(a) Sign conventions
1. The object is always placed to the left of the mirror.
2. All the distances parallel to the principal axis are always measured from the pole of the spherical mirror.
3. All the distances measured along the direction of incident light (along +ve x-axis), are considered to be positive.
4. Those distances measured opposite to the direction of incidence light (i.e. along -ve x-axis), are taken as negative.
5. The distances measured in upward direction, i.e. perpendicular to and above the principal axis (along +ve y-axis), are taken as positive.
6. The distances measured in the downward direction, (along -ve y-axis), i.e. perpendicular to and below the principal axis are taken as negative.
(b) u=-16cm, m=-3 for real But \(m=-\frac { v }{ u } =-3\)
v = 3u = 3 (-16) = -48 cm. 
Using mirror formula
\(\frac { 1 }{ f } =\frac { 1 }{ v } +\frac { 1 }{ u } \)
We get, \(\frac { 1 }{ f } =\frac { 1 }{ -48 } +\frac { 1 }{ -16 } \)
\(=\frac { 1 }{ -48 } -\frac { 1 }{ 16 } =\frac { -1-3 }{ 48 } =\frac { -4 }{ 48 } =\frac { -1 }{ 12 } \)
f=-12cm
(c) Negative sign of focal length indicated that mirror is concave in nature.

(a)Laws of refraction of light states that the incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane.
(ii)The second law of refraction of light is the Snell's Law of Refraction. It state that the ratio of sine of angle of refraction is a constant for given pair of medium.
\({sin\ i\over sin\ r}\)=Constant (n)
This constant (n) is called refractive index of the medium.
(i)When the light is going from vacuum to another medium, then the value of refractive index is called the absolute refractive index.
(ii)The ratio of speed of light in vacuum to the speed of light in a medium is called the absolute refractive index of the medium, i.e.,
Absolute refractive index (of medium)=\(Speed\ of\ light\ in\ vacuum(C)\over Speed\ of\ light\ in\ medium(V)\)
(b)n_A=2.0;
n_B=1.5;
v_B=2 x 10⁸m/s
(i)Speed of light in vacuum, c=?
\(n_B=1.5={c\over v_B}\) 
c=n_B
v_B=1.5 x 2 x 10⁸ m/s
=3 x 10⁸ m/s
(ii)For medium 'A': n_A=\(c\over v_A\)
\(\therefore\ v_A={c\over n_B}={3\times10^8m/s\over2}\)
=1.5 x 10⁸ m/s

(a) Laws of Refraction: 
(i) The first law of refraction of light states that the incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane. 
(ii) The second law of refraction of light is the Snell's Law of Refraction. It state that the ratio of sine of the angle of incidence to the sine of angle of refraction is a constant for a given pair of medium.
\(\frac { sin\quad i }{ sin\quad r } =Constant(n)\)
This constant (n) is called refractive index of the medium. 
(i) When the light is going from vacuum to another medium, then the value of refractive index is called the absolute refractive index. 
(ii) The ratio of speed of light in vacuum to the speed of light in a medium is called the absolute refractive index of that medium, 
i.e. 
Absolute refractive index (of a medium)
\(=\frac { Speed\quad of\quad light\quad in\quad vacuum(c) }{ Speed\quad of\quad light\quad in\quad medium(v) } \)
(b) \(_{ a }{ { n }_{ w } }=\frac { 4 }{ 3 } ,_{ a }{ { n }_{ g } }=\frac { 3 }{ 2 } \)
Speed of light in glass, v_g= 2 x 10⁸ m/s 
Speed of light in air, v_g= ? 
Speed of light in water, v_w =?
\(_{ a }{ { n }_{ w } }=\frac { { v }_{ a } }{ { v }_{ w } } \)
\(\Rightarrow \frac { 4 }{ 3 } =\frac { { v }_{ a } }{ { v }_{ w } } \)      ...(i)
\(_{ a }{ { n }_{ g } }=\frac { { v }_{ a } }{ { v }_{ g } } \)
\(\Rightarrow \frac { 3 }{ 2 } =\frac { { v }_{ a } }{ 2\times { 10 }^{ 8 } } \)

Putting the value of v_a in equation (i),
\(\frac { 4 }{ 2 } =\frac { 3\times { 10 }^{ 8 } }{ { v }_{ w } } \)
\(\Rightarrow { v }_{ w }=3\times { 10 }^{ 8 }\times \frac { 3 }{ 4 } =\frac { 9 }{ 4 } \times { 10 }^{ 8 }\)
=2.25x10⁸ m/s.

