UNIVERSITIES
OF MANCHESTER LIVERPOOL

LEEDS
SHEFFIELD** **AND BIRMINGHAM

Joint
Matriculation Board

General
Certificate of Education

**FURTHER
MATHEMATICS**

SPECIAL PAPER

FRIDAY**
**21 June 1963, 9.30-12.30

**Negligently presented or
slovenly work will be** penalized. *Answer ***six** *questions.*

**1.** (*a*)
The complex numbers _{} are represented by the points *A, B, *C
respectively in the Argand diagram. If

_{}

show that *AB *and *AC *are equal and perpendicular to each
other.

(*b*)* *State
and prove de Moivre’s theorem in the case when the index *n *is a positive
integer.

Show that the
roots of the equation

_{}

are _{} where *r *= 1.3,5,7,11,13, 15, 17.

Deduce that

_{}

**177 SP**. E *Turn over* * *

**2**. Find the number of ways in which *n *identical coins can be
placed (i) in two different boxes, (ii) in three different boxes, all the coins
being used and empty boxes being allowed in each case. Show that the number of
ways in which the coins can be placed in four different boxes is

_{ }

1f ** **_{n}*F _{r}*

* *_{n}*F _{r }=_{
n-1}F_{r }*

Use this relation to construct a table of values of _{n}*F _{r}* for

**3. **If
*s *is the length of arc of a curve *y *= *f(x) *measured from
an arbitrary point on it, show that

* _{}*

In a certain curve which passes
through the origin the length of arc satisfies the relation

_{}, (*a* > 0).

Show that the curve must lie between the lines *x*
= 0 and *x *= *a *and that it can be represented parametrically by
the equations

_{}*.*

* _{}*

Sketch the
complete curve in the range _{}

If
the curve makes a complete revolution about the line *x* = *a***, **show
that the area of the surface generated is * _{}*.

**177 SP.**

**3**

**4. **A
line of gradient** ***k *passes through the point *A*(0, *a*)
and cuts the fixed line *y *= *c, *where *c > *0, in the
point *P. *Find the coordinates of *P. *The line through *A *of
gradient** **1*/k *cuts the line *OP *in the point Q, O* *being
the origin. Show that for all values of *k *the point Q lies on the curve
whose equation is

* _{}*.

Describe how this curve changes
its character as the point *A *takes successive positions on the positive
part of the y-axis, stating the name of the curve for all values of *a* *>
*0, and for** ***a*** **= 0. Show in particular that when a = 5*c
*the curve is an ellipse with semi-axes *5c, *5c/2.

*5. *A body *A *of mass *M *is
connected by a light inextensible string to a vessel *B *containing
liquid, the mass of the vessel and liquid being *M*** **at time *t*
= 0. The string passes over a fixed smooth peg below which *A *and *B *hang
and the system is initially at rest. The liquid evaporates in such a way that
its mass decreases at a constant rate _{} per unit time. Find the acceleration of the
system at time *t, *and deduce that its speed is then

_{}.

Obtain an
approximation to this expression when _{}*t/M *is small, neglecting powers of *t
*higher than the second. Deduce that the distance through which *A* has
fallen at time t is *at*^{3 }approximately. where *a* is a
constant, and find the value of *a* in terms of *M, *_{}. and *g. *If the mass
which evaporates in one day is _{}, find the approximate distance
through which *A* has fallen at time t = 60 sec., taking the value of g to
be *32 *ft./sec^{2}.

**6.
**A particle** ***P*
of mass *m *moves in the *x,* *y*-plane under the action of
forces **R** and **S** which are respectively tangential and normal to
its path. These forces have magnitudes _{}and _{}* *respectively,
where _{}* *is
the speed of *P*, and** ***k, A *are constants. The tangential
force **R** opposes the motion and **S** is directed so that rotation
through a right angle from **R** to **S** is in the same sense as that
from the *x*-axis to the *y*-axis. Show that if *u, v *are the *x*-
and *y*-components of the velocity of *P *at time t, the equations of
motion are

* ** _{}*,

Deduce that

_{}

If
*u *= 0**, ***v *= *v _{0 }*when

_{}, _{}

7. Two particles move in a plane under the action only of the mutual
force between them. Prove that the sum of the components of their momenta in
any direction is constant.

Two atomic nuclei of
masses 3*m *and *m *move along perpendicular lines AO, BO with speeds
*u. *2*u *respectively and collide at O. On collision they are
transformed into two nuclei of masses *m* and 4*m *which move with
speeds _{},* *

_{}respectively along a line
_{}, where *OX* lies in
the right angle A*0B. *Show that tan AOX = 4/3*. *If at the collision
the total kinetic energy is decreased by_{}*, *find the possible values of _{}*, *_{}*.*

Show
also that in all cases in which the transformed nuclei move along _{} after the collision, the loss of energy at the
collision cannot exceed 3*mu ^{2}.*

177 **SP**.

* *

**8.** The
angular momentum *h *about the origin of a particle *P *of mass *m *moving
in a plane is given by

_{}

where *x*, *y* are the coordinates of *P *at*
*time *t. *Deduce that the moment about the origin of the resultant
force acting on *P *is equal to *dh/dt.*

A uniform rod of mass *M *and
length 2*a* is freely pivoted at its centre to a fixed point, and is
initially at rest in a horizontal position. A particle of mass *m. *released
from rest at a height *c *above the rod, strikes it at a distance *x*
from the pivot and adheres to it. Find the angular velocity of the rod
immediately after the impact. Show that the rod will subsequently make complete
revolutions if

* _{}* (1)

Show that this condition cannot de satisfied for any
value of *x* unless _{}.

If
*M > 3m, *show that the condition (1) cannot be satisfied unless* ** _{}*.

177
**SP**