For use only in Whitgift School. IGCSE Higher Sheets 3. IGCSE Higher

Transcription

1 IGCSE Higher Sheet H a-1 Formulae Sheet H3--03a- Formulae (x appearing twice) Sheet H a-3 Formulae (x appearing twice) Sheet H a-4 Formulae (x appearing twice and powers) Sheet H a-5 Formulae (x appearing twice and powers) Sheet H Proportion Sheet H Proportion Sheet H Proportion Sheet H Proportion Sheet H Inverse Proportion Sheet H Inverse Proportion Sheet H Inverse Proportion Sheet H Simultaneous Equations-Cancellation Sheet H Simultaneous Equations-Substitution Sheet H Simultaneous Equations-Problems Sheet H Simultaneous Equations-Problems Sheet H Simultaneous Equations Sheet H Simultaneous Equations-Graphs Sheet H Simultaneous Equations-Graphs Sheet H a-1Quadratic Factorisation-Solve Sheet H a-Quadratic Factorisation-Solve

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3 Sheet H a-1 Formulae 1. Make the following letters the subjects of the following formulae: (a) y x+ y = c (b) s V = s t (c) u r = t+ u (d) x D= y x (e) a s = a+ b (f ) p b= p+ q+ r (g) u v= u+ rt (h) R r = π R. Make the following letters the subjects of the following formulae: (a) t C = rt (b) t S = t d (c) a 1 b= ( a+ c) (d) t v= u 3t (e) y 3y 1 x= (f ) I A= P+ I 4 10 (g) R R r V = (h) r p= q+ I 5 3. Make the following letters the subjects of the following formulae: (a) x y = c x (b) q p= q r ( ) (c) t 1 p = ( s+ t) 4 (d) r C = π r (e) a ab d = c (f ) x c= a x+ b ( ) PRT (g) P R= 3 ( P+ Q) (h) R I = Make the following letters the subjects of the following formulae: (a) h A= πr + πrh (b) s v = u + as a (c) ( ) b s (e) x y = (f ) m A= π r( m n) x+ g 1 a s = ut + at (d) h T = r + h + s 5. Make the following letters the subjects of the following, more complicated, formulae: (a) a h+ a x+ d r = h( v a) + b (b) x = b c (c) w r t n r = + c (d) n + e= d w+ e c (e) T p h r = y (f ) u q= g a+ T y xu 6. Make u the subject of the formula D = ut + kt

4 Sheet H3--03a- Formulae (x appearing twice) 1. Make x the subject of the following formulae: (a) r+ mx= nx (b) ax+ b= cx+ d (c) d + bx m+ qx x= (d) n x= a p (e) A B ax = + C (f ) = c x x x+ b. Make the following letters the subjects of the following formulae: (a) p ap + b kq = c (b) q = e p q+ b (c) h uh + rs au + b = t (d) u = h cu+ d (e) A da + b rz = c (f ) z = y A z+ t (g) s as + 1 aw + m = t (h) w = s+ bw+ n (i) a a+ b 5 3y = x (j) y = y a r 3. Make x the subject of the following formulae: (a) 3x+ r e x = t (b) = x x f (c) p+ x x+ r = x (d) = q q x+ p 4. Make the following letters the subject of the following formulae: (a) w aw + y bw + u d a = (b) d = c c d d b (c) r r h 1 = b (d) h = t r+ a h (e) e Ae + B p qv = (f ) v = t Ce + D r sv

5 Sheet H a-3 Formulae (x appearing twice) 1. Make x the subject of the following formulae: (a) ax + b = cx (b) mx + n = x r (c) px + q c + bx = x (d) = x+ q s t (e) ax + b rx + v = e (f ) = p cx + d dx e. Make the following letters the subjects of the following formulae: (a) w wq + r b = a (b) Q + c = d w Q (c) R Rt 1 l + f = m (d) l + t = r Rt + 1 b 3. Make the following letters the subjects of the following formulae: (a) w aw + b ax = w + d (b) x p + = b c q (c) y y b z e a+ = c (d) z d + = g y z (e) n h+ n c+ n l fx s g+ = (f ) l tl+ = + pl fx n n f x

