Q. Solve for y: 4y + 8 = 24.
Show solution
Solution
Subtracting 8 from both sides gives 4y = 16, then dividing by 4 gives y = 4.
Correct Answer:
B
— 3
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Q. Solve for z: 2z/3 + 4 = 10.
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Solution
Subtract 4 from both sides: 2z/3 = 6. Multiply by 3: 2z = 18. Divide by 2: z = 9.
Correct Answer:
A
— 6
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Q. Solve the differential equation dy/dx + 2y = 4.
A.
y = 2 - Ce^(-2x)
B.
y = 2 + Ce^(-2x)
C.
y = 4 - Ce^(-2x)
D.
y = 4 + Ce^(2x)
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Solution
This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).
Correct Answer:
A
— y = 2 - Ce^(-2x)
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Q. Solve the differential equation dy/dx = 2x + 1.
A.
y = x^2 + x + C
B.
y = x^2 + 2x + C
C.
y = 2x^2 + x + C
D.
y = x^2 + C
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Solution
Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.
Correct Answer:
A
— y = x^2 + x + C
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Q. Solve the differential equation dy/dx = 2y + 3. (2023)
A.
y = Ce^(2x) - 3/2
B.
y = Ce^(-2x) + 3/2
C.
y = 3e^(2x)
D.
y = 2e^(2x) + C
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Solution
Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.
Correct Answer:
A
— y = Ce^(2x) - 3/2
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Q. Solve the differential equation dy/dx = 2y.
A.
y = Ce^(2x)
B.
y = 2Ce^x
C.
y = Ce^(x/2)
D.
y = 2x + C
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Solution
This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer:
A
— y = Ce^(2x)
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Q. Solve 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 + C
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Solution
Integrating both sides gives y = x^3 + C.
Correct Answer:
A
— y = x^3 + C
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Q. Solve the differential equation dy/dx = 5 - 2y.
A.
y = 5/2 + Ce^(-2x)
B.
y = 5/2 - Ce^(-2x)
C.
y = 2.5 + Ce^(2x)
D.
y = 2.5 - Ce^(2x)
Show solution
Solution
Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).
Correct Answer:
A
— y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation dy/dx = 6x.
A.
y = 3x^2 + C
B.
y = 6x^2 + C
C.
y = 2x^2 + C
D.
y = 3x + C
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Solution
Integrating gives y = 3x^2 + C, where C is the constant of integration.
Correct Answer:
A
— y = 3x^2 + C
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Q. Solve the differential equation dy/dx = 6x^2y.
A.
y = Ce^(2x^3)
B.
y = Ce^(3x^2)
C.
y = Ce^(6x^2)
D.
y = Ce^(x^6)
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Solution
This is a separable equation. Integrating gives y = Ce^(2x^3).
Correct Answer:
A
— y = Ce^(2x^3)
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Q. Solve the differential equation dy/dx = x^2 + y^2.
A.
y = x^3/3 + C
B.
y = x^2 + C
C.
y = x^2 + x + C
D.
y = Cx^2 + C
Show solution
Solution
This is a non-linear differential equation. The solution can be found using substitution methods.
Correct Answer:
A
— y = x^3/3 + C
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Q. Solve the differential equation dy/dx = y/x. (2023)
A.
y = Cx
B.
y = Cx^2
C.
y = C/x
D.
y = C ln(x)
Show solution
Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer:
A
— y = Cx
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Q. Solve the differential equation dy/dx = y^2.
A.
y = 1/(C - x)
B.
y = Cx
C.
y = C + x^2
D.
y = C - x
Show solution
Solution
This is a separable equation. Integrating gives y = 1/(C - x), where C is the constant.
Correct Answer:
A
— y = 1/(C - x)
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Q. Solve the differential equation y' = 3y + 6.
A.
y = Ce^(3x) - 2
B.
y = Ce^(3x) + 2
C.
y = 2e^(3x)
D.
y = 3e^(3x) + 2
Show solution
Solution
Using the integrating factor method, we find y = Ce^(3x) + 2.
Correct Answer:
B
— y = Ce^(3x) + 2
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Q. Solve the differential equation y' = 5 - 2y.
A.
y = 5/2 + Ce^(-2x)
B.
y = 5 + Ce^(-2x)
C.
y = 2 + Ce^(2x)
D.
y = 5/2 - Ce^(-2x)
Show solution
Solution
This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).
Correct Answer:
A
— y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation y' = 5y + 3.
A.
y = (3/5) + Ce^(5x)
B.
y = (5/3) + Ce^(5x)
C.
y = Ce^(5x) - 3
D.
y = Ce^(3x) + 5
Show solution
Solution
Using the integrating factor method, we find the solution y = (3/5) + Ce^(5x).
Correct Answer:
A
— y = (3/5) + Ce^(5x)
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Q. Solve the differential equation y'' + 4y = 0.
