A diving board is at a height of ' \(h\) ' from the water surface, A swimmer standing on this board thrown a stone vertically upward with a velocity \(16 \mathrm{~ms}^{-1}\). It reaches the water surface in a time of 5 s . In the next 0.2 s , the diver can hear the sound from water surface. The speed of sound is (acceleration due to gravity \(\mathrm{g}=10 \mathrm{~ms}^{-2}\) )

If the length of a stretched string is shortened by \(40 \%\) and the tension is increased by \(44 \%\), then the ratio of the final and initial fundamental frequencies is :

A toy car of mass \(100 \mathrm{~g}\) is moving with velocity of \((\hat{i}+2 \hat{j}-3 \hat{k}) m\), then the kinetic energy of the car is

The expressions below give current \(I\) through an electronic component as a function of applied potential \(V . I_0\) and \(V_0\) are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
(A) \(I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)\)
(B) \(I=I_0\left(e^{\frac{v}{2 V_0}}-1\right)\)
(C) \(I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)\)
(D) \(I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)\)

Two solid spheres \(A\) and \(B\) made of the same material have radii \(r_A\) and \(r_B\) respectively. Both the spheres are cooled from the same temperature under the conditions valid for Newton's law of cooling. The ratio of the rate of change of temperature of \(A\) and \(B\) is :

In a thermodynamic process if ' \(\Delta \mathrm{W}\) ' and ' \(\Delta \mathrm{U}\) ' are the work done and change in the internal energy of a system respectively, then

A rubber cube of side \(5 \mathrm{~cm}\) has one face fixed, while a tangential force \(1800 \mathrm{~N}\) is applied on its opposite face. If modulus of rigidity of rubber is \(2.4 \times 10^6 \mathrm{Nm}^{-2}\), then the lateral displacement of the strained face is ________

If each line of a pair of lines passing through origin is at a perpendicular distance of 4 units from the point \((3,4)\), then the equation of the pair of lines is

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

\(\sin ^4 \frac{\pi}{8}+\sin ^4 \frac{3 \pi}{8}+\sin ^4 \frac{5 \pi}{8}+\sin ^4 \frac{7 \pi}{8}=\)

A satellite is placed in a circular orbit around the earth at an altitude of \(1000 \mathrm{~km}\). The time period of the satellite in minutes is approximately
(mass of the earth \(=6 \times 10^{24} \mathrm{~kg}\), radius of the earth \(=6.4\) \(\times 10^6 \mathrm{~m}, \mathrm{G}=6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}\) )

The range of the real valued function \(f(x)=\frac{x^2+2 x-15}{2 x^2+13 x+15}\) is

A sphere of mass \(2 \mathrm{~kg}\) and diameter \(4.5 \mathrm{~cm}\) is attached to the lower end of a steel wire of \(2 \mathrm{~m}\) length and area of cross-section \(0.24 \times 10^{-6} \mathrm{~m}^2\). The wire is suspended from \(205 \mathrm{~cm}\) high ceiling of a room. When the system is made to oscillate as a simple pendulum, the sphere just grazes the floor at its lowest position. The velocity of the sphere at the lowest position is (Young's modulus of steel \(=2 \times 10^{11} \mathrm{Nm}^{-2}\) and acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )