# A-Level Maths Preparation

The following exercises cover vital skills from GCSE Maths and can help your A-Level Maths preparation. For successful study at A-Level, you must be highly proficient at applying these skills. You can become skilled by practising and getting help where necessary.

## Collecting Like Terms

Simplify each of these expressions as far as possible:

1. $$3x-2y+4y$$
2. $$5-3y-6y-2$$
3. $$5x+2y-4y-x^2$$
4. $$x^2+3x^2-4x^2+5x$$
5. $$2y^2-y(x-y)$$
6. $$8pq-9p^2-3pq$$
7. $$x^3-2x^2+x^2-4x+5x+7$$
8. $$3x(x-2)+4(3x-5)$$
9. $$7+3(x-1)$$
10. $$7b(a+2)-a(3b+3)$$
Solutions
1. $$3x+2y$$
2. $$3-9y$$
3. $$5x-2y-x^2$$
4. $$5x$$
5. $$3y^2-xy$$
6. $$5pq-9p^2$$
7. $$x^3-x^2+x+7$$
8. $$3x^2+6x-20$$
9. $$3x+4$$
10. $$4ab-3a+14b$$

## Solving Linear Equations

Find the value of $$x$$ in the following equations:

1. $$2x-5=11$$
2. $$8=7+3x$$
3. $$-7=2x-10$$
4. $$\frac{x}{5}=7$$
5. $$\frac{x}{10}=5$$
6. $$\frac{x}{2}=\frac{1}{3}$$
7. $$\frac{3x}{2}=-5$$
8. $$x-3=3x+7$$
9. $$5x-4=3-x$$
10. $$5x-16=16-2x$$
Solutions
1. $$x=8$$
2. $$x=\frac{1}{3}$$
3. $$x=\frac{3}{2}$$
4. $$x=35$$
5. $$x=50$$
6. $$x=\frac{2}{3}$$
7. $$x=-\frac{10}{3}$$
8. $$x=-5$$
9. $$x=\frac{7}{6}$$
10. $$x=\frac{32}{7}$$

## Expanding Brackets

Expand and collect like terms in each of the following:

1. $$(x+3)(x-2)$$
2. $$(x-5)(x-1)$$
3. $$(2x+y)(x-y)$$
4. $$(2x-3)(3x-1)$$
5. $$(2x+1)(x+2)$$
6. $$(x+2)(x-2)$$
7. $$2(x+3)(x-1)$$
8. $$(2x-3)(2x+3)$$
9. $$(5-x)(5+x)$$
10. $$(x+7)^2$$
Solutions
1. $$x^2+x-6$$
2. $$x^2-6x+5$$
3. $$2x^2-xy-y^2$$
4. $$6x^2-11x+3$$
5. $$2x^2+5x+2$$
6. $$x^2-4$$
7. $$2x^2+4x-6$$
8. $$4x^2-9$$
9. $$25-x^2$$
10. $$x^2+14x+49$$

## Factorising

Factorise each of the following. Note – you can check your answers by expansion:

1. $$x^2+3x+2$$
2. $$x^2-3x+2$$
3. $$x^2+5x+6$$
4. $$x^2+7x+6$$
5. $$x^2+4x$$
6. $$x^2-x-12$$
7. $$x^2-2x-3$$
8. $$2x^2+3x-5$$
9. $$x^2-9$$
10. $$4x^2-1$$
Solutions
• $$(x+1)(x+2)$$
• $$(x-1)(x-2)$$
• $$(x+2)(x+3)$$
• $$(x+6)(x+1)$$
• $$x(x+4)$$
• $$(x-4)(x+3)$$
• $$(x-3)(x+1)$$
• $$(2x+5)(x-1)$$
• $$(x+3)(x-3)$$
• $$(2x-1)(2x+1)$$

## Laws of Indices

Simplify the following:

