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CAT 2024 Slot 2 — QA Question 1

Forming a Quadratic Equation and Relation between roots and coefficientsEasy

The roots of α, β of the equation 3x2 + λx - 1 = 0, satisfy 1α2+1β2 = 15.

The value of (α3 + β3)2, is

Answer & solution

  • A

    1

  • 4

  • C

    9

  • D

    16

Solution

Medium

Use Vieta's formulas to write the symmetric expression 1α2+1β2\tfrac{1}{\alpha^2}+\tfrac{1}{\beta^2} in terms of α+β\alpha+\beta and αβ\alpha\beta, solve for λ\lambda, then evaluate α3+β3\alpha^3+\beta^3 via the same sums.

1

Read off the sum and product of roots. For 3x2+λx−1=03x^2+\lambda x-1=0:

α+β=−λ3,αβ=−13\begin{aligned} &\alpha+\beta=-\tfrac{\lambda}{3},\qquad \alpha\beta=-\tfrac{1}{3} \end{aligned}
2

Apply the given condition. Rewrite 1α2+1β2=α2+β2(αβ)2=(α+β)2−2αβ(αβ)2=15\tfrac{1}{\alpha^2}+\tfrac{1}{\beta^2}=\dfrac{\alpha^2+\beta^2}{(\alpha\beta)^2}=\dfrac{(\alpha+\beta)^2-2\alpha\beta}{(\alpha\beta)^2}=15.

λ29−2(−13)(−13)2=15⇒ λ29+2319=15(substitute step 1)⇒ λ2+6=15(multiply by 19)⇒ λ2=9\begin{aligned} &\frac{\tfrac{\lambda^2}{9}-2\left(-\tfrac13\right)}{\left(-\tfrac13\right)^2}=15\\ &\Rightarrow\ \frac{\tfrac{\lambda^2}{9}+\tfrac23}{\tfrac19}=15 \quad\text{(substitute step 1)}\\ &\Rightarrow\ \lambda^2+6=15 \quad\text{(multiply by }\tfrac19\text{)}\\ &\Rightarrow\ \lambda^2=9 \end{aligned}
3

Evaluate α3+β3\alpha^3+\beta^3. With λ2=9\lambda^2=9 we have (α+β)2=λ29=1(\alpha+\beta)^2=\tfrac{\lambda^2}{9}=1, and αβ=−13\alpha\beta=-\tfrac13. Use α3+β3=(α+β)[(α+β)2−3αβ]\alpha^3+\beta^3=(\alpha+\beta)\big[(\alpha+\beta)^2-3\alpha\beta\big].

α3+β3=(α+β)[1−3(−13)]⇒ α3+β3=(α+β) [ 1+1 ]=±2(since (α+β)2=1)⇒ (α3+β3)2=(±2)2=4\begin{aligned} &\alpha^3+\beta^3=(\alpha+\beta)\Big[1-3\left(-\tfrac13\right)\Big]\\ &\Rightarrow\ \alpha^3+\beta^3=(\alpha+\beta)\,[\,1+1\,]=\pm 2 \quad\text{(since }(\alpha+\beta)^2=1\text{)}\\ &\Rightarrow\ (\alpha^3+\beta^3)^2=(\pm 2)^2=4 \end{aligned}
(α3+β3)2=4(\alpha^3+\beta^3)^2=4

The sign of α+β\alpha+\beta is irrelevant because we square at the end, so you only need (α+β)2=1(\alpha+\beta)^2=1 and αβ=−13\alpha\beta=-\tfrac13.

CAT 2024 Slot 2 QA Q1: The roots of α, β of the equation 3x 2 + λx - 1 = 0, satisfy 1 α 2 + 1 β 2 = 15. — Solution | TheCATExam