### 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