Kilsyth Academy
National 5 Maths
Unit 3 Homework
Booklet Number
National 5 Homework
Exercise 16 Trig Graphs
Section A (Non-calculator)
1.
Simplify:
10 x 6
a5  a3
(d)
2x3
a2
Rationalise the denominator and simplify where possible:
2
2
(a)
(b)
3
8
(a) 3m 7  2m 2
2.
(b) 5 x 6  2 x 4 (c)
Section B (Knowledge)
Only use your calculator if you need to!
3.
Write down the period of the graph of y  4 sin 3x
4.
Write the equations of the following trigonometric functions:
(a)
(b)
5
90
180
–5
5.
Make a sketch of the following trigonometric functions, for 0  x  360  :
a) y  2 sin x
b) y  3 cos 2 x
c) y  10 sin( x  30)
Section C (Revision)
6.
Priscilla recorded the number of text messages she sent over 5 days:
14, 18, 16, 20, 17
Work out the mean and standard deviation.
7.
Solve, algebraically, these equations
a)
3x + 4y = 13
2x – 4y = 2
b)
2y + 3x = 9
5y + 2x = 17
National 5 Homework
Exercise 17 Trig Equations
Section A (Non-calculator)
1.
Break the brackets
(i) ( x  3)( x 2  4 x  1)
2.
Factorise
(i) 6a  3b
3.
Simplify the following surds: (a) 75
(ii) x 2  4
(iii) t 2  6t  16
(b) 3  12  27
Section B (Knowledge)
4.
Solve these trigonometric equations where 0 ≤ x ≤ 360, answers to 3 significant
figures –
a) sin x° = 0∙528
5.
c) cos x° = – 0∙714
Solve the following trigonometric equations in the range 0 ≤ x ≤ 360
a) 4 sin x   1
6.
b) tan x° = 1∙724
b) 7 cos x   4  0
c) 5tan x° + 3 = 0
d) 3sin x° + 2 = 0
Prove the following trigonometric identities:
a) tan Y cos Y  sin Y
b) 8  8 cos 2 x  8 sin 2 x
Section C (Mixed)
7.
A railway goes through an underground tunnel. The diagram below shows the
cross-section of the tunnel. It consists of part of a circle with a horizontal base.




The centre of the circle is O
XY is a cord of the circle
XY is 1.8 metres
The radius of the circle is 1.7 metres
Find the height of the tunnel.
National 5 Homework
Exercise 18 Quadratic Graphs
Section A (Non-calculator)
1.
Evaluate –
a) 80% of 72g
c)
b) t 2  t 7
5 7
d)

8 16
32
Section B (Knowledge)
2.
The diagram shows the parabola with equation y  kx 2
What is the value of k?
3.
The equation of the quadratic function
whose graph is shown below is of the form y
= (x + a)2 + b, where a and b are integers.
Write down the values of a and b.
8
6
4
2
0
-2
1
2
3
x
-1
4.
Sketch the graph y = (x - 2)(x + 4) on plain paper.
Mark clearly where the graph crosses the axes and state the coordinates of the turning
point.
5.
A parabola has equation y = (x + 3)2 − 5.
(a) Write down the equation of its axis of symmetry.
(b) Write down the coordinates of the turning point on the parabola and state whether
it is a maximum or minimum.
Section C (Revision)
6.
Work out the size of the side marked x in this triangle.
x
6 cm
75º
10·9 cm
National 5 Homework
Exercise 19 Quadratic Equations
Section A (Non-calculator)
1.
Evaluate –
a) 2½ % of £140
c)
128
b) 15 p 8  3 p 4
3 2 3
 
d)
4 5 8
Section B (Knowledge)
2.
Solve the following equations by factorising:
a) 5x2 – x = 0
b) y2 – 49 = 0
c) t2 + 8t + 15 = 0
2
2
d) w + w – 6 = 0
e) x – 6x + 8 = 0
f) 2y2 – 11y + 15 = 0
3.
Solve this equation algebraically:
4.
Use the quadratic formula to solve these equations giving your answer to 1 decimal
place:
a) x2 − 7x + 2 = 0
b) 3x2 + 2x − 4 = 0
c) 2y2 – 11y – 15 = 0
5.
Find where the following curve with equation
y = x2 − 4x − 12 cuts the x axis.
6.
By calculating the discriminant, determine the nature of the roots of the following
quadratic equations:
a) x2 + 6x + 9 = 0
b) 3x2 − 4x + 2 = 0
c) 4 – 2x − x2 = 0
y(y – 4) = 2y − 5
Section C (Mixed)
7.
AD is a diameter of a circle, centre O.
B is a point on the circumference of the circle.
The chord BD is extended to a point C, outside the circle.
Angle BOA = 98
DC = 9 centimetres.
The radius of the circle is 7 centimetres.
Calculate the length of AC.
x2 − 4x − 12
National 5 Homework
Exercise 21 Vectors
Section A (Non-calculator)
(a)
3x – 2 > 2x + 7
1.
Solve for x:
2.
Break the brackets and collect like terms:
b)
2x + 3 ≤ 6x – 9
(3x + 2)(x – 5) + 8x
Section B (Knowledge)
3.
The diagrams below show 2 directed line segments u and v.
Draw the resultant vector of 2u + v
  3
 
  4  , work out u – v .
 6 
 
4.
 4 
 
If u =   2  and v =
 5 
 
5.
If a = 
6.
The diagram below shows a model of the Pyramide du Louvre in Paris.
It is a square-based pyramid of height 8 centimetres.
Square OABC has sides of length 6 centimetres.
The coordinates of A are (6,0,0) and C lies on the y-axis.
  3
 and b =
 4 
 4 
  , calculate 3a  2b .
  1
D
(6,0,0)
Write down the coordinates of D.
7.
 3x 


u =  y6, v =
 2 z  1


 12 
 
  2  and u = v. Find x, y and z.
 9 
 
Section C (Revision)
8.
Change the subject of the formula to g:
(a) gx  t  s
2
9.
Show that the equation x + 3x + 5 = 0 has no real roots.
(b) h  f 2 g
National 5 Homework
Exercise 22 Algebraic Fractions
Section A (Non-calculator)
1.
Evaluate:
2.
Break the brackets:
3.
4.
(a) 60% of £8100
(b) 35% of £620
(c) 2% of £30,000
( x – 6 )( 3x + 5 )
Aidan invests £6000 into a savings account with the Clydesdale Bank. The bank
pays 3% interest per annum. Calculate how much interest Aidan receives after 1
year.
Find the equation of the straight line AB.
Section B (Knowledge)
5.
Simplify the following algebraic fractions:
6.
2y2
x  1x  2
12a 2 b
(a)
(b)
(c)
2
x  3x  1
3ab
4y
Factorise the numerator and/or the denominator, then simplify:
(a)
7.
x2
4x  8
(b)
x 2  16
( x  4)
(c)
x 2  2 x  15
5 x  25
Express each of the following as a single fraction and simplify where possible:
(a)
3 3k

2k 9
(b)
p 4

2 p
(c)
2 x 5x
3 7


(d)
4
3
m n
Section C (Revision)
8.
Fiona and Ross each book in at the Sleepwell Lodge.
(a) Fiona stays for 3 nights and has breakfast on 2 mornings. Her bill is £230.
Write down an algebraic equation to illustrate this
(b) Ross stays for 5 nights and has breakfast on 3 mornings. His bill is £380.
Write down an algebraic equation to illustrate this.
(c) Find the cost of one breakfast.