Q. Differentiate f(x) = 4x^2 + 3x - 5. (2019)
A.
8x + 3
B.
4x + 3
C.
2x + 3
D.
8x - 3
Show solution
Solution
Using the power rule, f'(x) = 8x + 3.
Correct Answer:
A
— 8x + 3
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Q. Differentiate f(x) = 4x^5 - 2x^3 + x. (2022)
A.
20x^4 - 6x^2 + 1
B.
20x^4 - 6x^2
C.
4x^4 - 2x^2 + 1
D.
5x^4 - 2x^2
Show solution
Solution
Using the power rule, f'(x) = 20x^4 - 6x^2 + 1.
Correct Answer:
A
— 20x^4 - 6x^2 + 1
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Q. Differentiate f(x) = ln(x^2 + 1). (2022)
A.
2x/(x^2 + 1)
B.
1/(x^2 + 1)
C.
2x/(x^2 - 1)
D.
x/(x^2 + 1)
Show solution
Solution
Using the chain rule, f'(x) = 2x/(x^2 + 1).
Correct Answer:
A
— 2x/(x^2 + 1)
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Q. Differentiate f(x) = x^2 * e^x. (2022)
A.
x^2 * e^x + 2x * e^x
B.
2x * e^x + x^2 * e^x
C.
x^2 * e^x + e^x
D.
2x * e^x
Show solution
Solution
Using the product rule, f'(x) = x^2 * e^x + 2x * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Differentiate f(x) = x^2 * ln(x).
A.
2x * ln(x) + x
B.
x * ln(x) + 2x
C.
2x * ln(x)
D.
x^2/x
Show solution
Solution
Using the product rule, f'(x) = 2x * ln(x) + x.
Correct Answer:
A
— 2x * ln(x) + x
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Q. Differentiate the function f(x) = ln(x^2 + 1).
A.
2x/(x^2 + 1)
B.
2/(x^2 + 1)
C.
1/(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. Differentiate the function f(x) = x^2 * e^x.
A.
x^2 * e^x + 2x * e^x
B.
2x * e^x + x^2 * e^x
C.
x^2 * e^x + e^x
D.
2x * e^x + e^x
Show solution
Solution
Using the product rule, f'(x) = (x^2)' * e^x + x^2 * (e^x)' = 2x * e^x + x^2 * e^x.
Correct Answer:
A
— x^2 * e^x + 2x * e^x
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Q. Evaluate the integral ∫ (3x^2 - 4) dx.
A.
x^3 - 4x + C
B.
x^3 - 2x + C
C.
3x^3 - 4x + C
D.
x^3 - 4x
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Solution
The integral evaluates to x^3 - 4x + C, where C is the constant of integration.
Correct Answer:
A
— x^3 - 4x + C
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Q. Evaluate the integral ∫ (4x^3 - 2x) dx.
A.
x^4 - x^2 + C
B.
x^4 - x^2
C.
x^4 - x^2 + 2C
D.
4x^4 - x^2 + C
Show solution
Solution
The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.
Correct Answer:
A
— x^4 - x^2 + C
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Q. Evaluate the integral ∫ (5x^4) dx.
A.
x^5 + C
B.
x^5 + 5C
C.
x^5 + 1
D.
5x^5 + C
Show solution
Solution
The integral is (5/5)x^5 + C = x^5 + C.
Correct Answer:
A
— x^5 + C
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Q. Evaluate the integral ∫(0 to 1) (1 - x^2) dx. (2022)
A.
1/3
B.
1/2
C.
2/3
D.
1
Show solution
Solution
∫(0 to 1) (1 - x^2) dx = [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer:
C
— 2/3
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Q. Evaluate the integral ∫(0 to π) sin(x) dx. (2021)
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Solution
∫(0 to π) sin(x) dx = [-cos(x)] from 0 to π = -(-1 - 1) = 2.
Correct Answer:
C
— 2
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Q. Evaluate the integral ∫(1 to 2) (3x^2 - 4) dx. (2019)
Show solution
Solution
∫(1 to 2) (3x^2 - 4) dx = [x^3 - 4x] from 1 to 2 = (8 - 8) - (1 - 4) = 3.
Correct Answer:
A
— 1
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Q. Evaluate the integral ∫(1 to 3) (3x^2 - 2) dx. (2019)
Show solution
Solution
∫(1 to 3) (3x^2 - 2) dx = [x^3 - 2x] from 1 to 3 = (27 - 6) - (1 - 2) = 20.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(1 to 4) (2x + 1) dx. (2021)
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Solution
∫(1 to 4) (2x + 1) dx = [x^2 + x] from 1 to 4 = (16 + 4) - (1 + 1) = 18.
Correct Answer:
B
— 12
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Q. Evaluate the integral ∫(2 to 3) (x^3 - 3x^2 + 2) dx. (2023)
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Solution
∫(2 to 3) (x^3 - 3x^2 + 2) dx = [x^4/4 - x^3 + 2x] from 2 to 3 = (81/4 - 27 + 6) - (16/4 - 8 + 4) = 1.
