Mathematics: Unit-wise Objective Question Bank

In Mathematics the one-mark items are one-step questions: a definition, a formula recalled correctly, a single substitution, a sign. They are also the items most often lost to carelessness rather than to ignorance. This chapter covers all twenty-one units of the Grade 8 textbook, grouped into ten sections. Each section opens with the formulas and facts the one-mark items keep returning to, then gives the items, then a key that shows the single step.

5.1 Sets, Whole Numbers, Rational and Irrational Numbers

Facts. n(A \cup B) = n(A) + n(B) - n(A \cap B). A set with n elements has 2^n subsets. Rational numbers can be written as \frac{p}{q} with q \neq 0; \sqrt{2}, \sqrt{3} and \pi are irrational. Every terminating or recurring decimal is rational. Zero is neither positive nor negative; it is a whole number but not a natural number. 0 has no multiplicative inverse.

1. If n(A) = 12, n(B) = 9 and n(A \cap B) = 4, then n(A \cup B) is [1]a17b21c25d13
2. The number of subsets of a set with 4 elements is [1]a4b8c16d32
3. Which of the following is an irrational number? [1]a0.25b\frac{3}{7}c\sqrt{9}d\sqrt{7}
4. The smallest whole number is [1]a-1b0c1dthere is none
5. The additive inverse of -\frac{5}{8} is [1]a\frac{8}{5}b-\frac{8}{5}c\frac{5}{8}d0
6. If A = \{1,2,3\} and B = \{3,4\}, then A \cap B is [1]a\{1,2,3,4\}b\{3\}c\{1,2\}d\emptyset
7. The decimal 0.\overline{6} expressed as a fraction is [1]a\frac{6}{10}b\frac{2}{3}c\frac{3}{5}d\frac{6}{11}
8. Which statement is true? [1]aevery integer is a whole numberbevery whole number is an integercevery rational number is an integerd\sqrt{2} is rational

Answers

1. (a) — 12+9-4 = 17.

2. (c) — 2^4 = 16.

3. (d) — \sqrt{9}=3 is rational; \sqrt{7} is not.

4. (b) — whole numbers start at 0.

5. (c) — the additive inverse changes the sign.

6. (b) — only 3 is common.

7. (b) — 0.\overline{6} = \frac{6}{9} = \frac{2}{3}.

8. (b) — -3 is an integer but not a whole number.

5.2 Ratio, Proportion and the Unitary Method

Facts. In the proportion a:b = c:d, ad = bc. To divide x in the ratio m:n, each share is \frac{m}{m+n}x and \frac{n}{m+n}x. In direct proportion, one quantity rises as the other rises; in inverse proportion, the product stays constant. Mean proportional between a and b is \sqrt{ab}.

1. If 3:5 = x:20, then x is [1]a10b12c15d4
2. Rs 480 divided between two friends in the ratio 3:5 gives the smaller share [1]aRs 160bRs 180cRs 240dRs 300
3. The mean proportional between 4 and 9 is [1]a6b6.5c13d36
4. If 6 workers build a wall in 10 days, 12 workers of the same capacity will take [1]a20 daysb10 daysc5 daysd2 days
5. 5 pens cost Rs 125. The cost of 8 such pens is [1]aRs 180bRs 200cRs 225dRs 250
6. The ratio 45 minutes : 2 hours in its simplest form is [1]a45:2b3:8c9:24d1:3
7. Which pair is in inverse proportion? [1]anumber of books and their costbspeed and time for a fixed distancecdistance and time at a fixed speeddside and perimeter of a square
8. If a:b = 2:3 and b:c = 4:5, then a:c is [1]a2:5b8:15c5:8d3:4

Answers

1. (b) — x = \frac{3\times20}{5} = 12.

2. (b) — \frac{3}{8}\times480 = 180.

3. (a) — \sqrt{4\times9} = 6.

4. (c) — inverse proportion: \frac{6\times10}{12} = 5 days.

5. (b) — one pen Rs 25, so 8\times25 = 200.

6. (b) — 45:120 = 3:8.

7. (b) — at fixed distance, speed × time is constant.

8. (b) — a:b:c = 8:12:15, so a:c = 8:15.

5.3 Profit and Loss, Simple Interest

Facts. Profit = S.P. - C.P.; profit per cent = \frac{\text{profit}}{\text{C.P.}}\times100. Discount is always on the marked price; VAT is charged on the price after discount. Simple interest I = \frac{PTR}{100}; amount A = P + I.

