UNIVERSITIES OF** **MANCHESTER LIVERPOOL

LEEDS SHEFFIELD AND BIRMINGHAM

**JOINT
MATRICULATION BOARD**

**________________________________**

GENERAL CERTIFICATE OF** **EDUCATION

**FURTHER MATHEMATICS—Paper I **

ADVANCED

Wednesday 2 June 1965 2-5

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

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

**1**. (a) The *rth *term of a series is _{}. Prove by induction, or
otherwise, that the sum of the first *n *terms of the series is _{}.

(*b*) * *The *r*th term of a series is * *_{}

Find the sum of the
first *n *terms of the series and deduce the sum to infinity.

**2**. By considering the differential coefficient
of the function _{}*, *show that the equation _{} has just one real root. Determine the two
consecutive integers between which the root is situated.

If _{} are the roots of the equation find the value
of

(i) _{}, (ii) _{}.

**85 ADV. ****BB** *Turn over*

**3**. (*a*) Find the modulus *r *and the argument _{} of

(i) _{}* * (ii)_{} _{}.

(*b*)* *The points *A, B, P *in the Argand
diagram correspond to the complex numbers a, *b, z *which satisfy the
relation

_{}

where *k *is a constant. Give a geometrical interpretation of this
relation, and find the form of the locus of *P *as *z *varies

**4. **(*a*)
Evaluate

_{}

and deduce the limiting value of the integral when *a*
tends to infinity through positive values.

(*b*)* *A surface of
revolution is formed by rotating about the *x*-axis the part of the
parabola _{} between *t *= 0 and** **** _{}. **Find the area of the surface.

**5.** The
straight line whose equations are

_{}

meets the
plane _{} at *B, *and *A *is the point (2,
0,-1) on the line. The foot of the perpendicular from *A *to the plane is *C.
*Find (i) the coordinates of *B *and *C*,* *(ii) the length
of *AC. *Show that the sine of the acute angle between *BA *and *BC
*is _{}.

The line *AC *is produced
to *D *so that *AC *= *2CD. *Find the coordinates of *D.*

85 ADV

**6. **A
force has components** ***X, Y *parallel to rectangular axes *Ox *and
*Oy *respectively and *(x, y*) is a point on the line of action. Show
that the force is equivalent to an equal force at the origin *O *together
with a couple. State the moment of the couple.

The components of three forces
in the plane are given at time *t*** **by

* ** _{} *and

_{},

and their lines of action pass respectively through * O *and the points (*a*, -*a*) and
(-3*a*, 2*a*) where *P*,*
** _{} *and

**7.** The power necessary to drive a car of mass

1650 lb. along a horizontal road at a steady speed of

60 ft. per sec. is 10 h.p. At a steady speed of 20 ft. per sec. the
necessary power is 2 h.p. Assuming the

resistance to
motion at *v *ft. per sec. to be _{}* *lb.
wt.. find the values of the constants *a* and *b.*

If the engine is turned off when
the speed of the car on the road is 60 ft. per sec. and the resistance remains
the same function of *v *as before, find, to the nearest foot, the
distance travelled while the speed falls to 20 ft. per,sec.

(Assume
that g = 32 ft. per sec^{2}.)

85 ADV. *Turn ove*r

**3**

**8**. A particle of mass *m *is
tied to one end of an elastic string of natural length *a* and modulus _{};* *the other end of
the string is attached to a point of a fixed smooth horizontal plane. The
particle is released from rest on the plane when the string is extended to a
length 2*a*. (i) If the subsequent motion is unresisted find the speed of
the particle when the string becomes slack. (ii) If when the extension of the
string is *x* the motion of the particle is resisted by a force 2*mn*
times the speed, show that

_{} _{}

Find *x *as a
function of *t*,** **and show that the speed of the particle at the
moment the string becomes slack is * *_{}.

**9. **A uniform solid cone has mass *M*,
height *h* and base-radius *r. *Show
by integration that its moment of inertia about a line through the vertex and
parallel to the base is

_{}

(It may be assumed
that the moment of inertia of a thin circular disc about a diameter is

mass x** **(radius)^{2}/4^{
}.*)*

The cone can swing freely under
gravity about a fixed smooth pivot at the vertex. If the cone is released from
rest with its axis horizontal, find the vertical component of the force on the
pivot when the axis is vertical.

85 ADV.