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7.6 Algebraic Fractions

EDEXCEL 1MA1 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Simplify fully: (6x^2y^3) / (9xy²)

(Total for Question 1 is 2 marks)

Simplify fully: (4x + 12) / (x + 3)

(Total for Question 2 is 2 marks)

Simplify fully: (x² + 2x - 15) / (x - 3)

(Total for Question 3 is 3 marks)

Simplify fully: [(x² - 4)/(x + 1)] * [(3x + 3)/(x + 2)]

(Total for Question 4 is 3 marks)

Simplify fully: (x² - 9)/(2x) divided by (x + 3)/(4x²)

(Total for Question 5 is 3 marks)

Write x/3 + (x - 1)/4 as a single fraction in its simplest form.

(Total for Question 6 is 3 marks)

Write 3/(x + 1) + 2/(x - 2) as a single fraction in its simplest form.

(Total for Question 7 is 4 marks)

Simplify fully: (x² - 9) / (x² + x - 6)

(Total for Question 8 is 3 marks)

Simplify fully: (2x + 1)/(x + 3) - (x - 2)/(x + 3)

(Total for Question 9 is 2 marks)

Solve (x + 1)/3 + (x - 2)/4 = 2. Give your answer as a fraction in its simplest form.

(Total for Question 10 is 4 marks)

Chloe bakes two cakes using a 4 kg bag of flour. The first cake uses (x + 2)/3 kg of flour and the second cake uses (2x - 1)/5 kg of flour. All of the flour in the bag is used between the two cakes.

(a)Form an equation in x for the total mass of flour used, in kg.(1)

(b)Solve your equation to find the value of x.(3)

(c)Hence find the mass of flour used in the first cake, giving your answer in kg correct to 2 decimal places.(2)

(Total for Question 11 is 6 marks)

Solve 5/(x + 2) = 3/(x - 1)

(Total for Question 12 is 4 marks)

Solve 3/x + 2/(x + 1) = 1. Give your solutions correct to 2 decimal places.

(Total for Question 13 is 5 marks)

Show that x/(x - 2) - (x - 3)/(x + 1) = (6x - 6) / [(x - 2)(x + 1)], where x ≠ 2 and x ≠ -1.

(Total for Question 14 is 4 marks)

Write 2/x + 3/(x + 2) - 1/(x - 1) as a single fraction in its simplest form.

(Total for Question 15 is 4 marks)

Solve (2x - 1)/(x - 3) - (x + 1)/(x + 2) = 1. Give your solution correct to 2 decimal places.

(Total for Question 16 is 5 marks)

Simplify fully: [1/x + 1/y] divided by [1/x - 1/y]. Give your answer as a single fraction in terms of x and y.

(Total for Question 17 is 4 marks)

Show that 4/(x² - 1) + 3/(x - 1) = (3x + 7) / [(x - 1)(x + 1)], where x ≠ 1 and x ≠ -1.

(Total for Question 18 is 4 marks)

Mark scheme · 7.6 Algebraic Fractions

Question 1

M1 cancels a common numerical or algebraic factor, e.g. 6/9 or x²/x or y³/y² seen
A1 2xy/3 oe cao
Answer: 2xy/3

Question 2

M1 factorises the numerator as 4(x + 3)
A1 4 cao (x ≠ -3)
Answer: 4

Question 3

M1 attempts to factorise x² + 2x - 15 as (x + 5)(x - 3)
A1 correct factorisation (x + 5)(x - 3) seen
A1 x + 5 oe cao (x ≠ 3)
Answer: x + 5

Question 4

M1 factorises x² - 4 as (x - 2)(x + 2) and 3x + 3 as 3(x + 1)
M1 cancels the common factors (x + 1) and (x + 2)
A1 3x - 6 oe (3(x - 2)) cao
Answer: 3x - 6

Question 5

M1 multiplies by the reciprocal of the second fraction: (x²-9)/(2x) * 4x²/(x+3)
M1 factorises x² - 9 as (x - 3)(x + 3) and cancels the common factor (x + 3)
A1 2x² - 6x oe (2x(x - 3)) cao
Answer: 2x² - 6x

Question 6

M1 uses a common denominator of 12
M1 correct numerator 4x + 3(x - 1)
A1 (7x - 3)/12 oe cao
Answer: (7x - 3)/12

