NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

In a $\Delta ABC$ , $(c + a + b)(a + b - c) = ab$ . The measure of the angle $C$ is:

Choose the correct answer:

Explanation

Solution

1. Simplify the given equation:

The equation is $(a + b + c)(a + b - c) = ab$ .

This is in the form $(X + c)(X - c) = ab$ , where $X = a + b$ .

Using the identity $(A + B)(A - B) = A^2 - B^2$ :

(a + b)^2 - c^2 = ab

a^2 + b^2 + 2ab - c^2 = ab

2. Rearrange the equation:

a^2 + b^2 - c^2 = ab - 2ab

a^2 + b^2 - c^2 = -ab

3. Use the Cosine Rule:

The formula for $\cos C$ is:

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

Substitute the value of $a^2 + b^2 - c^2$ from our simplified equation:

\cos C = \frac{-ab}{2ab}

\cos C = -\frac{1}{2}

4. Find the angle $C$ :

We know that $\cos(60^\circ) = \frac{1}{2}$ , and since the value is negative, the angle lies in the second quadrant:

C = \cos^{-1}\left(-\frac{1}{2}\right) = 120^\circ

To convert to radians:

C = 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}