If the greatest value of the term independent of 'x' in the
expansion of ( x sin α + a(cos α)/x )¹⁰ is 10!/(5!)², then the value of 'a' is equal to :

Solution

Maths sir · 4 steps

Pick the middle term, then write sin α cos α as 1/2 sin 2α to maximise it.

1. General term. Tᵣ₊₁ = ¹⁰Cᵣ (x sin α)¹⁰⁻ʳ((a cos α)/x)ʳ. The power of x is 10 - 2r, so the term is independent of x when r = 5.
2. Write the term. With r = 5, the term is ¹⁰C₅ a⁵ (sin α cos α)⁵. Since sin α cos α = 1/2 sin 2α, this becomes the expression below.

   T₆ = ¹⁰C₅ (a⁵ sin⁵ 2α)/32

3. Greatest value. Taking a positive, the term is largest when sin 2α = 1, so its greatest value is ¹⁰C₅ a⁵/32. Also note that ¹⁰C₅ = 10!/(5!)².
4. So. Setting ¹⁰C₅ a⁵/32 = 10!/(5!)² gives a⁵ = 32, so a = 2. The answer is (d).

The idea to keep. When sin α and cos α appear with the same power, write sin α cos α = 1/2 sin 2α. Then the greatest value comes from sin 2α = 1.

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