KVPY SA Exam 2016 Question Paper

Question : 63 of 80

Marks: +1, -0

Consider a semicircle of radius 1 unit constructed on the diameter AB, and let O be its centre. Let C be a point on AO such that AC : CO = 2:1. Draw CD perpendicular to AO with D on the semicircle. Draw OE perpendicular to AD with E on AD. Let OE and CD intersects at H. Then DH equals

$1/√5$
$1/√3$
$1/√2$
${√5-1}/2$

Validate

Solution:

'E' is mid point of AD.

\cos2θ=1/3⇒2\cos^2θ-1=1/3

⇒\cos^2θ=2/3

⇒\cosθ=√2/√3

∴\sinθ=√{1-2/3}=1/√3

{\ED}/1=1/√3

DH = ED sec θ

=1/√3×√3/√2=1/√2

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