Q. What is the solution to the equation y' = y(1 - y)?
A.
y = 1/(C - x)
B.
y = C/(C + x)
C.
y = C/(1 + Cx)
D.
y = C/(1 - Cx)
Show solution
Solution
This is a separable equation. Integrating gives y = C/(C + x).
Correct Answer:
B
— y = C/(C + x)
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Q. What is the solution to the first-order linear differential equation dy/dx + y = e^x?
A.
y = e^x + Ce^(-x)
B.
y = e^x - Ce^(-x)
C.
y = Ce^x - e^x
D.
y = Ce^(-x) + e^x
Show solution
Solution
Using the integrating factor e^x, we solve to get y = e^x + Ce^(-x).
Correct Answer:
A
— y = e^x + Ce^(-x)
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Q. What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?
Show solution
Solution
Setting 2k + 2 = 0 gives k = 2.
Correct Answer:
C
— 2
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Q. What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?
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Solution
Setting k(0) = 0^2 + 1 gives k = 1.
Correct Answer:
B
— 0
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Q. What is the value of k for which the function f(x) = { kx, x < 2; x^2, x >= 2 } is continuous at x = 2?
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Solution
Setting k(2) = 2^2 gives 2k = 4, thus k = 2.
Correct Answer:
C
— 4
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Q. What is the value of p for which the function f(x) = { 3x + p, x < 2; x^2 - 4, x >= 2 } is continuous at x = 2?
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Solution
Setting the two pieces equal at x = 2: 3(2) + p = 2^2 - 4. Solving gives p = -2.
Correct Answer:
A
— -1
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Q. What is the value of q for which the function f(x) = { 5 - q, x < 1; 3x + 2, x >= 1 } is continuous at x = 1?
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Solution
Setting the two pieces equal at x = 1: 5 - q = 3(1) + 2. Solving gives q = 0.
Correct Answer:
C
— 2
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Q. What is the value of the derivative of f(x) = ln(x^2 + 1) at x = 1?
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Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = 2/(1 + 1) = 1.
Correct Answer:
B
— 1/2
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Q. What is the value of the integral ∫(0 to 1) (3x^2 + 2)dx?
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Solution
Integral = [x^3 + 2x] from 0 to 1 = (1 + 2) - (0) = 3.
Correct Answer:
A
— 5/3
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Q. What is the value of the integral ∫(0 to 1) (3x^2)dx?
A.
1
B.
1/3
C.
1/2
D.
3/4
Show solution
Solution
The integral evaluates to [x^3] from 0 to 1 = 1^3 - 0^3 = 1.
Correct Answer:
A
— 1
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Q. What is the value of the integral ∫(1 to 2) (3x^2 - 2)dx?
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Solution
The integral evaluates to [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.
Correct Answer:
B
— 4
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Q. What is the value of the integral ∫(1 to 2) (x^2 + 2x)dx?
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Solution
The integral evaluates to 7.
Correct Answer:
A
— 7
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Q. What is the value of the integral ∫(1/(x^2 + 1))dx?
A.
tan^-1(x) + C
B.
sin^-1(x) + C
C.
cos^-1(x) + C
D.
ln(x) + C
Show solution
Solution
The integral evaluates to tan^-1(x) + C.
Correct Answer:
A
— tan^-1(x) + C
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Q. What is the value of the integral ∫_0^1 (3x^2 + 2x) dx?
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Solution
Evaluating the integral gives [x^3 + x^2]_0^1 = (1 + 1) - (0 + 0) = 2.
Correct Answer:
B
— 2
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Q. What is the value of the limit lim (x -> 1) (x^2 - 1)/(x - 1)?
A.
0
B.
1
C.
2
D.
Infinity
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Solution
Using L'Hôpital's Rule, the limit evaluates to 2.
Correct Answer:
C
— 2
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Q. What is the value of the limit: lim (x -> ∞) (1/x)?
A.
0
B.
1
C.
Infinity
D.
Undefined
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Solution
As x approaches infinity, 1/x approaches 0.
Correct Answer:
A
— 0
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Q. What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?
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Solution
Setting 2(1) + 1 = a and a = 2 for continuity.
Correct Answer:
B
— 2
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Q. What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
Show solution
Solution
Setting 2(3) + a = 5 gives a = -1.
Correct Answer:
C
— 2
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Q. What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
Show solution
Solution
Setting 4 = 2 gives a = 1 for continuity.
Correct Answer:
B
— 0
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Q. What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?
Show solution
Solution
Setting the left limit (k(1) = k) equal to the right limit (1 + 1 = 2), we find k = 2.
Correct Answer:
B
— 1
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Q. What value of m makes the function f(x) = { 3x + 1, x < 2; mx + 4, x = 2; x^2 - 1, x > 2 continuous at x = 2?
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Solution
Setting the left limit (3(2) + 1 = 7) equal to the right limit (2^2 - 1 = 3), we find m = 3.
