1. Consider the functions given below.
f(x) = 2x + 3
g(x) =
x
1
, x 0
(a) (i) ind (g ! f)(x) and write down the do"ain of the function.
(ii) ind (f ! g)(x) and write down the do"ain of the function.
(2)
(b) ind the coordinates of the #oint where the gra#h of y = f(x) and the gra#h of
y = (g
$1
! f ! g)(x) intersect.
(4)
(Total 6 marks)
2. %he diagra" below shows the gra#h of the function y = f(x), defined for all x
,
where b & a & 0.
Consider the function g(x) =
b a x f ) (
1
.
(a) ind the largest #ossible do"ain of the function g.
(2)
IB Questionbank Mathematics Higher Level 3rd edition 1
(b) 'n the a(es below, s)etch the gra#h of y = g(x). 'n the gra#h, indicate an* as*"#totes
and local "a(i"a or "ini"a, and write down their e+uations and coordinates.
(6)
(Total 8 marks)
3. %he +uadratic function f(x) = p + qx $ x
2
has a "a(i"u" value of , when x = 3.
(a) ind the value of p and the value of q.
(4)
(b) %he gra#h of f(x) is translated 3 units in the #ositive direction #arallel to the x-a(is.
.eter"ine the e+uation of the new gra#h.
(2)
(Total 6 marks)
4. %he diagra" shows the gra#h of y = f(x). %he gra#h has a hori/ontal as*"#tote at y = 2.
IB Questionbank Mathematics Higher Level 3rd edition 2
(a) 0)etch the gra#h of y =
) (
1
x f
.
(3)
(b) 0)etch the gra#h of y = x f(x).
(3)
(Total 6 marks)
5. 1 function is defined b* h(x) = 2e
x
x
x
,
e
1
. ind an e(#ression for h
$1
(x).
(Total 6 marks)
6. 2iven that Ax
3
+ Bx
2
+ x + 3 is e(actl* divisible b* (x + 1)(x $ 2), find the value of A
and the value of B.
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition 3
7. 0hown below are the gra#hs of y = f(x) and y = g(x).
4f (f
g)(x) = 3, find all #ossible values of x.
(Total 4 marks)
8. 0olve the e+uation 5
x$1
= 2
x
+ 6.
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition 4
9. %he gra#h of y =
cx b
x a
+
+
is drawn below.
(a) ind the value of a, the value of b and the value of c.
(4)
IB Questionbank Mathematics Higher Level 3rd edition 5
(b) 7sing the values of a, b and c found in #art (a), s)etch the gra#h of y =
x a
cx b
+
+
on the a(es below, showing clearl* all interce#ts and as*"#totes.
(4)
(Total 8 marks)
10. (a) 8(#ress the +uadratic 3x
2
$ 3x + , in the for" a(x + b)
2
+ c, where a, b, c
.
(3)
(b) .escribe a se+uence of transfor"ations that transfor"s the gra#h of y = x
2
to the gra#h of
y = 3x
2
$ 3x + ,.
(3)
(Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition 6
11. 1 function f is defined b* f(x) =
1
3 2
x
x
, x 1.
(a) ind an e(#ression for f
$1
(x).
(3)
(b) 0olve the e+uation 9f
$1
(x)9 = 1 + f
$1
(x).
(3)
(Total 6 marks)
12. ind the set of values of x for which 9x $ 19&92x $ 19.
(Total 4 marks)
13. :et g(x) = log
,
92log
3
x9. ind the #roduct of the /eros of g.
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition
14. (a) :et a & 0. .raw the gra#h of y =
2
a
x
for $a ; x ; a on the grid below.
(2)
(b) ind k such that
x
a
x k x
a
x
a
a
d
2
d
2
0
0
=
.
(5)
(Total 7 marks)
IB Questionbank Mathematics Higher Level 3rd edition !
15. %he diagra" below shows a solid with volu"e V, obtained fro" a cube with edge a & 1 when a
s"aller cube with edge
a
1
is re"oved.
diagram not to scale
:et x =
a
a
1
.
(a) ind V in ter"s of x.
(4)
(b) <ence or otherwise, show that the onl* value of a for which V = 5x is a =
2
, 1+
.
(4)
(Total 8 marks)
16. :et f be a function defined b* f(x) = x $ arctan x, x
.
(a) ind f(1) and f(
3
).
(2)
(b) 0how that f($x) = $f(x), for x
.
(2)
IB Questionbank Mathematics Higher Level 3rd edition "
(c) 0how that x $
2
=
) (
2
=
+ < < x x f
, for x
.
(2)
(d) ind e(#ressions for f>(x) and f?(x). <ence describe the behaviour of the gra#h of f at the
origin and @ustif* *our answer.
(8)
(e) 0)etch a gra#h of f, showing clearl* the as*"#totes.
(3)
(f) Austif* that the inverse of f is defined for all x
and s)etch its gra#h.
(3)
(Total 20 marks)
17. Bhen the function q(x) = x
3
+ kx
2
$ Cx + 3 is divided b* (x + 1) the re"ainder is seven ti"es the
re"ainder that is found when the function is divided b* (x + 2).
ind the value of k.
