A \(100\,m\) long wire having cross-sectional area \(6.25 \times 10^{-4}\,m ^2\) and Young's modulus is \(10^{10}\,Nm ^{-2}\) is subjected to a load of \(250\,N\), then the elongation in the wire will be :

A particle of charge \(q\) and mass \(m\) is projected from origin with an initial velocity \(\vec{v} = \left(\dfrac{v_0}{\sqrt{2}}\hat{x} + \dfrac{v_0}{\sqrt{2}}\hat{y}\right)\). There exists a uniform magnetic field \(\vec{B} = B_0 \hat{z}\) and a space varying electric field \(\vec{E} = E_0 e^{-\lambda x}\hat{x}\) within the region \(0 \leq x \leq L\). After travelling a distance such that \(x\)-coordinate has changed from \(x=0\) to \(x=L\), the change in the kinetic energy is _______.

A body of mass \(5\, kg\) under the action of constant force \(\overrightarrow F  = {F_x}\hat i + {F_y}\hat j\) has velocity at \(t\,= 0\, s\) as \(\overrightarrow v  = \left( {6\hat i - 2\hat j\,m/s} \right)\)  and at \(t\, = 10\,s\) as \(\overrightarrow v  = +6\hat j\,m/s\). The force \(\overrightarrow F \) is 

There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K. If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K , at steady state the pressure in the vessels will be (in kPa).

Voltage rating of a parallel plate capacitor is \(500\,V\). Its dielectric can withstand a maximum electric field of \({10^6}\,\frac{V}{m}\). The plate area is \(10^{-4}\, m^2\) . What is the dielectric constant if the capacitance is \(15\, pF\) ? (given \({ \in _0} = 8.86 \times {10^{ - 12}}\,{C^2}\,/N{m^2}\))

A force of \(- P \hat{k}\) acts on the origin of the coordinate system. The torque about the point \((2,-3)\) is \(P(a \hat{i}+b \hat{j})\), the ratio of \(\frac{a}{b}\) is \(\frac{x}{2}\). The value of \(x\) is

An object of mass \(5\, kg\) is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of \(10\, N\) throughout the motion. The ratio of time of ascent to the time of descent will be equal to \(\left[\right.\)Use \(g =10 \,ms ^{-2}]\)

The equations of the sides \(A B, B C \& C A\) of a triangle \(A B C\) are \(2 x+y=0, x+p y=21 a(a \neq 0)\) and \(x-y=3\) respectively. Let \(P(2, a)\) be the centroid of the triangle \(A B C\), then \((B C)^2\) is equal to

If the point of intersections of the ellipse \(\frac{ x ^{2}}{16}+\frac{ y ^{2}}{ b ^{2}}=1\) and the circle \(x ^{2}+ y ^{2}=4 b , b > 4\) lie on the curve \(y^{2}=3 x^{2},\) then \(b\) is equal to:

If \(\vec{A}\) and \(\vec{B}\) are two vectors satisfying the relation \(\vec{A} . \vec{B}=[\vec{A} \times \vec{B}]\). Then the value of \([\vec{A}-\vec{B}]\). will be :

If both the means and the standard deviation of \(50\) observations \(x_1, x_2, ………, x_{50}\) are equal to \(16\) , then the mean of \((x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2\) is

If \(b\) is the first term of an infinite \(G.P\) whose sum is five, then \(b\) lies in the interval

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is