REVISION 2 ALGEBRA55 sekolah bukit sion - igcse math revision mathematics (extended) 0580 igcse...

55 SEKOLAH BUKIT SION - IGCSE MATH REVISION

MATHEMATICS (EXTENDED) 0580 IGCSE MAY/JUNE 2020

REVISION 2

ALGEBRA


NOTES:


EXERCISE A 1. Factorise completely. 15p2 + 24pt

Answer: ………………………………………… [2] 2. Find r when (5)

$% = 125.

Answer: ………………………………………… [2] 3. Solve the simultaneous equations.

3𝑥 + 5𝑦 = 24 𝑥 + 7𝑦 = 56

Answer: x = ……………………………… [3]

Answer: y = ……………………………… [3]


4. y is inversely proportional to x2. When x = 4, y = 3. Find y when x = 5.

Answer: ………………………………………… [3] 5. Make w the subject of the formula.

𝑐 =4 + 𝑤𝑤 + 3

Answer: ………………………………………… [4] 6. For this question, 1 < x < 2. Write the following in order of size, smallest first.

5𝑥 5𝑥

𝑥5 𝑥 − 5

Answer: ………… < ……………<……………<………… [2]


7. x is a positive integer and 15x – 43 < 5x + 2. Work out the possible values of x.

Answer: ………………………………………… [3] 8. y varies directly as the square of (x – 3). y = 16 when x = 1. Calculate y when x = 10.

Answer: ………………………………………… [3] 9. Solve the inequality. 3y + 7 < 2 – y

Answer: ………………………………………… [2]

10. Make w the subject of the formula.

𝑡 = 2 −3𝑤𝑑

Answer: ………………………………………… [3]


11. The periodic time, T, of a pendulum varies directly as the square root of its length, l. T = 6 when l = 9. Find T when l = 25.

Answer: ………………………………………… [3] 12. (a) Find the value of 7p – 3q when p = 8 and q = –5.

Answer: ………………………………………… [2] (b) Factorise completely. 3uv + 9vw

Answer: ………………………………………… [2] 13. Simplify the following.

(a) (4𝑝𝑞8)9

Answer: ………………………………………… [2] (b) (16𝑥:);

<𝟒

Answer: ………………………………………… [2]


14. Solve the equation 2x2 + 6x – 3 = 0. Show your working and give your answers correct to 2 decimal places.

Answer: x = …………… or x = …………… [4] 15. Simplify fully.

𝑥8 − 𝑥 − 20

𝑥9 − 10𝑥8 + 25𝑥

Answer: ………………………………………… [5]


16. (a) A parallelogram has base (2x – 1) metres and height (4x – 7) metres. The area of the parallelogram is 1 m2.

(i) Show that 4x2 – 9x + 3 = 0. [3]

(ii) Solve the equation 4x2 – 9x + 3 = 0. Show all your working and give your answers correct to 2 decimal places.

Answer: x = …………… or x = …………… [4]

(iii) Calculate the height of the parallelogram.

Answer: ………………………………………… [1] (b) (i) Factorise x2 – 16.

Answer: ………………………………………… [1] (ii) Solve the equation 8?@9

?;A+ ?@AB

?C;DE= 2.

Answer: ………………………………………… [4]


17. (b) (i) Write the four missing terms in the table for sequences A, B, C and D. [3]

(ii) Which term in sequence D is equal to 500.

Answer: ………………………………………… [2]

(c) Simplify ?C;DE

8?C@F?;A

Answer: ………………………………………… [4] 18. (a) Simplify.

(i) (2x2y3)3

Answer: ………………………………………… [2]

(ii) G8F?HI;<%

Answer: ………………………………………… [3]


19. (a) Multiply and simplify. (3x – 2y)(2x + 5y)

Answer: ………………………………………… [3]

(b) Make h the subject of

(i) 𝑉 = 𝜋𝑟9 + 2𝜋𝑟8ℎ

Answer: ………………………………………… [2]

(ii) 𝑉 = √3ℎ

Answer: ………………………………………… [2] 20. Write as a single fraction in its simplest form. ?

