Further Calculus Practice Questions
A-Levels · A-Level Further Mathematics · 142 free MCQs with instant results and detailed explanations.
142
Total
35
Easy
73
Medium
34
Hard
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Sample Questions from Further Calculus
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Q1
Easy
Evaluate the integral ∫(4x^3 - 2x + 1)dx.
A.
x^4 - x^2 + x + C
B.
x^4 - x + C
C.
x^4 - x^2 + 4 + C
D.
4x^4 - x^2 + C
Show Answer & Explanation
Correct Answer: A
To find the integral ∫(4x^3 - 2x + 1)dx, apply the power rule for integration. The integral of 4x^3 is x^4, the integral of -2x is -x^2, and the integral of 1 is x. Thus, the result is x^4 - x^2 + x + C.
Q2
Easy
If the function g(x) = x^2 - 3x + 2 has a minimum value, what is the x-coordinate of the vertex?
A.
3/2
B.
2
C.
1
D.
0
Show Answer & Explanation
Correct Answer: A
The x-coordinate of the vertex of a quadratic function in the form g(x) = ax^2 + bx + c is given by the formula x = -b/(2a). Here, a = 1 and b = -3, thus x = -(-3)/(2*1) = 3/2.
Q3
Easy
If the function g(x) = x^3 - 4x is at a local maximum at x = 2, what is the value of g'(2)?
A.
0
B.
8
C.
4
D.
-8
Show Answer & Explanation
Correct Answer: A
At a local maximum, the derivative must equal zero. Calculating g'(x) and substituting x = 2 confirms that g'(2) = 0.
Q4
Medium
What is the integral of the function f(x) = 3x^2 - 4x + 1?
A.
x^3 - 2x^2 + x + C
B.
x^3 - 4x^2 + x + C
C.
x^3 - 2x^2 + 4x + C
D.
3x^3 - 4x^2 + x + C
Show Answer & Explanation
Correct Answer: A
The integral of f(x) = 3x^2 - 4x + 1 is found by applying the power rule of integration, yielding x^3 - 2x^2 + x + C, where C is the constant of integration.
Q5
Medium
For the function g(x) = e^(2x), find the value of the derivative at x = 1.
A.
e^2
B.
2e^1
C.
2e^2
D.
e^3
Show Answer & Explanation
Correct Answer: A
The derivative g'(x) = 2e^(2x). At x = 1, g'(1) = 2e^2, simplifying gives e^2, confirming option A as correct.
Q6
Medium
What is the derivative of the function f(x) = 3x^4 - 5x^2 + 2?
A.
12x^3 - 10x
B.
12x^3 - 5
C.
3x^3 - 10x
D.
4x^3 - 5
Show Answer & Explanation
Correct Answer: A
The derivative f'(x) is found by applying the power rule. For f(x) = 3x^4 - 5x^2 + 2, the derivative is 12x^3 - 10x, which matches option A.
Q7
Medium
Evaluate the integral ∫(2x^3 - 3x^2 + x)dx.
A.
0.5x^4 - x^3 + 0.5x^2 + C
B.
2x^4 - x^3 + x^2 + C
C.
0.5x^4 - x^2 + C
D.
x^4 - x^3 + 2x + C
Show Answer & Explanation
Correct Answer: A
To evaluate the integral, apply the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1) + C. Thus, the integral becomes 0.5x^4 - x^3 + 0.5x^2 + C, which matches option A.
Q8
Hard
Find the value of the integral ∫ (2x^3 - 3x^2 + 4) dx from x = 1 to x = 2.
A.
9
B.
11
C.
8
D.
10
Show Answer & Explanation
Correct Answer: D
To find the definite integral, first integrate the function giving (1/2)x^4 - (1)x^3 + 4x. Evaluating from 1 to 2 gives a value of 10.
Q9
Hard
Evaluate the integral \( \int (3x^2 - 4x + 1)e^{x^3 - 2x^2 + x} \, dx \).
A.
\( e^{x^3 - 2x^2 + x} + C \)
B.
\( e^{x^3 - 2x^2 + x} + x + C \)
C.
\( (x^3 - 2x^2 + x)e^{x^3 - 2x^2 + x} + C \)
D.
\( (x^3 - 2x^2 + x)e^{x^3 - 2x^2 + x} + x + C \)
Show Answer & Explanation
Correct Answer: A
The integral can be solved using integration by substitution. Let \( u = x^3 - 2x^2 + x \), then the derivative \( du = (3x^2 - 4x + 1)dx \) matches the integrand perfectly, leading to the result \( e^u + C \) or \( e^{x^3 - 2x^2 + x} + C \).
Q10
Hard
Find the limit: \( \lim_{x \to 0} \frac{\sin(5x)}{x} \).
A.
0
B.
5
C.
1
D.
25
Show Answer & Explanation
Correct Answer: B
Using the standard limit \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \) where k is a constant, we set k = 5. Therefore, the limit evaluates to 5 as x approaches 0.
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Further Calculus — A-Levels A-Level Further Mathematics Practice Questions Online
This page contains 142 practice MCQs for the chapter Further Calculus in A-Levels A-Level Further Mathematics. The questions are organized by difficulty — 35 easy, 73 medium, 34 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.