6 Sheet H a-4 Formulae (x appearing twice and powers) 1. Make x the subjects of the following formulae: (a) x + b= c (b) px + q= r (c) ax = b (d) = c x + u a x + b (e) = z (f ) = d v c (g) ax b x p = = t 3 a x b (h) r. Make the following letters the subjects of the following formulae: 5h a e( t+ y) (a) h = r (b) t = h b g (c) 4 3 m ( u+ p) r V = π r (d) u = h 3 a ( y+ q) x t (e) x = g (f ) y = r + k s h l a t (g) l T = (h) p = r g p+ f 3. Make the following letters the subjects of the following formulae: (a) u v = u + as (b) h S = 3π r + π rh 1 (c) r A= 4 π r (d) a s = ut+ at 3 l (e) l T = π (f ) r A= π r a g ( ) NB ( x y) + IS NOT THE SAME AS x + y 4. Make z the subject of the following formulae: (a) z = c (b) z = h g (c) z d = e (d) z+ t = E (e) az+ b = C ( f ) y z = r (g) (i) (k) 1 = (h) = A mz az b c ( ) z a mz + n = c (j) = p b q A ( hz k ) m = B (l) + v= u z n

7 Sheet H a-5 Formulae (x appearing twice and powers) 1. Make x the subject of the following formulae: rx + c (a) nx + p = mx (b) = d bx t px c+ bx (c) = sx (d) a = q x x h (e) a x + b = c (f ) = q p. Make the following letters the subjects of the following formulae: (a) A A+ s ar = t (b) R = b A R+ 1 (c) e ae + b ap = 1 (d) p = d ce + d bp + c (e) Q bq ax b = c (f ) x = d Q+ d c (g) y my + n n aq = q (h) q = c p bq+ m (i) a bt ak t = c (j) k = p t k + b 3. The time, T minutes, taken by the moon to eclipse the sun totally is given by the formula 1 rd T = d where d and D are the diameters (in km) of the moon and the sun v R respectively, r and R are the distances (in km) of the moon and the sun respectively from the earth and v is the speed of the moon in km/s. (a) (b) (c) Make d the subject of the formula. Make r the subject of the formula. Make R the subject of the formula.

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9 Sheet H Proportion 1. In the following tables, y is directly proportional to x, that is y = kx for some constant k. In each case find the value of k, then copy and complete the tables. (a) x (b) x y y 17 (c) x (d) x y 8 y The number of arrests, A, in a town is directly proportional to the number of policemen, P, on patrol. It is recorded that when there were 16 policemen on patrol the number of arrests was 4. (a) Write down an equation giving A in terms of P, having calculated the constant of proportionality. (b) How many arrests will be made when there are 4 policemen on patrol? (c) How many policemen are there on patrol if 7 arrests are made? 3. When a car is accelerating from rest at a constant rate its speed, v, is directly proportional to time, t. (a) Write down this statement using the symbol. (b) Rewrite this statement using the symbol =. After 5 seconds the car is travelling at 15 m/s. (c) How fast will the car be travelling after 7 seconds? (d) After how long will he be travelling at 4 m/s? 4. The cost, C, of a crash mat is directly proportional to its thickness, T. (a) Write down a relationship between C and T and a constant k. A crash mat which is 4cm thick costs 75. (b) How much does it cost to buy a crash mat which is 3cm thick? (c) How thick is the crash mat which costs 375? 5. It is known that s varies as t (i.e. that s is directly proportional to t). It is also known that s = 1 when t = 6. (a) Find a formula for s in terms of t, having calculated the constant of proportionality. (b) Find s when t = 16. (c) Find t when s = h is directly proportional to T and h = 63 when T = 15. (a) Find a formula for h in terms of T, having calculated the constant of proportionality. (b) Find h when T = 35. (c) Find T when h = u is directly proportional to t and it is known that u = 18 when t = 9. (a) Find a formula for u in terms of t, having calculated the constant of proportionality. (b) Find the increase in u when t increases from 10 to 17. (c) By how much has t decreased when u falls from 0 to 15? PTO

10 Sheet H Proportion (cont.) 8. It is known that M = kn where k is the constant of proportionality. It is also known that when M changes from 5 to 8, the value of n increases from r to r + 4. (a) Write down two equations involving r and k. (b) Solve these to find r and also find k. (c) Find a formula for M in terms of n. (d) Find n, as a fraction, when M = 5. (e) Find n, as a fraction, when M = 8.

