CAT 2022 Question Paper | Quant Slot 2

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

Video Explanation

Explanatory Answer

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.
AD² = AE² + ED²
AD² = \( \left( \frac { 3 \sqrt { 3 } } { 2 } \right) ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { 2 } \)
AD² = \( \frac { 27 } { 4 } + \frac { 1 } { 4 } \)
AD ² = 7
\( A D = \sqrt { 7 } \)

The answer is '\(\sqrt{7}\)'

Choice D is the correct answer.