**P. ****MATHS.**

**w. ****MECHS. ****I**

(0 26 1)

UNIVERSITIES OF
MANCHESTER LIVERPOOL

LEEDS SHEFFIELD AND
BIRMINGHAM

JOINT MATRICULATION
BOARD

GENERAL CERTIFICATE OF
EDUCATION

**PURE MATHEMATICS WITH
MECHANICS (0 26), PAPER 1**

ORDINARY

Wednesday
26 June 1963 9-30—12

**Negligently
presented or slovenly work will be penalized.**

*Mathematical
tables (green covers) will be provided.*

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

**1**. Three
motorists *A, B *and *C *start at the same time from a town X and
travel with constant speeds *V _{1} *m.p.h.,

Show that, if *D *meets *A, B *and *C *at
times that are in arithmetic progression, then *V _{1}, V_{2 }*and

Calculate the speeds of all four motorists if the
distance between *X* and *Y *is 70 miles and if *D *meets *A,
B *and *C 50, *60 and 70 min. respectively after leaving *Y.*

**[Turn over**

2

**2.** (a) Show that the term independent of x in the

expansion
of _{}is _{}

(*b) *Show
that, if _{} where * *_{}* *and *x* is not equal to 1, then

_{}

__ __

**3**. Prove that, if _{} then

_{} and _{}

Show that the
equation _{}* *has two unequal roots between 0 and * *2_{}if _{}

Show that in this case, if _{}and _{}are the roots of this equation,
then

* *

(i) _{}

(ii) _{}

**4. **Show that the straight lines whose equations are _{},
_{} and _{}* *form a right-angled triangle and find the
coordinates of the vertices of this triangle.

Show also that if
the triangle has an area of *5 *sq. units then _{}

3

**5***. *Find the equation of the straight line which intersects
the coordinate axes at (*a*, 0) and (0, *b).*

The straight line
_{} meets the axes of *x* and *y* in *A
*and *B *respectively, while the straight line _{}meets
these axes in *C *and *B *respectively. The line joining *0, *the
origin, to the mid-point of *CB *meets *AB *in *P, *while the
line drawn from *0 *perpendicular to *CB *meets *AB *in Q. Find the equations of the lines *OP,
OQ *and prove that *PQ *is of length _{} units.

**6. **The point** ***0 *is the
centre of the base of a right circular cone *C _{1 }*of given
volume

[The volume of a right circular cone of height *h
*units and base radius *r *units is
_{} cu. units.]

**[Turn
over**

*4*

**7**. (*a*) Find, from first
principles, the differential coefficient with respect to *x* of _{}.

*(b) *The
distance *s** *moved in a straight line by a particle in time *t *is given by * ** _{}, *where a,

**8**. Sketch the curve _{}, where *a* is a
positive number.

Find the area bounded by this
curve, the line *x* = 4*a*, and the coordinate axes.

Show also that the volume of the
solid of revolution generated when this area is rotated about the *y*-axis
is _{}cu. units.