Class 10 Discriminant and Nature of Roots — practice questions with answers
Algebra — Quadratics · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. What is the formula for the discriminant of the quadratic equation ax² + bx + c = 0?
Ab² - 4ac
Bb² + 4ac
C4ac - b²
Da² - 4bc
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Answer: A) b² - 4ac
The discriminant is denoted D = b² - 4ac. It decides the nature of the roots of ax² + bx + c = 0.
2. Which of these equations has two distinct real roots?
Ax² - 7x + 10 = 0
Bx² + 4 = 0
Cx² - 2x + 5 = 0
Dx² + 6x + 9 = 0
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Answer: A) x² - 7x + 10 = 0
For x² - 7x + 10: D = 49 - 40 = 9 > 0. The others give D = -16, -16 and 0.
3. The number of real roots of 2x² + 5x + 7 = 0 is:
A1
B0
C-1
D2
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Answer: B) 0
D = 25 - 4(2)(7) = 25 - 56 = -31 < 0, so there are 0 real roots.
4. Mohan claims that 2x² + 3x + 4 = 0 and 2x² + 3x - 4 = 0 both have no real roots. Is his claim correct?
ANo; the second one has real roots
BYes; both have no real roots
CNo; the first one has real roots
DBoth have equal roots
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Answer: A) No; the second one has real roots
First: D = 9 - 32 = -23 < 0 (no real roots). Second: D = 9 + 32 = 41 > 0 (real, distinct). So Mohan is wrong about the second.
5. A designer fits a parabolic arch modelled by y = x² - 2mx + (m² + m - 6). The arch just touches the x-axis (one point only) for some m. Find the value of m for which the curve touches the x-axis exactly once.
A7
B6
C5
D12
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Answer: B) 6
Touching once means equal roots, so D = 0: (-2m)² - 4(1)(m² + m - 6) = 0, i.e. 4m² - 4m² - 4m + 24 = 0, so -4m + 24 = 0, giving m = 6.