NIMCET 2008 — Mathematics PYQ

1. Identify the Boundaries

• Upper Boundary: The quadratic function $y_u=-x^2+2x+4$ is defined for the entire range $x\in [0,2]$.

• Lower Boundary: This is split into two parts:

  • From $x=0$ to $x=1$, the lower curve is $y_{l1}=\sqrt{x}$.

  • From $x=1$ to $x=2$, the lower curve is $y_{l2}=x^2$.

2. Set up the Definite Integrals

The total area $A$ is the integral of (Upper Curve - Lower Curve) over the specified intervals:

3. Calculate the First Integral ($I_1$)

For $x\in [0,1]$:

4. Calculate the Second Integral ($I_2$)

For $x\in [1,2]$:

Substituting the limits:

5. Total Area

Correct Option: (b) $\frac{19}{3}$ sq. units