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. Determine the maximum value of the function f(x) = -x^2 + 6x - 8. (2022)

A. 0
B. 4
C. 6
D. 8

Solution

The vertex is at x = -6/(2*(-1)) = 3. The maximum value is f(3) = -3^2 + 6*3 - 8 = 1.

Correct Answer: B — 4

Q. Determine the median of the following numbers: 9, 7, 5, 3, 1.

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

Solution

Arrange the numbers: 1, 3, 5, 7, 9. The median is the middle value, which is 5.

Correct Answer: A — 5

Q. Determine the median of the following set: 1, 2, 3, 4, 5, 6, 7, 8. (2020)

A. 4
B. 4.5
C. 5
D. 6

Solution

Arrange the numbers: 1, 2, 3, 4, 5, 6, 7, 8. The median is the average of the 4th and 5th numbers: (4 + 5) / 2 = 4.5.

Correct Answer: B — 4.5

Q. Determine the minimum value of f(x) = x^2 - 4x + 5. (2021)

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

Solution

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.

Correct Answer: A — 1

Q. Determine the minimum value of f(x) = x^2 - 4x + 7. (2021)

A. 3
B. 5
C. 4
D. 6

Solution

The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 7 = 3.

Correct Answer: A — 3

Q. Determine the minimum value of f(x) = x^2 - 6x + 10. (2019)

A. 2
B. 3
C. 4
D. 5

Solution

The minimum occurs at x = -b/(2a) = 6/(2*1) = 3. f(3) = 3^2 - 6(3) + 10 = 3.

Correct Answer: B — 3

Q. Determine the minimum value of the function f(x) = x^2 - 4x + 5.

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

Solution

The vertex occurs at x = 2. f(2) = 2^2 - 4*2 + 5 = 1. Thus, the minimum value is 1.

Correct Answer: A — 1

Q. Determine the minimum value of the function f(x) = x^2 - 4x + 6. (2020)

A. 2
B. 3
C. 4
D. 5

Solution

The function is a upward-opening parabola. The minimum occurs at x = -b/(2a) = 4/(2*1) = 2. f(2) = 2^2 - 4(2) + 6 = 2.

Correct Answer: A — 2

Q. Determine the mode of the following data: {1, 2, 2, 3, 4, 4, 4, 5, 5}.

A. 2
B. 3
C. 4
D. 5

Solution

The mode is 4, as it appears 3 times, more than any other number.

Correct Answer: C — 4

Q. Determine the moment of inertia of a solid sphere of mass M and radius R about an axis through its center.

A. 2/5 MR^2
B. 3/5 MR^2
C. 4/5 MR^2
D. MR^2

Solution

The moment of inertia of a solid sphere about an axis through its center is I = 2/5 MR^2.

Correct Answer: A — 2/5 MR^2

Q. Determine the nature of the lines represented by the equation 7x^2 + 2xy + 3y^2 = 0.

A. Parallel
B. Intersecting
C. Coincident
D. Perpendicular

Solution

The discriminant indicates that the lines intersect at two distinct points.

Correct Answer: B — Intersecting

Q. Determine the point at which the function f(x) = x^3 - 3x^2 + 4 has a local minimum.

A. (1, 2)
B. (2, 1)
C. (0, 4)
D. (3, 4)

Solution

Find f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. f''(2) = 6 > 0, so (2, 1) is a local minimum.

Correct Answer: A — (1, 2)

Q. Determine the point at which the function f(x) = |x - 1| is not differentiable.

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

Solution

The function |x - 1| is not differentiable at x = 1 due to a cusp.

Correct Answer: B — x = 1

Q. Determine the point at which the function f(x) = |x - 3| is not differentiable.

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

Solution

The function f(x) = |x - 3| is not differentiable at x = 3 because it has a sharp corner.

Correct Answer: C — x = 3

Q. Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

A. x = -2
B. x = 0
C. x = 2
D. x = -4

Solution

f(x) is not differentiable at x = -2 and x = 2, but is differentiable everywhere else.

