AP EAPCET 19-Aug-2021 Shift 1 Paper

Let p=

^

i

+a

^

j

+

^

k

q=

^

j

+a

^

k

⇒r=a

^

i

+

^

k

Volume of parallelopiped =[pqr]
=p.(q×r)
V=|

1	a	1
0	1	a
a	0	1

|
⇒V=a3−a+1
Differentiating V with respect to a

dV

da

=3a2−1
Again, differentiating w.r.t. a

d2V

da2

=6a
For stationary points

dV

da

=0
⇒3a2−1=0
ora2=

1

3

‌or‌a=±

1

√3

At,a=

1

√3

d2V

da2

=6.

1

√3

>0
∴ Volume is minimum at a=

1

√3