8+O?

9−F?

A

Answer: ………………………………………… [2]


21. The cost of 1 kg of tomatoes is $x and the cost of 1 kg of onions is $y. John pays a total of $10.70 for 10 kg of tomatoes and 4 kg of onions. Jao pays a total $10.10 for 8 kg of tomatoes and 6 kg of onions. Write down simultaneous equations and solve them to find x and y.

Answer: x = ……………………………… [3]

Answer: y = ……………………………… [6] 22. Solve 2x2 – 5x – 8 = 0. Give your answers correct to 2 decimal places. Show all working.

Answer: x =…………… or x = ……………… [4]


23. Rice costs $x per kilogram. Potatoes cost $(x + 1) per kilogram. The total cost of 12 kg of rice and 7 kg of potatoes is $31.70. Find the cost of 1 kg of rice.

Answer: ………………………………………… [3] 24. The cost of a small bottle of juice is $y. The cost of a large bottle of juice is $(y + 1). When Catriona spends $36 on small bottles only, she receives 25 more bottles than when she spends $36 on large bottles only.

(i) Show that 25y2 + 25y – 36 = 0. [3]

(ii) Factorise 25y2 + 25y – 36.

Answer: ………………………………………… [2]

(iii) Solve the equation 25y2 + 25y – 36 = 0.

Answer: ………………………………………… [1]

(iv) Find the cost of 1 small bottle of juice and 1 large bottle of juice.

Answer: ………………………………………… [1]


25. The diagrams show a sequence of dots and circles. Each diagram has one dot at the centre and 8 dots on each circle. The radius of the first circle is 1 unit. The radius of each new circle is 1 unit greater than the radius of the previous circle.

(a) Complete the table for diagrams 4 and 5. [4]

(b) (i) Write down, in terms of n, the number of dots in diagram n.

Answer: ………………………………………… [2]

(ii) Find n, when the number of dots in diagram n is 1097.

Answer: ………………………………………… [2]

(c) Write down, in terms of n and 𝜋, the area of the largest circle in (i) diagram n

Answer: ………………………………………… [1]

(ii) diagram 3n

Answer: ………………………………………… [1]

(d) Find, in terms of n and 𝜋, the total length of the circumference of the circles in diagram n.

Answer: ………………………………………… [2]


EXERICE B 1. Factorise completely. 12xy – 3x2

Answer: ………………………………………… [2] 8. Solve the inequality. 3x – 1 ≤ 11x + 2

Answer: ………………………………………… [2] 3. Factorise completely. ap + bp – 2a – 2b

Answer: ………………………………………… [2] 4. Write (27𝑥D8)

<% in simplest form.

Answer: ………………………………………… [2]


5. (a) Factorise x2 + x – 30.

Answer: ………………………………………… [2]

(c) Simplify.

(𝑥 − 5)(𝑥 + 4)𝑥8 + 𝑥 − 30

Answer: ………………………………………… [1] 6. t varies inversely as the square root of u. t = 3 when u = 4. Find t when u = 49.

Answer: ………………………………………… [3] 7. Write as a single fraction in its simplest form.

2𝑥 + 3 +

3𝑥 + 2

Answer: ………………………………………… [3]


8. Factorise completely. kp + 3k + mp + 3m

Answer: ………………………………………… [2] 9. The first five terms of a sequence are shown below. 13 9 5 1 –3 Find the nth term of this sequence.

Answer: ………………………………………… [2] 10. Solve the equation. 5(2y – 17) = 60.

Answer: ………………………………………… [2] 11. y is inversely proportional to x3. y = 5 when x = 2. Find y when x = 4.

Answer: ………………………………………… [3]


12. Use the quadratic equation formula to solve 2x2 + 7x – 3 = 0. Show all your working and give your answers correct to 2 decimal places.

Answer: x =…………… or x = ……………… [4] 13. Solve 6x + 3 < x < 3x + 9 for integer values of x.

