RD Chapter 8 Quadratic Equations Ex 8.7 Solutions
Question - 21 : - The sum of a number and its reciprocal is 17/4 , Find the number.
Answer - 21 : -
Question - 22 : - A two-digit number is such that the product of its digits is 8. When 18 is subtracted from the number, the digits interchange their places. Find the number.
Answer - 22 : -
Product of two digits = 8
Let units digit = x
Then tens digit = 8/x
Number = x + 10 x (8/x) = x + (80/x)
Question - 23 : - A two-digit number is such that the product of the digits is 12. When 36 is added to the number the digits interchange their places. Determine the number.
Answer - 23 : -
Question - 24 : - A two-digit number is such that the product of the digits is 16. When 54 is subtracted from the number, the digits are interchanged. Find the number.
Answer - 24 : -
Question - 25 : - Two numbers differ by 3 and their product is 504. Find the numbers. (C.B.S.E. 2002C)
Answer - 25 : -
Difference of two numbers = 3
Let first number = x
Then second number = x – 3
According to the condition,
x (x – 3) = 504
=> x² – 3x – 504 = 0
=> x² – 24x + 21x – 504 = 0
=> x (x – 24) + 21 (x – 24) = 0
=> (x – 24) (x + 21) = 0
Either x – 24 = 0, then x = 24
or x + 21 = 0, then x =-21
(i) If x = 24, then
First number = 24
and second number = 24 – 3 = 21
(ii) If x =-21, then
First number = -21
and second number = -21 – 3 = -24
Hence numbers are 24, 21 or -21, -24
Question - 26 : - Two numbers differ by 4 and their product is 192. Find the numbers. (C.B.S.E. 2000C)
Answer - 26 : -
Let first number = x
Then second number = x – 4
According to the condition,
x (x – 4) = 192
=> x² – 4x – 192 = 0
=> x² – 16x + 12x – 192 = 0
=> x (x – 16) + 12 (x – 16) = 0
=> (x – 16) (x + 12) = 0
Either x – 16 = 0, then x = 16
or x + 12 = 0, then x = -12
(i) If x = 16, then
First number = 16
and second number = 16 – 4 = 12
(ii) If x = -12, then
First number = -12
and second number = -12 – 4 = -16
Hence numbers are 16, 12 or -12, -16
Question - 27 : - A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number. (C.B.S.E. 1999C)
Answer - 27 : -
Let units digit of the number = x
and tens digit = y
Number = x + 10y
According to the given conditions,
x + 10y = 4 (x + y)
=> x + 10y = 4x + 4y
=> 10y – 4y = 4x – x
=> 3x = 6y
=>x = 2y …(i)
and x + 10y = 2xy ….(ii)
Substituting the value of x in (i)
2y + 10y = 2 x 2y x y
=> 12y = 4y2
=> 4y2 – 12y = 0
=> y2 – 3y = 0
=> y (y – 3) = o
Either y = 0, but it is not possible because y is tens digit number
or y – 3 = 0, then y = 3
x = 2y = 2 x 3 = 6
and number = x + 10y = 6 + 10 x 3 = 6 + 30 = 36
Question - 28 : - The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger, find the numbers. [CBSE 2014]
Answer - 28 : -
Let first large number = x
and smaller number = y
According to the condition,
x2 – y2 = 180 …(i)
and y2 = 8x
From (i) and (ii),
x2 – 8x – 180 = 0
=> x2 – 18x + 10x – 180 = 0
=> x (x – 18)+ 10 (x – 18) = 0
=> (x – 18) (x + 10) = 0
Either x – 18 = 0, then x = 18
or x + 10 = 0, then x = -10 But it is not possible being negative
x = 18
First number =18
Then second number y2 = 8x
y2 = 8 x 18 = 144 = (12)2
=> y = 12
Numbers are 18, 12
Question - 29 : - The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers. (C.B.S.E. 2005)
Answer - 29 : -
Sum of two numbers = 18
Let one number = x
Then second number = 18 – x
According to the condition,
=> x2 – 12x – 6x + 72 = 0
=> x (x – 12) – 6 (x – 12) = 0
=> (x – 12) (x – 6) = 0
Either x – 12 = 0, then x = 12
or x – 6 = 0, then x = 6
(i) If x = 12, then
First number = 12
Second number =18 – 12 = 6
(ii) If x = 6, then
First number = 6
Then second number = 18 – 6 = 12
Numbers are 6, 12
Question - 30 : - The sum of two numbers a and b is 15, and the sum of their reciprocals 1/a and 1/b is 3/10. Find the numbers a and b. (C.B.S.E. 2005)
Answer - 30 : -
or b – 5 = 0, then b = 5
(i) a = 15 – 10 = 5
(ii) or a = 15 – 5 = 10
Numbers are 5, 10 or 10, 5