11 Sheet H Proportion 1. y is directly proportional to the square of x, that is y = kx for some constant k. Find the value of k, then copy and complete the table. x 4 8 y y is directly proportional to the cube of x, that is y = kx for some constant k. Find the value of k, then copy and complete the table. x 3 8 y The mass (M) of a block is proportional to the cube of the side length (L). (a) Write down an equation involving M, L and a constant k. It is given that a block of side length 3cm has mass 54kg. (b) Calculate k. (c) What is the mass of a block of side length 8cm? (d) What is the side length of the block which has mass 50kg? 4. An object accelerates from rest at a constant rate. Its distance (s) is proportional to the square of its velocity (v). When it has travelled 45m it has a velocity of 15m/s. (a) Write down an equation involving s and v. (b) How far will it have gone when its velocity is 45m/s? (c) What is its velocity when it has travelled 45m? 5. When a cricket ball is dropped from the top of a building the time (t) it takes for it to fall to the ground is proportional to the square root of the height of the building (h). When the building is 19.6 metres high it takes seconds to reach the ground. (a) Write down an expression for t in terms of h, showing that the constant of proportionality is 0.45 (to sf). (b) Using the exact value of the constant, find how long will it take to fall from a building which is 44.1m tall. (c) Using the exact value of the constant, find how high is the building from which it takes 1 second for the cricket ball to reach the ground. PTO

12 Sheet H Proportion (cont.) 6. The energy, E, stored in spring is proportional to the square of the extension, e. (a) Write down the relationship between E, e and a constant k. (b) If the extension is 5cm, the energy stored is 150 joules then find k. (c) How much energy is stored when the extension is 3cm? (d) What is the extension in the spring when the energy stored is 384 joules. 7. It is known that the time, T, taken for a pendulum to swing back and forth once is proportional to the square root of its length, l. (a) If the time taken for a pendulum of length 9cm to swing back and forth once is.4 seconds then write down an equation involving T and l. (b) Find the time for a pendulum to swing back and forth once if the pendulum has length 16cm. (c) Find the length of the pendulum which takes 5.6 seconds to swing back and forth once. 8. The current I in an electrical circuit varies as the square root of the power P. If the current is 18 amps when the power is 5 watts, find the current when the power is 144 watts. 9. If a cube of side length 5cm has mass m kg then find, in terms of m, the mass of a cube of side length 10cm made of the same material.

13 Sheet H Proportion 1. It is given that the distance, s, travelled by an object is directly proportional to the square of the time, t, for which it has been travelling. It is found that s = 75 when t = 5. (a) Write down an equation for s in terms of t, having calculated the constant of proportionality. (b) Find the value of s when t = 7. (c) Find t when s = The mass, m, of an object is directly proportional to the cube of its side length, l. The mass of a cube with side length 3cm is found to be 16g. (a) Write down an equation for m in terms of l, having calculated the constant of proportionality. (b) Find the mass of the object with side length 7cm. (c) Find the side length of the object which has mass 961g. 3. Find k and copy and complete the table given that y = k x. x y The speed of a particle, v, is directly proportional to the square root of its potential energy, P. The potential energy of a particle travelling at 10m/s is found to be 400 joules. (a) Write down an equation for v in terms of P, having calculated the constant of proportionality. (b) Find the speed of the particle with potential energy 196 joules. (c) Find the potential energy of the particle with speed 18m/s. 5. Find k and copy and complete the table given that y 3 = kx. x y (a) If y is directly proportional to (b) If y is directly proportional to x then by what factor does y increase as x doubles? 3 x then by what factor does y increase as x doubles? 7. The table shows some values of x and y x y (a) Write down a relationship between x and y using the symbol. (b) Write down an equation involving x and y. (c) What will x be when y = 3993? PTO

14 Sheet H Proportion (cont.) 8. Find k in the following and fill in the gaps in the tables shown below: (a) y = kx x 7 15 y (b) y = kx x 1 7 y (c) y k x = 3 x 8 16 y 14 60

15 Sheet H Proportion 1. H is directly proportional to the cube root of r. When r = 7 it is known that H = 15. (d) Write down an equation for H in terms of r, having calculated the constant of proportionality. (e) Find H when r = (f) Find r when H = 0.. (a) If y is directly proportional to x then by what factor does y increase as x increases by a factor of 4? (b) If y is directly proportional to 3 x then by what factor does y increase as x increases by a factor of 1000? In the following y is directly proportional to one of the following: x, x, x, x or x. Find formulae for y in terms of x (as an equation y =...), having first calculated the constant of proportionality. Also fill in the gaps. (a) x y (b) x y (c) x y (d) x y 4 8 (e) x y

