$\def\D{\displaystyle}\def\frac{\dfrac}$

1 (CIE 2012, s, paper 12, question 6)

You must not use a calculator in this question.

(i) Express $\D \frac{8}{\sqrt{3}+1} $ in the form $\D a(\sqrt{3}-1),$ where $\D a$ is an integer. [2]

An equilateral triangle has sides of length $\D \frac{8}{\sqrt{3}+1}.$

(ii) Show that the height of the triangle is $\D 6 - 2\sqrt{3} .$ [2]

(iii) Hence, or otherwise, find the area of the triangle in the form $\D p\sqrt{3} - q,$ where $\D p$ and $\D q$ are integers. [2]

2 (CIE 2012, s, paper 21, question 2)

A cuboid has a square base of side $\D (2 + \sqrt{3} )$ cm and a volume of $\D (16 + 9\sqrt{3} )$ cm$\D ^3.$ Without using a calculator, find the height of the cuboid in the form $\D (a + b\sqrt{3} )$ cm, where $\D a$ and $\D b$ are integers. [4]

3 (CIE 2012, w, paper 11, question 7)

Do not use a calculator in any part of this question.

(a) (i) Show that $\D 3\sqrt{5} - 2 \sqrt{2}$ is a square root of $\D 53 - 12\sqrt{10}.$ [1]

(ii) State the other square root of $\D 53 - 12\sqrt{10}.$ [1]

(b) Express $\D \frac{6\sqrt{3}+7\sqrt{2}}{4\sqrt{3}+5\sqrt{2}}$ in the form $\D a + b \sqrt{6},$ where $\D a$ and $\D b$ are integers to be found. [4]

4 (CIE 2012, w, paper 12, question 6)

Using $\D \sin15^{\circ} =\frac{\sqrt{2}}{4}(\sqrt{3}-1)$ and without using a calculator, find the value of $\D \sin\theta$ in the form $\D a + b \sqrt{2},$ where $\D a$ and $\D b$ are integers. [5]

5 (CIE 2012, w, paper 23, question 3)

Without using a calculator, simplify $\D \frac{(3\sqrt{3}-1)^2}{2\sqrt{3}-3},$ giving your answer in the form $\D \frac{a\sqrt{3}+b}{3},$ where $\D a$ and $\D b$ are integers. [4]

6 (CIE 2013, s, paper 11, question 7)

Calculators must not be used in this question.

The diagram shows a triangle $\D ABC$ in which angle $\D A = 90^{\circ}.$ Sides $\D AB$ and $\D AC$ are $\D \sqrt{5} - 2$ and $\D \sqrt{5} + 1$ respectively. Find

(i) $\D \tan B$ in the form $\D a + b\sqrt{5},$ where $\D a$ and $\D b$ are integers, [3]

(ii) $\D \sec^2B$ in the form $\D c + d \sqrt{5},$ where $\D c$ and $\D d$ are integers. [4]

7 (CIE 2013, s, paper 22, question 5) Fig

The diagram shows a trapezium $\D ABCD$ in which $\D AD = 7$ cm and $\D AB =(4+\sqrt{5})$ cm. $\D AX$ is perpendicular to $\D DC$ with $\D DX = 2$ cm and $\D XC = x$ cm. Given that the area of trapezium $\D ABCD$ is $\D 15(\sqrt{5}+2)$ cm$\D ^2,$ obtain an expression for $\D x$ in the form $\D a + b \sqrt{5},$ where $\D a$ and $\D b$ are integers. [6]

8 (CIE 2013, w, paper 21, question 2)

Do not use a calculator in this question.

Express $\D \frac{(4\sqrt{5}-2)^2}{\sqrt{5}-1}$ in the form $\D p \sqrt{5} + q,$ where $\D p$ and $\D q$ are integers. [4]

9 (CIE 2014, s, paper 21, question 2)

Without using a calculator, express $\D 6(1+\sqrt{3})^{-2}$ in the form $\D a + b \sqrt{3},$ where $\D a$ and $\D b$ are integers to be found. [4]

10 (CIE 2014, s, paper 22, question 1)

Without using a calculator, express $\D \frac{(2+\sqrt{5})^2}{\sqrt{5}-1}$ in the form $\D a + b\sqrt{5},$ where $\D a$ and $\D b$ are constants to be found. [4]

11 (CIE 2014, s, paper 23, question 5)

Do not use a calculator in this question.

