past year functions stpm math m

7/24/2019 Past Year Functions STPM Math M

1/6

PAST YEAR QUESTION :CHAPTER 1 : FUNCTIONS

1. Functions f, g and h are defined by

.2

3x:h,x

2xx:g,

1:

xx

xxf +

+

+

a) state the domains of f and g. [2 marks]b) find the composite function gof and state its domain and range. [ marks]c) state the domain and range of h. [2 marks]d) state !hether h " g o f. #i$e a reason for your ans!er. [ 2%%&, 2 marks]

Ans : a) Df={ x : xR, x -1}, Dg={ x : xR, x 0},b) 3+2/x Dgof={ x : xR, x -1, x 0}, Rgof={ y : yR, y 3 },c) Dh={ x : xR, x 0}, Rh={ y : yR, y 3},

d)o!h f"x) and gof"x) ha#$ !h$ sa%$ &'($ b'! dff$&$n! do%ans and &ang$s, !h$&$fo&$ h gof

2. 'he functions f and g are defined by

f : xRx

x ,

1

( %*+

g : x .x,12 x

Find f g and its domain. [ 2%%-, marks]

Ans :1

2x1,D fog={x :xR , x

1

2}

3. Functions f and g are defined by

+

2

1x

12

:

fo&

x

xxf

,:2 cbxaxxg ++ !here a, b, and c are constants.

a) Find ff , and hence, determine the in$erse function of f. [ marks]

b) Find the $a/ues of a, b and c if2

2

)120

13

+=

x

xxfg

[ marks]

c) #i$en that p0x) " x 2 2, express h0x) " 2

22

2

x

x

in terms of f and p.[2%11,2 marks]

Ans : a) x, f -1:x x2x1

, xR , x1

2 b) a=1* =0, c=-1 c) h=f

. 'he graph of a function f is fo//o!s :

1


2/6

a) state the domain and range of f. [ 2marks] b) state !hether f is onetoone function or not. #i$e a reason for your ans!er.

[ 2%1%, 2marks] Ans : a) Do%an = { x: -3 x -1, -1 x 2, xR } Rang$ ={ y : -1 y 2, y R } b) .o! on$-!o-on$ f'nc!on* $ca's$ by 'sng !h$ ho&on!a( (n$ !$s!, !h$&$ a&$

%o&$ !han on$ n! of n!$&s$c!on b$!$$n !h$ ho&on!a( (n$ and !h$ g&ah of f*

. ketch on the same coordinate axes, the graphs of y " 2 x and y ".

x

12 +

[ marks]

4ence , so/$e the ine5ua/ity 2 x 6.

x

12 +

[2%%, marks] Ans : x -0*23

&. Find the so/ution set of the ine5ua/ity2x

7 x1

, !here x .% [2%%, 8marks]

Ans : { x : 0 x 1, 1 x 1 + 2 }

8. 9xpress )2)010

122 xx

x

+

+

in the form,

212 x

2

x

Ax

+

+

+

!here , ; and < are constants. [2%%&, 3marks]

Ans :

x

x2+1+

1

2x

-. 'he po/ynomia/ p0x)" x = a x 38 x 2ax = b has a factor x = 3 and, !hen di$idedby x3, has remainder &%. Find the $a/ues of a and b, and factorise p 0x) comp/ete/y.

[>marks]

?sing the substitution y ",

1

x so/$e the e5uation 12 y - y 3 8 y 2= 2 y = 1 " % [2%%&, 3

marks]

Ans : a=2, b=12, "x-1)"x+3)"x-2)"x+2) , y= 1, -1/3, , -1/2

2


3/6

>. Find the constants , ; , < and @ such that

3222

2

)10)1011)1)010

3

x

D

x

2

x

x

A

xx

xx

++

++

++

=

+

+

[ 2%%8, -

marks]Ans : A =1, =1, c = -1, D =-1

1%. 'he po/ynomia/ p0x ) " 2 x 3= x 2= A x k has factor 0 x = 1 ).a) find the $a/ues of k. [ 2marks]

b) factorise p0x) comp/ete/y. [ 2%%-, marks]

Ans : a) 3/2 b) "x+1)"2x+3)"x - 1

2)

11. Find the so/ution set of the ine5ua/ity

xx

33

1

>

[2%%-, 1%marks]

Ans : { x : 0 x 3, x 1 }

12. Find a// the $a/ues of x if y "x3

and y 0 x 2> ) " 2. [ 2%%>, > marks]

Ans : x = -4 o& 5

13. . @etermine the set of $a/ues of x satisfying the ine5ua/ity

1

1

1 +

+ xx

x

[ 2%%>, marks] Ans : { x : x -1 o& x 1 }

1. 'he po/ynomia/ p0x) " &x a x 3b x 2= 2-x = 12, !here a and b are rea/ constants,has factors 0 x = 2 ) and 0 x 2).

