UPSC NDA 2 2019 Mathematics Solved Paper

Given thaty2=2x....(i)
and y=x ...(ii)
The eq. (i) in a eq. of parabola in standard form and eq. y = x a st. line passing through origin having slope I. The point of intersection of (i) and (ii), we get
x2=2x
⇒x2−2x=0
⇒x(x−2)=0
either x=0 or x−2=0
or, x=2
∴ y=x  ∴y=2
Now, required area = area of shaded region
A=

2

∫

0

(y2−y1)dx=

2

∫

0

(√2x−x)dy
=

2

∫

0

(√2x

1

2

−x)dx
=(√2

x 3 2

3 2

−

x2

2

)02
=

2√2

3

(2)

3

2

−

4

2

=

8

3

−2=

8−6

3

=

2

3

square units