Q. Find the value of k for which the quadratic equation x^2 + kx + 16 = 0 has no real roots. (2020)
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Solution
The discriminant must be less than zero: k^2 - 4*1*16 < 0 leads to k < -8.
Correct Answer:
A
— -8
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Q. For the quadratic equation 2x^2 + 4x + 2 = 0, what is the value of the discriminant? (2020)
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Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.
Correct Answer:
A
— 0
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have equal roots, what should be the value of k? (2020)
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Solution
For equal roots, the discriminant must be zero: b^2 - 4ac = 0. Here, 4^2 - 4(2)(k) = 0 leads to k = 4.
Correct Answer:
A
— -4
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real and equal roots, what is the condition on k? (2020)
A.
k < 0
B.
k = 0
C.
k = 8
D.
k > 8
Show solution
Solution
For real and equal roots, the discriminant must be zero. Here, b^2 - 4ac = 0 gives 16 - 8k = 0, thus k = 8.
Correct Answer:
C
— k = 8
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Q. For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)
A.
k > 4
B.
k < 4
C.
k >= 4
D.
k <= 4
Show solution
Solution
The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.
Correct Answer:
D
— k <= 4
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Q. For the quadratic equation 2x^2 + 4x - 6 = 0, what is the value of the discriminant? (2020)
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Solution
The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.
Correct Answer:
A
— 16
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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have equal roots, what must be the value of k? (2019)
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Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.
Correct Answer:
C
— 4
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Q. For the quadratic equation 5x^2 + 3x - 2 = 0, what is the value of the roots using the quadratic formula? (2023)
A.
-1, 2/5
B.
1, -2/5
C.
2, -1/5
D.
0, -2
Show solution
Solution
Using the quadratic formula x = [-b ± √(b^2 - 4ac)] / 2a, we find the roots to be -1 and 2/5.
Correct Answer:
A
— -1, 2/5
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Q. For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)
A.
p > 2
B.
p < 2
C.
p = 2
D.
p >= 2
Show solution
Solution
The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.
Correct Answer:
D
— p >= 2
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Q. For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the condition on k? (2023)
A.
k < 1
B.
k > 1
C.
k >= 1
D.
k <= 1
Show solution
Solution
The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.
Correct Answer:
D
— k <= 1
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Q. For the quadratic equation x^2 + 6x + 9 = 0, what type of roots does it have? (2019)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 + 6x + k = 0 to have distinct roots, what must be the condition on k? (2020)
A.
k < 9
B.
k = 9
C.
k > 9
D.
k ≤ 9
Show solution
Solution
The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.
Correct Answer:
A
— k < 9
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Q. For the quadratic equation x^2 + 6x + k = 0 to have real roots, what must be the condition on k? (2020)
A.
k < 9
B.
k = 9
C.
k > 9
D.
k ≤ 9
Show solution
Solution
The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.
Correct Answer:
D
— k ≤ 9
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Q. For the quadratic equation x^2 + px + q = 0, if the roots are -2 and -3, what is the value of p? (2020)
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Solution
The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.
Correct Answer:
A
— 5
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Q. For the quadratic equation x^2 - 4x + 4 = 0, what type of roots does it have? (2019)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. For the quadratic equation x^2 - 6x + k = 0 to have one root equal to 3, what is the value of k? (2023)
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Solution
If one root is 3, then substituting x = 3 gives 3^2 - 6*3 + k = 0, leading to k = 9.
Correct Answer:
C
— 9
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Q. For the quadratic equation x^2 - 8x + 15 = 0, what are the roots? (2023)
A.
3 and 5
B.
2 and 6
C.
1 and 7
D.
4 and 4
Show solution
Solution
The roots can be found by factorization: (x - 3)(x - 5) = 0, hence the roots are 3 and 5.
Correct Answer:
A
— 3 and 5
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Q. For which value of k does the equation x^2 + kx + 16 = 0 have equal roots? (2019)
Show solution
Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0. Solving gives k = -8.
Correct Answer:
B
— -4
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Q. If one root of the quadratic equation x^2 + px + q = 0 is 3, and the other root is -1, what is the value of p? (2021)
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Solution
The sum of the roots is 3 + (-1) = 2, hence p = -2.