1. An article bought for Rs 800 is sold for Rs 920. The profit per cent is [1]a12 %b15 %c20 %d13 %
2. A shirt marked Rs 1,200 is sold at 10 % discount. Its selling price is [1]aRs 1,080bRs 1,100cRs 1,020dRs 1,320
3. The simple interest on Rs 5,000 for 2 years at 8 % per annum is [1]aRs 400bRs 800cRs 1,000dRs 80
4. If the cost price is Rs 500 and the loss is 12 %, the selling price is [1]aRs 440bRs 480cRs 560dRs 400
5. Rs 2,000 becomes Rs 2,300 in 3 years at simple interest. The rate per annum is [1]a3 %b5 %c10 %d15 %
6. VAT at 13 % on a bill of Rs 2,000 is [1]aRs 130bRs 200cRs 260dRs 2,260
7. Profit per cent is always calculated on the [1]aselling pricebmarked priceccost priceddiscount
8. In how many years will Rs 4,000 earn Rs 960 as simple interest at 6 % per annum? [1]a2b3c4d5

Answers

1. (b) — profit Rs 120; \frac{120}{800}\times100 = 15 %.

2. (a) — 1200 - 120 = 1080.

3. (b) — \frac{5000\times2\times8}{100} = 800.

4. (a) — 500 - 60 = 440.

5. (b) — I = 300; R = \frac{100\times300}{2000\times3} = 5 %.

6. (c) — 13 % of 2000 = 260.

7. (c) — on the cost price.

8. (c) — T = \frac{100\times960}{4000\times6} = 4 years.

5.4 Area and Volume

Facts. Rectangle A = lb; square A = l^2; triangle A = \frac{1}{2}bh; parallelogram A = bh; trapezium A = \frac{1}{2}(a+b)h; circle A = \pi r^2, circumference = 2\pi r. Cuboid: volume = lbh, total surface = 2(lb+bh+lh). Cube: volume = l^3, surface = 6l^2. Cylinder: volume = \pi r^2 h, curved surface = 2\pi rh. Take \pi = \frac{22}{7} unless told otherwise. 1 m³ = 1000 litres.

1. The area of a triangle with base 12 cm and height 9 cm is [1]a108 cm²b54 cm²c21 cm²d27 cm²
2. The volume of a cube of edge 5 cm is [1]a25 cm³b75 cm³c125 cm³d150 cm³
3. The circumference of a circle of radius 7 cm, with \pi = \frac{22}{7}, is [1]a22 cmb44 cmc154 cmd49 cm
4. The total surface area of a cube of edge 4 cm is [1]a16 cm²b64 cm²c96 cm²d24 cm²
5. The volume of a cylinder of radius 7 cm and height 10 cm is [1]a1540 cm³b220 cm³c440 cm³d770 cm³
6. A tank 2 m × 1 m × 0.5 m holds, when full, [1]a100 litresb500 litresc1000 litresd1500 litres
7. The area of a trapezium with parallel sides 8 cm and 12 cm and height 5 cm is [1]a50 cm²b100 cm²c40 cm²d60 cm²
8. If the side of a square is doubled, its area becomes [1]adoublebthree timescfour timesdunchanged

Answers

1. (b) — \frac{1}{2}\times12\times9 = 54 cm².

2. (c) — 5^3 = 125 cm³.

3. (b) — 2\times\frac{22}{7}\times7 = 44 cm.

4. (c) — 6\times4^2 = 96 cm².

5. (a) — \frac{22}{7}\times49\times10 = 1540 cm³.

6. (c) — 1 m³ = 1000 litres.

7. (a) — \frac{1}{2}(8+12)\times5 = 50 cm².

8. (c) — area varies as the square of the side.

5.5 Indices and Algebraic Expressions

Facts. a^m \times a^n = a^{m+n}; \frac{a^m}{a^n} = a^{m-n}; (a^m)^n = a^{mn}; a^0 = 1; a^{-n} = \frac{1}{a^n}; a^{1/2} = \sqrt{a}. (a+b)^2 = a^2+2ab+b^2; (a-b)^2 = a^2-2ab+b^2; a^2-b^2 = (a+b)(a-b).

1. 2^3 \times 2^4 equals [1]a2^7b2^{12}c4^7d2^{1}
2. The value of 5^0 + 3^0 is [1]a0b1c2d8
3. 3^{-2} equals [1]a-9b\frac{1}{9}c-\frac{1}{9}d9
4. (x+3)^2 expands to [1]ax^2+9bx^2+6x+9cx^2+3x+9dx^2+6x+6
5. x^2-16 factorises as [1]a(x-4)^2b(x+4)(x-4)c(x+16)(x-1)d(x+4)^2
6. \frac{a^7}{a^3} equals [1]aa^{10}ba^{21}ca^4da^{4/3}
7. \sqrt{64} written as a power of 2 is [1]a2^3b2^4c2^6d2^8
8. If 2^x = 32, then x is [1]a4b5c6d16

Answers

1. (a) — add the indices.

