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7.6D Algebraic Fractions: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Algebraic Fractions: Fluency and Exam Drill

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This worksheet opens with short fluency questions on cancelling common factors, adding, subtracting, multiplying and dividing algebraic fractions, before moving into exam-style questions that combine these skills with solving fractional equations, showing given identities and a real-life journey-time problem. It finishes with two grade-9 items: a compound fraction and a solve-and-reject question that tests whether a solution is actually valid once excluded values are considered.

Simplify fully (9x²)/(12x)

(Total for Question 1 is 1 mark)

Simplify fully (5x + 20)/(x + 4)

(Total for Question 2 is 2 marks)

Simplify fully (x² - 5x)/(3x)

(Total for Question 3 is 2 marks)

Simplify fully (x² - 16)/(x² + 4x)

(Total for Question 4 is 2 marks)

Write x/4 + x/6 as a single fraction in its simplest form.

(Total for Question 5 is 2 marks)

Write (x + 1)/3 - (x - 2)/5 as a single fraction in its simplest form.

(Total for Question 6 is 3 marks)

Simplify fully (2x/5) * (15/(4x²))

(Total for Question 7 is 2 marks)

Simplify fully (x²/6) divided by (x/3)

(Total for Question 8 is 2 marks)

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

(Total for Question 9 is 3 marks)

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

(Total for Question 10 is 3 marks)

Write 5/(x - 2) - 2/(x + 3) as a single fraction in its simplest form.

(Total for Question 11 is 3 marks)

Solve x/4 + x/6 = 10

(Total for Question 12 is 3 marks)

Solve 6/(x + 2) = 3

(Total for Question 13 is 3 marks)

Simplify fully (x² - x - 12)/(x² - 16)

(Total for Question 14 is 3 marks)

Solve (x + 3)/4 + (x - 2)/6 = 5/2

(Total for Question 15 is 3 marks)

Show that 3/(x - 1) + 4/(x + 2) = (7x + 2)/[(x - 1)(x + 2)], stating any value(s) of x for which the expression is not defined.

(Total for Question 16 is 4 marks)

Solve 5/(x + 1) = x - 3, giving all solutions.

(Total for Question 17 is 4 marks)

Priya drives 50 km from her home to a conference at an average speed of x km/h. Roadworks on the return journey reduce her average speed to (x - 5) km/h, and the return journey takes 30 minutes longer than the outward journey.

(a)Show that x² - 5x - 500 = 0.(3)

(b)Solve the equation to find the value of x, given that x is greater than 5.(3)

(Total for Question 18 is 6 marks)

A student is asked to simplify (x + 5)/(x² - 25) fully. The student writes: (x + 5)/(x² - 25) = 1/(x - 25).

(a)Identify the mistake the student has made.(1)

(b)Write down the correct simplified fraction, showing your working.(2)

(Total for Question 19 is 3 marks)

Simplify fully [(x² - 4)/(x + 3)] * [(2x + 6)/(x² - 5x + 6)]

(Total for Question 20 is 3 marks)

Consider the expression (3x² - 12)/(x² - x - 6).

(a)Simplify the expression fully.(3)

(b)Hence, or otherwise, find the value of the expression when x = 10.(2)

(Total for Question 21 is 5 marks)

Simplify fully [1/(x + 2) - 1/(x - 2)] divided by [1/(x + 2) + 1/(x - 2)]

(Total for Question 22 is 4 marks)

Solve 3/(x - 3) = 1 + 18/(x² - 9), stating clearly why one of your solutions must be rejected.

(Total for Question 23 is 4 marks)

Mark scheme · 7.6D Algebraic Fractions: Fluency and Exam Drill

Question 1

B1 3x/4 oe cao
Answer: 3x/4

Question 2

M1 factorises numerator: 5(x + 4)
A1 5 cao
Answer: 5

Question 3

M1 factorises numerator: x(x - 5)
A1 (x - 5)/3 oe cao
Answer: (x - 5)/3

Question 4

M1 factorises numerator (x - 4)(x + 4) and denominator x(x + 4)
A1 (x - 4)/x oe cao
Answer: (x - 4)/x

Question 5

M1 common denominator 12 used, e.g. 3x/12 + 2x/12
A1 5x/12 oe cao
Answer: 5x/12

Question 6

M1 common denominator 15 used
M1 numerator 5(x + 1) - 3(x - 2) formed and expanded
A1 (2x + 11)/15 oe cao
Answer: (2x + 11)/15

Question 7

M1 multiplies numerators and denominators: (2x * 15)/(5 * 4x²)
A1 3/(2x) oe cao
Answer: 3/(2x)

