Download Homework on analysing questionnaires – grade C

The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
HW2 – Grade C/B
Homework on nth term – Grade C/B
1.
The nth term of a sequence is given by the expression n2– 3
Write down the first three terms of the sequence.
...............................................................................................................................................
...............................................................................................................................................
Answer .................... , ..................... , ......................
(Total 2 marks)
2.
(a)
Write down the first three terms of the sequence whose nth term is given by 5n  3.
.....................................................................................................................................
.....................................................................................................................................
Answer........................................................................................................................
(2)
(b)
The letters p and q represent integers.
p is an odd number and q is an even number.
(i)
Which of these statements describes the number 2p + q?
always even
always odd
could be odd or even
Explain your answer.
...........................................................................................................................
...........................................................................................................................
(2)
(ii)
1
q is always even.
2
Give an example to show that Anna is wrong.
Anna says that p +
...........................................................................................................................
...........................................................................................................................
(1)
(Total 5 marks)
The Robert Smyth School
1
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
3.
(a)
p is an odd number.
Is 2p + 1 an odd number, an even number or could it be either?
Tick the correct box.
odd
even
either
(1)
(b)
The nth term of a sequence is 4n + 1
(i)
Write down the first three terms of the sequence.
...........................................................................................................................
...........................................................................................................................
Answer ..........................................................
(2)
(ii)
Is 122 a term in this sequence?
You must explain your answer.
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(1)
(Total 4 marks)
4.
A sequence of numbers is shown.
2
(a)
5
8
11
14
Find an expression for the nth term of the sequence.
......................................................................................................................................
......................................................................................................................................
Answer .........................................................................
(2)
(b)
Explain why 99 will not be a term in this sequence.
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
(2)
(Total 4 marks)
The Robert Smyth School
2
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
5.
A sequence of rectangular patterns is shown.
Pattern 1
(a)
Pattern 2
Pattern 3
Pattern 4
Pattern 5
Calculate the number of small squares in Pattern 20.
.....................................................................................................................................
Answer .........................................................................
(2)
(b)
Explain why the number of small squares in Pattern n is n(n + 2).
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
(2)
(Total 4 marks)
6.
Here is a number sequence.
1
(a)
3
6
10
15
Write down the next two numbers in the sequence.
......................................................................................................................................
Answer ………………………………….........
(2)
(b)
Describe a rule for continuing the sequence.
......................................................................................................................................
......................................................................................................................................
(1)
(c)
(i)
Work out the value of
2
x +x
when x = 5
............................................................................................................................
Answer ………………………………….........
(1)
(ii)
Factorise
2
x +x
............................................................................................................................
Answer ………………………………….........
(1)
The Robert Smyth School
3
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
Darren says that x2 + x is always even.
(iii)
Using your Answer to part (ii), or otherwise, explain why this is true.
............................................................................................................................
............................................................................................................................
............................................................................................................................
(2)
(Total 7 marks)
7.
Part of a number grid is shown below.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
The shaded shape is called L3 because it has the number 3 at the top.
The sum of the numbers in L3 is 26.
(a)
Calculate the sum of the numbers in L22
.....................................................................................................................................
Answer ………………………………….........
(1)
(b)
Fill in the empty squares of Ln
n
(2)
(c)
Write down an expression, in terms of n, for the sum of the numbers in Ln
Simplify your expression.
.....................................................................................................................................
.....................................................................................................................................
Answer ………………………………….........
(2)
(d)
If the sum of the numbers in Ln is 143, find the value of n.
.....................................................................................................................................
.....................................................................................................................................
Answer ………………………………….........
(2)
(Total 7 marks)
The Robert Smyth School
4
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
8.
Rearrange
y = mx + c
to make x the subject of the formula.
.....................................................................................................................................
.....................................................................................................................................
Answer x = ....................................................
(Total 4 marks)
9.
Make p the subject of the formula t = 5p + 40
.......................................................................................................................................
.......................................................................................................................................
Answer p = .................................................................
(Total 2 marks)
10.
Make r the subject of the formula
p = 3 + 2r
……………...........………………………….....…………………………………….
