Download the King’s Factor Year 13 further questions 4

the King’s Factor Year 13 further questions 4
1. Oxford and Cambridge Schools Examination Board Mathematics and Higher Mathematics
Paper II Pure Mathematics July 1965 question 3.
Prove that, if t = tan x and n ≥ 1,
d n
(t ) = n(tn−1 + tn+1 ) .
dx
Hence calculate the first five derivatives of tan x in terms of t and prove that
1
2
tan x = x + x3 + x5 + . . . .
3
15
2. [2007 STEP II question 3]
Z
By writing x = a tan θ, show that, for a 6= 0,
a2
1
1
x
dx = arctan + constant.
2
+x
a
a
π
2
Z
cos x
2 dx .
0 1 + sin x
i. Evaluate I.
(a) Let I =
ii. Use the substitution t = tan
Z
(b) Evaluate
0
1
1
x
2
Z
to show that
0
2
1−t
dt .
1 + 14t2 + t4
1
1
1
1 − t2
dt = I .
2
4
1 + 6t + t
2
the King’s Factor Year 13 further questions 4
3. [2007 STEP I question 9] [Mechanics]
A particle of weight W is placed on a rough plane inclined at an angle θ to the horizontal.
The coefficient of friction between the particle and the plane is µ. A horizontal force X acting
on the particle is just sufficient to prevent the particle from sliding down the plane; when a
horizontal force kX acts on the particle, the particle is about to slide up the plane. Both
horizontal forces act in the plane containing the line of greatest slope.
Prove that
(k − 1)(1 + µ2 ) sin θ cos θ = µ(k + 1)
and hence that k >
(1 + µ)2
.
(1 − µ)2
4. [2004 STEP I question 12] [Statistics]
In a certain factory, microchips are made by two machines. Machine A makes a proportion λ
of the chips, where 0 < λ < 1, and machine B makes the rest. A proportion p of the chips
made by machine A are perfect, and a proportion q of those made by machine B are perfect,
where 0 < p < 1 and 0 < q < 1. The chips are sorted into two groups: group 1 contains those
that are perfect and group 2 contains those that are imperfect.
In a large random sample taken from group 1 it is found that
Show that λ can be estimated as
2q
.
λ=
3p + 2q
2
5
were made by machine A.
Subsequently, it is found that the sorting process is faulty: there is a probability 41 that a
perfect chip is assigned to group 2 and a probability 14 that an imperfect chip is assigned to
group 1. Taking into account this additional information, obtain a new estimate of λ.
2