JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023A(2,6,2),B(−4,0,λ),C(2,3,−1) and D(4,5,0), ∣λ∣≤5 are the vertices of a quadrilateral ABCD. If its area is 18 square units, then 5−6λ is equal to:
Choose the correct answer:
- A.
11
(Correct Answer) - B.
10
- C.
9
- D.
8
11
Explanation
Solution
The area of a quadrilateral in 3D can be found using the cross product of its diagonals: Area=21∣AC×BD∣.
1. Find vectors for diagonals:
-
AC=(2−2)i^+(3−6)j^+(−1−2)k^=0i^−3j^−3k^
-
BD=(4−(−4))i^+(5−0)j^+(0−λ)k^=8i^+5j^−λk^
2. Cross Product:
AC×BD=(3λ+15)i^−24j^+24k^
3. Area calculation:
18=21(3λ+15)2+(−24)2+242
36=(3λ+15)2+1152
Square both sides: 1296=(3λ+15)2+1152
144=(3λ+15)2⟹3λ+15=±12
-
Case 1: 3λ+15=12⟹3λ=−3⟹λ=−1
-
Case 2: 3λ+15=−12⟹3λ=−27⟹λ=−9 (Rejected since ∣λ∣≤5)
4. Final Step:
Substitute λ=−1 into 5−6λ:
Explanation
Solution
The area of a quadrilateral in 3D can be found using the cross product of its diagonals: Area=21∣AC×BD∣.
1. Find vectors for diagonals:
-
AC=(2−2)i^+(3−6)j^+(−1−2)k^=0i^−3j^−3k^
-
BD=(4−(−4))i^+(5−0)j^+(0−λ)k^=8i^+5j^−λk^
2. Cross Product:
AC×BD=(3λ+15)i^−24j^+24k^
3. Area calculation:
18=21(3λ+15)2+(−24)2+242
36=(3λ+15)2+1152
Square both sides: 1296=(3λ+15)2+1152
144=(3λ+15)2⟹3λ+15=±12
-
Case 1: 3λ+15=12⟹3λ=−3⟹λ=−1
-
Case 2: 3λ+15=−12⟹3λ=−27⟹λ=−9 (Rejected since ∣λ∣≤5)
4. Final Step:
Substitute λ=−1 into 5−6λ: