Division Worksheet - A-Level - Intermediate
Division worksheet for A-Level / Vocational (Ages 16-18). Intermediate level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.
Advanced Division Techniques and Applications
A-Level Maths: Intermediate Division
Instructions: This worksheet focuses on division techniques and their applications in various mathematical contexts. You will encounter algebraic manipulation, problem-solving scenarios, and theoretical explorations. Ensure you show all workings clearly and use appropriate mathematical notation.
Algebraic Long Division
Perform algebraic long division on the following expressions. Show each step of your working and simplify your answers where possible.
1. Divide 2x^3 + 3x^2 - 5x + 6 by x - 2.
2. Divide 4x^4 - 8x^3 + x - 5 by 2x^2 - 1.
3. Divide x^3 - 6x^2 + 11x - 6 by x - 1.
Partial Fractions
Express the following rational functions as a sum of partial fractions. Show all steps and simplify your final answers.
1. \( \frac{3x + 5}{(x + 1)(x - 2)} \)
2. \( \frac{2x^2 + 7x + 3}{(x + 2)(x - 1)} \)
3. \( \frac{x^2 + 4x + 4}{(x - 1)^2(x + 3)} \)
Real-World Problem Solving
Solve the following division-related problems. Provide detailed solutions and justify your answers with appropriate reasoning.
1. A company produces 1200 units of a product in 5 days. If production is increased by 20%, how many units will be produced in 7 days?
2. A tank can be filled by two pipes in 3 hours and 4 hours respectively. How long will it take to fill the tank if both pipes are used simultaneously?
Division in Sequences and Series
Investigate the role of division in the following sequences and series. Provide full solutions and explanations.
1. Find the sum of the first 10 terms of the arithmetic series where the first term is 5 and the common difference is 3. Use division to find the average of these terms.
2. Determine the 15th term of the geometric sequence where the first term is 2 and the common ratio is 3.
Division in Calculus
Apply division techniques in solving the following calculus problems. Show all steps and provide clear justifications.
1. Differentiate the function \( f(x) = \frac{x^3 - 3x^2 + 4x - 5}{x - 1} \) using quotient rule.
2. Integrate the function \( \int \frac{2x^2 + 3x + 1}{x + 2} \, dx \) by first dividing the polynomial.
Proof and Reasoning
Prove the following statements using division. Provide a step-by-step logical argument.
1. Prove that if a number is divisible by 6, it is also divisible by 2 and 3.
2. Prove that the sum of two even numbers is always even using divisibility rules.
Activity 1: 2x^2 + 7x + 9 ; 2x^2 - 4x + 2 ; x^2 - 5x + 6
Activity 2: \( \frac{3}{x + 1} + \frac{5}{x - 2} \) ; \( \frac{2x + 3}{x + 2} + \frac{1}{x - 1} \) ; \( \frac{1}{(x - 1)} + \frac{3}{(x + 3)} \)
Activity 3: 2016 units ; 1.71 hours
Activity 4: Sum = 215 ; 14th term = 14348907
Activity 5: \( f'(x) = 3x^2 - 6x + 4 \) ; \( \int = x^2 + 3x + ln|x+2| + C \)
Activity 6: Proof using divisibility rules ; Proof using even number properties
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