CAT 2022 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 21 : The length of each side of an equilateral triangle \(\mathrm{ABC}\) is \(3 \mathrm{~cm}\). Let \(\mathrm{D}\) be a point on \(\mathrm{BC}\) such that the area of triangle \(\mathrm{ADC}\) is half the area of triangle \(\mathrm{ABD}\). Then the length of \(\mathrm{AD}\), in \(\mathrm{cm}\), is

- \(\sqrt{6}\)
- \(\sqrt{5}\)
- \(\sqrt{8}\)
- \(\sqrt{7}\)

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Let AE be the perpendicular dropped from A on to BC,

AE = \( \frac { \sqrt { 3 } } { 2 } 3 \).

E is the mid-point of BC.

EC = 1.5 cm

Ar(Δ ABD) : Ar(Δ ADC) = 2 : 1

BD : DC = 2 : 1

DC = 1cm

ED = EC - DC = 1.5 cm - 1 cm = 0.5 cm

Δ AED is a right-angled triangle.

AD^{2} = AE^{2} + ED^{2}

AD^{2} = \( \left( \frac { 3 \sqrt { 3 } } { 2 } \right) ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { 2 } \)

AD^{2} = \( \frac { 27 } { 4 } + \frac { 1 } { 4 } \)

AD ^{2} = 7

\( A D = \sqrt { 7 } \)

Choice D is the correct answer.

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