Higher Formal Homework – Functions and graphs of functions
1. The functions f x x x g x x( ) ( ) 2
3 1and are defined on the
set of positive real numbers (R+
(a) Evaluate ))2(( fg .
(b) Find a formula for i) f g x( )
ii) g f a( )
(c) Find a formula for the inverse function g x1
( ) .
2. On a suitable set of real numbers, functions f and h are defined by :
and h x
Find f h x( ) in its simplest form.
3. A function is defined as f x
Define the largest possible domain available to this function on the set of
4. The graph of y f x ( ) is shown opposite.
(a) Draw a sketch of y f x ( ) .
(b) Draw a sketch of y f x ( ) 1 .
(c) Draw a sketch of y f x ( )3 .
(d) Draw a sketch of y f x 1
( ) .
Higher Formal Homework – Exponentials and logs
1. Given that 22loglog2 yy xx find a relationship connecting x and
2. The rate of decomposition of an acid in a solution obeys the law
C e t
4 0 025
, where C is the concentration in millilitres of acid left
after t minutes.
(a) What is the intial concentration of acid ?
(b) Determine how long it takes for the concentration to reach 3ml ,
giving your answer to the nearest second.
(c) How long does it take to reduce to half its original concentration ?
3. Solve for x in 2 1 3 42 2log logx x .
4. Radioactive isotopes are used as indicators for bearing wear in high speed
The rate of decay of these isotopes is measured in terms of their
half-life , i.e. the time required for one half of the material to decay.
For a particular isotope the mass present at any time M a.m.u. (atomic
mass units), at time t hours is given by :
M M e kt
0 , where k is a constant and M0 is the initial mass.
Experiments have shown that at t 3 , M M 0 8 0 .
(a) Find the value of k correct to 3 decimal places.
(b) Hence evaluate the half-life of the isotope to the nearest minute.
5. From an experiment corresponding
replacements for P and Q were found.
log10 P was plotted against log10 Q and a
best fitting straight line drawn, as shown
in the diagram.
Find the values of n and k to one
0.00 0.20 0.40 0.60
Higher Formal Homework – The Wave Function
1. Express 8cosxo
in the form kcos(x + a)o
where k > 0 and
0 < a < 360o
2. Express 2sinx – 5cosx in the form ksin(x + 𝛼), 0 ≤ 𝛼 < 2π ans k > 0. (4)
3(a) Express sinx – 3cosx in the form ksin(x – 𝛼) where k > 0 and
0 ≤ 𝛼 < 2π. Find the values of k and . (4)
(b) Find the maximum value of 5 + sinx – 3cosx and state the value of x
for which this maximum value occurs. (2)
4. Solve the equation 2sinxo
= 2.5 in the interval 0 ≤ x < 360o
5. The displacement, d units, of a wave t seconds, is given by the formula
d = cos20to
(a) Express d in the form kcos(20t - 𝛼)o
, where k > 0 and 0 ≤ 𝛼 ≤ 360.
(b) Sketch the graph of d for 0 ≤ 𝑡 ≤ 18. (4)
(c) Find, correct to one decimal place, the values of t, 0 ≤ 𝑡 ≤ 18, for
which the displacement is 1.5 units. (3)
Higher Formal Homework – Addition Formulae
1. Solve the equation 26
2. Solve algebraically the equation
3. Solve algebraically the equation
3 2 2 0 0 360cos cos , .x x x
4. (a) Given that tan 5
, where 0
, find the
exact value(s) of sin and cos .
(b) Hence show that the exact value of sin2 53
5. The diagram opposite is a sketch of the tail - fin
of a model plane.
ED is parallel to AB.
ED = DB.
Show that the exact value of Cos ACE is
Cos ACE =
Higher Formal Homework – Vectors
1. Prove that the points E(2,-2,0) , F(1,4,-3) and G(-1,16,-9) are
collinear, and find the ratio EF : FG.
2. Two vectors are defined as V i j k1 3 2
~ ~ ~
and V i j k2 2 4
~ ~ ~
Show that these two vectors are perpendicular.
3. Vector v
is a unit vector parallel to the vector u
. Find v
4. The diagram below is a schematic of an elastic coupling used as part of a
cassette door mechanism. Relative to an origin O , points A , B and C
have coordinates (-2,1,-1) , (0,-2,3) and (-3,1,1) respectively .
Calculate the size of angle ABC .
5. A cuboid is placed on coordinate axes as shown.
The dimensions of the cuboid are in the ratio OA : AB : BF = 4 : 1 : 2
The point F has coordinates (12, p, q) as shown.
G F (12, p, q)
(a) Establish the values of p and q .
(b) Write down the coordinates of the points A and C and hence
calculate the size of ACF .