(i) Show that $\D (2\sqrt{2}+4)^2-8(2\sqrt{2}+3)=0.$ [2]

(ii) Solve the equation $\D (2\sqrt{2}+3)x^2-(2\sqrt{2}+4)x+2=0,$ giving your answer in the form $\D a + b\sqrt{2} $ where $\D a$ and $\D b$ are integers. [3]

12 (CIE 2014, w, paper 21, question 9)

Integers $\D a$ and $\D b$ are such that $\D (a+ 3\sqrt{5} )^2+ a- b\sqrt{5}= 51.$ Find the possible values of $\D a$ and the corresponding values of $\D b.$ [6]

1 (CIE 2012, s, paper 12, question 6)

You must not use a calculator in this question.

(i) Express $\D \frac{8}{\sqrt{3}+1} $ in the form $\D a(\sqrt{3}-1),$ where $\D a$ is an integer. [2]

An equilateral triangle has sides of length $\D \frac{8}{\sqrt{3}+1}.$

(ii) Show that the height of the triangle is $\D 6 - 2\sqrt{3} .$ [2]

(iii) Hence, or otherwise, find the area of the triangle in the form $\D p\sqrt{3} - q,$ where $\D p$ and $\D q$ are integers. [2]

2 (CIE 2012, s, paper 21, question 2)

A cuboid has a square base of side $\D (2 + \sqrt{3} )$ cm and a volume of $\D (16 + 9\sqrt{3} )$ cm$\D ^3.$ Without using a calculator, find the height of the cuboid in the form $\D (a + b\sqrt{3} )$ cm, where $\D a$ and $\D b$ are integers. [4]

3 (CIE 2012, w, paper 11, question 7)

Do not use a calculator in any part of this question.

(a) (i) Show that $\D 3\sqrt{5} - 2 \sqrt{2}$ is a square root of $\D 53 - 12\sqrt{10}.$ [1]

(ii) State the other square root of $\D 53 - 12\sqrt{10}.$ [1]

(b) Express $\D \frac{6\sqrt{3}+7\sqrt{2}}{4\sqrt{3}+5\sqrt{2}}$ in the form $\D a + b \sqrt{6},$ where $\D a$ and $\D b$ are integers to be found. [4]

4 (CIE 2012, w, paper 12, question 6)

Using $\D \sin15^{\circ} =\frac{\sqrt{2}}{4}(\sqrt{3}-1)$ and without using a calculator, find the value of $\D \sin\theta$ in the form $\D a + b \sqrt{2},$ where $\D a$ and $\D b$ are integers. [5]

5 (CIE 2012, w, paper 23, question 3)

Without using a calculator, simplify $\D \frac{(3\sqrt{3}-1)^2}{2\sqrt{3}-3},$ giving your answer in the form $\D \frac{a\sqrt{3}+b}{3},$ where $\D a$ and $\D b$ are integers. [4]

6 (CIE 2013, s, paper 11, question 7)

Calculators must not be used in this question.

The diagram shows a triangle $\D ABC$ in which angle $\D A = 90^{\circ}.$ Sides $\D AB$ and $\D AC$ are $\D \sqrt{5} - 2$ and $\D \sqrt{5} + 1$ respectively. Find

(i) $\D \tan B$ in the form $\D a + b\sqrt{5},$ where $\D a$ and $\D b$ are integers, [3]

(ii) $\D \sec^2B$ in the form $\D c + d \sqrt{5},$ where $\D c$ and $\D d$ are integers. [4]

7 (CIE 2013, s, paper 22, question 5) Fig

The diagram shows a trapezium $\D ABCD$ in which $\D AD = 7$ cm and $\D AB =(4+\sqrt{5})$ cm. $\D AX$ is perpendicular to $\D DC$ with $\D DX = 2$ cm and $\D XC = x$ cm. Given that the area of trapezium $\D ABCD$ is $\D 15(\sqrt{5}+2)$ cm$\D ^2,$ obtain an expression for $\D x$ in the form $\D a + b \sqrt{5},$ where $\D a$ and $\D b$ are integers. [6]

8 (CIE 2013, w, paper 21, question 2)

Do not use a calculator in this question.

Express $\D \frac{(4\sqrt{5}-2)^2}{\sqrt{5}-1}$ in the form $\D p \sqrt{5} + q,$ where $\D p$ and $\D q$ are integers. [4]

9 (CIE 2014, s, paper 21, question 2)

Without using a calculator, express $\D 6(1+\sqrt{3})^{-2}$ in the form $\D a + b \sqrt{3},$ where $\D a$ and $\D b$ are integers to be found. [4]

10 (CIE 2014, s, paper 22, question 1)

Without using a calculator, express $\D \frac{(2+\sqrt{5})^2}{\sqrt{5}-1}$ in the form $\D a + b\sqrt{5},$ where $\D a$ and $\D b$ are constants to be found. [4]

11 (CIE 2014, s, paper 23, question 5)

Do not use a calculator in this question.

