Q. For the function f(x) = x^2 - 4x + 5, find the vertex.
A.
(2, 1)
B.
(2, 5)
C.
(4, 1)
D.
(4, 5)
Show solution
Solution
The vertex is at x = -b/(2a) = 4/2 = 2. f(2) = 2^2 - 4(2) + 5 = 1, so the vertex is (2, 1).
Correct Answer:
A
— (2, 1)
Learn More →
Q. For the function f(x) = x^2 - 6x + 8, find the x-coordinate of the vertex.
Show solution
Solution
The x-coordinate of the vertex is given by x = -b/(2a) = 6/(2*1) = 3.
Correct Answer:
B
— 3
Learn More →
Q. For the function f(x) = x^3 - 3x^2 + 2, find the points where it is not differentiable.
A.
None
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
As a polynomial, f(x) is differentiable everywhere, hence no points of non-differentiability.
Correct Answer:
A
— None
Learn More →
Q. For the function f(x) = x^3 - 3x^2 + 4, find the points where it is not differentiable.
A.
None
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
The function is a polynomial and is differentiable everywhere, hence there are no points where it is not differentiable.
Correct Answer:
A
— None
Learn More →
Q. For the function f(x) = x^3 - 3x^2 + 4, find the value of x where f is not differentiable.
Show solution
Solution
The function is a polynomial and is differentiable everywhere, so there is no such x.
Correct Answer:
A
— 0
Learn More →
Q. For the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where f is differentiable.
Show solution
Solution
f(x) is a polynomial and is differentiable everywhere. The x-coordinate can be any real number.
Correct Answer:
C
— 1
Learn More →
Q. For the function f(x) = x^3 - 6x^2 + 9x, find the critical points.
A.
x = 0, 3
B.
x = 1, 2
C.
x = 2, 3
D.
x = 3, 4
Show solution
Solution
First, find f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives (x - 3)(x - 1) = 0, so critical points are x = 1 and x = 3.
Correct Answer:
A
— x = 0, 3
Learn More →
Q. For the function f(x) = x^3 - 6x^2 + 9x, find the intervals where the function is increasing.
A.
(-∞, 0)
B.
(0, 3)
C.
(3, ∞)
D.
(0, 6)
Show solution
Solution
f'(x) = 3x^2 - 12x + 9. The critical points are x = 1 and x = 3. The function is increasing on (1, 3) and (3, ∞).
Correct Answer:
B
— (0, 3)
Learn More →
Q. For the function f(x) = x^4 - 8x^2 + 16, find the coordinates of the inflection point.
A.
(0, 16)
B.
(2, 0)
C.
(4, 0)
D.
(2, 4)
Show solution
Solution
Find f''(x) = 12x^2 - 16. Setting f''(x) = 0 gives x^2 = 4, so x = ±2. f(2) = 0, thus the inflection point is (2, 0).
Correct Answer:
B
— (2, 0)
Learn More →
Q. For the function f(x) = x^4 - 8x^2 + 16, find the intervals where the function is increasing.
A.
(-∞, -2)
B.
(-2, 2)
C.
(2, ∞)
D.
(-2, ∞)
Show solution
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x(x^2 - 4) = 0, so x = -2, 0, 2. Test intervals: f' is positive in (-2, ∞).
Correct Answer:
D
— (-2, ∞)
Learn More →
Q. For the function f(x) = { x^2 + 1, x < 0; 2x + b, x = 0; 3 - x, x > 0 to be continuous at x = 0, what is b?
Show solution
Solution
Setting the left limit (0 + 1 = 1) equal to the right limit (3 - 0 = 3), we find b = 1.
Correct Answer:
B
— 0
Learn More →
Q. For the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 }, what is the value of f(1)?
Show solution
Solution
By definition, f(1) = 3, as given in the piecewise function.
Correct Answer:
C
— 3
Learn More →
Q. For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.
Show solution
Solution
Setting f(1-) = f(1+) and f'(1-) = f'(1+) gives k = 2 for differentiability.
