**MATHMAA**

KVPY2013solution5 contains the solutions of the questions 10.

10) In the figure given below, ABCDEF is a regular hexagon of side length 1, AFPS and ABQR are squares . Then the ratio Area(APQ)/Area(SRP) equals

A) \(\frac{\sqrt{2}+1}{2}\) B) \(\sqrt{2}\) C)\(\frac{3\sqrt{3}}{4}\) D) 2

Solution :

From the figure we know the values AB = BC = CD = DE = EF = AF =1 and ABQR, AFPS are squares then AB = BQ = QR = AR = 1 = AF = FP = PS = AS and AQ, AP are diagonals of length \(\sqrt{2}\) each.

In a regular polygon of sides n , each interior angle = \(\frac{(n-2)*\Pi}{n}\) .

In a regular hexagon each interior angle = (6-2)* 180/6 = 120 degrees.

\(\angle FAB = 120^{\circ}\)

\(\angle RAB = 90^{\circ}=\angle FAS\) as ABQR and AFPS are squares.

\(\angle FAB = \angle RAB + \angle FAS - \angle RAS\)

120 = 90 + 90 - \(\angle RAS\)

\(\angle RAS = 60^{\circ}\) means triangle RAS is equvilateral triangle of side 1.

SR = 1.

Similarly we prove PS = 1 .

\(\angle RAS = \angle RAQ + \angle PAS - \angle PAQ\)

60 = 45 + 45-\(\angle PAQ\)

\(\angle PAQ = 30^{\circ}\)

\(\angle ASR = 60^{\circ} and \angle ASP = 90^{\circ}\) then \(\angle RSP= 30\)

\(\frac{ar\triangle APQ}{ar\triangle SRP}\)=\(\frac{\frac{1}{2}AP*AQ*\sin (30^{\circ})}{\frac{1}{2}*RS*PS*\sin 30^{\circ}}\)

= \(\frac{\sqrt{2}*\sqrt{2}}{1*1}\) = 2

Hence option D is the correct answer.

11) A person X is running around a circular track completing one round
every 40 seconds. Another person Y running in the opposite direction
meets X every 15 seconds. The time, expressed in seconds, taken by Y to
complete one round is

A) 12.5 B) 24 C) 25 D) 55

Solution :

X takes 40 seconds to complete one round. In 15 seconds it completes 15/40 =3/8 part of the circle.

Y meets X in the opposite direct in every 15 seconds.

If X completes 3/8 part in 15 seconds then Y completes 5/8 parts in the opposite direction.

Time takes to complete 1 part (round the circle) by Y = \(\frac{15}{\frac{5}{8}}\)

= \(\frac{15*8}{3}\) =24 seconds

Therefore Y takes 24 seconds to complete one round.

Hence the option B is the correct answer.

12) The least positive integer n for which

\(\sqrt{n+1}\) - \(\sqrt{n-1}\) < 0.2 is

A) 24 B) 25 C) 26 D) 27

Solution :

Rationalize left side, we get

\(\frac{2}{\sqrt{n+1}+\sqrt{n-1}}< 0.2\)

\(\sqrt{n+1}+\sqrt{n-1}\)> 10

\(\sqrt{n+1}\) > 10 - \(\sqrt{n-1}\)

squaring on both sides we get,

\(\sqrt{n-1}\) >4.9

again squaring on both sides we get,

n> \( 4.9^{2} +1\) =25.01

So the least positive integer n = 26 (since we can estimate this by taking 5^2+1)

Therefore the option is (C).

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