10TH CBSE Maths QUIZ – Quadratic Equations 10 MCQ Tests × 10 Questions = 100 Questions

Class 10 Maths – Quadratic Equations

10 MCQ Tests × 10 Questions = 100 Questions

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Test 1

Q1. The quadratic equation x² − 4x + k = 0 has distinct real roots when:

A. k = 4 B. k > 4 C. k = 16 D. k < 4

Correct Answer: D. k < 4

Q2. The discriminant of ax² + bx + c = 0 is:

A. a² − 4bc B. b² − 4ac C. c² − 4ab D. b² + 4ac

Correct Answer: B. b² − 4ac

Q3. If the discriminant of a quadratic equation is zero, its roots are:

A. Non-real B. Real and distinct C. Real and equal D. Always irrational

Correct Answer: C. Real and equal

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Q4. If one zero of x² + 3x + k is 2, then k is:

A. 10 B. −10 C. −7 D. −2

Correct Answer: B. −10

Q5. For ax² + bx + c = 0 to have two distinct real roots, we need:

A. D < 0 B. D = 0 C. D > 0 D. D ≥ 0 only

Correct Answer: C. D > 0

Q6. The roots of x² − 3x − m(m + 3) = 0 are:

A. m, m + 3 B. −m, m + 3 C. m, −(m + 3) D. −m, −(m + 3)

Correct Answer: B. −m, m + 3

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Q7. The equation 4x² − 20x + 25 = 0 has:

A. No real roots B. Real distinct roots C. Real equal roots D. Three roots

Correct Answer: C. Real equal roots

Q8. If x = 2 is a root of 3x² − px − 2 = 0, p equals:

A. 1 B. 5 C. 3 D. 0

Correct Answer: B. 5

Q9. The roots of x² + x − p(p + 1) = 0 are:

A. p, p + 1 B. −p, p + 1 C. p, −(p + 1) D. −p, −(p + 1)

Correct Answer: C. p, −(p + 1)

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Q10. For x² − ax + 1 = 0 to have two distinct real roots:

A. |a| = 2 B. |a| < 2 C. |a| > 2 D. |a| = 0

Correct Answer: C. |a| > 2

Test 2

Q1. A quadratic equation is an equation of degree:

A. 1 B. 2 C. 3 D. 4

Correct Answer: B. 2

Q2. For x² − 5x + 6 = 0, the roots are:

A. 1 and 6 B. 2 and 3 C. −2 and −3 D. 5 and 6

Correct Answer: B. 2 and 3

Q3. The sum of roots of ax² + bx + c = 0 is:

A. b/a B. −b/a C. c/a D. −c/a

Correct Answer: B. −b/a

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Q4. The product of roots of ax² + bx + c = 0 is:

A. −b/a B. b/a C. c/a D. −c/a

Correct Answer: C. c/a

Q5. The equation 9x² + 6x + 1 = 0 has:

A. Distinct real roots B. Equal real roots C. No real roots D. Three roots

Correct Answer: B. Equal real roots

Q6. The least positive value of k for which 2x² + kx − 4 = 0 has rational roots is:

A. 1 B. 2 C. 3 D. 4

Correct Answer: B. 2

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Q7. If the roots are 5 and −2, the quadratic equation is:

A. x² − 3x + 10 = 0 B. x² − 3x − 10 = 0 C. x² + 3x − 10 = 0 D. x² + 3x + 10 = 0

Correct Answer: B. x² − 3x − 10 = 0

Q8. A quadratic equation has non-real roots when:

A. D = 0 B. D > 0 C. D < 0 D. D ≥ 0

Correct Answer: C. D < 0

Q9. If one root is 3 and the sum of roots is zero, the equation is:

A. x² + 9 = 0 B. 9x² − 1 = 0 C. x² − 9 = 0 D. 9x² + 1 = 0

Correct Answer: C. x² − 9 = 0

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Q10. The roots of x² − 4px + 4p² − q² = 0 are:

