secondary 1 algebra

7/25/2019 Secondary 1 Algebra

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SECONDARY 1 ALGEBRA

1. Simplify the following, showing your working clearly.

(a) 4 c+3 d2+5d10

(b) w5v2

7w5 v

31


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2. (a) Factorise the following completely.

(i) mx+3mxy

(ii) 2axbx6 ay+3by

(b) Solve the following equation.5

(x1

)2

(1

x )=4

Answers (ai) !!!!!!!!!!!!!!! " 1 #

(aii) !!!!!!!!!!!!!!! " 2 #

(b) !!!!!!!!!!!!!!! " 2 #

$. % book costs &2.' an a maga*ine costs &+.. f -len wants to buy q books

an (q+3) maga*ines, write own an simplify an algebraic epression for the

amount, in ollars, that he has to pay for the books an maga*ines.


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Answer $

!!!!!!!!!!!!!!

" 2 #

/. 0rite own an simplify an algebraic epression for each of the following

(a) he prouct of3

x an5

y(b) Subtract the sum of 3ab an 5a+ba from 107 ab

Answers (a) !!!!!!!!!!!!!!!!! " 1 #

(b) !!!!!!!!!!!!!!!!! " $ #

'. (a) Factorise pqkq

(b) ence, fin the value of 332045232045 without the use of a

calculator.


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Answers (a) !!!!!!!!!!!!!!!! " 1 #

(b) !!!!!!!!!!!!!!!! " 2 #

3. he base of a triangle,y, is / times its height.

(a) 0hat is its area in terms ofy4

(b) Fin the area of the triangle if the base is 16 cm.

Answers (a) !!!!!!!!!!!!

cm

2

" 2 #

(b) !!!!!!!!!!!!

cm2

" 1 #

+. (a) % piece of wire is cut into 2 equal parts. 5ne part is bent to form a triangle

with sies 4x cm, (x22) cm an 2x cm while the other part is

use to form a rectangle with breath (x2) cm an length twice as

long as its breath.(i) 0rite own an simplify an epression for the

(a) perimeter of the triangle " 1 #

(b) perimeter of the rectangle " 1 #

(ii)

ence, fin the value of x . " 2 #


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6. Solve the following equations

(a) 2x+33

x=12 (x+1 )

2 " $ #

(b) x+72(x7)=

1 14 " $ #


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7. 8press the following wor statements in terms of algebraic epressions.

(a) % 21 to the prouct of $xany.

(b) Subtract $cfrom the quotient of aivie by b.

(c) 9ultiply / by two thirs of d.

%ns (a) !!!!!!!!!!!!!!!!!!

"1#

%ns (b) !!!!!!!!!!!!!!!!!!

"1#

%ns (c) !!!!!!!!!!!!!!!!!!

"1#

1. (a) Subtract (

+2$ + yx) from (

1$63 yx).

(b) 8valuateyx

yx

+

when/=x

an$=y

.


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%ns (a) !!!!!!!!!!!!!!!!!!

"2#

%ns (b) !!!!!!!!!!!!!!!! "2#

11. Simplify

(a)

)7/($ ba+,

(b)

)311($1 dcc ,

(c)

gfe

gef

fe

2$

/

'

2 11167''

.

%ns (a) !!!!!!!!!!!!!!!!!!

"1#


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%ns (b) !!!!!!!!!!!!!!!!!!

"2#

%ns (c) !!!!!!!!!!!!!!!!!!

"2#

12. 0rite algebraic epressions for each of the following statements.(a) he cost ofxlitres of petrol at &1.7 per litre.

(b) alf the prouct of man the square of v.

%ns

wer (a) & !!!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!!!! "1#

1$. 9rs Fiels sol pens at '' cents each an markers at &2 each. She sol / pens an markers

altogether an collecte a total of &'7.+.

(a) f 9rs Fiels solxpens, epress in terms ofxfor the number of markers that she has sol. "1#

(b) Form an equation inxconnecting the cost of pens an that of the markers that were sol by 9rs

Fiels. "2#

(c) Solve the above equation to fin the number of markers. "/#


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1/. Factorise the following

(a)

)+(2)+($ xyayx +

(b)

$222$662 xyyxyx ++

(c)

yzxyxzx 2+77$ 2 +


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%ns (a) !!!!!!!!!!!!!!!!!!

