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GCSE · Number: Surds

N1 Higher Surds: Simplifying, Multiplying and Dividing

AQA 8365 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Simplify each surd fully, giving your answer in the form ksqrt(n) where n has no square factors.

(a)√50(1)

(b)√72(1)

(c)√45(1)

(Total for Question 1 is 3 marks)

Simplify each surd fully, giving your answer in the form ksqrt(n) where n has no square factors.

(a)√98(1)

(b)√180(1)

(c)√300(1)

(Total for Question 2 is 3 marks)

Work out each product, giving your answer in its simplest form.

(a)√3 x √12(2)

(b)√5 x √20(2)

(Total for Question 3 is 6 marks)

Work out each quotient, giving your answer in its simplest form.

(a)√48 / √3(2)

(b)√72 / √8(2)

(Total for Question 4 is 6 marks)

Simplify each expression fully by collecting like surds.

(a)3sqrt(2) + 5sqrt(2) - √2(2)

(b)√12 + √27(2)

(c)√50 - √8 + √18(2)

(Total for Question 5 is 6 marks)

Expand and simplify each expression.

(a)√2(3 + √2)(2)

(b)(√3 + 1)(√3 - 1)(2)

(c)(2 + √5)(2 - √5)(2)

(Total for Question 6 is 6 marks)

Expand and simplify each expression, giving your answer in the form a + bsqrt(n).

(a)(√5 + 2)(√5 + 3)(3)

(b)(2sqrt(3) - 1)(√3 + 4)(3)

(c)(√7 + √2)(√7 - √2)(3)

(Total for Question 7 is 9 marks)

Rationalise the denominator of each expression, simplifying fully.

(a)1 / √3(2)

(b)5 / √2(2)

(Total for Question 8 is 6 marks)

Rationalise the denominator of each expression, simplifying fully.

(a)1 / (√5 + 2)(3)

(b)3 / (√7 - 2)(3)

(c)(√2 + 1) / (√2 - 1)(3)

(Total for Question 9 is 9 marks)

A right-angled triangle has legs of length √20 cm and √5 cm.

Diagram NOT accurately drawn

(a)Show that the hypotenuse of the triangle has length 5 cm.(3)

(b)Find the exact area of the triangle.(3)

(Total for Question 10 is 6 marks)

A rectangle has sides of length (3 + √2) cm and (3 - √2) cm.

Diagram NOT accurately drawn

(a)Calculate the area of the rectangle, giving your answer as an integer.(3)

(b)Find the perimeter of the rectangle.(2)

(Total for Question 11 is 5 marks)

Show that (2 + √3)² = 7 + 4sqrt(3).

(Total for Question 12 is 3 marks)

Given that x x √2 = √8 + √18, find the value of the integer x.

(Total for Question 13 is 4 marks)

Which of the following is equivalent to √75?

A) 3sqrt(5)
B) 5sqrt(3)
C) 15sqrt(5)
D) 25sqrt(3)

(Total for Question 14 is 1 mark)

A is the point (2sqrt(3), 1) and B is the point (√3, 4). Find the exact length of AB, giving your answer in the form ksqrt(3).

(Total for Question 15 is 3 marks)

Simplify (5 + √3) / (2 - √3) fully, giving your answer in the form a + bsqrt(3), where a and b are integers.

(Total for Question 16 is 4 marks)

Prove, using proof by contradiction, that √2 is an irrational number.

(Total for Question 17 is 5 marks)

Consider the expression (√x + 2) / (√x - 2), where x is greater than 4.

(a)Rationalise the denominator of the expression, giving your answer in terms of x.(3)

(b)Hence, or otherwise, find the value of the expression when x = 9.(3)

(Total for Question 18 is 6 marks)

A trapezium has parallel sides of length (4 + √5) cm and (4 - √5) cm, and height √5 cm.

Diagram NOT accurately drawn

(a)Find the exact area of the trapezium, giving your answer in the form ksqrt(5).(3)

(b)Given that each sloped side of the trapezium has length (√5 + 1) cm, find the exact perimeter of the trapezium.(2)

(Total for Question 19 is 5 marks)

Given that (a + bsqrt(2))(3 - √2) = 11 + √2, where a and b are integers, find the value of a and the value of b.

