The displacement and the increase in the velocity of a moving particle in the time interval of \(t\) to \((t+1) \mathrm{s}\) are \(125 \mathrm{~m}\) and \(50 \mathrm{~m} / \mathrm{s}\), respectively. The distance travelled by the particle in \((\mathrm{t}+2)^{\mathrm{th}} \mathrm{s}\) is __________ \(\mathrm{m}\).

Assuming the experimental mass of \({}^{12}_{6}C\) as \(12\text{ u}\), the mass defect of \({}^{12}_{6}C\) atom is _______ \(\text{MeV}/c^2\).
(Mass of proton \(= 1.00727\text{ u}\), mass of neutron \(= 1.00866\text{ u}\), \(1\text{ u} = 931.5\text{ MeV}/c^2\) and \(c\) is the speed of the light in vacuum).

Arrange the following in the ascending order of wavelength \((\lambda)\) :
(A) Microwaves \(\left(\lambda_1\right)\)
(B) Ultraviolet rays \(\left(\lambda_2\right)\)
(C) Infrared rays \(\left(\lambda_3\right)\)
(D) X-rays \(\left(\lambda_4\right)\)
Choose the most appropriate answer from the options given below :

An observer moves towards a stationary source of sound with a velocity equal to one-fifth of the velocity of sound. The percentage change in the frequency will be \(\dots \;\)%

A parallel plate capacitor was made with two rectangular plates, each with a length of \(l=3 \mathrm{~cm}\) and breath of \(\mathrm{b}=1 \mathrm{~cm}\). The distance between the plates is \(3 \mu \mathrm{~m}\). Out of the following, which are the ways to increase the capacitance by a factor of 10 ?
A. \(l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
B. \(l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}\)
C. \(l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}\)
D. \(l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}\)
E. \(l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
Choose the correct answer from the options given below:

The energy levels of an hydrogen atom are shown below. The transition corresponding to emission of shortest wavelength is

\(5\, beats/ second\) are heard when a turning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either \(0.95\,m\) or \(1\,m\) . The frequency of the fork will be ... \(Hz\)

Let \(\mathrm{M}\) and \(\mathrm{m}\) respectively be the maximum and minimum values of the function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) in \(\left[0, \frac{\pi}{2}\right]\), Then the value of \(\tan (\mathrm{M}-\mathrm{m})\) is equal to:

The term independent of \(x\) in the binomial expansion of \(\left( {1 - \frac{1}{x} + 3{x^5}} \right){\left( {2{x^2} - \frac{1}{x}} \right)^8}\) is

Tho damnin of tho finction \(\cos ^{-1}\left(\frac{2 \sin ^{-1}\left(\frac{1}{4 x^{2}-1}\right)}{\pi}\right)\) is

Substance \(A\) has atomic mass number \(16\) and half life of \(1\) day. Another substance \(B\) has atomic mass number \(32\) and half life of \(\frac{1}{2}\) day. If both \(A\) and \(B\) simultaneously start undergo radio activity at the same time with initial mass \(320\,g\) each, how many total atoms of \(A\) and \(B\) combined would be left after \(2\) days \(.........\times 10^{24}\)

The slope of tangent at any point \((x, y)\) on a curve \(y = y ( x )\) is \(\frac{x^2+y^2}{2 x y},x > 0\). If \(y(2)=0\), then a value of \(y(8)\) is

A series combination of resistor of resistance \(100\,\Omega\), inductor of inductance \(1\,H\) and capacitor of capacitance \(6.25\,\mu F\) is connected to an ac source. The quality factor of the circuit will be \(.............\).