CUET PG 2023 — Mathematics PYQ

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Question

CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

If $\overset{n}{^}_{1}, \overset{n}{^}_{2}$ are two unit vectors and $θ$ is the angle between them, then $cos \frac{θ}{2}$ is equal to

Choose the correct answer:

Explanation

Maana ki $\hat{n}_1$ aur $\hat{n}_2$ do unit vectors hain. Inke sum ka magnitude nikaalne ke liye hum niche diye gaye formula ka use karte hain:

|\hat{n}_1 + \hat{n}_2|^2 = |\hat{n}_1|^2 + |\hat{n}_2|^2 + 2|\hat{n}_1||\hat{n}_2|\cos\theta

Chunki $\hat{n}_1$ aur $\hat{n}_2$ unit vectors hain, isliye $|\hat{n}_1| = 1$ aur $|\hat{n}_2| = 1$ :

|\hat{n}_1 + \hat{n}_2|^2 = 1^2 + 1^2 + 2(1)(1)\cos\theta

|\hat{n}_1 + \hat{n}_2|^2 = 2 + 2\cos\theta

Ab hum $2$ ko common le lete hain:

|\hat{n}_1 + \hat{n}_2|^2 = 2(1 + \cos\theta)

Trigonometry ki identity $1 + \cos\theta = 2\cos^2\dfrac{\theta}{2}$ ka use karne par:

|\hat{n}_1 + \hat{n}_2|^2 = 2 \left( 2\cos^2\frac{\theta}{2} \right)

|\hat{n}_1 + \hat{n}_2|^2 = 4\cos^2\frac{\theta}{2}

Dono sides ka square root (vargmool) lene par:

|\hat{n}_1 + \hat{n}_2| = 2\cos\frac{\theta}{2}

Isliye, $\cos\dfrac{\theta}{2}$ ki value hogi:

\cos\frac{\theta}{2} = \frac{1}{2}|\hat{n}_1 + \hat{n}_2|

Final Answer:

The value of $\cos \dfrac{\theta}{2}$ is:

\mathbf{\frac{1}{2}|\hat{n}_1 + \hat{n}_2|}

Choose the correct answer:

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