(a) 1/4 and 1/3
In order to find a rational number lying between two rational numbers, first of all, make the denominators of both the numbers same.
To do that, first we find the LCM of the denominators of both the numbers. 
LCM (4, 3) = 12

Now we divide the LCM by denominator of first number, and we get a quotient.
12/4 = 3
Now multiply the numerator and denominator of first number by this quotient.
(1×3)/(4×3) = 3/12

Now we divide the LCM by denominator of second number, and we get another quotient.
12/3 = 4
Now multiply the numerator and denominator of second number by this quotient.
(1×4)/(3×4) = 4/12

Now our numbers are: 3/12 and 4/12

Now we consider the numerators of both the numbers which are 3 and 4. Since there is no integer between 3 and 4, we will have to apply one more step. 
We multiply by the numerators and denominators of both the numbers by a Natural number like 5 or 10 etc in order to get rational numbers.

Let's multiply by 10:
Numbers = (3×10)/(12×10) and (4×10)/(12×10)
Numbers = 30/120 and 40/120 

Now we can write the rational numbers between 30/120 and 40/120 as 
31/120, 32/120, 33/120, 34/120, 35/120, 36/120, 37/120, 38/120, 39/120

We can choose any of them as per the requirement. 

(b) Not mentioned clearly

(c) -1 and 1/2
Let's repeat the same procedure as mentioned in part (a)

LCM (1,2) = 2

For first number, quotient = LCM/1 = 2/1 = 2
First number = (-1×2)/(1×2) = -2/2

For second number, quotient = LCM/2 = 2/2 = 1
Second Number = (1×1)/(2×1) = 1/2

Now our numbers are -2/2 and 1/2.
Now we consider the numerators of both the numbers which are -2 and 1. So -1 and 0 are the intergers between -2 and 1.

So rational numbers between -2/2 and 1/2 are -1/2, 0
We can choose any them as required.