NIMCET 2014 — Mathematics PYQ

1. Using the Given Equation:

We are given the equation:

Squaring both sides to simplify the expression:

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

2. Rearranging with Fundamental Identities:

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

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

3. Evaluating the Target Expression:

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

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

4. Substituting Equation (1):

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

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

5. Final Result:

Taking the square root of both sides:

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

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