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. Find the focus of the parabola given by the equation y^2 = 12x.
A.
(3, 0)
B.
(0, 3)
C.
(0, 6)
D.
(6, 0)
Show solution
Solution
The standard form of a parabola is y^2 = 4px. Here, 4p = 12, so p = 3. The focus is at (p, 0) = (3, 0).
Correct Answer:
C
— (0, 6)
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Q. Find the general solution of dy/dx = 3x^2. (2020)
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x + C
Show solution
Solution
Integrating 3x^2 gives y = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Find the general solution of the differential equation dy/dx = 2y.
A.
y = Ce^(2x)
B.
y = 2Ce^x
C.
y = Ce^(x/2)
D.
y = 2x + C
Show solution
Solution
This is a separable equation. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer:
A
— y = Ce^(2x)
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Q. Find the general solution of the differential equation dy/dx = 3x^2.
A.
y = x^3 + C
B.
y = 3x^3 + C
C.
y = x^2 + C
D.
y = 3x^2 + C
Show solution
Solution
Integrating both sides gives y = (3/3)x^3 + C = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Find the general solution of the differential equation dy/dx = 4y.
A.
y = Ce^(4x)
B.
y = 4Ce^x
C.
y = Ce^(x/4)
D.
y = 4Ce^(x)
Show solution
Solution
This is a separable differential equation. Integrating gives y = Ce^(4x), where C is the constant.
Correct Answer:
A
— y = Ce^(4x)
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Q. Find the general solution of the differential equation dy/dx = y.
A.
y = Ce^x
B.
y = Ce^(-x)
C.
y = Cx
D.
y = C/x
Show solution
Solution
This is a separable equation. Integrating gives ln|y| = x + C, hence y = Ce^x.
Correct Answer:
A
— y = Ce^x
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Q. Find the general solution of the differential equation y'' - 5y' + 6y = 0.
A.
y = C1 e^(2x) + C2 e^(3x)
B.
y = C1 e^(3x) + C2 e^(2x)
C.
y = C1 e^(x) + C2 e^(2x)
D.
y = C1 e^(4x) + C2 e^(5x)
Show solution
Solution
The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Correct Answer:
B
— y = C1 e^(3x) + C2 e^(2x)
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Q. Find the general solution of the equation cos(2x) = 0.
A.
x = (2n+1)π/4
B.
x = nπ/2
C.
x = (2n+1)π/2
D.
x = nπ
Show solution
Solution
The general solution is x = (2n+1)π/4, where n is any integer.
Correct Answer:
A
— x = (2n+1)π/4
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Q. Find the general solution of the equation dy/dx = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = Ce^(x^3/3)
D.
y = Ce^(x^2)
Show solution
Solution
This is a separable equation. Separating and integrating gives y = Ce^(x^3).
Correct Answer:
A
— y = Ce^(x^3)
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Q. Find the general solution of the equation sin(x) + sin(2x) = 0.
A.
x = nπ
B.
x = nπ/2
C.
x = (2n+1)π/4
D.
x = nπ/3
Show solution
Solution
Factoring gives sin(x)(1 + 2cos(x)) = 0, leading to x = nπ or cos(x) = -1/2.
Correct Answer:
A
— x = nπ
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Q. Find the general solution of the equation sin(x) + √3 cos(x) = 0.
A.
x = (2n+1)π/3
B.
x = (2n+1)π/6
C.
x = nπ
D.
x = (2n+1)π/4
Show solution
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer:
A
— x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) + √3cos(x) = 0.
A.
x = (2n+1)π/3
B.
x = nπ
C.
x = (2n+1)π/4
D.
x = nπ + π/6
Show solution
Solution
The general solution is x = (2n+1)π/3, where n is an integer.
Correct Answer:
A
— x = (2n+1)π/3
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Q. Find the general solution of the equation sin(x) = -1/2.
A.
x = 7π/6 + 2nπ
B.
x = 11π/6 + 2nπ
C.
x = 7π/6, 11π/6
D.
Both 1 and 2
Show solution
Solution
The general solutions are x = 7π/6 + 2nπ and x = 11π/6 + 2nπ.
Correct Answer:
D
— Both 1 and 2
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Q. Find the general solution of the equation sin(x) = sin(2x).
A.
x = nπ
B.
x = nπ/3
C.
x = nπ/2
D.
x = nπ/4
Show solution
Solution
Using the identity sin(a) = sin(b) gives x = nπ or x = (2n+1)π/3.
Correct Answer:
A
— x = nπ
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Q. Find the general solution of the equation sin(x) = sin(π/4).
A.
x = nπ + (-1)^n π/4
B.
x = nπ + π/4
C.
x = nπ + 3π/4
D.
x = nπ + π/2
Show solution
Solution
The general solution is x = nπ + (-1)^n π/4, where n is any integer.
