Two identical glass $\left({\mu }_g=3/2\right)$ equi-convex lenses of focal length $f$ each are kept in contact. The space between the two lenses is filled with water $\left({\mu }_w=4/3\right)$. The focal length of the combination is

(a)  $f/3$             (b) $f$

(c) $\frac{4f}{3}$               (d) $\frac{3f}{4}$

Match the corresponding entries of Column 1 with Column 2. [Where m is the magnification produced by the mirror]

Column 1                         Column 2 
A. m=-2                     a. Convex mirror   
B. m=-1/2                  b. Concave mirror
C. m=+2                    c. Real image
D. m=+1/2                 d. Virtual Image

(1)A->a and c;B->a and d; C->a and b; D->c and d

(2)A->a and d; B->b and c; C->b and d; D-> b and c

(3)A->c and d; B->b and d;C->b and c;D->a and d

(4)A->b and c; B->b and c; C->b and d; D->a and d

The angle of incidence for a ray of light at a refracting surface of a prism is 45°. The angle of prism is 60°. If the ray suffers minimum deviation through the prism, the angle of deviation and refracting index of the material of the prism respectively are

(a)30°,$\sqrt{2}$

(b)45°,$\sqrt{2}$

(c)30°,$\frac{1}{\sqrt{2}}$

(d)45°,$\frac{1}{\sqrt{2}}$

A ray of light is incident at an angle of incidence, \(i\), on one face of a prism of angle A (assumed to be small) and emerges normally from the opposite face. If the refractive index of the prism is \(\mu\), the angle of incidence \(i\), is nearly equal to:
1. \(\mu A\)  
2. \(\dfrac{\mu A}{2}\) 
3. \(\frac{A}{\mu}\)
4. \(\frac{A}{2\mu}\)                                 

A convex lens A of focal length \(20~\text{cm}\) and a concave lens \(B\) of focal length \(5~\text{cm}\) are kept along the same axis with the distance \(d\) between them. If a parallel beam of light falling on \(A\) leaves \(B\) as a parallel beam, then distance \(d\) in \(\text{cm}\) will be:
1. \(25\)                         
2. \(15\) 
3. \(30\)                         
4. \(50\)

The refractive index of the material of a prism is $\sqrt{\mathrm{2 }}$and the angle of the prism is \(30^\circ.\) One of the two refracting surfaces of the prism is made a mirror inwards with a silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if the angle of incidence on the prism is:

A beam of light from a source \(L\) is incident normally on a plane mirror fixed at a certain distance \(x\) from the source. The beam is reflected back as a spot on a scale placed just above the source \(L\). When the mirror is rotated through a small angle \(\theta,\) the spot of the light is found to move through a distance \(y\) on the scale. The angle \(\theta\) is given by:

A thin prism having refracting angle \(10^\circ\) is made of glass of a refractive index \(1.42\). This prism is combined with another thin prism of glass with a refractive index \(1.7\). This combination produces dispersion without deviation. The refracting angle of the second prism should be:
1. \(6^{\circ}\)
2. \(8^{\circ}\)
3. \(10^{\circ}\)
4. \(4^{\circ}\)

An air bubble in a glass slab with a refractive index \(1.5\) (near-normal incidence) is \(5\) cm deep when viewed from one surface and \(3\) cm deep when viewed from the opposite surface. The thickness (in cm) of the slab is: