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 function f(x) = sin(x) + cos(x), find the x-coordinate of the maximum point in the interval [0, 2π].

A. π/4
B. 3π/4
C. 5π/4
D. 7π/4

Solution

f'(x) = cos(x) - sin(x). Setting f'(x) = 0 gives tan(x) = 1, so x = π/4 + nπ. In [0, 2π], the maximum occurs at x = 3π/4.

Correct Answer: B — 3π/4

Q. For the function f(x) = sin(x) + cos(x), what is f'(π/4)? (2023)

A. 0
B. √2
C. 1
D. √2/2

Solution

f'(x) = cos(x) - sin(x). At x = π/4, f'(π/4) = cos(π/4) - sin(π/4) = √2/2 - √2/2 = 0.

Correct Answer: B — √2

Q. For the function f(x) = sin(x), what is f'(π/2)? (2021)

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

Solution

f'(x) = cos(x); f'(π/2) = cos(π/2) = 0.

Correct Answer: B — 1

Q. For the function f(x) = x^2 + 2x + 1, what is f'(x)?

A. 2x + 1
B. 2x + 2
C. 2x
D. x + 1

Solution

f'(x) = 2x + 2.

Correct Answer: B — 2x + 2

Q. For the function f(x) = x^2 + 2x + 3, find the point where it is not differentiable.

A. x = -1
B. x = 0
C. x = 1
D. It is differentiable everywhere

Solution

The function is a polynomial and is differentiable everywhere.

Correct Answer: D — It is differentiable everywhere

Q. For the function f(x) = x^2 + 2x, find the local maximum. (2022)

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

Solution

f'(x) = 2x + 2. Setting f'(x) = 0 gives x = -1. f(-1) = 1.

Correct Answer: A — -1

Q. For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?

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

Solution

Setting the derivative f'(-1) = 0 gives k = 1 for differentiability.

Correct Answer: C — 1

Q. For the function f(x) = x^2 - 2x + 1, find the slope of the tangent line at x = 1.

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

Solution

f'(x) = 2x - 2. Thus, f'(1) = 2(1) - 2 = 0.

Correct Answer: A — 0

Q. For the function f(x) = x^2 - 4x + 4, find the point where it is not differentiable.

A. x = 0
B. x = 2
C. x = 4
D. It is differentiable everywhere

Solution

As a polynomial, f(x) is differentiable everywhere, including at x = 2.

Correct Answer: D — It is differentiable everywhere

Q. For the function f(x) = x^2 - 4x + 5, find the minimum value.

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

Solution

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

Correct Answer: B — 2

Q. For the function f(x) = x^2 - 4x + 5, find the vertex.

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

Solution

The vertex is at x = -b/(2a) = 4/2 = 2. f(2) = 2^2 - 4(2) + 5 = 1, so the vertex is (2, 1).

Correct Answer: A — (2, 1)

Q. For the function f(x) = x^2 - 6x + 10, what is the minimum value? (2020)

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

Solution

The vertex is at x = 6/2 = 3. The minimum value is f(3) = 3^2 - 6*3 + 10 = 1.

Correct Answer: B — 3

Q. For the function f(x) = x^2 - 6x + 8, find the x-coordinate of the vertex.

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

Solution

The x-coordinate of the vertex is given by x = -b/(2a) = 6/(2*1) = 3.

Correct Answer: B — 3

Q. For the function f(x) = x^3 - 3x + 2, find the points of discontinuity.

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

Solution

f(x) is a polynomial function and is continuous everywhere, hence no points of discontinuity.

Correct Answer: A — None

Q. For the function f(x) = x^3 - 3x^2 + 2, find the points where it is not differentiable.

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

Solution

As a polynomial, f(x) is differentiable everywhere, hence no points of non-differentiability.

Correct Answer: A — None

Q. For the function f(x) = x^3 - 3x^2 + 4, find the points where it is not differentiable.

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

Solution

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

Correct Answer: A — None

Q. For the function f(x) = x^3 - 3x^2 + 4, find the value of x where f is not differentiable.

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

Solution

The function is a polynomial and is differentiable everywhere, so there is no such x.

Correct Answer: A — 0

Q. For the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where f is differentiable.

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

Solution

f(x) is a polynomial and is differentiable everywhere. The x-coordinate can be any real number.

Correct Answer: C — 1

Q. For the function f(x) = x^3 - 6x^2 + 9x, find the critical points.

A. x = 0, 3
B. x = 1, 2
C. x = 2, 3
D. x = 3, 4

Solution

First, find f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 3)(x - 1) = 0, so critical points are x = 1 and x = 3.

Correct Answer: A — x = 0, 3

Q. For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.

A. (-∞, 0)
B. (0, 3)
C. (3, ∞)
D. (0, 6)

Solution

f'(x) = 3x^2 - 12x + 9. The critical points are x = 1 and x = 3. The function is increasing on (1, 3) and (3, ∞).

Correct Answer: B — (0, 3)

Q. For the function f(x) = x^3 - 6x^2 + 9x, find the local minima. (2022)

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

Solution

f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1, 3. f(2) = 1 is a local minimum.

Correct Answer: B — (2, 1)

Q. For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.

A. (0, 16)
B. (2, 0)
C. (4, 0)
D. (2, 4)

Solution

Find f''(x) = 12x^2 - 16. Setting f''(x) = 0 gives x^2 = 4, so x = ±2. f(2) = 0, thus the inflection point is (2, 0).

Correct Answer: B — (2, 0)

Q. For the function f(x) = x^4 - 8x^2 + 16, find the intervals where the function is increasing.

A. (-∞, -2)
B. (-2, 2)
C. (2, ∞)
D. (-2, ∞)

Solution

f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x(x^2 - 4) = 0, so x = -2, 0, 2. Test intervals: f' is positive in (-2, ∞).

Correct Answer: D — (-2, ∞)

Q. For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) continuous at x = 1?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

The left limit as x approaches 1 is 3, the right limit is 1, and f(1) = 3. Since the limits do not match, f(x) is discontinuous at x = 1.

Correct Answer: B — No

Q. For the function f(x) = { x^2 + 1, x < 0; 2x + b, x = 0; 3 - x, x > 0 to be continuous at x = 0, what is b?

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

Solution

Setting the left limit (0 + 1 = 1) equal to the right limit (3 - 0 = 3), we find b = 1.

Correct Answer: B — 0

Q. For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) continuous at x = 0?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

The left limit as x approaches 0 is 0, the right limit is 1, and f(0) = 0. Since the limits do not match, f(x) is discontinuous at x = 0.

Correct Answer: B — No

Q. For the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 }, what is the value of f(1)?

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

Solution

By definition, f(1) = 3, as given in the piecewise function.

Correct Answer: C — 3

Q. For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.

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

Solution

Setting f(1-) = f(1+) and f'(1-) = f'(1+) gives k = 2 for differentiability.

Correct Answer: B — 1

Q. For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) continuous at x = 2?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

At x = 2, left limit is 4 and right limit is 4, but f(2) = 4. Hence, f(x) is continuous at x = 2.

Correct Answer: B — No

Q. For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuous at x = 3, the value of f(3) must be:

A. 6
B. 9
C. 3
D. 12

Solution

For continuity, f(3) must equal the limit as x approaches 3, which is 9.

Correct Answer: B — 9

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