Let a⃗ = î + 5ĵ + α k̂, b⃗ = î + 3ĵ + β k̂ and c⃗ = - î + 2ĵ - 3k̂ be three vectors such that, | b⃗ × c⃗ | = 5√3 and a⃗ is perpendicular to b⃗ . Then the greatest amongst the values of | a⃗ |² is _____________.

Solution

Maths sir · 4 steps

Two values of β mean two values of α: check both before picking the greatest.

1. Expand the cross product. With b⃗ = (1, 3, β) and c⃗ = (-1, 2, -3), the cross product is b⃗ × c⃗ = (-9 - 2β) î + (3 - β) ĵ + 5 k̂. Since |b⃗ × c⃗| = 5√3, its square is 75.

   (9 + 2β)² + (3 - β)² + 25 = 75

2. Solve for beta. Expanding gives 5β² + 30β + 115 = 75, so β² + 6β + 8 = 0. This factors as (β + 2)(β + 4) = 0, so β = -2 or β = -4.
3. Use perpendicularity. Since a⃗ · b⃗ = 0, we get 1 + 15 + αβ = 0, so αβ = -16. For β = -2, α = 8. For β = -4, α = 4.
4. So. |a⃗|² = 1 + 25 + α², which is 26 + 64 = 90 or 26 + 16 = 42. The greatest value is 90.

The common slip: 42. Stopping at one root, β = -4, gives α = 4 and |a⃗|² = 42, which is the smaller value, not the greatest.

The idea to keep. When a question asks for the greatest value, the condition usually has two roots. Carry both roots through to the end and compare the final values.

Answer from the NTA final answer key. Solution by PrepWiser.