MILLBURN ACADEMY
MATHS DEPARTMENT
HIGHER MATHS
HOMEWORK BOOKLET
The Straight Line
1)
A quadrilateral has vertices A(–1, 8), B(7, 12),
C(8, 5) and D(2, –3) as shown in the diagram.
(a)
(b)
(c)
2)
3)
Find the equation of diagonal BD.
The equation of diagonal AC is x 3y 23.
Find the coordinates of E, the point of
intersection of the diagonals.
(i)
Find the equation of the perpendicular
bisector of AB.
(ii)
Show that this line passes through E.
Triangle PQR has vertex P on the x-axis,
as shown in the diagram.
Q and R are the points (4, 6) and (8, –2)
respectively.
The equation of PQ is 6x – 7y 18 0.
(a)
State the coordinates of P.
(b)
Find the equation of the altitude of
the triangle from P.
(c)
The altitude from P meets the line
QR at T. Find the coordinates of T.
The vertices of triangle ABC are A(7, 9),
B(–3, –1) and C(5, –5) as shown in the
diagram.
The broken line represents the perpendicular
bisector of BC.
(a)
Show that the equation of the
perpendicular bisector of BC is
y 2x – 5.
(b)
Find the equation of the median
from C.
(c)
Find the coordinates of the point
of intersection of the perpendicular
bisector of BC and the median from C.
2
3
5
1
3
4
4
3
3
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Quadratics
1)
For what values of k does the equation x2 – 5x + (k + 6) = 0 have equal roots?
2)
a)
b)
Given f(x) = x2 + 2x – 8, express f(x) in the form (x + a)2 – b.
State the minimum value of the function f(x).
3
2
1
y
3)
The diagram shows a sketch of a
parabola passing through (-1,0),
(o,p) and (p,0).
(0,p)
•
a) Show that the equation of the
parabola is y = p + (p – 1)x – x2.
b) For what value of p will the line
y = x + p be a tangent to this curve?
4)
3
(p,0)
(-1,0)
•
O
•
Show that the equation (1 – 2k)x2 – 5kx – 2k = 0 has real roots for all integer
values of k.
Write f(x) = x2 + 6x + 11 in the form (x + a)2 + b.
Hence or otherwise sketch the graph of y = f(x).
x 3
5
5)
a)
b)
2
2
6)
Show that the line with equation y = 2x + 1 does not intersect the parabola with equation
y = x2 + 3x + 4.
5
7)
Prove that the roots of the equation 2x2 + px – 3 = 0 are real for all values of p.
4
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Polynomials
Given (x + 2) is a factor of 2x3 + x2 + kx + 2, find the value of k.
Hence solve the equation 2x3 + x2 + kx + 2= 0, when k takes this value.
1)
(a)
(b)
2)
Given (x – 2) and (x + 3) are factors of f (x) = 3x3 + 2x2 + cx + d
Find the values of c and d.
5
The graph of f (x) = 2x3 – 5x2 – 3x + 1 has a root between 0 and 1.
Find the value of this root to one decimal place.
3
f (x) = 6x3 – 5x2 – 17x + 6
(a) Show (x – 2) is a factor of f (x).
(b) Express f (x) in its fully factorised form.
2
2
3)
4)
5)
6)
7)
f (x) = x3 – x2 – 5x – 3
(a) (i)
Show (x + 1) is a factor of f (x)
(ii) Hence or otherwise factorise f (x) fully.
(b) One of the turning points of the graph lies on the x-axis.
Write down the coordinates of this turning point.
f (x) = 2x3 – 7x2 + 9
Show (x – 3) is a factor of f (x) and factorise f (x) fully.
(a)
(b)
Show x = -1 is a solution of x3 + px2 + px + 1 = 0
Hence find the range of values of p for the equation to have real roots.
3
2
5
1
5
1
7
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Composite Functions
1)
2)
Given f (x) = 2x – 1 & g(x) = 3 – 2x
(a)
Find an expression for k(x) = f (g(x))
(b)
Find an expression for h(k (x)) given h(x) = ¼ (5 – x)
2
(c)
What can you say about h(x) and k (x) and why?
1
(a)
(i)
Find an expression for f (x + 1) given f (x) = 4x2 – 3x + 5
2
(ii)
Find an expression for f (x – 1) given f (x) = 4x2 – 3x + 5
2
(iii)
Show that ½ [f (x + 1) – f (x – 1)] simplifies to
1
(i)
Find an expression for g(x + 1) given g(x) = 2x2 + 7x – 8
2
(ii)
Find an expression for g(x – 1) given g(x) = 2x2 + 7x – 8
2
(iii)
Show that ½ [g(x + 1) – g(x – 1)] simplifies to
1
(b)
(c)
3)
2
8x – 3
4x + 7
Hence if h(x) = 3x2 + 5x – 1, find an expression ½ [h(x + 1) – h(x – 1)]
2
Given f (x) = 2x and g(x) = sin(x) + cos(x) find an expression for
(a)
f (g(x))
(b)
g(f (x))
4)
Find an expression for f (f (x)) given 𝑓(𝑥) =
5)
Find f (g(x)) given 𝑓(𝑥) =
6)
Given
𝑓(𝑥) =
1
𝑥+2
1
𝑥2
−4
&
1, 1
𝑥
1−𝑥
& 𝑔(𝑥) =
1
𝑥
−2
3
3
g(x) = 2x + 1
(a)
Find g(f (x))
3
(b)
State a suitable domain.
2
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Differentiation 1
2
, find 𝑓 ′ (4).
