Millburn Academy
Higher Maths
Homework Booklet
Higher Homework Exercise 1

1.
2.
Given that f(x) = x² + 2x – 8, express f(x) in the
form (x + a)² - b.
Hence state the minimum value of f(x).
Find the size of the angle aº
that the line joining the
points A(0,-1) and B(4,2)
makes with the positive
direction of the x-axis.
(3)
• B(4,2)
aº
• A(0,-1)
(3)

3.
3x² + 6x – 10 is expressed in the form 3(x + p)² + q.
What is the value of q?
A
B
C
D
-9
-10
-13
-16
(2)
Total
(8)
Higher Homework Exercise 2



1.
Find the equation of the straight line which is parallel
to the line with equation 2x + 3y = 5 and which passes
through the point (2,-1).
(3)
2.
a) Write f(x) = x² + 6x + 11 in the form (x + a)² + b.
b) Hence or otherwise sketch the graph of y = f(x).
(2)
(2)
3.
The line 2y = 3x + 6 meets the y-axis at C. The gradient
of the line joining C to A(4,-3) is:
2
A
-3
B
3
2
C
-2
D
2
3
3
(2)
Total
(9)
Higher Homework Exercise 3




1.
2.
Find the equation of the altitude through A in the
triangle with vertices A(7,8), B(6,2) and C(-5,2).
(3)
ABCD is a parallelogram. A, B and C have coordinates
(2,3), (4,7) and (8,11). Find the equation of DC.
(3)
Show that the function f(x) = 2x² + 8x – 3 can be
written in the form f(x) = a(x + b)² + c where a, b and c
are constants.
(3)
3b. Hence, or otherwise, find the coordinates of the
turning point of the function f.
(1)
3a.
4.
Given that f(x) = 2x and g(x) = 4x + 1, then
f(g(x)) equals
A
B
C
D
6x + 1
8x + 1
8x + 2
8x² + 1
(2)
Total
(12)
Higher Homework Exercise 4


1.
(3)
Express f(x) = x² - 4x + 5 in the form f(x) = (x – a) ² + b .
(2)
2b. On the same diagram sketch:
i) the graph of y = f(x)
ii) the graph of y = 10 – f(x)
(4)
2a.
2c.


Find the equation of the line which is parallel to the
line with equation 2x + 3y = 5 and passes through the
point (2,-1).
3.
Find the range of values for which y = 10 – f(x)
is positive.
If f ( x) 
1
x 2
2
and
g ( x)  x  3 . Find
f(g(x)) in its simplest form.
4.
(1)
(3)
The equation of the straight line through the points
(1,-2) and (-3,4) is:
A
B
C
D
3x + 2y = -1
3x – 2y = 7
2x + 3y = -4
2x – 3y = 8
(2)
Total
(15)



Higher Homework Exercise 5
1.
2.
(2)
Given that f ( x)  x  22 , find f ' (4)
(5)
On the same axes sketch the graphs of:
a) y  log 5 x
(2)
b) y  log5 ( x  2)
Mark clearly 2 points on each graph.
(2)
x
3.
4.

Sketch the graph of y  3x . Mark clearly 2 points
on the graph.
5.
a) Find the equation of the straight line between the
points A(-1,5) and B(3,1).
b) Find the size of the angle which AB makes with
the positive direction of the x-axis.
(2)
(2)
3 ,f’(x) equals:
x2
f ( x) 
A
3
2x
B

3
x
C

6
x
D

6
x3
(2)
Total
(17)
Higher Homework Exercise 6



1.
2.
Find the equation of the tangent to the curve
y = x³ - 9x² + 20x – 8 at x = 1.
(4)
The point A has coordinates (7,4). The straight lines with
equations x + 3y + 1 = 0 and 2x + 5y = 0 intersect at B.
a) Find the gradient of AB.
(3)
b) Hence show that AB is perpendicular to only one of
these two lines.
(5)
3.
Given that f(x) = 2√x , then f ’(4) equals
A
B
C
-2
½
4
D
¼
(2)
Total
(14)
Higher Homework Exercise 7



1.
The point P(-1,7) lies on the curve with equation y = 5x² + 2.
Find the equation of the tangent to the curve at P.
(3)
2.
Two functions f and g are defined by f(x) = 2x + 3 and
g(x) = 2x – 3, where x is a real number.
a) Find expressions for i) f(g(x))
ii) g(f(x))
b) Determine the least possible value of the product
f(g(x)) × g(f(x))
3.
Part of the graph of y = f(x) is shown in the diagram.
On separate diagrams sketch the graphs of:
i) y = f(x + 1)
ii) y = -2f(x)
Indicate on each graph the images of O, A, B, C and D.
(3)
(2)
(5)
C(6,4)
B
O

