GCSE - New College Leicester

May 2013, 4HR Q9: 4 MarksALGEBRA PROBLEMS[ESTIMATED TIME: 80 minutes] GCSE

(+ IGCSE) EXAM QUESTION PRACTICE

9

*P42932A0924* Turn over

9

In the isosceles triangle ABC, AB = AC angle B = (3x + 32)! angle C = "#$ – 2x)!

Work out the value of x. Show clear algebraic working.

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Total for Question 9 is 4 marks)

A

B C"#$ – 2x)!(3x + 32)!

Diagram NOT accurately drawn

1. [4 marks]

x = ............................

Questions compiled by:@Maths4Everyone

Contains questions which have been reproduced with the kindpermission of Pearson Education Limited UK

4

*P41038A0420*

2 Rectangle A has a width of x metres and a height of (x + 2) metres. Rectangle B has a width of 2x metres and a height of 4x metres.

(x + 2)

x

4x

2x

A B

The perimeter of rectangle A is equal to the perimeter of rectangle B.

(i) Use this information to write down an equation in x.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) Find the value of x.

x = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



2. [4 marks]

-1-

Jane invests £300 at a simple interest rate of 4.5% per year.

At the end of each year Jane gives the interest to a charity.

Work out the least number of years it will take for the total amount given to the charity to be greater than £50

57 The diagram shows a shape, PQRSTU. All the corners are right angles. The lengths of four of the sides are given in terms of a and b.

Find an expression, in terms of a and b, for (i) PU, ........................................................ (ii) PQ. ........................................................

57 xxx. ........................................................

(2)

57 xxx.

1'#F'+

G.#&Q

V

!"#$%&'()#*%! ;>:(%'K7:

F(0@7:%F&&%;L*C;B%;LD%=>70/.'(0E

L:./7%G'>:%,(0@7:0%.(%/67%0A,870%A:'K.)7)E

B'>%->0/%@:./7%)'@(%,<<%/67%0/,970%.(%G'>:%@':M.(9E

NE+ 0M'+2%#8(#$+*M-P*+#+*M#='N+!"#$%&'

+ S..+5M'+@-(&'(*+#('+(%8M5+#&8.'*;

+ 0M'+.'&85M*+-?+?-,(+-?+5M'+*%2'*+#('+8%F'&+%&+5'($*+-?+(+#&2+);

+ \%&2+#&+'"=('**%-&N+%&+5'($*+-?+(+#&2+)N+?-(

+ 6%7+ !&N

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

% 6%%7+ !";

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ON

H;'/,<%P%-,:M0J

Q

b

3a

U T

RSa – b

3a + 2b

P

3. [3 marks]

NOK

0C'+.'&ADC*F+%&+7$F+-6+DC'+*%2'*+-6+#+D(%#&A.'+#('+8% ^ Y9F+8[% @ U9+#&2+8>% @ <9H

0C'+4'(%$'D'(+-6+DC'+D(%#&A.'+%*+>: 7$H

T-(O+-,D+DC'+M#.,'+-6+%H

% ]+ HHHHHHHHHHHHHHHHHHHHHHHH

C;&/,<)O)-,:M0E

N!K ;'('+%*+#+6#%(+[c*%2'2+*4%&&'(H

LD*+*%2'*+#('+.#I'..'2+<F+>+#&2+[+#*+*C-N&H

8#9 Q%*C#+%*+A-%&A+D-+*4%&+DC'+*4%&&'(+DN%7'H

T-(O+-,D+DC'+4(-I#I%.%D/+DC#D+%D+N%..+.#&2+-&+<+I-DC+D%$'*H

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

CDE

#$%&$'(%)*

E>?U<<5Q \ ;>:')&F7:

PNO

% ^ Y

[% @ U

>% @ <

4. [3 marks]

a = ........................

