Surds - Worksheets, Questions and Revision

20 original exam-style questions - 6 pages of questions with a full mark scheme - free printable PDF.

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7.1 Surds

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Work out the value of each of the following.

(a)√16(1)

(b)√3 x √3(1)

(c)√2 x √8(1)

(Total for Question 1 is 3 marks)

Simplify each surd fully.

(a)√28(1)

(b)√72(1)

(c)√200(1)

(d)√45(1)

(Total for Question 2 is 4 marks)

Simplify fully, showing your working: 3sqrt(2) + √50 - √8

(Total for Question 3 is 3 marks)

Expand and simplify each expression.

(a)√3(5 + √3)(2)

(b)√5(√5 - 2sqrt(2))(2)

(Total for Question 4 is 4 marks)

Expand and simplify (2 + √3)(4 - √3)

(Total for Question 5 is 4 marks)

Show that (√2 + 1)² = 3 + 2sqrt(2)

(Total for Question 6 is 3 marks)

Rationalise the denominator: 5/√2, giving your answer in its simplest form.

(Total for Question 7 is 2 marks)

Rationalise the denominator and simplify fully: 12/√8

(Total for Question 8 is 3 marks)

The diagram shows right-angled triangle ABC, with the right angle at B. AB = √18 cm and BC = √2 cm. Find 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 9 is 3 marks)

A rectangle has length (3 + √2) cm and width (3 - √2) cm. Show that the area of the rectangle is 7 cm².

(Total for Question 10 is 3 marks)

Solve algebraically 2x² = 24, giving your solutions in the form ± ksqrt(n), where k and n are integers.

(Total for Question 11 is 3 marks)

Using exact trigonometric values, show that sin(60) x cos(30) = 3/4

(Total for Question 12 is 3 marks)

Simplify fully √48 - √12 + √3

(Total for Question 13 is 3 marks)

Expand and simplify (√5 + 2)(√5 - 3)

(Total for Question 14 is 4 marks)

A rectangle has length √20 cm and width √5 cm. Find the perimeter of the rectangle, giving your answer in the form ksqrt(n) cm.

(Total for Question 15 is 3 marks)

A square has an area of 45 cm². Find the length of one side of the square, giving your answer in simplest surd form.

(Total for Question 16 is 2 marks)

Rationalise the denominator and simplify fully: 4/(3 + √2)

(Total for Question 17 is 4 marks)

Simplify (5 + √3)/(√3 - 1) fully, writing your answer in the form a + bsqrt(3), where a and b are integers.

(Total for Question 18 is 4 marks)

The area of a rectangle is (7 + 3sqrt(5)) cm². The width of the rectangle is (2 + √5) cm. Find the length of the rectangle, giving your answer in the form a + bsqrt(5), where a and b are integers.

(Total for Question 19 is 5 marks)

Show that (4 + √2)/(3 - √2) can be written in the form a + bsqrt(2), where a and b are integers, and find the values of a and b.

(Total for Question 20 is 5 marks)

Mark scheme · 7.1 Surds

Question 1

(a) B1 4 cao
(a) Answer: 4
(b) B1 3 cao
(b) Answer: 3
(c) B1 4 cao
(c) Answer: 4

Question 2

(a) B1 2sqrt(7) oe cao
(a) Answer: 2sqrt(7)
(b) B1 6sqrt(2) oe cao
(b) Answer: 6sqrt(2)
(c) B1 10sqrt(2) oe cao
(c) Answer: 10sqrt(2)
(d) B1 3sqrt(5) oe cao
(d) Answer: 3sqrt(5)

Question 3

M1 √50 = 5sqrt(2)
M1 √8 = 2sqrt(2)
A1 6sqrt(2) cao
Answer: 6sqrt(2)

Question 4

(a) M1 5sqrt(3) + √3 x √3 obtained
(a) A1 5sqrt(3) + 3 oe cao
(a) Answer: 5sqrt(3) + 3
(b) M1 √5 x √5 - √5 x 2sqrt(2) obtained
(b) A1 5 - 2sqrt(10) oe cao
(b) Answer: 5 - 2sqrt(10)