D
A(1,-2)
4.
If f(x) = 4x³ + 5, then f ’(2) equals:
A
B
C
D
21
26
48
53
(2)
Total
(15)
Higher Homework Exercise 8

1.
2.
Find the equation of the tangent to the curve with equation
y = 5x³ - 6x² at the point where x = 1 .
(4)
A rectangular beam is to be cut from a cylindrical log of
diameter 20cm.
The diagram shows a cross section of the log and beam where
the beam has a breadth of w cm and a depth of d cm.
d cm
w cm


3.
4.
The strength S of the beam is given by S = 1.7w(400 - w²).
Find the dimensions of the beam for maximum strength.
(5)
Find the x coordinate of each point on the curve
y = 2x³ - 3x²- 12x + 20 at which the tangent is parallel to
the x-axis.
(4)
The tangent to the curve y = x³ - 1 at the point (1,0) is:
A
B
C
D
y = 3x²(x – 1)
y = 3(x – 1)
y = ¼(x – 1)
y=x–1
(2)
Total
(15)
Higher Homework Exercise 9

1.
f (x) = x3 – x2 – 5x – 3.
a) Show that (x + 1) is a factor of f (x) .
b) Hence or otherwise factorise f (x) fully.
(5)
2.
Find the value of k such that the equation
kx2 + kx + 6 = 0, k  0, has equal roots.
(4)

3.
Here are two statements about the roots of the
equation x2 + x + 1 = 0:
(1) the roots are equal;
(2) the roots are real.
Which of the following is true?
A
B
C
D
Neither statement is correct.
Only statement (1) is correct.
Only statement (2) is correct.
Both statements are correct.
(2)
Total (11)
Higher Homework Exercise 10

1. A function f is defined on the set of real numbers
by f(x) = x3 – 3x + 2.
(a) Find the coordinates of the stationary points on
the curve y = f(x) and determine their nature.
(6)
(b) (i) Show that (x – 1) is a factor of x3 – 3x + 2.
(ii) Hence or otherwise factorise x3 – 3x + 2 fully.
(5)
(c) State the coordinates of the points where the curve
with equation y = f(x) meets both the axes and hence
sketch the curve.
(4)

2. The maximum value of the function f(x) = x2 – 4x + 3
in the interval – 1  x  4 is
A
B
C
D
0
3
-1
8
(2)
Total (17)
Higher Homework Exercise 11
1.
(a) Show that x = -1 is a solution of the cubic equation
x3 + px2 + px + 1 = 0.
(1)
(b) Hence find the range of values of p for which all
the roots of the cubic equation are real.
(7)
2.
The curve y = f(x) is such that
dy
dx
= 4x – 6x2. The curve
passes through the point (-1 , 9). Express y in terms of
x.
(4)



3.
What is the solution of the equation 2sinx - √3 = 0
where π ≤ x ≤ π ?
2
A
π
6
B
2π
3
C
3π
4
D
5π
6
(2)
Total (14)
Higher Homework Exercise 12

1.
2.
The diagram shows a sketch of the graph of
y = x3 – 4x2 + x + 6.
(a) Show that the graph cuts the x-axis at (3,0).
(b) Hence or otherwise find the coordinates of A.
(c) Find the shaded area.
(1)
(3)
(5)
The circles with equations (x - 3)2 + (y – 4)2 = 25 and
x2 + y2 – kx – 8y – 2k = 0 have the same centre.
Determine the radius of the larger circle.
(5)

3.
The x-axis is a tangent to a circle with centre (-7,6) as
shown.
What is the equation of the circle?
A (x + 7)2 + (y – 6)2 = 1
B (x + 7)2 + (y – 6)2 = 49
C (x - 7)2 + (y + 6)2 = 36
D (x + 7)2 + (y – 6)2 = 36
(2)
Total (16)
Higher Homework Exercise 13
1a) Write down the centre and calculate the radius of the circle
with equation x2 + y2 + 8x + 4y – 38 = 0.
(2)
2
2
b) A second circle has equation (x – 4) + (y – 6) = 26.
Find the distance between the centres of these two circles
and hence show that the circles intersect.
(4)
c) The line with equation y = 4 – x is a common chord
passing through the points of intersection of the two circles.
Find the coordinates of the points of intersection of the two
circles.
(5)






