The aperture diameter of a telescope is \(5\; \mathrm{m} .\) The separation between the moon and the earth is \(4 \times 10^{5} \;\mathrm{km} .\) With light of wavelength of \(5500\;\mathring A\), the minimum separation between objects on the surface of moon, so that they are just resolved, is close to......\(m\)

The graph shows variation of stopping potential \(V_o\) with the frequency \(\nu\) of the incident radiation for three photosensitive metals \(X_1\), \(X_2\) and \(X_3\). Which metal will give out electrons with greater kinetic energy, for the same wavelength of incident radiation?

Consider a uniform wire of mass \(M\) and length \(L\). It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is :

The Boolean expression \(\mathrm{Y}=\mathrm{A} \overline{\mathrm{B}} \mathrm{C}+\overline{\mathrm{A}} \overline{\mathrm{C}}\) can be realised with which of the following gate configurations.
A. One 3-input AND gate, 3 NOT gates and one 2input OR gate, One 2-input AND gate,
B. One 3-input AND gate, 1 NOT gate, One 2input NOR gate and one 2-input OR gate
C. 3-input OR gate, 3 NOT gates and one 2-input AND gate
Choose the correct answer from the options given below.

The ratio of average electric energy density and total average energy density of electromagnetic wave is:

A wire of cross sectional area \(A\), modulus of elasticity \(2 \times 10^{11} \mathrm{Nm}^{-2}\) and length \(2 \mathrm{~m}\) is stretched between two vertical rigid supports. When a mass of \(2 \mathrm{~kg}\) is suspended at the middle it sags lower from its original position making angle \(\theta=\frac{1}{100}\) radian on the points of support. The value of \(A\) is _______ \(\times 10^{-4} \mathrm{~m}^2\) (consider \(\mathrm{x}<\mathrm{L}\) ). (given: \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

A capacitor is connected to a \(20\, {V}\) battery through a resistance of \(10\, \Omega .\) It is found that the potential difference across the capacitor rises to \(2\, {V}\) in \(1\, \mu {s}\). The capacitance of the capacitor is \(....\,\mu {F}\) Given : \(\ln \left(\frac{10}{9}\right)=0.105\)

If \({I_1} = \int\limits_0^1 {{e^{ - x}}} {\cos ^2}\,x\,dx\,;\,{I_2} = \int\limits_0^1 {{e^{ - {x^2}}}} {\cos ^2}\,x\,dx\) and \(\,{I_3} = \int\limits_0^1 {{e^{ - {x^3}}}} dx\) ; then

Let the solution curve \(y = y ( x )\) of the differential equation \(\quad \frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{\frac{3}{2}}} y=2 x\) \(\exp \frac{x^3-\tan ^{-1} x^3}{\sqrt{(1+x)^6}}\) pass through the origin. Then \(y (1)\) is equal to:

If \(2\,cos\,\theta  + sin\, \theta \, = 1\) \(\left( {\theta  \ne \frac{\pi }{2}} \right)\) , then \(7\, cos\,\theta + 6\, sin\, \theta \) is equal to

With reference to the observations in photo-electric effect, identify the correct statements from below: \(A.\) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light. \(B.\) The value of saturation current increases on moving the source of light away from the metal surface. \(C.\) The maximum kinetic energy of photo-electrons decreases on decreasing the power of LED (light emitting diode) source of light. \(D.\) The immediate emission of photo-electrons out of metal surface can not be explained by particle nature of light/electromagnetic waves. \(E.\) Existence of threshold wavelength can not be explained by wave nature of light/electromagnetic waves. Choose the correct answer from the options given below:

The de Broglie wavelength of an oxygen molecule at \(27^{\circ} C\) is \(x \times 10^{-12} m\). The value of \(x\) is (take Planck's constant \(=6.63 \times 10^{-34} J . s\), Boltzmann constant \(=1.38 \times 10^{-23} J / K\), mass of oxygen. Molecule \(\left.=5.31 \times 10^{-26} kg\right)\).

If \(a+\alpha=1, b+\beta=2\) and \(\operatorname{af}(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}, x \neq 0,\) then the value of expression \(\frac{ f ( x )+ f \left(\frac{1}{ x }\right)}{ x +\frac{1}{ x }}\) is ..... .