MAH-CET 2025 — Mathematics PYQ

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Question

MAH-CET 2025 — Mathematics PYQ

MAH-CET | Mathematics | 2025

If $cos^{- 1} x - cos^{- 1} \frac{y}{2} = α$ then $4 x^{2} - 4 x y + y^{2}$ is equal to

Choose the correct answer:

Explanation

Solution

Step A: Identity apply karna

Di gayi equation:

\cos^{-1} x - \cos^{-1} \frac{y}{2} = \alpha

Formula ke hisaab se $A = x$ aur $B = \frac{y}{2}$ rakhte hain:

\cos^{-1} \left( x \cdot \frac{y}{2} + \sqrt{1-x^2}\sqrt{1-\left(\frac{y}{2}\right)^2} \right) = \alpha

Step B: Cosine ko right side le jana

\frac{xy}{2} + \sqrt{1-x^2}\sqrt{1-\frac{y^2}{4}} = \cos \alpha

Ab root waale term ko ek taraf rakhte hain:

\sqrt{1-x^2}\sqrt{1-\frac{y^2}{4}} = \cos \alpha - \frac{xy}{2}

Step C: Dono sides ka Square karna

Roots hatane ke liye square karte hain:

\left(1-x^2\right)\left(1-\frac{y^2}{4}\right) = \left(\cos \alpha - \frac{xy}{2}\right)^2

Isse expand karte hain:

1 - \frac{y^2}{4} - x^2 + \frac{x^2y^2}{4} = \cos^2 \alpha - 2(\cos \alpha)\left(\frac{xy}{2}\right) + \frac{x^2y^2}{4}

Step D: Simplify karna

$\frac{x^2y^2}{4}$ dono sides se cancel ho jayega:

1 - \frac{y^2}{4} - x^2 = \cos^2 \alpha - xy \cos \alpha

Ab terms ko rearrange karte hain:

x^2 - xy \cos \alpha + \frac{y^2}{4} = 1 - \cos^2 \alpha

Humein pata hai ki $1 - \cos^2 \alpha = \sin^2 \alpha$ :

x^2 - xy \cos \alpha + \frac{y^2}{4} = \sin^2 \alpha

Step E: Poori equation ko 4 se multiply karna

Humein $4x^2 - 4xy \cos \alpha + y^2$ ki value chahiye:

4 \left( x^2 - xy \cos \alpha + \frac{y^2}{4} \right) = 4 \sin^2 \alpha

4x^2 - 4xy \cos \alpha + y^2 = 4 \sin^2 \alpha