Question 8

M1 multiplies by the reciprocal: (x²/6) * (3/x)
A1 x/2 oe cao
Answer: x/2

Question 9

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

Question 10

M1 common denominator x(x + 1) used
M1 numerator 4(x + 1) + 3x formed and expanded
A1 (7x + 4)/(x² + x) oe cao
Answer: (7x + 4)/(x² + x)

Question 11

M1 common denominator (x - 2)(x + 3) used
M1 numerator 5(x + 3) - 2(x - 2) formed and expanded
A1 (3x + 19)/(x² + x - 6) oe cao
Answer: (3x + 19)/(x² + x - 6)

Question 12

M1 multiplies every term by the common denominator 12: 3x + 2x = 120
M1 collects to 5x = 120
A1 x = 24 cao
Answer: x = 24

Question 13

M1 multiplies both sides by (x + 2): 6 = 3(x + 2)
M1 expands and rearranges: 3x = 0
A1 x = 0 cao
Answer: x = 0

Question 14

M1 factorises numerator: (x - 4)(x + 3)
M1 factorises denominator: (x - 4)(x + 4)
A1 (x + 3)/(x + 4) oe cao
Answer: (x + 3)/(x + 4)

Question 15

M1 multiplies every term by the common denominator 12: 3(x + 3) + 2(x - 2) = 30
M1 expands and collects: 5x + 5 = 30
A1 x = 5 cao
Answer: x = 5

Question 16

M1 common denominator (x - 1)(x + 2) used
M1 numerator 3(x + 2) + 4(x - 1) expanded correctly
A1 (7x + 2)/[(x - 1)(x + 2)] cso, answer printed
B1 states x not equal to 1 and x not equal to -2
Answer: (7x + 2)/[(x - 1)(x + 2)] (shown); x not equal to 1 or -2

Question 17

M1 cross multiplies: 5 = (x - 3)(x + 1)
M1 expands and rearranges to x² - 2x - 8 = 0
M1 factorises (x - 4)(x + 2) = 0
A1 x = 4 and x = -2, both values required
Answer: x = 4 or x = -2

Question 18

(a) M1 forms time expressions 50/x (outward) and 50/(x - 5) (return)
(a) M1 sets up 50/(x - 5) - 50/x = 0.5 and multiplies through by x(x - 5)
(a) A1 x² - 5x - 500 = 0 cso, answer printed
(a) Answer: x² - 5x - 500 = 0 (shown)
(b) M1 factorises (x - 25)(x + 20) = 0
(b) A1 x = 25 and x = -20 found
(b) A1 rejects x = -20 (speed must be positive/x > 5); x = 25 km/h cao, with units
(b) Answer: x = 25 km/h

Question 19

(a) B1 identifies that the student has not factorised x² - 25 as (x - 5)(x + 5) before cancelling, instead incorrectly cancelling the 5 in the numerator with the 25 in the denominator, oe
(a) Answer: The student has not factorised x² - 25 as (x - 5)(x + 5); the 5 and 25 cannot be cancelled directly.
(b) M1 factorises x² - 25 = (x - 5)(x + 5)
(b) A1 1/(x - 5) oe cao
(b) Answer: 1/(x - 5)

Question 20

M1 factorises x² - 4 = (x - 2)(x + 2), 2x + 6 = 2(x + 3) and x² - 5x + 6 = (x - 2)(x - 3)
M1 cancels common factors (x + 3) and (x - 2)
A1 2(x + 2)/(x - 3) oe cao
Answer: 2(x + 2)/(x - 3)

Question 21

(a) M1 factorises numerator: 3(x - 2)(x + 2)
(a) M1 factorises denominator: (x - 3)(x + 2)
(a) A1 3(x - 2)/(x - 3) oe cao
(a) Answer: 3(x - 2)/(x - 3)
(b) M1 substitutes x = 10 into the simplified expression (or the original), e.g. 3(8)/7
(b) A1 24/7 oe cao
(b) Answer: 24/7

Question 22

M1 combines the numerator over a common denominator (x + 2)(x - 2) to get -4/(x² - 4)
M1 combines the denominator over a common denominator (x + 2)(x - 2) to get 2x/(x² - 4)
M1 divides the two fractions (multiplies by the reciprocal), cancelling (x² - 4)
A1 -2/x oe cao
Answer: -2/x

Question 23

M1 multiplies every term by (x² - 9) = (x - 3)(x + 3) to clear denominators
M1 expands and rearranges to x² - 3x = 0
M1 factorises x(x - 3) = 0
A1 x = 0 only cao, with x = 3 correctly rejected because it makes (x - 3) and (x² - 9) equal to zero, so the original equation is not defined there
Answer: x = 0 (x = 3 is rejected as the equation is undefined there)

Algebraic Fractions: Fluency and Exam Drill - Worksheets, Questions and Revision