UPSC NDA Math Model Paper 4

Given that,
Hypotenuse of right angle tringle = 10 cm.
From the diagram By using Pythagoras theorem
⇒x2+y2=102
⇒y=√100−x2 ....…. (1)
Area of tringle
⇒ A=

1

2

x×y
⇒A=

1

2

x×√100−x2
To maximize area we have to find stationary point where f’(x) = 0
So differentiate A with respect to x
⇒

dA

dx

=

1

2

[√100−x2+x ×

−2x

2√100−x2

]
⇒

dA

dx

=0
⇒

1

2

[√100−x2+x×

−x

√100−x2

]=0
⇒

100−2x2

√100−x2

=0
⇒100−2x2=0
⇒x=√50 …. (2)
Now we will check second derivative polarity.
⇒

dA

dx

=

100−2x2

√100−x2

While putting x=√50
⇒

d2A

dx2

<0
So at x=√50 it gives local maxima.
Now, put x=√50 in Area
⇒Amax=

1

2

x×√100−x2
⇒Amax=50∕2=25cm2