Correct Answer:
B
— 1
Learn More →
Q. For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuous at x = 3, the value of f(3) must be:
Show solution
Solution
For continuity, f(3) must equal the limit as x approaches 3, which is 9.
Correct Answer:
B
— 9
Learn More →
Q. For the function f(x) = |x - 2| + |x + 3|, find the point where it is not differentiable.
Show solution
Solution
The function is not differentiable at x = -3 and x = 2, but the first point of interest is -3.
Correct Answer:
A
— -3
Learn More →
Q. For what value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Show solution
Solution
Setting 1^3 - 3(1) + b = 2(1) + 1 gives b = 2.
Correct Answer:
B
— 1
Learn More →
Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?
Show solution
Solution
To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.
Correct Answer:
B
— 0
Learn More →
Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?
Show solution
Solution
The function is a polynomial and is differentiable for all real numbers, hence any value of a works.
Correct Answer:
B
— 0
Learn More →
Q. For which value of a is the function f(x) = x^2 - ax + 2 differentiable at x = 1?
Show solution
Solution
Setting the derivative f'(1) = 0 gives a = 1 for differentiability.
Correct Answer:
B
— 1
Learn More →
Q. For which value of a is the function f(x) = x^2 - ax + 4 differentiable at x = 2?
Show solution
Solution
f(x) is a polynomial and is differentiable for all a, hence any value of a works.
Correct Answer:
A
— 0
Learn More →
Q. For which value of a is the function f(x) = x^3 - 3ax + 2 differentiable at x = 1?
Show solution
Solution
Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.
Correct Answer:
B
— 1
Learn More →
Q. For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
Show solution
Solution
Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.
Correct Answer:
B
— 1
Learn More →
Q. For which value of a is the function f(x) = { 2x + a, x < 0; x^2 + 1, x >= 0 continuous at x = 0?
Show solution
Solution
Setting a = 1 gives continuity at x = 0.
Correct Answer:
B
— 0
Learn More →
Q. For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >= 2 continuous at x = 2?
Show solution
Solution
Setting 3(2) + a = 4(2) - 1 gives a = 1.
Correct Answer:
A
— -1
Learn More →
Q. For which value of a is the function f(x) = { ax + 1, x < 0; 2, x = 0; 3x - 1, x > 0 } continuous at x = 0?
Show solution
Solution
Setting ax + 1 = 2 at x = 0 gives a = 2.
Correct Answer:
A
— -1
Learn More →
Q. For which value of a is the function f(x) = { ax + 1, x < 0; 2x + a, x >= 0 } continuous at x = 0?
Show solution
Solution
Setting the two pieces equal at x = 0 gives 1 = a, hence a = 1.
Correct Answer:
A
— -1
Learn More →
Q. For which value of a is the function f(x) = { ax + 2, x < 1; 3, x >= 1 } continuous at x = 1?
Show solution
Solution
Setting ax + 2 = 3 at x = 1 gives a = 1.
Correct Answer:
B
— 2
Learn More →
Q. For which value of a is the function f(x) = { x^2 + a, x < 1; 3, x >= 1 } continuous at x = 1?
Show solution
Solution
To ensure continuity at x = 1, we set limit as x approaches 1 from left (1 + a) equal to f(1) = 3, thus a = 2.
Correct Answer:
B
— 2
Learn More →
Q. For which value of a is the function f(x) = { x^2 - a, x < 0; 2x + 1, x >= 0 } continuous at x = 0?
Show solution
Solution
Setting the two pieces equal at x = 0 gives -a = 1, so a = -1.
Correct Answer:
A
— -1
Learn More →
Q. For which value of a is the function f(x) = { x^2 - a, x < 1; 3x - 2, x >= 1 } continuous at x = 1?
Show solution
Solution
Setting the two pieces equal at x = 1 gives 1 - a = 1. Thus, a = 0.
Correct Answer:
C
— 2
Learn More →
Showing 331 to 360 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!
