Paper I — Mathematics

Quadratic Equations and Inequalities

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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Core5 mapped NDA questions

Build the idea

Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.

Concept

A quadratic equation ax²+bx+c=0 (a≠0) has two roots given by the quadratic formula. The discriminant D = b²−4ac decides the nature of the roots. Inequalities are solved by finding critical points and testing sign intervals.

Sum and product of roots relate directly to the coefficients.

Key Points & Formulas

Roots: x = [−b ± √(b²−4ac)] / 2a.
D>0: real & distinct; D=0: real & equal; D<0: complex conjugate roots.
Sum of roots = −b/a; product of roots = c/a.
If roots are α,β, the equation is x² − (α+β)x + αβ = 0.

Worked Example

Q. Find the value of k for which x² − kx + 9 = 0 has equal roots.

Equal roots ⇒ discriminant D = 0.

D = k² − 4(1)(9) = k² − 36 = 0.

k² = 36 ⇒ k = ±6.

Common traps

Reading the prompt too quickly and missing a qualifier.
Memorising the result without checking when it applies.
Skipping units, signs, chronology or the final reasonableness check.

NDA shortcuts

State the governing rule before substituting values.
Eliminate impossible options before doing full calculation.

One-screen revision sheet

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Core definition
Most-used rule
One representative example

Past-paper bank

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5 NDA questions

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Question 1

NDA Practice 2026

The values of $m$ for which $x^2-mx+4=0$ has equal roots are

Question 2

NDA I 2025

If one root of the equation $x^2 - kx + k = 0$ exceeds the other by $2\sqrt{3}$ , then which one of the following is a value of $k$ ?

Question 3

NDA II 2024

Which one of the following is a root of the equation?

Question 4

NDA I 2024

Under which one of the following conditions does the equation (cosβ − 1)x² + (cosβ)x + sinβ = 0 in x have a real root for β ∈ [0, π]?

Question 5

NDA I 2024

If a, b and c (a > 0, c > 0) are in GP, then consider the following in respect of the equation ax² + bx + c = 0:
1. The equation has imaginary roots.
2. The ratio of the roots of the equation is 1 : ω where ω is a cube root of unity.
3. The product of roots of the equation is (b²/a²).
Which of the statements given above are correct?

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