CAT 2024 Question Paper | Quant Slot 2

CAT Questions

CAT 2024 Quant was dominated by Algebra followed by Arithmetic. 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 16 The roots $\alpha, \beta$ of the equation $3 x^2+\lambda x-1=0$, satisfy $\frac{1}{\alpha^2}+\frac{1}{\beta^2}=15$. The value of $\left(\alpha^3+\beta^3\right)^2$, is

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Video Explanation

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Explanatory Answer

$\begin{array}{l}\text { Sum of the roots: } \alpha+\beta=-\lambda / 3 \text {. } \\ \text { Product of the roots }=\alpha \beta=-1 / 3 \text {. } \\ 1 / \alpha^2+1 / \beta^2=15 \text {. } \\ \alpha^2+\beta^2 /(\alpha \beta)^2=15 \\ \alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta=\left(\lambda^2 / 9\right)-2(-1 / 3)=\left(\lambda^2 / 9\right)+(2 / 3)=\left(\lambda^2+6\right) / 9 \text {. } \\ \left(\lambda^2+6\right) /(9 * 1 / 9)=15 \text {. } \\ \lambda^2+6=15 \text {. } \\ \lambda^2=9 \text {. } \\ \lambda=+3 \text { or }-3 \text {. } \\ a^3+\beta^3=(\alpha+\beta)^3-3(\alpha \beta)(\alpha+\beta) . \\ \alpha^3+\beta^3=\left(-\lambda^3 / 27\right)-3(-1 / 3)(-\lambda / 3)=\left(-\lambda^3 / 27\right)-(\lambda / 3)=-\left(\lambda^3 / 27+\lambda / 3\right) . \\ \text { Substituting } \lambda=+3 \text { in }\left(\alpha^3+\beta^3\right)^2 \text { we get }(-2)^2=4 \text {. } \\ \text { Substituting } \lambda=-3 \text { in }\left(\alpha^3+\beta^3\right)^2 \text { we get }(+2)^2=4 \text {. }\end{array}$

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The question is "The roots $\alpha, \beta$ of the equation $3 x^2+\lambda x-1=0$, satisfy $\frac{1}{\alpha^2}+\frac{1}{\beta^2}=15$. The value of $\left(\alpha^3+\beta^3\right)^2$, is"

Hence, the answer is '4'

Choice C is the correct answer.

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