The time taken for the amplitude of vibrations of a damped oscillator to drop to \(25 \%\) of its initial value is \(t\). The time taken for its mechanical energy to drop to \(50 \%\) of its initial mechanical energy is

The device used to detect infrared radiations is

Two tuning forks \(X\) and \(Y\) are of frequencies \(280 \mathrm{~Hz}\) and \(284 \mathrm{~Hz}\). A third tuning fork \(Z\) is of unknown frequency. When \(X\) and \(Z\) are sounded together certain beats are heard per second. When \(Y\) and \(Z\) are sounded together beat frequency is found to be thrice as great. The frequency of \(Z\) is

The dot product of unit vectors \(\hat{n}_1\) and \(\hat{n}_2\) that are parallel to \(5 \hat{i}+12 \hat{j}\) and \(3 \hat{i}+4 \hat{j}\) respectively is

A change of $0.04$ volts takes place between the base and the emitter when an input signal is connected to the common emitter transistor amplifier. As a result, $20 \mathrm{\mu A}$ change takes place in the base current and a change of $2 \mathrm{mA}$ takes place in the collector current. The input resistance and $\mathrm{AC}$ current gain are

A body is projected at \(t=0\) with a velocity \(10 \mathrm{~ms}^{-1}\) at an angle of \(60^{\circ}\) with the horizontal. The radius of curvature of its trajectory at \(t=1 s\) is \(R\). Neglecting air resistance and taking acceleration due to gravity \(g=10 \mathrm{~ms}^{-2}\), the value of \(R\) is :

A gas is expanded from an initial state to a final state along a path on a $P-V$ diagram. The path consists of (i) an isothermal expansion of work $50 \mathrm{J}$, (ii) an adiabatic expansion and (iii) an isothermal expansion of work $20 \mathrm{J}$. If the internal energy of gas is changed by $-30 \mathrm{J}$, then the work done by gas during adiabatic expansion is

The heat evolved for the rise of water when one end of the capillary tube of radius \(r\) is immersed vertically into water is (Assume surface tension \(=T\) and density of water to be p)

The binomial coefficients which are in decreasing order are

The curve $f\left(x\right)=e^x\sin x$ is defined in the interval $[0,2\pi ].$ The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is

The voltage applied to an electron microscope to produce electrons of wavelength \(0.50 Å\) is

A bomb at rest explodes into three pieces of equal masses. If two pieces move perpendicular to each other, each with a speed \(v\) then the speed of the third piece is

Match the following
List - I
A) The equation of line passing through \((4,3)\) whose \(\mathrm{X}\)-intercept is twice its Y-intercept
B) The equation of the line passing through the centroid and circumcentre of \(\triangle A B C\) with vertices \(\mathrm{A}(1,1), \mathrm{B}(3,3), \mathrm{C}(6,-6)\)
C) The equation of the line whose \(\mathrm{X}\)-intercept is \((-3 / 5)\)
III) \(x+2 y+\sqrt{2}=0\) and is perpendicular to \(x-y+2=0\)
D) The equation of the line whose distance from the origin is 2 and the normal drawn from the origin makes an angle \(45^{\circ}\) with the positive direction of \(\mathrm{X}\)-axis
List - II
I) \(x+y-2 \sqrt{2}=0\)
II) \(7 x+23 y-8=0\)
IV) \(x+2 y-10=0\)
IV) \(x+2 y-10=0\)
V) \(5 x+5 y+3=0\)
The correct answer is