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 \(......\,\%\)

A particle of mass \(100\,g\) is projected at time \(t =0\) with a speed \(20\,ms ^{-1}\) at an angle \(45^{\circ}\) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time \(t=2\,s\) is found to be \(\sqrt{ K }\,kg\,m ^2 / s\). The value of \(K\) is \(............\) \(\left(\right.\) Take \(\left.g =10\,ms ^{-2}\right)\)

An ac source is connected in given series \(LCR\) circuit. The rms potential difference across the capacitor of \(20 \mu \mathrm{F}\) is _______\(v\). [\(\mathrm{V}=50 \sqrt{2} \sin 100 \mathrm{t}\) volt]

If the angular momentum of a planet of mass \(m\), moving a round the Sun in a circular orbit its \(L\), about the center of the Sun, its areal velocity is

Resistance are connected in a meter bridge circuit as shown in the figure. The balancing length \(l_{1}\) is \(40\,cm\). Now an unknown resistance \(x\) is connected in series with \(P\) and new balancing length is found to be \(80\,cm\) measured from the same end. Then the value of \(x\) will be \(.......\Omega\)

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis \(A B\) as shown in figure is \(\sqrt{\frac{8}{x}}\). The value of \(x\) is _______.

A girl standing on road holds her umbrella at \(45^{\circ}\) with the vertical to keep the rain away. If she starts running without umbrella with a speed of \(15 \sqrt{2} \; kmh ^{-1}\), the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is ...........  \(kmh ^{-1}\)

In terms of resistance \(R\) and time \(T\), the dimensions of ratio \(\frac{\mu } {\varepsilon }\) of the permeability \(\mu \) and permittivity \(\varepsilon \)  is

A system of three charges are placed as shown in the figure. If  \(D > > d,\) the potential energy of the system is best given \(b\)

A bullet of mass \(0.1\,kg\) moving horizontally with speed \(400\,ms ^{-1}\) hits a wooden block of mass \(3.9\,kg\) kept on a horizontal rough surface. The bullet gets embedded into the block and moves \(20\,m\) before coming to rest. The coefficient of friction between the block and the surface is \(........\) \(\left(\text { Given } g=10 \,ms ^2\right. \text { ) }\)

For a transverse wave travelling along a straight line, the distance between two peaks (crests) is \(5 \,m ,\) while the distance between one crest and one trough is \(1.5 \,m\) The possible wavelengths (in \(m\) ) of the waves are

A thin convex lens \(L\) (refractive index \(= 1.5\)) is placed on a plane mirror \(M\). When a pin is placed at \(A\), such that \(OA = 18\, cm\), its real inverted image is formed at \(A\) itself, as shown in figure. When a liquid of refractive index \({\mu _1}\) is put between the lens and the mirror, the pin has to be moved to \(A’\). such that \(OA’ = 27\,cm\), to get its inverted real image at \(A'\) itself. The value of \({\mu _1}\) will be