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Question

Match List I with the List II and select the correct answer using the code given below the lists :

List IList II(A)The possible value of a if r=(^i+^j)+λ(^i+2^j^k)(P) 4and r=(^i+2^j)+μ(^i+^j+a^k) are not consistent,where λ and μ are scalars, is(B)The angle between vectors a=λ^i3^j^k and(Q) 2b=2λ^i+λ^j^k is acute, whereas vector bmakes an obtuse angle with the axes of coordinates.Then λ can be(C)The possible value of a such that 2^i^j+^k,(R)    1^i+2^j+(1+a)^k and 3^i+a^j+5^k are coplanar, is(D)If A=2^i+λ^j+3^k,B=2^i+λ^j+^k,C=3^i+^j(S)    2and A+λB is perpendicular to C,then |2λ| is(T)    3  

Which of the following is the only CORRECT combination?
  1. (A)(P),(Q),(R) 
  2. (A)(P),(Q),(S),(T)
  3. (B)(P),(S)
  4. (B)(S),(T) 


Solution

The correct option is B (A)(P),(Q),(S),(T)
(A) Given equations are consistent if
(^i+^j)+λ(^i+2^j^k)=(^i+2^j)+μ(^i+^j+a^k)
1+λ=1μ, 1+2λ=2+μ, λ=aμ
λ=13 and μ=13
a=1
So, for inconsistency, a1
(A)(P),(Q),(S),(T)

(B) a=λ^i3^j^k 
b=2λ^i+λ^j^k
Angle between a and b is acute.
ab>0
2λ23λ+1>0
(2λ1)(λ1)>0
λ(,12)(1,)

Also, b makes an obtuse angle with the axes. Therefore,
b.^i<0λ<0
b.^j<0λ<0
Hence, λ can be 4,2
(B)(P),(Q)

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