I.G.C.S.E. Functions & Vectors Index: Please click on the question number you want
Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 You can access the solutions from the end of each question
Question 1 The following functions Æ , g and h are defined as follows:
: 4 1x xâ âÆ , 2 2: 3
xg x ââ , 4:h x x
â .
Find a. ( )7Æ , ( )7âÆ , ( )17Æ . b. g(4), g(-3), ( )12g . c. h(10), h(-4), ( )18h â Click here to read the solution to this question Click here to return to the index
Solution to question 1 a. : 4 1x xâ âÆ ( ) ( )4 1 28 1 27 77 = â = â =Æ ( ) ( ) 27 74 2 1 91 8= â = â ââ = ââÆ ( ) ( )1 17 7 37474 1 1= â = = ââÆ
b. 2 2: 3
xg x ââ
( ) 2 2 16 2 14( 24
33 4)
4 3 3
g â
â
= = = =
( )2 2 9 2 7( ) 3 3 3
3 1 3
3 2g â
â
= = = =
â
â
( )12g . ( ) 2 1 7
41 4 1 2
2
2 2( ) 3 3
7 3 12
g â
â â
= â= = =
c. 4:h x x
â
( ) 4 210)
5 (
10 h = =
( ) 4( 14) 4
h â
= = ââ
( )18h â ( ) 1 8 1
8
4( ) 32h â â
= = â
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Question 2 For the following functions : 2 3x xâ âÆ and 2: 2g x x xâ â , find a. the value of x such that ( ) 3x = âÆ b. the values of x such that ( ) 3g x = Click here to read the solution to this question Click here to return to the index
Solution to question 2 a. : 2 3x xâ âÆ
( ) 3x = âÆ â2 3 3 3
5 5
3
x x
x
â = â
â = â
=
b. 2: 2g x x xâ â ( ) 3g x = â 2
2
2 3 2 3 0
x x x x
â =
â â =
product = -3 sum = -2 factors -3, 1
( ) ( ) ( )( )
2 3 0 3 3
3 0
3 1 0
x x x x
x
x x
x
â =
â + â =
â +
â
=
+
â 3x = or 1x = â Click here to read the question again Click here to return to the index
Question 3 Given that the function Æ is defined as : x ax bâ +Æ , where a and b are constants. If ( )2 3=Æ and ( )4 9â =Æ . Click here to read the solution to this question Click here to return to the index
Solution to question 3
: x ax bâ +Æ ( ) 32 =Æ â 12 .3a b+ = â¦
( )4 9â =Æ â 24 .9a b+ =â â¦
Solving equations 1 and 2 simultaneously we have 1 2â 1.
2. 2 3 4 9 6 6
a b a b a
+ =
â + =
= â
⦠â¦
â 1a = â Substitute 1a = â into equation 1 we have
( )2 3 2
5
1 3
b b b
+ =
â + =
=
â
Check in equation 2 we have ( ) ( )4 5 91 45â + = + =â Hence 1a = â and 5b = . Click here to read the question again Click here to return to the index
Question 4
Given the functions ( ) 3 2l x x= â , ( ) 22 4m x x= + and ( ) 4n x x
= find
a. ( )lm x b. ( )ml x c. ( )lmn x d. ( )2lmn â . Click here to read the solution to this question Click here to return to the index
Solution to question 4
( ) 3 2l x x= â , ( ) 22 4m x x= + and ( ) 4n x x
=
a. ( ) ( ) ( ) ( )2 2 22 22 4 2 4 2 33 2 6 1 2 1 52 6 0lm x l xx x x x= = â = + â = + =+ ++ b. ( ) ( ) ( ) ( )
( ) 2
2
2
2 2
2 4 2 9 12 4 4
18
3 2
6 3 424 8
3 2
24 18 24 12 x
xml x m x x
x xx x x
x= = + = â + +
= â + â+ = â + =
â
+
â
c. ( ) ( )
2 32 324 44 4 962 4 3 2 12 2
96 96 101 2
0 44 5
x lmn x lm
x
l l
x x x x
xx x x         
= = + = = â = + â         ï£ ï£¸ ï£ ï£¸ ï£ ï£¸
+ + ï£ ï£¸ï£¯  
+ +
+ = = =
d. ( ) ( )( ) 2 24
2 2
3 5
4 4
lmn â + 
â
 = â=â
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Question 5
Given the functions : 7 3x xâ âÆ and 3 5: 4
xg x ââ find the inverse
functions ( )-1 xÆ and ( )1g xâ . Click here to read the solution to this question Click here to return to the index
Solution to question 5
: 7 3x xâ âÆ Let 7 3
3 7 3
7
y x y x y x
= â
+ =
+ =
Interchange the x and y to give ( )-1 3 7
x x= +Æ
3 5:
4 xg x ââ
Let 3 5 4
4 3 5 4 5 3 4 5
3
xy
y x y x y x
â
=
= â
+ =
+ =
Interchange the x and y to give ( )1 4 5 3
g x xâ = +
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Question 6 OABC is a parallelogram where OA a=
"""# # and OC c=
"""# # .
