A-Level Mathematics Course | Edexcel & AQA Board | Complete Fall Semester 2025/2026 Updated Study Pack | 270+ Verified Exam & Application Questions with Correct Answers & Detailed Rationales | Comprehensive Pure, Statistics & Mechanics A-Level Math Exam P

A-Level Mathematics Course | Edexcel & AQA Board |
Complete Fall Semester 2025/2026 Updated Study
Pack | 270+ Verified Exam & Application Questions with
Correct Answers & Detailed Rationales |
Comprehensive Pure, Statistics & Mechanics A-Level
Math Exam Prep Guide

Question 1
What is the quadratic formula used to find the roots of a quadratic equation?
−𝑏±√𝑏 2 −4𝑎𝑐
Correct Answer: 𝑥 = 2𝑎
Rationale: This formula provides the solutions for any quadratic equation of the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.


Question 2

How do you differentiate the function 𝑓(𝑥) = 𝑥 3 − 5𝑥 2 + 6𝑥?
Correct Answer: 𝑓 ′ (𝑥) = 3𝑥 2 − 10𝑥 + 6
Rationale: Differentiation applies the power rule to each term of the polynomial.


Question 3

What is the integral of 𝑓(𝑥) = 3𝑥 2 ?

Correct Answer: ∫ 3𝑥 2 𝑑𝑥 = 𝑥 3 + 𝐶
𝑥 𝑛+1
Rationale: The integral of 𝑥 𝑛 is + 𝐶.
𝑛+1




Question 4
Define the term "asymptote."
Correct Answer: An asymptote is a line that a graph approaches but never touches or
intersects.
Rationale: Asymptotes help describe the behavior of functions at extreme values.


Question 5
What is the formula for the area of a circle?
Correct Answer: 𝐴 = 𝜋𝑟 2
Rationale: This formula calculates the area enclosed by a circle with radius 𝑟.

,Question 6

Solve the system of equations: 2𝑥 + 3𝑦 = 6and 𝑥 − 𝑦 = 2.
Correct Answer: 𝑥 = 3, 𝑦 = 0
Rationale: Solving the system gives the intersection point of the two lines.


Question 7

What is the derivative of 𝑓(𝑥) = sin⁡(𝑥)?
Correct Answer: 𝑓 ′ (𝑥) = cos⁡(𝑥)
Rationale: The derivative of the sine function is the cosine function.


Question 8

What is the formula for the sum of the first 𝑛natural numbers?
𝑛(𝑛+1)
Correct Answer: 𝑆𝑛 = 2
Rationale: This formula calculates the total when adding up natural numbers from 1 to
𝑛.


Question 9
sin⁡(𝑥)
Find the limit of lim⁡𝑥→0 .
𝑥

Correct Answer: 1
Rationale: This limit is fundamental in calculus and is often used in the derivation of
other limits.


Question 10
What is the equation of a line in slope-intercept form?
Correct Answer: 𝑦 = 𝑚𝑥 + 𝑏
Rationale: Here, 𝑚is the slope and 𝑏is the y-intercept of the line.


Question 11
What is the formula for the volume of a cylinder?

,Correct Answer: 𝑉 = 𝜋𝑟 2 ℎ
Rationale: This formula calculates the volume using the radius 𝑟and height ℎ.


Question 12

Differentiate the function 𝑓(𝑥) = 𝑒 𝑥 .
Correct Answer: 𝑓 ′ (𝑥) = 𝑒 𝑥
Rationale: The derivative of the exponential function 𝑒 𝑥 is itself.


Question 13
What is the relationship between the roots of a quadratic equation and its
coefficients?
𝑏 𝑐
Correct Answer: The sum of the roots is − and the product of the roots is .
𝑎 𝑎
Rationale: These relationships come from Vieta's formulas.


Question 14

What is the integral of cos⁡(𝑥)?

Correct Answer: ∫ cos⁡(𝑥) 𝑑𝑥 = sin⁡(𝑥) + 𝐶
Rationale: The integral of cosine is sine plus a constant of integration.


Question 15
What does it mean for a function to be continuous at a point?

Correct Answer: A function is continuous at a point if the limit as 𝑥approaches that
point equals the function's value at that point.
Rationale: Continuity ensures there are no breaks or jumps in the graph of the function.


Question 16

Find the roots of the equation 𝑥 2 − 4𝑥 + 3 = 0.
Correct Answer: 𝑥 = 1,3
Rationale: Factoring or using the quadratic formula reveals the solutions.


Question 17

What is the derivative of 𝑓(𝑥) = ln⁡(𝑥)?

, 1
Correct Answer: 𝑓 ′ (𝑥) = 𝑥
Rationale: The derivative of the natural logarithm function is the reciprocal of 𝑥.


Question 18
What is the area of a triangle with a base of 6 and a height of 4?
1
Correct Answer: 𝐴 = 2 × 6 × 4 = 12
Rationale: The area formula for a triangle uses base and height.


Question 19
What is the quadratic equation in standard form?

Correct Answer: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Rationale: This is the standard representation of a quadratic equation.


Question 20
What is the formula for the circumference of a circle?
Correct Answer: 𝐶 = 2𝜋𝑟
Rationale: This formula calculates the distance around a circle using its radius 𝑟.


Question 21

What is the integral of 𝑥 𝑛 ?
𝑥 𝑛+1
Correct Answer: ∫ 𝑥 𝑛 𝑑𝑥 = + 𝐶for 𝑛 ≠ −1
𝑛+1
Rationale: This is the power rule for integration.


Question 22
What does the Fundamental Theorem of Calculus state?
Correct Answer: It states that differentiation and integration are inverse processes.
Rationale: This theorem connects the two major concepts in calculus.


Question 23
What is the derivative of 𝑓(𝑥) = tan⁡(𝑥)?