1.
The graph below shows y = a cos (bx) + c.
Find the value of a, the value of b and the value of c.
(Total 4 marks)
2.
a=3
c=2
2
=3
b
(M1)
2
(= 2.09)
3
A1
period =
b=
A1
A1
[4]
IB Questionbank Mathematics Higher Level 3rd edition
1
3.
The diagram below s
1.
Consider the functions given below.
f(x) = 2x + 3
1
x
g(x) = , x 0
(a)
(i)
Find (g f)(x) and write down the domain of the function.
(ii)
Find (f g)(x) and write down the domain of the function.
(2)
(b)
Find the coordinates of the point where the graph
1.
Three Mathematics books, five English books, four Science books and a dictionary are to be placed on a students
shelf so that the books of each subject remain together.
(a)
In how many different ways can the books be arranged?
(4)
(b)
In how many of th
1.
Consider the matrix A =
(a)
cos 2
sin 2
sin
cos
, for 0 < < 2.
Show that det A = cos .
(3)
(b)
2
Find the values of for which det A = sin .
(3)
(Total 6 marks)
2.
(a)
det A = cos 2 cos + sin 2 sin
= cos (2 )
M1A1
A1
Note: Allow use of double angle
1.
sin 2
Show that 1 cos 2 = tan .
(a)
(2)
(b)
Hence find the value of cot 8 in the form a + b 2 , where a, b
.
(3)
(Total 5 marks)
2.
sin 2
2 sin cos
1 cos 2 1 2 cos 2 1
(a)
M1
Note: Award M1 for use of double angle formulae.
2 sin cos
2 cos 2
sin
=
1.
(a)
x
Find the first three terms of the Maclaurin series for ln (1 + e ).
(6)
lim
(b)
Hence, or otherwise, determine the value of
2 ln(1 e x ) x ln 4
x2
x 0
.
(4)
(Total 10 marks)
2.
dy
x 2 y 2
dx
Consider the differential equation
where y = 1 when x =
1.
th
The sum, Sn, of the first n terms of a geometric sequence, whose n term is un, is given by
7n an
7n
Sn =
(a)
, where a > 0.
Find an expression for un.
(2)
(b)
Find the first term and common ratio of the sequence.
(4)
(c)
Consider the sum to infinity
1.
sin x 2 sin x sin
3
3 , show that 11 tan x = a + b 3 ,
If x satisfies the equation
+
where a, b
.
(Total 6 marks)
2.
sin x sin x cos cos x sin
3
3
3
sin x cos cos x sin 2 sin x sin
3
3
3
1
3
3
sin x cos x 2 sin x
2
2
2
dividing by cos x and
1.
The diagram below shows the graph of the function y = f(x), defined for all x
where b > a > 0.
,
1
Consider the function g(x) = f ( x a ) b .
(a)
Find the largest possible domain of the function g.
(2)
(b)
On the axes below, sketch the graph of y = g(
1.
(a)
Given that A =
cos
sin
sin
cos
2
, show that A =
cos 2
sin 2
sin 2
cos 2
.
(3)
(b)
Prove by induction that
cos n
n
A = sin n
sin n
cos n
, for all n
+
.
(7)
(c)
1
Given that A is the inverse of matrix A, show that the result in part (
Chapter 7: Review Test
1.
3
The function f(x) = 4x + 2ax 7a, a
by (x a).
(a)
leaves a remainder of 10 when divided
Find the value of a.
(3)
(b)
Show that for this value of a there is a unique real solution to the equation f(x) = 0.
(2)
(Total 5 marks)
3
1.
e
Using integration by parts, show that
(a)
x
0
cos xdx
e
0
x
sin xdx
.
(5)
(b)
Find the value of these two integrals.
(6)
(Total 11 marks)
e
x
cos xdx e x sin x
sin x ddx (e
0
x
)dx
(a)
x
x
since e 0 as x and sin x is bounded e sin x 0 as x
(or a
1.
4
3
2
Consider the polynomial p(x) = x + ax + bx + cx + d, where a, b, c, d
.
Given that 1 + i and 1 2i are zeros of p(x), find the values of a, b, c and d.
(Total 7 marks)
2.
(a)
3
Solve the equation z = 2 + 2i, giving your answers in modulusargument
1.
METHOD 1
1 + i is a zero 1 i is a zero
1 2i is a zero 1 + 2i is a zero
2
(x (1 i)(x (1 + i) = (x 2x + 2)
2
(x (1 2i)(x (1 + 2i) = (x 2x + 5)
2
2
p(x) = (x 2x + 2) (x 2x + 5)
4
3
2
= x 4x + 11x 14x + 10
a = 4, b = 11, c = 14, d = 10
(A1)
(A1)
(M1)A1
A1
1.
Show that the points (0, 0) and ( 2 ,
common tangent.
2 ) on the curve e(x + y) = cos (xy) have a
(Total 7 marks)
2.
(a)
2
Differentiate f(x) = arcsin x + 2 1 x , x [1, 1].
(3)
(b)
Find the coordinates of the point on the graph of y = f(x) in [1, 1],
1.
Attempt at implicit differentiation
dy
dy
y
1 sin( xy ) x
(x+y)
dx
dx
e
M1
A1A1
M1
let x = 0, y = 0
dy
1
0
dx = 0
e
dy
dx = 1
A1
2 , y 2
let x =
dy
dy
y
1 sin( 2 ) x
0
dx
=0
e dx
dy
so dx = 1
A1
since both points lie on the line y = x
1.
(a)
x
Find the first three terms of the Maclaurin series for ln (1 + e ).
(6)
x
lim
2 ln(1 e ) x ln 4
x2
x 0
(b)
Hence, or otherwise, determine the value of
.
(4)
(Total 10 marks)
dy
x 2 y 2
dx
2.
Consider the differential equation
where y = 1 when x =
1.
ln x
2
Consider the function f(x) = x , 0 < x < e .
(a)
(i)
Solve the equation f(x) = 0.
(ii)
Hence show the graph of f has a local maximum.
(iii)
Write down the range of the function f.
(5)
(b)
Show that there is a point of inflexion on the graph and
x
1.
(a)
(i)
f(x) =
1 ln x
1
ln x
x
x2
M1A1
2
= x
so f(x) = 0 when ln x = 1, i.e. x = e
(ii)
A1
f(x) > 0 when x < e and f(x) < 0 when x > e
hence local maximum
R1
AG
Note: Accept argument using correct second derivative.
1
y e
(iii)
A1
1
(1 ln x)2 x
x
x
1.
The graph below shows y = a cos (bx) + c.
Find the value of a, the value of b and the value of c.
(Total 4 marks)
2.
a=3
c=2
2
period = b = 3
2
b = 3 (= 2.09)
A1
A1
(M1)
A1
[4]
IB Questionbank Mathematics Higher Level 3rd edition
1
3.
The diagram below