ABBASI MOHAMMED ASIM Page: 1 Ch7: Equations and Inequalities. 1. Solve the simultaneous equations 2y + 3x = 6, x = 4y + 16.

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1 Ch7: Equations and Inequalities 1. Solve the simultaneous equations 2y + 3x = 6, x = 4y Answer x = y = 2. Solve the equation 5(x ) = Answer x = 3. Solve the simultaneous equations 2x + 3y = 4, y = 2x 12. Answer x = y = 4. B ( x + 1) cm A ( x + 6) cm D ( x + 2) cm C NOT TO SCALE In triangle ABC, the line BD is perpendicular to AC. AD = (x + 6) cm, DC = (x + 2) cm and the height BD = (x + 1) cm. The area of triangle ABC is 40 cm 2. ABBASI MOHAMMED ASIM Page: 1 mdasimabbasi@yahoo.co.in

2 (i) Show that x 2 + 5x 36 = 0. Answer (i)... (ii) Solve the equation x 2 + 5x 36 = 0. Answer (ii) x =. or x = (iii) Calculate the length of BC. Answer (iii) BC =.. cm 5. Amira takes 9 hours 25 minutes to complete a long walk. 113 (i) Show that the time of 9 hours 25 minutes can be written as hours. 12 Answer (i) (ii) She walks (3y + 2) kilometres at 3 km/h and then a further (y + 4) kilometres at 2 km/h. Show that the total time taken is 9 y 16 6 hours Answer (ii). 9 y (iii) Solve the equation = Answer (iii) y = (iv) Calculate Amira s average speed, in kilometres per hour, for the whole walk. Answer (iv) km/h 6. f(x) = 2x 1 g(x) = x h(x) = 2 x (a) Find the value of 1 (i) f, 2 Answer (a)(i).. (ii) g( 5), Answer (a)(ii). ABBASI MOHAMMED ASIM Page: 2 mdasimabbasi@yahoo.co.in

3 (iii) h( 3). Answer (a)(iii) (b) Find the inverse function f 1 (x). Answer (b) f 1 (x) =.. (c) g(x) = z. Find x in terms of z. Answer (c) x =... (d) Find gf(x), in its simplest form. Answer (d) gf(x) =.. (e) h(x) = 512. Find the value of x. Answer (e) x =. (f) Solve the equation 2f(x) + g(x) = 0, giving your answers correct to 2 decimal places. Answer (f) x =. or x =. [5] (g) Sketch the graph of (i) (ii) y = f(x), y = g(x). y y O x O x (i) y = f( x ) (ii) y = g( x) ABBASI MOHAMMED ASIM Page: 3 mdasimabbasi@yahoo.co.in

4 7. Solve the inequality 2 x 5 x Answer Find the co-ordinates of the point of intersection of the straight lines 2x + 3y = 11, 3x 5y = 12. Answer (.., ) 9. A student played a computer game 500 times and won 370 of these games. He then won the next x games and lost none. He has now won 75% of the games he has played. Find the value of x. Answer x =.. [4] 10. Solve these simultaneous equations. x + 2y 18 = 0 3x 4y 4 = 0 Answer x =.. y = x Solve the inequality <. 7 5 Answer 12. Solve these simultaneous equations. x + 3y 11 = 0 3x 4y 7 = 0 Answer x =.. ABBASI MOHAMMED ASIM Page: 4 mdasimabbasi@yahoo.co.in

5 y.= (i) Factorise x 2 x 20. (ii) Solve the equation x 2 x 20 = Solve the equation 3x 2 2x 2 = 0. Show all your working and give your answers correct to 2 decimal places. [4] 15. C x cm A y ( x + 4) cm B NOT TO SCALE (a) When the area of triangle ABC is 48 cm 2, (i) show that x 2 + 4x 96 = 0, (ii) solve the equation x 2 + 4x 96 = 0, (iii) write down the length of AB. 1 (b) When tan y =, find the value of x. 6 (c) When the length of AC is 9 cm, (i) show that 2x 2 + 8x 65 = 0, ABBASI MOHAMMED ASIM Page: 5 mdasimabbasi@yahoo.co.in

6 (ii) solve the equation 2x 2 + 8x 65 = 0, (iii) (Show your working and give your answers correct to 2 decimal places.) calculate the perimeter of triangle ABC. [4] 16. P Q y cm X ( y + 2) cm (2 y 1) cm ( y + 1) cm R S NOT TO SCALE In the diagram PQ is parallel to RS. PS and QR intersect at X. PX = y cm, QX = (y + 2) cm, RX = (2y 1) cm and SX = (y + 1) cm. (i) Show that y 2 4y 2 = 0. (ii) Solve the equation y 2 4y 2 = 0. (iii) Show all your working and give your answers correct to two decimal places. Write down the length of RX. [4] 17. A packet of sweets contains chocolates and toffees. (a) There are x chocolates which have a total mass of 105 grams. Write down, in terms of x, the mean mass of a chocolate. (b) There are x + 4 toffees which have a total mass of 105 grams. Write down, in terms of x, the mean mass of a toffee. ABBASI MOHAMMED ASIM Page: 6 mdasimabbasi@yahoo.co.in

7 (c) The difference between the two mean masses in parts (a) and (b) is 0.8 grams. Write down an equation in x and show that it simplifies to x 2 + 4x 525 = 0. [4] (d) (i) Factorise x 2 + 4x 525. (ii) Write down the solutions of x 2 + 4x 525 = 0. (e) Write down the total number of sweets in the packet. (f) Find the mean mass of a sweet in the packet. 18. D iagram 1 D iagram 2 D iagram 3 The first three diagrams in a sequence are shown above. The diagrams are made up of dots and lines. Each line is one centimetre long. (a) Make a sketch of the next diagram in the sequence. (b) The table below shows some information about the diagrams. Diagram n Area x Number of dots p y Number of one centimetre lines q z (i) Write down the values of p and q. (ii) Write down each of x, y and z in terms of n. [4] (c) The total number of one centimetre lines in the first n diagrams is given by the expression n 3 fn gn. ABBASI MOHAMMED ASIM Page: 7 mdasimabbasi@yahoo.co.in

8 (i) Use n = 1 in this expression to show that 10 f + g =. (ii) Use n = 2 in this expression to show that 32 4 f 2 g. 3 (iii) Find the values of f and g. (iv) Find the total number of one centimetre lines in the first 10 diagrams. 19. Each year a school organises a concert. In 2007, the number of tickets sold was 210. Adult tickets were $2.60 each and student tickets were $1.40 each. The total amount received from selling the 210 tickets was $480. How many student tickets were sold? [4] ABBASI MOHAMMED ASIM Page: 8 mdasimabbasi@yahoo.co.in

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education www.xtremepapers.com MATHEMATICS 0580/04, 0581/04 Paper 4 (Extended) SPECIMEN PAPER (New mat)