Major Competitive Exams

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Major Competitive Exams

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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!

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Q. For the polynomial x^3 - 3x^2 + 3x - 1, what is the value of the sum of the roots? (2019)

A. 1
B. 3
C. 0
D. 2

Solution

The sum of the roots is given by -b/a = 3/1 = 3.

Correct Answer: B — 3

Q. For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?

A. All roots are real
B. All roots are complex
C. One root is real
D. Two roots are real

Solution

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

Correct Answer: A — All roots are real

Q. For the quadratic equation 2x^2 + 4x + 2 = 0, what is the value of the discriminant? (2020)

A. 0
B. 4
C. 8
D. 16

Solution

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.

Correct Answer: A — 0

Q. For the quadratic equation 2x^2 + 4x + k = 0 to have equal roots, what should be the value of k? (2020)

A. -4
B. 0
C. 4
D. 8

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

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

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

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

Solution

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

Correct Answer: D — k <= 4

Q. For the quadratic equation 2x^2 + 4x - 6 = 0, what is the value of the discriminant? (2020)

A. 16
B. 4
C. 0
D. 36

Solution

The discriminant D = b^2 - 4ac = 4^2 - 4(2)(-6) = 16 + 48 = 64.

Correct Answer: A — 16

Q. For the quadratic equation 2x^2 - 4x + k = 0 to have equal roots, what must be the value of k? (2019)

A. 0
B. 2
C. 4
D. 8

Solution

For equal roots, the discriminant must be zero: (-4)^2 - 4*2*k = 0. Solving gives k = 4.

Correct Answer: C — 4

Q. For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?

A. k >= 0
B. k <= 0
C. k >= 2
D. k <= 2

Solution

The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.

Correct Answer: C — k >= 2

Q. For the quadratic equation 2x^2 - 8x + 6 = 0, what is the value of the discriminant?

A. 4
B. 16
C. 8
D. 0

Solution

The discriminant D = b^2 - 4ac = (-8)^2 - 4*2*6 = 64 - 48 = 16.

Correct Answer: A — 4

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

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

Q. For the quadratic equation ax^2 + bx + c = 0, if a = 1, b = -3, and c = 2, what are the roots?

A. 1 and 2
B. 2 and 1
C. 3 and 0
D. 0 and 3

Solution

The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.

Correct Answer: A — 1 and 2

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

Solution

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

Correct Answer: D — p >= 2

Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the nature of the roots?

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0, indicating that the roots are real and equal.

Correct Answer: B — Real and equal

Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the vertex of the parabola?

A. (-1, 0)
B. (-1, 1)
C. (0, 1)
D. (1, 1)

Solution

The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).

Correct Answer: A — (-1, 0)

Q. For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

A. < 0
B. ≥ 0
C. ≤ 0
D. > 0

Solution

The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.

Correct Answer: A — < 0

Q. For the quadratic equation x^2 + 2x + k = 0 to have real roots, what must be the minimum value of k? (2020)

A. -1
B. 0
C. 1
D. 2

Solution

The discriminant must be non-negative: 2^2 - 4*1*k >= 0, thus k <= 1.

Correct Answer: A — -1

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

Solution

The discriminant must be non-negative: 2^2 - 4*1*k >= 0 leads to k <= 1.

Correct Answer: D — k <= 1

Q. For the quadratic equation x^2 + 4x + 4 = 0, what is the nature of the roots?

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.

Correct Answer: B — Real and equal

Q. For the quadratic equation x^2 + 4x + k = 0 to have equal roots, what must be the value of k? (2022)

A. 4
B. 8
C. 16
D. 0

Solution

For equal roots, the discriminant must be zero: 4^2 - 4*1*k = 0, thus k = 4.

Correct Answer: B — 8

Q. For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:

A. 0
B. 1
C. 2
D. 3

Solution

The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.

Correct Answer: A — 0

Q. For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?

A. k >= 4
B. k <= 4
C. k > 0
D. k < 0

Solution

The discriminant must be non-negative: 4^2 - 4*1*k >= 0, which simplifies to k <= 4.

Correct Answer: A — k >= 4

Q. For the quadratic equation x^2 + 6x + 8 = 0, what are the roots?

A. -2 and -4
B. -4 and -2
C. 2 and 4
D. 0 and 8

Solution

Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.

Correct Answer: B — -4 and -2

Q. For the quadratic equation x^2 + 6x + 9 = 0, what is the nature of the roots?

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

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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

Solution

The discriminant D = 6^2 - 4*1*9 = 0, indicating real and equal roots.

Correct Answer: B — Real and equal

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

Solution

The discriminant must be positive: 6^2 - 4*1*k > 0, which simplifies to k < 9.

Correct Answer: A — k < 9

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

Solution

The discriminant must be non-negative: 6^2 - 4(1)(k) ≥ 0, which gives k ≤ 9.

Correct Answer: D — k ≤ 9

Q. For the quadratic equation x^2 + mx + n = 0, if the roots are 2 and 3, what is the value of n?

A. 5
B. 6
C. 7
D. 8

Solution

The product of the roots is n = 2 * 3 = 6.

Correct Answer: B — 6

Q. For the quadratic equation x^2 + px + q = 0, if the roots are -2 and -3, what is the value of p? (2020)

A. 5
B. 6
C. 7
D. 8

Solution

The sum of the roots is -(-2) + -(-3) = 5, hence p = 5.

Correct Answer: A — 5

Q. For the quadratic equation x^2 + px + q = 0, if the roots are 1 and -3, what is the value of p?

A. 2
B. -2
C. 3
D. -3

Solution

The sum of the roots is 1 + (-3) = -2, hence p = -2.

Correct Answer: A — 2

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