1.
Expand and simplify 3(x + 2) + 2(x − 1).
[2 marks]
- Expand then collect like terms
- Final answer: 5x + 4
JD SCIENCE
Algebra
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Expand (x + 3)(x + 2).
[2 marks]
- FOIL
- Final answer: x² + 5x + 6
3.
Factorise x² + 5x + 6.
[2 marks]
- Two numbers that multiply to c term and add to x term
- Final answer: (x + 3)(x + 2)
4.
Factorise fully 6x² + 9x.
[2 marks]
- Highest common factor 3x
- Final answer: 3x(2x + 3)
5.
Solve 5x − 2 = 18.
[2 marks]
- Add, then divide
- Final answer: x = 4
6.
Solve 3x + 7 = 5x − 3.
[3 marks]
- Collect x terms on one side
- Final answer: x = 5
7.
Solve the inequality 2x + 1 < 11.
(a) [2 marks]
- x < 5
(b) [1 mark]
- Open circle at the critical value, arrow to the left
8.
Solve the simultaneous equations: y = 2x + 1 and 3x + y = 11.
[3 marks]
- Substitute y into the second equation
- Final answer: x = 2, y = 5
9.
Solve x² − 5x + 6 = 0.
[3 marks]
- Factorise (x − 2)(x − 3)
- Final answer: x = 2 or x = 3
10.
Solve x² + 4x − 2 = 0. Give answers in surd form.
[3 marks]
- Quadratic formula / complete the square
- Final answer: x = −2 ± √6
11.
Complete the square for x² + 6x + 5.
[3 marks]
- Half of 6 is 3
- Final answer: (x + 3)² − 4
12.
The nth term of a sequence is 3n + 2. Find the first 3 terms and the 20th term.
[3 marks]
- Substitute n = 1, 2, 3, 20
- Final answer: 5, 8, 11; 20th = 62
13.
Here is a sequence: 5, 8, 11, 14, … Find the nth term.
[2 marks]
- 3n + 2
14.
Rearrange v = u + at to make t the subject.
[2 marks]
- Subtract u, divide by a
- Final answer: t = (v − u)/a
15.
Rearrange A = 2πr(r + h) to make h the subject.
[3 marks]
- Divide by 2πr then subtract r
- Final answer: h = A/(2πr) − r
16.
Simplify (3x²y)³.
[2 marks]
- Cube each factor; multiply indices
- Final answer: 27x⁶y³
17.
Simplify (8x⁶)¹/³.
[2 marks]
- Cube root of 8 is 2; divide index by 3
- Final answer: 2x²
18.
f(x) = 3x − 1. Find f(4) and f⁻¹(x).
[3 marks]
- Swap x and y, rearrange
- Final answer: f(4) = 11; f⁻¹(x) = (x + 1)/3
19.
Solve 2^{x} = 32.
[2 marks]
- 32 = 2⁵
- Final answer: x = 5
20.
The line L has gradient 3 and passes through (0, 2). Write the equation of L.
[2 marks]
- y = mx + c
- Final answer: y = 3x + 2
21.
Find the equation of the line through (2, 5) and (6, 13).
[3 marks]
- Gradient = 8/4 = 2, then substitute a point
- Final answer: y = 2x + 1
22.
Expand (2x − 3)².
[2 marks]
- (2x − 3)(2x − 3)
- Final answer: 4x² − 12x + 9
23.
Simplify (x² − 9)/(x − 3), x ≠ 3.
[2 marks]
- Difference of two squares
- Final answer: x + 3
24.
Amira solves 3(x − 2) = 12 and writes x − 2 = 12, so x = 14.
(a) [1 mark]
- Did not divide 12 by 3 / expanded incorrectly
(b) [2 marks]
- x − 2 = 4
- x = 6
25.
Solve |2x − 1| = 7.
[3 marks]
- 2x − 1 = 7 or 2x − 1 = −7
- Final answer: x = 4 or x = −3
26.
The quadratic graph y = x² − 4x + 3 crosses the x-axis at A and B. Find the coordinates of A and B.
[3 marks]
- Factorise (x − 1)(x − 3)
- Final answer: (1, 0) and (3, 0)
27.
Sketch y = (x + 2)(x − 4), showing intercepts.
[4 marks]
- x-intercepts −2 and 4
- y-intercept −8
- U-shaped parabola
28.
Work out the 5th triangular number and show that the nth triangular number is n(n + 1)/2 for n = 5.
[2 marks]
- 1+2+3+4+5 = 15 and 5×6/2 = 15
- Final answer: 15
29.
Solve 4x² = 64.
[2 marks]
- x² = 16
- Final answer: x = 4 or x = −4
30.
If p is directly proportional to q and p = 5 when q = 2, find p when q = 8.
[3 marks]
- p = kq, k = A/2, p = 4A
- Final answer: 20
31.
Simplify 2(x + 4) − (x − 4).
[2 marks]
- Watch the minus sign in front of the second bracket
- Final answer: x + 12
32.
On a AQA paper, a 3-mark algebra question usually needs a method mark. Write two things an examiner looks for when you solve a quadratic.
[2 marks]
- Correct factorisation or formula substitution
- Both solutions, including the negative root if it exists