Answer: ………………………………………… [4]


14. The mass, m, of a sphere varies directly with the cube of its radius, r. m = 160 when r = 2. Find m when r = 5.

Answer: ………………………………………… [3] 15. Find the value of 2x + y for the simultaneous equations.

3x + 5y = 48 2x – y = 19

Answer: ………………………………………… [4]


16. Write as a single fraction in its simplest fraction.

𝑥 + 3𝑥 − 3 −

𝑥 − 1𝑥 + 1

Answer: ………………………………………… [4] 17. (a) Solve 3n + 23 < n + 41.

Answer: ………………………………………… [2] (b) Factorise completely ab + bc + ad + cd.

Answer: ………………………………………… [2]


18. (a)

𝑦 = Q8 +4𝑥

Find y when x = 2. Give your answer correct to 4 decimal places.

Answer: ………………………………………… [2]

(a) Rearrange 𝑦 = S8+ A? to make x the subject.

Answer: ………………………………………… [4]


19.

The diagrams show a sequence of stars made of lines and dots.

(a) Complete the table for Star 5, Star 7 and Star n. [4]

(b) The sums of the number of dots in two consecutive stars are shown in the table.

Find the sum of the number of dots in

(i) Star 10 and Star 11

Answer: ………………………………………… [1]

(ii) Star n and Star (n + 1)

Answer: ………………………………………… [1]

(iii) Star (n + 7) and Star (n + 8)

Answer: ………………………………………… [1]


(c) The total number of dots in the first n stars is given by the expression 5n2 + 6n. (i) Show that this expression is correct when n = 3.

Answer: ………………………………………… [2]

(ii) Find the total number of dots in the first 10 stars.

Answer: ………………………………………… [1]

(d) The total number of dots in the first n stars is 5n2 + 6n. The number of dots in the (n + 1)th star is 10(n + 1) + 1. Add these two expressions to show that the total number of dots in the first (n + 1) stars is

5(n + 1)2 + 6(n + 1). You must show each step of your working. [4]


20. Paul buys a number of large sacks of fertilizer costing $x each. He spends $27.

(a) Write down, in terms of x, an expression for the number of large sacks which Paul buys.

Answer: ………………………………………… [1]

(b) Rula buys a number of small sacks of fertilizer. Each small sack costs $2 less than a large sack. Rula spends $25. Write down, in terms of x, an expression for the number of small sacks which Rula buys.

Answer: ………………………………………… [1]

(c) Rula buys 4 more sacks than Paul. Write down an equation in x and show that it simplifies to 2x2 – 3x – 27 = 0. [4]

(d) Solve 2x2 – 3x – 27 = 0.

Answer: x = ……...… or x =………… [3]

(e) Calculate the number of sacks which Paul buys.

Answer: ………………………………………… [1]


21. (a) Write as a single fraction.

(i) OA−8?

O

Answer: ………………………………………… [2]

(ii) A?@9

+ 8?;D9

Answer: ………………………………………… [3] 22. Solve the simultaneous equations.

9x – 2y = 12 3x + 4y = –10

Answer: x = ……………………………… [3]

Answer: y = ……………………………… [3]


23. Simplify.

7𝑥 + 212𝑥8 + 9𝑥 + 9

Answer: ………………………………………… [4] 24.

(a) Solve 2(3x – 7) = 13.

Answer: ………………………………………… [3]

(b) Solve by factorising x2 – 7x + 6 = 0.

Answer: x = ……...… or x =………… [3]

(c) Solve. 3𝑥 − 25 +

𝑥 + 210 = 4

Answer: ………………………………………… [4]


25.

(b) 12 = 1 12 + 22 = 5 12 + 22 + 33 = 14 12 + 22 + 33 + 42 = 30 12 + 22 + 33 + 42 + ………… + n2 = 𝑎𝑛9 + 𝑏𝑛8 + X

E

Work out the values of a and b.