1. $$b\times 5b^5$$
2. $$3c^2\times 2c^5$$
3. $$b^2c\times bc^3$$
4. $$2n^6\times\left(-6n^2\right)$$
5. $$8n^8\div\left(2n^3\right)$$
6. $$d^{11}\div d^9$$
7. $$\left(a^3\right)^2$$
8. $$\left(-d^4\right)^3$$
9. $$\left(25g^{12}\right)^\frac{1}{2}$$
10. $${\left(64h^{-3}\right)}^\frac{1}{3}$$
Solutions
1. $$5b^6$$
2. $$6c^7$$
3. $$b^3 c^4$$
4. $$-12n^8$$
5. $$4n^5$$
6. $$d^2$$
7. $$a^6$$
8. $$-d^{12}$$
9. $$5g^6$$
10. $$4h^{-1}$$

Solve the following equations:

1. $$x^2+10-7x=0$$
2. $$15-x^2-2x=0$$
3. $$x^2-3x=4$$
4. $$12-7x+x^2=0$$
5. $$2x-1+3x^2=0$$
6. $$x\left(x+7\right)+6=0$$
7. $$2x^2-4x=0$$
8. $$x\left(4x+5\right)=-1$$
9. $$2-x=3x^2$$
10. $$6x^2+3x=0$$
Solutions
1. $$x=2\ \mathrm{or}\ x=5$$
2. $$x=-5\ \mathrm{or}\ x=3$$
3. $$x=-1\ \mathrm{or}\ x=4$$
4. $$x=3\ \mathrm{or}\ x=4$$
5. $$x=-1\ \mathrm{or}\ x=\frac{1}{3}$$
6. $$x=-6\ \mathrm{or}\ x=-1$$
7. $$x=0\ \mathrm{or}\ x=2$$
8. $$x=-1\ \mathrm{or}\ x=-\frac{1}{4}$$
9. $$x=-1\ \mathrm{or}\ x=\frac{2}{3}$$
10. $$x=-\frac{1}{2}\ \mathrm{or}\ x=0$$

## Simultaneous Equations

Solve the following pairs of equations. Remember – you can check your answer by substituting your $$x$$ and $$y$$ into the equations to see if they balance:

1. \left\{\begin{align*}x+y&=12\\x-y&=6\end{align*}\right.
2. \left\{\begin{align*}2x+y&=10\\x-y&=2\end{align*}\right.
3. \left\{\begin{align*}4x+y&=10\\3x+y&=9\end{align*}\right.
4. \left\{\begin{align*}2x+y&=7\\3x+y&=10\end{align*}\right.
5. \left\{\begin{align*}2x+3y&=19\\2x+y&=9\end{align*}\right.
Solutions
1. $$x=9,\ y=3$$
2. $$x=4,\ y=2$$
3. $$x=1,\ y=6$$
4. $$x=3,\ y=1$$
5. $$x=2,\ y=5$$

## Changing the Subject

Make $$x$$ the subject of each of these formulae. Hint: when $$x$$ appears in more than one place in the formula, collect the terms involving $$x$$ on one side of the equation and move the other terms to the other side. Then factorise out the common factor of $$x$$ before making $$x$$ the subject.

1. $$y=7x-1$$
2. $$y=\frac{x+5}{4}$$
3. $$4y=\frac{x}{3}-2$$
4. $$y=\frac{4\left(3x-5\right)}{9}$$
5. $$ax+3=bx+c$$
6. $$3\left(x+a\right)=k\left(x-2\right)$$
7. $$y=\frac{2x+3}{5x-2}$$
8. $$\frac{x}{a}=1+\frac{x}{b}$$
Solutions
1. $$x=\frac{y+1}{7}$$
2. $$x=4y-5$$
3. $$x=12y+6$$
4. $$x=\frac{3}{4}y+\frac{5}{3}$$
5. $$x=\frac{c-3}{a-b}=\frac{3-c}{b-a}$$
6. $$x=-\frac{3a+2k}{3-k}=\frac{3a+2k}{k-3}$$
7. $$x=\frac{3+2y}{5y-2}$$
8. $$x=\frac{ab}{b-a}$$