2. (c) — 1+1 = 2.

3. (b) — 3^{-2} = \frac{1}{3^2}.

4. (b) — (a+b)^2 = a^2+2ab+b^2.

5. (b) — difference of two squares.

6. (c) — subtract the indices.

7. (a) — \sqrt{64} = 8 = 2^3.

8. (b) — 2^5 = 32.

5.6 Algebraic Fractions and Equations

Facts. Simplify a fraction by factorising and cancelling common factors — never cancel across a + sign. To add fractions, take the LCM of the denominators. A linear equation keeps its balance: whatever is done to one side is done to the other. The graph of y = mx + c is a straight line of slope m cutting the y-axis at c.

1. \frac{x^2-9}{x+3} simplifies to [1]ax-3bx+3cx-9dx^2-3
2. The solution of 3x - 7 = 11 is [1]a4b6c18d\frac{4}{3}
3. \frac{1}{x} + \frac{1}{y} equals [1]a\frac{1}{x+y}b\frac{x+y}{xy}c\frac{2}{xy}d\frac{xy}{x+y}
4. The slope of the line y = 4x - 5 is [1]a-5b4c\frac{4}{5}d5
5. The line y = 2x + 3 cuts the y-axis at [1]a(0,2)b(0,3)c(3,0)d(2,0)
6. If \frac{x}{4} = \frac{9}{6}, then x is [1]a6b8c12d13.5
7. The value of \frac{2a}{3} - \frac{a}{6} is [1]a\frac{a}{2}b\frac{a}{3}c\frac{3a}{6}d\frac{a}{6}
8. The solution of \frac{x}{2} + 3 = 7 is [1]a2b5c8d20

Answers

1. (a) — \frac{(x+3)(x-3)}{x+3} = x-3.

2. (b) — 3x = 18, x = 6.

3. (b) — common denominator xy.

4. (b) — the coefficient of x.

5. (b) — put x = 0.

6. (a) — x = \frac{4\times9}{6} = 6.

7. (a) — \frac{4a-a}{6} = \frac{3a}{6} = \frac{a}{2}.

8. (c) — \frac{x}{2} = 4, x = 8.

5.7 Lines, Angles and Plane Figures

Facts. Angles on a straight line add to 180°; angles at a point add to 360°. Vertically opposite angles are equal. With parallel lines, corresponding and alternate angles are equal and co-interior angles are supplementary. Triangle angle sum 180°; exterior angle = sum of the two opposite interior angles. Quadrilateral angle sum 360°. Each interior angle of a regular n-gon is \frac{(n-2)180°}{n}; the exterior angles add to 360°.

1. Two angles on a straight line are x and 3x. The value of x is [1]a30°b45°c60°d90°
2. In a triangle two angles are 55° and 65°. The third angle is [1]a50°b60°c70°d120°
3. Each interior angle of a regular hexagon is [1]a90°b108°c120°d135°
4. Vertically opposite angles are always [1]asupplementarybcomplementarycequaldright angles
5. The exterior angle of a triangle is 110° and one opposite interior angle is 45°. The other is [1]a55°b65°c70°d35°
6. The sum of the exterior angles of any convex polygon is [1]a180°b360°c540°ddepends on the number of sides
7. A quadrilateral with both pairs of opposite sides parallel but no right angle is a [1]atrapeziumbparallelogramcrectangledkite
8. Two parallel lines are cut by a transversal and one co-interior angle is 70°. The other is [1]a70°b20°c110°d140°

Answers

1. (b) — 4x = 180°, x = 45°.

2. (b) — 180° - 120° = 60°.

3. (c) — \frac{(6-2)180°}{6} = 120°.

4. (c) — equal.

5. (b) — 110° - 45° = 65°.

6. (b) — always 360°.

7. (b) — a parallelogram.

8. (c) — co-interior angles are supplementary.

5.8 Congruency, Similarity and Solid Objects

Facts. Congruent figures have the same shape and size; tests: SSS, SAS, ASA, RHS. Similar figures have the same shape; corresponding sides are proportional and corresponding angles equal. For a solid: faces F, vertices V, edges E satisfy Euler's relation F + V - E = 2. A cube has 6 faces, 8 vertices, 12 edges; a triangular prism 5 faces, 6 vertices, 9 edges; a square pyramid 5 faces, 5 vertices, 8 edges.

