Mathematics: Model Question Sets and Predicted Questions

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Mathematics: Model Question Sets and Predicted Questions

Three full papers to the grid — ten one-mark items, ten two-mark questions, five four-mark questions, two hours. In Mathematics the clock matters more than in any other paper, because a long answer cannot be guessed. Give Group A no more than twelve minutes, and keep the last ten for checking: a sign, a unit and a transferred digit are the three commonest ways to lose marks you had already earned.

Rough work belongs on the paper

Do not rub out working. If the final number is wrong but the method is visible, the step marks still come. If the working is erased, a wrong answer earns nothing.

7.1 Model Set 1

Group A [10×1=10]

  1. If n(A) = 15, n(B) = 10 and n(A\cap B) = 6, find n(A \cup B).
  2. Write the value of 7^0 + 2^{-1}.
  3. Find the simple interest on Rs 1,000 for 1 year at 5 % per annum.
  4. Write the formula for the volume of a cylinder.
  5. The area of a square is 49 cm². Write its perimeter.
  6. State the quadrant in which (-2,-7) lies.
  7. Write the bearing of due south.
  8. Find the mode of 3, 4, 4, 6, 9.
  9. Write the number of edges of a cube.
  10. If 2:x = 6:15, find x.

Group B [10×2=20]

  1. If A = \{2,4,6,8\} and B = \{4,8,12\}, find A \cup B and A \cap B.
  2. Simplify: \frac{a^6 \times a^2}{a^5}.
  3. Divide Rs 900 in the ratio 4:5.
  4. Find the selling price of an article bought for Rs 2,400 and sold at a loss of 8 %.
  5. Find the area of a trapezium with parallel sides 9 cm and 15 cm and height 8 cm.
  6. Solve: 5x - 3 = 2x + 12.
  7. Two angles of a triangle are 38° and 72°; find the third angle.
  8. Find the interior angle of a regular octagon.
  9. Find the image of (3,-4) under reflection in the y-axis.
  10. Find the median of 11, 7, 15, 9, 20, 13.

Group C [5×4=20]

  1. In a group of 50 students, 32 like Mathematics, 28 like Science and 15 like both. Find how many like at least one subject and how many like neither, and show the result in a Venn diagram.
  2. A bag is marked Rs 2,500. The shopkeeper gives a 12 % discount and then charges 13 % VAT. Find the amount paid by the customer.
  3. A rectangular plot is 24 m long and 18 m wide. Find the cost of fencing it at Rs 250 per metre and the cost of levelling it at Rs 60 per m².
  4. Simplify \frac{x^2-16}{x^2-x-12} and state the values of x for which it is undefined.
  5. Find the mean of the following distribution using mid-values: 0–10: 4, 10–20: 9, 20–30: 12, 30–40: 10, 40–50: 5.
Model Set 1 — answers

Group A. 1. 19. 2. 1 + 0.5 = 1.5. 3. Rs 50. 4. V = \pi r^2 h. 5. 28 cm. 6. third quadrant. 7. 180°. 8. 4. 9. 12. 10. x = 5.

Group B. 11. A\cup B = \{2,4,6,8,12\}; A\cap B = \{4,8\}. 12. a^{8-5} = a^3. 13. One part = 100; shares Rs 400 and Rs 500. 14. Loss = 192; S.P. = Rs 2,208. 15. \frac{1}{2}(9+15)\times8 = 96 cm². 16. 3x = 15, x = 5. 17. 180° - 110° = 70°. 18. \frac{6\times180°}{8} = 135°. 19. (-3,-4). 20. Ordered: 7,9,11,13,15,20; median = \frac{11+13}{2} = 12.

Group C. 21. n(M\cup S) = 32+28-15 = 45 like at least one; 50-45 = 5 like neither. Venn: 17 only Mathematics, 15 both, 13 only Science, 5 outside. 22. Discount = 300, price = 2200; VAT = 286; amount = Rs 2,486. 23. Perimeter = 2(24+18) = 84 m; fencing = 84\times250 = Rs 21,000. Area = 432 m²; levelling = 432\times60 = Rs 25,920. 24. \frac{(x+4)(x-4)}{(x-4)(x+3)} = \frac{x+4}{x+3}; undefined at x = 4 and x = -3. 25. \sum f = 40; \sum fx = 4(5)+9(15)+12(25)+10(35)+5(45) = 20+135+300+350+225 = 1030; mean = \frac{1030}{40} = 25.75.

