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120

Marks total

Chapters covered

OCR (Cambridge University Press & Assessment) · United Kingdom

STEP 3 1996

1996

14 questions worth 120 marks in total. Each part is annotated: check the hint if you are stuck, or reveal the worked solution once you have committed to an approach.

Define $cosh x$ and $sinh x$ in terms of exponentials and prove, from your definitions, that $cosh^{4} x - sinh^{4} x = cosh 2 x$ and $cosh^{4} x + sinh^{4} x = \frac{1}{4} cosh 4 x + \frac{3}{4} .$ Find $a_{0}, a_{1}, \dots, a_{n}$ in terms of $n$ such that $cosh^{n} x = a_{0} + a_{1} cosh x + a_{2} cosh 2 x + \dots + a_{n} cosh n x .$ Hence, or otherwise, find expressions for $cosh^{2 m} x - sinh^{2 m} x$ and $cosh^{2 m} x + sinh^{2 m} x,$ in terms of $cosh k x,$ where $k = 0, \dots, 2 m .$
Trigonometry
For all values of $a$ and $b,$ either solve the simultaneous equations $x + y + a z x + a y + z 2 x + y + z = 2 = 2 = 2 b$ or prove that they have no solution.
Proof & Number Theory
Find $\int_{0}^{θ} \frac{1}{1 - a cos x} d x,$ where $0 < θ < π$ and $- 1 < a < 1.$

Hence show that $\int_{0}^{\frac{1}{2} π} \frac{1}{2 - a cos x} d x = \frac{2}{4 - a ^{2}} tan^{- 1} \frac{2 + a}{2 - a},$ and also that $\int_{0}^{\frac{3}{4} π} \frac{1}{2 + cos x} d x = \frac{π}{2} .$
Integration
Find the integers $k$ satisfying the inequality $k ⩽ 2 (k - 2) .$

Given that $N$ is a strictly positive integer consider the problem of finding strictly positive integers whose sum is $N$ and whose product is as large as possible. Call this largest possible product $P (N) .$ Show that $P (5) = 2 \times 3, P (6) = 3^{2}, P (7) = 2^{2} \times 3, P (8) = 2 \times 3^{2}$ and $P (9) = 3^{3} .$

Find $P (1000)$ explaining your reasoning carefully.
Proof & Number Theory
Show, using de Moivre's theorem, or otherwise, that $tan 7 θ = \frac{t ( t ^{6} - 21 t ^{4} + 35 t ^{2} - 7 )}{7 t ^{6} - 35 t ^{4} + 21 t ^{2} - 1},$ where $t = tan θ .$
By considering the equation $tan 7 θ = 0,$ or otherwise, obtain a cubic equation with integer coefficients whose roots are $tan^{2} (\frac{π}{7}), tan^{2} (\frac{2 π}{7}) and tan^{2} (\frac{3 π}{7})$ and deduce the value of $tan (\frac{π}{7}) tan (\frac{2 π}{7}) tan (\frac{3 π}{7}) .$
Find, without using a calculator, the value of $tan^{2} (\frac{π}{14}) + tan^{2} (\frac{3 π}{14}) + tan^{2} (\frac{5 π}{14}) .$
Complex Numbers
Let $S$ be the set of matrices of the form $(a a a a),$ where $a$ is any real non-zero number. Show that $S$ is closed under matrix multiplication and, further, that $S$ is a group under matrix multiplication.
Let $G$ be a set of $n \times n$ matrices which is a group under matrix multiplication, with identity element $E .$ By considering equations of the form $BC = D$ for suitable elements $B,$ $C$ and $D$ of $G$ , show that if a given element $A$ of $G$ is a singular matrix (i.e. $det A = 0$ ), then all elements of $G$ are singular. Give, with justification, an example of such a group of singular matrices in the case $n = 3.$
Proof & Number Theory
If $x + y + z = α,$ $x y + y z + z x = β$ and $x y z = γ,$ find numbers $A, B$ and $C$ such that $x^{3} + y^{3} + z^{3} = A α^{3} + B α β + C .$ Solve the equations $x + y + z x^{2} + y^{2} + z^{2} x^{3} + y^{3} + z^{3} = 1 = 3 = 4.$
The area of a triangle whose sides are $a, b$ and $c$ is given by the formula $area = s (s - a) (s - b) (s - c)$ where $s$ is the semi-perimeter $\frac{1}{2} (a + b + c) .$ If $a, b$ and $c$ are the roots of the equation $x^{3} - 16 x^{2} + 81 x - 128 = 0,$ find the area of the triangle.
Coordinate Geometry & Conics
A transformation $T$ of the real numbers is defined by $y = T (x) = \frac{a x - b}{c x - d},$ where $a, b, c$ , $d$ are real numbers such that $a d \neq = b c$ . Find all numbers $x$ such that $T (x) = x .$ Show that the inverse operation, $x = T^{- 1} (y)$ expressing $x$ in terms of $y$ is of the same form as $T$ and find corresponding numbers $a^{'}, b^{'}, c^{'}$ , $d^{'}$ .

Let $S_{r}$ denote the set of all real numbers excluding $r$ . Show that, if $c \neq = 0,$ there is a value of $r$ such that $T$ is defined for all $x \in S_{r}$ and find the image $T (S_{r}) .$ What is the corresponding result if $c = 0$ ?

If $T_{1},$ given by numbers $a_{1}, b_{1}, c_{1}, d_{1},$ and $T_{2},$ given by numbers $a_{2}, b_{2}, c_{2}, d_{2}$ are two such transformations, show that their composition $T_{3},$ defined by $T_{3} (x) = T_{2} (T_{1} (x)),$ is of the same form.

