NIMCET 2009 — Mathematics PYQ

1. Use the Property of Odd Functions

We know that ${\sin }^{-1}(-x)=-{\sin }^{-1}x$. Substituting this into the given equation:

2. Rearrange the Equation

Move all ${\sin }^{-1}x$ terms to one side:

3. Apply Trigonometric Functions to Both Sides

Take the cosine of both sides:

Using the property $\cos ({\cos }^{-1}\theta )=\theta$ and the fact that cosine is an even function ($\cos (-\theta )=\cos \theta$):

4. Use the Double Angle Formula

Recall the identity $\cos (2\theta )=1-2{\sin }^2\theta$. Let $\theta ={\sin }^{-1}x$, which implies $\sin \theta =x$.

Substituting this into the identity:

5. Form the Quadratic Equation

Simplify the resulting equation:

6. Compare with Options

The equation we derived is $2x^2-x=0$.

Looking at the provided options:

• (a) $2x^2-x+2=0$ (Incorrect)

• (b) $2x^2-3x=0$ (Incorrect)

• (c) $2x^2+x-1=0$ (Incorrect)

Since $2x^2-x=0$ does not match any of the first three options exactly, we select the "None of these" category.

Final Answer:

The correct option is (d) None of these.