(i) Snell's law states that, "the ratio of sine of angle of incidence to the sine of angle of refraction is constant for a given pair of media." 
\(\frac { sin\quad i }{ sin\quad r } \)=Constant
The value of constant for a ray of light passing from air into a particular medium is called the refractive index of the medium. 
Refractive index (of a medium) = \(\frac { Speed\quad of\quad light\quad in\quad vacuum }{ Speed\quad of\quad light\quad in\quad medium } \)
(ii) Refractive index of medium 'a' with respect to 'b'=\(_{ b }{ { n }_{ a } }=\frac { 2 }{ 3 } \)
\(_{ b }{ { n }_{ a } }=\frac { Speed\quad of\quad light\quad in\quad medium\quad 'b' }{ Speed\quad of\quad light\quad in\quad medium\quad 'a' } \)
\(=\frac { { v }_{ b } }{ { v }_{ a } } =\frac { 2 }{ 3 } \)
Refractive index of medium 'b' with respect to 'c' =\(_{ c }{ { n }_{ b } }=\frac { 4 }{ 3 } \)
\(_{ c }{ { n }_{ b } }=\frac { Speed\quad of\quad light\quad in\quad medium\quad 'c' }{ Speed\quad of\quad light\quad in\quad medium\quad 'b' } \)
\(=\frac { { v }_{ c } }{ { v }_{ b } } =\frac { 4 }{ 3 } \)
Refractive index of medium 'c' with respect to 'a' =\(_{ a }{ { n }_{ c } }=?\)
\(_{ a }{ { n }_{ c } }=\frac { Speed\quad of\quad light\quad in\quad medium\quad 'a' }{ Speed\quad of\quad light\quad in\quad medium\quad 'c' } \)
\(=\frac { { v }_{ a } }{ { v }_{ c } } =?\)
\(\because \quad \frac { { v }_{ b } }{ { v }_{ a } } =\frac { 2 }{ 3 } \Rightarrow { v }_{ a }=\frac { 3 }{ 2 } { v }_{ b }\)
\(\because \quad \frac { { v }_{ c } }{ { v }_{ b } } =\frac { 4 }{ 3 } \Rightarrow { v }_{ c }=\frac { 4 }{ 3 } { v }_{ b }\)

When light is going from vacuum to another medium, then the value of refractive index is called absolute refractive index to the medium. 
The absolute refractive index of diamond is 2.42. It means that the speed of light in diamond is 1/2.42 times the speed of light in vacuum. 
As the refractive indices increase, speed of the light decreases in the medium. 
(i) The refractive index of medium D is maximum (1.65). So the speed of light in medium D is minimum. 
(ii) The refractive index of medium A is minimum (1.33). So the speed of light in medium A is maximum. 
BⁿC=?           ...Refractive index of medium C with respect to medium B
\({ n }_{ B }=\frac { Speed\ of\ light\ in\ vaccum }{ Speed\ of\ light\ in\ medium\ B } \)
\(1.44=\frac { v }{ { v }_{ B } } \)
\(\Rightarrow { v }_{ B }=\frac { v }{ 1.44 } \quad \quad ...(i)\)
\({ n }_{ C }=\frac { Speed\quad of\quad light\quad in\quad vaccum }{ Speed\quad of\quad light\quad in\quad medium\quad C } \)
\(1.52=\frac { v }{ { v }_{ C } } \)
\(\Rightarrow { v }_{ C }=\frac { v }{ 1.52 } \quad \quad ...(ii)\)
\({ B }_{ C }^{ n }=\frac { Speed\quad of\quad light\quad in\quad medium\quad B }{ Speed\quad of\quad light\quad in\quad medium\quad C } \)
\({ B }_{ C }^{ n }=\frac { { v }_{ B } }{ { v }_{ C } } \)
\(\Rightarrow { B }_{ C }^{ n }=\frac { \frac { v }{ 1.44 } }{ \frac { v }{ 1.52 } } \quad \quad \quad ...[From(i)\quad and\quad (ii)]\)

=1.055=1.06

(a) The lens is concave 
(b) Optical centre is the point on the lens through which a ray of light passes undeviated. 
(c) Convex lens Focal length,f= +20 cm 
Object distance, u = -30 cm 
Image distance, v = ? 
Magnification, m = ? 
Nature of the image = ? 
According to lens formula:
\(\frac { 1 }{ u } =\frac { 1 }{ v } -\frac { 1 }{ f } \)
\(\Rightarrow \frac { 1 }{ v } -\frac { 1 }{ -30 } =\frac { 1 }{ 20 } \)
\(\Rightarrow \frac { 1 }{ v } -\frac { 1 }{ 30 } =\frac { 1 }{ 20 } \)
\(\Rightarrow \frac { 1 }{ v } =\frac { 1 }{ 20 } -\frac { 1 }{ 30 } =\frac { 3-2 }{ 60 } =\frac { 1 }{ 60 } \)
v=+60cm
\(\therefore \quad m=\frac { v }{ u } =\frac { 60 }{ -30 } =-2\)
Nature. The +ve sign of v shows that the image is formed on the right side of the convex lens, so the image formed is real.