Answer: a = ……………………………… [3]

Answer: b = ……………………………… [6]


EXERICE C 1. Find the value of y when x = 6. Give your answer as a mixed number in its simplest form.

𝑦 =2𝑥8 +

𝑥8

2

Answer: ………………………………………… [2] 2. Solve the equation.

𝑛 − 82 = 11

Answer: ………………………………………… [2] 3. Make x the subject of the formula.

y = (x – 4)2 + 6


2𝑥 −

2𝑥 + 1

Answer: ………………………………………… [3]


5. Factorise completely.

(a) ax + ay + bx + by

Answer: ………………………………………… [2]

(b) 3(x – 1)2 + (x – 1)

Answer: ………………………………………… [2] 6. Solve the inequality for positive integer values of x.

21 + 𝑥5 > 𝑥 + 1

Answer: ………………………………………… [4] 7. (a) (28A)

<Z = 𝑝A

Find the value of p.

Answer: ………………………………………… [2]

(b) Simplify [C@[C

[<Z×[

<Z

Answer: ………………………………………… [3]


8. Solve the simultaneous equations.

2x – y = 7 3x + y = 3

Answer: ………………………………………… [2] 9. V = D

9Ah

(a) Find V when A = 15 and h = 7.

Answer: ………………………………………… [1]

(b) Make h the subject of the formula.

Answer: ………………………………………… [2]

10. Solve the equation. 32𝑥 +

1𝑥 + 1 = 0

Answer: ………………………………………… [3]


11. w varies inversely as the square root of x. When x = 4, w = 4. Find w when x = 25.

Answer: ………………………………………… [3] 12. Factorise completely.

(a) 4p2q – 6pq2

Answer: ………………………………………… [2]

(b) u + 4t + ux + 4tx

Answer: ………………………………………… [2] 13. (a) Simplify (3125𝑡D8O)

<] .

Answer: ………………………………………… [2] (b) Find the value of p when 3^ = D

_.

Answer: ………………………………………… [1]

(c) Find the value of w when 𝑥F8 ÷ 𝑥a = 𝑥:.

Answer: ………………………………………… [1]


14. Simplify.

𝑥8 + 6𝑥 − 73𝑥 + 21

Answer: ………………………………………… [4]

15. 32 25 18 11 4 These are the first 5 terms of a sequence. Find

(a) the 6th term

Answer: ………………………………………… [1]

(b) the nth term

Answer: ………………………………………… [2]

(c) which term is equal to –332.


15a3 – 5ab

Answer: ………………………………………… [2] 17. Simplify.

3x2y3 × x4y

Answer: ………………………………………… [2]


18. Solve the inequality.

5t + 23 < 17 – 2t

Answer: ………………………………………… [2] 19. y varies as the cube root of (x + 3). When x = 5, y = 1. Find the value of y when x = 340.

Answer: ………………………………………… [3] 20. (a) Factorise 3x2 + 2x – 8.

Answer: ………………………………………… [2] (b) Solve the equation 3x2 + 2x – 8 = 0.

Answer: ………………………………………… [1]


21. Robbie pays $10.80 when he buys 3 notebooks and 4 pencils. Paniz pays $14.50 when she buys 5 notebooks and 2 pencils. Write down simultaneous equations and use them to find the cost of a notebook and the cost of a pencil.

Answer: Cost of a notebook: ………………… [5] Cost of a pencil: ………………… [5]

22. Simplify.

𝑥8 − 3𝑥 + 2𝑥8 + 3𝑥 − 10

Answer: ………………………………………… [4]


23. The distance a train travels on a journey is 600 km.

(a) Write down an expression, in terms of x, for the average speed of the train when (i) the journey takes x hours,

Answer: ………………………………………… [1]

(ii) the journey takes (x + 1) hours

Answer: ………………………………………… [1]

(b) The difference between the average speeds in part (a)(i) and part (a)(ii) is 20 km/h. (i) Show that x2 + x – 30 = 0.

