UPSC NDA Math Model Paper 7

→

a

=

^

i

+

^

j

+

^

k

,

→

b

=2

^

i

−4

^

k

,

→

c

=

^

i

+λ

^

j

+3

^

k

are coplanar
As we know that, if three non-zero vectors

→

a

,

→

b

and

→

c

are coplanar if and only if [a‌‌b‌‌c]=0
We also know that,

→

a

.(

→

b

×

→

c

)=|

a1	a2	a3
b1	b2	b3
c1	c2	c3

| =[

a

b

c

]
Here, a1=1,b1=1,c1=1,a2=2,b2=0, c2=−4,a3=1,b3=λ and c3=3
⇒[

a

b

c

]=|

1	2	1
1	0	λ
1	−4	3

| =0
∴ 1(4λ) - 2(3 - λ) + 1(- 4) = 0
∴ 4λ - 6 + 2λ - 4 = 0
∴ λ = 5/3
∴ option (D) is the correct answer.