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CAT 2019 Slot 1 — QA Question 3

Basics of Mensuration/PrismEasy

If the rectangular faces of a brick have their diagonals in the ratio 3 : 2√3 : √15, then the ratio of the length of the shortest edge of the brick to that of its longest edge is?

Answer & solution

Correct answer: 1 : √3

  • A

    √3 : 2

  • B

    2 : √5

  • 1 : √3

  • D

    √2 : √3

Solution

Easy

A cuboid with edges x,y,zx,y,z has three distinct face diagonals: x2+y2\sqrt{x^2+y^2}, y2+z2\sqrt{y^2+z^2}, z2+x2\sqrt{z^2+x^2}. Square the given ratios to get three equations in x2,y2,z2x^2,y^2,z^2, solve, then compare the smallest edge to the largest.

1

Write the face-diagonal equations. Let the diagonals be 3k, 23 k, 15 k3k,\,2\sqrt3\,k,\,\sqrt{15}\,k and square each.

x2+y2=9k2(1)y2+z2=12k2(2)z2+x2=15k2(3)\begin{aligned} &x^2+y^2 = 9k^2 \quad\text{(1)}\\ &y^2+z^2 = 12k^2 \quad\text{(2)}\\ &z^2+x^2 = 15k^2 \quad\text{(3)} \end{aligned}
2

Add all three. Each square appears twice.

2(x2+y2+z2)=36k2[(1)+(2)+(3)]⇒ x2+y2+z2=18k2(4)\begin{aligned} &2(x^2+y^2+z^2) = 36k^2 \quad\text{[(1)+(2)+(3)]}\\ &\Rightarrow\ x^2+y^2+z^2 = 18k^2 \quad\text{(4)} \end{aligned}
3

Solve for each edge. Subtract (1), (2), (3) from (4) in turn.

z2=18k2−9k2=9k2⇒z=3k[(4)-(1)]x2=18k2−12k2=6k2⇒x=6 k[(4)-(2)]y2=18k2−15k2=3k2⇒y=3 k[(4)-(3)]\begin{aligned} &z^2 = 18k^2-9k^2 = 9k^2 \Rightarrow z = 3k \quad\text{[(4)-(1)]}\\ &x^2 = 18k^2-12k^2 = 6k^2 \Rightarrow x = \sqrt6\,k \quad\text{[(4)-(2)]}\\ &y^2 = 18k^2-15k^2 = 3k^2 \Rightarrow y = \sqrt3\,k \quad\text{[(4)-(3)]} \end{aligned}
4

Compare shortest to longest. Edges are 3 k<6 k<3k\sqrt3\,k < \sqrt6\,k < 3k, so shortest =3 k=\sqrt3\,k, longest =3k=3k.

shortestlongest=3 k3k=13\begin{aligned} &\frac{\text{shortest}}{\text{longest}} = \frac{\sqrt3\,k}{3k} = \frac{1}{\sqrt3} \end{aligned}
shortest:longest=1:3\text{shortest}:\text{longest} = 1:\sqrt3

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CAT 2019 Slot 1 QA Q3: If the rectangular faces of a brick have their diagonals in the ratio 3 : 2&radic;3 : &radic;15, then the rati — Solution | TheCATExam