(i) Show that $\D (2\sqrt{2}+4)^2-8(2\sqrt{2}+3)=0.$ [2]

(ii) Solve the equation $\D (2\sqrt{2}+3)x^2-(2\sqrt{2}+4)x+2=0,$ giving your answer in the form $\D a + b\sqrt{2} $ where $\D a$ and $\D b$ are integers. [3]

12 (CIE 2014, w, paper 21, question 9)

Integers $\D a$ and $\D b$ are such that $\D (a+ 3\sqrt{5} )^2+ a- b\sqrt{5}= 51.$ Find the possible values of $\D a$ and the corresponding values of $\D b.$ [6]

13 (CIE 2015, s, paper 11, question 2)

The diagram shows the triangle $A B C$ where angle $B$ is a right angle, $A B=(4+3 \sqrt{2}) \mathrm{cm}$, $B C=(8+5 \sqrt{2}) \mathrm{cm}$ and angle $B A C=\theta$ radians. Showing all your working, find

(i) $\tan \theta$ in the form $a+b \sqrt{2}$, where $a$ and $b$ are integers,[2]

(ii) $\sec ^{2} \theta$ in the form $c+d \sqrt{2}$, where $c$ and $d$ are integers.[3]

14 (CIE 2015, s, paper 22, question 3)

The diagram shows the right-angled triangle $A B C$, where $A B=(6+3 \sqrt{5}) \mathrm{cm}$ and angle $B=90^{\circ}$. The area of this triangle is $\left(\frac{36+15 \sqrt{5}}{2}\right) \mathrm{cm}^{2}$.

(i) Find the length of the side $B C$ in the form $(a+b \sqrt{5}) \mathrm{cm}$, where $a$ and $b$ are integers.[3]

(ii) Find $(A C)^{2}$ in the form $(c+d \sqrt{5}) \mathrm{cm}^{2}$, where $c$ and $d$ are integers.$[2]$

15 (CIE 2015, w, paper 13, question 4)

The diagram shows triangle $A B C$ with side $A B=(4 \sqrt{3}+1) \mathrm{cm}$. Angle $B$ is a right angle. It is given

that the area of this triangle is $\frac{47}{2} \mathrm{~cm}^{2}$.

(i) Find the length of the side $B C$ in the form $(a \sqrt{3}+b) \mathrm{cm}$, where $a$ and $b$ are integers.[3]

(ii) Hence find the length of the side $A C$ in the form $p \sqrt{2} \mathrm{~cm}$, where $p$ is an integer.[2]

16 (CIE 2015, w, paper 23, question 4)

Solve the following simultaneous equations, giving your answers for both $x$ and $y$ in the form $a+b \sqrt{3}$, where $a$ and $b$ are integers.

$2 x+y=9$

$\sqrt{3} x+2 y=5$

17 (CIE 2016, march, paper 22, question 6)

The diagram shows two parallelograms that are similar. The base and height, in centimetres, of each parallelogram is shown. Given that $x$, the height of the smaller parallelogram, is $\frac{p+q \sqrt{3}}{6}$, find the value of each of the integers $p$ and $q$.

18 (CIE 2016, s, paper 12 , question 4)

Do not use a calculator in this question.

Find the positive value of $x$ for which $(4+\sqrt{5}) x^{2}+(2-\sqrt{5}) x-1=0$, giving your answer in the form $\frac{a+\sqrt{5}}{b}$, where $a$ and $b$ are integers.

19 (CIE 2016, s, paper 21, question 5)

Do not use a calculator in this question.

(a) Express $\frac{\sqrt{8}}{\sqrt{7}-\sqrt{5}}$ in the form $\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are integers.

(b) Given that $28+p \sqrt{3}=(q+2 \sqrt{3})^{2}$, where $p$ and $q$ are integers, find the values of $p$ and of $q$. [3]

20 (CIE $2016, \mathrm{w}$, paper 21 , question 2)

Without using a calculator, find the integers $a$ and $b$ such that $\frac{a}{\sqrt{3}+1}+\frac{b}{\sqrt{3}-1}=\sqrt{3}-3$. [5]

21 (CIE 2016, w, paper 23, question 1)

Without using a calculator, show that $\frac{\sqrt{5}+3 \sqrt{3}}{\sqrt{5}+\sqrt{3}}=\sqrt{k}-2$ where $k$ is an integer to be found.

22 (CIE 2016, w, paper 23, question 5)

In the triangle $A B C$ shown above, $A C=\sqrt{3}+1, B C=\sqrt{3}-1$ and angle $A C B=60^{\circ}$.

