Surds: Fluency and Exam Drill - Worksheets, Questions and Revision

24 original exam-style questions - 5 pages of questions with a full mark scheme - free printable PDF.

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7.1D Surds: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 65 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Surds: Fluency and Exam Drill

Original content written for Revision Library. Styled on Edexcel 1MA1 GCSE Mathematics conventions.

This is a non-calculator practice set for Edexcel 1MA1 topic 7.1 (Surds). Work through the fluency questions first to secure the core skills of simplifying, combining and rationalising surds, then attempt the exam-style questions, finishing with two grade 9 stretch questions. Show all your working clearly, since method marks are available even when a final answer is not fully correct. Give all answers in their simplest surd form unless told otherwise.

Simplify √18 fully.

(Total for Question 1 is 1 mark)

Simplify √50 fully.

(Total for Question 2 is 1 mark)

Simplify √75 fully.

(Total for Question 3 is 1 mark)

Simplify √32 fully.

(Total for Question 4 is 1 mark)

Simplify √80 fully.

(Total for Question 5 is 1 mark)

Work out √3 * √12.

(Total for Question 6 is 1 mark)

Work out √20 * √5.

(Total for Question 7 is 1 mark)

Work out √48 / √3.

(Total for Question 8 is 1 mark)

Simplify 3sqrt(2) + 5sqrt(2).

(Total for Question 9 is 1 mark)

Simplify 7sqrt(5) - 2sqrt(5).

(Total for Question 10 is 1 mark)

Simplify √8 + √2 fully.

(Total for Question 11 is 2 marks)

Simplify √27 - √3 fully.

(Total for Question 12 is 2 marks)

Rationalise the denominator of 1/√5, giving your answer in its simplest form.

(Total for Question 13 is 2 marks)

Rationalise the denominator of 6/√3 fully.

(Total for Question 14 is 2 marks)

Expand and simplify (3 + √5)(5 - √5).

(Total for Question 15 is 3 marks)

Show that (6 + √7)(6 - √7) = 29.

(Total for Question 16 is 2 marks)

Rationalise the denominator of 1/(4 + √3), giving your answer in the form (a - √b)/c, where a, b and c are integers.

(Total for Question 17 is 3 marks)

A rectangular garden bed has length (6 + √11) m and width (6 - √11) m. Find the area of the garden bed, giving your answer as an integer.

(Total for Question 18 is 3 marks)

Simplify fully √90 + √40 - √10, showing each step of your working.

(Total for Question 19 is 3 marks)

In triangle ABC, the angle at B is 90 degrees. AB = √21 cm and BC = √54 cm. Work out the length of AC, giving your answer in the form ksqrt(n) cm, where k and n are integers.

Diagram NOT accurately drawn

(Total for Question 20 is 4 marks)

Show that (2 + √5)² = 9 + 4sqrt(5).

(Total for Question 21 is 3 marks)

A student writes down the statement √16 + √9 = √16 + 9.

(a)Show, by evaluating both sides, that the statement above is not correct.(2)

(b)By simplifying √20 + √5 and √20 + 5 separately, show that √a + √b is not equal to √a + b in general.(3)

(Total for Question 22 is 5 marks)

Show that √7 + 2sqrt(10) = √5 + √2.

(Total for Question 23 is 3 marks)

Simplify (5 + 2sqrt(2))/(3 - 2sqrt(2)) fully, giving your answer in the form p + qsqrt(2), where p and q are integers.

(Total for Question 24 is 4 marks)

Mark scheme · 7.1D Surds: Fluency and Exam Drill

Question 1

B1 3sqrt(2) oe cao
Answer: 3sqrt(2)

Question 2

B1 5sqrt(2) oe cao
Answer: 5sqrt(2)

Question 3

B1 5sqrt(3) oe cao
Answer: 5sqrt(3)

Question 4

B1 4sqrt(2) oe cao
Answer: 4sqrt(2)

Question 5

B1 4sqrt(5) oe cao
Answer: 4sqrt(5)

