Let α, β (α ≠ β) be the values of m , for which the equations x+y+z=1 , x+2y+4z=m and x+4y+10z=m² have infinitely many solutions. Then the value of Σₙ₌₁¹⁰ (n^α+n^β) is equal to :

Solution

Maths sir · 4 steps

Eliminate x first. The two leftover equations must be the same line, and that fixes m.

1. Eliminate x. Subtract the first equation from the second and from the third. This gives y + 3z = m - 1 and 3y + 9z = m² - 1.
2. Same left side. The left side of the second equation is exactly 3 times the first. For infinitely many solutions the right sides must match the same way. Otherwise the equations contradict and there is no solution.

   m² - 1 = 3(m - 1) ⇒ m² - 3m + 2 = 0 ⇒ m = 1, 2

3. Add the powers. With α = 1 and β = 2, the sum is Σ n + Σ n² = (10 × 11)/2 + (10 × 11 × 21)/6 = 55 + 385.
4. So. The value is 55 + 385 = 440, which is option (d).

The idea to keep. When the coefficient determinant is zero, reduce the system. Infinitely many solutions needs the right sides in the same ratio as the left sides. A mismatch means no solution.

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