1.
The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a
standard deviation of 0.2 kg.
(a)
Determine the probability that a fish that is caught weighs less than 1.4 kg.
(1)
(b)
John catches 6 fish. Calculate the probabi
1.
Attempt at implicit differentiation
dy
dy
+ y
1 + = sin( xy ) x
(x+y)
dx
dx
e
let x = 0, y = 0
dy
1 +
0
dx = 0
e
dy
dx = 1
M1
A1A1
M1
A1
2 , y = 2
let x =
dy
dy
+ y
1 + = sin( 2 ) x
0
dx
=0
e dx
dy
so dx = 1
A1
since both points lie on t
1.
(x + y )
Show that the points (0, 0) and ( 2 , 2 ) on the curve e
= cos (xy) have a
common tangent.
(Total 7 marks)
2.
1
(9 + 8 x 2 x 4 )
The curve C has equation y = 8
.
(a)
dy
Find the coordinates of the points on C at which dx = 0.
(4)
(b)
The tange
1.
(a)
A1
Note: Award A1 for intercepts of 0 and 2 and a concave down curve
in the given domain
Note: Award A0 if the cubic graph is extended outside the domain [0, 2].
2
(b)
kx( x + 1)(2 x)dx = 1
(M1)
0
Note: The correct limits and = 1 must be seen but
1.
The random variable X has probability density function f where
kx( x + 1)(2 x), 0 x 2
0,
otherwise.
f(x) =
(a)
Sketch the graph of the function. You are not required to find the coordinates of the
maximum.
(1)
(b)
Find the value of k.
(5)
(Total 6 mar
1.
Events A and B are such that P(A) = 0.3 and P(B) = 0.4.
(a)
Find the value of P(A B) when
(i)
A and B are mutually exclusive;
(ii)
A and B are independent.
(4)
(b)
Given that P(A B) = 0.6, find P(A | B).
(3)
(Total 7 marks)
2.
In a class of 20 students
1.
(a)
using the law of total probabilities:
0.1p + 0.3(1 p) = 0.22
0.1p + 0.3 0.3p = 0.22
0.2p = 0.08
0.08
p = 0.2 = 0.4
p = 40% (accept 0.4)
(b)
0.4 0.1
required probability = 0.22
2
= 11 (0.182)
(M1)
A1
A1
M1
A1
[5]
2.
(a)
A1A1
Note: Award A1 for a dia
1.
An influenza virus is spreading through a city. A vaccination is available to protect against the
virus. If a person has had the vaccination, the probability of catching the virus is 0.1; without the
vaccination, the probability is 0.3. The probability
1.
(a)
(i)
A2
Note: Award A1 for correct sin x, A1 for correct sin 2x.
Note: Award A1A0 for two correct shapes with 2 and/or 1 missing.
Note: Condone graph outside the domain.
(ii)
(iii)
sin 2x = sin x, 0 x 2
2 sin x cos x sin x = 0
sin x (2 cos x 1) = 0
1.
(a)
(i)
Sketch the graphs of y = sin x and y = sin 2x, on the same set of axes,
for 0 x 2 .
(ii)
Find the x-coordinates of the points of intersection of the graphs in the
domain 0 x 2 .
(iii)
Find the area enclosed by the graphs.
(9)
1
(b)
Find the val
x
1.
(a)
(i)
f(x) =
1 ln x
1
ln x
x
x2
M1A1
2
=x
so f(x) = 0 when ln x = 1, i.e. x = e
(ii)
A1
f(x) > 0 when x < e and f(x) < 0 when x > e
hence local maximum
R1
AG
Note: Accept argument using correct second derivative .
1
y e
(iii)
A1
1
(1 ln x)2 x
x
x
1.
ln x
2
Consider the function f(x) = x , 0 < x < e .
(a)
(i)
Solve the equation f(x) = 0.
(ii)
Hence show the graph of f has a local maximum.
(iii)
Write down the range of the function f.
(5)
(b)
Show that there is a point of inflexion on the graph and
1.
A skydiver jumps from a stationary balloon at a height of 2000 m above the ground.
1
0.2t
Her velocity, v m s , t seconds after jumping, is given by v = 50(1 e ).
(a)
Find her acceleration 10 seconds after jumping.
(3)
(b)
How far above the ground is s