1. Which test does not prove two triangles congruent? [1]aSSSbSAScAAAdRHS
2. A cube has edges numbering [1]a6b8c12d16
3. In two similar triangles, corresponding sides are [1]aequalbproportionalcperpendiculardunrelated
4. A solid with 5 faces and 6 vertices has edges numbering [1]a8b9c10d12
5. Two triangles have sides 3,4,5 and 6,8,10 cm. They are [1]acongruentbsimilar but not congruentcneitherdidentical
6. The number of faces of a square pyramid is [1]a4b5c6d8
7. If two figures are congruent, they are [1]asimilar as wellbnever similarcsimilar only if squaresdequal in area but not in shape
8. The RHS test applies only to [1]aequilateral trianglesbright-angled trianglescisosceles trianglesdall triangles

Answers

1. (c) — equal angles fix the shape, not the size.

2. (c) — 12 edges.

3. (b) — proportional.

4. (b) — E = F+V-2 = 5+6-2 = 9.

5. (b) — sides in the ratio 1:2, so similar, not congruent.

6. (b) — four triangular faces and one square base.

7. (a) — congruent figures are similar with ratio 1.

8. (b) — right angle, hypotenuse and one side.

5.9 Coordinates, Transformation, Tessellation, Bearing and Scale Drawing

Facts. In (x,y), x is measured along the horizontal axis. Quadrant signs: I (+,+), II (-,+), III (-,-), IV (+,-). Distance from the origin = \sqrt{x^2+y^2}. Reflection in the x-axis: (x,y) \to (x,-y); in the y-axis: (x,y) \to (-x,y). Translation by \binom{a}{b}: (x,y) \to (x+a, y+b). Rotation through 180° about the origin: (x,y) \to (-x,-y). A regular polygon tessellates if its interior angle divides 360° exactly — so triangle, square and hexagon do. Bearing is measured clockwise from north, written with three figures.

1. The point (-3, 5) lies in quadrant [1]aIbIIcIIIdIV
2. The image of (4, -2) after reflection in the x-axis is [1]a(-4,-2)b(4,2)c(-4,2)d(2,4)
3. The distance of (6,8) from the origin is [1]a10b14c48d\sqrt{14}
4. Which regular polygon does not tessellate by itself? [1]aequilateral trianglebsquarecregular pentagondregular hexagon
5. The bearing of due west is [1]a090°b180°c270°d360°
6. The image of (2,3) under translation \binom{-1}{4} is [1]a(1,7)b(3,-1)c(-2,12)d(1,-1)
7. On a map of scale 1:50{,}000, 3 cm represents [1]a150 mb1.5 kmc15 kmd150 km
8. Rotating (5,-1) through 180° about the origin gives [1]a(-5,1)b(5,1)c(-1,5)d(1,-5)

Answers

1. (b) — negative x, positive y.

2. (b) — the sign of y changes.

3. (a) — \sqrt{36+64} = 10.

4. (c) — its interior angle 108° does not divide 360°.

5. (c) — 270° clockwise from north.

6. (a) — (2-1,\,3+4).

7. (b) — 3\times50{,}000 = 150{,}000 cm = 1.5 km.

8. (a) — both signs change.

5.10 Statistics

Facts. Mean = \frac{\sum x}{n}; for a frequency table, mean = \frac{\sum fx}{\sum f}. Median is the middle value of the ordered data — for even n, the mean of the two middle values. Mode is the most frequent value. Range = largest - smallest. In a pie chart, each item's angle = \frac{\text{its frequency}}{\text{total}}\times360°.

1. The mean of 4, 7, 9, 12 is [1]a7b8c8.5d9
2. The median of 3, 7, 9, 11, 15 is [1]a7b9c11d10
3. The mode of 2, 3, 3, 5, 7, 3, 8 is [1]a2b3c5d8
4. The range of 12, 5, 20, 8 is [1]a8b12c15d20
5. The median of 4, 8, 10, 14 is [1]a8b9c10d12
6. In a pie chart of 60 students, the sector for 15 students has an angle of [1]a60°b90°c120°d150°
7. If the mean of 5 numbers is 12, their sum is [1]a12b17c60d2.4
8. Which average is most affected by a single very large value? [1]ameanbmediancmodedrange

Answers

1. (b) — \frac{32}{4} = 8.

2. (b) — the third of five ordered values.

3. (b) — 3 occurs three times.

4. (c) — 20-5 = 15.

5. (b) — \frac{8+10}{2} = 9.

6. (b) — \frac{15}{60}\times360° = 90°.

7. (c) — 5\times12 = 60.

8. (a) — the mean uses every value, so an outlier pulls it.