Algebra

Simplify 3a + 2b + a - b.

• A4a + b✓
• B4a + 3b
• C3a + 2b
• D5ab

Answer: Collect like terms: 3a + a = 4a, and 2b - b = b. So 3a + 2b + a - b = 4a + b.

Expand 3(x + 4).

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

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

Solve x + 5 = 12.

• Ax = 17
• Bx = 7✓
• Cx = -7
• Dx = 2.4

Answer: Subtract 5 from both sides: x + 5 - 5 = 12 - 5, so x = 7.

Solve 2x = 16.

• Ax = 18
• Bx = 8✓
• Cx = 32
• Dx = 14

Answer: Divide both sides by 2: 2x ÷ 2 = 16 ÷ 2, so x = 8.

If y = 3x + 2, what is y when x = 4?

• A10
• B9
• C12
• D14✓

Answer: Substitute x = 4: y = 3(4) + 2 = 12 + 2 = 14.

Simplify 5x × 2y.

• A10xy✓
• B5xy × 2
• C10x + y
• D7xy

Answer: 5x × 2y = (5 × 2) × (x × y) = 10xy.

Expand 2(3x - 5).

• A6x - 5
• B3x - 10
• C5x - 5
• D6x - 10✓

Answer: 2 × 3x = 6x, and 2 × (-5) = -10. So 2(3x - 5) = 6x - 10.

Solve 3x - 2 = 10.

• Ax = 12
• Bx = 8
• Cx = 2.67
• Dx = 4✓

Answer: Add 2 to both sides: 3x = 12. Divide by 3: x = 4.

Factorise 6a + 9.

• A6(a + 1.5)
• B3(2a + 3)✓
• C2(3a + 4.5)
• D3(a + 3)

Answer: Find the highest common factor: HCF(6, 9) = 3. So 6a + 9 = 3(2a + 3).

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

• A12
• B13
• C14✓
• D15

Answer: The sequence increases by 3 each time (2→5, 5→8, 8→11). So the next term is 11 + 3 = 14.

What is the nth term formula for the sequence 3, 6, 9, 12, ...?

• An + 2
• Bn + 3
• C3n✓
• D3n + 3

Answer: Each term is 3 times its position: 1st term = 3(1) = 3, 2nd term = 3(2) = 6, 3rd term = 3(3) = 9. The nth term is 3n.

Expand (x + 2)(x + 3).

• Ax² + 6x + 5
• B2x + 5
• Cx² + 5x + 6✓
• Dx² + 6

Answer: x(x+3) + 2(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6.

Solve 4x + 3 = 2x + 11.

• Ax = 2
• Bx = 4✓
• Cx = 14
• Dx = 8

Answer: Subtract 2x from both sides: 2x + 3 = 11. Subtract 3: 2x = 8. Divide by 2: x = 4.

Simplify 12x² ÷ 3x.

• A9x
• B12x
• C4x✓
• D4x²

Answer: 12x² ÷ 3x = (12 ÷ 3) × (x² ÷ x) = 4 × x = 4x.

What is the 5th term in the sequence 1, 4, 7, 10, ...?

• A13✓
• B14
• C15
• D16

Answer: The sequence increases by 3. The terms are: 1st = 1, 2nd = 4, 3rd = 7, 4th = 10, 5th = 13.

Factorise 8a - 12.

• A4(2a - 3)✓
• B2(4a - 6)
• C8(a - 1.5)
• D4(2a - 12)

Answer: HCF(8, 12) = 4. So 8a - 12 = 4(2a - 3).

If p = 2q - 3, what is p when q = 5?

• A13
• B-1
• C7✓
• D10

Answer: Substitute q = 5: p = 2(5) - 3 = 10 - 3 = 7.

Expand 5(2x + 4) - 3x.

• A10x + 1
• B7x + 20✓
• C10x + 20
• D7x + 40

Answer: 5(2x + 4) = 10x + 20. Then 10x + 20 - 3x = 7x + 20.

Solve 2x + 5 = 3x - 1.

• Ax = -4
• Bx = 4
• Cx = 6✓
• Dx = -6

Answer: Subtract 2x: 5 = x - 1. Add 1: 6 = x, so x = 6.

What is the nth term formula for 5, 10, 15, 20, ...?

• An + 5
• B10n - 5
• C5n✓
• D5n + 5

Answer: Each term is 5 times its position. 1st = 5(1) = 5, 2nd = 5(2) = 10, etc. The nth term is 5n.

Solve 2x + 5 = 13.

• Ax = 9
• Bx = 6
• Cx = 4✓
• Dx = 2

Answer: 2x + 5 = 13. Subtract 5: 2x = 8. Divide by 2: x = 4.

Solve 3x - 7 = 11.

• Ax = 6✓
• Bx = 4
• Cx = 3
• Dx = 5

Answer: 3x - 7 = 11. Add 7: 3x = 18. Divide by 3: x = 6.

Solve x/4 + 3 = 8.