1'#O'+

N.#&Q

W

!"#$%&#'%#&%! ;>:')&K7:

F'0@7:)F%%);L*C;B);LD)=>70/.&'0E

L:./7)G&>:),'0@7:0).')/67)0A,870)A:&K.(7(E

B&>)->0/)@:./7)(&@'),<<)/67)0/,970).')G&>:)@&:M.'9E

NE+ 3K-P+5K#5

+

IE+ S&8'.-,+K#*+!+*P''5*;

+ B'+'#5*+Y+-?+5K'*'+*P''5*;

+ B'+=,5*+#..+5K'+*P''5*+K'+K#*+.'?5+%&5-+#+N#8;

+ 6%7+ :%&#+K#*+W+5%$'*+#*+$#&/+*P''5*+#*+5K'+&,$N'(+5K#5+S&8'.-,+=,5+%&5-+5K'+N#8;

+ + :%&#+K#*+WH+*P''5*;

+ + [*'+5K%*+%&?-($#5%-&+5-+P(%5'+2-P&+#&+'R,#5%-&+%&+!;

+ + + + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

+ 6%%7+ 3-.O'+/-,(+'R,#5%-&+5-+?%&2+5K'+O#.,'+-?+!;

+ + + + !+^++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

ON

H;&/,<)P)-,:M0J

32 ÷

95 = 1

51

OI

H;&/,<)Q)-,:M0J

5. [5 marks]

-2-

52. Paper clips are sold in small boxes and in large boxes. There is a total of 1115 paper clips in 4 small boxes and 5 large boxes. There is a total of 530 paper clips in 3 small boxes and 2 large boxes. Work out the number of paper clips in each small box and in each large box. ........................................................

(5)


1 A shop sells packets of envelopes. There are 5 envelopes in a small packet. There are 20 envelopes in a large packet. There is a total of T envelopes in x small packets and y large packets. Write down a formula for T in terms of x and y. ........................................................

(3)


9 Dan, Harry and Regan sell cars. Dan sells x cars. Harry sells 5 more cars than Dan. Regan sells twice as many cars as Dan. Write an expression, in terms of x, for the mean number of cars Dan, Harry and Regan sell. ........................................................

(2)


N. Amita, Monica and Rita are three sisters. Monica is x years old.

Amita is 3 years older than Monica. Rita is twice the age of Monica. If the mean age of the three sisters is 12, how old is Amita? ........................................................

(2)


6. [5 marks]

....................................................

71+Q1!

()+.X

A

!"#$%&'()$#$!

EM!! I1*,+.8')+2!,$)1%!#+Q1!4$3,#!$ */!+.3!#1$8#,!M$!]!JN */9

! Z-/1!-L!,#1%1!,$)1%!+21!'%13!,-!L-2/!+!%#+&19! "#1!%#+&1!$%!=!,$)1%!4$31!+.3!A!,$)1%!#$8#9

! M+N! E2$,1!3-4.!16&21%%$-.%T!$.!,12/%!-L!$T!L-2!,#1!4$3,#!+.3!#1$8#,!-L!,#$%!%#+&19

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4$3,#!^!999999999999999999999999999999999999999999999999999999999999999999999999999!*/

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#1$8#,!^!99999999999999999999999999999999999999999999999999999999999999999999999999!*/DEF

! M(N! "#1!4$3,#!+.3!,#1!#1$8#,!-L!,#$%!%#+&1!+21!1Y'+)9!

! ! M$N! E2$,1!3-4.!+.!1Y'+,$-.!$.!$9

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

! ! M$$N! Z-)Q1!0-'2!1Y'+,$-.!,-!L$.3!,#1!Q+)'1!-L!$9

! $!^!999999999999999999999D!F QE

D<'0-=*R*.-;O1F

_$+82+/!IL<+**'2+,1)0!32+4.

$

$ ] J

_$+82+/!IL<+**'2+,1)0!32+4.

4$3,#

#1$8#,

7. [6 marks]

x = .......................(4)

Leaveblank

4

*N24647A0416*

2. (a) Factorise 3x2 – 2x

........................... (1)

(b) Expand y3(y – 4)

........................... (2)

(c) Here is a formula used in physics.

v = u + at

Find the value of t when v = 30, u = 5 and a = 10

t = ...........................(2)

3. Arul had x sweets. Nikos had four times as many sweets as Arul.

(a) Write down an expression, in terms of x, for the number of sweets Nikos had.

