A thin cylindrical rod of length \(10\,cm\) is placed horizontally on the principle axis of a concave mirror of focal length \(20\,cm\). The rod is placed in a such a way that mid point of the rod is at \(40\,cm\) from the pole of mirror. The length of the image formed by the mirror will be \(\frac{x}{3} cm\). The value of \(x\) is \(............\).

A rectangular loop of sides \(12 \mathrm{~cm}\) and \(5 \mathrm{~cm}\), with its sides parallel to the \(x\)-axis and \(y\)-axis respectively moves with a velocity of \(5 \mathrm{~cm} / \mathrm{s}\) in the positive \(\mathrm{x}\) axis direction, in a space containing a variable magnetic field in the positive \(z\) direction. The field has a gradient of \(10^{-3} \mathrm{~T} / \mathrm{cm}\) along the negative \(\mathrm{x}\) direction and it is decreasing with time at the rate of \(10^{-5} \mathrm{~T} / \mathrm{s}\). If the resistance of the loop is \(6 \mathrm{~m} \Omega\), the power dissipated by the loop as heat is _______ \(\times 10^{-9} \mathrm{~W}\).

In the figure shown, what is the current (in Ampere) drawn from the battery ? You are given \(R_1 = 15\,\Omega \)\(,R _2 = 10\,\Omega ,\)\( R_3 = 20\,\Omega ,\)\( R_4 = 5\,\Omega ,\)\(R_5 = 25\,\Omega ,\)\(R_6 = 30\,\Omega , \)\(E = 15\,V\)

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through \(O\) (the centre of mass) and \(O ^{\prime}(\) corner point \()\) is 

In \(LC\) circuit the inductance \(\mathrm{L}=40\; \mathrm{mH}\) and capacitance \(\mathrm{C}=100\; \mu \mathrm{F} .\) If a voltage \(\mathrm{V}(\mathrm{t})=10 \sin (314 \mathrm{t})\) is applied to the circuit, the current in the circuit is given as 

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason \(R\) Assertion \(A:\) The beam of electrons shows wave nature and exhibit interference and diffraction. Reason \(R\) : Davisson Germer Experimentally verified the wave nature of electrons. In the light of the above statements. Choose the most appropriate answer from the options given below :

A spherical body of mass \(100 \mathrm{~g}\) is dropped from a height of \(10 \mathrm{~m}\) from the ground. After hitting the ground, the body rebounds to a height of \(5 \mathrm{~m}\). The impulse of force imparted by the ground to the body is given by : _______. (given \(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\) )

Consider a circular coil of wire carrying constant current \(I\), forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by \(\phi_{i} .\) The magnetic flux through the area of the circular coil area is given by \(\phi_{0} .\) Which of the following option is correct?

Let \(a, b, c \in \{1, 2, 3, 4\}\). If the probability, that \(ax^2 + 2\sqrt{2}\,bx + c > 0\) for all \(x \in \mathbb{R}\), is \(\dfrac{m}{n}\), \(\gcd(m, n) = 1\), then \(m + n\) is equal to _______.

The first overtone frequency of an open organ pipe is equal to the fundamental frequency of a closed organ pipe. If the length of the closed organ pipe is \(20 \,cm\). The length of the open organ pipe is........\(cm\).

If the distance of the earth from Sun is \(1.5 \times 10^6\,km\). Then the distance of an imaginary planet from Sun, if its period of revolution is \(2.83\) years is \(.............\times 10^6\,km\)

If tangents are drawn to the ellipse \(x^2 + 2y^2 = 2\) at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is