FIND THE LEAST NUMBER TO BE ADDED TO GET A PERFECT SQUARE

Problem 1 :

The smallest number added to 680621 to make the sum a perfect square is :

(a) 4     (b)  5    (c)  6   (d) 8

Solution :

(825)² is the nearest perfect square.

825² = 825 x 825 ==> 680625

The difference between 680625 and 680621 is 

680625 - 680621 = 4

So, 4 should be added to the given number to get a perfect square.

Problem 2 :

What is the smallest number to be added to 1000 to make the resulting figure as perfect square.

Solution :

(32)² is the nearest perfect square.

32² = 32 x 32 ==> 1024

The difference between 1024 and 1000 is 

1024 - 1000 = 24

So, 24 should be added to the given number to get a perfect square.

Problem 3 :

If √1369 + √(0.0615) + x = 37.25, then x is equal to 

(a) 10^-1         (b) 10^-2           (c) 10^-3         (d) None

Solution :

√1369 + √(0.0615) + x = 37.25

37 + √(0.0615) + x = 37.25

√(0.0615) + x = 37.25 - 37

√(0.0615) + x = 0.25

Take square on both sides.

(0.0615) + x = (0.25)²

(0.0615) + x = (25/100)²

(0.0615) + x = (1/4) ⋅ (1/4)

0.0615 + x = 0.0625

x = 0.0625 - 0.0615

x = 0.001

x = 1/1000

x = 10^-3

Problem 4 :

The sum of two numbers is 22 and the sum of their square is 404, then the product of the numbers is 

Solution :

Let the two numbers be x and y.

x + y = 22

x² + y² = 404

(x + y)² = x² + y² + 2xy

Applying the given values, we get

(22)² = 404 + 2xy

484 = 404 + 2xy

2xy = 484 - 404

2xy = 80

xy = 80/2

xy = 40

Problem 5 :

Solution :

Problem 6 :

If y = 5, then what is the value of 10y √y³ - y² ?

a) 50 √2     b)  100   c)  200√5    d)  500

Solution :

Given that y = 5

Applying the value of y, we get

= 10(5) √5³ - 5²

= 50√(125 - 25)

= 50√100

= 50√(10 x 10)

= 50(10)

= 500

So, option d is correct.

Problem 7 :

If 52/x = √(169/289), then the value of x is

a) 52     b)  58   c)  62    d)  68

Solution :

52/x = √(169/289)

52/x = √(13 ⋅ 13) / (17 ⋅ 17)

52/x = 13/17

17(52) = 13x

x = 17(52) / 13

x = 17(4)

= 68

So, option d is correct.

Problem 8 :

If √1369 + √(0.0615 + x) = 37.25, then x is equal to 

a) 10^-1        b)  10^-2       c)  10^-3      d) None

Solution :

√1369 + √(0.0615 + x) = 37.25

√(37 ⋅ 37) + √(0.0615 + x) = 37.25

37 + √(0.0615 + x) = 37.25

√(0.0615 + x) = 37.25 - 37

√(0.0615 + x) = 0.25

0.0615 + x = (0.25)²

0.0615 + x = 0.0625

x = 0.0625 - 0.0615

x = 0.001

x = 1/1000

x = 10^-3 

So, option c is correct.

Problem 9 :

If √(0.04 x 0.4 x a)  = 0.004 x 0.4 x √b, then a / b is :

a) 16 x 10^-3        b)  16 x 10^-4       c)  16 x 10^-5      d) None

Solution :

√(0.04 x 0.4 x a)  = 0.004 x 0.4 x √b

√0.04 x √0.4 x √a  = 0.004 x 0.4 x √b

√a/√b = (0.004 x 0.4) / (√0.04 x √0.4)

Simplifying √0.04, we get

= √0.04 x (100/100)

= √(4/100)

= 2/10

= 0.2

√0.4 = √0.4 x (10/10)

= √(4/10)

= 2√10

√a/√b = (0.004 x 0.4) / (0.2 x (2/√10))

√a/√b = (0.004 x 0.4) / (0.4/√10))

√a/√b = 0.004 √10

Squaring on both sides, we get

a/b = (0.004)²√10²

a/b = 0.00016

= 16/100000

= 16/10⁵

= 16 x 10^-5

So, option c is correct.

Problem 10 :

If 28√x + 1426 = 3/4 of 2872

a) 576        b)  676       c)  1296      d) 144

Solution :

28√x + 1426 = 3/4 of 2872

28√x + 1426 = (3/4) ⋅ 2872

28√x + 1426 = 3 ⋅ 718

28√x + 1426 = 2154

28√x = 2154 - 1426

28√x = 728

√x = 728/28

√x = 26

x = 26²

x = 676

So, option b is correct.