……………...........………………………….....…………………………………….
……………...........………………………….....…………………………………….
……………...........………………………….....…………………………………….
Answer r = ………………………………….
(Total 2 marks)
11.
Make c the subject of the formula
d=
c
+e
5
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
...............................................................................................................................................
Answer .......................................
(Total 2 marks)
The Robert Smyth School
5
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
1.
–2, 1, 6
B2
–1 each error or emission.
Ignore extra terms
12  3, 22  3, 32  3 is B1
[2]
2.
(a)
2, 7, 12
B2
B1 for 2 correct
(b)
(i)
Always even
B1
2 × odd = even,
even + even = even
(ii)
B1
e.g. 3 + ½ (4) = 5
look for 1 2 (multiple of 4)
B1
[5]
3.
(a)
Odd
B1
Any clear indication
(b)
(i)
5, 9, 13
– 1 each error or omission
1, 5, 9 scores B1
9, 13, 17 scores B1
B2
(ii)
No and valid reason
eg 121 is in the sequence
(all) terms are odd
122 is even
121 ÷ 4 is not an integer
Note: “No” can be implied
B1
[4]
4.
(a)
3n – 1
B2
oe
B1 for any of the following:
3n (+c)
n=×3–1
nth = × 3 – 1
nth × 3 – 1
n3 – 1
(b)
Complete explanation
eg 2, 5, 8… not multiples of 3
eg 98 and 101 are in the sequence
eg 3n – 1 = 99 does not give a whole number
eg n = 33.3…
eg 100 is not a multiple of 3
eg 99 is a multiple of 3
Part explanation B1
The Robert Smyth School
B2
6
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
eg 101 is in the sequence
eg 98 is the nearest
SC1 for correctly using their answer from (a) provided linear
but not n + 3
[4]
5.
(a)
(b)
20 × 22
M1
440
A1
n squares across or n + 2 squares high
n wide or n along
n + 2 up or length n + 2
B1
n(n + 2) for area
B1
[4]
Multiply them for area/total number of squares
6.
(a)
21
B1
28
B1
(b)
Valid explanation
Accept:
Add 1, then add 2. ...etc
Differences increasing by 1
Triangle numbers
Adding on one more
Do not accept:
Adding on one
Adding on an extra number
+6 +7
B1
(c)
(i)
30
B1
(ii)
x(x + 1)
B1
(iii)
Valid explanation
Odd × even = even scores B1
Even × odd = even scores B1
If x is odd, x2 is odd, odd + odd = even
and if x is even, x2 is even,
even + even = even scores B2 or Bl
for each part
One numerical eg 22 + 2 = 6 B1
Two numerical Bl
One odd and one even with eg both
even B2
B2
[7]
The Robert Smyth School
7
The Robert Smyth School
Mathematics Faculty
Sequences
Innovation & excellence
7.
(a)
83
(b)
n+8
B1
B1
n+9
B1
Allow one mark if transposed
(c)
(d)
n + their (n + 8) + their (n + 9)
Provided n ± k in both boxes
M1
3n + 17
A1
143 – 17 or 126 seen
T & I  42 gets 2 marks
M1
42
A1
Embedded answer Ml A0
[7]
8.
y – c = mx
B1
( y – c)
=x
m
B1
must be in correct order
y
– c gets B0B0
m
[2]
9.
t – 40 = 5p or t ÷ 5 = p + 8
oe condone one sign error in rearranging to isolate the term in
p, for M1
M1
(t – 40) ÷ 5 = p or t ÷ 5 – 8 = 9
t – 40 ÷ 5 = p (without brackets) scores M1 only
t – 40 ÷ 5 = p, seen without working, implies M1
A1
[2]
10.
p – 3 = 2r or p/2 = 3/2 + r
or
M1
(p + 3)/2 or (3 – p)/2
(p – 3)/2 or p/2 – 3/2
oe
A1
[2]
11.
c
=d–e
5
M1
or 5d = c + 5e o.e
c = 5(d –e)
A1
or 5d – 5e
[2]
The Robert Smyth School
8