((\mu)\) is equal to

Consider a tank made of glass (refractive index \(1.5\)) with a thick bottom. It is filled with a liquid of refractive index \(\mu \). A student finds that, irrespective of what the incident angle \(I\) (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of \(\mu \) is

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

The half life period of radioactive element \({x}\) is same as the mean life time of another radioactive element \(y.\) Initially they have the same number of atoms. Then:

A person pushes a box on a rough horizontal platform surface. He applies a force of \(200\, N\) over a distance of \(15\, m .\) Thereafter, he gets progressively tired and his applied force reduces linearly with distance of \(100\, N\). The total distance through which the box has been moved is \(30\, m\). What is the work done by the person during the total movement of the box\(........J\)\(?\)

Hysteresis loops for two magnetic materials \(A\) and \(B\) are given below These materials are used to make magnets for elecric generators, transformer core and electromagnet core. Then it is proper to use

Light ray incident along a vector \(\vec{AO}\) \((\vec{AO} = 2\hat{i}-3\hat{j})\) emerges out along vector \(\vec{OB}\) \((\vec{OB} = C\hat{i}-4\hat{j})\) as shown in the figure below. The value of \(C\) is ________.

Two identical positive charges \(Q\) each are fixed at a distance of ' \(2 a\) ' apart from each other. Another point charge qo with mass ' \(m\) ' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge \(q_{0}\) executes \(SHM\). The time period of oscillation of charge \(q_{0}\) will be.

Suppose \(\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}\). Then the value of \(\alpha\) is \(............\)

The value of the integral \(\displaystyle\int_0^\infty \dfrac{\log_e(x)}{x^2 + 4}\,dx\) is:

If for \(3 \leq r \leq 30\), \(\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}\), then \(m\) equals:

If \(\lim \limits_{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}\) then the value of \(k\) is

\(\mathop {\lim }\limits_{x \to 0} \frac{{{{(27 + x)}^{_{\frac{1}{3}}}} - 3}}{{9 - {{(27 + x)}^{\frac{2}{3}}}}}\)  equals.