2) For what range of values of k does the equation
x2 + y2 + 4kx – 2ky – k – 2 = 0 represent a circle?
(5)
3) The diagram shows a circle, centre (2,5) and a tangent
drawn at the point (7,9). What is the equation of this
tangent?
A y–9=-
5
4
(x – 7)
B y+9=-
4
5
(x + 7)
C y–7=
D y+9=
4
5
(x – 9)
5
4
(x + 7)
(2)
Total (18)
Higher Homework Exercise 14
1)
The diagram shows the graphs of
a cubic function y = f ( x ) and its
derived function y = f ‘ (x).
Both graphs pass through the point (0,6).
The graph of y = f ‘ (x) also passes through
the points (2,0) and (4,0).
(a) Given that f ‘ (x) is of the form k (x – a)(x – b):
(i) write down the values of a and b;
(ii) find the value of k.
(3)
(b) Find the equation of the graph of the cubic function
y = f ( x ).
(4)









2) Solve the equation sin 2xo = 6 cos xo for 0
3)
4)

x

360.
(4)
Show that the line with equation y = 2x + 1 does not
intersect the parabola with equation y = x2+ 3x +4.
(5)
A sequence is generated by the recurrence relation
u n+1 = 0.4 u n – 240.
What is the limit of this sequence as n   ?
A -800
B - 400
C 200
D 400
(2)
Total (18)
Higher Homework Exercise 15
1.
2.
Solve the equation cos2xo + 2sin xo = sin2xo in the
interval 0  x < 360.
(5)
The parabola shown in the diagram
has equation y = 32 – 2x2.
The shaded area lies between the lines
y = 14 and y = 24.
Calculate the shaded area.
(8)



3.
The diagram shows part of the graph of a quadratic
function y = f (x).
The graph has an equation of the form y = k(x – a)(x – b).
What is the equation of the graph?
A y = 3(x – 1)(x – 4)
B y = 3(x + 1)(x + 4)
C y = 12(x – 1)(x – 4)
D y = 12(x + 1)(x + 4)
(2)
Total (15)
Higher Homework Exercise 16
1.
(a) Find the equation of AB, the perpendicular bisector of
the line joining the points P(-3,1) and Q(1,9).
(4)
(b) C is the centre of a circle
passing through P and Q.
Given that QC is parallel to
the y-axis, determine the
A
equation of the circle.
(c )The tangents at P and Q
intersect at T.
Write down
(i) the equation of the
tangent at Q.
(ii) the coordinates of T.
2.
y
Q(1,9)
•
(3)
•C
•
P(-3,1)
B
O
x
(2)
Calculate the shaded area enclosed between the parabolas
with equations y = 1 + 10x - 2x2 and
y = 1 + 5x - x2.
y
y = 1 + 10x – 2x²
y = 1 + 5x – x²
O
x
(6)

3.
The diagram shows part of the graph of a function with
equation y = f (x).
Which of the following diagrams shows the graph with
equation y = - f(x – 2)?
(2)
Total (17)
Higher Homework Exercise 17
1.
The point Q divides the line joining P(-1,-1,0) to R(5,2,-3)
in the ratio 2:1. Find the coordinates of Q
(3)
2.
The diagram shown shows a square based pyramid
height 8 units. Square OACB has a side length of 6 units.
The coordinates of A and D are (6,0,0) and (3,3,8).
C lies on the y-axis.
z

D(3,3,8)
C
O
3.
y
B
x
A (6,0,0)
a)
Write down the coordinates of B.
(1)
b)
Determine the components of DA and DB
(2)
A curve has equation y = x - 16 , x>0
√x
Find the equation of the tangent at the point where x = 4
(6)
4.

Given that (x - 2) and (x + 3) are factors of f(x) where
f(x) = 3x3 + 2x2 + cx + d, the values of c and d are:
A c = 19
B c = -19
C c = 19
D c = -19
d=6
d = -6
d = -6
d=6
(2)
Total (14)
Higher Homework Exercise 18
1.

The diagram shows two vectors a and b, with |a| = 3 and
|b| = 2√2.
These vectors are inclined at an angle of 45o to each other.
a) Evaluate i) a.a
ii) b.b
iii) a.b
a
45o
b
b) Another vector p is defined by p = 2a + 3b
Evaluate p.p and hence write down |p|
2.
(2)
(4)
Two sequences are defined by the recurrence relations
Un+1 = 0.2Un + p,
Vn+1 = 0.6Vn + q,
U0 = 1 and
V0 = 1
a) Explain why each of these sequences has a limit.
b) If both sequences have the same limit, express p in
terms of q.
(3)
A curve has equation y = 2x3 + 3x2 + 4x – 5.
Prove that this curve has no stationary points.
3.
4.