........................... (1)

Nikos gave 6 of his sweets to Arul. Now they both have the same number of sweets.

(b) Use this information to form an equation in x.

................................................................................. (2)

(c) Solve your equation to find the number of sweets that Arul had at the start.

........................... (2) Q3

(Total 5 marks)

Q2

(Total 5 marks)8. [5 marks]

Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


Leaveblank

4

*N24647A0416*


........................... (1)


........................... (2)


v = u + at


t = ...........................(2)



........................... (1)



................................................................................. (2)


........................... (2) Q3

(Total 5 marks)

Q2


8. [5 marks]

Leave blank

6

*H34884A0624*

5. Cups cost x dollars each. Mugs cost (x + 2) dollars each.

(a) Write down an expression, in terms of x, for the total cost of 12 cups and 6 mugs.

............................................................................... dollars(2)

(b) The total cost of 12 cups and 6 mugs is 57 dollars. Work out the cost of 1 cup.

......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

Leave blank

6

*H34884A0624*



............................................................................... dollars(2)


......................... dollars(2) Q5

(Total 4 marks)

9. [4 marks]

............................ dollars(2)

9. [4 marks]

............................ dollars(2)

1'#O'+

N.#&Q

[

!"#$%&'(&,)*!

QE+ >'@5#&8,.#(+5%.'*+K#O'+P%25K+6%+`+D7+@$+#&2+K'%8K5+6Z+%+H+F7+@$;

5x – 2

x + 1Diagram NOTaccurately drawn

+ 3-$'+-?+5K'*'+5%.'*+#('+,*'2+5-+?-($+#+.#(8'+('@5#&8.';

+ 0K'+.#(8'+('@5#&8.'+%*+^+5%.'*+P%2'+#&2+W+5%.'*+K%8K;

Diagram NOTaccurately drawn

width

height

+ 0K'+='(%$'5'(+-?+5K'+.#(8'+('@5#&8.'+%*+X[+@$;

+ 6#7+ T(%5'+2-P&+#&+'R,#5%-&+%&+%;

+ +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

HPJ

+ 6N7+ 3-.O'+5K%*+'R,#5%-&+5-+?%&2+5K'+O#.,'+-?+%;

+ %+_++;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

HPJ OQ

H;&/,<)S)-,:M0J

10. [6 marks]

x = .......................(3)

9


9 7KH�GLDJUDP�VKRZV�D�SDUDOOHORJUDP�ABCD. ,Q�WKH�GLDJUDP��DOO�WKH�DQJOHV�DUH�LQ�GHJUHHV�

Work out the value of x DQG�WKH�YDOXH�RI�y.

x = .. . . . . . . . . . . . . . . . . . . . . . . . . . .

y = .. . . . . . . . . . . . . . . . . . . . . . . . . . .


Do NOT write in this space.

Diagram NOT DFFXUDWHO\�GUDZQ

A B

D C

4x – 30

x�� x��y

11. [4 marks]

-1-




57 Kalinda buys x packs of currant buns and y boxes of iced buns. There are 6 currant buns in a pack of currant buns. There are 8 iced buns in a box of iced buns. Kalinda buys a total of T buns. Write down a formula for T in terms of x and y. ........................................................

(2)


N. There are 30 sweets in a box. x of the sweets are blue. The rest of the sweets are green. Aaron takes at random a sweet from the box. Write down an expression, in terms of x, for the probability that Aaron takes a green sweet. ........................................................

(2)


N. Ben is x cm tall. Kieran is 8 cm taller than Ben. Bianca is 2 cm shorter than Ben. Write an expression, in terms of x, for the mean of their heights in centimetres. Give your answer in its simplest form. ........................................................

(3)


12. [3 marks]

............................

11


11 The diagram shows a right-angled triangle and a rectangle.

The area of the triangle is twice the area of the rectangle.