Find in terms of a
# and c
# only
a. AB
"""# b. BC
"""# c. CA
"""# d. BO
"""# .
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a #
c #
C B
AO
Solution to question 6 OABC is a parallelogram where OA a=
"""# # and OC c=
"""# # .
a. AB OC c= =
"""# """# #
b. BC OA aâ= â =
"""# """# #
c. CA CO OA c a a c= + â= â + =
"""# """# " # # #"" ##
d. BO BC CO a c= + â= â
"""# """# "" #" ##
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a #
c #
C B
AO
Question 7 OPQR is a trapezium where 3RQ OP= and M is the point on PQ such that
2PM MQ= .
Given that OP p= """# #
and OR r= """# #
find in terms of p #
and r #
only
a. PR """#
b. RQ """#
c. PQ """#
d. PM """"#
e. MR """"#
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M
p #
r #
R Q
PO
Solution to question 7 OPQR is a trapezium where 3RQ OP= and M is the point on PQ such that
2PM MQ= .
OP p= """# #
and OR r= """# #
a. PR PO OR p r r p= + â= â + =
"""# """# " # # #"" ##
b. 3 3RQ OP p= =
"""# """ ##
c. 23PQ PO OR RQ p p rr p= + + = â + + = +
"""# """# """# """# # # # ##
d. ( )1 1 2 3 3
2 1 3 3
pP PQ r rM p= = + = + """"# """# # # ##
e. ( )2 2 4 22 3 23 3 3 3 3
5 3 3
MR MQ QR PQ QR p r p rp p pr= + = + = + â + â= â = """"# """"# """# """# """# # # # # # # ##
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M
p #
r #
R Q
PO
Question 8
Given the following vectors 2 3
a â 
=   ï£ ï£¸
# and
4 5
b  =   âï£ ï£¸
# . Find the following giving
your answers exactly, wherever appropriate. a. a b+
## b. a bâ
## c. 3 2a bâ
## d. b
# e. 3 2a bâ
##
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Solution to question 8
2 3
a â 
=   ï£ ï£¸
# and
4 5
b  =   âï£ ï£¸
# .
a. 24
5 2 2 3
a b â   
+ = + =    â
    â ï£¸ï£ ï£ ï£ï£¸ 
##
b. 64
5 8 2 3
a b â   
â = â =    âï£ ï£¸ ï£
â
 
   ï£
##
c. 2 4 6 8
3 2 3 2 3 5 9 10
14 19
a b â â       
â = â = â =        â âï£ ï£¸ ï£ ï£¸ ï£ ï£¸ ï£
â    ï£ ï£¸ï£¸
##
d. ( )224 5 4116 25b = + â = + =# e. ( )2 23 2 14 19 196 3 576 51a bâ = â + = + =## Click here to read the question again Click here to return to the index