Coordinate geometry, sequences and the binomial expansion practice problems A level Maths, Topic 2
Straight lines and circles, arithmetic and geometric sequences and series, and the binomial expansion, including for any rational power. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.
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Key equations for coordinate geometry, sequences and the binomial expansion
| (x − a)2 + (y − b)2 = r2 | circle, centre (a, b), radius r |
|---|---|
| m1m2 = −1 | perpendicular gradients |
| un = a + (n − 1)d, Sn = n/2(2a + (n − 1)d) | arithmetic |
| S∞ = a/(1 − r), |r| < 1 | geometric sum to infinity |
A formula booklet is provided in the exam. Answers within 2.5% are marked correct, and you can type fractions such as 1/6. What the booklet gives you, and what it leaves out, is in our A level Maths formula booklet guide.
Warm-up questions
Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.
Hint
Perpendicular gradients multiply to −1.
Worked solution
- 2 × m = −1.
- m = −1/2.
Hint
un = a + (n − 1)d.
Worked solution
- u20 = 5 + 19 × 3.
- = 62.
Hint
Compare with (x − a)2 + (y − b)2 = r2.
Worked solution
- a = 2 and b = −1, and r2 = 25 so r = 5.
- The answer is A.
Exam-style questions
Questions at the level of the A level Maths exam. The ones marked Challenge combine more than one idea.
Hint
Sn = n/2(2a + (n − 1)d).
Worked solution
- S20 = 10 × (10 + 57).
- = 670.
Hint
S∞ = a/(1 − r).
Worked solution
- 12/(1 − 0.25) = 12/0.75.
- = 16.
Hint
The x3 term is 5C3(2x)3.
Worked solution
- 5C3 = 10 and 23 = 8.
- Coefficient = 80.
Hint
Use Pythagoras on the differences.
Worked solution
- √(62 + 82) = √100.
- = 10.
Hint
Use n(n − 1)/2! (3x)2 with n = ½.
Worked solution
- (½)(−½)/2 = −1/8.
- −1/8 × 9 = −9/8.
Exam question practice
One original exam-style question in the style of an A level paper. Answer it in full, with working, before you open the mark scheme: M marks are for method, A marks for accurate answers that follow from it, and B marks for correct statements on their own.
Exam-style question style · 7 points · about 8 minutes
A circle C has centre (3, −2) and passes through the point P(6, 2).
- (a) Find the radius of C.
- (b) Write down an equation for C.
- (c) Find an equation of the tangent to C at P, in the form ax + by + c = 0.
Write your full answer on paper first, then open the scoring guide and tick each point you earned.
Scoring guide and model answer
Part (a)
r = √((6 − 3)2 + (2 + 2)2) = √25 = 5.
Part (b)
(x − 3)2 + (y + 2)2 = 25.
Part (c)
Gradient of radius = 4/3, so tangent gradient = −3/4. y − 2 = −3/4(x − 6), giving 3x + 4y − 26 = 0.
Your mark: 0 of 7
Common mistakes in coordinate geometry, sequences and the binomial expansion
- Reading the centre of (x + 1)2 as (1, …) instead of (−1, …).
- Using the sum to infinity when |r| ≥ 1.
- Forgetting to raise the coefficient inside the bracket, as in (2x)3 = 8x3.
- Using the binomial series outside its range of validity.
Keep going
Revise the ideas behind these questions:
Then try real exam questions: A level Maths past papers, with official scoring guidelines. · All A level Maths units