What is {{AB}}{ {C}} equal to ?

Correct Answer is : 50350/3350 m

Correct Answer is : 50350/3350 m

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UPSC NDA 2 2022 Mathematics Solved Paper

Question Numbers: 40-42

ABC is a triangular plot with A

B = 16

m, B

C = 10

m and C

A = 10

m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45° at the vertex B.

Question : 41 of 120

Marks: +1, -0

What is

\frac{\text{AB}}{\sin \text{C}}

equal to ?

Solution:

Concept:

Use the cosine rule to find

\cos C

from the side lengths, then compute

\sin C

using

\sin^2 C + \cos^2 C = 1

. Finally, divide side

AB

\sin C

Explanation:

In the given triangle,

a = BC = 10

cm,

b = AC = 10

cm, and

c = AB = 16

cm.

By cosine rule:

\cos C = \frac{a^2 + b^2 - c^2}{2ab}

Substitute:

\cos C = \frac{10^2 + 10^2 - 16^2}{2 \times 10 \times 10} = \frac{100+100-256}{200} = \frac{-56}{200} = -\frac{7}{25}

Then

\sin C = \sqrt{1 - \cos^2 C} = \sqrt{1 - \left(-\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}

Thus

\frac{AB}{\sin C} = \frac{16}{\frac{24}{25}} = 16 \times \frac{25}{24} = \frac{400}{24} = \frac{50}{3}

Answer:

\frac{AB}{\sin C} = \frac{50}{3}

m (Option B).

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