KVPY SA Exam 05-Nov-2017 Question Paper

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Question : 5 of 80

Marks: +1, -0

Let

A_1A_2A_3....A_9

be a nine-sided regular polygon with side length 2 units. The difference between the lengths of the diagonals

A_1A_5

and

A_2A_4

equals

$2 +√12$
$√ 12 – 2$
6
2

Solution:

∠A_1OA_2={2π}/9=40°

\cos{2π}/9={x^2+x^2-4}/{2x^2}

x^ 2 \cos 40°) = x^2 -2

x^ 2 (1 -\cos 40°) = 2

x=1/{\sin20°}

.....(i)
Now

\cos {8π}/7={x^2+x^2-(A_1A_5)^2}/{2x^2}

(in

∆A_1OA_5

)

(A_1A_5 ) ^2 = 2x^2(1-\cos{8π}/9)

= 2x^2 (1 -\cos160°)

= 4x^2 \sin^2 80°

A_1A_5 = 2x \sin 80°

.....(ii)
Similarly in

∆A_2OA_4

A_2A_4 = 2x \sin 40°

.....(iii)
(ii) – (iii)

A_1A_5 -A_2A_4 = 2x (\sin80°-\sin40°)

= 2 (using (i))

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