Question 6:
--- 6(a) ---
6(a) | 1
kxn1
1
 
n1
0 | M1: attempting to integrate
xn xn1
and putting equal to 1,
ignore limits
A1: correct integration | M1A1
k = n + 1 | n1
A1: k = n + 1 Do not accept
1n1 | A1
(b) | 1
kxn2
1
 kxn1dx
 
0 n2
0 | 1
M1: Writing or using  kxn1dx,
0
ignore limits. Allow  1 kxxn dx
0
Allow substitution of their k
kxn2
A1: correct integration
n2 | M1A1
n1
=
n2 | A1: correct answer only- must be in
terms or n | A1cao
(c) | kxn3
1
 kxn2dx
 
0 n3 | M1: Attempting to integrate
 1 kxn2dx, xn2 xn3 , ignore
0
limits. Do not allow substitution of k
if it has x in it. This must be on its
own with no extra bits added on. | M1
A1cao
n1
=
n3 | A1: correct answer only
k
SC if they have as answer to
n2
k
part(b) award A1 for
n3
(d) | 2
3 3 3
Var (X) =    =
5 4 80 | M1: using
“their(c)”- [“their(b)”]2 with n = 2 or
correct Var(X)
2
Using  1 kx4dx   1 kx3dx  for
 
0 0
Var(X) | M1
Var (3X) = 9 Var (X) | M1: for writing or using 9 Var (X) or
32Var(X) | M1
A1cso
27
= oe or 0.3375 or 0.338
80 | A1: cso
NB: If there is a fully correct table award full marks.
P(10) = 0.2, P(20) = 0.4 and P(50) = 0.4 | B1: using P(10) = 0.2 (p) P(20) =
0.4(q) and P(50) = 0.4(r) may be seen
in calculations or implied by
a correct probability. | B1
Median 10, 20, 50 | B1: three correct medians and no
extras. | B1
P(Median 10) =
0.23 +30.220.4+30.220.4
or
0.23 +30.220.8 | M1: allow if (pqr)=1 and use
p3+3p2q+3p2r
or
p3+3p2(qr)
1 6 6
look for  
125 125 125 | See
below
for how
to award
P(Median 50) =
0.43 30.420.2+30.420.4
+
or
0.43 +30.420.6 | M1: allow if (pqr)=1 and use
r3+3r2 p+3r2q
or
r3+3r2(pq)
8 12 24
Look for  
125 125 125
P(Median 20) =
30.20.42 +60.20.40.4+0.43
+
30.420.4 | M1: allow if (pqr)=1 and use
3pq2+6pqr+q3+
3q2r
12 24 8 24
  
125 125 125 125
How to award the M marks – Allow the use of 1, 2 and 5 for the medians for the
method marks
M1 any correct calculation (implied by correct answer) for P(m = 10) or
P(m = 20) or P(m = 50)
M1 any 2 correct calculations (implied by 2 correct answers) P(m = 10) or
P(m = 20) or P(m = 50)
M1 any 3 correct calculations (implied by 3 correct answers) for P(m = 10) and P(m =
20) and P(m = 50) or
3 probabilities that add up to 1 providing it is 1 – their 2 other calculated
1 2 2
probabilities. Do not allow
5 5 5
NB if they do not have a correct answer their working must be clear including the
addition signs.
median 10 20 50
0.104 0.544 0.352
13 68 44
Or Or Or
125 125 125 | A1: awrt any 1 correct
A2: awrt all 3 correct
These do not need to be in a table as
long as the correct probablity is with
the correct median(10, 20 & 50)
NB: Do Not allow the use of 1,2 and
5 for the medians for the A marks | A2
median | 10 | 20 | 50
0.104 | 0.544 | 0.352
13
Or
125 | 68
Or
125 | 44
Or
125
PMT
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