(Total for Question 20 is 4 marks)

Mark scheme · N1 Higher Surds: Simplifying, Multiplying and Dividing

Question 1

(a) B1 5sqrt(2) cao
(a) Answer: 5sqrt(2)
(b) B1 6sqrt(2) cao
(b) Answer: 6sqrt(2)
(c) B1 3sqrt(5) cao
(c) Answer: 3sqrt(5)

Question 2

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

Question 3

(a) M1 combines to √36 oe
(a) A1 6 cao
(a) Answer: 6
(b) M1 combines to √100 oe
(b) A1 10 cao
(b) Answer: 10
(c) M1 6sqrt(18) oe (multiplies integer parts and surd parts)
(c) A1 18sqrt(2) cao
(c) Answer: 18sqrt(2)

Question 4

(a) M1 √48/3 = √16 oe
(a) A1 4 cao
(a) Answer: 4
(b) M1 √72/8 = √9 oe
(b) A1 3 cao
(b) Answer: 3
(c) M1 5 x √45/5 = 5sqrt(9) oe
(c) A1 15 cao
(c) Answer: 15

Question 5

(a) M1 collects coefficients 3 + 5 - 1
(a) A1 7sqrt(2) cao
(a) Answer: 7sqrt(2)
(b) M1 √12 = 2sqrt(3) and √27 = 3sqrt(3) oe
(b) A1 5sqrt(3) cao
(b) Answer: 5sqrt(3)
(c) M1 simplifies at least two of the three surds correctly (5sqrt(2), 2sqrt(2), 3sqrt(2))
(c) A1 6sqrt(2) cao
(c) Answer: 6sqrt(2)

Question 6

(a) M1 3sqrt(2) + 2 oe (unsimplified expansion)
(a) A1 3sqrt(2) + 2 cao
(a) Answer: 3sqrt(2) + 2
(b) M1 3 - 1 oe (difference of two squares used)
(b) A1 2 cao
(b) Answer: 2
(c) M1 4 - 5 oe
(c) A1 -1 cao
(c) Answer: -1

Question 7

(a) M1 all four terms of the expansion correct: 5 + 3sqrt(5) + 2sqrt(5) + 6
(a) M1 collects rational terms (5+6=11) and surd terms (3sqrt(5)+2sqrt(5)=5sqrt(5))
(a) A1 11 + 5sqrt(5) cao
(a) Answer: 11 + 5sqrt(5)
(b) M1 all four terms correct: 6 + 8sqrt(3) - √3 - 4
(b) M1 collects rational terms (6-4=2) and surd terms (8sqrt(3)-√3=7sqrt(3))
(b) A1 2 + 7sqrt(3) cao
(b) Answer: 2 + 7sqrt(3)
(c) M1 recognises difference of two squares
(c) M1 7 - 2
(c) A1 5 cao
(c) Answer: 5

Question 8

(a) M1 multiplies numerator and denominator by √3
(a) A1 √3/3 oe cao
(a) Answer: √3/3
(b) M1 multiplies numerator and denominator by √2
(b) A1 5sqrt(2)/2 oe cao
(b) Answer: 5sqrt(2)/2
(c) M1 simplifies √12 to 2sqrt(3) or rationalises directly
(c) A1 √3 cao
(c) Answer: √3

Question 9

(a) M1 multiplies numerator and denominator by (√5 - 2)
(a) M1 denominator simplifies to 5 - 4 = 1
(a) A1 √5 - 2 cao
(a) Answer: √5 - 2
(b) M1 multiplies numerator and denominator by (√7 + 2)
(b) M1 denominator simplifies to 7 - 4 = 3
(b) A1 √7 + 2 cao
(b) Answer: √7 + 2
(c) M1 multiplies numerator and denominator by (√2 + 1)
(c) M1 denominator simplifies to 2 - 1 = 1 and numerator expanded to 2 + 2sqrt2 + 1
(c) A1 3 + 2sqrt(2) cao
(c) Answer: 3 + 2sqrt(2)