Question 6

B1 6 cao
Answer: 6

Question 7

B1 10 cao
Answer: 10

Question 8

B1 4 cao
Answer: 4

Question 9

B1 8sqrt(2) oe cao
Answer: 8sqrt(2)

Question 10

B1 5sqrt(5) oe cao
Answer: 5sqrt(5)

Question 11

M1 √8 = 2sqrt(2) identified
A1 3sqrt(2) oe cao
Answer: 3sqrt(2)

Question 12

M1 √27 = 3sqrt(3) identified
A1 2sqrt(3) oe cao
Answer: 2sqrt(3)

Question 13

M1 multiplies numerator and denominator by √5
A1 √5/5 oe cao
Answer: √5/5

Question 14

M1 multiplies numerator and denominator by √3, giving 6sqrt(3)/3
A1 2sqrt(3) cao
Answer: 2sqrt(3)

Question 15

M1 at least 3 of the 4 terms 15, -3sqrt(5), 5sqrt(5), -5 correct
A1 correct unsimplified expansion 15 - 3sqrt(5) + 5sqrt(5) - 5 (all 4 terms right)
A1 10 + 2sqrt(5) oe cao
Answer: 10 + 2sqrt(5)

Question 16

M1 correct expansion 36 - 6sqrt(7) + 6sqrt(7) - 7 (or 36 - 7)
A1 cso - 36 - 7 = 29 shown
Answer: 29 (given)

Question 17

M1 multiplies numerator and denominator by (4 - √3)
M1 denominator simplifies to 16 - 3 = 13
A1 (4 - √3)/13 oe cao
Answer: (4 - √3)/13

Question 18

M1 sets up area = (6 + √11)(6 - √11)
M1 36 - 11 evaluated (from 36 - 6sqrt(11) + 6sqrt(11) - 11 or difference of two squares)
A1 25 m² cao
Answer: 25 m²

Question 19

M1 √90 = 3sqrt(10)
M1 √40 = 2sqrt(10)
A1 4sqrt(10) oe cao
Answer: 4sqrt(10)

Question 20

M1 uses Pythagoras: AC² = AB² + BC²
M1 AC² = 21 + 54 (= 75)
M1 AC = √75 oe
A1 AC = 5sqrt(3) cm cao
Answer: AC = 5sqrt(3) cm

Question 21

M1 (2+√5)(2+√5) = 4 + 2sqrt(5) + 2sqrt(5) + 5 (or equivalent expansion)
A1 combines to 9 + 4sqrt(5)
A1 cso - fully shown, matches printed answer
Answer: 9 + 4sqrt(5) (given)

Question 22

(a) B1 left-hand side evaluated: √16 + √9 = 4 + 3 = 7
(a) B1 right-hand side evaluated: √16 + 9 = √25 = 5, and 7 is not equal to 5 so the statement is incorrect
(a) Answer: 7 is not equal to 5, so the statement is false
(b) M1 √20 = 2sqrt(5), so √20 + √5 = 3sqrt(5)
(b) A1 √20 + 5 = √25 = 5 evaluated
(b) A1 cso - since 3sqrt(5) is not equal to 5, confirms √a + √b is not equal to √a + b in general
(b) Answer: 3sqrt(5) is not equal to 5, confirming the general result

Question 23

M1 squares the right-hand side: (√5 + √2)² = 5 + 2sqrt(5)√2 + 2
A1 simplifies to 7 + 2sqrt(10), matching the expression under the root
A1 cso - since both sides are positive and (√5+√2)² = 7+2sqrt(10), concludes √7+2sqrt(10) = √5+√2
Answer: √5 + √2 (given)

Question 24

M1 multiplies numerator and denominator by (3 + 2sqrt(2))
M1 numerator simplifies to 15 + 10sqrt(2) + 6sqrt(2) + 8 (= 23 + 16sqrt(2))
A1 denominator simplifies to 9 - 8 = 1
A1 23 + 16sqrt(2) cao (p = 23, q = 16)
Answer: 23 + 16sqrt(2) (p = 23, q = 16)

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Surds: Fluency and Exam Drill - Worksheets, Questions and Revision

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7.1D Surds: Fluency and Exam Drill

Surds: Fluency and Exam Drill

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