Question 7

M1 uses the common denominator (x + 1)(x - 2)
M1 correct numerator 3(x - 2) + 2(x + 1)
A1 numerator simplified to 5x - 4 oe
A1 (5x - 4)/(x² - x - 2) oe (denominator expanded or left factorised)
Answer: (5x - 4)/(x² - x - 2)

Question 8

M1 factorises the numerator as (x - 3)(x + 3)
M1 factorises the denominator as (x + 3)(x - 2)
A1 (x - 3)/(x - 2) oe cao (x ≠ -3, x ≠ 2)
Answer: (x - 3)/(x - 2)

Question 9

M1 combines the fractions over the common denominator (x + 3): (2x + 1) - (x - 2) all over (x + 3)
A1 1 cao (x ≠ -3)
Answer: 1

Question 10

M1 multiplies every term by 12 (the LCM of 3 and 4)
M1 expands both brackets correctly: 4(x + 1) + 3(x - 2)
M1 collects terms to reach 7x - 2 = 24
A1 x = 26/7 oe (awrt 3.71)
Answer: x = 26/7 (awrt 3.71)

Question 11

(a) B1 correct equation (x + 2)/3 + (2x - 1)/5 = 4 oe
(a) Answer: (x + 2)/3 + (2x - 1)/5 = 4
(b) M1 multiplies every term by 15 (the LCM of 3 and 5)
(b) M1 expands and collects terms to reach 11x + 7 = 60 oe
(b) A1 x = 53/11 oe (awrt 4.82)
(b) Answer: x = 53/11 (awrt 4.82)
(c) M1 substitutes their value of x into (x + 2)/3
(c) A1 awrt 2.27 kg ft
(c) Answer: 2.27 kg (awrt)

Question 12

M1 cross multiplies: 5(x - 1) = 3(x + 2)
M1 expands both sides correctly: 5x - 5 = 3x + 6
M1 collects terms to reach 2x = 11
A1 x = 5.5 oe (11/2)
Answer: x = 5.5

Question 13

M1 multiplies through by x(x + 1)
M1 expands to 3(x + 1) + 2x = x(x + 1)
M1 rearranges to x² - 4x - 3 = 0
M1 uses the quadratic formula (or completed square) with a = 1, b = -4, c = -3
A1 x = 4.65 or x = -0.65 (both values, awrt 2 dp)
Answer: x = 4.65 or x = -0.65 (awrt)

Question 14

M1 writes both fractions over the common denominator (x - 2)(x + 1)
M1 expands x(x + 1) = x² + x and (x - 3)(x - 2) = x² - 5x + 6
A1 combines to give a numerator of 6x - 6
A1 cso: fully correct working leading to (6x - 6)/[(x - 2)(x + 1)]
Answer: (6x - 6)/[(x - 2)(x + 1)] (shown)

Question 15

M1 identifies the common denominator x(x + 2)(x - 1)
M1 writes each numerator over the common denominator correctly
A1 numerator simplified to 4x² - 3x - 4
A1 (4x² - 3x - 4)/[x(x + 2)(x - 1)] oe, fully correct single fraction
Answer: (4x² - 3x - 4)/[x(x + 2)(x - 1)]

Question 16

M1 multiplies both sides by (x - 3)(x + 2)
M1 expands (2x - 1)(x + 2) and (x + 1)(x - 3) correctly
M1 expands (x - 3)(x + 2) on the right-hand side and recognises the x² terms cancel
M1 rearranges the linear equation to 6x = -7
A1 x = -7/6 oe (awrt -1.17)
Answer: x = -7/6 (awrt -1.17)

Question 17

M1 writes the numerator as (y + x)/(xy)
M1 writes the denominator as (y - x)/(xy)
M1 divides the two fractions, cancelling the common factor xy
A1 (x + y)/(y - x) oe
Answer: (x + y)/(y - x)

Question 18

M1 factorises x² - 1 as (x - 1)(x + 1)
M1 writes 3/(x - 1) as 3(x + 1)/[(x - 1)(x + 1)]
A1 combines numerators to give 4 + 3(x + 1) = 3x + 7
A1 cso: fully correct working leading to (3x + 7)/[(x - 1)(x + 1)]
Answer: (3x + 7)/[(x - 1)(x + 1)] (shown)

Algebraic Fractions - Worksheets, Questions and Revision