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Calculus Practice Questions

HSC/SSC (Bangladesh) · HSC Mathematics · 144 free MCQs with instant results and detailed explanations.

144
Total
39
Easy
73
Medium
32
Hard

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Sample Questions from Calculus

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Q1
Easy
If f(x) = x^3 - 3x + 2, what is f(1)?
A. 0
B. 1
C. 2
D. 3
Show Answer & Explanation
Correct Answer: A
To find f(1), substitute x = 1 into the function: f(1) = (1)^3 - 3(1) + 2 = 1 - 3 + 2 = 0.
Q2
Easy
What is the integral of the function f(x) = 2x with respect to x?
A. x^2 + C
B. x + C
C. 2x + C
D. 2x^2 + C
Show Answer & Explanation
Correct Answer: A
The integral of f(x) = 2x with respect to x is found using the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C. Here, ∫2x dx = (2/2)x^(2) + C = x^2 + C.
Q3
Easy
What is the limit of the function f(x) = (2x^2 - 3)/(x - 1) as x approaches 1?
A. -1
B. 1
C. 0
D. Undefined
Show Answer & Explanation
Correct Answer: D
As x approaches 1, the function f(x) = (2(1)^2 - 3)/(1 - 1) leads to a division by zero in the denominator, making the limit undefined.
Q4
Medium
If the function f(x) = x^3 - 3x^2 + x has a local maximum, what is the value of x at that maximum?
A. 0
B. 1
C. 2
D. 3
Show Answer & Explanation
Correct Answer: C
To find local maxima, set the derivative f'(x) = 0. Here, f'(x) = 3x^2 - 6x + 1. Solving for x gives critical points, and testing x = 2 shows it is a local maximum.
Q5
Medium
The area bounded by the curve y = x^2 and the x-axis from x = 0 to x = 3 is calculated as which of the following?
A. 9
B. 18
C. 27
D. 36
Show Answer & Explanation
Correct Answer: A
The area under y = x^2 from 0 to 3 is found by evaluating ∫(x^2) dx from 0 to 3: [(1/3)x^3] from 0 to 3 = (1/3)(27) - (1/3)(0) = 9.
Q6
Medium
If the second derivative of a function is f''(x) = 12x - 4, what is the point of inflection?
A. 0
B. 1/3
C. 1/4
D. 2/3
Show Answer & Explanation
Correct Answer: B
A point of inflection occurs where the second derivative is zero or undefined. Setting 12x - 4 = 0 gives x = 1/3 as the point of inflection.
Q7
Medium
Evaluate the limit: lim x→2 of (x^2 - 4)/(x - 2).
A. 0
B. 2
C. 4
D. Undefined
Show Answer & Explanation
Correct Answer: C
The limit can be simplified by factoring the numerator as (x-2)(x+2). Canceling (x-2) gives lim x→2 (x + 2) = 4.
Q8
Hard
If the function f(x) = x^3 - 6x^2 + 9x - 2 is minimized at x = a, what is the value of a?
A. 2
B. 1
C. 3
D. 0
Show Answer & Explanation
Correct Answer: A
To find the minimum, we first find the derivative, f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 results in x^2 - 4x + 3 = 0, which factors to (x-1)(x-3) = 0, giving critical points x = 1 and x = 3. Evaluating f at these points shows f(2) is less than both, indicating x = 2 is the minimum.
Q9
Hard
Evaluate the integral ∫(2x^3 - 3x^2 + 4) dx from x = 1 to x = 2. What is the result?
A. 4
B. 5
C. 6
D. 7
Show Answer & Explanation
Correct Answer: B
The integral evaluates to F(2) - F(1), where F(x) is the antiderivative of the integrand. F(x) = (1/2)x^4 - x^3 + 4x. Calculating F(2) = 8 - 8 + 8 = 8 and F(1) = 1/2 - 1 + 4 = 3. Therefore, the integral evaluates to 8 - 3 = 5.
Q10
Hard
If f(x) = x^3 - 6x^2 + 9x, what are the critical points of the function?
A. (0, 0)
B. (1, 4)
C. (2, 0)
D. (3, 0)
Show Answer & Explanation
Correct Answer: C
To find the critical points, we first need to find the derivative f'(x) = 3x^2 - 12x + 9. Setting f'(x) to 0 and solving for x gives us x = 2. Plugging x = 2 back into the function, we get f(2) = 0, hence the critical point is (2, 0).

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Calculus — HSC/SSC (Bangladesh) HSC Mathematics Practice Questions Online

This page contains 144 practice MCQs for the chapter Calculus in HSC/SSC (Bangladesh) HSC Mathematics. The questions are organized by difficulty — 39 easy, 73 medium, 32 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.