7.2 Model Set 2

Group A [10×1=10]

  1. Write the number of subsets of \{a,b,c\}.
  2. Simplify: (3^2)^2.
  3. Write the formula for simple interest.
  4. Find the volume of a cube of edge 3 cm.
  5. Write the sum of the interior angles of a quadrilateral.
  6. Find the distance of (3,4) from the origin.
  7. Write the image of (x,y) under rotation of 180° about the origin.
  8. Find the range of 4, 11, 7, 20.
  9. Write the number of faces of a square pyramid.
  10. A profit of Rs 150 on a cost price of Rs 1,000 is what percentage?

Group B [10×2=20]

  1. Write any two differences between congruent and similar figures.
  2. Find the value of x if 2^{x+1} = 64.
  3. If 8 books cost Rs 1,120, find the cost of 14 such books.
  4. Find the compound ratio of 2:3 and 5:4.
  5. The circumference of a circle is 44 cm; find its radius, taking \pi = \frac{22}{7}.
  6. Factorise: x^2 + 9x + 20.
  7. Find the slope and the y-intercept of y = -3x + 7.
  8. Verify Euler's relation for a square pyramid.
  9. Find the sector angle for 18 out of 90 in a pie chart.
  10. Solve: \frac{x}{3} - 2 = 4.

Group C [5×4=20]

  1. A sum of Rs 20,000 is invested at 8 % per annum simple interest. Find the interest for 3 years and the total amount, and find how long the same sum would take to earn Rs 8,000.
  2. A cylindrical drum of radius 0.7 m and height 1 m is to be painted on its curved surface and both circular ends. Find the total area painted, taking \pi = \frac{22}{7}.
  3. In the figure described: two parallel lines are cut by a transversal; one of the angles so formed is 115°. Find all the other seven angles with reasons.
  4. The vertices of a triangle are A(1,1), B(5,1), C(1,4). Find the image of the triangle under reflection in the x-axis and state the area of the original triangle.
  5. The daily wages of 50 workers are: 200–300: 8, 300–400: 15, 400–500: 17, 500–600: 10. Find the mean daily wage.
Model Set 2 — answers

Group A. 1. 2^3 = 8. 2. 3^4 = 81. 3. I = \frac{PTR}{100}. 4. 27 cm³. 5. 360°. 6. 5 units. 7. (-x,-y). 8. 16. 9. 5. 10. 15 %.

Group B. 11. Congruent figures have the same shape and size, with corresponding sides equal; similar figures have the same shape only, with corresponding sides proportional and angles equal. 12. 2^{x+1} = 2^6, so x+1 = 6 and x = 5. 13. One book Rs 140, so 14 books = Rs 1,960. 14. \frac{2}{3}\times\frac{5}{4} = \frac{10}{12} = 5:6. 15. 2\pi r = 44 \Rightarrow r = \frac{44\times7}{2\times22} = 7 cm. 16. (x+4)(x+5). 17. Slope -3, intercept 7, i.e. the point (0,7). 18. F = 5, V = 5, E = 8: 5+5-8 = 2. 19. \frac{18}{90}\times360° = 72°. 20. \frac{x}{3} = 6, x = 18.

Group C. 21. I = \frac{20000\times3\times8}{100} = Rs 4,800; amount = 20000 + 4800 = Rs 24,800. For I = 8000: T = \frac{100\times8000}{20000\times8} = 5 years. 22. Curved surface = 2\pi rh = 2\times\frac{22}{7}\times0.7\times1 = 4.4 m². Two ends = 2\pi r^2 = 2\times\frac{22}{7}\times0.49 = 3.08 m². Total = 7.48 m². 23. The 115° angle has: its vertically opposite angle 115°; the two angles on the straight line with it 65° each; on the other parallel line, the corresponding angle 115°, its vertically opposite 115°, and the remaining two 65° each. Reasons: vertically opposite angles equal, angles on a straight line supplementary, corresponding angles equal between parallels. 24. Reflection in the x-axis, (x,y)\to(x,-y): A'(1,-1), B'(5,-1), C'(1,-4). The triangle is right-angled at A with base AB = 4 and height AC = 3, so area = \frac{1}{2}\times4\times3 = 6 square units (the image has the same area). 25. Mid-values 250, 350, 450, 550. \sum fx = 8(250)+15(350)+17(450)+10(550) = 2000+5250+7650+5500 = 20400; mean = \frac{20400}{50} = Rs 408.