(1)
Using triangle PQR, as shown,
find the exact value of cos 2x.
A
-
R
3
11
B
4√7
11
C
45
√11
D
4
√11
(5)
√7
P
x
2
Q
(2)
Total (17)
Higher Homework Exercise 19
dy
1.
A curve for which dx = 3sin(2x) passes through the
5
point( 12
π, √3). Find y in terms of x.
2. a)
b)




(4)
Given that x + 2 is a factor of 2x3 + x2 + kx + 2
find the value of k
(3)
Hence solve the equation 2x3 + x2 + kx + 2 = 0 when
k takes this value.
(2)
Solve the equation sin 2x – cos x= 0 in the interval
0≤x≤360
(4)
3.
4. Given f(x) = x2 + 2x – 8 f(x) in the form (x+a)2- b is:
A
(x + 2)2 - 8
B
(x + 2)2 - 12
C
(x + 1)2 - 9
D
(x + 1)2 - 10
(2)
Total (15)
Higher Homework Exercise 20
1. a) Write √3sin xo + cos xo in the form k sin (x + a) o
where k>0 and 0≤a≤360
(4)
b) Hence find the maximum value of
5 + √3sin xo + cos xo and determine the
corresponding value of x in the interval 0≤x≤360
(2)

2. Find

3
√x - 1
(4) √x
dx
3. PQRST is a regular hexagon of side 2 units.
PQ, QR and RS represent vectors a, b and c respectively.
U
Find the value of a.(b + c)
T
S
P
a
c
Q
b
R
(3)
4. Which of the following expressions is equal to 2 cos x + 2π
3

A
B
C
D
2 cos x - 1
2 cos x + 1
√3 cos x + sin x
-cos x - √3 sin x
(2)
Total (15)
Higher Homework Exercise 21
1.
Find the stationary points on the curve with equation
y = x3 - 9x2 + 24x – 20 and justify their nature.


2. Find

x2 – 5
x√x
dx
(7)
(4)
3. Find the maximum value of cos x – sin x and the value
of x for which it occurs in the interval 0 ≤ x ≤ 2π.
(6)
4. For what value of t are the vectors u =
perpendicular?

A
t
-2
3
and v =
2
10
t
4
B 5
C 3 45
D
-1
(2)
Total (19)
Higher Homework Exercise 22
1. Two sequences are generated by the recurrence relations
un+1 = aun + 10 and vn+1 = a2vn + 16.
The two sequences approach the same limit as n→∞.
Determine the value of a and evaluate the limit.
(5)


2. Evaluate log52 + log520 – log54
3.
The parabola shown crosses
the x-axis at (0,0) and (4,0)
and has a maximum at (2,4).
The shaded area is bounded by
the parabola and, the x-axis and
x = k.
(a) Find the equation of the parabola
(b) Hence show that the shaded area,
A, is given by
A = - 1 k3 + 2k2 3
(3)
y
4
O
2
(2)
(3)
16
3
4. 3x2 + 6x – 10 is expressed in the form 3(x + p)2 + q.
What is the value of q?

A
-9
B
-10
C -13
D
-16
(2)
Total (15)
k
4
x
Higher Homework Exercise 23
1. If f(x) = cos(2x) – 3 sin(4x), find the exact value of f ’
π
6
(4)

2. Solve the equation log2(x+1) – 2log2(3) = 3
(4)
3. The intensity, I, of light is reduced as it passes through a
-kt
filter according to the law It = I0e
where I0 is the
initial intensity and It is the intensity after passing
through a filter of thickness t cm.
k is a constant.
a) A filter of thickness 4cm reduces the intensity from
120 candle-power to 90 candle-power. Find the value
of k.
(4)
b) The light is passed through a filter of thickness 10cm.
Find the percentage reduction in its intensity
(3)
4. Simplify log2(x+1) – 2 log23

log2 x+1
9
B log2(x - 8)
A
C log2(x - 2)
D
log26(x + 1)
(2)
Total (17)
Higher Homework Exercise 24



1. Find the solution(s) of the equation
sin2p – sin p + 1 = cos2p for π < p < π.
2
(5)
2. Find x if 4 log x 6 – 2 log x 4 = 1
(3)
3. The graph illustrates the law y = kxn.
If the straight line passes through
A(0.5,0) and B(0,1), find the values
of k and n
log5y
B(0,1)
O
A(0.5,0)
)
(4)
4. If 3k e4, find an expression for k.
A k 3√4e

B k 
e4
3
C k 4
loge3
D k 1
loge3
(2)
Total (14)
log5x