AS Mathematics Assignment 2
Due Date: 7th Oct 2011
NAME…………………………… GROUP…………………………..
Instructions to Students
All questions must be attempted. You should present your solutions on file paper and submit them with this
cover sheet.
(Work submitted without a cover sheet complete with name will not be marked). All work is
to be submitted either to your module teacher or the Faculty Office by 4.15pm on the due date above.
TEACHER ASSESSED QUESTIONS
SUBJECT SPECIFIC SKILLS
1. Presentation : written work presented legibly using standard mathematical notation
Grade
Evidence/Comment :
12345
2. Communication: written solutions presented in a logical and coherent manner ( showing
workings clearly)
Grade
Evidence/Comment :
12345
3. Problem solving: identifying and using appropriate mathematical techniques
Grade
Evidence/Comment :
12345
4. Competence in using algebraic techniques
Grade
Evidence/Comment :
12345
5. Competence in using graphical calculator
Grade
Evidence/Comment :
12345
NOTE: GRADE 1 = BRIILIANT /SORTED
GRADE 5 = YET TO DEMONSTRATE THIS PARTICULAR SKILL
1.
Evaluate:
a)
15 85 
2.
Simplify:
a)
17
187
3.
Factorise:
a)
c)
2
3
b)
c)
b)
5
14  2
6x 3 y  9x 2 y 2
b)
20 x 3  5x
x 2  5x  84
d)
6 x 2  5x  25
5 x  625
5x  3  1
4.
Solve: a)
5.
Simplify the following expressions:
6.
65 2
b)
a)
(3x  5 y ) 2  ( 2 x  4 y ) 2
b)
( 2 x  6)( 3x  1)  ( x  3)( 6 x  2)
Rationalise the denominator of
5

1
19  3
c)
1 79 
2 x  3  4 x 1
1
2
(9 marks)
(6 marks)
(5 marks)
(6 marks)
(4 marks)

(4 marks)
7 . Solve each of the following quadratic equations, if possible
(a)
2x² – x – 3 = 0
(b)
3x² – 2x + 4 = 0
(c)
x² + 5x – 1 = 0
[9]
8
(a) Write the quadratic expression x² + 2x + 5 in the form A(x + B)² + C.
(b) Hence write down the coordinates of the minimum point of the graph
y = x² + 2x + 5
(c) Find the discriminant of the quadratic equation x² + 2x + 5 = 0.
(d) What does this discriminant tell you about the solutions of the equation
x² + 2x + 5 = 0?
(e) Sketch the graph of y = x² + 2x + 5, and explain how this confirms your
answer to (d).
[9]
9
Solve the simultaneous equations
(a) 2x + 3y = –7
(b)
5x – 2y = 11
x + 2y = 13
x² – y² = 9
[8]
10 Solve the following inequalities.
(a) 2x + 3 < 1 – x
(b) 3(y – 1)  5y – 8
[6]
H6FC Maths Department Lower 6th Assignment 2
11.
a) Solve, giving your answer to 3 decimal places, the equation
4 x 2 – 20x – 122 = 0
b) Hence sketch the curve y = 4 x 2 – 20x – 122 , showing clearly where the curve cuts
the coordinate axes.
(7 marks)
12.
Express (4 –
7 )(5 + 2 7 ) in the form a + b 7 , where a and b are integers.
(Total 3 marks)
13.
Express each of the following in the form p + q2, where p and q are rational:
(a)
(3 – 2)2;
(2)
(b)
1
(3 – 2 ) 2
.
(2)
(Total 4 marks)
14.
(a)
Express x2 – 12x + 40 in the form (x – p)2 + q
(2)
(b)
Hence, or otherwise, find the least value of x2 – 12x + 40.
(1)
(Total 3 marks)
15.
The diagram shows the graph of y = f (x), where f (x) = x2 + 6x + 1.
y
O
(a)
x
Express f (x) in the form (x + m)2 + n, where m and n are integers.
(2)
(b)
Solve the equation f (x) = 0, giving your answers in the form p + q 2 , where p and q are
integers.
(3)
(Total 5 marks)
H6FC Maths Department Lower 6th Assignment 2
16.
The quadratic equation
x2 + (3 – k)x + 5 – k2 = 0
is to be considered for different values of the constant k.
(a)
When k = 7:
(i)
show that x2 – 4x – 44 = 0;
(1)
(ii)
find the roots of this equation, giving your answers in the form a + b 3 , where a and b
are integers.
(2)
(b)
When the quadratic equation x2 + (3 – k)x + 5 – k2 = 0 has equal roots:
(i)
show that 5k2 – 6k – 11 = 0
(3)
(ii)
hence find the possible values of k.
(2)
(Total 8 marks)
17.
(a)
Solve the equation
2x2 + 32x + 119 = 0.
Write your answers in the form p + q 2 , where p and q are rational numbers.
(3)
(b)
(i)
Express
2x2 + 32x + 119
in the form
2 (x + m)2 + n.
where m and n are integers.
(2)
(ii)
Hence write down the minimum value of
2x2 + 32x + 119.
(1)
(Total 6 marks)
H6FC Maths Department Lower 6th Assignment 2
18.
(a)
Solve the equation
2x2 – 12x +17 = 0,
giving your answers in surd form.
(3)
(b)
Show that the equation
2x2 – 12x + 21 = 0
has no real roots.
(2)
(c)
Find the value of p for which the equation.
2x2 – 12x + p = 0
has equal roots.
(2)
19.
Calculate the coordinates of the points of intersection of the graphs which have equations
x + 2y = 5 and xy = 3
(Total 4 marks)
H6FC Maths Department Lower 6th Assignment 2