The following exercises are intended to get you thinking about mathematics again before starting the EMB course. They are all based on the level of maths expected at Key Stage 4 Higher and GCSE. If you encounter any problems, please discuss these with your College supervisor.

**(1) Approximations.**

(i) Round off the number 4384556 to 2 significant figures.

(ii) Round off the number 4384556 to 3 significant figures.

(iii) Round off the number 0.000033678 to 2 significant figures and write the result in scientific notation.

(iv) Round off the number 3.14159265 to 3 decimal places.

**(2) Fractions and percentages.**

(i) Evaluate (1/2) + (1/3) .

(ii) Evaluate (1/2) × (1/3).

(iii) Evaluate (1/2) ÷ (1/3).

(iv) Change (3/20) to a decimal.

(v) Change 7 % to a decimal.

(vi) Change (1/8) to a percentage.

(vii) What is the percent increase of a rise in temperature from 80° to 100°C?

(Work this out for both the Kelvin and the Celsius temperature scales.)

**(3) Order of operations.**

Simplify the following expressions:

(i) 6 + 4 × 3^{2}.

(ii) 3^{2} + 6(4 + 1).

(iii) 12 – 2(8 + 2) + 5.

(iv) 8[3(3^{2} – 8) + 1].

(v) 6{4[2(3 + 2) – 8] – 8}.

(vi) 6(12 + 8) ÷ 2 + 1.

** **

**(4) Algebraic expressions.**

(i) Simplify the expression *x *+ 5 + 3*x *– 6 + 9*x *+ 3 – 4.

(ii) Simplify the expression 5*y *^{2} – 4*y *+ 8 – *y* ^{2 }+ 7*y *+ 5.

(iii) Multiply out the expression (3*x *+ 5)^{ 2} and simplify the result.

(iv) Simplify the expression 4(*xy *^{3})^{ 5}.

(v) Factorise the expression 8*x *^{3} – 12*x *^{2}

(vi) Factorise the expression 2*x *^{2} – 8*x *+ 6.

(vii) Evaluate if *x *= 2 and *y *= 6.

**(5) Solving equations.**

(i) Solve the following equation for *x*: 5*x *+ 7 = 2*x *+ 13.

(ii) Solve the following equation for *y*: (4/7)*y* + 6 = 18.

(iii) Solve the following equation for *x*: *x *^{2} – 6*x *+ 8 = 0.

(iv) Solve the following inequality for *x*: 4*x *– 3 < *x *– 6.

(v) Solve the following equation for *x*: *x *^{2} – *x *– 4 = 0.

(vi) Solve the following equation for *x*: *x *^{3} + 2*x *^{2} – *x *– 2 = 0.

(vii) Solve the following equation for *a*: (*a*-1)/3 + (*a*+2)/6 = 2.

**(6) Series.**

(i) The *n*th term in an arithmetic series of numbers, *t _{n}*, can be expressed using the formula

* t _{n}*

*=*

*t*

_{1}+ (

*n*−1)

*d*, where

*t*

_{1}is the first term and

*d*is the common difference between consecutive terms. Use this formula to determine the value of the 12th term in the following sequence: 3, 8, 13, 18, 23, … .

(ii) The *n*th term in a geometric series of numbers, *t _{n}*, can be expressed using the formula

* t _{n }*=

*t*

_{1}×

*r*

^{n}^{−}

^{1}, where

*t*

_{1}is the first term and

*r*is the common ratio. If

*t*

_{1}= 1 and

*r*= 3, evaluate the first six terms of the series.

(iii) Work out a formula for the *n*th term of the following series: 32, 302, 3002, 30002, … .

**(7) Coordinates and graphs.**

For the points *A *and *B*, which have the (*x*, *y*) coordinates (1, 2) and (4, 6), respectively:

(i) Determine the length of the line that connects *A *and *B*.

(ii) Determine the slope of the line that connects *A *and *B*.

(iii) Determine the equation of the straight line that goes through *A *and *B*, along with the coordinates of the points where this line intersects with the *x *and *y* axes.

(iv) Determine the equation of the line parallel to *AB *that goes through the origin.

**(8) Trigonometry and geometry.**

(i) Evaluate sin(30°), sin(120°), sin(210°), sin(300°) and .

(ii) If a triangle has two angles equal to 110° and 20°, determine the size of the third angle.

(iii) Let *ABC *be a triangle whose angles at *A *and *B *are 30° and 45°. If the side opposite angle *B *has length 9 cm, find the lengths of the remaining sides and the size of the angle at *C*.

(iv) Calculate the surface area and the volume of a cube of side 4 cm.

**Need some help?**

There are many excellent web sites for revising mathematics at this level, including:

http://www.bbc.co.uk/schools/gcsebitesize/maths/