NATIONAL 5 MATHEMATICS. Unit 1 Assessment Revision

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1 NATIONAL 5 MATHEMATICS Unit 1 Assessment Revision Volume, Changing the Subject, Equations & Inequations, Simultaneous Equations, Fractions & Percentages, Arcs & Sectors Mr Miscandlon gw13miscandlondavid@glow.sch.uk

2 Volume Basic B1. Find the volume of a sphere for the following values of r and d. (give your answers correct to 3 significant figures) r (a) r = 10cm (f) d = 18cm (b) r = 5cm (g) r = 80mm (c) d = m (h) d = 55cm (d) r = 00mm (i) r = 3 5m (e) d = 11cm (j) d = 48cm B. Find the volume of a cone for the following values of r and h. (give your answers correct to 3 significant figures) (a) r = 5cm h = 14cm (b) r = 7cm h = 5cm (c) r = 3cm h = cm (d) r = 1cm h = 7cm h r B3. A cone has a base radius of 9cm and a slant height of 15cm. Calculate the volume of the cone. B4. A pyramid has a square base of side 4cm and a vertical height of 7cm. Calculate the volume of the pyramid correct to significant figures. Extension 6 6 cm E1. An ice cream is shaped like the one in the diagram. The overall height of is 11 7 cm. The height of the cylinder is 3 cm. The diameter of the cone and cylinder is 6 6 cm. 3 cm Calculate the volume of ice cream to 3 significant figures cm

3 E. A company that produces bins uses the design of a cylindrical base with a hemispherical lid. If the total height of the bin is 60cm and the radius of the bin is 14cm, calculate the total volume of the bin in litres correct to 3 significant figures. 60cm 14cm E3. A glass candle holder is in the shape of a cuboid with a cone removed. The cuboid measures 4cm by 4cm by 6cm. The cone has a diameter of 3cm and a height of 5cm. Calculate the volume of glass in the candle holder. 6 cm 4 cm 4 cm E4. For the Christmas market a confectioner has created a chocolate Santa. It consists of a solid hemisphere topped by a solid cone. Both have diameter 5cm and the height of the cone is 4cm as shown in the diagram. 4cm 5cm Calculate the volume of chocolate required to make one chocolate Santa, giving your answer correct to 3 significant figures. E5. The diameter of an ordinary snooker ball is 5 5cm. Calculate the volume of a snooker ball giving your answer correct to 3 significant figures.

4 E6. A dessert is in the shape of a truncated cone [a cone with a slice taken from the top]. The radius of the base is 4 1cm and is 1 6 cm at the top. The other dimensions are shown in the diagram. 3 7cm 5 6cm Calculate the volume of the dessert.

5 Changing the Subject Basic B1. Make x the subject of each formula below. (a) y = ax + b (b) y = mx + c (c) t = s r (d) p = qx + r (e) m = f 3n (f) a = b + cx (g) k = h mx (h) d = 3b + cx (i) g = kc hx B. Change the subject of each formula to the letter shown in brackets. (a) P = 4l (l) (b) V = IR (I) (c) S = DT (T) (d) A = lb (b) (e) C = d (d) (f) G = UT (U) (g) v = u + at (t) (h) P = l + b (l) (i) H = xy + 5m (y) Extension E1. Change the subject of each formula to the letter shown in brackets. (a) v = u + as (s) (b) v = u + as (u) (c) V = rh (h) (d) V = rh (r) A (e) r = (g) k = 1 x (i) ar = b ( p 4) (A) (f) L = a (a) 4yz (p) (h) x = t (y) (b) (j) st = A( 3y) (A) (k) R = A( 3y) (x) (l) na = t ( n 1) (m) d = n (1 n R r (n) (n) 1 r ) (n) (R) a ( b) (o) d = 4 (a)