(i) Write down an equation for x.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(ii) Find the area of the rectangle. Show clear algebraic working.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm2


9 cm

(8x + 4) cm (10 – x) cm

7 cmDiagram NOT accurately drawn

13. [7 marks]

.................................. cm2

-7-

46


In the diagram, all angles are in degrees. Angle AOB is a right angle. Angle AOC = Angle BOC. Work out the value of x.

29


D

O N

OT

WRI

TE IN

TH

IS A

REA

D

O N

OT

WRI

TE IN

TH

IS A

REA

D

O N

OT

WRI

TE IN

TH

IS A

REA

27

In the diagram, all angles are in degrees.

Angle AOB is a right angle. Angle AOC = Angle BOC.

Work out the value of x.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



A

O6x – 21

4x + 31

B

C

PMT

14. [3 marks]

x = ............................

-3-

N. Tara has 3 dogs and 4 cats. The dogs have a mean weight of x kg.

The cats have a mean weight of y kg. Write down an expression, in terms of x and y, for the mean weight of all 7 of Tara’s pets. ........................................................

(2)


15 Steph's weight in kilograms is 2x + 9 Kyle's weight in kilograms is 4x − 7 Write down an expression, in terms of x, for the mean of their weights. ........................................................

(2)


N. Barney has the same number of sweets as Millie. Barney gives 15 of his sweets to Millie. Millie now has 4 times as many sweets as Barney. Work out the total number of sweets that Barney and Millie have. ........................................................

(4)


19 Stephanie is x years old. Tobi is twice as old as Stephanie. Ulrika is 3 years younger than Tobi. The sum of all their ages is 52 years. Work out the age of Ulrika. ........................................................

(5)


15. [2 marks]

............................ kg

-8-

40 Here is a rectangle.


All measurements are in centimetres. The area of the rectangle is 242 cm2. Find the value of w.

C

D A B

6.4 cm

5.8 cm

72º x

3x + 5 7x – 3

w

16. [5 marks]

............................ cm

-3-

N. Tara has 3 dogs and 4 cats. The dogs have a mean weight of x kg.

The cats have a mean weight of y kg. Write down an expression, in terms of x and y, for the mean weight of all 7 of Tara’s pets. ........................................................

(2)


15 Steph's weight in kilograms is 2x + 9 Kyle's weight in kilograms is 4x − 7 Write down an expression, in terms of x, for the mean of their weights. ........................................................

(2)


N. Barney has the same number of sweets as Millie. Barney gives 15 of his sweets to Millie. Millie now has 4 times as many sweets as Barney. Work out the total number of sweets that Barney and Millie have. ........................................................

(4)


19 Stephanie is x years old. Tobi is twice as old as Stephanie. Ulrika is 3 years younger than Tobi. The sum of all their ages is 52 years. Work out the age of Ulrika. ........................................................

(5)


17. [4 marks]

............................

-4-

5. Vicky makes 8 purses and 9 key rings to sell for charity. The price of a purse will be twice as much as the price of a key ring. Vicky wants to get a total of exactly £40 when she sells all the purses and all the key rings. Work out the price Vicky needs to charge for each purse and for each key ring. ........................................................

(4)


21. Asha and Lucy are selling pencils in a school shop. They sell boxes of pencils and single pencils. Asha sells 7 boxes of pencils and 22 single pencils. Lucy sells 5 boxes of pencils and 2 single pencils. Asha sells twice as many pencils as Lucy. Work out how many pencils there are in a box. ........................................................

(4)


23 Dan has some marbles. Ellie has twice as many marbles as Dan. Frank has 15 marbles. Dan, Ellie and Frank have a total of 63 marbles. How many marbles does Dan have? ........................................................

(3)


33 Hilary, Imogen and Jeeha are playing a game with cards. Imogen has 3 cards more than Hilary. Jeeha has twice as many cards as Imogen. They have a total of 53 cards. Work out how many cards Hilary has. ........................................................

(4)


18. [4 marks]

purse ............................ key ring ............................

