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UPSC Maths Study Materials

1. The average of 5 nos. is 46 and that of the first four of them is 45. The fifth number is
Sol:
The fifth number = total of 5 numbers - total of first four numbers
= 5 X 46 - 4 X 45
= 230 - 180 = 50
The 5th number is 50. 

2. The average of n numbers is x.When 36 is substracted from two of the numbers, the new average becomes (x-8). The value of n is
Sol:
The average of ‘n’ numbers = x
Total of ‘n’ numbers = nx
According to this question,
nx – 36-36 = n(x-8)
nx – 72 = nx-8n
8n = 72
N = 72/8 = 9
The value of n is 9. 

3. Four years ago the average age of A, B and C was 25 years. Five years ago the average age of and C was 20 years. A’s present ages is
Sol:
Four years ago,
Average of A+B+C = 25 years
Now the age of A+B+C = 3 X 25 + 12 = 75 + 112 = 87 years.
Five years ago,
Average of B+C = 20 years
Now, the age of B+C = 2 X 20 + 10 = 40 + 10 = 50 years
A’s present age = (A+B+C) is present ages – (B+C) is present ages
= 87 – 50 = 37 years. 

4. A person goes to a place at the speed of 20 km/hr and returns at the speed of 12 km/hr. His average speed for the journey is
Sol:
Average speed = (2xy/x+y) km/hr.
X = 20km/hr. Y = 12 km/hr.
Average speed = 2 X 200 X 12 / 20 + 12
= 2 X 20 X 12 / 32
= 15 km/hr. 

5. The average age of 20 boys in a class is 12 years. What will be the average age of the class of 5 new boys take admission whose average age is 7 years?
Sol:
Total ages of 20 boys = 20 X 12 = 240 years
Total ages of 5 new boys = 5 X 7 = 35 years
In the class, 25 boys total ages = 240 + 35 = 275 years
Average age of 25 boys = 275/25 = 11 years

7
3
4
5
6
?
2
8
3
33
10
33





6. ? = 9
Sol:
7 X 5 = 35 – 2 = 33
6 X 3 = 18 – 10 = 8
4 X 9 = 36 – 3 = 33
5
7
99
4
3

3
9
?
7
5

7
6
114
5
2




2.
? = 164
Sol:
5 square = 25
7 square = 49
4 square = 16
3 square = 9
So total = 99 same for the another two also.

8. If x = 8 + 2√15, the value of √x + 1/√x is
Sol: X = 8 + 2√15
2 2
= (√5) + (√3) + 2√5 √3
2
= (√5 + √3)
√x = √5 + √3 and 1/√x
= 1/√5 + √3 X √5 - √3 / √5 - √3
= √5 - √3 / 2
√x + 1/ √x = √5 + √3 + √5 - √3 / 2
= 2√5 + 2√3 + √5 - √3 / 2
= 3√5 / 2 + √3 /2

9. A boy walked 3 km towards south. Then he turned towards west and walked for 5 km. Then he turned towards north and walked for 6 km. Now he turned towards east and walked for 5 km.
What is the distance between the boy and the starting point?
Sol : 3 kms.

10. The ratio of the velocities of the hour hand and the minute hand of a clock is
Sol:
Velocity = distance / time
Velocity of hour hand / Velocity of minute hand
= 360 / 12 X 60 = 360 X 60 / 360 X 12 X 60 = ½
360 / 60