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Question

The angle between the lines whose direction cosines are given by the equations 3l+m+5n=0,6mn2nl+5lm=0

A
θ=cos1(16)
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B
θ=cos1(13)
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C
θ=cos1(12)
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D
θ=cos1(18)
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Solution

The correct option is D θ=cos1(16)
Given,
3l+m+5n=0
m=5n3l ………..(1)
and 6mn2nl+5lm=0
6(5n3l)n2nl+5l(5n3l)=0 [from (1)]
30n218nl2nl25nl15l2=0
30n245nl15l2=0
15(2n2+3nl+l2)=0
2n2+3nl+l2=0
2n2+2nl+nl+l2=0
2n(n+l)+l(n+l)=0
(n+1)(2n+l)=0
l=n or 2n+l=0
l=2n
When l=n,
m=5n3(n)
=5n+3n
=2n
l:m:n=n:2n:n
=1:2:1
When l=2n
m=5n3(2n)
=5n+6n
=n
l:m:n=2n:n:n
=2:1:1
The direction ratios are (1,2,1) and (2,1,1)
Angle between the lines whose direction cosines are given by the equations.
3l+m+5n=0 and 6mn2nl+5lm=0
θ=cos1⎪ ⎪⎪ ⎪(1)×(2)+(2)×1+1×1(1)2+(2)2+12(2)2+12+12⎪ ⎪⎪ ⎪
=cos1{22+11+4+14+1+1}
=cos1{166}
=cos1(16).

1210304_1315905_ans_b0db6348c02f4179bcd3cda79389c1d2.jpg

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