Shown in the figure is rigid and uniform one meter long rod \(AB\) held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass \('m'\) and has another weight of mass \(2m\) hung at a distance of \(75\, cm\) from \(A\). The tension in the string at \(A\) is\(....mg\)

The temperature of an ideal gas in \(3-\)dimensions is \(300\, {K}\). The corresponding de-Broglie wavelength of the electron approximately at \(300\,{K}\), is \(....\,nm\) \(\left\lfloor{m}_{e}=\text { mass of electron }=9 \times 10^{-31} \,{kg}\right.\) \({h}=\text { Planck constant }=6.6 \times 10^{-34} {Js}\) \(\left.{k}_{{B}}=\text { Boltzmann constant }=1.38 \times 10^{-23}\, {JK}^{-1}\right]\)

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 \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

When a rubber ball is taken to a depth of \(......\,{m}\) in deep sea, its volume decreases by \(0.5\, \%\) (The bulk modulus of rubber \(=9.8 \times 10^{8}\, {Nm}^{-2}\) Density of sea water \(=10^{3} {kgm}^{-3}\) \(\left.{g}=9.8\, {m} / {s}^{2}\right)\)

The angular momentum of an electron in a hydrogen atom is proportional to _______. (Where \(\mathrm{r}\) is the radius of orbit of electron)

A rigid massless road of length \(3l\) has tow masses attached at each end as shown in the figure. The rod is pivoted at point \(P\) on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be

A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point \(P\) which lies on the centre of the semicircle ?

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius \(3\) is

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear if \(\vec{a}+5 \vec{b}\) is collinear with \(\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{b}}+6 \overrightarrow{\mathrm{c}}\) is collinear with \(\overrightarrow{\mathrm{a}}\) and \(\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}\), then \(\alpha+\beta\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \(\operatorname{cosec}^{2} x d y+2 d x=(1+y \cos 2 x) \operatorname{cosec}^{2} x d x\), with \(y\left(\frac{\pi}{4}\right)=0\). Then, the value of \((y(0)+1)^{2}\) is equal to :

A soap bubble is blown to a diameter of \(7 \mathrm{~cm}\). \(36960 \mathrm{erg}\) of work is done in blowing it further. If surface tension of soap solution is \(40 \mathrm{dyne} / \mathrm{cm}\) then the new radius is ______ \(\mathrm{cm}\). Take \(:\left(\pi=\frac{22}{7}\right)\).

Let \(A\) be the sum of the first \(20\) terms and \(B\) be the sum of the first \(40\) terms of the series  \({1^2} + 2 \cdot {2^2} + {3^2} + 2 \cdot {4^2} + {5^2} + .\;.\;.\;.\).If \(B - 2A = 100\lambda \) then \(\lambda \) is equal to : 

Projectiles \(A\) and \(B\) are thrown at angles of \(45^{\circ}\) and \(60^{\circ}\) with vertical respectively from top of a \(400 \mathrm{~m}\) high tower. If their ranges and times of flight are same, the ratio of their speeds of projection \(v_A: v_B\) is _______.