6 Equations and Inequations Basic B1. 6. Multiply out the brackets and solve : (a) (x + 5) = 1 (b) 5 (y + 7) = 45 (c) 3 (a + 6) = 36 (d) 6 (x + 4) = 54 (e) 4 (x + 9) = 48 (f) 3 (c + 8) = Solve : (a) 6y + 3 = y + 18 (b) 5a + 7 = a + 15 (c) 9c + 5 = c + 1 (d) 10x + 1 = 4x + 19 (e) 5b + 3 = b + 9 (f) 7n + 6 = 3n Solve : (a) 6y 3 = 3y + 15 (b) 5a 9 = a + 15 (c) 9c 8 = 4c + 1 (d) 10 1 = 4x + 5 (e) 5b 3 = b + 9 (f) 3n 10 = n + (g) 7 14 = 3x + (h) 6c 13 = 3c + 59 (i) 7y 16 = y + 34 (j) 15a 8 = 3a + 76 (k) 516 = 5x + 84 (l) 6n 3 5 = 3n + 4 (m) b + 13 = 9b 7 (n) 3x + 1 = 4 4 (o) x + 5 = 3 5 (p) 5y + 4 = 0y 6 (q) a + 6 = 13a 18 (r) 10x + 40 = (s) 8y + 3 = 19y 74 (t) b + = 5b 16 (u) + x = (v) 13x + 11 = 0 38 (w) 4x + 10 = 9 50 (x) 10a + 13 = 0a Solve each of the following inequations. (a) 3a 17 a (b) 7(x 3) 8x 7 (c) (5 p 1) 7p 18 (d) 40 3k 8 k (e) 3 5( m ) ( m 7) (f) 3( y 4) 1 4(4 y) (g) (3 4h) 13 15h (h) 3( x ) (1 x) Solve each of the following inequations. (a) a a (b) 14 3x x 6 (c) 3( p ) 5p 10 (d) 16 3k 0 k (e) 7( d ) ( d 1) (f) (y 1) 8 10(1 y) (g) 4 (3 4h) 1 h (h) 3( y ) (1 3y) 7

7 Simultaneous Equations Basic B1. (a) Copy and complete the tables below. Table 1 : y = 9 x x y 6 Table : y = 1 x 5 7 y 1 (b) Plot the points from table 1. Join them carefully with a straight line. (c) Plot the points from table on the same graph. Join them with a straight line. (d) Write down the coordinates of the points where the lines cross. B. Solve the following pairs of simultaneous equations: (a) x + y = 4 (b) x + y = 9 (c) x + y = 7 y = y = 5 y = 3 (d) x + y = 1 (e) x + y = 3 (f) x + y = 1 y = 3 y = 9 y = 9 B3. Solve the following pairs of simultaneous equations: x 4y 4 4a 3b 18 e 7 f 6 (a) 7x y 4 (b) a 6b 6 (c) 8e 5 f 38 5x y x 3y 10 4 p 3q 1 (d) 3x y 3 (e) 3x 6y 18 (f) 8p 5q 1 B4. Solve the following pairs of simultaneous equations: 3y 17 a 3b 6 e f 1 (a) 3x y 4 (b) 3a b 8 (c) 5e f 0 5x 3y 7 x 5y 14 p 3q 6 (d) 4x y 0 (e) y 5 (f) 4 p q 8 B5. Solve the following pairs of simultaneous equations: (a) 5x + y = 9 (b) 4x + 5y = 7 (c) 5x + y = 14 3y = 4 7 3y = 4 4 5y = (d) 3x + y = 16 (e) 8 3y = 19 (f) 5x + 3y = 19 x + 3y = 13 3 y = 1 7 4y = 43

8 Extension E1. Two numbers are such that three times the larger plus twice the smaller is equal to 31 and the sum of twice the smaller plus the larger is 13. Find the two numbers. E. Four chocolate bars and six packets of crisps together cost Ten chocolate bars and three packets of crisps cost Form simultaneous equations and solve them to find the cost of each packet of crisps and each bar of chocolate. E3. Four sandwiches and 3 hot-dogs cost Two sandwiches and 4 hot-dogs cost 6. Form simultaneous equations and solve them to find the cost of each sandwich and hotdog. E4. At Smith s Stationers, the cost of a ruler and a pencil together is 57p. The ruler costs 3p more than the pencil. Find the cost of each. E5. Look at the two rectangles opposite. The smaller one has a perimeter of 60cm. The larger one has a perimeter of twice the smaller. (a) Form two equations and solve them simultaneously to find the values of x and y. 8y (b) Hence calculate the area of the smaller rectangle. x 3x y E6. A van is carrying eight identical boxes and five identical parcels. (a) (b) If 3 boxes and parcels weigh a total of kg and 4 boxes and 3 parcels weigh 30kg, find the weight of an individual box and a single parcel. What is the total weight carried by the van?

