Major Competitive Exams MCQ & Objective Questions
Major Competitive Exams play a crucial role in shaping the academic and professional futures of students in India. These exams not only assess knowledge but also test problem-solving skills and time management. Practicing MCQs and objective questions is essential for scoring better, as they help in familiarizing students with the exam format and identifying important questions that frequently appear in tests.
What You Will Practise Here
Key concepts and theories related to major subjects
Important formulas and their applications
Definitions of critical terms and terminologies
Diagrams and illustrations to enhance understanding
Practice questions that mirror actual exam patterns
Strategies for solving objective questions efficiently
Time management techniques for competitive exams
Exam Relevance
The topics covered under Major Competitive Exams are integral to various examinations such as CBSE, State Boards, NEET, and JEE. Students can expect to encounter a mix of conceptual and application-based questions that require a solid understanding of the subjects. Common question patterns include multiple-choice questions that test both knowledge and analytical skills, making it essential to be well-prepared with practice MCQs.
Common Mistakes Students Make
Rushing through questions without reading them carefully
Overlooking the negative marking scheme in MCQs
Confusing similar concepts or terms
Neglecting to review previous years’ question papers
Failing to manage time effectively during the exam
FAQs
Question: How can I improve my performance in Major Competitive Exams?Answer: Regular practice of MCQs and understanding key concepts will significantly enhance your performance.
Question: What types of questions should I focus on for these exams?Answer: Concentrate on important Major Competitive Exams questions that frequently appear in past papers and mock tests.
Question: Are there specific strategies for tackling objective questions?Answer: Yes, practicing under timed conditions and reviewing mistakes can help develop effective strategies.
Start your journey towards success by solving practice MCQs today! Test your understanding and build confidence for your upcoming exams. Remember, consistent practice is the key to mastering Major Competitive Exams!
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
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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 negative, what is the condition for k? (2023)
A.
k > 1
B.
k < 1
C.
k > 0
D.
k < 0
Show solution
Solution
For both roots to be negative, k must be greater than 0.
Correct Answer:
C
— 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
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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 + 2x + k = 0 has roots that are equal, what is the value of k?
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Solution
For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.
Correct Answer:
D
— -2
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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
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Solution
The discriminant is 0, indicating one real root (a repeated root).
Correct Answer:
B
— One real root
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Q. If the quadratic equation x^2 + 4x + c = 0 has one root equal to -2, what is the value of c?
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Solution
If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.
Correct Answer:
A
— 0
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Q. If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?
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Solution
Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.
Correct Answer:
B
— 4
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Q. If the quadratic equation x^2 + 5x + 6 = 0 is solved, what is the product of the roots? (2022)
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Solution
The product of the roots is given by c/a = 6/1 = 6.
Correct Answer:
A
— 6
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Q. If the quadratic equation x^2 + 5x + k = 0 has one root as 2, what is the value of k? (2019)
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Solution
Substituting x = 2 into the equation gives 2^2 + 5(2) + k = 0, leading to 4 + 10 + k = 0, thus k = -14.
Correct Answer:
A
— 6
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Q. If the quadratic equation x^2 + 6x + 9 = 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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Q. If the quadratic equation x^2 + 6x + 9 = 0 is solved, what is the nature of the roots?
A.
Real and distinct
B.
Real and equal
C.
Complex
D.
None of the above
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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 + 6x + k = 0 has roots -2 and -4, what is the value of k?
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Solution
Using Vieta's formulas, k = (-2)(-4) = 8.
Correct Answer:
B
— 12
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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?
A.
k > 9
B.
k < 9
C.
k = 9
D.
k < 0
Show solution
Solution
For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.
Correct Answer:
A
— k > 9
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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)
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Solution
For both roots to be positive, k must be greater than 9 (since the sum of roots is -b/a and must be positive).
Correct Answer:
C
— 4
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Q. If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)
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Solution
For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.
Correct Answer:
C
— 8
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Q. If the quadratic equation x^2 + bx + 9 = 0 has roots 3 and -3, what is the value of b?
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Solution
The sum of the roots is 3 + (-3) = 0, so b = -0.
Correct Answer:
C
— -6
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Q. If the quadratic equation x^2 + kx + 16 = 0 has equal roots, what is the value of k?
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Solution
For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k = -8.
Correct Answer:
A
— -8
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Q. If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the condition on k?
A.
k^2 >= 64
B.
k^2 < 64
C.
k > 16
D.
k < 16
Show solution
Solution
For real roots, the discriminant must be non-negative: k^2 - 4*1*16 >= 0, leading to k^2 >= 64.
Correct Answer:
A
— k^2 >= 64
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Q. If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)
A.
k > 8
B.
k < -8
C.
k > -8
D.
k < 8
Show solution
Solution
The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.
Correct Answer:
B
— k < -8
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Q. If the quadratic equation x^2 + kx + 9 = 0 has no real roots, what is the condition on k?
A.
k < 6
B.
k > 6
C.
k < 0
D.
k > 0
Show solution
Solution
The discriminant must be less than zero: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6.
Correct Answer:
B
— k > 6
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of m?
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Solution
Using Vieta's formulas, m = -(1 + (-3)) = 2.
Correct Answer:
A
— 2
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 1 and -3, what is the value of n?
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Solution
Using Vieta's formulas, the product of the roots is n = 1 * (-3) = -3.
Correct Answer:
A
— -3
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Q. If the quadratic equation x^2 + mx + n = 0 has roots 2 and -3, what is the value of m + n?
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Solution
Using Vieta's formulas, m = -(-1) = 1 and n = 2*(-3) = -6, thus m + n = 1 - 6 = -5.
Correct Answer:
B
— 5
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Q. If the quadratic equation x^2 + px + q = 0 has roots 2 and 3, what is the value of p + q?
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Solution
Using Vieta's formulas, p = -(2 + 3) = -5 and q = 2*3 = 6. Thus, p + q = -5 + 6 = 1.
Correct Answer:
C
— 7
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Q. If the quadratic equation x^2 + px + q = 0 has roots 2 and 3, what is the value of p?
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Solution
The sum of the roots is -p = 2 + 3 = 5, so p = -5.
Correct Answer:
A
— -5
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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and -2, what is the value of p + q? (2023)
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Solution
Using Vieta's formulas, p = -(3 + (-2)) = -1 and q = 3 * (-2) = -6. Therefore, p + q = -1 - 6 = -7.
Correct Answer:
B
— 5
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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and -2, what is the value of p? (2020)
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Solution
Using the sum of roots formula, p = -(3 + (-2)) = -1. Therefore, p = 1.
Correct Answer:
C
— 5
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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and 4, what is the value of p + q? (2023)
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Solution
Using Vieta's formulas, p = -(3 + 4) = -7 and q = 3 * 4 = 12. Therefore, p + q = -7 + 12 = 5.
Correct Answer:
B
— 12
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Q. If the quadratic equation x^2 - 10x + 25 = 0 is solved, what is the value of x? (2022)
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Solution
The equation can be factored as (x - 5)^2 = 0, giving the root x = 5.
Correct Answer:
A
— 5
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