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Mixed practice · Upper Secondary · G3 Mathematics

Final Push Mini Check

Seven original mixed questions for a short final review, with complete worked solutions.

At a glance

Level
Upper Secondary · G3 Mathematics
Focus
Mixed review
Time
About 20–30 minutes
Format
Seven questions

Complete practice set

Show your working. Give non-exact numerical answers to 3 significant figures unless otherwise stated. Questions 1–3 use familiar methods; in 4–5, choose the approach; in 6, question the obvious answer; and in 7, organise a multi-step problem.

Download the complete PDF ↓

Questions

  1. Question 01

    A quadratic curve has equation y = x² + bx + c. Its x-intercepts are −2 and 6. Find its minimum point.

  2. Question 02

    Two fair six-sided dice are rolled. A student records the larger score (recording that score once if the dice agree). Find the probability that the recorded score is 4.

  3. Question 03

    Before rounding to the nearest cent, a bill has a 9% service charge and then a 5% discount on the resulting amount. The calculated amount is $93.195. Find the original bill b. Is the net change a 4% increase? Explain.

  4. Question 04

    AB and DE are parallel segments with lengths 6 cm and 9 cm. Lines AD and BE cross at C, with C between A and D and between B and E. Given that AC = 8 cm, find CD. Give geometrical reasons for your working.

    Original worksheet diagram: AB is parallel to ED; lines AD and BE intersect at C. AB is labelled 6 cm and ED 9 cm.

  5. Question 05

    In triangle ABC, AB = 12 cm, AC = 8 cm and angle ABC = 30°. Find all possible values of angle ACB. Explain why the information does not determine a unique triangle. Do not calculate any side lengths.

  6. Question 06

    (a) A cyclist travels the first half of a route at 12 km/h and the second half at 18 km/h. Find the average speed. (b) Rice costing 12/kg and rice costing 18/kg are bought in equal dollar amounts. Find the average cost per kilogram of the combined purchase. Explain why 15 is not the answer to either part.

  7. Question 07

    A school is arranging an outing for n pupils, where n is an integer from 100 to 140. Two packages are available. A: Coaches seat 36 and cost $150 each. Admission costs $12 per pupil, reduced to $10 for every pupil if at least 120 attend. B: Each package costs $590 and includes a 40-seat coach and admission for up to 40 pupils. Partly filled coaches cost the full amount. (a) Find the cheaper option for 117 pupils. (b) Find all values of n for which A is cheaper than B. Justify your answer. (c) With a budget of $1900, find the greatest number of pupils who can attend and state the package.

Worked solutions

Try each question before opening its solution. In print, all solutions are shown.

Question 01 · reveal solution
  1. Step 1

    The intercepts give the factorised form directly: y = (x + 2)(x − 6).

  2. Step 2

    = x² − 4x − 12 = (x − 2)² − 16.

  3. Step 3

    The square term is smallest when x = 2.

Answer: (2, −16)

If this broke, look here first: Start with the x-intercepts → factors → turning point.

Question 02 · reveal solution
  1. Step 1

    For the recorded score to be 4, both dice must be at most 4, but they cannot both be at most 3.

  2. Step 2

    Number of favourable outcomes = 4² − 3² = 7.

  3. Step 3

    There are 7 favourable outcomes from 36 equally likely outcomes.

Answer: 7/36

If this broke, look here first: “Largest score is 4” means both dice ≤ 4, but not both ≤ 3.

Question 03 · reveal solution
  1. Step 1

    0.95(1.09b) = 93.195.

  2. Step 2

    b = 93.195 ÷ (0.95 × 1.09) = 90.

  3. Step 3

    The overall multiplier is 1.09 × 0.95 = 1.0355, so it is not a 4% increase.

Answer: Original bill = $90; net increase = 3.55%

If this broke, look here first: Turn each percentage into a multiplier and apply them in order.

Question 04 · reveal solution
  1. Step 1

    ∠ACB = ∠DCE (vertically opposite angles) and ∠CAB = ∠CDE (alternate angles). Therefore triangles ACB and DCE are similar.

  2. Step 2

    AC/CD = AB/DE = 6/9, so 8/CD = 6/9.

  3. Step 3

    CD = 12 cm.

Answer: CD = 12 cm

If this broke, look here first: Establish the equal angles before using the lengths.

Question 05 · reveal solution
  1. Step 1

    Using the sine rule: sin C / 12 = sin 30° / 8, so sin C = 3/4.

  2. Step 2

    The calculator gives C = 48.590…°.

  3. Step 3

    Since sine is also positive in the second quadrant, C may also be 180° − 48.590…° = 131.409…°.

  4. Step 4

    Both values leave a positive third angle when added to 30°, so both triangles are possible.

Answer: Angle ACB = 48.6° or 131°

If this broke, look here first: Check whether a second angle is possible after applying the sine rule.

Question 06 · reveal solution
  1. Step 1

    (a) Take a 36 km route. The time is 18/12 + 18/18 = 2.5 h, and average speed = 36/2.5 = 14.4 km/h.

  2. Step 2

    (b) Spend $36 on each type. The mass is 36/12 + 36/18 = 5 kg, and average cost = 72/5 kg = 14.40/kg.

  3. Step 3

    The quantities are not equally weighted, so the arithmetic mean 15 is not valid in either part.

Answer: (a) 14.4 km/h; (b) $14.40/kg

If this broke, look here first: Work out the correct weighting before choosing a formula.

Question 07 · reveal solution
  1. Step 1

    (a) For 117 pupils, A needs four coaches and B needs three packages. Cost of A = 4(150) + 117(12) = $2004. Cost of B = 3(590) = $1770. Package B is cheaper.

  2. Step 2

    For part (b), split the range wherever a coach count or the admission price changes: after 108 pupils, at 120 pupils, and after 120 pupils.

    Pupils nCost A ($)Cost B ($)Cheaper
    100 to 108450 + 12n1770A
    109 to 119600 + 12n1770B
    12018001770B
    121 to 140600 + 10n2360A
  3. Step 3

    Therefore A is cheaper for 100 ≤ n ≤ 108 or 121 ≤ n ≤ 140.

  4. Step 4

    (c) Above 120 pupils, B costs $2360 and exceeds the budget. Under A: 600 + 10n ≤ 1900, so n ≤ 130. At n = 130, A costs exactly $1900. Hence the greatest possible number is 130 pupils, using package A.

Answer: (a) B; (b) 100 ≤ n ≤ 108 or 121 ≤ n ≤ 140; (c) 130 pupils, package A

If this broke, look here first: Split into cases whenever a coach count or admission price changes.

Transcribed from Mentora O-Level E-Math Final Push, 9 pages. The downloadable PDF retains the original authored page layout.

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