A particle of mass \(M\) originally at rest is subjected to a force whose direction is constant but magnitude varies with time according to the relation \(F=F_{0}\left(1-\left(\frac{t-T}{T}\right)^{2}\right)\) Where \(F_{0}\) and \(T\) are constants. The force acts only for the time internal \(2 T\). The velocity \(v\) of the particle after time \(2 {T}\) is -

Two soap bubbles coalesce to form a single bubble. If \(V\) is the subsequent change in volume of contained air and \(S\) change in total surface area, \(T\) is the surface tension and \(P\) atmospheric pressure, then which of the following relation is correct?

When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by __________ %.

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

A cylindrical wire of mass \((0.4 \pm 0.01)\,g\) has length \((8 \pm 0.04)\,cm\) and radius \((6 \pm 0.03)\,mm\).The maximum error in its density will be \(......\,\%\)

When a diode is forward biased, it has a voltage drop of \(0.5\, V.\) The safe limit of current through the diode is \(10\, mA\). If a battery of emf \(1.5\, V\) is used in the circuit, the value of minimum resistance to be connected in series with the diode so that the current does not exceed the safe limit is\(.....\Omega\)

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point \(P\) is \(\frac{x \sigma}{\epsilon_0}\). The value of \(x\) is _______. (all quantities are measured in \(SI\) units).

Lines are drawn parallel to the line \(4x -3y + 2 = 0\) at a distance \(\frac {3}{5}\) from the origin. Then which one of the following points lies on any of these lines?

The coefficient of \({x^5}\) in the expansion of \({{{x^2} + 1} \over {({x^2} + 4)(x - 2)}}\) is

Consider a circle \(C\) which touches the \(y\)-axis at \((0,6)\) and cuts off an intercept \(6 \sqrt{5}\) on the \(x\) axis. Then the radius of the circle \(C\) is equal to:

Let \(N\) denotes the sum of the numbers obtained when two dice are rolled. If the probability that \(2^{ N } < N !\) is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(4 m -3 n\) is equal to \(......\).

let \(y = y\left( x \right)\) be the solution of the differential equation \(\sin x\frac{{dy}}{{dx}} + ycos\;x = 4x\;\), \(x \in \left( {0,\pi } \right)\) . If  \(y\left( {\frac{\pi }{2}} \right) = 0\) then \(y\left( {\frac{\pi }{6}} \right) = .\;.\;..\;\) .

A boy thrown a ball into air at \(45^{\circ}\) from the horizontal to land it on a roof of a building of height H . If the ball attains maximum height in 2 s and lands on the building in 3 s after launch, then value of H is _________ m .
\(\left( g =10 m / s ^2\right)\)