The set of all α, for which the vectors a⃗=α t î+6 ĵ-3 k̂ and b⃗=t î-2 ĵ-2 α t k̂ are inclined at an obtuse angle for all t ∈ ℝ, is

Solution

Maths sir · 4 steps

Obtuse means dot product below zero, so this quadratic in t must stay negative for every real t.

1. Dot product. The angle is obtuse when a⃗·b⃗ < 0. Multiply matching components: a⃗·b⃗ = α t² - 12 + 6α t.

   α t² + 6α t - 12 < 0 for all t ∈ ℝ

2. Case alpha zero. If α = 0, the expression is just -12, which is negative for every t. So α = 0 works.
3. Case alpha nonzero. A quadratic is negative for all t only if it opens downward, so α < 0, and its discriminant is negative, so it never touches the axis. That gives 36α² + 48α < 0, that is, 12α(3α + 4) < 0, so -4/3 < α < 0.
4. So. Putting both cases together gives α ∈ (-4/3, 0], which is option (b).

The idea to keep. When a condition must hold for all real t, make it a quadratic and check two things: the sign of the leading coefficient and the discriminant. Test leading coefficient zero separately.

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