7.3 Model Set 3

Group A [10×1=10]

  1. Write \{x : x is a factor of 12\} in listing form.
  2. Simplify: \frac{5^7}{5^5}.
  3. Write the marked price if the discount of 10 % is Rs 90.
  4. Write the formula for the total surface area of a cuboid.
  5. Find the third angle of a triangle whose two angles are 90° and 37°.
  6. Write the coordinates of the origin.
  7. Which regular polygon has an interior angle of 108°?
  8. Find the mean of 6, 8, 10, 12.
  9. Write the formula for the area of a circle.
  10. Convert the ratio 1.5:2 into its simplest whole-number form.

Group B [10×2=20]

  1. If n(A) = 20, n(B) = 16 and n(A\cup B) = 30, find n(A\cap B).
  2. Solve: 3(x-2) = 2(x+4).
  3. Find the cost price if the selling price is Rs 1,045 at a gain of 10 %.
  4. A car travels 240 km in 4 hours. How far will it travel in 7 hours at the same speed?
  5. Find the curved surface area of a cylinder of radius 3.5 cm and height 10 cm.
  6. Expand: (2x+5)^2.
  7. Find the number of sides of a regular polygon whose exterior angle is 24°.
  8. Find the image of (−2,5) under the translation \binom{3}{-2}.
  9. Find the mode and range of 2, 5, 5, 7, 9, 9, 9, 12.
  10. Write any two properties of a rhombus.

Group C [5×4=20]

  1. In a survey of 80 households, 45 read a newspaper, 50 watch television news and 25 do both. Find how many use at least one source, how many use neither, and draw the Venn diagram.
  2. A trader bought 120 kg of rice at Rs 85 per kg. He sold 70 kg at Rs 95 per kg and the rest at Rs 80 per kg. Find his total profit or loss and the profit or loss per cent.
  3. A room 5 m long, 4 m wide and 3 m high has one door of 2 m × 1 m and two windows each 1.5 m × 1 m. Find the area of the four walls to be papered.
  4. Solve \frac{2x+1}{3} + \frac{x-1}{2} = 3 and check your answer.
  5. The marks obtained by 60 students are: 0–20: 5, 20–40: 12, 40–60: 20, 60–80: 15, 80–100: 8. Find the mean mark and the modal class.
Model Set 3 — answers

Group A. 1. \{1,2,3,4,6,12\}. 2. 5^2 = 25. 3. Rs 900. 4. 2(lb+bh+lh). 5. 53°. 6. (0,0). 7. regular pentagon. 8. 9. 9. A = \pi r^2. 10. 3:4.

Group B. 11. n(A\cap B) = 20+16-30 = 6. 12. 3x-6 = 2x+8 \Rightarrow x = 14. 13. C.P. = \frac{1045}{1.10} = Rs 950. 14. \frac{240}{4} = 60 km/h, so 60\times7 = 420 km. 15. 2\pi rh = 2\times\frac{22}{7}\times3.5\times10 = 220 cm². 16. 4x^2+20x+25. 17. \frac{360°}{24°} = 15 sides. 18. (1,3). 19. Mode 9; range 12-2 = 10. 20. All four sides equal; diagonals bisect each other at right angles (opposite angles equal, opposite sides parallel).