(a) Optical centre of the lens. It is a point within the lens that lies on the principal axis through which away of light passes undeflected. 
= f=-20cm  h₁ = 4 cm 
v = -10 cm u=?
h₂=?
\(\frac { 1 }{ f } =\frac { 1 }{ v } -\frac { 1 }{ u } \)
\(\Rightarrow \frac { 1 }{ -20 } -\frac { 1 }{ 10 } -\frac { 1 }{ u } \)
\(\Rightarrow \frac { 1 }{ u } =\frac { 1 }{ 10 } +\frac { 1 }{ 20 } \)
\(\Rightarrow \frac { 1 }{ u } -\frac { 1 }{ 10 } +\frac { 1 }{ 20 } \)
\(\Rightarrow \frac { 1 }{ u } =\frac { -2+1 }{ 20 } \)
\(\Rightarrow \frac { 1 }{ u } =\frac { -1 }{ 20 } \)
Now,
\(\frac { { h }_{ 2 } }{ { h }_{ 1 } } =\frac { v }{ u } \)
\(\Rightarrow \frac { { h }_{ 2 } }{ 4 } =\frac { -10 }{ -20 } \)
\(\Rightarrow { h }_{ 2 }=\frac { 10 }{ 20 } \times 4=2cm\)
h₂=2cm

(a) The distance between optical centre and focus of the lens is called focal length of a spherical lens. 
(b) Diverging lens: Concave lens Focal -30 cm
Object distance, u =? 
Height of the object, h₁ = 5 cm 
Image distance, v = - 15 cm 
Image size, h₂ = ? 
According to the lens formula,
\(\frac { 1 }{ f } =\frac { 1 }{ v } +\frac { 1 }{ u } \)
\(\frac { 1 }{ -30 } =\frac { -1 }{ -15 } +\frac { 1 }{ u } \)
\(\Rightarrow \frac { 1 }{ u } =\frac { -1 }{ 15 } +\frac { 1 }{ 30 } =\frac { -2+1 }{ 30 } =\frac { -1 }{ 30 } \)
u=-30cm
\(\frac { { h }_{ 2 } }{ { h }_{ 1 } } =\frac { -v }{ u } \)
\(\Rightarrow \frac { { h }_{ 2 } }{ 5 } =\frac { -15 }{ -30 } \)
\(\Rightarrow { h }_{ 2 }=\frac { 5 }{ 2 } =2.5cm\)

(i) Yes.If a convex lens of focal length 10cm is covered one half with a black paper, it can produce an image of the complete object between F₂ and 2F₂.The rays of light coming from the object get refracted by the upper half of the lens. The image formed will be real, inverted and diminished.

(ii)Object height, h₁=4cm
Focal length, f=+20cm
Object distance, u=-15cm
Image distance, v=?
Image height, h₂=?
By lens formula,
\({1\over f}={1\over v}-{1\over u}\)
\(\Rightarrow\ {1\over v}={1\over f}+{1\over u}={1\over +20}+{1\over -15}={1\over 20}-{1\over 15}\)
\(\Rightarrow\ {1\over V}={3-4\over 60}={-1\over 60}\)
v=-60cm
Negative sign of v shows that the image is virtual.

It is defined as the ability of a lens to bend the rays of light it is given by the reciprocal of focal length in metre. Its unit is dioptre. The lens of focal length + 50 cm is a convex lens and the lens of focal length - 50 cm is a concave lens.
Their powers are
P1 = 1/F1 = 1/+0.5 = 2D
P2 = 1/F2 = 1/-0.5 = -2D 

The following are the laws of refraction of light.
(i)The incident ray, the refracted ray and the normal to the interface of two transparent media at the point of incidence, all lie in the same plane.
(ii)The ratio of sine of angle of incidence to the sine of angle of refraction is a constant, for the light of a given pair of media. This law is also known as Snell's law of refraction. The ray diagram is as shown. As seen in the refracted ray are in the same plane.