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. Determine the critical points of f(x) = x^4 - 8x^2.
A.
x = 0, ±2
B.
x = ±4
C.
x = ±1
D.
x = 2
Show solution
Solution
f'(x) = 4x^3 - 16x = 4x(x^2 - 4). Critical points are x = 0, ±2.
Correct Answer:
A
— x = 0, ±2
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Q. Determine the critical points of the function f(x) = x^2 - 4x + 4. (2022)
Show solution
Solution
f'(x) = 2x - 4; Setting f'(x) = 0 gives x = 2 as the critical point.
Correct Answer:
C
— 2
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Q. Determine the critical points of the function f(x) = x^3 - 6x^2 + 9x.
A.
(0, 0)
B.
(1, 4)
C.
(2, 0)
D.
(3, 0)
Show solution
Solution
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 1)(x - 3) = 0, so critical points are x = 1 and x = 3.
Correct Answer:
D
— (3, 0)
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Q. Determine the derivative of f(x) = 1/x.
A.
-1/x^2
B.
1/x^2
C.
1/x
D.
-1/x
Show solution
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer:
A
— -1/x^2
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Q. Determine the derivative of f(x) = ln(x^2 + 1).
A.
2x/(x^2 + 1)
B.
1/(x^2 + 1)
C.
2/(x^2 + 1)
D.
x/(x^2 + 1)
Show solution
Solution
Using the chain rule, f'(x) = (1/(x^2 + 1)) * (2x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Determine the derivative of f(x) = x^2 * e^x.
A.
e^x * (x^2 + 2x)
B.
e^x * (2x + 1)
C.
2x * e^x
D.
x^2 * e^x
Show solution
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x * (x^2 + 2x).
Correct Answer:
A
— e^x * (x^2 + 2x)
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Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
A.
3x^2 - 4
B.
3x^2 + 4
C.
x^2 - 4
D.
3x^2 - 7
Show solution
Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
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Q. Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)
A.
5x^4 - 9x^2 + 2
B.
5x^4 - 9x + 2
C.
5x^4 - 3x^2 + 2
D.
5x^4 - 3x^3
Show solution
Solution
Using the power rule, f'(x) = 5x^4 - 9x^2 + 2.
Correct Answer:
A
— 5x^4 - 9x^2 + 2
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Q. Determine the distance between the points (-1, -1) and (2, 2).
Show solution
Solution
Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[(2 + 1)² + (2 + 1)²] = √[9 + 9] = √18 = 3√2 ≈ 4.24.
Correct Answer:
C
— 5
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Q. Determine the distance between the points (0, 0) and (0, 8).
Show solution
Solution
Using the distance formula: d = √[(0 - 0)² + (8 - 0)²] = √[0 + 64] = √64 = 8.
Correct Answer:
A
— 8
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Q. Determine the distance between the points (1, 2) and (4, 6). (2022)
Show solution
Solution
Using the distance formula: d = √[(4 - 1)² + (6 - 2)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. Determine the distance between the points (2, 3) and (2, -1).
Show solution
Solution
Using the distance formula: d = √[(2 - 2)² + (-1 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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Q. Determine the distance between the points (2, 3) and (5, 7). (2020)
Show solution
Solution
Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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Q. Determine the distance from the point (1, 2) to the line 2x + 3y - 6 = 0. (2023)
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Solution
Using the formula for distance from a point to a line, the distance is |2(1) + 3(2) - 6| / sqrt(2^2 + 3^2) = 1.
Correct Answer:
B
— 2
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Q. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.
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Solution
Using the formula for distance from a point to a line, d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), we find d = |2(3) + 3(4) - 12| / sqrt(2^2 + 3^2) = 3.
Correct Answer:
B
— 3
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Q. Determine the equation of the circle with center (2, -3) and radius 5.
A.
(x - 2)² + (y + 3)² = 25
B.
(x + 2)² + (y - 3)² = 25
C.
(x - 2)² + (y - 3)² = 25
D.
(x + 2)² + (y + 3)² = 25
Show solution
Solution
Equation of circle: (x - h)² + (y - k)² = r² => (x - 2)² + (y + 3)² = 5² = 25.
Correct Answer:
A
— (x - 2)² + (y + 3)² = 25
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Q. Determine the equation of the line that passes through the points (0, 0) and (3, 9).
