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JEE Main 202628 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
28 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2026 · 28 January · Morning Shift · Mathematics · Quadratic Equation

Let S=\ x^ 3 +a x^ 2 +b x+c: a, b, c ∈ ~N. and .a, b, c ≤ 20\ be a...

Let S={x3+ax2+bx+c:a,b,c∈ Nanda,b,c≤20}\mathrm{S}=\left\{x^{3}+a x^{2}+b x+c: a, b, c \in \mathrm{~N}\right. and \left.a, b, c \leq 20\right\} be a set of polynomials. Then the number of polynomials in S, which are divisible by x2+2x^{2}+2, is

Options

  1. A10
  2. B20
  3. C6
  4. D120

Correct answer

A. 10

Step-by-step solution

If x2+2x^2 + 2 divides x3+ax2+bx+cx^3 + ax^2 + bx + c, write x3+ax2+bx+c=(x2+2)(x+k)x^3 + ax^2 + bx + c = (x^2 + 2)(x + k).
Expanding: x3+kx2+2x+2kx^3 + kx^2 + 2x + 2k.
Comparing coefficients: a=ka = k, b=2b = 2, c=2ac = 2a.
Constraints: a∈Na \in \mathbb{N}, a≤20a \leq 20, and c=2a≤20⇒a≤10c = 2a \leq 20 \Rightarrow a \leq 10.
So a∈{1,2,3,...,10}a \in \{1, 2, 3, ..., 10\}, giving 10 polynomials.
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