Algebra

Simplify the expression 5a + 3a − 2a.

• A6a✓
• B4a
• C6
• D10a

Answer: Collect like terms: 5a + 3a − 2a = 6a.

Solve for x: x + 7 = 12.

• A19
• B5✓
• C7
• D−5

Answer: Subtract 7 from both sides: x = 12 − 7 = 5.

Expand 3(x + 4).

• Ax + 12
• B3x + 7
• C3x + 4
• D3x + 12✓

Answer: Multiply each term in the bracket by 3: 3 × x + 3 × 4 = 3x + 12.

Solve for x: 2x − 3 = 11.

• A4
• B7✓
• C8
• D14

Answer: Add 3: 2x = 14. Divide by 2: x = 7.

If y = 4x and x = 3, what is y?

• A7
• B43
• C12✓
• D1

Answer: Substitute x = 3: y = 4 × 3 = 12.

What is the next term in the sequence 2, 5, 8, 11, …?

• A12
• B14✓
• C13
• D15

Answer: The sequence increases by 3 each time, so 11 + 3 = 14.

Simplify x × x × x.

• Ax³✓
• B3x
• Cx²
• D3

Answer: Multiplying x by itself three times gives x³ (x cubed).

Factorise fully: 6x + 9.

• A2(3x + 9)
• B6(x + 9)
• C3(2x + 9)
• D3(2x + 3)✓

Answer: The highest common factor of 6 and 9 is 3: 6x + 9 = 3(2x + 3).

Expand and simplify (x + 3)(x + 4).

• Ax² + 12
• Bx² + 7x + 7
• Cx² + 7
• Dx² + 7x + 12✓

Answer: Multiply every term in the first bracket by every term in the second: x² + 4x + 3x + 12. Collecting the two x terms gives x² + 7x + 12. Multiplying only the firsts and only the lasts misses the x terms completely.

Expand and simplify (x - 5)(x + 2).

• Ax² - 10
• Bx² + 3x - 10
• Cx² - 7x - 10
• Dx² - 3x - 10✓

Answer: The four products are x², +2x, -5x and -10. Collecting +2x and -5x gives -3x, so the expansion is x² - 3x - 10.

Factorise fully: 12x² + 18x.

• A6x(2x + 3)✓
• B3x(4x + 6)
• C6(2x² + 3x)
• D2x(6x + 9)

Answer: The highest common factor of 12x² and 18x is 6x, and taking it out leaves 2x + 3. The other versions do multiply back correctly but still have a common factor left inside the bracket, so they are not fully factorised.

Factorise x² + 8x + 15.

• A(x + 1)(x + 15)
• B(x - 3)(x - 5)
• C(x + 3)(x + 5)✓
• D(x + 4)(x + 4)

Answer: You need two numbers that multiply to 15 and add to 8. Those numbers are 3 and 5, giving (x + 3)(x + 5). Choosing 1 and 15 multiplies correctly but adds to 16.

Solve for x: 5x - 3 = 2x + 12.

• Ax = 3
• Bx = 9
• Cx = 5✓
• Dx = 15

Answer: Subtracting 2x from both sides leaves 3x - 3 = 12. Adding 3 gives 3x = 15, and dividing by 3 gives x = 5.

Solve for x: 4(x - 2) = 20.

• Ax = 3
• Bx = 5.5
• Cx = 5
• Dx = 7✓

Answer: Dividing both sides by 4 gives x - 2 = 5, so x = 7. Expanding as 4x - 2 instead of 4x - 8 is the usual mistake and leads to 5.5.

Solve the inequality 3x + 2 > 14.

• Ax > 4✓
• Bx < 4
• Cx > 12
• Dx ≥ 4

Answer: Subtracting 2 from both sides gives 3x > 12, then dividing both sides by 3 gives x > 4. The inequality sign only turns round when you multiply or divide by a negative number.

On a number line an open (unfilled) circle is drawn at -2 and the line is shaded to the right. Which inequality does this show?

• Ax ≥ -2
• Bx > -2✓
• Cx < -2
• Dx ≤ -2

Answer: An unfilled circle means the value at that point is excluded, and shading to the right means larger values, so this shows x > -2. A filled circle would have included -2 itself.

Find the nth term of the sequence 5, 8, 11, 14, ...

• An + 3
• B3n
• C3n + 5
• D3n + 2✓

Answer: The terms go up by 3 each time, so the rule begins with 3n. When n = 1 that gives 3, but the first term is 5, so 2 must be added: 3n + 2.

The nth term of a sequence is 4n - 1. What is the 10th term?

• A41
• B39✓
• C40
• D3

Answer: Substituting n = 10 gives 4 × 10 - 1, which is 40 - 1 = 39. Leaving off the -1 gives 40.

Use the formula v = u + at with u = 5, a = 3 and t = 4. What is the value of v?

• A32
• B17✓
• C12
• D60

Answer: Multiplication comes before addition, so at = 3 × 4 = 12 and then v = 5 + 12 = 17. Adding u and a first and multiplying by t gives 32, which breaks the order of operations.

What is the gradient of the straight line with equation y = 4x - 7?

• A-7
• B7
• C1/4
• D4✓

Answer: In the form y = mx + c the gradient is the number multiplying x, so the gradient here is 4. The -7 is where the line crosses the y-axis.

A straight line has a gradient of 2 and crosses the y-axis at (0, -3). What is its equation?

• Ay = 2x + 3
• By = -3x + 2
• Cy = 3x - 2
• Dy = 2x - 3✓

Answer: Using y = mx + c, the gradient m is 2 and the y-intercept c is -3, which gives y = 2x - 3. Swapping the gradient and the intercept round is the usual slip.

Two numbers satisfy both x + y = 10 and x - y = 4. What are the values of x and y?

• Ax = 7, y = 3✓
• Bx = 5, y = 5
• Cx = 6, y = 4
• Dx = 8, y = 2

Answer: Adding the two equations removes y and gives 2x = 14, so x = 7. Putting x = 7 back into x + y = 10 gives y = 3. Every other pair adds to 10 but does not have a difference of 4.