Q. For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 + 1, x > 1 continuous at x = 1?
Show solution
Solution
Setting the left limit (2(1) + 1 = 3) equal to the right limit (1^2 + 1 = 2), we find b = 3.
Correct Answer:
B
— 2
Learn More →
Q. For which value of b is the function f(x) = { x^2 - 1, x < 1; b, x = 1; 3x - 2, x > 1 continuous at x = 1?
Show solution
Solution
Setting limit as x approaches 1 gives b = 2 for continuity.
Correct Answer:
C
— 2
Learn More →
Q. For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >= 2 } continuous at x = 2?
Show solution
Solution
Setting the two pieces equal at x = 2 gives us 0 = 2b + 2. Solving for b gives b = -1.
Correct Answer:
B
— 4
Learn More →
Q. For which 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 + b = 2 gives b = 4 for continuity.
Correct Answer:
A
— 0
Learn More →
Q. For which value of c is the function f(x) = { 3x + c, x < 1; 2x^2, x >= 1 continuous at x = 1?
Show solution
Solution
Setting 3(1) + c = 2(1)^2 gives c = -1 for continuity.
Correct Answer:
B
— 0
Learn More →
Q. For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?
Show solution
Solution
Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.
Correct Answer:
C
— 3
Learn More →
Q. For which value of c is the function f(x) = { x^2 - c, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Show solution
Solution
Setting x^2 - c = 2x + 1 at x = 1 gives c = 2.
Correct Answer:
C
— 2
Learn More →
Q. For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c continuous at x = c?
Show solution
Solution
Setting x^2 = 2x + 1 at x = c gives c = 2.
Correct Answer:
C
— 2
Learn More →
Q. For which value of m is the function f(x) = { 3x + m, x < 1; 2, x = 1; mx + 1, x > 1 continuous at x = 1?
Show solution
Solution
Setting 3 + m = 2 and 2 = m + 1 gives m = 1 for continuity.
Correct Answer:
B
— 0
Learn More →
Q. For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?
Show solution
Solution
Setting the two pieces equal at x = 0 gives us p = -1.
Correct Answer:
B
— 0
Learn More →
Q. If f(x) = 1/(x-1), what is the point of discontinuity?
A.
x = 0
B.
x = 1
C.
x = -1
D.
x = 2
Show solution
Solution
The function is discontinuous at x = 1 because it leads to division by zero.
Correct Answer:
B
— x = 1
Learn More →
Q. If f(x) = x^2 - 4, what is the limit of f(x) as x approaches 2?
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
The limit as x approaches 2 is f(2) = 2^2 - 4 = 0.
Correct Answer:
C
— 4
Learn More →
Q. If f(x) = x^3 - 3x + 2, then f(x) is continuous at:
A.
All x
B.
x = 0
C.
x = 1
D.
x = -1
Show solution
Solution
f(x) is a polynomial function and is continuous for all x.
Correct Answer:
A
— All x
Learn More →
Q. If f(x) = { 2x + 3, x < 0; kx + 1, x >= 0 } is continuous at x = 0, what is the value of k?
A.
-3/2
B.
1/2
C.
3/2
D.
2
Show solution
Solution
Setting the two pieces equal at x = 0: 3 = k(0) + 1. Solving gives k = -3/2.
Correct Answer:
A
— -3/2
Learn More →
Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?
Show solution
Solution
For continuity at x = 0, we need the left limit (1) to equal k. Thus, k = 1.
Correct Answer:
A
— 1
Learn More →
Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 }, what value of k makes f continuous at x = 0?
Show solution
Solution
To be continuous at x = 0, k must equal the limit from the left, which is 1.
Correct Answer:
B
— 1
Learn More →
Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuous at x = 0, k must be:
Show solution
Solution
For continuity at x = 0, k must equal the limit as x approaches 0, which is 1.
Correct Answer:
B
— 1
Learn More →
Q. If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x = 0, find k.
Show solution
Solution
For continuity at x = 0, we need 1 = 2, thus k must be 1.