Find necessary and sufficient conditions on the numbers $a, b, c, d$ for $T^{2}$ , the composition of $T$ with itself, to be the identity. Hence, or otherwise, find transformations $T_{1}, T_{2}$ and their composition $T_{3}$ such that $T_{1}^{2}$ and $T_{2}^{2}$ are each the identity but $T_{3}^{2}$ is not.
Complex Numbers
A particle of mass $m$ is at rest on top of a smooth fixed sphere of radius $a$ . Show that, if the particle is given a small displacement, it reaches the horizontal plane through the centre of the sphere at a distance $a (55 + 423) /27$ from the centre of the sphere.

[Air resistance should be neglected.]
Mechanics
Two rough solid circular cylinders, of equal radius and length and of uniform density, lie side by side on a rough plane inclined at an angle $α$ to the horizontal, where $0 < α < π /2$ . Their axes are horizontal and they touch along their entire length. The weight of the upper cylinder is $W_{1}$ and the coefficient of friction between it and the plane is $μ_{1}$ . The corresponding qunatities for the lower cylinder are $W_{2}$ and $μ_{2}$ respectively and the coefficient of friction between the two cylinders is $μ$ . Show that for equilibrium to be possible:
$W_{1} ⩾ W_{2}$ ;
$μ ⩾ \frac{W _{1} + W _{2}}{W _{1} - W _{2}}$ ;
$μ_{1} ⩾ (\frac{2 W _{1} cot α}{W _{1} + W _{2}} - 1)^{- 1} .$
Find the similar inequality to (iii) for $μ_{2}$ .
Mechanics
A smooth circular wire of radius $a$ is held fixed in a vertical plane with light elastic strings of natural length $a$ and modulus $λ$ attached to the upper and lower extremities, $A$ and $C$ respectively, of the vertical diameter. The other ends of the two strings are attached to a small ring $B$ which is free to slide on the wire. Show that, while both strings remain taut, the equation for the motion of the ring is $2 ma \ddot{θ} = λ (cos θ - sin θ) - m g sin θ,$ where $θ$ is the angle $∠ C A B$ .

Initially the system is at rest in equilibrium with $sin θ = \frac{3}{5}$ . Deduce that $5 λ = 24 m g$ .

The ring is now displaced slightly. Show that, in the ensuing motion, it will oscillate with period approximately $10 π \frac{a}{91 g} .$
Mechanics
It has been observed that Professor Ecks proves three types of theorems: 1, those that are correct and new; 2, those that are correct, but already known; 3, those that are false. It has also been observed that, if a certain of her theorems is of type $i$ , then her next theorem is of type $j$ with probability $p_{ij}^{()},$ where $p_{ij}^{()}$ is the entry in the $i$ th row and $j$ th column of the following array: $0.3 0.2 0.1 0.3 0.4 0.3 0.4 0.4 0.6 .$ Let $a_{i},$ $i = 1, 2, 3$ , be the probability that a given theorem is of type $i$ , and let $b_{j}$ be the consequent probability that the next theorem is of type $j$ .
Explain why $b_{j} = a_{1}^{()} p_{1 j}^{()} + a_{2}^{()} p_{2 j}^{()} + a_{3}^{()} p_{3 j}^{()} .$
Find values of $a_{1}^{()}, a_{2}^{()}$ and $a_{3}^{()}$ such that $b_{i} = a_{i}$ for $i = 1, 2, 3.$
For these values of the $a_{i}$ find the probabilities $q_{ij}^{()}$ that, if a particular theorem is of type $j$ , then the preceding theorem was of type $i$ .
Probability & Statistics
Let $X$ be a random variable which takes only the finite number of different possible real values $x_{1}, x_{2}, \dots, x_{n} .$ Define the expectation $E (X)$ and the variance $var (X)$ of $X$ . Show that , if $a$ and $b$ are real numbers, then $E (a X + b) = a E (X) + b$ and express $var (a X + b)$ similarly in terms of $var (X)$ .

Let $λ$ be a positive real number. By considering the contribution to $var (X)$ of those $x_{i}$ for which $∣ x_{i} - E (X) ∣ ⩾ λ,$ or otherwise, show that $P (∣ X - E (X) ∣ ⩾ λ) ⩽ \frac{var ( X )}{λ ^{2}} .$ Let $k$ be a real number satisfying $k ⩾ λ .$ If $∣ x_{i} - E (X) ∣ ⩽ k$ for all $i$ , show that $P (∣ X - E (X) ∣ ⩾ λ) ⩾ \frac{var ( X ) - λ ^{2}}{k ^{2} - λ ^{2}} .$
Probability & Statistics
Whenever I go cycling I start with my bike in good working order. However if all is well at time $t$ , the probability that I get a puncture in the small interval $(t, t + δ t)$ is $α δ t .$ How many punctures can I expect to get on a journey during which my total cycling time is $T$ ?

When I get a puncture I stop immediately to repair it and the probability that, if I am repairing it at time $t$ , the repair will be completed in time $(t, t + δ t)$ is $β δ t .$ If $p (t)$ is the probability that I am repairing a puncture at time $t$ , write down an equation relating $p (t)$ to $p (t + δ t)$ , and derive from this a differential equation relating $p^{'} (t)$ and $p (t) .$ Show that $p (t) = \frac{α}{α + β} (1 - e^{- (α + β) t})$ satisfies this differential equation with the appropriate initial condition.

Find an expression, involving $α, β$ and $T$ , for the time expected to be spent mending punctures during a journey of total time $T$ . Hence, or otherwise, show that, the fraction of the journey expected to be spent mending punctures is given approximately by $\frac{α T}{2} if (α + β) T is small,$ and by $\frac{α}{α + β} if (α + β) T is large.$
Probability & Statistics