"1#

%ns (b) !!!!!!!!!!!!!!!!!!

"2#

%ns (c) !!!!!!!!!!!!!!!!!!

"2#

1'. f a: ;2, t: $, u: /, v: ;', evaluate the following.

(a)t

uv

(b)

2

2

1atut+

%nswer (a) !!!!!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!!!! "2#

13. Solve the following equations.

(a)

3/)'$(' += xxx.


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%ns (a) !!!!!!!!!!!!!!!!!! "2#

(b)'

/$

/

12'

=

xx

.

%ns !!!!!!!!!!!!!!!!!!

"$#

1+. (a) Solve the following equations.

(i)xx $

+

2

$=

"2#

(ii)

/$'

)+($=

y

y

"$#

(b) -iven that12

$

+

=m

lk

, fin mwhen k: 7 an l: $. "$#


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16. (a) Factorise /x2; 13x completely.

(b) Factorise 'p2; /qr< 'pr; /pqby grouping.


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%nswer (a) !!!!!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!!!! "2#

17. (a) Subtract1'$ 2 + xx

from3/+ 2$ ++ xxx

, leaving your answer in the simplest form.

(b) Simplify/

2

$

1

+ xx

.

%nswer (a) !!!!!!!!!!!!!!!!!!!!!! "2#

(b) !!!!!!!!!!!!!!!!!!!!!! "$#

2. (a) 8pan an simplify

)$(2)12($ + xx. "2#

(b) Simplify

( )[ ]ababa 2/$6 ++. "$#

(c) Simplify$

)1(/2

+

x

, epress your answer as a single fraction. "2#


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21. Simplify the following

(a)abba 3$2

(b)

y

xz

$

1'


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%nswer (a) "2#

(b) "1#

22. Simplify

(a)

xxyxyx /2'7 ++

(b)

( )pqp '2$ +

%nswer (a) "1#

(b) "2#

2$. (a) Subtract

$2+ ++ xyfrom

/$1 + xy.

(b) Simplify2/

$

3

' uuu+

.


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%nswer (a) "2#

(b) "2#

2/. -iven that12=a

,$=b

an1=c

, fin the value ofbac 2

%nswer "2#

2'. Simplify'

$

$

' +

xx

.

%nswer "2#2+. Solve the following equations


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(a)7'2 =+x

(b)

+$

/ =x

%nswer (a) x: "2#

(b) x: "$#

26. Factorise

(a)xx /6$3 2

(b)

( ) ( )/$/3 +++ xxx

(c)

pyqyqxpx 22 +

%nswer (a) "1#


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(b) "2#

(c) "$#29. Simplify

(a)

( )+3 + x "1#

(b)( ) ( )2$$'32 +++ xx

.

"2#

30. Simplify the following epressions.

(a)

)/$(3 aaa ,

(b)

1)2$(')$($ ++ yxyx,

(c)/

'/

$

2$ qpqp

.


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Answer (a) =========..=. "1#

(b) =======.==..= "2#

(c) ====...====== "2#

31.

>ames went to a supermarket which sells / apples at &xan $ honeyews at &

( )/6 x

. e bought 6 apples an $ honeyews.(a) -iving your answers in ollars, write own in terms ofx, the cost of each

(i) apple

(ii) honeyew

"1#

"1#

(b) 0rite own an simplify, in terms ofx, an epression for the total amount ofmoney that >ames has to pay for the apples an honeyews that he bought.

-ive your answer in ollars.

"2#

(c) >ames pai the shopkeeper &2 an receive a change of &/. Form an equation,

in terms ofx, an fin the value ofx.

"$#


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32.

(a) Solve

2/

2$

=+x

.

"$#

(b) he iagram below shows a rectanglePQRSin whichPQ:

'

/

$k

cm,

PS:

+1

$

2k

cm an QR:

+ $

2

1k

cm. Fin the value of k.

cm1

$

2

+k

cm'/

$

k

cm$2

1

+k

"$#

SP

RQ


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33

. Subtract

)3$'12( 2$ ++ xxxfrom

)$'/1/1( 2$/ ++ xxxx.

Answer ===========."2#

34

.

?erek bought some egg tarts an buns from a bakery. he number of egg tarts bought is x.

he number of buns bought is ' more than the number of egg tarts.

