A block of mass \(200\, g\) is kept stationary on a smooth inclined plane by applying a minimum horizontal force \(F =\sqrt{ x }N\) as shown in figure. The value of \(x =.....\)

\(5\,g\) and radius \(1\,cm\) is fixed to a thin stick \(AB\) of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about \(AB\) at \(25\) rotations per second in \(5\,s\) is close to

A homogeneous solid cylindrical roller of radius \(R\) and mass \(M\) is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is

A granite rod of \(60\ cm\) length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is \(2.7 \times 10^3 \) \(kg/m^3\) and its Young's modulus is \(9.27 \times 10^{10}\) \(Pa\) What will be the fundamental frequency of the longitudinal vibrations .... \(kHz\) ?

In a semiconductor, the number density of intrinsic charge carriers at \(27^{\circ} {C}\) is \(1.5 \times 10^{16} \,/ {m}^{3}\). If the semiconductor is doped with impurity atom, the hole density increase to \(4.5 \times 10^{22}\, / {m}^{3}\). The electron density in the doped semiconductor is \(.....\,\times 10^{9} / {m}^{3}\)

The current \(i\) in the network is.....\(A\)

If \(\overrightarrow{ A }=(2 \hat{ i }+3 \hat{ j }-\hat{ k }) \;m\) and \(\overrightarrow{ B }=(\hat{ i }+2 \hat{ j }+2 \hat{ k })\; m\). The magnitude of component of vector \(\overrightarrow{ A }\) along vector \(\vec{B}\) will be \(......m\).

The integral \(\int_{\frac{\pi }{{12}}}^{\frac{\pi }{4}} {\frac{{8\,\cos \,2x}}{{{{\left( {\tan \,x + \cot \,x} \right)}^3}}}\,dx} \) equals

Consider a frame that is made up of two thin massless rods \(AB\) and \(AC\) as shown in the figure. \(A\) vertical force \(\overrightarrow{ P }\) of magnitude \(100 \;N\) is applied at point \(A\) of the frame. Suppose the force is \(\overrightarrow{ P }\) resolved parallel to the arms \(AB\) and \(AC\) of the frame. The magnitude of the resolved component along the arm \(AC\) is \(xN\). The value of \(x\), to the nearest integer, is ............ [Given : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

The value of \(\alpha\) for which \(4 \alpha \int\limits_{-1}^{2} \mathrm{e}^{-\alpha \mathrm{|x|} } \mathrm{d} \mathrm{x}=5,\) is 

Let \(C\) be the circle with centre at \((1, 1)\) and radius \(= 1\). If  \(T\) is the circle centred at \((0, y),\) passing through origin and touching the circle \(C\) externally, then the radius of \(T\) is equal 

Let A and B be two distinct points on the line \(\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}\). Both A and B are at a distance \(2 \sqrt{17}\) from the foot of perpendicular drawn from the point \((1,2,3)\) on the line L . If O is the origin, then \(\overrightarrow{O A} \cdot \overrightarrow{O B}\) is equal to:

The function \(f : R \rightarrow R\) defined by \(f(x)=\lim _{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is continuous for all \(x\) in.