Correct Answer: A — x = -2

Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)

A. (1, 3)
B. (2, 2)
C. (0, 6)
D. (3, 0)

Solution

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.

Correct Answer: A — (1, 3)

Q. Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6.

A. (1, 3)
B. (2, 2)
C. (3, 1)
D. (0, 6)

Solution

f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x = 0 and x = 2. The point of inflection is at (1, 3).

Correct Answer: A — (1, 3)

Q. Determine the point of inflection for the function f(x) = x^4 - 4x^3 + 6x^2.

A. (1, 3)
B. (2, 2)
C. (3, 1)
D. (0, 0)

Solution

Find f''(x) = 12x^2 - 24x + 12. Setting f''(x) = 0 gives x = 1 and x = 2. Testing intervals shows a change in concavity at x = 1.

Correct Answer: A — (1, 3)

Q. Determine the point of intersection of the lines y = 2x + 1 and y = -x + 4.

A. (1, 3)
B. (2, 5)
C. (3, 7)
D. (4, 9)

Solution

Setting 2x + 1 = -x + 4 gives 3x = 3, hence x = 1. Substituting back gives y = 3.

Correct Answer: A — (1, 3)

Q. Determine the point where the function f(x) = 2x^3 - 9x^2 + 12x has a local minimum. (2023)

A. (1, 5)
B. (2, 0)
C. (3, 3)
D. (4, 4)

Solution

Set f'(x) = 0. f'(x) = 6x^2 - 18x + 12 = 0 gives x = 2. f(2) = 0.

Correct Answer: C — (3, 3)

Q. Determine the point where the function f(x) = 4x - x^2 has a maximum. (2022)

A. (0, 0)
B. (2, 4)
C. (1, 3)
D. (3, 3)

Solution

The maximum occurs at x = 2, found by setting f'(x) = 4 - 2x = 0. f(2) = 4(2) - (2^2) = 4.

Correct Answer: B — (2, 4)

Q. Determine the points where f(x) = x^3 - 3x is not differentiable.

A. x = 0
B. x = 1
C. x = -1
D. Nowhere

Solution

The function is a polynomial and is differentiable everywhere, hence nowhere.

Correct Answer: D — Nowhere

Q. Determine the points where the function f(x) = x^4 - 4x^3 is not differentiable.

A. x = 0
B. x = 1
C. x = 2
D. None

Solution

The function is a polynomial and is differentiable everywhere. Thus, there are no points where it is not differentiable.

Correct Answer: D — None

Q. Determine the product of the roots of the equation x² + 6x + 8 = 0. (2023)

A. 8
B. 6
C. 4
D. 2

Solution

The product of the roots is given by c/a = 8/1 = 8.

Correct Answer: A — 8

Q. Determine the product of the roots of the equation x² + 6x + 9 = 0. (2021)

A. 9
B. 6
C. 3
D. 0

Solution

The product of the roots is c/a = 9/1 = 9.

Correct Answer: A — 9

Q. Determine the roots of the equation x² + 2x - 8 = 0. (2023)

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

Solution

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

Correct Answer: A — -4 and 2

Q. Determine the roots of the equation x² + 6x + 9 = 0. (2023)

A. -3
B. 3
C. 0
D. -6

Solution

This is a perfect square: (x + 3)² = 0, hence the root is x = -3.

Correct Answer: A — -3

Q. Determine the scalar product of the vectors (0, 1, 2) and (3, 4, 5).

A. 10
B. 11
C. 12
D. 13

Solution

Scalar product = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

Correct Answer: B — 11

Q. Determine the scalar product of the vectors A = (1, 1, 1) and B = (2, 2, 2).

A. 3
B. 4
C. 6
D. 8

Solution

A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: C — 6

Q. Determine the scalar product of the vectors A = (2, 2, 2) and B = (3, 3, 3).

A. 12
B. 18
C. 6
D. 9

Solution

A · B = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.

Correct Answer: A — 12

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