Question

1.A two digit number is four times the sum and three time the product of its digits .find the numbers.

2.if the sum of first n even natural number is 420.find the value of n.

3.A chess board contains 64equal squares and the area of each square is 6.25cm².a border round the board is 2cm wide. Find the length of the side of the chess board.

Answer 1

1)
A two digit positive integer can be expressed as 10x+y, where x is the tens digit and y is the ones digit. For example, 37 = 3(10)+7.

So, 10x+y = 4(x+y) and 10x+y = 3xy

10x+y = 4x+4y

6x = 3y

y = 2x

Since y = 2x and 10x+y = 3xy, we have 10x+2x=3x(2x)

12x = 6x2 6x2-12x = 0

6x(x-2) = 0 x = 0 or x = 2

Since x can't be zero, x = 2

y = 2x = 4




2)
Even natural no:s form an A.P :-

2,4,6,8 .....

a = 2 , d = 2 , Sn = 420

Sn = n/2 [ 2a + (n-1)d]
420 = n/2 [ 4 + 2n - 2 ]
840 = n [ 2 + 2n ]

2n²+ 2n = 840

2 [ n² + n ] = 840

n² + n = 420

n² + n - 420 = 0
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n = -b ± √ b² - 4ac / 2a

= -1 ± √1 - 4 x 1 x -420 = 2

= -1 ± √1681 / 2

= -1 ± 41 /2

= -1 + 41/2 = 40/2 = 20

n = 20





3)
side length of the square board= x
Area of one square in the chess board = 6.25 cm²
Area of 64 squares = 64 x 62.5
(x – 4)² = 400
(x – 4) = √400
(x – 4) = √20 x 20
x – 4 = 20
x = 20 + 4
x = 24 cm