To complete a certain work A alone would take m times as many days as B and C together; B would take n times as many days as A and C together; C alone would take p times as many days as A and B together. Show that the number of days in which each do it alone are as . Prove also .

Solve the equation
 

Solve
 
 

Prove that

Prove if a, b, c are unequal and in harmonic progression, then
 

Prove if a, b, c are in harmonic progression, then
 

If the sum of m terms of an arithmetic progression is equal to sum of either the next n terms or the next p terms, prove that
 

Prove that

Eliminate x, y, z from the equations
 
 
 
 

A geometric and harmonic progression have the same p^th, q^th, r^th terms a, b, c respectively. Show that

Solve the equation  

Solve the equations
 

Solve the equation
 

Find the value of

A waterman rows a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. Find the rate of the stream.

If α, β are roots of  and γ, δ are roots of  , show that .

In the binomial expansion of , the sum of 5^th and 6^th terms is zero. Then    equals

a)

b)

c)

d)

The sum of the series   is

a)

b)

c) 0

d)

If the coefficient in powers of x of the function  is
 , then  is

a)

b)

c)

d)

For the natural numbers m, n if
  and  

then (m, n) is

a) (35, 20)

b) (45, 35)

c) (35, 45)

d) (20, 45)

The value of  is

a)

b)

c)

d)

The coefficient of the   and  terms in the binomial expansion of  are in arithmetic progression then m and r satisfy the relation

a)

b)

c)

d)

If the coefficient of  in  equals the coefficient of  in  , then a and b satisfy the relation

a)

b)  1

c)

d)