Correct Answer:
B
— 2
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Q. Evaluate the integral ∫(2x + 3) dx from 1 to 2.
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Solution
The integral evaluates to [x^2 + 3x] from 1 to 2, which gives (4 + 6) - (1 + 3) = 8 - 4 = 4.
Correct Answer:
B
— 7
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Q. Evaluate the integral ∫(2x + 3) dx. (2021)
A.
x^2 + 3x + C
B.
x^2 + 3x
C.
2x^2 + 3x + C
D.
2x^2 + 3x
Show solution
Solution
The integral of (2x + 3) is (2x^2/2) + 3x + C = x^2 + 3x + C.
Correct Answer:
A
— x^2 + 3x + C
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Q. Evaluate the integral ∫(sin x)dx. (2022)
A.
-cos x + C
B.
cos x + C
C.
sin x + C
D.
-sin x + C
Show solution
Solution
The integral of sin x is -cos x + C.
Correct Answer:
A
— -cos x + C
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Q. Evaluate the integral ∫(x^2 - 2x + 1) dx. (2022)
A.
(1/3)x^3 - x^2 + x + C
B.
(1/3)x^3 - x^2 + C
C.
(1/3)x^3 - 2x + C
D.
(1/3)x^3 - x^2 + x
Show solution
Solution
The integral of (x^2 - 2x + 1) is (1/3)x^3 - x^2 + x + C.
Correct Answer:
A
— (1/3)x^3 - x^2 + x + C
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
A.
5, Continuous
B.
0, Not continuous
C.
5, Not continuous
D.
0, Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5, and the function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.
A.
5, Continuous
B.
0, Continuous
C.
5, Not Continuous
D.
0, Not Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
A.
5, Continuous
B.
5, Discontinuous
C.
0, Continuous
D.
0, Discontinuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0 if defined as f(0) = 5.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit lim (x -> 0) (sin(x)/x) = 1. Since the limit exists and equals the function value at x = 0, it is continuous.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit is 1, and if we define f(0) = 1, then f(x) is continuous at x = 0.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)
A.
0, Yes
B.
0, No
C.
6, Yes
D.
6, No
Show solution
Solution
lim (x -> 3) (x^2 - 9)/(x - 3) = lim (x -> 3) (x + 3) = 6. The function is not defined at x = 3, hence not continuous.
Correct Answer:
C
— 6, Yes
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Q. Evaluate the limit lim x→2 (x^2 - 4)/(x - 2).
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Using L'Hôpital's Rule, lim x→2 (x^2 - 4)/(x - 2) = lim x→2 (2x)/(1) = 4.
Correct Answer:
C
— 4
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Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.
Correct Answer:
A
— 0 degrees
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Q. Find the angle between the vectors A = i + j and B = i - j.
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
135 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Showing 151 to 180 of 1593 (54 Pages)
Mathematics (NDA) MCQ & Objective Questions
Mathematics plays a crucial role in the NDA exam, as it tests your analytical and problem-solving skills. Practicing Mathematics (NDA) MCQ and objective questions is essential for scoring better in this competitive environment. By focusing on practice questions, you can identify important questions and enhance your exam preparation effectively.
What You Will Practise Here
Algebra: Understanding equations, inequalities, and functions.
Geometry: Key concepts of shapes, angles, and theorems.
Trigonometry: Important ratios, identities, and applications.
Statistics: Basics of mean, median, mode, and standard deviation.
Probability: Fundamental principles and problem-solving techniques.
Calculus: Introduction to limits, derivatives, and integrals.
Mensuration: Formulas for areas and volumes of various shapes.
Exam Relevance
The Mathematics (NDA) syllabus is relevant not only for the NDA exam but also for various other competitive exams like CBSE, State Boards, NEET, and JEE. In these exams, you will often encounter multiple-choice questions that test your understanding of mathematical concepts. Common question patterns include direct application of formulas, problem-solving scenarios, and conceptual understanding, making it essential to practice regularly.
Common Mistakes Students Make
Misinterpreting the question: Students often overlook key details in the problem statement.
Formula errors: Forgetting or misapplying mathematical formulas can lead to incorrect answers.
Calculation mistakes: Simple arithmetic errors can cost valuable marks.
Neglecting units: Failing to consider units in problems involving measurements.
Rushing through questions: Students may skip steps or fail to double-check their work under time pressure.
FAQs
Question: What are the best ways to prepare for Mathematics (NDA) MCQs?Answer: Regular practice with objective questions, understanding key concepts, and solving previous years' papers are effective strategies.
Question: How can I improve my speed in solving Mathematics (NDA) questions?Answer: Time yourself while practicing and focus on solving simpler problems quickly to build speed and confidence.
Start solving Mathematics (NDA) MCQs today to test your understanding and boost your confidence for the exams. Remember, consistent practice is the key to success!
