Distance between object and its image \(\left( \text{magnified by } -\frac{1}{3}\right)\) is 30 cm. The focal length of the mirror used is \(\left(\frac{x}{4}\right) \mathrm{cm}\),
where magnitude of value of \(x\) is _________.

In the given figure of meter bridge experiment, the balancing length \(AC\) corresponding to null deflection of the galvanometer is \(40\,cm\). The balancing length, if the radius of the wire \(AB\) is doubled, will be \(....cm\)

Intensity of sunlight is observed as \(0.092\, {Wm}^{-2}\) at a point in free space. What will be the peak value of magnetic field at that point? \(\left(\sigma_{0}=8.85 \times 10^{-12}\, {C}^{2} \,{N}^{-1} \,{m}^{-2}\right.\) )

Three rays of light, namely red (R), green (G) and blue (B) are incident on the face \(PQ\) of a right angled prism PQR as shown in figure. The refractive indices of the material of the prism for red, green and blue wavelength are \(1.27, 1.42\) and \(1.49\) respectively. The colour of the ray(s) emerging out of the face \(PR\) is:

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Consider a ferromagnetic material :
Assertion (A) : The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains.
Reason (R) : At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature.
In the light of the above statements, choose the correct answer from the options given below :

In the figure shown below, the charge on the left plate of the \(10\, \mu F\) capacitor is \(-30\,\mu C.\) The charge on the right place of the \(6\,\mu F\) capacitor is.....\(\mu C\)

A capacitor of capacitance \(500\,\mu F\) is charged completely using a dc supply of \(100\,V\). It is now connected to an inductor of inductance \(50\,mH\) to form an \(LC\) circuit. The maximum current in \(LC\) circuit will be \(.........A\).

The number of elements in the relation \( R=\{(x,y): 4x^{2}+y^{2}<52, x, y\in Z\} \) is

In a class of \(60\) students, \(40\) opted for \(NCC,\,30\) opted for \(NSS\) and \(20\) opted for both \(NCC\) and \(NSS.\) If one of these students is selected at random, then the probability that the student selected has opted neither for \(NCC\) nor for \(NSS\) is

A ray of light \(AO\) in vacuum is incident on a glass slab at angle \(60^o\) and refracted at angle \(30^o\) along \(OB\) as shown in the figure. The optical path length of light ray from \(A\) to \(B\) is

For the function \(\mathrm{f}(\mathrm{x})=(\cos \mathrm{x})-\mathrm{x}+1, \mathrm{x} \in \mathbb{R}\), between the following two statements (\(S1\)) \(f(x)=0\) for only one value of \(x\) is \([0, \pi]\). (\(S2\)) \(\mathrm{f}(\mathrm{x})\) is decreasing in \(\left[0, \frac{\pi}{2}\right]\) and increasing in \(\left[\frac{\pi}{2}, \pi\right] .\)

Let \(y = y(x)\) be the solution of the differential equation, \(x\frac{{dy}}{{dx}} + y = x\,{\log _e}\,x,\,\left( {x > 1} \right)\) If \(2y(2) = log_e\, 4 -1\), then \(y(e)\) is equal to

Let the shortest distance between the lines \(L : \frac{ x -5}{-2}=\frac{ y -\lambda}{0}=\frac{ z +\lambda}{1}, \lambda \geq 0\) and \(L _1: x +1= y -\) \(1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\), then which of the following is NOT possible?