Question

If a ϵ Z and the equation (x – a) (x – 10) + 1 = 0 has integral roots, then the values of a are

$
  {\text{A}}{\text{. 10,8}} \\
  {\text{B}}{\text{. 12,10}} \\
  {\text{C}}{\text{. 12,8}} \\
  {\text{D}}{\text{. 18,10}} \\
$

Answer 1

Hint – To find the values of a, we solve the given equation. And it can only take integral values. Z represents the set of integers.

Complete Step-by-Step solution:
Given data, (x – a) (x – 10) + 1 = 0
⟹(x – a) (x – 10) = -1
⟹ {(x – a) = 1 and (x -10) = -1} or {(x – a) = -1 and (x -10) = 1}
(Because it can only take integral values)

I.e.,
(x – a) = 1 and (x -10) = -1 (x – a) = -1 and (x -10) = 1
⟹(x – 10) = -1 ⟹(x -10) = 1
⟹x = 10 -1 ⟹x = 10 +1
⟹x = 9 ⟹x = 11

Substitute x values in the respective second terms,

⟹(x – a) = 1 ⟹(x –a) = -1
⟹9 – a = 1 ⟹11 – a = -1
⟹a = 8 ⟹a = 12

Hence the values of ‘a’ are 8, 12.
Option C is the correct answer.

Note: In order to solve this type of problems the key is to know to assign values (can only be integral values). An integral value is a whole number as opposed to a fraction such as 3, 6, 8, 15, 1284. Integral means consisting of a whole number or an undivided quantity.