Formal Homeworks

1. 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 x1
( ) .
2. On a suitable set of real numbers, functions f and h are defined by :
f x
x
( ) 

2
3 3
and h x
x
( )  
1
12
Find  f h x( ) in its simplest form.
3. A function is defined as f x
x
( ) 

1
2 1
.
Define the largest possible domain available to this function on the set of
real numbers.
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
( ) .

2. Higher Formal Homework – Exponentials and logs
1. Given that 22loglog2  yy xx find a relationship connecting x and
y .
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
machinery.
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.

3. 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
decimal place.
1.00
1.80
2.60
0.00 0.20 0.40 0.60
log10Q
log10P

4. Higher Formal Homework – The Wave Function
1. Express 8cosxo
– 6sinxo
in the form kcos(x + a)o
where k > 0 and
0 < a < 360o
.
(4)
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
– 3cosxo
= 2.5 in the interval 0 ≤ x < 360o
. (8)
5. The displacement, d units, of a wave t seconds, is given by the formula
d = cos20to
+ √3sin20to
.
(a) Express d in the form kcos(20t - 𝛼)o
, where k > 0 and 0 ≤ 𝛼 ≤ 360.
(4)
(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)

5. Higher Formal Homework – Addition Formulae
1. Solve the equation 26
0for,062sin22 )(   
2. Solve algebraically the equation
.3600for07sin62cos  xxx 
3. Solve algebraically the equation
3 2 2 0 0 360cos cos , .x x x 
    
4. (a) Given that tan  5
9
, where 0  

2
, find the
exact value(s) of sin and cos .
(b) Hence show that the exact value of sin2 53
7
  .
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
given as
Cos ACE =
3 10
10
.
.
A B
C
DE
6cm
10cm
6cm

6. 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
~
 










2
1
1
. 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.
A(-2,1,-1)
B(0,-2,3)
C(-3,1,1)
D
G F (12, p, q)
x
z
A
BC
E
O
y

7. (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 .