(b) If the product xyz be equal to unity, show that

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18. Two men A and B have the same total number of oranges and apples, but A has as many oranges as B has apples. All the fruit is sold, the oranges at 4 for 3d., the apples at 3 for 2d. On the whole they fetched £1 14s., but A has got 2d. more than B. Find how many oranges and how many apples A had at first.

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Stages 2, 3, and 4, and Honours in Division I.

INSTRUCTIONS.

You may take the Stage 2, or Stage 3, or Stage 4, or Honours, but you must confine yourself to one of them.

A table of logarithms must NOT be used in working these papers. All the logarithms needed will be found on page 58.

The figures in descriptive geometry should not only be constructed with ruler and compasses, but the construction should in all cases be explained and its accuracy demonstrated.

The examination in this subject lasts for three and a half hours.

Stage 2.

You are not to

The paper is divided into three sections. answer more than three questions in any one section, and not more thau eight questions altogether. No candidate will be allowed to pass who fails to obtain marks in any one of the three sections.

u 56905.

D

A.

21. Show how to divide a given straight line into two parts, so that the rectangle under the whole line and one part may be equal to the square on the other part.

Draw a line, say about 2 in. in length, and then, by rule and compasses, draw another line equal to three times the given line diminished by 5 times the given line.

(20) 22. Show that the tangent at any point of a circle and the radius through the point are perpendicular to one another.

Draw two circles with their centres 3 inches apart, the radius of one being 1 inches and of the other 1 inch. Find by a geometrical construction the points of contact of the four common tangents to these two circles.

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23. Show how to cut off from a given circle a segment containing a given angle.

ABC is a triangle inscribed in a circle, and the line bisecting the angle A cuts the circumference in D; show that BC is parallel to the tangent at D.

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24. Define supplementary angles, and show that the opposite angles of any quadrilateral inscribed in a circle are supplementary.

Show also that the angles subtended at the centre of a circle by two opposite sides of a circumscribed quadrilateral are supplementary.

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25. Draw a triangle ABC having a right angle at C, and, without drawing the circle, mark the points in which AB will be cut by the circle which passes through C and through the middle points of BC and CA. Prove that you have marked the points correctly.

(20)

26. A and B are given points on the same side of a given straight line CD; show how to find a point P in CD so that the angle APB may be the greatest possible.

(25)

28. Show that the square root of an expression of the form a+b, where b is a surd, can be found as the sum of two simple surds, if a2 — b is a square number.

Find the square root of 112/30, and show that

find

(b) Three consecutive numbers are multiplied together;
a number which, added to the product, will make the sum
Verify your result when the three numbers

a perfect cube.

are 17, 18, 19.

and show that a root of (i) is the cube of a root of (ii).
(b) Show that the difference of the roots of the equation
x2 + px + q = 0

is equal to the difference of the roots of the equation
x2 + 3px + 2p2 + q = 0.

(20)

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31. A man invests 2,000l. in Consols; afterwards he sells part of them at an advance of 27. per 1007. of stock and thereby realises 1,8047. He has 3007. of stock left; at what price per 1007. of stock did he make the original investment?

(25)

32. (a) Show that the difference between the square of any number of two digits, and the square of the number obtained by reversing the digits, is 99 times the difference of the squares of the digits.

(b) Solve the simultaneous equations :

{9x2 + bxy - 4y2 = 1;

3x - 2y + 1 = 0.

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C.

33. (a) Define the logarithm of a number to a given base, and write down (i) the logarithm of 16 to the base 2, (ii) the logarithm of 3 to the base 9.

(b) Prove that

log mn = log m + log n, and that log (m3) = 3 log m. Using logarithms given in the Table on page 58, find

the numerical value of a when 10′′ = 620.

(c) Calculate the numerical value of

25

27 × (14·4).

(20)

34. (a) Define a radian, and state what is the value of a right angle in radians. Calculate the value of 132° 20′ 10′′ in

(b) Find the values of A, between 0° and 90°, which satisfy

the equation

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Calculate the

36. (a) By ruler and compasses construct an angle whose cosine
is also an angle whose cosine is
:
sine of the latter angle to three places of decimals.
(b) Show that the numerical value of

diminishes from 2 to 0, as A increases from 0° to 180°,
and illustrate your answer by a diagram.

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37. (a) In a triangle ABC the lines drawn from A and C. perpendicular to the opposite sides, intersect in O. If the angle A is acute, show that

Also draw a diagram in which A is an obtuse angle, and establish the corresponding expression for OA in that

case.

(b) Show that in any triangle the product of a side and the
sines of the two adjacent angles is the same, whichever
side be taken.

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38. When two sides of a triangle, and the angle they include are given, show how to calculate the remaining angles of the triangle.

If the two sides are 7,235 ft. and 4,635 ft. respectively and if the included angle is 78° 26', find the remaining angles of the triangle.

Stage 3.

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The paper is divided into three sections. You are not to answer more than three questions in any one section, and not more than eight questions altogether. No candidate will be allowed to pass who fails to obtain marks in any one of the three sections.

1

A.

41. A circle is drawn within the angle BAC, formed by two straight lines AB, AC. Show how to draw the circles which touch AB and AC, and with which the given circle has internal contact. Give a carefully drawn diagram.

(35) 42. Show how to divide a given straight line into parts having given ratios to each other.

Find by a geometrical construction the point which divides a given straight line in the ratio of 3 to 5.

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43. ABC, DEF are two triangles in which the angles ABC, DEF are equal, and the angles ACB, DFE are supplementary; show that AB: DE = AC : DF.

If two triangles have one angle of the one equal to one angle of the other, and the sides about one other angle in each proportional, so that the sides opposite equal angles correspond, show that the remaining angles are either equal or supplementary.

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44. Show that the areas of similar triangles are to one another in the duplicate ratio of their homologous sides.

Show that similar triangles are to each other in the same ratio as the areas of their inscribed circles.

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45. Given two right-angled triangles, show how to construct a right-angled triangle which shall have the same height as that of one of the given triangles, and whose area shall be a mean proportional to the areas of the given triangles.

(40) 46. A parallelogram ABCD is formed of four rods, freely jointed at the corners; the rods BA, BC are prolonged through A and C; O is a fixed point in the prolongation of BA, P is a point in AD, and Q is a point in the prolongation of BC, such that O, P, Q are in a straight line. Show that if P is made to describe any plane figure, Q will trace out a similar figure.

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B.

47. If m and n denote positive integers, state what is the meaning of a", and why am × an is equal to am+n.

What is the meaning of a? Explain how you are justified in giving it that meaning.

(a) Simplify

(a - b3 c − )*.

(b) Find the square root of

a2 + 4axa + 6a3 + 12a − § x + 9a
− x2 + 9a - 3 + 4x.

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