JEE Main Maths: Basic Mathematics (Polynomials) questions with solutions

Q1. The polynomials P(x) = k*x³ + 3x² - 3 and Q(x) = 2x³ - 5x + k leave equal remainders when each is divided by (x - 4). Find k.

1. 2
2. 1
3. 0
4. -1

Answer: 1

By the Remainder Theorem the two remainders are P(4) and Q(4); equating them yields a linear equation in k.

Q2. Let P(x) = 2x³ + a x² + b x + c with a, b, c integers. Given P(sqrt(3)) = 10 - 2 sqrt(3), find the value of 3a + c.

1. 0
2. 4
3. -2
4. 10

Answer: 10

Plug in sqrt(3), group integer terms and sqrt(3) terms, then equate to 10 - 2 sqrt(3). The rational equation gives 3a + c = 10.