UPSC NDA Math Model Paper 9

Equation of curve ⇒y=x2
Equation of line ⇒y=2x+3
Line intersect the curve
∴x2=2x+3
⇒x2−2x−3=0
⇒x2−3x+x−3=0
⇒x(x−3)+1(x−3)=0
⇒(x−3)(x+1)=0⇒x=3,−1
So the line cut the curve at x=3 and x=−1
Now area of the region bounded by curve = area below the line – area below curve
∴ Area =

∫

−1

3(2x+3−x2)
=[

2x2

2

+3x−

x3

3

]−13 =[32+3(3)−

33

3

−((−1)2−3−(

−1

3

))]
=[9+9−9−1+3−

1

3

]
=[8+

8

3

]
=

32

3

‌sq units
Hence, option (2) is correct.