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CAT 1996 — QA Question 20

Forming a Quadratic Equation and Relation between roots and coefficientsEasy
Passage / Data

Direction: Answer the questions based on the following information.

A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. But, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs. 1,148, but the inventory reduced by 54.

Given the quadratic equation x2 – (A – 3)x – (A – 2), for what value of A will the sum of the squares of the roots be zero?

Answer & solution

  • A

    -2

  • B

    3

  • C

    6

  • None of these

Solution

If we write the given equation in the conventional form, i.e. ax2 + bx + c, a = 1, b = – (A – 3), i.e. (3 – A) and c = –(A – 2), i.e. (2 – A). 

Let the roots of this equation be α and β. So the sum of the squares of the roots = α2 + β2 = (α + β)2 – 2αβ.

Now (α + β) = Sum of the roots = -ba =(A-3)1=(A-3) 

and αβ = Product of the roots = ca=(2-A)1= (2 - A).

Hence, α2 + β2 = (A – 3)2 – 2(2 – A) = A2 – 4A + 5 = 0.

Discriminant of this equation is less than 0, hence there no real value of A possible.

Hence, option (d).

CAT 1996 QA Q20: Given the quadratic equation x 2 – (A – 3)x – (A – 2), for what value of A will the su — Solution | TheCATExam