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UPSC NDA II 2014 Mathematics Paper

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For the next three (03) items that follow:

The line

2y=3x+12

cuts the parabola

4y=3x^2

[NDA II 2014]

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Question : 66 of 120

Marks: +1, -0

What is the area enclosed by the parabola, the line and the y-axis in the first quadrant?

7 square unit
14 square unit
20 square unit
21 square unit

Solution:

Area

\left| \int\limits_{-3}^{4} \left( \frac{3}{4}x^{2} - \frac{3}{2}x - 6 \right) dx \right|

\left| \left[ \frac{3}{4}\cdot\frac{x^{3}}{3} - \frac{3}{2}\cdot\frac{x^{2}}{2} - 6x \right]_{-2}^{4} = \left[ \frac{4^{3}}{4} - \frac{3}{2}\cdot\frac{4^{2}}{2} - 6\times 4 \right] - \left[ \frac{0^{3}}{4} - \frac{3}{2}\times\frac{0^{2}}{2} - 6(0) \right] \right|

= \left| 16-12-24+0 \right| = \left| -20 \right| = 20

square units

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