Answer:

Answer: ………………………………………… [3] (ii) Find the average speed of the train for the journey in part (a)(ii).

Show all your working.

Answer: ………………………………………… [4]


24. (a) (i) Show that the equation F?@A

+ 8?;98

= 1 can be simplified to 2𝑥8 + 3𝑥 − 6 = 0.

[3]

(ii) Solve the equation 2𝑥8 + 3𝑥 − 6 = 0.

Show all your working and give your answers correct to 2 decimal places.

Answer: x =…………… or x = ……………… [4]

(b) The total surface area of a cone with radius x and slant height 3x is equal to the area of a circle with radius r. Show that r = 2x. [The curved surface area, A, of a cone with radius r and slant height l is A = 𝜋rl.]

Answer:

[4]


25.

The first three diagrams in a sequence are shown above. Diagram 1 shows an equilateral triangle with sides of length 1 unit. In Diagram 2, there are 4 triangles with sides of length ½ unit. In Diagram 3, there are 16 triangles with sides of length ¼ unit.

(a) Complete this table for Diagrams 4, 5, 6 and n. [6]

(b) (i) Complete this table for the number of the smallest triangles in Diagrams 4, 5 and 6.

(ii) Find the number of the smallest triangles in Diagram n, giving your answers as a power of 2.

Answer: ………………………………………… [1]

(c) Calculate the number of the smallest triangles in the diagram where the smallest triangles have sides of length D

D8: unit.

Answer: ………………………………………… [2]


EXERICE D 1. Expand and simplify. 𝑥(2𝑥 + 3) + 5(𝑥 − 7)

Answer: ………………………………………… [2] 2. Simplify. 6𝑢𝑤;9 × 4𝑢𝑤E

Answer: ………………………………………… [2] 3. Find the nth tem of each sequence.

(a) 4, 8, 12, 16, 20, …….

Answer: ………………………………………… [1]

(b) 11, 20, 35, 56, 83, …….

Answer: ………………………………………… [2] 4. p is inversely proportional to the square of (q + 4). p = 2 when q = 2. Find the value of p when q = −2.

Answer: ………………………………………… [3]


5. Solve the simultaneous equations. You must show all your working.

5𝑥 + 2𝑦 = −2 3𝑥 − 5𝑦 = 17.4


(a) 𝑦𝑝 + 𝑦𝑡 + 2𝑥𝑝 + 2𝑥𝑡

Answer: ………………………………………… [2]

(b) 7(ℎ + 𝑘)8 − 21(ℎ + 𝑘)

Answer: ………………………………………… [2] 7. Find the value of x.

81x = 3.

Answer: ………………………………………… [1] 8. Solve. 5(𝑤 + 4 ×109) = 6 ×10A

Answer: ………………………………………… [2]


9. 5, 11, 21, 35, 53, …

Find the nth term of this sequence.

Answer: ………………………………………… [2] 10.

(a) ABCD is a square. Find the value of x.

Answer: ………………………………………… [1]

(b) Square ABCD and isosceles triangle EFG have the same perimeter.

Work out the length of FG.


3𝑥 + 2 −

42𝑥 − 5

Answer: ………………………………………… [3]


12. f(x) = x2 + 4x – 6

(a) f(x) can be written in the form (x + m)2 + n. Find the value of m and the value of n.

Answer: ………………………………………… [2]

(b) Use your answer in part (a) to find the positive solution to x2+ 4x – 6 = 0.

Answer: ………………………………………… [2] 13. Factorise 9x2 – 6x completely.

Answer: ………………………………………… [2] 14. Factorise 2x2 – 5x – 3.

Answer: ………………………………………… [2] 15. Solve the equation. 3(x + 4) = 2(4x – 1)

Answer: ………………………………………… [3]


16. Simplify.

(a) 12x12 ÷ 3x3

Answer: ………………………………………… [2]

(b) (256𝑦8OE)<e

Answer: ………………………………………… [1] 17. Solve the equation. 2x2 + x – 2 = 0 Show your working and give your answers correct to 2 decimal places.

