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Mixed skills
10 original practice questions · Untimed · Fractions and exact decimals accepted
1. The Venn diagram gives counts in each region. How many members are in A∪B?
Open full-size diagram ↗2. What is the coefficient of x when (3x+2)(x+3) is expanded?
algebra-methods-01-13. A circle has radius 20 cm. A chord is at perpendicular distance 12 cm from the centre. Find the chord length in cm.
circle-theorems-04-24. Find the larger root of x²−9x+18=0.
Optional written extension · self-assessment only
For x²−9x+18=0: (a) factor and give both roots; (b) identify the larger root; (c) if x and x−2 are rectangle lengths, decide whether either root must be rejected and justify.
Show your reasoning for each part. This task is not automatically scored.
5. For a fixed job, time is inversely proportional to the number of equally productive workers. 4 workers need 12 hours. How many hours would 8 workers need?
proportion-04-26. Find the lowest common multiple of 24 and 36.
number-structure-04-47. A sequence is defined by u_n=n²+3n for n=1,2,3,... . Find u_5−u_4.
quadratics-sequences-04-18. A sphere has radius 5 cm. Its surface area is cπ cm². Find c.
Answer: __________________
solids-and-scale-02-39. What is the greatest integer x satisfying 2x+3<10?
algebra-methods-04-110. Write x²+10x+7=(x+5)²+k. What is k?
quadratics-sequences-02-3Answers & reasoning
1. 21 · Choice B
Add both set counts. The overlap was counted twice, so subtract it once. The result counts every member of either set once.
sets-functions-01-12. 11 · Choice C
The x terms come from the two cross-products. They are 9x and 2x. Add their coefficients.
algebra-methods-01-13. 32 · Choice D
The perpendicular from the centre bisects the chord; use Pythagoras for half the chord and double. Using the stated quantities: 2×√(20^2-12^2) = 32. Check the domain, units and whether the answer is an exact value, bound or estimate.
circle-theorems-04-24. 6 · Choice B
Factor as (x−3)(x−6)=0. Set each factor to zero. Compare 3 and 6.
Optional written extension · self-assessment only
For x²−9x+18=0: (a) factor and give both roots; (b) identify the larger root; (c) if x and x−2 are rectangle lengths, decide whether either root must be rejected and justify.
- Finds (x−3)(x−6) (1 points)
- Uses zero-product reasoning to obtain 3 and 6 (1 points)
- Identifies 6 as the larger root (1 points)
- Recognises both roots give positive x and x−2, so neither is rejected by positivity alone (1 points)
(x−3)(x−6)=0, so x=3 or x=6.
The larger root is 6.
At 3 the lengths are 3 and 1; at 6 they are 6 and 4. Both are positive, so further context would be needed to select only one.
5. 6 · Choice C
The number of workers times hours stays constant. The total is 48 worker-hours. Divide by the new number of workers.
proportion-04-26. 72 · Choice C
List multiples or combine prime factors at their greatest powers. 72 is divisible by each given number. No smaller positive common multiple exists.
number-structure-04-47. 12 · Choice C
Evaluate the two specified terms separately. u_5=40 and u_4=28. Subtract in the requested order.
quadratics-sequences-04-18. 100
The surface-area formula is 4πr²; distinguish area from volume. Using the stated quantities: 4×5^2 = 100. Check the domain, units and whether the answer is an exact value, bound or estimate.
solids-and-scale-02-39. 3 · Choice D
Isolate x to obtain x<3+1/2. The greatest integer below this boundary is 3. Substitute that integer to check the strict inequality.
algebra-methods-04-110. -18 · Choice D
The square expands to x²+10x+25. The constants on both sides must agree. k=7−25.
quadratics-sequences-02-3Original parameterized practice, independently checked by computation. Educator review pending. Not an official exam paper.