(Total 5 marks)
18. 1 function is defined as f(x) =
x k
, with k & 0 and x D 0.
(a) 0)etch the gra#h of y = f(x).
(1)
(b) 0how that f is a one-to-one function.
(1)
(c) ind the inverse function, f
$1
(x) and state its do"ain.
(3)
(d) 4f the gra#hs of y = f(x) and y = f
$1
(x) intersect at the #oint (5, 5) find the value of k.
(2)
IB Questionbank Mathematics Higher Level 3rd edition 1#
(e) Consider the gra#hs of y = f(x) and y = f
$1
(x) using the value of k found in #art (d).
(i) ind the area enclosed b* the two gra#hs.
(ii) %he line x = c cuts the gra#hs of y = f(x) and y = f
$1
(x) at the #oints E and F
res#ectivel*. 2iven that the tangent to y = f(x) at #oint E is #arallel to the tangent to
y = f
$1
(x) at #oint F find the value of c.
(9)
(Total 16 marks)
19. Bhen 3x
,
$ ax + b is divided b* x $ 1 and x + 1 the re"ainders are e+ual. 2iven that a, b
,
find
(a) the value of a;
(4)
(b) the set of values of b.
(1)
(Total 5 marks)
20. Consider the function f, where f(x) = arcsin (ln x).
(a) ind the do"ain of f.
(3)
(b) ind f
$1
(x).
(3)
(Total 6 marks)
21. %he real root of the e+uation x
3
$ x + 5 = 0 is $1.CG3 to three deci"al #laces.
.eter"ine the real root for each of the following.
(a) (x $ 1)
3
$ (x $ 1) + 5 = 0
(2)
IB Questionbank Mathematics Higher Level 3rd edition 11
(b) 6x
3
$ 2x + 5 = 0
(3)
(Total 5 marks)
22. 1 tangent to the gra#h of y = ln x #asses through the origin.
(a) 0)etch the gra#hs of y = ln x and the tangent on the sa"e set of a(es, and hence find the
e+uation of the tangent.
(11)
(b) 7se *our s)etch to e(#lain wh* ln x ;
e
x
for x & 0.
(1)
(c) 0how that x
e
; e
x
for x & 0.
(3)
(d) .eter"ine which is larger, =
e
or e
=
.
(2)
(Total 17 marks)
23. %he functions f and g are defined asH
f (x) =
, e
2
x
x 0
g (x) =
. 3 ,
3
1
+
x
x
(a) ind h (x) where h (x) = g I f (x).
(2)
(b) 0tate the do"ain of h
J1
(x).
(2)
IB Questionbank Mathematics Higher Level 3rd edition 12
(c) ind h
J1
(x).
(4)
(Total 8 marks)
24. %he #ol*no"ial P(x) = x
3
+ ax
2
+ bx + 2 is divisible b* (x +1) and b* (x J 2).
ind the value of a and of b, where a, b .
(Total 6 marks)
25. :et f (x) =
2 ,
2
5
+
x
x
and g (x) = x J 1.
4f h = g I f, find
(a) h (x)K
(2)
(b) h
J1
(x), where h
J1
is the inverse of h.
(4)
(Total 6 marks)
26. Bhen f(x) = x
5
+ 3x
3
+ px
2
$ 2x + q is divided b* (x $ 2) the re"ainder is 1,,
and (x + 3) is a factor of f(x).
ind the values of p and q.
(Total 6 marks)
27. ind all values of x that satisf* the ine+ualit*
1
1
2
<
x
x
.
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition 13
28. %he #ol*no"ial f(x) = x
3
+ 3x
2
+ ax + b leaves the sa"e re"ainder when divided b* (x 2) as
when divided b* (x +1). ind the value of a.
(Total 6 marks)
29. %he functions f and g are defined b* f H x
e
x
, g H x
x + 2.
Calculate
(a) f
$1
(3) L g
$1
(3)K
(3)
(b) (f ! g)
$1
(3).
(3)
(Total 6 marks)
30. (a) 0how that p = 2 is a solution to the e+uation p
3
+ p
2
$ ,p 2 = 0.
(2)
(b) ind the values of a and b such that p
3
+ p
2
$ ,p 2 = (p 2)(p
2
+ ap + b).
(4)
(c) <ence find the other two roots to the e+uation p
3
+ p
2
,p 2 = 0.
(3)
(d) 1n arith"etic se+uence has p as its co""on difference. 1lso, a geo"etric se+uence has p
as its co""on ratio. Moth se+uences have 1 as their first ter".
(i) Brite down, in ter"s of p, the first four ter"s of each se+uence.
(ii) 4f the su" of the third and fourth ter"s of the arith"etic se+uence is e+ual to the
su" of the third and fourth ter"s of the geo"etric se+uence, find the three #ossible
values of p.
(iii) or which value of p found in (d)(ii) does the su" to infinit* of the ter"s of the
geo"etric se+uence e(istN
IB Questionbank Mathematics Higher Level 3rd edition 14
(iv) or the sa"e value p, find the su" of the first 20 ter"s of the arith"etic se+uence,
writing *our answer in the for" a +
c b
, where a, b, c
.
(13)
(Total 22 marks)
IB Questionbank Mathematics Higher Level 3rd edition 15