16 Sheet H Inverse Proportion 1. It is known that the force, F, experienced by an object going round a circle is inversely proportional to the radius, r, of the circle. It is calculated that the force experienced by the object going around a circle of radius 15m is 8000N. (a) Find an expression for F in terms of r having first calculated the constant of proportionality. (b) Use this to find the force experienced by the object when going around a circle of radius 4m. (c) Find also the radius of the circle if the force is 1000N.. Fill in the following given that y is inversely proportional to x (a) x 5 10 y (b) x y The volume V of a given mass of gas varies inversely as the pressure P. When V = m 3, P = 400N/ m. (a) Find the volume when the pressure is 00N/ m. (b) Find the pressure when the volume is 16m When a force is applied to a block of mass M kg it produces an acceleration, a m/s. It is found that a is inversely proportional to M. The force produces an acceleration of 6 m/s when it acts on a block of 40kg. (a) If the mass of the block 15kg what acceleration will the force produce? (b) If the force acts on a block and produces an acceleration of 4m/s what is the mass of the block? 5. The resistance, R, in a wire of fixed length is inversely proportional to the square of the diameter, d. The resistance is 0.09 ohms when the diameter is 15mm. (a) Find an expression for R in terms of d having first calculated the constant of proportionality. (b) Find the resistance when the diameter is 9mm It is known that the quantity P is inversely proportional to r. It is also found that P = 8 when r = 5. (a) Find an equation to express P in terms of r having first found the constant of proportionality. (b) Use this to find P when r = 10. (c) Find also the value of r when P = The force of attraction F between two magnets is inversely proportional to the square of the distance d between them. When the magnets are 3cm apart the force of attraction is 1N. (a) What is the attractive force when they are 1cm apart? (b) How far apart are they if the attractive force is 7N?

17 Sheet H Inverse Proportion 1. It is known that the quantity Q is inversely proportional to the square root of t. It is also found that Q = 1 when t = 9. (d) Find an equation to express Q in terms of t having first found the constant of proportionality. (e) Use this to find Q when t = 16. (f) Find also the value of t when Q = 18. k. Find k and copy and complete the table given that y =. x x 3 10 y The light intensity, l, is measured at a distance d away from a lamp. It is found that l 1. It is observed that l = 180 when d = 7. d (a) Write down an equation involving l and d, having first found the constant of proportionality. (b) Find the value of l when d =. (c) Find the value of d when l = In a set of similar shapes the length L is inversely proportional to the cube root of the 3 volume V. When the length is 10cm, the volume is 7000 cm. (a) Find an equation for L in terms of V having first calculated the constant of proportionality. (b) 3 Find the length of the shape which has a volume of 1000 cm. (c) Find the volume of a shape which has a length of 1cm. 5. It is found that when x is 5, y is 8 and that when x is 10, y is. Given that y is inversely 3 3 proportional to one of the following: x, x, x, x or x, find the connection between y and x. (Write your answer as y =...) 6. The number of coins, N, with diameter dcm and with a fixed thickness can be made from a k given volume of metal can be found by using the formula N = d where k is a constant. (a) Given that 3600 coins of diameter cm can be made from the volume of metal, find the value of k. (b) Calculate how many coins of diameter 1.5cm can be made from an equal volume of metal. k (c) Rearrange the formula N = d to make d the subject. (d) 500 coins are to be made from an equal volume of metal. Calculate the diameter of these coins. PTO

18 Sheet H Inverse Proportion (cont.) 7. (a) If y is inversely proportional to x then by what factor does y increase / decrease when x doubles? (b) If y is inversely proportional to x then by what factor does y increase / decrease when x doubles?