1)
Given that 𝑓(𝑥) = √𝑥 +
2)
A curve has equation 𝑦 = 𝑥 −
3)
𝑥2
16
√𝑥
5
, 𝑥 > 0.
Find the equation of the tangent at the point where 𝑥 = 4.
6
The point P(x,y) lies on the curve with equation 𝑦 = 6𝑥 2 − 𝑥 3 .
a)
Find the value of x for which the gradient of the tangent at P is 12.
b)
Hence find the equation of the tangent at P.
5
2
𝑑𝑦
4)
Given that 𝑦 = √3𝑥 2 + 2, find
5)
Given that 𝑦 = 3 sin(𝑥) + cos(2𝑥), find
.
3
6)
Differentiate (1 + 2 sin 𝑥)4 with respect to x.
2
𝑑𝑥
.
3
𝑑𝑦
𝑑𝑥
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Higher Maths Homework – Trig: Radians and Equations
Questions 1, 3, 4 & 5 should be done WITHOUT a calculator.
𝜋
5𝜋
1)
What is the exact value of sin( ) - cos ( ) ?
3
4
2)
What is the solution of the equation 4 sin 𝑥 − √7 = 0 where
3)
Functions f and g are defined on suitable domains by 𝑓(𝑥) = sin 𝑥 and
𝑔(𝑥) = 2𝑥.
Find expressions for
a) 𝑓(𝑔(𝑥))
b) 𝑔(𝑓(𝑥))
4)
3
𝜋
2
≤𝑥≤𝜋?
3
1
1
The diagram shows the graph of a cosine function from 0 to 𝜋.
a) State the equation of the graph.
2
a) The line with equation 𝑦 = −√3
intersects this graph at points A and B.
Find the coordinates of B.
5)
Find all the values of x in the interval 0 ≤ 𝑥 ≤ 2𝜋 for which tan2 𝑥 = 3.
4
4
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Higher Maths Homework – Vectors
1)
VABCD is a pyramid with a rectangular base ABCD.
Relative to some appropriate axes,
⃗⃗⃗⃗⃗
𝑉𝐴 represents -7i – 13j – 11k
⃗⃗⃗⃗⃗
𝐴𝐵 represents 6i + 6j – 6k
⃗⃗⃗⃗⃗
𝐴𝐷 represents 8i – 4j + 4k
2)
3)
4)
K divides BC in the ratio 1:3.
Find ⃗⃗⃗⃗⃗
𝑉𝐾 in component form.
3
Vectors u and v are defined by u = 3i + 2j and v = 2i – 3j + 4k.
Determine whether or not u and v are perpendicular to each other.
2
The diagram shows a cuboid
OABC, DEFG.
F is the point (8, 4, 6).
P divides AE in the ratio 2:1.
Q is the midpoint of CG.
(a) State the coordinates of P and Q.
(b) Write down the components of
⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗
𝑃𝑄
𝑃𝐴
(c) Find the size of angle QPA.
D, E and F have coordinates (10, –8, –15), (1, –2, –3) and (–2, 0, 1) respectively.
(a) (i) Show that D, E and F are collinear.
(ii) Find the ratio in which E divides DF.
(b) G has coordinates (k, 1, 0).
Given that DE is perpendicular to GE, find the value of k.
2
2
5
4
4
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Higher Maths Homework – Differentiation 2
1)
2)
3)
4)
5)
The graph of the cubic function y = f (x) is shown
in the diagram. There are turning points at (1,1)
and (3,5). Sketch the graph of y = f ‘(x).
3
A company spends x thousand pounds a
year on advertising and this results in a
profit of P thousand pounds.
A mathematical model, illustrated in the
diagram, suggests that P and x are related
by P = 12x3 – x4 for 0 ≤ x ≤ 12.
Find the value of x which gives the maximum
profit.
5
A function is defined by f (x) = (2x – 1)5.
Find the coordinates of the stationary point on the graph with equation
y = f (x) and determine its nature.
7
A function f is defined by the formula f (x) = 3x – x3.
(a) Find the exact values where the graph of y = f (x) meets the x- and y-axes.
(b) Find the coordinates of the stationary points of the function and
determine their nature.
(c) Sketch the graph of y = f (x).
2
7
1
A householder has a garden in the shape
of a right-angled isosceles triangle.
It is intended to put down a section of
rectangular wooden decking at the side
of the house, as shown in the diagram.
(a) (i) Find the exact value of ST.
(ii) Given that the breadth of the decking is x metres, show that the area
of the decking, A square metres, is given by
𝐴 = (10√2) 𝑥 − 2𝑥 2
(b) Find the dimensions of the decking which maximises its area.
3
5
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Trig - Addition & Double Angle Formulae Homework
Questions 1, 3, 4 & 5 should be done WITHOUT a calculator.
1)
7𝜋
𝜋
𝜋
7𝜋
a) Using the fact that = + , find the exact value of sin ( ).
12
3
4
12
b) Show that sin(A + B) + sin(A – B) = 2sin A cos B.
𝜋
𝜋
𝜋
c) i) Express in terms of and .
12
3
4
7𝜋
𝜋
ii) Hence or otherwise find the exact value of sin ( ) + sin ( ) .
12
12
3
2
4
2)
Solve 2cos 2x – 5cos x – 4 = 0 for 0 < x < 2π.
5
3)
A right-angled triangle has sides and angles as shown in the diagram.
What is the exact value of sin 2a?
2
1
4)
If the exact value of cos x is
, find the exact value of cos 2x.
5)
The diagram shows two right-angled
triangles with sides and angles given.
What is the exact value of sin (p + q)?
√5
3
3
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