Correct Answer:
D
— 4
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Q. Which of the following functions is continuous at x = 2?
A.
f(x) = 1/x
B.
f(x) = x^2 - 4
C.
f(x) = sin(1/x)
D.
f(x) =
.
x
.
Show solution
Solution
f(x) = x^2 - 4 is a polynomial function and is continuous everywhere, including at x = 2.
Correct Answer:
B
— f(x) = x^2 - 4
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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }
A.
Continuous
B.
Not continuous
C.
Depends on k
D.
None of the above
Show solution
Solution
At x = 2, f(2) = 0 and limit from left is 0, limit from right is also 0. Hence, it is continuous.
Correct Answer:
A
— Continuous
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Q. Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }
A.
Continuous
B.
Not continuous
C.
Depends on k
D.
None of the above
Show solution
Solution
To check continuity at x = 2, we find the left limit (4), right limit (4), and f(2) (4). All are equal, so f(x) is continuous.
Correct Answer:
A
— Continuous
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Q. Which of the following functions is continuous everywhere?
A.
f(x) = 1/x
B.
f(x) = x^2
C.
f(x) = sin(x)
D.
f(x) =
.
x
.
Show solution
Solution
f(x) = x^2 is a polynomial function and is continuous everywhere.
Correct Answer:
B
— f(x) = x^2
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Q. Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }
A.
f(1) = 1
B.
f(1) = 0
C.
f(1) = 2
D.
f(1) = 3
Show solution
Solution
Check continuity and differentiability at x = 1 by equating left and right derivatives.
Correct Answer:
A
— f(1) = 1
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Q. Which of the following functions is differentiable everywhere?
A.
f(x) =
B.
x
C.
D.
f(x) = x^2
.
f(x) = sqrt(x)
.
f(x) = 1/x
Show solution
Solution
f(x) = x^2 is a polynomial and differentiable everywhere.
Correct Answer:
B
— x
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Q. Which of the following functions is not continuous at x = 0?
A.
f(x) = x^3
B.
f(x) = e^x
C.
f(x) = 1/x
D.
f(x) = ln(x)
Show solution
Solution
The function f(x) = 1/x is not defined at x = 0, hence it is not continuous there.
Correct Answer:
C
— f(x) = 1/x
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Q. Which of the following functions is not continuous at x = 1?
A.
f(x) = x^2
B.
f(x) = 1/x
C.
f(x) = sin(x)
D.
f(x) = { x, x < 1; 2, x >= 1 }
Show solution
Solution
The function has a jump discontinuity at x = 1, hence it is not continuous.
Correct Answer:
D
— f(x) = { x, x < 1; 2, x >= 1 }
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Q. Which of the following functions is not differentiable at x = 0? f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0.
A.
f(x)
B.
g(x) =
C.
x
D.
.
h(x) = x^3
.
k(x) = x^2
Show solution
Solution
The function f(x) is not differentiable at x = 0 due to the oscillatory nature of sin(1/x) as x approaches 0.
Correct Answer:
A
— f(x)
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Showing 541 to 570 of 574 (20 Pages)
Calculus MCQ & Objective Questions
Calculus is a vital branch of mathematics that plays a significant role in various school and competitive exams. Mastering calculus concepts not only enhances your problem-solving skills but also boosts your confidence during exams. Practicing MCQs and objective questions is essential for effective exam preparation, as it helps you identify important questions and strengthens your understanding of key topics.
What You Will Practise Here
Limits and Continuity
Differentiation and its Applications
Integration Techniques and Fundamental Theorem of Calculus
Applications of Derivatives in Real Life
Definite and Indefinite Integrals
Area Under Curves and Volume of Solids of Revolution
Common Functions and Their Derivatives
Exam Relevance
Calculus is a crucial topic in the CBSE curriculum and is also featured prominently in State Board exams, NEET, and JEE. Students can expect questions that test their understanding of limits, derivatives, and integrals. Common question patterns include solving problems based on real-life applications, finding maxima and minima, and evaluating integrals. Familiarity with these patterns through practice questions will help you excel in your exams.
Common Mistakes Students Make
Confusing the concepts of limits and continuity.
Misapplying differentiation rules, especially for composite functions.
Overlooking the importance of the constant of integration in indefinite integrals.
Failing to interpret the meaning of derivatives in real-world scenarios.
Neglecting to check the domain of functions when solving problems.
FAQs
Question: What are the key formulas I should remember for calculus? Answer: Important formulas include the power rule, product rule, quotient rule for differentiation, and basic integration formulas like ∫x^n dx = (x^(n+1))/(n+1) + C.
Question: How can I improve my speed in solving calculus MCQs? Answer: Regular practice with timed quizzes and focusing on understanding concepts rather than rote memorization can significantly improve your speed.
Start solving practice MCQs today to test your understanding and solidify your calculus knowledge. Remember, consistent practice is the key to success in your exams!
