Higher Maths Course Problems
Units 1,2,3
1. If f ′(x) =
1
and f(1) = 1, find an expression for f(x).
2x2
2. The diagram shows part of the graph of a function f.
On separate diagrams sketch the graphs of
a) f(x+3)
b) -f(x) c) 2 + f(x-1)
y
(0,1.5)
x
(2,0)
0
(1,-2)
(3,-2)
3. Solve the equation 4sin2x = 5cosx , 0º < x < 360º.
4. A mushroom bed contains some 4800 mushrooms. Each
morning 35% of the mushrooms are picked. By the end of the day
some 600 new mushrooms become ready for picking. Set up a
recurrence relation and use it to determine
a) How many mushrooms will there be in the bed at the end of
one week?
b) Estimate how many mushrooms there will be in the bed after
one year.
5. P is the point with coordinates (sin(x+30)°,cos(x-30)°) and Q
has coordinates (sin(x-30)°,cos(x+30)°). Find, in its simplest form,
an expression for the gradient of the line PQ.
6. The line joining the points (2,3) and (8,k) is perpendicular to
the line with equation 2y – 3x + 5 = 0. Find the value of k.
7. The circle with equation x2 + y2 + 11x + 7y + 10 = 0 cuts the x
axis at the points P and Q. Find the length of the line PQ.
8. If p = 6k and q = 3 j + 4k , are the position vectors of points P
and Q respectively, find the exact value of cosPOQ.
9. State the range of the function y = 4 + 3sin 2 x .
10. A rectangular box without a lid is made from 200cm2 of metal.
Its base measures 2x cm by 3x cm.
a) Find the height of the box in terms of x and show that the
volume V is given by V = 120 x −
18 3
x.
5
Show that the maximum volume occurs when x =
10
and find
3
this maximum volume.
11. The centre of a circle lies on the line 2x + y = 0.
The lines y = 1 and y = 7 are tangents to this circle. Find the
equation of the circle.
12. Find the x coordinates of the two points on the curve
y = x3 – x2 – 4x + 2 where the gradient of the tangent is 1.
13. Show that the line x + y = 10 is a tangent to the circle
x2 + y2 - 2x – 10y + 18 = 0 and find the coordinates of the
point of contact.
14.
The yearly operating costs C of a small manufacturing company
9300
+ 10 x , where x is the average stock
x
9300
held throughout the year. ₤
is the yearly ordering costs,
x
are given by C = 30000 +
₤10x is the yearly cost of holding stock and ₤30000 is a fixed
yearly operating cost.
a) Find the value of x which minimises C.
b) What integral value of x should the company aim at
maintaining, to keep ordering costs low as well as the total
operating costs?
15. A, B and C are the points (0,1,2), (1,1,1) and (3,2,0)
respectively. D is the point on AC dividing AC in the ratio 1:2.
a) Find the coordinates of D.
b) Find BA , BD and BA . BD
c) Hence or otherwise, find the size of angle ABD.
16. Consider the function f(x) = 4sin(x-50)° = 0.
Write down the maximum and minimum values
of f and the values of x at which these max/min values occur,
for x in the interval 0º < x < 360º.
Solve the equation f(x) - 3 = 0.
17.
x
x
x
x
x
x
8cm
x
x
x
15cm
From a rectangular piece of cardboard, small squares of side x are
cut from the corners. The ends are then folded up to form the box
shown on the right above.
a) Show that the volume of the box is given by
V = 4 x 3 − 46 x 2 + 120 x
b) Find x so that this volume will be a maximum.
18. Given A(0,6), B(4,2) and C(-6, -2) are the vertices of a
triangle. Find
a) The equation of side AB
b) The equation of the altitude from A
c) The equation of the median from B.