Let f(x) be a polynomial of degree 3 such that
f(k) = - 2/k for k = 2, 3, 4, 5. Then the value of 52 - 10f(10) is equal to :

Solution

Maths sir · 4 steps

Multiply by x to clear the fraction: xf(x) + 2 is a quartic with four known roots.

1. Build a new polynomial. Let g(x) = xf(x) + 2. Since f has degree 3, g has degree 4. For k = 2, 3, 4, 5 we get g(k) = k(-2/k) + 2 = 0, so these are four roots of g.

   xf(x) + 2 = a(x-2)(x-3)(x-4)(x-5)

2. Find a. Put x = 0. The left side becomes 2, and the right side becomes a(-2)(-3)(-4)(-5) = 120a. So a = 1/60.
3. Put x = 10. 10f(10) + 2 = 1/60 × 8 × 7 × 6 × 5 = 1680/60 = 28. So 10f(10) = 26.
4. So. 52 - 10f(10) = 52 - 26 = 26.

The common slip: 24. Treating g(10) = 28 as 10f(10) and forgetting to subtract the 2 gives 52 - 28 = 24, not 26.

The idea to keep. When a polynomial takes values like c/k at several points, multiply by x and shift so those points become roots. Then x = 0 fixes the leading coefficient.

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