HCU MCA 2013 Paper

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Question : 71 of 75

Marks: +1, -0

The vectors

λi + j + k, i + λj - k \and 2i - j + λk

are coplanar, if

\I.

λ

= - 1

\II.

λ

=

{1±√5}/2

\III.

λ

= 0
Which of the following is correct?

$\I.$ and $\II.$
$\II.$ and $\III.$
$\I.$ and $\III.$
Only $\I$

Solution:

(d) Let a =

λi + j + k

b =

i + λj - k

c = 2i - j + λk
If these vectors are coplanar,
then

\;\;\;\;\;\;\;

[a b c] = 0
⇒

\;\;\;\;\;\;\;|\tableλ,1,1;1,λ,-1;2,-1,λ|=0

Expanding along

\R_1,

λ ( λ ^ { 2 } - 1 ) - 1 ( λ + 2 ) + 1 ( - 1 - 2 λ ) = 0

⇒ \;\;\;\;\;\;\;\;\;\;\;\;λ ^ { 3 } - λ - λ - 2 - 1 - 2 λ = 0

⇒ \;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\λ ^ { 3 } - 4 λ - 3 = 0

⇒\;\;λ ^ { 2 } (λ + 1 ) - λ ( λ + 1 ) - 3 ( λ + 1 ) = 0

⇒\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;( λ + 1 ) ( λ ^ { 2 } - λ - 3 ) = 0

∴\;\;\;\;\;\;\;\;\;\;\;\λ + 1 = 0

and

λ ^ { 2 } - λ - 3 = 0

⇒\;\;\;\λ = - 1

and

λ = { 1 ± √ { 1 + 12 } }/ { 2 } = { 1± √ { 13 } } /{ 2 }

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