19 Sheet H Inverse Proportion 1. A stone is dropped from a tall cliff. The time, t, for which it has been falling is directly proportional to the square root of the distance d that it has fallen in that time. (a) Use the following table showing distance against time to find an equation to express t in terms of d, having first found the constant of proportionality. t d (b) Hence copy and complete the table.. It is known that H is inversely proportional to the cube root of u. It is also known that H = 1 cm when u = 7 mm. (a) Find an equation to express H in terms of u, having first found the constant of proportionality. (b) Find H when u = 64mm. (c) Find u when H = 18 cm. 3. In the following tables find the value of k, then copy and complete the tables. (a) (b) y y = kx x 6 11 y = kx x y (c) k y = x x y k (d) y = x x 5 10 y

20 Sheet H Simultaneous Equations-Cancellation 1. Solve the following simultaneous equations: (a) 3x+ 4y = 17 (b) 5p+ q = 8 (c) 7v+ 6w= 46 5x+ 4y = 3 5 p+ 3q = 37 5v+ 6w= 38 (d) 5a+ 3b = 13 (e) 4c+ 5d = 17 (f ) 9x+ y = 31 7a+ 6b = 0 8c+ 3d = 7 3x+ y = 11 (g) 11p+ q = 63 (h) 5g + 3h = 7 (i) 9m+ 7n = 43 6 p+ 4q = 46 4g + 5h = 3 5m+ n = ( j) 8b+ 3c = 46 (k) 6x+ 5y = 13 (l) 8r+ 5s = 41 5b+ c = 9 7x+ y = 19 9r+ 11s = 30. Solve the following simultaneous equations: (a) x+ y = 11 (b) 3u+ v = 10 (c) 11p+ 3q = 71 3x y = 14 7u v = 9 5p q = 37 (d) 9a+ b = 41 (e) 7 p 3q = 15 (f ) 13b 7c = 47 5a 4b = 33 5p+ q = 19 7b 9c = Solve the following simultaneous equations: (a) 3x y = 19 (b) 3a 7b = 1 (c) 6r 5s = 38 5x+ 3y = 19 7a+ b = 39 5r s = 3 4. Solve the following simultaneous equations (to 3sf): (a) x+ 7y = 10 (b) 6a+ 5b = 11 (c) 9g + 7h = 45 3x+ y = 7 5a b = 5 5g h = 31

21 Sheet H Simultaneous Equations-Substitution 1. Solve the following equations (by first of all rearranging them): (a) 5x+ y = 0 (b) a = 3b+ 31 3y = 7x+ 1 5a+ 4b = 0 (c) x = 7y+ 16 (d) 11p q = 49 3x+ 4y = 53 8 p = 3q+ 48 (e) 3a+ 5b = 7 (f ) a+ 4b+ 3 = 5 5a = b 9 3b 5a = 10. Solve the following equations (by the method of substitution): (a) x+ 3y = 30 (b) a = 3b+ 1 y = 3x 1 7a+ b = 53 (c) c = 7d (d) 10 p+ 11q = 1 3c 4d = 8 p = 5q 4 (e) a 9b = 73 (f ) a+ 3b = 13 b = 3 a a = 3+ 5b (g) w+ 7x = 41 (h) 3t 5s = 170 w= 3x+ 1 t = 4 7s 3. Solve the following equations (by first of all rearranging them and then substituting): (a) 5m+ 3n = 7 (b) p+ 7q = 3 m+ n = 7 p q = 6 (c) u+ v = 7 (d) 3p+ q = 1 u+ 3v = 11 p 3q+ 4= 0 (e) 7r + s = 17 (f ) 5x 7y+ 5 = 6 r 3s = 9 x y 9= 0

22 Sheet H Simultaneous Equations-Problems 1. A man buys 5 first class tickets and second class tickets which cost him 46. Another man buys first class and 3 second class which cost him 149. Let the price of a first class ticket be x and the price of a second class ticket be y. (a) Write down a pair of simultaneous equations involving x and y. (b) Find x and y.. Two numbers p and q (where p is the bigger number) are such that their sum is 95 and their difference is 1. (a) Write down a pair of simultaneous equations involving p and q. (b) Find p and q. 3. A man buys three student tickets and five adult tickets which costs him 6. Another man buys seven student tickets and three adult tickets which cost him 71. Let the price of a student ticket be s and the price of an adult ticket be a. (a) Write down a pair of simultaneous equations involving s and a. (b) Find s and a. 4. A bag contains a collection of p and 5p coins. The total amount of money in the bag is 1.60 and there are fifty coins in total. Let t be the number of p coins and f be the number of 5p coins. (a) Write down two simultaneous equations involving t and f. (b) Find t and f. 5. A theatre has 60 rows of seats. Some of the rows have 30 seats, the rest of them have 35 seats and the theatre holds 015 people Let x be the number of rows with 30 seats and y be the number of rows with 35 seats. By solving the set of simultaneous equations find x and y. 6. A wallet contains three times as many 5 notes as 10 notes. The total amount of money in the wallet is 100. Find the number of each type of note in the wallet (by solving simultaneous equations). 7. A man is 5 years older than his son. He is, at present, six times older than his son. Let the man s age be m and the son s age be s. (a) Write down two simultaneous equations involving m and s. (b) Find m and s by the method of substitution.