9 Fractions Basics B1. Calculate: (a) 7 8 (b) 5 8 (c) 9 7 (d) (e) 8 5 (f) 6 5 (g) 9 7 (h) 8 3 B. Calculate: (a) (b) (c) (d) (e) (f) (g) (h) 4 6 B3. Calculate: (a) (b) (c) (d) (e) (f) (g) 5 1 (h) 3 9 Extension 5 E1. A twenty one metre length of fabric is cut into 8 metre pieces. (a) (b) How many pieces can be cut? What length of fabric would be left over? 3 E. A triangle has base E3. A rectangle measures 4 3 cm and height 5 cm. Calculate its area. metres by 3 5 metres. Calculate its area.

10 Percentages Basics B1. In 1990 the world population was estimated to be 5300 million, and was increasing at the rate of 1 7% per annum. What will the population be in the year 00? (answer to significant figures) B. Peter buys a car for If it depreciates at the rate of 0% per annum, how much will he be able to sell it for in 3 years time? B3. Brian buys a new car costing It depreciates in value by 30% in the first year and by 0% each year after that. How much will he be able to trade it in for in 3 years time B4. These amounts have been reduced by 15%. What was the original amount? (a) 85 (b) 1 5 mm (c) (d) 5 5 litres (e) 357 miles (f) 435 m B5. A gym s membership has increased by 17% over the past year. It now has 585 members. How many members did it have a year ago? B6. The number of school pupils not wearing school uniform has decreased by 7% since the start of last year. There are now 4 pupils not wearing school uniform.

11 Arcs and Sectors Basics B1. Calculate the length of the arc in each diagram below, giving your answer correct to 1d.p. A E (a) (b) D (c) O 90 o 8cm B C 14mm 140 o 35 o O O m F B. Calculate the perimeter of each sector in Question 1. Giving your answers correct to 1 d.p. B3. Calculate the area of the sector in each diagram below, giving your answer correct to 3 significant figures. A E (a) (b) (c) D O Extension 90 o 8cm B C 14mm 140 o 35 o O O m F E1. Two congruent circles overlap to form the symmetrical shape shown below. Each circle has a diameter of 1 cm and have centres at B and D. A D B C Calculate the area of the shape.

12 E. A sector of a circle with radius 6cm is shown opposite. O Angle AOB x If the exact area of the sector is 4 square centimetres, calculate the size of the angle marked x. 6cm A B E3. A hand fan is made of wooden slats with material on the outer edge. 9cm 15cm 105 (a) (b) Calculate the area of material needed for the hand fan. Calculate the perimeter of the shaded area in the diagram above. E4. O 7 o The area of the shaded sector is 5ˑ04 cm. Calculate the area of the circle. P Q E5. A grandfather clock has a pendulum which travels along an arc of a circle, centre O. The arm length of the pendulum is 60cm. The pendulum swings from position OA to OB. The length of the arc AB is 1cm. O Calculate the size of angle AOB to the nearest degree. A B

13 Summary

19 Answers Volume B1. (a) 4190cm³ (b) 65400cm³ (c) 4 19m³ (d) mm³ (e) 697cm³ (f) 3050cm³ (g) mm³ (h) 87100cm³ (i) 180m³ (j) 57900cm³ B. (a) 366cm³ (b) 180cm³ (c) 07cm³ (d) 1060cm³ B3. 100cm³ B4. 37cm³ E1. 06cm 3 E cm 3 E cm³ E cm³ E cm³ E cm³ Changing the Subject y b B1. (a) a (b) y c m (c) t r s (d) p r q (e) m 3n f (f) a b c (g) h k m (h) d 3b c (i) kc g h P l B. (a) 4 (b) V I R (c) T S D (d) b A l (e) C d (f) G U T (g) t v u a P b l (h) (i) y H 5m x v u s E1. (a) a (b) u v as (c) V h r (d) r V h (e) ( 3) a L A r (f) 6 (g) p 4k 4 (h) tx y 4z (i) b x 4a r (j) A st 3y (k) R 3A y A 1 n (l) a 1 (m) t n t d r1 r R (n) r r1 (o) a 4d b

