Three identical spheres each of diameter \(2 \sqrt{3} \mathrm{~m}\) are kept on a horizontal surface such that each sphere touches the other two spheres. If one of the sphere is removed, then the shift in the position of the centre of mass of the system is

A convex lens has its radii of curvature equal. The focal length of the lens is \(f\). If it is divided vertically into two identical plano-convex lenses by cutting it, then the focal length of the plano-convex lens is ( \(\mu=\) the refractive index of the material of the lens)

In the following circuit, the output \(Y\) becomes zero for the input combinations.

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 \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

An alternating current is given by \(\mathrm{i}=(3 \sin \omega t+4 \cos \omega t)\)
A. The \(r m s\) current will be

When a force $\overrightarrow{F}=17-2x+6x^2 \mathrm{N}$ acts on a body of mass $2 \mathrm{kg}$ and displaces it from $\mathrm{x}= 0 \mathrm{m}$ to $x=8 \mathrm{m}$, the work done is

Match the following (Take the relative strength of the strongest fundamental forces in nature as one)


The correct match is

A bimetallic strip is made of aluminium and steel \(\left(\alpha_{\mathrm{Al}}>\alpha_{\text {steel }}\right)\). On heating, the strip will

If \(\tan \theta+\tan 2 \theta+\sqrt{3} \tan \theta \tan 2 \theta=\sqrt{3}\), then the general values of \(\theta\) are

The set of all real values of \(x\) satisfying the inequality \(\frac{7 x^2-5 x-18}{2 x^2+x-6} \lt 2\) is

Which of the following statements is incorrect with reference to alkaline earth metals?

If \(A=\left[\begin{array}{ccc}1 & 1 & a+1 \\ 1 & a+1 & 1 \\ a+1 & 1 & 1\end{array}\right]\) is not an invertible matrix, then the sum of all the values of \(a\) is

The pole of the straight line \(9 x+y-28=0\) with respect to the circle \(2 x^2+2 y^2-3 x+5 y-7=0\) is