Correct Answer:
A
— x = nπ + (-1)^n π/4
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Q. Find the general solution of the equation y' = 3x^2y.
A.
y = Ce^(x^3)
B.
y = Ce^(3x^3)
C.
y = C/x^3
D.
y = Cx^3
Show solution
Solution
This is a separable equation. Integrating gives y = Ce^(x^3).
Correct Answer:
A
— y = Ce^(x^3)
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Q. Find the general solution of the equation y' = 3y + 2.
A.
y = (C - 2/3)e^(3x)
B.
y = Ce^(3x) - 2/3
C.
y = 2/3 + Ce^(3x)
D.
y = 3x + C
Show solution
Solution
This is a first-order linear differential equation. The integrating factor is e^(-3x).
Correct Answer:
B
— y = Ce^(3x) - 2/3
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Q. Find the general solution of the equation y' = 5y + 3.
A.
y = Ce^(5x) - 3/5
B.
y = Ce^(5x) + 3/5
C.
y = 3/5 + Ce^(-5x)
D.
y = 5x + C
Show solution
Solution
The integrating factor method gives the general solution y = Ce^(5x) - 3/5.
Correct Answer:
A
— y = Ce^(5x) - 3/5
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Q. Find the general solution of the equation y'' - 3y' + 2y = 0.
A.
y = C1 e^(2x) + C2 e^(x)
B.
y = C1 e^(x) + C2 e^(2x)
C.
y = C1 e^(3x) + C2 e^(0)
D.
y = C1 e^(0) + C2 e^(3x)
Show solution
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r-1)(r-2)=0. Thus, the general solution is y = C1 e^(x) + C2 e^(2x).
Correct Answer:
B
— y = C1 e^(x) + C2 e^(2x)
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Q. Find the general solution of the equation y'' - 5y' + 6y = 0.
A.
y = C1 e^(2x) + C2 e^(3x)
B.
y = C1 e^(3x) + C2 e^(2x)
C.
y = C1 e^(x) + C2 e^(2x)
D.
y = C1 e^(4x) + C2 e^(5x)
Show solution
Solution
The characteristic equation is r^2 - 5r + 6 = 0, giving roots 2 and 3. Thus, y = C1 e^(2x) + C2 e^(3x).
Correct Answer:
B
— y = C1 e^(3x) + C2 e^(2x)
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Q. Find the HCF of 60, 72, and 90.
Show solution
Solution
The HCF of 60, 72, and 90 is 18, as it is the largest number that divides all three.
Correct Answer:
B
— 12
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Q. Find the integral of (1/x) dx.
A.
ln
B.
x
C.
+ C
D.
x + C
.
1/x + C
.
e^x + C
Show solution
Solution
The integral of (1/x) is ln|x| + C, where C is the constant of integration.
Correct Answer:
A
— ln
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Q. Find the integral of (2x + 1)^3 dx. (2019)
A.
(1/4)(2x + 1)^4 + C
B.
(1/3)(2x + 1)^4 + C
C.
(1/5)(2x + 1)^4 + C
D.
(1/2)(2x + 1)^4 + C
Show solution
Solution
Using substitution, the integral is (1/4)(2x + 1)^4 + C.
Correct Answer:
A
— (1/4)(2x + 1)^4 + C
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Q. Find the integral of (2x + 3)dx. (2022)
A.
x^2 + 3x + C
B.
x^2 + 3x + 1
C.
x^2 + 3 + C
D.
2x^2 + 3x + C
Show solution
Solution
Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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Q. Find the integral of cos(2x)dx. (2023)
A.
(1/2)sin(2x) + C
B.
sin(2x) + C
C.
(1/2)cos(2x) + C
D.
2sin(2x) + C
Show solution
Solution
The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.
Correct Answer:
A
— (1/2)sin(2x) + C
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Q. Find the integral of cos(x) with respect to x. (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
Show solution
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of cos(x). (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
Show solution
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of cos(x)dx. (2023)
A.
sin(x) + C
B.
-sin(x) + C
C.
cos(x) + C
D.
-cos(x) + C
Show solution
Solution
The integral of cos(x) is sin(x) + C.
Correct Answer:
A
— sin(x) + C
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Q. Find the integral of e^(2x) dx.
A.
(1/2)e^(2x) + C
B.
2e^(2x) + C
C.
e^(2x) + C
D.
(1/2)e^(x) + C
Show solution
Solution
The integral of e^(2x) is (1/2)e^(2x) + C, where C is the constant of integration.
Correct Answer:
A
— (1/2)e^(2x) + C
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Q. Find the integral of e^x dx. (2022)
A.
e^x + C
B.
e^x
C.
x e^x + C
D.
ln(e^x) + C
Show solution
Solution
The integral of e^x is e^x + C.
Correct Answer:
A
— e^x + C
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