20 Equations and Inequations B1. (a) 1 (b) (c) 6 (d) 5 (e) 3 (f) B. (a) 3 (b) (c) (d) 3 (e) (f) 3 B3. (a) 6 (b) 6 (c) 4 (d) 1 (e) 4 (f) 6 (g) 4 (h) 4 (i) 8 (j) 7 (k) 5 (l) 5 (m) 5 (n) 16 (o) 15 (p) (q) (r) 4 (s) 7 (t) 6 (u) (v) 7 (w) 1 (x) 40 B4. (a) a 3 (b) 1 (c) p (d) k 3 (e) m 7 (f) y 9 (g) h 1 (h) 1 5 B5. (a) a 3 (b) (c) p (d) k (e) 38 d (f) 9 10 y (g) h 0 (h) 3 11 y 9 Simultaneous Equations B1. (a) Tables completed Table 1: 9, 6, Table : 1, 4, 6 (b) and (c) (d) (5, 4) Graphs drawn B. (a) x = 3 and y = 1 (b) x = 7 and y = (c) x = 5 and y = (d) x = and y = 1 (e) x = 6 and y = 3 (f) x = 4 and y = 5 (g) x = 3 and y = (h) x = 11 and y = 3 (i) x = 8 and y = 10 B3. (a) x = and y = 5 (b) a = 3 and b = (c) e = 6 and f = (d) x = 1 and y = 3 (e) x = and y = (f) p = and q = 3 (g) g = 4 and h = 3 (h) x = 3 and y = 4 (i) u = and v = 3 (j) x = 4 and y = 1 (k) p = 5 and q = (l) a = 4 and b = B4. (a) x = and y = 5 (b) a = 3 and b = 1 (c) e = and f = 5 (d) x = 1 and y = 4 (e) x = 3 and y = 4 (f) p = 3 and q = 4 (g) g = 8 and h = 5 (h) x = 7 and y = (i) u = and v = (j) x = 4 and y = 1 (k) p = 1 and q = (l) a = 8 and b = B5. (a) x = 1 and y = (b) x = 3 and y = 1 (c) x = and y = (d) x = 5 and y = 1 (e) x = 5 and y = 7 (f) x = 5 and y = (g) x = 3 and y = 3 (h) x = 4 and y = 3 (i) x = and y = 3 5 (j) x = 0 5 and y = (k) x = 1and y = (l) x = 5 and y = 0 E1. 9 and E. Chocolate costs 40p and crisps cost 30p

21 E3. Sandwich costs 1.0 and hotdog costs 90p. E4. Ruler costs 49p and pencil costs 6p. E5. (a) 4x + 4y = 60; 6x + 16y = 10; x = 1 and y = 3 (b) 144cm² E6. (a) Box weighs 6kg and parcel weighs kg. (b) 58kg Fractions B1. (a) (b) (c) (d) B. (a) (b) B3. (a) (b) 35 9 (c) 3 (d) 4 1 (c) (d) (e) 9 40 (f) 7 30 (g) 63 (h) 5 4 (e) (f) (e) 7 3 (f) (g) 11 5 (h) 3 4 E4. (a) 33 (b) 3 8 m E m E m (g) (h) 3 Percentages B million B B B4. (a) 100 (b) 50mm (c) 75 (d) 30 litres (e) 40 miles (f) 51 m B B Arcs and Sectors B1. (a) 1.6cm (b) 34.mm (c) 1.m B. (a) 44.6cm (b) 6.mm (c) 5.m B3. (a) 50 cm² (b) 39mm² (c) 1 m² E1. 06 cm² E. 40 o E3. (a) 173cm² (b) 56 5cm E4. 5 1cm² E5. 0 o

NATIONAL 5 MATHEMATICS. Unit 1 Assessment Revision. Volume, Changing the Subject, Simultaneous Equations, Fractions & Percentages

NATIONAL 5 MATHEMATICS. Unit 1 Assessment Revision. Volume, Changing the Subject, Simultaneous Equations, Fractions & Percentages NATIONAL 5 MATHEMATICS Unit 1 Assessment Revision Volume, Changing the Subject, Simultaneous Equations, Fractions & Percentages Mr Miscandlon gw13miscandlondavid@glow.sch.uk Volume Basic B1. Find the volume