Question 5

M1 correct expansion: 8 - 2sqrt(3) + 4sqrt(3) - 3 (or equivalent four terms)
M1 surd terms combined: -2sqrt(3) + 4sqrt(3) = 2sqrt(3)
A1 constant terms combined: 8 - 3 = 5
A1 5 + 2sqrt(3) oe cao
Answer: 5 + 2sqrt(3)

Question 6

M1 (√2+1)(√2+1) = 2 + √2 + √2 + 1 (or equivalent expansion)
M1 combines to 3 + 2sqrt(2)
A1 cso - fully shown, matches printed answer
Answer: 3 + 2sqrt(2) (given)

Question 7

M1 multiplies numerator and denominator by √2
A1 5sqrt(2)/2 oe cao
Answer: 5sqrt(2)/2

Question 8

M1 √8 simplified to 2sqrt(2) (or 12/√8 x √8/√8 = 12sqrt(8)/8)
M1 further simplification (e.g. 6/√2 or 24sqrt(2)/8)
A1 3sqrt(2) cao
Answer: 3sqrt(2)

Question 9

M1 AC² = 18 + 2 (= 20)
M1 AC = √20 oe
A1 AC = 2sqrt(5) cm cao
Answer: AC = 2sqrt(5) cm

Question 10

M1 (3+√2)(3-√2) = 9 - 3sqrt(2) + 3sqrt(2) - 2 (or equivalent expansion)
M1 the 3sqrt(2) terms are recognised to cancel to zero
A1 cso - 9 - 2 = 7 cm² shown
Answer: 7 cm² (given)

Question 11

M1 x² = 12
M1 x = √12 (or -√12) seen
A1 x = ±2sqrt(3) cao (both values)
Answer: x = 2sqrt(3) or x = -2sqrt(3)

Question 12

B1 sin(60) = √3/2 stated
B1 cos(30) = √3/2 stated
A1 cso - √3/2 x √3/2 = 3/4 shown
Answer: 3/4 (given)

Question 13

M1 √48 = 4sqrt(3)
M1 √12 = 2sqrt(3)
A1 3sqrt(3) cao
Answer: 3sqrt(3)

Question 14

M1 correct expansion: 5 - 3sqrt(5) + 2sqrt(5) - 6 (or equivalent four terms)
M1 surd terms combined to -√5
A1 constant terms combined to -1
A1 -1 - √5 oe cao
Answer: -1 - √5

Question 15

M1 √20 = 2sqrt(5)
M1 perimeter = 2(2sqrt(5) + √5) oe
A1 6sqrt(5) cm cao
Answer: 6sqrt(5) cm

Question 16

M1 side = √45 seen
A1 3sqrt(5) cm cao
Answer: 3sqrt(5) cm

Question 17

M1 multiplies numerator and denominator by (3 - √2)
M1 denominator simplifies to 9 - 2 = 7
A1 numerator simplifies to 12 - 4sqrt(2)
A1 (12 - 4sqrt(2))/7 oe cao
Answer: (12 - 4sqrt(2))/7

Question 18

M1 multiplies numerator and denominator by (√3 + 1)
M1 denominator simplifies to 3 - 1 = 2
A1 numerator simplifies to 8 + 6sqrt(3)
A1 4 + 3sqrt(3) cao (a = 4, b = 3)
Answer: 4 + 3sqrt(3) (a = 4, b = 3)

Question 19

M1 length = (7 + 3sqrt(5))/(2 + √5) set up
M1 multiplies numerator and denominator by (2 - √5)
M1 denominator simplifies to 4 - 5 = -1
A1 numerator simplifies to -1 - √5
A1 1 + √5 cm cao (a = 1, b = 1)
Answer: length = 1 + √5 cm (a = 1, b = 1)

Question 20

M1 multiplies numerator and denominator by (3 + √2)
M1 denominator simplifies to 9 - 2 = 7
A1 numerator simplifies to 14 + 7sqrt(2)
A1 cso - simplifies to 2 + √2
A1 a = 2 and b = 1 stated
Answer: 2 + √2 (a = 2, b = 1)

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Surds - Worksheets, Questions and Revision

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7.1 Surds

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Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

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Question 2

Question 3

Question 4

Question 5

Question 6

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Question 10

Question 11

Question 12

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Question 15

Question 16

Question 17

Question 18

Question 19

Question 20