A. 2p + q, 2p + q B. p + 2q, p − 2q C. 2p − q, 2p − q D. 2p + q, 2p − q

Correct Answer: D. 2p + q, 2p − q

Test 3

Q1. The equation x² − 8x + k = 0 has distinct real roots when:

A. k = 16 B. k > 16 C. k = 8 D. k < 16

Correct Answer: D. k < 16

Q2. For equal roots of 2x² + kx + 2 = 0, k is:

A. 2 B. −2 C. ±4 D. ±8

Correct Answer: C. ±4

Q3. The equation 9x² + 12x + 4 = 0 has:

A. Real distinct roots B. Real equal roots C. No real roots D. More than two roots

Correct Answer: B. Real equal roots

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Q4. The roots of 2x² − x − 6 = 0 are:

A. 2, −3/2 B. −2, 3/2 C. 3, −2 D. 1/2, −6

Correct Answer: A. 2, −3/2

Q5. If x = 3 is a root of 3x² + (k − 1)x + 9 = 0, then k is:

A. −11 B. 11 C. −13 D. 13

Correct Answer: A. −11

Q6. The discriminant of x² − 6ax + 6a² = 0 is:

A. 4a² B. 12a² C. 2a² D. 6a²

Correct Answer: B. 12a²

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Q7. If a quadratic equation has equal roots, then:

A. b² = 4ac B. b² = ac C. b² = 2ac D. b² = −ac

Correct Answer: A. b² = 4ac

Q8. The graph of a quadratic polynomial is generally a:

A. Straight line B. Circle C. Parabola D. Hyperbola

Correct Answer: C. Parabola

Q9. If D > 0, the quadratic equation has:

A. No real roots B. Two equal real roots C. Two distinct real roots D. One non-real root

Correct Answer: C. Two distinct real roots

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Q10. A root of a quadratic equation is a value of x that:

A. Makes the equation undefined B. Satisfies the equation C. Makes the coefficient zero D. Must be positive

Correct Answer: B. Satisfies the equation

Test 4

Q1. The perimeter of a rectangle is 82 m and area is 400 m². Its breadth is:

A. 25 m B. 20 m C. 16 m D. 9 m

Correct Answer: C. 16 m

Q2. The two numbers whose sum is 27 and product is 182 are:

A. 14 and 15 B. 12 and 13 C. 13 and 14 D. 12 and 15

Correct Answer: C. 13 and 14

Q3. A train travels 360 km. If its speed were 5 km/h more, it would take 1 hour less. Its actual speed is:

A. 48 km/h B. 40 km/h C. 36 km/h D. 45 km/h

Correct Answer: B. 40 km/h

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Q4. A right triangle has hypotenuse 29 cm and perimeter 70 cm. Its other sides are:

A. 20 cm and 21 cm B. 18 cm and 23 cm C. 15 cm and 26 cm D. 14 cm and 27 cm

Correct Answer: A. 20 cm and 21 cm

Q5. For roots 2p+q and 2p−q, their sum is:

A. 2p B. 4p C. 2q D. 4q

Correct Answer: B. 4p

Q6. For roots 2p+q and 2p−q, their product is:

A. 4p² − q² B. 4p² + q² C. 2p² − q² D. p² − 4q²

Correct Answer: A. 4p² − q²

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Q7. The expression (x−2p)² − q² = 0 can be factored using:

A. a²+b² B. a²−b² C. (a+b)² D. (a−b)³

Correct Answer: B. a²−b²

Q8. The equation (2x−1)² − 4x² + 5 = 0 simplifies to:

A. A quadratic equation B. A linear equation C. A cubic equation D. An identity

Correct Answer: B. A linear equation

Q9. For 500 bananas divided among x students, the source obtains x as:

A. 50 B. 100 C. 125 D. 250

Correct Answer: B. 100

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Q10. In the train problem, the negative algebraic solution is rejected because speed:

A. Must be zero B. Cannot be negative C. Must be irrational D. Must be even

Correct Answer: B. Cannot be negative

Test 5

Q1. The equation x² + 2x + 2 = 0 has:

A. Two distinct real roots B. Two equal real roots C. No real roots D. One positive root

Correct Answer: C. No real roots

Q2. For 16x² + 4kx + 9 = 0 to have equal roots, k equals:

A. ±3 B. ±4 C. ±6 D. ±9

Correct Answer: C. ±6

Q3. If x = 1 is a common root of ax² + ax + 3 = 0 and x² + x + b = 0, then ab is:

A. 6 B. 3 C. −3 D. 3.5

Correct Answer: B. 3

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Q4. Which is a quadratic equation?