Correct Answer:
B
— 2
Learn More →
Q. If f(x) = { x^2 + 1, x < 0; kx + 3, x = 0; 2x - 1, x > 0 is continuous at x = 0, find k.
Show solution
Solution
For continuity at x = 0, we need 1 = 3 and 1 = -1 + 3k, solving gives k = 1.
Correct Answer:
C
— 1
Learn More →
Q. If f(x) = { x^2, x < 0; 2x + 3, x >= 0 }, find f(0).
A.
0
B.
3
C.
1
D.
undefined
Show solution
Solution
At x = 0, we use the second piece: f(0) = 2(0) + 3 = 3.
Correct Answer:
B
— 3
Learn More →
Q. If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x = 0, find k.
Show solution
Solution
To ensure continuity at x = 0, we set k(0) + 1 = 0^2, leading to k = 2.
Correct Answer:
C
— 1
Learn More →
Q. If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.
Show solution
Solution
Setting the two pieces equal at x = 1 gives 1 = k + 1, hence k = 0.
Correct Answer:
B
— 1
Learn More →
Q. If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what is the value of f(2)?
Show solution
Solution
For continuity at x = 2, f(2) must equal the limit from both sides, which is 4.
Correct Answer:
B
— 4
Learn More →
Q. If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what is the value of k?
Show solution
Solution
For continuity at x = 3, we need limit as x approaches 3 from left (9) to equal f(3) = k, thus k = 9.
Correct Answer:
C
— 6
Learn More →
Q. If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3, what is k?
Show solution
Solution
For continuity at x = 3, we need k to equal the limit from both sides, which is 9.
Correct Answer:
C
— 8
Learn More →
Q. If f(x) is continuous on [a, b], which of the following must be true?
A.
f(a) = f(b)
B.
f(x) takes every value between f(a) and f(b)
C.
f(x) is increasing
D.
f(x) is decreasing
Show solution
Solution
By the Intermediate Value Theorem, a continuous function on a closed interval takes every value between f(a) and f(b).
Correct Answer:
B
— f(x) takes every value between f(a) and f(b)
Learn More →
Q. Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.
Correct Answer:
A
— Yes
Learn More →
Q. Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
A.
Yes
B.
No
C.
Depends on x
D.
Not defined
Show solution
Solution
Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.
Correct Answer:
A
— Yes
Learn More →
Q. Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Correct Answer:
A
— Yes
Learn More →
Q. Is the function f(x) = |x|/x continuous at x = 0?
A.
Yes
B.
No
C.
Depends on direction
D.
None of the above
Show solution
Solution
The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.
Correct Answer:
B
— No
Learn More →
Showing 61 to 90 of 124 (5 Pages)
Continuity MCQ & Objective Questions
Understanding the concept of "Continuity" is crucial for students preparing for school exams and competitive tests in India. Mastering this topic not only enhances your conceptual clarity but also boosts your confidence in tackling objective questions. Practicing MCQs and important questions on Continuity can significantly improve your exam performance, making it essential for effective exam preparation.
What You Will Practise Here
Definition and properties of continuity
Types of continuity: point continuity and interval continuity
Continuity of functions and their graphical representations
Intermediate Value Theorem and its applications
Limits and their role in establishing continuity
Common functions that exhibit continuity
Real-life applications of continuous functions
Exam Relevance
The topic of Continuity is frequently featured in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that assess their understanding of continuity through MCQs that may involve identifying continuous functions, applying the Intermediate Value Theorem, or solving problems related to limits. Familiarity with common question patterns will help you tackle these effectively.
Common Mistakes Students Make
Confusing continuity with differentiability
Overlooking the importance of limits in determining continuity
Misinterpreting graphical representations of continuous functions
Neglecting to check endpoints in interval continuity
Failing to apply the Intermediate Value Theorem correctly
FAQs
Question: What is the definition of continuity in mathematics?Answer: Continuity refers to a function being unbroken or uninterrupted over an interval, meaning small changes in input result in small changes in output.
Question: How can I determine if a function is continuous at a point?Answer: A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function's value.
Start solving practice MCQs on Continuity today to enhance your understanding and prepare effectively for your exams. Remember, consistent practice is key to mastering this important topic!