(a) 0rite own, in terms ofx , an epression for the number of buns ?erek bought.

(b) 8ach egg tart an bun costs 6 cents an 3 cents respectively.

Fin, in its simplest form in terms ofx

, an epression for the total cost of the egg tarts an

buns ?erek bought.

(c)f ?erek pai a total of &12.6 for all the egg tarts an buns purchase, fin the number of

egg tarts an the number buns he bought respectively.


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Answer (a) =====...=== buns "1#

(b) & ==========. "1#

(c) =.= egg tarts, .== buns "2#

35

.

Factorise each of the following.

(a)

22 613 xyyx ,

(b)

qcqdpdpc /1362 +.

Answer (a) =========...=. "1#

(b) =========== "2#

36. Solve

1/)1(2)12($' =+ xxx. "2#


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BA

C

E

BA

C

E

BA

C

E

BA

C

E

37

.

(a) @enson thinks of a number, multiplies the number by ' an subtracts the result from 36.

he answer is the same if he as / to the number an multiplies the result by $.

Fin the number. "2#

(b)Simplify

2222 32' yxyx ++completely. "2#


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(c)-iven that

72

'6=

yx

xy

, fin the value of

y

x2$

. "2#

38. -iven thatx: 2 any: ;$, fin the value of(a) 'x; /y

(b) xy2

%nswer (a) !!!!!!!!!!!!!!!!!!!!

(b) !!!!!!!!!!!!!!!!!!!!

"2#

"2#

39. (a) Simplify the epression 2k; $(2 ; k) < /


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(b) Solve the following equations

(i)

/'=

p

,

(ii)

'1

$ 2 =

t t

.

%nswer (a) !!!!!!!!!!!!!!!!!!!!

(b) (i)p =!!!!!!!!!!!!!!!

(b) (ii) t =!!!!!!!!!!!!!!!

"2#

"1#

"$#

40.

(a) Simplify

)$2(2)12($ + aa "2#

(b) Factori*e

yzxzxyx $21+ + "2#


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41. Solve the following

(a)( ) ( ) 13121$/ =+ xxx

(b)/

11

2

1

$

/=

kk

"$#

"$#

42. %lice isxyears ol. om, her brother, is 7 years oler than %lice. heir mother is $

times as ol as %lice. heir father is twice as ol as om.

(a) 0rite own epressions, in terms ofx, for

(i) omAs age,

(ii) their motherAs age,

(iii) their fatherAs age,

(b) he sum of the ages of the four members of the family is 1$7. (i) 0rite own the equation satisfie byx.

(ii) Solve the equation to fin the value ofx.

"1#

"1#

"1#

"2#

"1#


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/$. Factorise completely

qpqpq + 12

. "$#


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//. (i) 8pan an simplify

abbabbaa +++ )2()(. "2#

(ii) ence, using suitable substitution for aand b fro (i),

evaluate

$2)$/($)$2(2 +++without the use of calculator. "2#

/'. Simplify3

12

/

2 +

xx

. "/#

/3. -iven the formulaR

QP

++=

1

2

fin

(i) the value of

when$=P

,

/=Qan

2=RB "2#

(ii) the values ofP

when1$=

,2/=Q

an1=R

. "$#


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/+. Solve for the value ofxfor the following equations.

(a)/

11

$

2=+

x

B "$#

(b)

)1(2

$

)1($

6

+=

+ xx. "/#


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/6

.

(a) Simplify $p; 2($


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/7

.

(a)

Solve

'$

'2=

+p

(b

) -iven that

$ '11 bca =+, fin the value of aif c: 12 an b: C/

%nswer (a) =====..=====..=....."2#(b) =====..=====.==.."2#


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'

.

8press the following as a single fraction in its simplest form.

(a)

yy

2'

)'$(2+

.

(b) 2

1$

3

$'

xx

.

%nswer (a) =====..=====..=....."2#(b) =====..=====.==.."$#

'1. Simplify the following algebraic epressions.


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(a)

pqpqp 11'+3/ ++

(b)

)2(/)1($ ++ xx

Answer (a) !!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!! "2#

'2. Solve the following equations.