A.
y = 3x
B.
y = 2x
C.
y = 3x + 1
D.
y = x + 1
Show solution
Solution
The slope m = (9 - 0) / (3 - 0) = 3. The equation is y = 3x.
Correct Answer:
A
— y = 3x
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Q. Determine the equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1.
A.
y = 3x - 2
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = x + 3
Show solution
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 4). The tangent line is y - 4 = 4(x - 1) => y = 4x - 4 + 4 => y = 4x - 2.
Correct Answer:
A
— y = 3x - 2
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Q. Determine the family of curves represented by the equation x^2 - y^2 = c, where c is a constant.
A.
Circles
B.
Ellipses
C.
Hyperbolas
D.
Parabolas
Show solution
Solution
The equation x^2 - y^2 = c represents a family of hyperbolas with varying values of c.
Correct Answer:
C
— Hyperbolas
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Q. Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1.
A.
Circles
B.
Ellipses with varying axes
C.
Hyperbolas
D.
Parabolas
Show solution
Solution
The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with varying semi-major and semi-minor axes.
Correct Answer:
B
— Ellipses with varying axes
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Q. Determine the family of curves represented by the equation y = ax^2 + bx + c.
A.
Parabolas
B.
Circles
C.
Ellipses
D.
Straight lines
Show solution
Solution
The equation y = ax^2 + bx + c represents a family of parabolas with varying coefficients a, b, and c.
Correct Answer:
A
— Parabolas
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Q. Determine the family of curves represented by the equation y = ax^3 + bx.
A.
Cubic functions
B.
Quadratic functions
C.
Linear functions
D.
Exponential functions
Show solution
Solution
The equation y = ax^3 + bx represents a family of cubic functions where a and b are constants.
Correct Answer:
A
— Cubic functions
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Q. Determine the family of curves represented by the equation y = ax^3 + bx^2 + cx + d.
A.
Cubic functions
B.
Quadratic functions
C.
Linear functions
D.
Exponential functions
Show solution
Solution
The equation y = ax^3 + bx^2 + cx + d represents a family of cubic functions.
Correct Answer:
A
— Cubic functions
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Q. Determine the family of curves represented by the equation y = e^(kx) for varying k.
A.
Exponential curves
B.
Linear functions
C.
Quadratic functions
D.
Logarithmic functions
Show solution
Solution
The equation y = e^(kx) represents a family of exponential curves with varying growth rates determined by k.
Correct Answer:
A
— Exponential curves
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Q. Determine the family of curves represented by the equation y = k/x, where k is a constant.
A.
Hyperbolas
B.
Circles
C.
Ellipses
D.
Parabolas
Show solution
Solution
The equation y = k/x represents a family of hyperbolas with varying values of 'k'.
Correct Answer:
A
— Hyperbolas
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Q. Determine the family of curves represented by the equation y = kx^2, where k is a constant.
A.
Circles
B.
Ellipses
C.
Parabolas
D.
Hyperbolas
Show solution
Solution
The equation y = kx^2 represents a family of parabolas that open upwards or downwards depending on the sign of k.
Correct Answer:
C
— Parabolas
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Q. Determine the focus of the parabola defined by the equation x^2 = 12y.
A.
(0, 3)
B.
(0, -3)
C.
(3, 0)
D.
(-3, 0)
Show solution
Solution
The equation x^2 = 4py gives 4p = 12, hence p = 3. The focus is at (0, p) = (0, 3).
Correct Answer:
A
— (0, 3)
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Q. Determine the focus of the parabola given by the equation x^2 = 8y.
A.
(0, 2)
B.
(0, 4)
C.
(2, 0)
D.
(4, 0)
Show solution
Solution
The standard form of the parabola is x^2 = 4py. Here, 4p = 8, so p = 2. The focus is at (0, p) = (0, 2).
Correct Answer:
B
— (0, 4)
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Q. Determine the hybridization of the central atom in BF3.
A.
sp
B.
sp2
C.
sp3
D.
dsp3
Show solution
Solution
Boron in BF3 is sp2 hybridized, forming three equivalent sp2 hybrid orbitals.
Correct Answer:
B
— sp2
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Q. Determine the hybridization of the central atom in O3.
A.
sp
B.
sp2
C.
sp3
D.
dsp3
Show solution
Solution
The central atom in ozone (O3) is sp2 hybridized, forming a resonance structure.
Correct Answer:
B
— sp2
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