A. x² + 1 = (2 − x)² + 3 B. x³ − x² = (x − 1)³ C. 2x² + 3 = (5 + x)(2x − 3) D. None of these

Correct Answer: B. x³ − x² = (x − 1)³

Q5. The sum of roots of 2x² − 3x + 6 = 0 is:

A. 3/2 B. −3/2 C. 2 D. 3

Correct Answer: A. 3/2

Q6. The product of roots of 3x² + 8x + 2 = 0 is:

A. 8/3 B. 2/3 C. −2/3 D. 3/2

Correct Answer: B. 2/3

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Q7. If x² − kx + 1 = 0 has no real roots, then:

A. k < −2 B. k > 2 C. −2 < k < 2 D. k = 2

Correct Answer: C. −2 < k < 2

Q8. The roots of x² − 7x + 12 = 0 are:

A. 3 and 4 B. −3 and −4 C. 3 and −4 D. −3 and 4

Correct Answer: A. 3 and 4

Q9. The quadratic formula for ax² + bx + c = 0 is:

A. (−b ± √(b²−4ac))/2a B. (b ± √(b²+4ac))/2a C. (−b ± √(b²+4ac))/a D. (b ± √(b²−4ac))/2c

Correct Answer: A. (−b ± √(b²−4ac))/2a

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Q10. If D = 0, the graph of the quadratic polynomial:

A. Does not meet x-axis B. Touches x-axis at one point C. Cuts x-axis at two points D. Is a straight line

Correct Answer: B. Touches x-axis at one point

Test 6

Q1. If the roots are rational, the discriminant is typically:

A. Negative B. Zero only C. A non-negative perfect square D. Always 1

Correct Answer: C. A non-negative perfect square

Q2. Which equation has irrational roots?

A. x² − 5x + 6 = 0 B. 4x² − 7x + 3 = 0 C. 6x² − 3x − 5 = 0 D. 2x² + 3x − 2 = 0

Correct Answer: C. 6x² − 3x − 5 = 0

Q3. Which equation has equal roots?

A. x² − 3x + 4 = 0 B. 2x² − 2x + 1 = 0 C. 5x² − 10x + 1 = 0 D. 9x² + 6x + 1 = 0

Correct Answer: D. 9x² + 6x + 1 = 0

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Q4. Which equation has two distinct real roots?

A. x² + 3x + 1 = 0 B. −x² + 3x − 3 = 0 C. 4x² + 8x + 4 = 0 D. 3x² + 6x + 4 = 0

Correct Answer: A. x² + 3x + 1 = 0

Q5. The value of p when x² + px + q has roots twice those of 4x² − 5x − 6 is:

A. 5/2 B. −5/2 C. −5/4 D. 5/4

Correct Answer: B. −5/2

Q6. If one root of 2x² + ax + 6 = 0 is 2, then a is:

A. 7 B. −7 C. 14 D. −14

Correct Answer: B. −7

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Q7. If x = 2 is a root of 3x² − px − 2 = 0, p is:

A. 1 B. 5 C. 3 D. 0

Correct Answer: B. 5

Q8. For equal roots of kx² − 12x + 4 = 0, k is:

A. 4 B. 8 C. 9 D. 12

Correct Answer: C. 9

Q9. If 4x² − 12x + 9 = 0, the roots are:

A. Distinct real B. Repeated/equal real C. Non-real D. One real and one imaginary

Correct Answer: B. Repeated/equal real

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Q10. A quadratic equation ax² + bx + c = 0 has two distinct real roots if:

A. b²−4ac < 0 B. b²−4ac = 0 C. b²−4ac > 0 D. a = 0

Correct Answer: C. b²−4ac > 0

Test 7

Q1. The condition for real and equal roots is:

A. D > 0 B. D = 0 C. D < 0 D. D = 1

Correct Answer: B. D = 0

Q2. The condition for no real roots is:

A. D > 0 B. D = 0 C. D < 0 D. D ≥ 0

Correct Answer: C. D < 0

Q3. The condition for two distinct real roots is:

A. D > 0 B. D = 0 C. D < 0 D. D ≤ 0

Correct Answer: A. D > 0

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Q4. If the roots of x² − 3x + k = 0 are equal, k is:

A. 3/4 B. 9/4 C. 4/9 D. 9

Correct Answer: B. 9/4

Q5. If roots are α and β, the quadratic equation is:

A. x² + (α+β)x + αβ = 0 B. x² − (α+β)x + αβ = 0 C. x² − αβx + α+β = 0 D. x² + αβx − (α+β) = 0

Correct Answer: B. x² − (α+β)x + αβ = 0

Q6. If roots are 5 and −2, their sum is:

A. −7 B. 7 C. 3 D. −3

Correct Answer: C. 3

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Q7. If roots are 5 and −2, their product is:

A. 10 B. −10 C. 7 D. −7

Correct Answer: B. −10

Q8. If p = −7 and q = 12 in x² + px + q = 0, roots are:

A. 3 and 4 B. 3 and −4 C. −3 and −4 D. −3 and 4

Correct Answer: A. 3 and 4

Q9. A quadratic equation whose one root is 3 can be:

A. x² − 5x − 6 = 0 B. x² − 6x − 6 = 0 C. x² − 5x + 6 = 0 D. x² + 6x − 5 = 0

Correct Answer: C. x² − 5x + 6 = 0

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Q10. The roots of x² − 4x + 4 = 0 are:

A. 2 and 2 B. 4 and 0 C. −2 and −2 D. 1 and 4

Correct Answer: A. 2 and 2

Test 8

Q1. If the roots are α and β, then α + β equals:

A. −b/a B. b/a C. c/a D. −c/a

Correct Answer: A. −b/a

Q2. If the roots are α and β, then αβ equals:

A. −b/a B. b/a C. c/a D. −c/a

Correct Answer: C. c/a

Q3. If α = β for ax² + bx + c = 0, then α equals:

A. −b/a B. −b/(2a) C. b/(2a) D. c/a

Correct Answer: B. −b/(2a)

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Q4. If a quadratic has repeated roots, its graph touches the x-axis:

A. At two points B. At one point C. At no point D. At three points

Correct Answer: B. At one point

Q5. The equation x² + 4x + 4 = 0 has root:

A. −2 B. 2 C. 4 D. −4

Correct Answer: A. −2

Q6. The equation 3x² + 17x − 30 = 0 has:

A. No real roots B. Equal roots C. Distinct real roots D. Only one root

Correct Answer: C. Distinct real roots

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Q7. A quadratic polynomial has irrational roots when D is:

A. Positive and a perfect square B. Positive but not a perfect square C. Zero D. Negative

Correct Answer: B. Positive but not a perfect square

Q8. The standard form of a quadratic equation is:

A. ax + b = 0 B. ax² + bx + c = 0, a ≠ 0 C. ax³ + bx² + c = 0 D. a/x + b = 0

Correct Answer: B. ax² + bx + c = 0, a ≠ 0

Q9. The coefficient of x² in a quadratic equation cannot be:

A. 1 B. −2 C. 0 D. 5

Correct Answer: C. 0

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Q10. The discriminant helps determine the:

A. Degree only B. Nature of roots C. Sum only D. Product only

Correct Answer: B. Nature of roots

Test 9

Q1. A rectangle has perimeter 82 m and area 400 m². Its dimensions are:

A. 25 m × 16 m B. 20 m × 20 m C. 40 m × 10 m D. 32 m × 18 m

Correct Answer: A. 25 m × 16 m

Q2. Two numbers have sum 27 and product 182. They are:

