Train \(A\) and train \(B\) are running on parallel tracks in the opposite directions with speeds of \(36\, km / hour\) and \(72 \,km / hour,\) respectively. A person is walking in train \(A\) in the direction opposite to its motion with a speed of \(1.8\, km / hr\). Speed (in \(ms ^{-1}\) ) of this person as observed from train \(B\) will be close to (take the distance between the tracks as negligible)

A bullet is fired into a fixed target looses one third of its velocity after travelling \(4 \mathrm{~cm}\). It penetrates further \(\mathrm{D} \times 10^{-3} \mathrm{~m}\) before coming to rest. The value of \(D\) is:

A gas mixture consists of \(2\) moles of oxygen and 4 moles of neon at temperature \(T\). Neglecting all vibrational modes, the total internal energy of the system will be \(...........\,RT\) 

A body moves on a frictionless plane starting from rest. If \(\mathrm{S}_{\mathrm{n}}\) is distance moved between \(\mathrm{t}=\mathrm{n}-1\) and \(\mathrm{t}\) \(=\mathrm{n}\) and \(\mathrm{S}_{\mathrm{n}-1}\) is distance moved between \(\mathrm{t}=\mathrm{n}-2\) and \(t=n-1\), then the ratio \(\frac{S_{n-1}}{S_n}\) is \(\left(1-\frac{2}{x}\right)\) for \(n\) \(=10\). The value of \(x\) is _______.

In an expression a \(\times 10^{\mathrm{b}}\) _______.

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 force of \(40\, N\) acts on a point \(B\) at the end of an \(L-\) shaped object, as shown in the figure. The angle \(\theta\) that will produce maximum moment of the force about point \(A\) is given by

\(\begin{array}{c|c|c} A & B & Y \\\hline 0 & 0 & 1 \\0 & 1 & 1 \\1 & 0 & 0 \\1 & 1 & 1\end{array}\)
To obtain the given truth table, following logic gate should be placed at G :

A square loop is carrying a steady current \(I\) and the magnitude of its magnetic dipole moment is \(m\). If this square loop is changed to a circular loop and it carries the same current, the magnitude of the magnetic dipole moment of circular loop will be

Two tangent lines \(l_{1}\) and \(l_{2}\) are drawn from the point \((2,0)\) to the parabola \(2 y^{2}=-x\). If the lines \(l_{1}\) and \(l_{2}\) are also tangent to the circle \((x-5)^{2}+y^{2}=r\), then \(17 r\) is equal to.

Let \(P\) be a moving point on the circle \(x^2 + y^2 - 6x - 8y + 21 = 0\). Then, the maximum distance of \(P\) from the vertex of the parabola \(x^2 + 6x + y + 13 = 0\) is equal to:

The integral \(80 \int_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta\) is equal to :

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg , kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is