JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023Let S be the set of all values of θ∈[−π,π] for which the system of linear equations
x+y+3z=0
−x+(tanθ)y+7z=0
x+y+(tanθ)z=0
has non-trivial solution. Then π120θ∈S∑θ is equal to
Choose the correct answer:
- A.
20
(Correct Answer) - B.
40
- C.
30
- D.
10
20
Explanation
Since, the given system has a non-trivial solution, So, Δ=0
⟹Δ=1−11amp;1amp;tanθamp;1amp;3amp;7amp;tanθ=0
⇒1(tan2θ−7)−1(−tanθ−7)+3(−1−tanθ)=0
⇒tan2θ−7+tanθ+7−3−3tanθ=0
⇒tanθ(tanθ−3)+1(tanθ−3)=0
⇒tanθ=3ortanθ=−1
∴θ={3π,3−2π,4−π,43π}
So, π120θ∈S∑θ=π120{124π−8π−3π+9π}
=π120[122π]=20
Explanation
Since, the given system has a non-trivial solution, So, Δ=0
⟹Δ=1−11amp;1amp;tanθamp;1amp;3amp;7amp;tanθ=0
⇒1(tan2θ−7)−1(−tanθ−7)+3(−1−tanθ)=0
⇒tan2θ−7+tanθ+7−3−3tanθ=0
⇒tanθ(tanθ−3)+1(tanθ−3)=0
⇒tanθ=3ortanθ=−1
∴θ={3π,3−2π,4−π,43π}
So, π120θ∈S∑θ=π120{124π−8π−3π+9π}
=π120[122π]=20