(a)

2+3$ =a

(b)

6$

12

=+

x

x

Answer (a) !!!!!!!!!!!!!!!!!!!!! "2#


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(b) !!!!!!!!!!!!!!!!!!!!! "2#

'$. 8press the following as a single fraction in its simplest form.

(a)3

'

'

2

$

xxx+

(b)

( )

/

12$

'

/

+ xx

Answer (a) !!!!!!!!!!!!!!!!!!!!!! "2#

(b) !!!!!!!!!!!!!!!!!!!!!! "$#

'/. -iven that

( )2$a

abcP=

, fin the value ofPif2=a

,1=b

an'=c

.


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Answer !!!!!!!!!!!!!!!!!!!!!!! "2#

''. Subtract

)'/$( 2 yxyx +from the sum of

)32$( 2 yxyx +an

)22( $ yxyx +.

Answer !!!!!!!!!!!!!!!!!!!! "$#

'3. Factorise the following algebraic epressions.

(a)

bcaba 133/ +

(b)

)2$()$2(2 xzxy +++

(c)

srsqprpq + 216$3


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Answer (a) !!!!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!!! "1#

(c) !!!!!!!!!!!!!!!!!!!!! "2#

'+. Subtract

$2 ' $x x +

from the sum of

$ 21 $

6 'x x x+ +

an

2 12/

x x +

.

%nswer !!!!!!!!!!!!!!!!!!!!!!!!! "2#

'6. 8pan an simplify)72($2/)'$(62+ +++ yxyyyxy.


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%nswer !!!!!!!!!!!!!!!!!!!!!!!!! "2#

'7. f2=a

,1=b

,$=c

, an/=d

evaluate

dabbc 2$ $/

(b)

2$ c

cd

b

bc +

+

%nswer (a) !!!!!!!!!!!!!!!! "2#

(b) !!!!!!!!!!!!!!!! "2#

3. Factorise each of the following

(a)

)'2(+)2'( xy!yx

(b)

sqpqsrpr $'1/1/ +


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%nswer (a) !!!!!!!!!!!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!!!!!!!!!!! "2#

31. Simplify the following algebraic fractions, epressing as a single fraction

$

22

2/

)$$(+

3

)+2('

xxx

(b)

( )3

+$/

'

$2

+ ww


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%nswer (a) !!!!!!!!!!!!!!!!!!!! "$#

(b) !!!!!!!!!!!!!!!!!!!! "$#

32. Solve the following equations

(a)2$

/

1+

'

+=

xx (b)

$

)/2(2

/

)1($

'

/+ =

+

rrr


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%nswer (a)=x!!!!!!!!!! "2#

(b)=r

!!!!!!!!!! "$#

3$

.-iven that

1'

12 2

++

=b

b"

, calculate the value of" when b:7

1

.

8press your answer as

(a) a fraction ,

(b) a ecimal ,correct to 1 significant figure.

"2#

"1#

3/

. Simplify

( )[ ]pmpm 23$'/2 .

"2#


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3'

.

Factorise the following

(a)

kpkmkb $31//2/ +,

(b)xzmxzgmg 13/12$ +

.

"1#

"2#

33

. 8press

( )

'

$/

2

/$

+

23 xxx

+

as a single algebraic fraction in its simplest form. "$#


42/55

3+

.

Solve the following equation

(a)

( ) ww '7$2'3

1+=+

.

(b)

( ) ( )2$1+2/12' =+ mmm.

"$#

"$#


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36

.

Fiona bought 2' hanphone pouches an pencil cases. he cost of each pencil case is &6.

8ach hanphone pouch cost twice as much as each pencil case. She spent a total of &23/.

Det the number of pencil cases bought bex.

(a) 8press the number of hanphone pouches bought in terms ofx.

(b) Form an equation inxan solvex.

(c) ence, fin the number of hanphone pouches an pencil cases Fiona bought respectively.

"1#

"$#

"2#


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37. 8press an simplify the following statements algebraically.

(a) Subtractsfrom the prouct of $pan ;2q.

(b) % the square of ato the quotient of 1'a#ivie by $a.

(c) Subtract 2k; '!from the sum of 3k< 2!; 1 an ;+k; !< '.

