NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $\sin x + a \cos x = b$ , then $|a \sin x - \cos x|$ is:

Choose the correct answer:

Explanation

Solution

1. Using the Given Equation:

We are given the equation:

\sin x + a \cos x = b \quad \dots

Squaring both sides to simplify the expression:

(\sin x + a \cos x)^2 = b^2 \quad \dots

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$ :

\sin^2 x + a^2 \cos^2 x + 2a \sin x \cos x = b^2 \quad \dots

2. Rearranging with Fundamental Identities:

Using the identity $\sin^2 x = 1 - \cos^2 x$ :

(1 - \cos^2 x) + a^2 \cos^2 x + 2a \sin x \cos x = b^2 \quad \dots

(a^2 - 1) \cos^2 x + 2a \sin x \cos x = b^2 - 1 \quad \dots

From this, we can isolate the term $2a \sin x \cos x$ :

2a \sin x \cos x = b^2 - 1 + (1 - a^2) \cos^2 x \quad \dots(1) \quad \dots

3. Evaluating the Target Expression:

Let $k = |a \sin x - \cos x|$ . To find $k$ , we first find $k^2$ :

k^2 = (a \sin x - \cos x)^2 \quad \dots

k^2 = a^2 \sin^2 x + \cos^2 x - 2a \sin x \cos x \quad \dots

Substitute $\sin^2 x = 1 - \cos^2 x$ into the equation:

k^2 = a^2 (1 - \cos^2 x) + \cos^2 x - 2a \sin x \cos x \quad \dots

k^2 = a^2 + (1 - a^2) \cos^2 x - 2a \sin x \cos x \quad \dots

4. Substituting Equation (1):

Now, replace $2a \sin x \cos x$ with the value derived in the first step:

k^2 = a^2 + (1 - a^2) \cos^2 x - [b^2 - 1 + (1 - a^2) \cos^2 x] \quad \dots

Simplifying the expression (the terms with $\cos^2 x$ cancel out):

k^2 = a^2 - b^2 + 1 \quad \dots

5. Final Result:

Taking the square root of both sides:

k = \sqrt{a^2 - b^2 + 1} \quad \dots

Therefore, $|a \sin x - \cos x| = \sqrt{a^2 - b^2 + 1}$ .

Correct Option: 2 ( $\sqrt{a^2 - b^2 + 1}$ )