Iit foundation grades 9 & 10 - sample questions

1. Evelyn Learning Systems
Class 9, Topic- Logarithms
Question:
If 2 2
7x y xy  , then prove that  
1
log log log
3 2
x y
x y
 
  
 
. Given that 0, 0x y  .
Solution:
Using the identity 
2 2 2
2a b a b ab    , we simplify 2 2
7x y xy  as shown:
 
 
2 2
2 2
2
2
2
7
2 7 2
9
9
3
3
x y xy
x y xy xy xy
x y xy
x y
xy
x y
xy
x y
xy
 
    
  

 
 
  
 
 
  
 
Consider the equation
3
x y
xy
 
 
 
Take log on both sides of the above equation:
 
log log
3
1
log log
3 2
1
log log log
3 2
x y
xy
x y
xy
x y
x y
 
 
 
 
  
 
 
   
 

2. Evelyn Learning Systems
Class 10. Topic- Arithmetic Progression
Question:
If a, b, c are in AP, then prove that      2 2 2
, ,a b c b c a c a b   are also in AP.
Solution:
To prove that      2 2 2
, ,a b c b c a c a b   are in AP, we need to show that the following
equation holds true:
       
     
     
2 2 2 2
2 2 2
2 2 2
2
2
b c a a b c c a b b c a
b c a c a b a b c
c a b a b c b c a
      
     
     
In order to solve this question, we will simply show that the above equation holds true.
2 2
2 2 2 2
2 2 2 2
LHS c a b a b c
c a c b a b a c
a b c b a c c a
2 2
b a c ac a c …….(i)
Since a, b, c are in AP, we get
2
2
b a c b
b a c
a c
b
  
 


Substitute the above back in Equation (i):
2 2
2 2
2 2
3
2
2
2
LHS
2
2
2
2
2
4
2
2
RHS
b a c ac a c
a c
a c ac a c
a c
a c ac
a c
a c a c
b a c
b a c

3. Evelyn Learning Systems
Hence,      2 2 2
, ,ca b c b c a a b   are in AP.