UNIVERSITIES OF** **MANCHESTER LIVERPOOL

LEEDS SHEFFIELD AND BIRMINGHAM

**JOINT
MATRICULATION BOARD**

**________________________________**

GENERAL CERTIFICATE OF** **EDUCATION

**MATHEMATICS—Paper
I**

ADVANCED

Friday 4 June 1965 9.30-12.30

**Careless and untidy ****work ****will be penalized.**

*Answer
***seven** *questions.*

**83 ADV** EE T*urn over*

**1.** (*a*) Solve for *x* and *y* the simultaneous
equations

_{} and _{}

where *a* and *b *are positive.

if *x *and *y* also satisfy the relation

_{}

prove that
_{}

(*b*)* *If
*x* is real and * *

_{}

show that *k *cannot lie between certain limits, and find these
limits.

**2.** (*a*) Express the following
complex numbers in the form a *+ ib, *where
*a* and *b *are real:

(i) _{}

(ii) _{}*, *where *c *is real.

(*b*)* *If:
_{}* *and _{}*, *where *x***, ***y, a,
b *are rea1, prove that

_{}

By solving the
equation

_{}

for _{}*, *or otherwise. express each of
the four roots of the equation in the form *x+ iy.*

83 ADV.

**3**. (*a*) Find (i) the sum and (ii) the product of the first *n *terms
of the geometric sequence 18, 6, 2,...

(*b*)* *If _{} , show that _{} provided

that *y* is either positive or less than -1.

Use the
expansions of _{} and _{}to show that

_{}

for *y *> 0 and for *y* < -1.

From the first three terms of this series, calculate _{} giving your answer to three decimal places.

**4.** (a) Prove that, in the triangle *ABC.*

* *_{}

Show that, if cot *A.
*cot *B, *cot *C *are in arithmetic progression, then so are a^{2},
b* ^{2}, *c

(*b*)* *Find
the solutions between -180^{0} and 180^{0} of the equation

_{}

giving your answers correct to the nearest tenth of a degree.

83 ADV *Turn over*

**5.**
(a) Show that the equation of the tangent at the origin to the circle * ** _{}*is

The circle * _{}*cuts
a circle

point (2, 0), find its equation. -

(*b*)* *Show
that the equation of the tangent to the parabola _{} at the’ point * *_{}is

_{}

Find the equation of the common
tangent to the parabolas

_{} and _{}.

**6.** Show that the equation of the
normal to the ellipse

_{}

** **

at the point * *_{} is

* *_{}

If *J* and *K *are the feet of the
perpendiculars from the centre *O *to the normal and the tangent at *P *respectively.
find the lengths of *OJ *and *OK. *and deduce that the area of the
rectangle *OJPK *is

* _{}*

If the normal meets *Ox *at
*G* and *Oy *at *H. *show that the locus of the mid-point of *GH
*as _{} varies is an ellipse.

**7.** (*a*) Find the values of *x* for
which the function

_{}

has
its maximum and minimum values.

Find
the coordinates of the maximum and minimum points, and of the point of
inflexion on the graph of the function. Sketch the graph.

(*b*)* *Find the volume generated by the rotation
through one revolution about the x-axis of the region between the x-axjs and
that part of the curve _{} for which _{}and *y *> 0.

8. (a) A tower stands on level ground. A man
observes that the top of the tower is at an elevation _{} to the horizontal from a fixed point on the ground. He
walks a distance *x *straight towards the tower, and finds that the
elevation is now_{}.
Express the height *h *of the tower in terms of _{}, *x*
and _{}.

Find *dx/d*_{}* *in
terms of *h *and _{} and show that, if _{} increases by _{}** **as *x*** **increases by *z*. where _{} and *z* are small, then

_{}

(*b*)* *A curve is described by the equation

_{}

Obtain
an expression for *dy/dx.*

If (*a*, *b*)* *is the point of
the curve where *dy/dx *= 0, prove that _{}* *and
hence find *a* and *b.*

**9. **(a) Prove that

_{}

(*b*)** **Evaluate

_{}

(*c*)* *By means of the substitution _{}*, *or otherwise, evaluate

_{}