Question 10

(a) M1 uses Pythagoras: hyp² = (sqrt20)² + (sqrt5)²
(a) M1 simplifies to hyp² = 20 + 5 = 25
(a) A1 hyp = √25 = 5 cso (answer given)
(a) Answer: 5 cm (given)
(b) M1 Area = 1/2 x √20 x √5
(b) M1 combines surds to 1/2 x √100
(b) A1 5 cm² cao
(b) Answer: 5 cm²

Question 11

(a) M1 sets up (3+sqrt2)(3-sqrt2)
(a) M1 uses difference of two squares to reach 9 - 2
(a) A1 7 cm² cao
(a) Answer: 7 cm²
(b) M1 adds the four sides: 2(3+sqrt2) + 2(3-sqrt2) oe
(b) A1 12 cm cao
(b) Answer: 12 cm

Question 12

M1 writes (2+sqrt3)(2+sqrt3) and expands to 4 + 2sqrt3 + 2sqrt3 + 3
M1 collects rational terms (4+3=7) and surd terms (2sqrt3+2sqrt3=4sqrt3)
A1 7 + 4sqrt(3) cso (answer given, all steps must be shown)
Answer: 7 + 4sqrt(3) (given)

Question 13

M1 simplifies √8 to 2sqrt(2)
M1 simplifies √18 to 3sqrt(2)
M1 combines right-hand side to 5sqrt(2)
A1 x = 5 cao
Answer: x = 5

Question 14

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

Question 15

M1 finds the differences: x-difference = √3 (or -√3), y-difference = -3 (or 3)
M1 uses AB = √(sqrt3)² + (-3)² = √3+9 = √12
A1 2sqrt(3) cao (units not required)
Answer: 2sqrt(3)

Question 16

M1 multiplies numerator and denominator by (2 + √3)
M1 expands numerator to 10 + 5sqrt3 + 2sqrt3 + 3
M1 simplifies denominator to 4 - 3 = 1
A1 13 + 7sqrt(3) cao
Answer: 13 + 7sqrt(3)

Question 17

B1 assumes for contradiction that √2 is rational, so √2 = a/b where a and b are integers with no common factor and b is not 0
M1 squares both sides to obtain 2 = a²/b², so a² = 2b²
A1 deduces a² is even, so a itself must be even, so a = 2k for some integer k
M1 substitutes to get 4k² = 2b², so b² = 2k², so b² is even and hence b is even
A1 states that a and b are both even, contradicting the assumption that a/b was in a form with no common factor, so √2 cannot be written as a/b: √2 is irrational
Answer: √2 is irrational (proof by contradiction)

Question 18

(a) M1 multiplies numerator and denominator by (√x + 2)
(a) M1 expands numerator to x + 4sqrt(x) + 4 and denominator to x - 4
(a) A1 (x + 4sqrt(x) + 4) / (x - 4) oe cao
(a) Answer: (x + 4sqrt(x) + 4) / (x - 4)
(b) M1 substitutes x = 9 and uses √9 = 3
(b) M1 evaluates numerator = 9 + 12 + 4 = 25 and denominator = 9 - 4 = 5 (or uses (3+2)/(3-2))
(b) A1 5 cao
(b) Answer: 5

Question 19

(a) M1 adds the parallel sides: (4+sqrt5)+(4-sqrt5) = 8
(a) M1 Area = 1/2 x 8 x √5
(a) A1 4sqrt(5) cm² cao
(a) Answer: 4sqrt(5) cm²
(b) M1 sums all four sides: (4+sqrt5)+(4-sqrt5)+2(sqrt5+1)
(b) A1 10 + 2sqrt(5) cm oe cao
(b) Answer: 10 + 2sqrt(5) cm

Question 20

M1 expands the brackets: 3a - a*sqrt2 + 3b*sqrt2 - 2b
M1 collects rational and surd parts separately: (3a - 2b) + (3b - a)√2
M1 equates rational parts (3a - 2b = 11) and surd parts (3b - a = 1) and solves simultaneously
A1 a = 5 and b = 2 cao
Answer: a = 5, b = 2

Higher Surds: Simplifying, Multiplying and Dividing - Worksheets, Questions and Revision

N1 Higher Surds: Simplifying, Multiplying and Dividing

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Question 20