Question: 11 -

If the mean of a, b, c is M and ab + bc + ca = 0, then the mean of a^2+b^2+c^2/3 is :

M

2 (M/3)

M^2

3 M^2

Answer:

3 M^2

Solution:

We have : ( a + b + c) / 3 = M   or (a + b + c) = 3M.

(a+b+c)^2=(3M)^2=9M^2 
and
(a+b+c)^2<=>a^2+b^2+c^2+2(ab+bc+ca)=9M^2
 
a^2+b^2+c^2=9M^2   ,(ab+bc+ca=0)     

Required mean =(a^2+b^2+c^2/3)=9M^2/3=3M^2

Question: 12 -

The average of 7 consecutive numbers is 20. The largest of these numbers is :

24

21

23

22

Answer:

23

Solution:

Let the numbers be x, x + 1, x + 2,  x + 3, x + 4, x + 5 and x + 6,                            Then   (x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6)) / 7 = 20.

Question: 13 -

There are seven consecutive positive even numbers. The average of the square of the first and the last number is 712. Find the 2nd largest number.

22

26

25

30

Answer:

30

Solution:

Let, Seven consecutive even number= x, x+2, x+4, x+6, x+8, x+10, x+12  
712=(x^2+(x+12)^2)/2           
1424=x2+x2+144+24x
2x2+24x-1280=0
x2+12x-640=0
x2+(32-20)x-640=0
x2+32x-20x-640=0
x(x+32)-20(x+32)=0
(x+32)(x-20)=0
x=-32,20
2nd largest number= x+10
=20+10=30 

Question: 14 -

The average weight of 18 workers is 50 kg. A new worker replaced an old worker whose weight is 45 kg. Hence, the average weight of the workers increased by 1 kg. What is the ratio between the old workman and new workman? 

5:7

3:4

23:35

11:12

Answer:

5:7

Solution:

Total of 18 workers= 18×50=900
New average=51 years

51×18=900-45+x
918-900+45=x
x=63
Ratio=45:63
=5:7

Question: 15 -

The average price of three items of furniture is Rs. 15000. If their prices are in the ratio 3:5:7, the price of the cheapest item is :

 6000

8888

9000

7000

Answer:

9000

Solution:

Let their prices be 3x,  5x and 7x. Then, 3x + 5x + 7x = (15000 * 3) or x = 3000.  Cost of cheapest item = 3x = Rs. 9000.