CAT 2021 Question Paper | Quant Slot 2

CAT 2021 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 19 : Suppose one of the roots of the equation a x² - b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³ then |a| equals

Video Explanation

Explanatory Answer

For a quadratic equation, a x² + b x + c = 0.
the roots are given by x = \\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a})
Since the roots, a, b, c are all rational and one of the roots, 2 + √ 3 is irrational, the other root of the equation will also be irrational and a conjugate of 2 + √ 3.
That is, the other root is, 2 - √ 3.
Sum of roots is given by \\frac{-b}{a}) = (2 + √ 3) + (2 - √ 3) = 4.
And Product of roots is given by \\frac{c}{a}) = (2 + √ 3) (2 - √ 3) = 2² - (√ 3)² = 4 - 3 = 1
But here the quadratic equation is a x² - b x + c = 0.
Therefore,
\\frac{b}{a}) = 4 and \\frac{c}{a}) = 1
b = 4a; c = a.
We are also given that b = c³
4a = a³
4 = a²
a = ∓2
|a| = 2

The question is " Suppose one of the roots of the equation a x² - b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³ then |a| equals "

Hence, the answer is '2'

Choice A is the correct answer.

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