%ns a) !!!!!!!!!!!!!!!!!!!!!!!! "1#

b) !!!!!!!!!!!!!!!!!!!!!!!! "2#

c) !!!!!!!!!!!!!!!!!!!!!!!! "2#

+. (a) -iven thatx: 2 any: 1, evaluatex2$ .2y;/

x

.

(b) -iven the formula

1

$ 1

mS

n

=

+, fin mif n: $, S: 1.


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%ns a) !!!!!!!!!!!!!!!!!!!!!!!! "1#

b) !!!!!!!!!!!!!!!!!!!!!!!! "2#

71. Simplify the following expressions.

(a)

aaaa 23276 ++.

(b)xyyx 3')2$(/ +

.

(c)'

2

$

2 qpqp

+

.


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Ans: a) ________________________ [2

b) ________________________ [2

c) ________________________ [!

+2. Factorise the following epressions.

(a)

)1(2)1($ xyx.

(b)xyxwyw + 2613

.

%ns a) !!!!!!!!!!!!!!!!!!!!!!!! "1#

b) !!!!!!!!!!!!!!!!!!!!!!!! "2#

+$. Solve the following equations.

(a)x

72+=

.

(b)

)1($

2)1'(/ += xx

.

(c)2

21$

'+= xx

.


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(a) !!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!! "2#

(c) !!!!!!!!!!!!!!!!!! "2#

+3. Solve forxin the following equations.

a. $x; / : 11

b.

+( /) 2( /)x x+ =

(a) !!!!!!!!!!!!!!!!!! "1#

(b) !!!!!!!!!!!!!!!!!! "$#

++. Simplify

cb

ba

b

ac

c

ab'

22

6

1'

3

'$


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!!!!!!!!!!!!!!!!!! "2#

+6. 8press6

)'($7

2

$/ +

yy

as a single fraction.

!!!!!!!!!!!!!!!!!! "2#


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+7. (a) Solve the equation

)2($1++' mm =.

(b) f

yxy

zyx

2

1

$

$2=

++

, finxwheny: / anz: 1.

(a) !!!!!!!!!!!!!!!!!!! "2#

(b) !!!!!!!!!!!!!!!!!!! "$#

6. Solve the following equations.

(a)xx /

$

/'

+=

+

(b)+12

7

=+x


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(a) !!!!!!!!!!!!!!!!!!! "2#

(b) !!!!!!!!!!!!!!!!!!! "2#

61. Factorise $y ; /w* ; 3* < 2wy.

!!!!!!!!!!!!!!!!!!!! "2#

62. >acie isxyears ol. >aen, her brother, is 11 years oler than >acie. heir mother is three times as

ol as >acie. >aen is half their fatherAs age.

a. 0rite own epressions, in terms ofxfor

i. >aenAs age,

ii. their fatherAs age.

b. he sum of the ages of the four members of the family is 1+$.

i. !"owthat the sum of the ages of the four members of the family can be written as

+x< $$ : 1+$.

ii. Solve the equation to fin the value ofx.

iii. ence fin the motherAs age when >aen was born.


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(a) i. !!!!!!!!!!!!!!!!!!! "1#

(a) ii. !!!!!!!!!!!!!!!!!!! "1#

(b) i. #or$ing% as s"own. "1#

(b) ii. !!!!!!!!!!!!!!!!!!! "1#

(b) iii. !!!!!!!!!!!!!!!!!!! "1#

83. 8press as a single fraction in its simplest form

(a) +

1$

'

++

xx

,

(b) 3

)1($

$

/$ xx

.

years old

years old


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%ns (a) "2#

(b) "2#

84. Factorise completely

(a) $$ab< 7b,

(b) 1/ab % 1/a < 'b% ' .

%ns (a) "1#

(b) "2#

85. Simplify

(a) +m; $n< 13+m ; 7n,

(b) ( ) ( )26/2'1'

'

$ xx

.


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%ns (a) "2#

(b) "2#

63. Simplify

(a)

( ) ( ) + + 2 ! " ! ! 2a b b a b,

(b)

( )+ + +

2 11 ! 1

2 ! "

xx x

.

Answer (a)========.= "2#

(b)=========. "2#87. Solve

(a) ( ) '+2+$ = x,

"2#

(b)6$

'

+2=

+x

x

,

"2#

(c)

3

'

/

2

$

)$(2=

+

xx

.

"$#


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secondary 1 algebra

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