RD Chapter 8 Quadratic Equations Ex 8.7 Solutions
Question - 31 : - The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers. [CBSE 2012]
Answer - 31 : -
Sum of two numbers = 9
Let first number = x
Then second number = 9 – x
According to the condition,
By cross multiplication
18 = 9x – x2
=> x2 – 9x + 18 = 0
=> x2 – 6x – 3x + 18 = 0
=> x (x – 6) – 3 (x – 6) = 0
=> (x – 6) (x – 3) = 0
Either x – 6 = 0, then x = 6
or x – 3 = 0, then x = 3
Numbers are 6 and (9 – 6) = 3, or 3 and (9 – 3) = 6
Numbers are 3, 6
Question - 32 : - Three consecutive positive integers are such that the sum of the square of the’ first and the product of other two is 46, find the integers. [CBSE 2010]
Answer - 32 : -
Let first number = x
Then second number = x + 1
and third number = x + 2
According co the condition,
(x)2 + (x+ 1) (x + 2) = 46
x2 + x2 + 3x + 2 = 46
=> 2x2 + 3x + 2 – 46 = 0
=> 2x2 + 3x – 44 = 0
=> 2x2 + 11x – 8x – 44 = 0
=> x (2x + 11) – 4 (2x + 11) = 0
=> (2x + 11) (x – 4) = 0
Either 2x + 11 = 0, then x = −11/2 which is not possible being fraction
or x – 4 = 0, then x = 4
Numbers are 4, 5, 6
Question - 33 : - The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers. [CBSE 2010]
Answer - 33 : -
Let smaller number = x
Then larger number = 2x – 5
According to the condition,
(2x – 5)2 – x2 = 88
=> 4x2 – 20x + 25 – x2 – 88 = 0
=> 3x2 – 20x – 63 = 0
=> 3x2 – 27x + 7x – 63 = 0
=> 3x (x – 9) + 7 (x – 9) = 0
=> (x – 9) (3x + 7) = 0
Either x – 9 = 0, then x = 9
or 3x + 7 = 0, then x = −73 which is not possible
Smaller number = 9
and greater number = 2x – 5 = 2 x 9 – 5 = 18 – 5 = 13
Hence numbers are 13, 9
Question - 34 : - The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers. [NCERT]
Answer - 34 : -
Question - 35 : - Find two consecutive odd positive integers, sum of whose squares is 970.
Answer - 35 : -
Let two consecutive positive integers be x and x + 2
A.T.Q.,
(x)2 + (x + 2)2 = 970
=> x2 + x2 + 4x + 4 – 970 = 0
=> 2x2 + 4x – 966 = 0
=> x2 + 2x – 483 = 0 (Dividing by 2)
=> x2 + 23x – 21x – 483 = 0
=> x (x + 23) – 21 (x + 23) = 0
=> (x – 21) (x + 23) = 0
Either x – 21 = 0 or x + 23 = 0
x = 21 or x = – 23 (rejected being -ve)
As integers should be +Ve
x = 21 and x + 2 = 21 + 2 = 23
Hence integers are 21, 23
Question - 36 : - The difference of two natural numbers is 3 and the difference of their reciprocals is 3/28. Find the numbers.
Answer - 36 : -
y(y + 7) – 4(y + 7) = 0
(y – 4) (y + 7) = 0
y – 4 = 0 or y + 7 = 0
y = 4 or y = -7 (rejected being natural no.)
When y = 4, x = 3 + 4 = 7 [From (ii)]
Number are 7, 4
Question - 37 : - The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
Answer - 37 : -
Let two consecutive positive integers be x and x + 2
A.T.Q.,
(x)2 + (x + 2)2 = 394
x2 + x2 + 4x + 4 – 394 = 0
2x2 + 4x – 390 = 0
x2 + 2x – 195 = 0 (Dividing by 2)
x2 + 15x – 13x – 195 = 0
x (x + 15) – 13 (x + 15) = 0
(x – 13) (x + 15) = 0
Either x – 13 = 0 or x + 15 = 0
x = 13 or x = -15 (rejected)
Number should be x = 13 and x = 13 + 2 = 15
or x = -15 and x = -15 + 2 = -13
Hence odd numbers are 13, 15 or -15, -13
Question - 38 : - The sum of the squares of two consecutive multiple of 7 is 637. Find the multiples. [ICSE 2014]
Answer - 38 : -
Let first multiple of 7 = 7x
Then second = 7x + 7
(7x)2 + (7x + 7) = 637
49x2 + 49x2 + 98x + 49 = 637
98x2 + 98x + 49 – 637 = 0
98x2 + 98x – 588 = 0
x2 + x – 6 = 0 (dividing by 98)
x2 + 3x – 2x – 6 = 0
x (x + 3) – 2 (x + 3) = 0
(x + 3) (x – 2) = 0
Either x + 3 = 0, then x = -3, but not possible being negative
or x – 2 = 0, then x = 2
Numbers will be 14, 21
Question - 39 : - The sum of the squares of two consecutive even numbers is 340. Find the numbers. [CBSE 2014]
Answer - 39 : -
Let first even number = 2x
Then second number = 2x + 2
(2x)2 + (2x + 2)2 = 340
4x2 + 4x2 + 8x + 4 – 340 = 0
8x2 + 8x – 336 = 0
x2 + x – 42 = 0 (Dividing by 8)
x2 + 7x – 6x – 42 = 0
x (x + 7) – 6 (x + 7) = 0
=> (x + 7) (x – 6) = 0
Either x + 7 = 0, then x = -7 but not possible being negative
or x – 6 = 0, then x = 6
Numbers are 12, 14
Question - 40 : - The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is 29/20. Find the original fraction.
Answer - 40 : -