23 Sheet H Simultaneous Equations-Problems 1. A man had 1.55 in his pocket made up of p and 5p coins. If he had 40 coins in his pocket then find the number of p coins he had. (First write down simultaneous equations involving t and f where t is the number of p coins and f is the number of 5p coins).. A boy buys 4 tickets in stand A and five tickets in stand B for a football match which costs him 144. Another boy buys six tickets in stand A and seven tickets in stand B and these cost him 07. If the stand A tickets all cost A and the stand B tickets all costs B then write down simultaneous equations involving A and B. Solve these to find A and B. 3. A cinema has 30 rows of seating, some of the rows have 5 seats and the rest have only 18 seats. If the cinema holds 659 people then find the number of the different types of rows. (First write down simultaneous equations involving x, the number of rows with 5 seats and y, the number of rows with 18 seats. 4. A man buys three ties and two shirts which cost him 13. Another man buys four ties and three shirts which cost him 190. If the cost of a tie is t and the cost of a shirt is s then find t and s (using simultaneous equations). 5. A man buys 3 first class tickets and 5 second class tickets for a plane journey. These cost him 164. A woman buys 7 first class tickets and 3 second class tickets for the same journey. These cost her 187. (a) If x is the cost of a first class ticket and y is the cost of a second class ticket write down two equations involving x and y. (b) Solve these equations to find x and y. 6. Mrs Brown buys 3 tickets for adults and 5 tickets for children at Logoland which cost her 34. Mr Green buys 5 tickets for adults and 7 tickets for children which cost him 5. (a) If x is the cost of the adult ticket, and y is the cost of the child ticket, write down equations involving x and y. (b) Solve these equations to find x and y. 7. A man buys seven seats for a concert at full price and three at a reduced price. These cost him 0. A woman buys four at the full price and five at the reduced price which costs her 45 less than the man. If the price of a full price ticket is x and the price of a reduced ticket is y then: (a) Write down two simultaneous equations involving x and y. (b) Solve these to find x and y. 8. Solve the following simultaneous equations (by the most efficient method): (a) 5a+ b = 1 (b) 4c 3d = 31 3a 5b = 5 5c d = 37 (c) 5y+ 3x = 41 (d) 6y+ 7x = 70 y = 3x+ 1 3y = 5x+ 1 (e) 3p+ 4q+ 13 = 0 (f ) 4s+ 11r = 5 q = 3 p s = 3r+ 13

24 Sheet H Simultaneous Equations 1. Solve the following simultaneous equations (by the most efficient method): (a) 3v+ 7w= 3 (b) 3p+ q = 11 5v = 3w+ 7q = 3 11p. A shops sells two types of pens, one type costs 5 and the other costs 7. One day it sells 17 pens and received 109. By first writing down simultaneous equations find how many of each pen it sold. 3. In a sale a bookshop was selling all its hard backs at the same price and all its paper backs at the same price. A woman bought 7 hard backs and 5 paper backs which cost her A man bought 11 hard backs and 7 paper backs which cost him Find the price of a hard back and the price of a paper back in the sale (by solving the relevant simultaneous equations). 4. A mother is six times older than her daughter. Let the mother s age be m and the daughter s age be d. (a) Write down an equation involving m and d. (b) Write down an expression for the mother s age in two years time. (c) Write down an expression for the daughter s age in two years time. In two years time, the mother is five times older than her daughter. (d) Write down a second equation involving m and d. (e) Solve the equations of (a) and (d) to find the present age of the mother and daughter. 5. An airline sold twice as many second class tickets as it sold first class tickets for a certain flight. The second class tickets cost 15 and the first class tickets cost 0. The total cost of the tickets was 17,390. Let f represent the number of first class tickets and s represent the number of second class tickets. (a) Write down two equations involving f and s. (b) Solve these equations to find f and s. 6. Solve the following simultaneous equations: (a) 5x + 3y = 5 7x + y = 18 (b) 3p q = 3 11p + 5q = 1 (c) b = 3a + 1 3a + 7b = 79 (d) s = 5r 3 9r + 4s = A man buys three second class and seven first class tickets for a flight which costs him 165. Another man buys two second class and five first class tickets which costs him (a) If the cost of a first class ticket is x and the cost of a second class ticket is y then write down two simultaneous equations involving x and y. (b) Solve these to find the cost of the first and second class tickets.

