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7. A circular target consists of a circular bull's-eye of two feet

radius, surrounded by three rings, each of uniform width. The areas of the bull's-eye and the rings are all the same.

Find the radial width of each of the rings. 8. If the length of sto of the circumference of a sphere is 69

miles, find the area of the surface in square miles. 9. The radius of a spherical iron ball, supposed to be solid, is 6

inches, and its weight is 228 pounds. If iron is bulk for bulk seven times as heavy as water, and one cubic foot of water weighs 624 lbs., find the size of the flaw in the ball. Take 7 as equal to 3?].

ALGEBRA.

TWO HOURS AND A HALF.

1. Simplify the following, without actual multiplication

✓ {(a+b)(-y)-2(a2-62) (x2-y)+(a-6)(x+y).

a

[ocr errors]
[ocr errors]

(1

[ocr errors]
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2. Given

+
btcx-b

x remainder;
72
да - да

78

C.C prove that this remainder is

74 3. Solve the equations (i.)

1 *-xy=15,

yaxy=10. (ii.)

x-xy=2a, 1 y2 + xy=3a.

2+2y+3=0, (iii.) 4x-3y+z=0,

x?+y++z=3. 4. Extract the square root of 30—12 v6. Also find the rational

part of the cube root of 99—70 V2, having given that the

irrational part of the cube root is –2 V 2. 5. If p arithmetic means are inserted between a and l, find the

qth of those means, and shew that the pth of q means, subtracted from the qth of p means, gives a remainder (l-a (1-P) (+p+1)= ((+1)(p+1)}.

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6. If sty+z=0, prove that the three symmetrical quantities, of

which y2 + z2 — 22 is one, are inversely proportional to

X, y and z.

а

7. If a rational, integral function of x vanishes when x=a, it is

divisible by x—a without remainder. Prove this theorem,

and explain why the words in italics are necessary.
8. P bicycles from A towards B at twelve miles an hour, but

breaks down and, at once going on, completes his journey
at two miles an hour. Q meanwhile walks from B to X
at four miles an hour. If P and Q both start and finish
simultaneously, find how far apart they were when the
breakdown occurred, the whole distance from A to B
being fifteen miles.

GEOMETRY.

TWO HOURS AND A HALF.

1. The angles at the base of an isosceles triangle are equal to

one another, and, if the equal sides are produced, the

angles on the other side of the base are also equal. 2. Describe a triangle, having given the base, the difference of

the other sides and the difference of the angles at the

base.
3. If a triangle is constructed having two sides equal to the

diagonals of a quadrilateral and the included angle equal
to the angle between these diagonals, the triangle and

quadrilateral are equal in area.
4. Divide a straight line so that the rectangle contained by the

whole and one part may be equal to the square on the

other part.

5. If a straight line is divided as proposed in the last question,

shew that the rectangle contained by the sum and difference
of the parts is equal to the rectangle contained by the

parts.
6. If a straight line touches a circle, and from the point of contact

a straight line is drawn cutting the circle, the angles
between the tangent and secant are equal to the angles in
the alternate segments of the circle.

7. The circle DAE touches the circle BAC internally at A, and the

chord BC of the circle BAC touches the circle DAE at D.

Shew that the angle BAD is equal to the angle CAD. 8. Inscribe a regular pentagon in a given circle. 9. Prove that any equilateral figure which is inscribed in a circle

is also equiangular.

TRIGONOMETRY:

TWO HOURS AND A HALF.

A +–)(0–7.

tan

=

1. Prove the formula for expanding sin(A+B) and simplify

sin(A +90°), cos(--A—180°), tan 2. In any triangle the sines of the angles are proportional to the

opposite sides. 3. If AD be drawn perpendicular to BC in the triangle ABC,

prove that BD-CD=a cosec A sin(C-B). 4. Simplify

1+sin 2A +cos 4A+sin 6A
cos 2A +sin

4A
5. Find tan 30°, tan 15°, tan 22°30'.
6. Solve the equations

(i.) tan(+a)=cot(0-3). (ii.) cos 0+cos 20=

1

V2

7. Prove the formula which expresses the cosine of the halfangle of a triangle in terms of the sides, and prove that

? =.

becog24

++etc=s.

8. Two rocks bear E. and N.E. from a ship. A chart shews that

one rock is 2 miles S.E. of the other. How far is the

ship from the rocks? 9. Find the approximate area of the disc of water visible from a

balloon one mile above the water in mid-ocean, neglecting refraction, and taking the earth-radius as 4000 miles.

W. E. SMITH LTD..

PRINTERS TO THE UNIVERSITY OF SYDNEY,

Bridge STREET, SYDNEY.

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