A vector \(\vec A \) is rotated by a small angle \(\Delta \theta\) radian \(( \Delta \theta << 1)\) to get a new vector \(\vec B\) In that case \(\left| {\vec B - \vec A} \right|\) is

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is

A transistor is used in common-emitter mode in an amplifier circuit. When a signal of \(10\, mV\) is added to the base-emitter voltage, the base current changes by \(10 \mu A\) and the collector current changes by \(1.5 \,mA\). The load resistance is \(5 \,k \Omega\). The voltage gain of the transistor will be

The self produced \(emf\) of a coil is \(25\,volts\). When the current in it is changed at uniform rate from \(10\,A\) to \(25\, A\) in \(1\,s\), the change in the energy of the inductance is......\(J\)

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 : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the minimum thickness of transparent film to be coated for the maximum transmission of Green light of wavelength 550 nm . [Assume that the light is incident nearly perpendicular to the glass surface.]

Two bodies are projected from ground with same speeds \(40 ms ^{-1}\) at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of \(60^{\circ}\), with horizontal then sum of the maximum heights, attained by the two projectiles, is \(.......m\). (Given \(g =10 ms ^{-2}\) )

The ratio of the coefficient of \(x^{15}\) to the term independent of \(x\) in the expansion of \({\left( {{x^2} + \frac{2}{x}} \right)^{15}}\) is

Let \(a\) be a positive real number such that \(\int_{0}^{a} e^{x-[x]} d x=10 e-9\) where \([x]\) is the greatest integer less than or equal to \(x\). Then \(a\) is equal to:

Let \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) be a solution curve of the differential equation \((y+1) \tan ^{2} x d x+\tan x d y+y d x=0\) \(x \in\left(0, \frac{\pi}{2}\right)\). If \(\lim _{x \rightarrow 0+} x y(x)=1\), then the value of \(\mathrm{y}\left(\frac{\pi}{4}\right)\) is :

A bob of mass \(m\) is suspended at a point \(O\) by a light string of length \(l\) and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity \(v_0\) at the point ' A ', the string becomes slack when, the bob reaches at the point ' D '. The ratio of the kinetic energy of the bob at the points \(B\) and \(C\) is ______ -.

If \({ }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_n=11: 21\), then \(n^2+n+15\) is equal to:

A particle executes simple harmonic motion with an amplitude of \(5\, cm\). When the particle is at \(4\, cm\) from the mean position, the magnitude of its velocity is \(SI\,units\) is equal to that of its acceleration. Then, its periodic time in seconds is