JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023If a point P(α,β,γ) satisfying (αβγ)298amp;10amp;3amp;4amp;8amp;8amp;8=(000) lies on the plane 2x+4y+3z=5, then 6α+9β+7γ is equal to:
Choose the correct answer:
- A.
-1
- B.
511
- C.
45
- D.
11
(Correct Answer)
11
Explanation
Solution:
-
Matrix Multiplication se equations:
-
2α+9β+8γ=0 ---(i)
-
10α+3β+4γ=0 ---(ii)
-
8α+8β+8γ=0⟹α+β+γ=0 ---(iii)
-
-
α,β,γ ka ratio:
Eq (iii) se γ=−α−β. Ise (i) mein rakhte hain:
2α+9β+8(−α−β)=0⟹−6α+β=0⟹β=6α
Toh, γ=−α−6α=−7α.
Point P ho gaya: (α,6α,−7α).
-
Plane ki equation mein rakhne par:
2(α)+4(6α)+3(−7α)=5
2α+24α−21α=5⟹5α=5⟹α=1
Isliye: α=1,β=6,γ=−7.
-
Final Calculation:
6α+9β+7γ=6(1)+9(6)+7(−7)
=6+54−49=11
Correct Option: (4)
Explanation
Solution:
-
Matrix Multiplication se equations:
-
2α+9β+8γ=0 ---(i)
-
10α+3β+4γ=0 ---(ii)
-
8α+8β+8γ=0⟹α+β+γ=0 ---(iii)
-
-
α,β,γ ka ratio:
Eq (iii) se γ=−α−β. Ise (i) mein rakhte hain:
2α+9β+8(−α−β)=0⟹−6α+β=0⟹β=6α
Toh, γ=−α−6α=−7α.
Point P ho gaya: (α,6α,−7α).
-
Plane ki equation mein rakhne par:
2(α)+4(6α)+3(−7α)=5
2α+24α−21α=5⟹5α=5⟹α=1
Isliye: α=1,β=6,γ=−7.
-
Final Calculation:
6α+9β+7γ=6(1)+9(6)+7(−7)
=6+54−49=11
Correct Option: (4)