JEE Advanced Maths: Polynomials and Equations questions with solutions

Q1. Let p(x) be a cubic polynomial with leading coefficient 1 satisfying p(1) = 1, p(2) = 4, and p(3) = 9. Which of the following statements is/are correct?

1. p(4) = 22
2. p(6/5) = (6/5)²
3. p(x) = x² holds for exactly two values of x
4. p(x) = 0 has a root in the interval (0, 1)

Answer: p(4) = 22

Setting g(x) = p(x) - x² gives a monic cubic vanishing at x = 1, 2, 3, so g(x) = (x-1)(x-2)(x-3). All statements can be checked from this explicit form.

Q2. Let alpha, beta, gamma, delta be the four roots of x⁴ - x³ - x² - 1 = 0. Define p(x) = x⁶ - x⁵ - x³ - x² - x. Then which of the following values cannot be the value of p(alpha) + p(beta) + p(gamma) + p(delta)?

1. 4
2. 5
3. 6
4. 7

Answer: 7

Using alpha⁴ = alpha³ + alpha² + 1, compute alpha⁵ and alpha⁶ in terms of lower powers, substitute into p(alpha), and simplify to get p(alpha) = alpha² - alpha + 1. Summing over all four roots and using Vieta's formulas gives the total sum = 6, which is always fixed — so 4, 5, and 7 cannot be achieved.