Answer: x =……………… or x =……………… [4] 18 (a) Jamil, Kiera and Luther collect badges. Jamil has x badges. Kiera has 12 badges more than Jamil. Luther has 3 times as many badges as Kiera. Altogether they have 123 badges. Form an equation and solve it to find the value of x.

Answer: ………………………………………… [3]


(b) Find the integer values of t which satisfy the inequalities.

4𝑡 + 7 < 39 ≤ 7𝑡 + 2

Answer: ………………………………………… [3]

(c) Solve the following equations.

(i) 8D;??@9

= 4

Answer: ………………………………………… [3] (ii) 3𝑥8 + 7𝑥 − 5 = 0

Show all your working and give your answers correct to 2 decimal places.

Answer: ………………………………………… [4]


19. (a) Make x the subject of the formula.

𝐴 − 𝑥 =𝑥𝑟𝑡

Answer: ………………………………………… [4]

(b) Find the value of a and the value of b when x2 – 16x + a = (x + b)2.

Answer: ………………………………………… [3]

(d) Write as a single fraction in its simplest form.

6

𝑥 − 4 −5

3𝑥 − 2

Answer: ………………………………………… [3]


20. On the first part of a journey, Alan drove a distance of x km and his car used 6 litres of fuel.

The rate of fuel used by the car was EBB? litres per 100 km.

(a) Alan drove another (x + 20) km and his car used another 6 litres of fuel. (i) Write down an expression, in terms of x, for the rate of fuel used by his car on this part

of the journey. Give your answer in litres per 100 km.

Answer: ………………………………………… [1]

(ii) On this part of the journey, the rate of fuel used by the car decreased by 1.5 litres per 100 km. Show that x2 + 20x – 8000 = 0. [4]

(b) Solve the equation x2 + 20x – 8000 = 0.

Answer: ………………………………………… [3]

(c) Find the rate of fuel used by Alan’s car for the complete journey.

Give your answer in litres per 100 km.

Answer: ………………………………………… [1]


21. Expand and simplify.

3𝑥(𝑥 − 2) − 2𝑥(3𝑥 − 5)

Answer: ………………………………………… [3]

22. (a)Factorise the following completely.

(i) 6𝑤 + 3𝑤𝑦 − 4𝑥 − 2𝑥𝑦

Answer: ………………………………………… [2]

(ii) 4𝑥8 − 25𝑦8

Answer: ………………………………………… [2]

(b) Simplify.

G DE_hZI;%C

Answer: ………………………………………… [2]

23. n is an integer.

(i) Explain why 2n – 1 is an odd integer

Answer: ………………………………………………………………………………

……………………………………………………………………………………… [1]

(ii) Write down, in terms of n, the next odd number after 2n – 1.

Answer: ………………………………………… [1]

(iii) Show that the difference between the squares of two consecutive odd numbers

multiple of 8.

[3]


24. (a) The total surface area of a cone is given by the formula A = 𝜋rl + 𝜋r2. (i) Find A when r = 6.2cm and l = 10.8cm.

Answer: ………………………………………… [2] (ii) Rearrange the formula to make l the subject.

Answer: ………………………………………… [2] (b) (i) Irina walks 10 km at 4 km/h and then a further 8 km at 5 km/h. Calculate Irina’s average speed for the whole journey.

Answer: ………………………………………… [3] (ii) Dariella walks x km at 5 km/h and then runs (x + 4) km at 10 km/h. The average speed of this journey is 7 km/h. Find the value of x. Show all your working.

Answer: ………………………………………… [5]


25. The first four terms of sequences A, B, C and D are shown in the table.

(a) Complete the table. [8]

(b) Which term in sequence A is equal to 9E9F

?

Answer: ………………………………………… [2]

(c) Which term in sequence D is equal 725?

Answer: ………………………………………… [2]

REVISION 2 ALGEBRA55 sekolah bukit sion - igcse math revision mathematics (extended) 0580 igcse...

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