Find the $a/ues of a and b, and hence, factorise p0x) comp/ete/y. [ 2%%>, 8

marks] Ans : a = 5, b = 25, 6"x) = "x+2)"x-2)"2x-3)"3x+1)

1. 'he po/ynomia/ p0x)"2x 8x 3=x2=ax = b, !here a and b are rea/ constants, isdi$isib/e by 2x 2= x 1.

a) find a and b. [ marks] b) for these $a/ues of a and b, determine the set of $a/ues of x such that p0x) %

[ 2%1%, marks]

Ans : a) a = 4, b = -7 b) { x : -1 x 1

2, xR }

3


4/6

1&. Find the set of $a/ues of x satisfying the ine5ua/ity 2x 11+ x

. [ 2%11, & marks]Ans : { x 8 x 2 }

18. 'he po/ynomia/ p0x) " ax3=bx2x=3, !here a and b are constants, has a factor 0x=1).

Bhen p0x) is di$ided by 0x2), it /ea$es a reminder of >.a) find the $a/ues of a and b, and hence, factorise p0x) comp/ete/y. [ & marks]

b) find the set of $a/ues of x !hich satisfies%

3

)0

x

x+

[ marks]

c) by comp/eting the s5uare, find the minimum $a/ue of3,x,

3

)0

x

x+

and the $a/ueof x at !hich it occurs. [ 2%11, marks]

Ans : a) a= 2, b=-7, b) { x : x 1,x1

2, x 3} c) %n #a('$ 0s -4/9 h$n x = -1/

1-. Cf),20/og2/og3)0/og

2 ax

ax aaa =

express x in terms of a. [ 2%%8, & marks] Ans : x = a

1>. #i$en that,

2

1a%!here),210/og

/og

23/og)30/og

2

3 ==y

xxy

[ 2%1%, marks]Ans : x = 25, y = 421. o/$e the e5uation /nx = /n 0x=2) " 1 [ 2%11, marks]

Ans : x = 1+e1

22. ketch, on the same coordinate axes, the cur$es y " & e xand y " e x, and find thecoordinates of the points of intersection. [ 2%%-, 8 marks]

Ans : " (n7, 1 )

23. #i$en that 2 x x 2is a factor of D0x) " ax3x2=bx 2. Find the $a/ues of a and b.4ence, find the set of $a/ues of x for !hich D0x) is negati$e. [ 2%12, & marks]

Ans : a = -2, b=7, { x : -2 x 1

2, x>1 }

2. Functions f and gof are defined by f0x)" e x=2 and 0gof)0x) " Ex. For a// x %.a) find the function g, and state its domain, [ marks]

b) determine the $a/ue of 0fog)0e3) [ 2%12, 2marks]

Ans : a) g"x) = lnx2 Dg={ x : x e2 } b) $ 3


5/6

2. o/$e the simu/taneous e5uationsx

y

log9 )"

3

4and 0 log3x (lo g3y )=1

[2%12, -marks]

Ans : x= 13

, y=19

, x=9,y=3

2&. 'he function f is defined by f : x2x, for x 1

2.

a) find f 1, state its domain, [ marks]

b) find the coordinates of the point of intersection of graphs f and f 1. [ 3 marks]c) sketch, on the same coordinates axes, the graphs of f and f 1. [ 2%12, 3 marks]

Ans : a) f1 (x )=

1

2+

x+

1

4

, Df1={x :x1

4

} b) "2,2)

28. 'he function f is defined by f0x) "1

2(exex) , where xR .

a) sho! that f has an in$erse. [ 3 marks]b) find the in$erse function of f, and state its domain. [ 2%13, 8 marks]

Ans : b)x+x2+1

f1 (x )=ln

Df1={x ; xR }

2-. @etermine the set of $a/ues of x satisfying the ine5ua/ity x+4 3

x .

[ 'DG 2%1, & marks]Ans : { x : x -2- 7 o& 0 x 2+7 }

2>. Functions f and g are defined byf0x) "x2+ x= 2,x ,

g0x) " 3

3+x, x 3,xR .

a) ketch the graph of f, and find its range. [marks]

b) ketch the graph of g, and sho! that g is a onetoone function. [3 marks]

c) #i$e a reason !hy g1

exists. Find g1

and state its domain. [marks]d) #i$e a reason !hy gof exists. Find gHf, and state its domain.['DG 2%1, marks]

Ans : a) Rf = { y : y 2 }

b) g s on$-!o-on$ b$ca's$ !h$ (n$ y=;, ; 0 c'!s !h$ g&ah a! on$ on! on(y*

c) g-1 $xs!s b$ca's$ g s on$-!o-on$ f'nc!on* g-1"x)= 3

x3,D

g1={x :x 0}

d) RfD g , Dgofis R


6/6

3%. continuous function f is defined byx31, 1 x

past year functions stpm math m

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