-9-

36 ABCD is a trapezium. STUV is a rectangle.


All measurements are in centimetres. The two shapes have the same perimeter. Work out the length of ST. 5 marks

18

*P45828A01824*

16 ABCD is a trapezium. STUV is a rectangle.

All measurements are in centimetres.

The two shapes have the same perimeter.

Work out the length of ST.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cm



A

9x – 2 CD

B7x – 2

5x 5x

S T

V U

4x

5x + 5

19. [5 marks]

............................ cm

-10-

29 Here are two straight lines, ABCDE and PQ Diagram NOT accurately drawn

In the diagrams all the lengths are in cm. AE = 2PQ. Find an expression, in terms of x, for the length of DE. Give your answer in its simplest form.

8

*P44019A0816*

7 (a) Simplify 2e + 3f – e + 4f

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(2)

(b) Expand 5(2c + 3d)

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)

(c) Here are two straight lines, ABCDE and PQ.

In the diagrams all the lengths are in cm. AE = 2PQ. Find an expression, in terms of x, for the length of DE. Give your answer in its simplest form.

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .cm(4)


x x + 3 2xA B C D E

3x – 2P Q

Diagrams NOT accurately drawn

20. [4 marks]

............................ cm

-1-




57 Kalinda buys x packs of currant buns and y boxes of iced buns. There are 6 currant buns in a pack of currant buns. There are 8 iced buns in a box of iced buns. Kalinda buys a total of T buns. Write down a formula for T in terms of x and y. ........................................................

(2)


N. There are 30 sweets in a box. x of the sweets are blue. The rest of the sweets are green. Aaron takes at random a sweet from the box. Write down an expression, in terms of x, for the probability that Aaron takes a green sweet. ........................................................

(2)


N. Ben is x cm tall. Kieran is 8 cm taller than Ben. Bianca is 2 cm shorter than Ben. Write an expression, in terms of x, for the mean of their heights in centimetres. Give your answer in its simplest form. ........................................................

(3)


21. [2 marks]

............................

-4-

52. Paper clips are sold in small boxes and in large boxes. There is a total of 1115 paper clips in 4 small boxes and 5 large boxes. There is a total of 530 paper clips in 3 small boxes and 2 large boxes. Work out the number of paper clips in each small box and in each large box. ........................................................

(5)


1 A shop sells packets of envelopes. There are 5 envelopes in a small packet. There are 20 envelopes in a large packet. There is a total of T envelopes in x small packets and y large packets. Write down a formula for T in terms of x and y. ........................................................

(3)


9 Dan, Harry and Regan sell cars. Dan sells x cars. Harry sells 5 more cars than Dan. Regan sells twice as many cars as Dan. Write an expression, in terms of x, for the mean number of cars Dan, Harry and Regan sell. ........................................................

(2)


N. Amita, Monica and Rita are three sisters. Monica is x years old.

Amita is 3 years older than Monica. Rita is twice the age of Amita. If the mean age of the three sisters is 15, how old is Amita? ........................................................

(2)


22. [3 marks]

............................ years

-4-

5. Vicky makes 8 purses and 9 key rings to sell for charity. The price of a purse will be twice as much as the price of a key ring. Vicky wants to get a total of exactly £40 when she sells all the purses and all the key rings. Work out the price Vicky needs to charge for each purse and for each key ring. ........................................................

(4)


21. Asha and Lucy are selling pencils in a school shop. They sell boxes of pencils and single pencils. Asha sells 7 boxes of pencils and 22 single pencils. Lucy sells 5 boxes of pencils and 2 single pencils. Asha sells twice as many pencils as Lucy. Work out how many pencils there are in a box. ........................................................

(4)


23 Dan has some marbles. Ellie has twice as many marbles as Dan. Frank has 15 marbles. Dan, Ellie and Frank have a total of 63 marbles. How many marbles does Dan have? ........................................................

(3)


33 Hilary, Imogen and Jeeha are playing a game with cards. Imogen has 3 cards more than Hilary. Jeeha has twice as many cards as Imogen. They have a total of 53 cards. Work out how many cards Hilary has. ........................................................

(4)


23. [4 marks]

............................

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