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CAT 1997 — QA Question 18

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

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

If the roots x1 and x2 of the quadratic equation x2 − 2x + c = 0 also satisfy the equation 7x2 – 4x1 = 47, then which of the following is true?

Answer & solution

  • c = -15

  • B

    x1 = -5, x2 = 3

  • C

    x1 = 4.5, x2 = -2.5

  • D

    None of these

Solution

We know that the sum of the roots = -ba.

Hence, x1 + x2 = 2. Now we have two equations, viz. x1 + x2 = 2 ...(i)
and 7x2 – 4x1 = 47 ...(ii)

Solving these two equations, we get x1 = –3 and x2 = 5. Since it does not satisfy options (b) and (c), we will verify it for option (a). 

The product of the roots = (–3) × 5 = –15, ca in our case is c. Hence, c = –15.

Alternately,
Put values of x1, x2 in equation (ii). Do not match. So put c = –15 in equation (i) to get the roots of equation. After finding the roots of equation (i), check whether they satisfy equation (ii) or not. 

The roots (5, –3) satisfy the equation (ii) so answer is (a).

Hence, option (a).

CAT 1997 QA Q18: If the roots x 1 and x 2 of the quadratic equation x 2 − 2x + c = 0 also satisfy the equation 7x 2 &ndas — Solution | TheCATExam