Solution

Q.1 ABC is an Isoceles triangle with AB=AC. A circle through B touching AC at the middle point intersects AB at P. Then AP:AB is:

Solution: First Draw the figure

Now The Informations are AB=AC and D is the middle point of AC so AD=DC=AC/2=AB/2 (Since AB=AC)

Since AD is the tangent to the Circle so 

AD^2= AP*AB

(AC/2)^2 = AP*AB

(AB/2)^2 = AP*AB

AB^2 / 4 = AP*AB

By Dividing both sides by AB^2 we get

1/4 = AP/AB

So the answer is AP:AB= 1:4

Q2. A,B,C and D purchase a gift worth Rs 60. A pays 1/2 of what others are paying, B pays 1/3rd of what others are paying and C pays 1/4th of what others are paying. What is the amount paid by D?

Solution: A+B+C+D=60 -----------> (Equation 1)

Now According to Question A= 1/2 (B+C+D) ------> B+C+D = 2A 

Put the value of B+C+D in Equation 1

A+2A =60 ------> 3A=60-------> A=20

Now again according to Question 

B=1/3 (A+C+D)--------> A+C+D= 3B

By Putting the Value of A+C+D in Equation 1

we get    B+3B= 60------> 4B=60 ---------> B=15

Now again accoring to Question 

C= 1/4 (A+B+D) ---------> A+B+D = 4C

By Putting the Value of A+B+D in equation 1 

We get C+4C = 60------> 5C= 60 ------> C=12

Since A+B+C+D= 60

20+15+12+D=60------> D=60-47=13

Answer : D=13