1.
Solve x² − 7x − 6 = 0. [3]
JD SCIENCE
Pure — algebra and functions
Answer all questions. Show working. Answers are in a separate file on the Worksheets page — do not look at them until you have finished.
2.
Express 2x² + 8x + 3 in the form 2(x + p)² + q. [3]
3.
Solve |3x − 1| = 5. [3]
4.
Solve the inequality x² − 5x + 6 > 0. [3]
5.
f(x) = 4x − 3, g(x) = x². Find fg(x) and gf(x). [3]
6.
Find the inverse of f(x) = (x − 1)/(x + 2), x ≠ −2. [4]
7.
The remainder when x³ + ax² − 4x + 1 is divided by (x − 1) is 3. Find a. [3]
8.
(x − 2) is a factor of x³ − 3x² + kx − 4. Find k and fully factorise. [4]
9.
State the factor theorem. [2]
10.
Simplify (x³ − 8)/(x − 2). [2]
11.
Solve 2 ln x = ln 9. [2]
12.
Solve log₁₀(x + 1) + log₁₀(x − 1) = 1. [4]
13.
Sketch y = |x − 2| + 1, showing the vertex and intercepts. [3]
14.
The function f(x) = x³ − 3x + 1 has a turning point. Find dy/dx and the stationary points. [4]
15.
Partial fractions: express (5x + 3)/((x + 1)(x − 2)) in partial fractions. [4]
16.
Binomial expansion of (1 + 2x)⁵ up to the term in x³. [3]
17.
Find the term independent of x in (x + 2/x)⁶. [3]
18.
State the conditions for the binomial expansion of (1 + x)^n when n is not a positive integer. [2]
19.
Solve 3^{2x} − 12 × 3^x + 27 = 0. [4]
20.
The modulus of a complex number is not required at this spec for all boards — instead solve 2^{x+1} = 5. [3]
21.
f(x) = 2x + 1, domain x ≥ 0.
(a) Find the range of f. [1]
(b) Find f⁻¹(x) and its domain. [2]
22.
Solve simultaneously y = x + 1 and x² + y² = 25. [4]
23.
Describe the transformation that maps y = f(x) to y = 2f(x − 3) + 1. [3]
24.
Prove that (1 + sin x)/(cos x) + (cos x)/(1 + sin x) = 2/cos x. [4]
25.
Find the set of values of k for which kx² + 4x + k = 0 has real roots. [4]
26.
Explain why a function must be one-to-one to have an inverse, and how a domain restriction can fix y = x². [3]
27.
Expand (2 − 3x)⁴. [3]
28.
Solve x + 4/x = 5, x ≠ 0. [3]
29.
A OCR 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]
30.
Given f(x) = x³ − 4x, find f(2 + h) − f(2) and hence the derivative from first principles at x = 2. [4]
31.
The roots of 2x² − 5x − 3 = 0 are α and β. Find α + β and αβ. [2]
32.
State two common mistakes on a 4-mark algebra rearrangement and how to avoid them. [2]