Correct Answer:
A
— 2
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Q. If one root of the quadratic equation x^2 - 4x + k = 0 is 2, what is the value of k? (2021)
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Solution
Substituting x = 2 into the equation gives 2^2 - 4*2 + k = 0, which simplifies to k = 4.
Correct Answer:
C
— 4
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Q. If one root of the quadratic equation x^2 - 7x + k = 0 is 3, what is the value of k? (2023)
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Solution
Using the root 3 in the equation: 3^2 - 7*3 + k = 0, we get k = 6.
Correct Answer:
A
— 6
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Q. If the quadratic equation ax^2 + bx + c = 0 has roots p and q, what is the value of p + q? (2020)
A.
-b/a
B.
b/a
C.
c/a
D.
-c/a
Show solution
Solution
By Vieta's formulas, the sum of the roots p + q = -b/a.
Correct Answer:
A
— -b/a
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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what are the roots? (2022)
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Solution
The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.
Correct Answer:
A
— -1
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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the nature of its roots? (2019)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the nature of the roots? (2022)
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
Show solution
Solution
The discriminant is 0, indicating that the roots are real and equal.
Correct Answer:
B
— Real and equal
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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the value of x? (2023)
Show solution
Solution
The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.
Correct Answer:
A
— -1
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Q. If the quadratic equation x^2 + 2x + k = 0 has roots 1 and -3, what is the value of k? (2022)
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Solution
The product of the roots is 1 * (-3) = -3, hence k = -3.
Correct Answer:
A
— -3
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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition on k? (2023)
A.
k > 0
B.
k < 0
C.
k >= 0
D.
k <= 0
Show solution
Solution
For both roots to be negative, k must be greater than 0.
Correct Answer:
A
— k > 0
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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, what is the condition on k? (2019)
A.
k < 0
B.
k > 0
C.
k < 4
D.
k > 4
Show solution
Solution
For both roots to be positive, k must be less than 4 (from the condition of the sum and product of roots).
Correct Answer:
C
— k < 4
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Q. If the quadratic equation x^2 + 4x + 4 = 0 is solved, what is the nature of its roots? (2019)
A.
Two distinct real roots
B.
One real root
C.
Two complex roots
D.
No roots
Show solution
Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer:
B
— One real root
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Showing 1 to 30 of 70 (3 Pages)
Quadratic Equations MCQ & Objective Questions
Quadratic equations are a fundamental part of mathematics that play a crucial role in various school and competitive exams. Mastering this topic not only enhances your problem-solving skills but also boosts your confidence in tackling objective questions. Practicing MCQs related to quadratic equations helps you identify important questions and improves your exam preparation significantly.
What You Will Practise Here
Understanding the standard form of quadratic equations.
Solving quadratic equations using factorization, completing the square, and the quadratic formula.
Identifying the nature of roots using the discriminant.
Graphical representation of quadratic functions and their properties.
Application of quadratic equations in real-life problems.
Common word problems involving quadratic equations.
Important theorems related to quadratic equations.
Exam Relevance
Quadratic equations are frequently featured in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that require them to solve equations, analyze graphs, and apply concepts to real-world scenarios. Common question patterns include multiple-choice questions that test both theoretical understanding and practical application of quadratic equations.
Common Mistakes Students Make
Confusing the signs when applying the quadratic formula.
Misinterpreting the discriminant and its implications on the nature of roots.
Overlooking the importance of checking solutions in word problems.
Failing to simplify equations properly before solving.
Neglecting to graph the equations accurately, leading to incorrect conclusions.
FAQs
Question: What is the standard form of a quadratic equation?Answer: The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Question: How do I determine the nature of the roots of a quadratic equation?Answer: The nature of the roots can be determined using the discriminant (D = b² - 4ac). If D > 0, there are two distinct real roots; if D = 0, there is one real root; and if D < 0, the roots are complex.
Now is the time to boost your understanding of quadratic equations! Dive into our practice MCQs and test your knowledge to excel in your exams.