A.
y = C1 cos(2x) + C2 sin(2x)
B.
y = C1 e^(2x) + C2 e^(-2x)
C.
y = C1 cos(x) + C2 sin(x)
D.
y = C1 e^(x) + C2 e^(-x)
Show solution
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer:
A
— y = C1 cos(2x) + C2 sin(2x)
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Q. Solve the differential equation y'' - 3y' + 2y = 0.
A.
y = C1e^(2x) + C2e^(x)
B.
y = C1e^(x) + C2e^(2x)
C.
y = C1e^(-x) + C2e^(-2x)
D.
y = C1e^(3x) + C2e^(x)
Show solution
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).
Correct Answer:
B
— y = C1e^(x) + C2e^(2x)
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Q. Solve 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^(2x) + C2 e^(x)
Show solution
Solution
The characteristic equation is r^2 - 5r + 6 = 0, which factors to (r - 2)(r - 3) = 0, giving the solution y = C1 e^(2x) + C2 e^(3x).
Correct Answer:
B
— y = C1 e^(3x) + C2 e^(2x)
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Q. Solve the equation 2sin(x) + √3 = 0 for x in the interval [0, 2π].
A.
5π/3
B.
π/3
C.
2π/3
D.
4π/3
Show solution
Solution
Rearranging gives sin(x) = -√3/2, so x = 4π/3 and x = 5π/3.
Correct Answer:
A
— 5π/3
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Q. Solve the equation 2sin(x) - 1 = 0 for x in the interval [0, 2π].
A.
π/6
B.
5π/6
C.
π/2
D.
7π/6
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Solution
The solution is x = π/2.
Correct Answer:
C
— π/2
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Q. Solve the equation 3cos^2(x) - 1 = 0.
A.
x = π/3, 2π/3
B.
x = π/4, 3π/4
C.
x = 0, π
D.
x = π/6, 5π/6
Show solution
Solution
Rearranging gives cos^2(x) = 1/3, so x = π/3 and 2π/3.
Correct Answer:
A
— x = π/3, 2π/3
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Q. Solve the equation 3sin(x) - 4 = 0 for x in the interval [0, 2π].
A.
π/6
B.
π/3
C.
2π/3
D.
5π/6
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Solution
The solution is x = π/3.
Correct Answer:
B
— π/3
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Q. Solve the equation cos(x) + sin(x) = 1 for x in the interval [0, 2π].
A.
π/4
B.
π/2
C.
3π/4
D.
0
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Solution
The only solution is x = π/2.
Correct Answer:
B
— π/2
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Q. Solve the equation cos(x) = -1/2 for x in the interval [0, 2π].
A.
2π/3, 4π/3
B.
π/3, 5π/3
C.
π/2, 3π/2
D.
0, π
Show solution
Solution
The solutions are x = 2π/3 and x = 4π/3 in the interval [0, 2π].
Correct Answer:
A
— 2π/3, 4π/3
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Q. Solve the equation dy/dx = y^2 - x.
A.
y = sqrt(x + C)
B.
y = x + C
C.
y = 1/(C - x)
D.
y = x - C
Show solution
Solution
This is a separable equation. Separating variables and integrating gives y = 1/(C - x).
Correct Answer:
C
— y = 1/(C - x)
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Q. Solve the equation sin(2x) = 0 for x in the interval [0, 2π].
A.
0, π, 2π
B.
π/2, 3π/2
C.
π/4, 3π/4
D.
π/6, 5π/6
Show solution
Solution
The solutions are x = 0, π, 2π.
Correct Answer:
A
— 0, π, 2π
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Q. Solve the equation sin(2x) = 1 for x in the interval [0, 2π].
A.
π/4
B.
3π/4
C.
π/2
D.
5π/4
Show solution
Solution
The equation sin(2x) = 1 gives 2x = π/2 + 2nπ, hence x = π/4 + nπ/2. In [0, 2π], the solutions are π/4 and 5π/4.
Correct Answer:
C
— π/2
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Q. Solve the equation sin(2x) = √3/2 for x in the interval [0, 2π].
A.
π/12
B.
5π/12
C.
7π/12
D.
11π/12
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Solution
The solutions are x = π/12, 5π/12, 7π/12, and 11π/12.
Correct Answer:
A
— π/12
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Q. Solve the equation sin(3x) = 0 for x in the interval [0, 2π].
A.
0, π, 2π
B.
0, π/3, 2π/3
C.
0, π/2, π
D.
0, π/4, π/2
Show solution
Solution
The solutions are x = 0, π, 2π, and x = nπ/3 for n = 0, 1, 2, 3, 4, 5.
Correct Answer:
A
— 0, π, 2π
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Showing 16591 to 16620 of 31669 (1056 Pages)
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!
