Two vectors \(\vec A\) and \(\vec B\) have equal magnitudes. The magnitude of \((\vec A + \vec B)\) is \(‘n’\) times the magnitude of \((\vec A - \vec B)\). The angle between \( \vec A\) and \(\vec B\) is

The half-life of a radioactive substance is \(T\). The time taken, for disintegrating \(\frac{7}{8}\) th part of its original mass will be

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

If the binding energy of ground state electron in a hydrogen atom is \(13.6\,eV\), then, the energy required to remove the electron from the second excited state of \(Li ^{2+}\) will be: \(x \times 10^{-1}\,eV\). The value of \(x\) is \(...........\)

A convex lens of focal length 30 cm is placed in contact with a concave lens of focal length 20 cm. An object is placed at 20 cm to the left of this lens system. The distance of the image from the lens in cm is ________

The current in an inductor is given by \(\mathrm{I}=(3 \mathrm{t}+8)\) where \(t\) is in second. The magnitude of induced emf produced in the inductor is \(12 \mathrm{mV}\). The selfinductance of the inductor _______ \(\mathrm{mH}\).

A force acts on a \(2\,kg\) object so that its position is given as a function of time as \(x= 3t^2 + 5.\) What is the work done by this force in first \(5\,seconds\) ? ................ \(\mathrm{J}\)

If \(\int_0^{\frac{\pi }{2}} {\frac{{\cot \,x}}{{\cot \,x + \cos ec\,x}}} dx = m\left( {\pi  + n} \right)\), then \(m.n\) is equal to

The measured value of the length of a simple pendulum is \(20 \mathrm{~cm}\) with \(2 \mathrm{~mm}\) accuracy. The time for \(50\) oscillations was measured to be \(40\) seconds with \(1\) second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is \(\mathrm{N} \%\). The value of \(\mathrm{N}\) is _______.

A particle moves from the point \(\left( {2.0\hat i + 4.0\hat j} \right)\,m\), at \(t = 0\) with an initial velocity \(\left( {5.0\hat i + 4.0\hat j} \right)\,m{s^{ - 1  }}\). It is acted upon by a constant force which produces a constant acceleration \(\left( {4.0\hat i + 4.0\hat j} \right)\,m{s^{ - 2}}\). What is the distance of the particle from the origin at time \(2\,s\)

Two loudspeakers ( \(L_1\) and \(L_2\) ) are placed with a separation of 10 m , as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant volume. A voice recorder, initially at point \(A\), at equidistance to both loud speakers, is moved by 25 m along the line \(A B\) while monitoring the audio signal. The measured signal was found to undergo 10 cycles of minima and maxima during the movement. The frequency of the input signal is __________ Hz
(Speed of sound in air is \(324 m / s\) and \(\sqrt{5}=2.23\) )

If the line \(y =4+ kx , k >0\), is the tangent to the parabola \(y = x - x ^{2}\) at the point \(P\) and \(V\) is the vertex of the parabola, then the slope of the line through \(P\) and \(V\) is

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of \(20 \mathrm{~cm}\) from its center is \(1.5 \times 10^{-5} \  \mathrm{Tm}\). The magnetic moment of the dipole is _______ \(\mathrm{Am}^2\). (Given : \(\frac{\mu_0}{4 \pi}=10^{-7} \  \mathrm{TmA}^{-1}\) )