Group C. 21. At least one = 45+50-25 = 70; neither = 80-70 = 10. Venn: 20 newspaper only, 25 both, 25 television only, 10 outside. 22. Cost = 120\times85 = Rs 10,200. Sales = 70\times95 + 50\times80 = 6650 + 4000 = Rs 10,650. Profit = Rs 450; profit per cent = \frac{450}{10200}\times100 = 4.41 % (2 d.p.). 23. Four walls = 2h(l+b) = 2\times3\times9 = 54 m². Openings = 2 + 2(1.5) = 5 m². Area to be papered = 54 - 5 = 49 m². 24. Multiply by 6: 2(2x+1) + 3(x-1) = 18 \Rightarrow 4x+2+3x-3 = 18 \Rightarrow 7x = 19, so x = \frac{19}{7}. Check: \frac{2(19/7)+1}{3} + \frac{(19/7)-1}{2} = \frac{45/7}{3} + \frac{12/7}{2} = \frac{15}{7} + \frac{6}{7} = 3. ✓ 25. Mid-values 10, 30, 50, 70, 90. \sum fx = 5(10)+12(30)+20(50)+15(70)+8(90) = 50+360+1000+1050+720 = 3180; mean = \frac{3180}{60} = 53 marks. Modal class = 40–60, with the highest frequency 20.

7.4 Predicted questions: ranked practice priorities

What "predicted" means here

Nothing below is a forecast of a particular paper, and no local level's question is reproduced. These are practice priorities, ranked from the specification grid and from the shape of the syllabus: a topic ranks higher when it can be asked cleanly at the size the grid allows, when it has a fixed method that is quick to mark, and when the textbook gives it a full chapter. Work the top tier until the method is automatic.

7.4.1 Tier A — rehearse until automatic

Rank Topic and likely form Marks Why it ranks here
1 Profit and loss with discount and VAT, in one chain 4 Four steps, one answer — the classic long question of this paper.
2 Simple interest, with the formula rearranged for P, T or R 4 or 2 Four question forms from a single formula.
3 Area and volume: walls and ceiling of a room, cylinder in litres, cost of plastering 4 A real-life application with units to carry, which the grid rewards.
4 Sets: n(A \cup B) with a Venn diagram and a "neither" count 4 One formula, one diagram, one subtraction — easy to mark in steps.
5 Statistics: mean from a frequency table, median, mode, pie-chart angles 4 Table-based and mechanical; nearly always present.
6 Unitary method: workers and days, pipes and hours, speed and time 4 or 2 Direct and inverse variation in one question.
7 Algebra: factorise and simplify an algebraic fraction, state excluded values 4 Tests identities and the cancelling rule together.
8 Linear equations, including fractional equations and two-number problems 4 or 2 Solve-and-check is a complete four-mark answer.

7.4.2 Tier B — second pass

Rank Topic and likely form Marks Why it ranks here
9 Angles: triangle and polygon angle sums, parallel lines with a transversal 2 or 4 Reasons carry the marks, and every reason is a named rule.
10 Indices: simplification and solving a^x = b 2 Short, exact, and quick to mark.
11 Ratio and proportion: sharing in a ratio, compound ratio, mean proportional 2 or 4 Appears in both groups in most papers.
12 Coordinates: distance, midpoint, plotting a triangle and its area 4 A diagram plus two formulas.
13 Transformation: reflection and translation of a triangle 4 The image is easy to state and to check.
14 Solid objects: Euler's relation, surface area and volume of a cuboid or cube 2 One relation, one substitution.

7.4.3 Tier C — a quick pass before the paper

Rank Topic and likely form Marks Why it ranks here
15 Rational and irrational numbers: classify with a reason 1 or 2 Mostly objective items.
16 Tessellation: which regular polygons tessellate and why 2 One idea — the interior angle must divide 360°.
17 Bearing and scale drawing: three-figure bearings, back bearing, map scale 2 or 4 Short but regularly examined; practise the conversion.
18 Congruency and similarity: name the test, find a missing side 2 A named test and a scale factor.
Formula card

Write every formula from Tier A on a single card: profit per cent, discount, VAT, I=\frac{PTR}{100}, areas, volumes, n(A\cup B), mean from a table, pie-chart angle. Read the card once a day. Most lost marks in this paper come from a half-remembered formula, not from arithmetic.

After the third set

Add up where your marks went, not just how many. If Group C questions 21–25 cost you more than six marks in total, the fix is to practise one four-mark question a day from chapter 6 — written out in full, with the formula line, until the four steps are automatic.