If the constant term in the binomial expansion of
( √x - k/x² )¹⁰ is 405, then |k| equals :

Solution

Maths sir · 4 steps

Write the general term, add up the powers of x, and set that total to zero.

1. General term. In (√x - k/x²)¹⁰ the (r+1)th term is Tᵣ₊₁ = C(10, r)(√x)¹⁰⁻ʳ(-k/x²)ʳ.

   Tᵣ₊₁ = C(10, r)(-k)ʳ x^((10-r)/2 - 2r)

2. Set power to zero. For the constant term, (10-r)/2 - 2r = 0. Multiply by 2 to get 10 - r - 4r = 0, so r = 2.
3. Use the value 405. The constant term is C(10, 2)(-k)² = 45k². Setting 45k² = 405 gives k² = 9.
4. So. k = ± 3, so |k| = 3, which is option (a).

The idea to keep. For a constant term, or any particular power of x, write Tᵣ₊₁, combine all the powers of x into one exponent, and solve for r. The coefficient then gives the unknown.

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