CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

The question is about Roots of Equation. How you would relate sum and product of roots of two quadratic equations. Framing and solving equations forms the basis of quants. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 10: Equation x² + 5x – 7 = 0 has roots a and b. Equation 2x² + px + q = 0 has roots a + 1 and b + 1. Find p + q?

1. 6
2. 0
3. -16
4. 2

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Quadratic Equations: Knowledge about the product and sum of the roots is important to solve this one!!

Given, x² + 5x – 7 = 0 has roots a and b. We know that,
Sum of roots in a quadratic equation = a+b = \\frac{(-5)}{1}\\) = -5.
Product of the roots = ab = \\frac{(-7)}{1}\\) = -7.

Now, The second equation 2x² + px + q = 0 has roots a + 1 and b + 1.
Sum of the roots = a+1+b+1 = a+b+2 = \\frac{(-p)}{2}\\) = -5 +2 = -3 = \\frac{(-p)}{2}\\) => -p = -6 => p =6.
Product of the roots = (a+1)(b+1) = ab+a+b+1 = \\frac{q}{2}\\). We know the values of ab and a+b. Substituting this, we get, -7+(-5)+1 = \\frac{q}{2}\\) => -11 = \\frac{q}{2}\\)=> q =-22.

Hence, p = 6 and q = -22. => p+q = 6 +(-22) = -16.

The question is "Equation x² + 5x – 7 = 0 has roots a and b. Equation 2x² + px + q = 0 has roots a + 1 and b + 1. Find p + q?"

Hence the answer is "-16"

Choice C is the correct answer.