1.
Solve x² − 5x − 6 = 0.
[3 marks]
- Show both roots
- Final answer: Use quadratic formula / factorise if possible
JD SCIENCE
Pure — algebra and functions
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Express 2x² + 8x + 3 in the form 2(x + p)² + q.
[3 marks]
- Complete the square
- Final answer: 2(x + 2)² − 5
3.
Solve |3x − 1| = 5.
[3 marks]
- Two cases
- Final answer: x = 2 or x = −4/3
4.
Solve the inequality x² − 5x + 6 > 0.
[3 marks]
- Critical values 2 and 3, sketch
- Final answer: x < 2 or x > 3
5.
f(x) = 2x − 3, g(x) = x². Find fg(x) and gf(x).
[3 marks]
- Order of composition
- Final answer: fg(x) = 2x² − 3; gf(x) = (2x − 3)²
6.
Find the inverse of f(x) = (x − 1)/(x + 2), x ≠ −2.
[4 marks]
- Swap x and y, rearrange
- Final answer: f⁻¹(x) = (1 + 2x)/(1 − x), x ≠ 1
7.
The remainder when x³ + ax² − 4x + 1 is divided by (x − 1) is 3. Find a.
[3 marks]
- f(1) = 3
- Final answer: a = 5
8.
(x − 2) is a factor of x³ − 3x² + kx − 4. Find k and fully factorise.
[4 marks]
- f(2)=0 → 8 − 12 + 2k − 4 = 0 → 2k = 8 → k = 4
- Then factorise
- Final answer: k = 2; (x − 2)²(x + 1) wait check
9.
State the factor theorem.
[2 marks]
- If f(a) = 0 then (x − a) is a factor
10.
Simplify (x³ − 8)/(x − 2).
[2 marks]
- Difference of cubes
- Final answer: x² + 2x + 4
11.
Solve 2 ln x = ln 9.
[2 marks]
- ln x² = ln 9
- Final answer: x = 3 (x > 0)
12.
Solve log₁₀(x + 1) + log₁₀(x − 1) = 1.
[4 marks]
- log((x²−1))=1 so x²−1=10
- Final answer: x = √11
13.
Sketch y = |x − 2| + 1, showing the vertex and intercepts.
[3 marks]
- Vertex (2, 1)
- V shape
- y-intercept 3
14.
The function f(x) = x³ − 3x + 1 has a turning point. Find dy/dx and the stationary points.
[4 marks]
- Set derivative = 0
- Final answer: dy/dx = 3x² − 3; x = ±1
15.
Partial fractions: express (5x + 3)/((x + 1)(x − 2)) in partial fractions.
[4 marks]
- A/(x+1) + B/(x−2) = (2/3)/(x+1) + (13/3)/(x−2)
- Final answer: A=2/3? Let's compute: 5x+3 = A(x−2)+B(x+1). x=2: 13=3B, B=13/3; x=−1: −2=−3A, A=2/3
16.
Binomial expansion of (1 + 2x)⁵ up to the term in x³.
[3 marks]
- C(5,k)(2x)^k
- Final answer: 1 + 10x + 40x² + 80x³
17.
Find the term independent of x in (x + 2/x)⁶.
[3 marks]
- (6 choose r) x^{6−r} 2^r x^{−r}
- Final answer: C(6,3)×1³×2³ = 160
18.
State the conditions for the binomial expansion of (1 + x)^n when n is not a positive integer.
[2 marks]
- |x| < 1
19.
Solve 3^{2x} − 12 × 3^x + 27 = 0.
[4 marks]
- Quadratic in 3^x
- Final answer: 3^x = 3 or 9 so x = 1 or 2
20.
The modulus of a complex number is not required at this spec for all boards — instead solve 2^{x+1} = 5.
[3 marks]
- Take logs
- Final answer: x = log₂(5/2) or ln5/ln2 − 1
21.
f(x) = 2x + 1, domain x ≥ 0.
(a) [1 mark]
- f(x) ≥ 1
(b) [2 marks]
- (x − 1)/2
- x ≥ 1
22.
Solve simultaneously y = x + 1 and x² + y² = 25.
[4 marks]
- Substitute
- Final answer: x = 3, y = 4 or x = −4, y = −3
23.
Describe the transformation that maps y = f(x) to y = 2f(x − 3) + 1.
[3 marks]
- Translation 3 right
- Stretch ×2 parallel to y-axis
- Translation 1 up
24.
Prove that (1 + sin x)/(cos x) + (cos x)/(1 + sin x) = 2/cos x.
[4 marks]
- Identity proof — each step must be justified
- Final answer: Combine over a common denominator and use 1 − sin²x = cos²x
25.
Find the set of values of k for which kx² + 4x + k = 0 has real roots.
[4 marks]
- Discriminant 16 − 4k² ≥ 0
- Final answer: k ≤ −2 or k ≥ 2, and k ≠ 0 if needed for quadratic
26.
Explain why a function must be one-to-one to have an inverse, and how a domain restriction can fix y = x².
[3 marks]
- Each output from one input
- Restrict x ≥ 0 so it is one-to-one
27.
Expand (2 − 3x)⁴.
[3 marks]
- Binomial
- Final answer: 16 − 96x + 216x² − 216x³ + 81x⁴
28.
Solve x + 4/x = 5, x ≠ 0.
[3 marks]
- Multiply through by x
- Final answer: x = 1 or x = 4
29.
A AQA Pure paper often awards method marks for a correct first line. Write the first line you should write when using the factor theorem to test (x + 3).
[2 marks]
- f(−3) = …
30.
Given f(x) = x³ − 4x, find f(2 + h) − f(2) and hence the derivative from first principles at x = 2.
[4 marks]
- Limit as h→0 of [12h + 6h² + h³]/h
- Final answer: f'(2) = 8
31.
The roots of 2x² − 5x − 3 = 0 are α and β. Find α + β and αβ.
[2 marks]
- −b/a and c/a
- Final answer: 5/2 and −3/2
32.
State two common mistakes on a 4-mark algebra rearrangement and how to avoid them.
[2 marks]
- Not changing the sign when crossing the equals
- Dividing only one term