(i) Without using a calculator, show that the length of $A B=\sqrt{6}$.[3]

(ii) Show that angle $C A B=15^{\circ}$.[2]

(iii) Without using a calculator, find the area of triangle $A B C$.[2]

The diagram shows a trapezium made from a rectangle and a right-angled triangle. The dimensions, in centimetres, of the rectangle and triangle are shown. The area, in square centimetres, of the trapezium is $13+5 \sqrt{5}$. Without using a calculator, find the value of $x$ in the form $p+q \sqrt{5}$, where $p$ and $q$ are integers.[5]

24 (CIE 2017, s, paper 13, question 4)

The diagram shows an isosceles triangle $A B C$, where $A B=A C$. The point $M$ is the mid-point of $B C$. Given that $A M=3+2 \sqrt{5}$ and $B C=4+6 \sqrt{5}$, find, without using a calculator,

(i) the area of triangle $A B C$,[2]

(ii) $\tan A B C$, giving your answer in the form $\frac{a+b \sqrt{5}}{c}$ where $a, b$ and $c$ are positive integers.[3]

25 (CIE 2017, s, paper 21, question 2)

Do not use a calculator in this question.

(a) Show that $\sqrt{24} \times \sqrt{27}+\frac{9 \sqrt{30}}{\sqrt{15}}$ can be written in the form $a \sqrt{2}$, where $a$ is an integer.[3]

(b) Solve the equation $\sqrt{3}(1+x)=2(x-3)$, giving your answer in the form $b+c \sqrt{3}$, where $b$ and are integers.[3]

26 (CIE 2017, s, paper 22, question 2)

Without using a calculator, express $\left(\frac{1+\sqrt{5}}{3-\sqrt{5}}\right)^{-2}$ in the form $a+b \sqrt{5}$, where $a$ and $b$ are integers. [5]

27 (CIE $2017, \mathrm{w}$, paper 21, question 8$)$

Given that $z=a+(a+3) \sqrt{3}$ and $z^{2}=79+b \sqrt{3}$, find the value of each of the integers $a$ and $b$. $[6]$

28 (CIE 2017,w, paper 23, question 3) Find integers $p$ and $q$ such that $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3 \sqrt{3}$. 29 (CIE 2018, s, paper 11, question 10) Do not use a calculator in this question.

(a) Simplify $\frac{5+6 \sqrt{5}}{6+\sqrt{5}}$.

(b) Show that $3^{0.5} \times(\sqrt{2})^{7}$ can be written in the form $a \sqrt{b}$, where $a$ and $b$ are integers and $a>b$.

(c) Solve the equation $x+\sqrt{2}=\frac{4}{x}$, giving your answers in simplest surd form.[4]

Answers

1.(i) $\D a=4$

(ii)

(iii) $\D 16\sqrt{3}-24$

2. $\D 4-\sqrt{3}$

3.(a)(i)

(ii) $\D -3\sqrt{5}+2\sqrt{2}$

(b) $\D -1+\sqrt{6}$

4. $\D 6-4\sqrt{2}$

5. $\D \frac{38\sqrt{3}}{3}$

6.(i) $\D 7+3\sqrt{5}$

(ii) $\D 95+42\sqrt{5}$

7. $\D 4+3\sqrt{5}$

8. $\D 17\sqrt{5}+1$

9. $\D 6-3\sqrt{3}$

10. $\D \frac{29}{4}+\frac{13}{4}\sqrt{5}$

11. $\D 2-\sqrt{2}$

12. $\D a=-3,2:b=-18,12$

13. (i) $1+2 \sqrt{2}$

(ii) $10-4 \sqrt{2}$

14. (i) $1+2 \sqrt{5}$

(ii) $102+40 \sqrt{5}$

15. (i) $4 \sqrt{3}-1$

(ii) $7 \sqrt{2}$

16. $x=4+\sqrt{3}$

$y=1-2 \sqrt{3}$

17. $p=-27, q=23$

18. $\frac{7+\sqrt{5}}{22}$

19. (a) $\sqrt{14}+\sqrt{10}$

(b) $q=\pm 4, p=\pm 16$

20. $a=4, b=-2$

21. $k=15$

22. (iii) $\sqrt{3} / 2$

23. $1+\sqrt{5}$

24. (i) $36+13 \sqrt{5}$

(ii) $\frac{24+5 \sqrt{5}}{41}$

25. (a) $9 \sqrt{2}$

(b) $15+8 \sqrt{3}$

26. $9-4 \sqrt{5}$

27. $a=2, b=20$

28. $p=5, q=2$

29. $\sqrt{5}, 8 \sqrt{6}, x=\sqrt{2},-2 \sqrt{2}$

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