A. 12 and 15 B. 13 and 14 C. 10 and 17 D. 11 and 16

Correct Answer: B. 13 and 14

Q3. A train's actual speed in the 360 km problem is:

A. 36 km/h B. 40 km/h C. 45 km/h D. 48 km/h

Correct Answer: B. 40 km/h

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Q4. The area of the right triangle with sides 20 cm, 21 cm, 29 cm is:

A. 180 cm² B. 200 cm² C. 210 cm² D. 250 cm²

Correct Answer: C. 210 cm²

Q5. For roots 2p+q and 2p−q, their sum is:

A. 2p B. 4p C. 2q D. 4q

Correct Answer: B. 4p

Q6. For roots 2p+q and 2p−q, their product is:

A. 4p² − q² B. 4p² + q² C. 2p² − q² D. p² − 4q²

Correct Answer: A. 4p² − q²

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Q7. The expression (x−2p)² − q² = 0 can be factored using:

A. a²+b² B. a²−b² C. (a+b)² D. (a−b)³

Correct Answer: B. a²−b²

Q8. The equation (2x−1)² − 4x² + 5 = 0 simplifies to:

A. A quadratic equation B. A linear equation C. A cubic equation D. An identity

Correct Answer: B. A linear equation

Q9. For 500 bananas divided among x students, the source obtains x as:

A. 50 B. 100 C. 125 D. 250

Correct Answer: B. 100

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Q10. In the train problem, the negative algebraic solution is rejected because speed:

A. Must be zero B. Cannot be negative C. Must be irrational D. Must be even

Correct Answer: B. Cannot be negative

Test 10

Q1. For 8x² + 3kx + 2 = 0 to have equal roots, the discriminant must be:

A. 1 B. 0 C. 8 D. 64

Correct Answer: B. 0

Q2. The equation 9x² + 3kx + 4 = 0 has equal roots for:

A. k = ±2 B. k = ±4 C. k = ±6 D. k = ±8

Correct Answer: B. k = ±4

Q3. The equation 4x² − 12x + 9 = 0 is:

A. Not quadratic B. A quadratic with repeated roots C. A quadratic with no real roots D. A cubic

Correct Answer: B. A quadratic with repeated roots

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Q4. If D < 0, the roots are:

A. Real and equal B. Real and distinct C. Complex/non-real D. Always rational

Correct Answer: C. Complex/non-real

Q5. If D > 0 and is a perfect square, roots can be:

A. Real rational and distinct B. Equal only C. Non-real D. Never real

Correct Answer: A. Real rational and distinct

Q6. Which equation has no real roots?

A. −4x² + 7x − 4 = 0 B. −4x² + 7x − 2 = 0 C. −2x² + 5x − 2 = 0 D. 3x² + 6x + 2 = 0

Correct Answer: A. −4x² + 7x − 4 = 0

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Q7. Which equation has rational roots?

A. x² + x − 1 = 0 B. x² − 5x + 6 = 0 C. 6x² − x + 11 = 0 D. 3x² + 2x + 2 = 0

Correct Answer: B. x² − 5x + 6 = 0

Q8. Which equation has irrational roots?

A. x² − 5x + 6 = 0 B. 4x² − 7x + 3 = 0 C. 6x² − 3x − 5 = 0 D. 2x² + 3x − 2 = 0

Correct Answer: C. 6x² − 3x − 5 = 0

Q9. Which equation has equal roots?

A. x² − 3x + 4 = 0 B. 2x² − 2x + 1 = 0 C. 5x² − 10x + 1 = 0 D. 9x² + 6x + 1 = 0

Correct Answer: D. 9x² + 6x + 1 = 0

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Q10. Which equation has two distinct real roots?

A. x² + 3x + 1 = 0 B. −x² + 3x − 3 = 0 C. 4x² + 8x + 4 = 0 D. 3x² + 6x + 4 = 0

Correct Answer: A. x² + 3x + 1 = 0

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