**M.B**

** I**

(O
25*, *B,I)

UNIVERSITIES
OF MANCHESTER LIVERPOOL

LEEDS
SHEFFIELD AND BIRMINGHAM

JOINT
MATRICULATION BOARD

GENERAL
CERTIFICATE OF EDUCATION

**MATHEMATICS (O 25)**

**SYLLABUS B, PAPER I**

ORDINARY

Monday
17 June 1963 9-30—12

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

*Mathematical tables and one sheet of graph paper will be
provided.*

*Answer ***all ***questions in Section ***A ***and any ***three ***questions from Section *B.

*In each question necessary details of working, including
rough work, must be shown with the answer *

**Section
A**

**A 1**. (a) To build a motorway costs half a million pounds per mile. Find the
cost per yard, in pounds, correct to the nearest pound.

*(b*)* *Solve
the equation

_{}

** **

(*c) *In
a triangle *ABC, AB = *AC = 20 cm.

*BC *=
32 cm. Calculate _{}*C.*

**[Turn over**

** A 2**. (*a) *Find the angle whose sine is

_{}

(*b*)* *Calculate the side of a regular polygon
of 40 sides which is inscribed in
a circle of radius 8 cm.

(*c*)* *Find *x*, where _{}

**A 3**. (*a*)* *The angle *A *is obtuse and

_{}

Without using tables, find tan *A.*

(*b*)* *Simplify

_{}

(*c*)* *A
tangent from a point *P *to a circle with centre *O *touches the
circle at *T. *The line joining *O *to *P *cuts the circle at *S
*and _{}*TPS
*= 20°.
Calculate the obtuse angle of triangle *TPS.*

**A 4. **(*a*)* *Through how many degrees does the hour
hand of a clock rotate in *x* minutes?

(*b*)* *Four
angles of a pentagon are 30°,. 88°, 112° and 145*°. *Find the fifth angle.

(*c*)* *If _{} , and *y *= 0 when *x* = -1, find *y *in terms of *x*.

**A 5**. (*a) *An isosceles
right-angled triangle is equal in area to a circle whose radius is _{} in. Calculate the length, of one of the equal
sides of the triangle, taking _{} as _{} .

(*b*)* *A
man is allowed two-ninths of a sum of money tax-free and pays tax at 7s. 9d. in the £ on the remainder.
Find the sum if the tax payable is £217.

**A 6**., (*a*)* *. Calculate the largest angle of a triangle with sides 4, *5 *and 6
cm.

* *

.
(*b*)* *A
quantity *h* is equal to _{} Find the. percentage increase in *h *when
*a *is increased by 25 per cent, *b *by 8 per cent and *c *by 20
per cent..

**Section
B **

*Answer ***three** *questions from this section.*

**B 7.**

The inscribed circle touches *AB *at *D. *Write,
down the value of each of *AD *and *DB *in terms of the radius of
this circle and hence calculate the radius.

**B 8**. *ABCD *is a parallelogram and X, *Y *are the mid-points of *AD,
BC *respectively. *AC *meets *BX *at *P *and *XY *at *Q**.*

(i) Prove that *P *is a point of trisection of *AC.*

(ii) Find the areas of *QYC, ABP *and *PBYQ *as
fractions of the area of *ABC.*

**[Turn over**

** B 9. **Draw the graph of
_{}* *from
*x* = 1 to

* x*= 5*, *taking 1 in. as the
unit of both *x* and *y.*

From
your graph estimate two roots of the equation ** **** _{}. **Calculate the gradient of the

** B** **10. **A factory makes cylindrical pencils 9
in. long and of radius _{} in. The graphite core is cylindrical, of
radius _{} in., and it is surrounded by wood. During
manufacture there is a wastage of 12 per. cent of graphite and 20 per cent of
wood. Taking _{} as _{}, find how many pencils may be made from 50 cu.** **ft. of graphite and
calculate the volume of wood required.

**B 11. **A 100 h.p. racing car travels at
a steady speed of 40 metres per sec. in a race of 300 miles. Taking 8 km. as *5
*miles, calculate the time in hours and minutes to complete the course.

Calculate the time taken for a
200 h.p. car to complete the course, assuming that the speed is proportional to
the square root of the horse-power. Give your answer to the nearest minute.