25 Sheet H Simultaneous Equations-Graphs 1. By looking at points of intersection of the straight lines shown on the graph below, solve the following simultaneous equations. (a) y x 5 = 0 (b) y x = 5 y x = 1 y+ x = 1 (c) y+ x = 1 (d) 3y+ 5x = 30 y 1= x y x = 1 (e) x + y 1 = 0 (f ) y = 1 x y+ 39 = 7x y 7x = 39 y = 7x 39 3y + 5x = 30 1 y = 6 x y = x + 5 y = x +1 y = 1 x. (a) Copy and complete the following tables: y = 3x y = 8 x 3 y + x = 1 x 1 3 x 0 4 x y y y (b) (c) Draw a set of axes with x and y from - to 8, using 1cm per unit and draw on it the curves y = 3x, y = 8 x and 3 y + x = 1. Use your graph to solve the following simultaneous equations: (i) y 3x = and y + x = 8 (ii) y + x = 8 and 3 y + x = 1

26 Sheet H Simultaneous Equations-Graphs 1. (a) Draw a set of axes with x from 8 to 8 and y from 10 to 10. (b) On this set of axes draw the following lines: 3x + y = 6 x + y = 6 x + y = 8 y x = 4 y = x (c) Use the above graphs to solve the following sets of simultaneous equations: (i) 3x + y = 6 and y = x (ii) x + y = 8 and x + y = 6 (iii) y x = 4 and x + y = 6 (iv) y = x and x + y = 6. y + 4 x + 7 = 0 y x = 4 x + y = 5 y = x 1 Use the above graph to solve the simultaneous equations : (a) x + y = 5 (b) y = x + 4 y = x 1 y + 4x + 7 = 0 (c) y = 7 4x y = 5 x (d) y 4 = x y + 7 = 4x

27 Sheet H a-1Quadratic Factorisation-Solve 1. Solve the following equations: (a) x + 9x+ 0 = 0 (b) x + 7x+ 1 = 0 x + x+ = x + x+ = (c) (d) x + x+ = x x+ = (e) (f ) x x+ = x x+ = (g) (h) x x+ = x x = (i) ( j) 6 0. Solve the following equations: ( a) x 8x 0= 0 (b) x x 15= 0 x + x = x + x = (c) (d) x x = x + x = (e) 0 0 (f ) 35 0 x x= x + x= (g) 6 0 (h) 0 x = x = (i) 4 0 ( j) Solve the following equations: (a) x + 7x+ 1 = 0 (b) y + 13y+ = 0 () (d) c m m = a a+ = (e) (f ) 1 0 z z = z + z+ = (g) (h) c + c+ = t t+ = (i) 6 0 ( j) 11 0 r r = t + t = (k) 3 0 (l) 0 w w= k + k =

28 Sheet H a-Quadratic Factorisation-Solve 1. Solve the following equations: (a) x + 7x+ 5 = 0 (b) y + 13y+ 11= 0 m m = a a+ = (c) (d) z z = a + a+ = (e) (f ) c + c+ = g g+ = (g) (h) z + z+ = h h = (i) 1 0 ( j) Solve the following equations: (a) t 18t + 81 = 0 (b) 8z + 19z+ 6 = 0 r r = t + t = (c) 6 0 (d) 11 0 w w= x = (e) 3 0 (f ) y + y = k + k = (g) 14 0 (h) 0 1 u u = y y = (i) (j) 0 3. Solve the following equations: (a) x 3x = (b) x + 30 = 11x (c) x 6x 7 (d) u u 10 + = = + (e) r r (f ) 4q 0q 9 = = 4. Solve the following equations: (a) hh ( 4) + 1 = 6 (b) 4 zz ( 1) = 3 (c) tt ( 4) 14 = t (d) ( y+ 1)( y+ 4) + = 0 (e) ( e+ )( e ) = 3 e (f ) ( r 3)( r ) = 6r 4 ( y )( y ) ( t )( t ) (g) = 1 (h) = 3