• Ax = 16
• Bx = 20✓
• Cx = 12
• Dx = 5

Answer: x/4 + 3 = 8. Subtract 3: x/4 = 5. Multiply by 4: x = 20.

Which is the solution to 2x + 3 > 11?

• Ax > 3
• Bx > 7
• Cx > 4✓
• Dx > 5

Answer: 2x + 3 > 11. Subtract 3: 2x > 8. Divide by 2: x > 4.

Make x the subject of y = 4x - 7.

• Ax = (y - 7)/4
• Bx = 4y + 7
• Cx = (y + 7)/4✓
• Dx = y/4 + 7

Answer: Add 7 to both sides: y + 7 = 4x. Then divide both sides by 4: x = (y + 7)/4.

Solve 5x - 2 ≤ 18.

• Ax ≤ 3
• Bx ≤ 5
• Cx ≤ 4✓
• Dx ≤ 6

Answer: 5x - 2 ≤ 18. Add 2: 5x ≤ 20. Divide by 5: x ≤ 4.

Solve the simultaneous equations: x + y = 7 and x - y = 3.

• Ax = 4, y = 3
• Bx = 6, y = 1
• Cx = 2, y = 5
• Dx = 5, y = 2✓

Answer: Adding the equations: (x + y) + (x - y) = 7 + 3, so 2x = 10 and x = 5. Then 5 + y = 7 gives y = 2. So x = 5, y = 2.

Factorise x² + 7x + 12.

• A(x + 2)(x + 6)
• B(x + 1)(x + 12)
• C(x - 3)(x - 4)
• D(x + 3)(x + 4)✓

Answer: Look for two numbers that multiply to give 12 and add to give 7. Those numbers are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

Solve 2x + y = 8 and x - y = 1.

• Ax = 2, y = 4
• Bx = 3, y = 2✓
• Cx = 5, y = -2
• Dx = 4, y = 1

Answer: Add the equations: 3x = 9, so x = 3. Substitute into x - y = 1: 3 - y = 1, so y = 2.

What is the nth term for the sequence 5, 8, 11, 14, ...?

• A3n
• B3n + 2✓
• Cn + 4
• D2n + 3

Answer: The sequence increases by 3 each time. When n = 1, the term is 5, so 3(1) + 2 = 5. When n = 2, 3(2) + 2 = 8. So the nth term is 3n + 2.

What is the nth term for the sequence 2, 5, 10, 17, ...?

• An² + 1✓
• Bn² + n
• C2n²
• D2n + 1

Answer: Differences: 3, 5, 7 (increasing by 2). These are quadratic. Testing: n² + 1: when n=1, 2; n=2, 5; n=3, 10; n=4, 17. This matches!

A function machine multiplies by 3 then adds 5. If the input is 4, what is the output?

• A12
• B17✓
• C20
• D27

Answer: Input: 4. Multiply by 3: 4 × 3 = 12. Add 5: 12 + 5 = 17.

A function machine divides by 2 then subtracts 3. If the output is 2, what was the input?

• A4
• B5
• C10✓
• D14

Answer: Work backwards from the output 2: add 3 to get 5, then multiply by 2 to get 10. The input was 10.

The nth term of a sequence is 5n + 2. What is the 10th term?

• A50
• B7
• C52✓
• D57

Answer: Substitute n = 10 into 5n + 2: 5 × 10 + 2 = 50 + 2 = 52.

Which graph represents y = 2x + 1?

• AA straight line with gradient 2 and y-intercept -1
• BA straight line with gradient 1 and y-intercept 2
• CA straight line with gradient -2 and y-intercept 1
• DA straight line with gradient 2 and y-intercept 1✓

Answer: The equation y = 2x + 1 is in the form y = mx + c, where m = 2 (gradient) and c = 1 (y-intercept).

What is the gradient of the line y = -3x + 5?

• A-5
• B3
• C5
• D-3✓

Answer: In y = mx + c, the gradient is m. Here, m = -3.

What is the y-intercept of the line y = 4x - 7?

• A7
• B-4
• C4
• D-7✓

Answer: In y = mx + c, the y-intercept is c. Here, c = -7.

A pen costs p pence. A ruler costs 30 pence more than a pen. Write an expression for the total cost, in pence, of 4 pens and 1 ruler.

• A4p + 30
• B4p + 30p
• C5p + 120
• D5p + 30✓

Answer: Four pens cost 4p and the ruler costs p + 30, so the total is 4p + p + 30 = 5p + 30 pence.

Solve 2(x + 3) = 14.

• Ax = 11
• Bx = 5.5
• Cx = 4✓
• Dx = 7

Answer: Divide both sides by 2: x + 3 = 7. Then subtract 3 from both sides: x = 4. Expanding first gives 2x + 6 = 14, which leads to the same answer.

Factorise 4x + 8.

• A4(x + 2)✓
• B2(2x + 4)
• C8(x + 1)
• D4(x + 1)

Answer: 4x + 8 = 4(x + 2). The HCF is 4.