Let [t] denote the greatest integer ≤ t. If the constant term in the expansion of (3 x²-1/(2 x⁵))⁷ is α, then [α] is equal to ___________.

Solution

Maths sir · 4 steps

Write the general term, make the power of x zero, then take the floor carefully.

1. General term. The (r+1)th term of (3x² - 1/2x⁵)⁷ multiplies the coefficients and the powers of x from both parts.

   Tᵣ₊₁ = C(7, r)(3x²)⁷⁻ʳ(-1/2x⁵)ʳ = C(7, r) 3⁷⁻ʳ(-1/2)ʳ x¹⁴⁻⁷ʳ

2. Find r. The power of x is 2(7-r) - 5r = 14 - 7r. For a constant term, 14 - 7r = 0, so r = 2.
3. Compute the term. α = C(7, 2) × 3⁵ × (-1/2)² = 21 × 243 × 1/4 = 5103/4 = 1275.75. The sign is positive because r is even.
4. So. [α] = [1275.75] = 1275, because the greatest integer function always rounds down.

The common slip: 5103. Leaving out the factor (1/2)² from the second part gives 21 × 243 = 5103. The 1/2 is raised to the power r just like x⁵.

The idea to keep. For a constant term, find the power of x in the general term and set it to zero. Carry the numbers and signs of both parts along, and remember that [t] always rounds down.

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