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SET I.

EUCLID.

1. Any two sides of a triangle are together greater than the third

side.

If D be a point on the base BC of the triangle ABC, prove that AD is less than the greater of the two sides AB and AC, and if E be a point on either of these two sides, prove that DE

is less than the greatest side of the triangle. 2. The complements of the parallelograms which are about the

diameter of any parallelogram, are equal to one another. 3. If a straight line be divided into two equal parts and also into two

unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of

section, is equal to the square on half the line. 4. Divide a given straight line into two parts, so that the rectangle

contained by the whole and one of the parts may be equal to the

square on the other part. 5. The angle at the centre of a circle is double of the angle at the

circumference on the same base, that is, on the same arc.

A triangle ABC is inscribed in a circle of which O is the centre and the arc BC is bisected at D.

Prove that the angle ADO is half the difference of the angles

ABC and ACB, 6. If from any point without a circle there be drawn two straight

lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on the line which meets the circle, the line which meets the circle

shall touch it. 7. Describe a circle about a given triangle.

Given the radius of the circumscribing circle, the length of one of the sides, and the area of the triangle, construct the

triangle. 8. Inscribe an equilateral and equiangular quindecagon in a given

circle.

or

9. The sides about the equal angles of triangles which are equiangular

to one another are proportionals; and those which are opposite to the equal angles are hoinologous sides, that is, are the antecedents the

consequents of the ratios. The chords AB and CD in the circle ABCD are produced towards B and D respectively to meet in the point E, and through E, the line EF is drawn parallel to AŰ to meet CB produced in F

Prove that EF is a mean proportional between FB and FC. 10. If four straight lines be proportionals the similar rectilinear

figures similarly described upon them shall be proportionals.

ALGEBRA.

1. When (m) and (n) are whole numbers, express the result of the

division of a" by a"; (1) when (m) is greater than (n); (2) when (m) is less than (n), and trace the steps in the theory

1 of indices from which it is inferred that a-" represents

and

ani

that añ represents ņā.

Express by an arithmetical fraction (88 +44) x 16% 2. Multiply (2} + 2y3 + 32}) by (z 2 – 2,1 323). 3. Divide (c + y+ (x + 2) + (y + 2) + 2(x + y) (x + 2)

+ 2 (2 + y) (y + 2) +2 (a + 2) (y+z) by (x + y + z). 4. Express a' (c – b) + b? (a – c) + c (b − a) as the product of three

simple binomial factors. 5. Reduce to its lowest terms

(a? b) (x? – ) – 4abxy
(a? 62) (ac2 + y2) + 2 (a? + b2) xy

b 6. Prove

+
(a - b)(c – a) (ac + a) (a - b)(b - c)(a + b)

+

(6 - 0) (0 - a) (x + c)

a

с

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=

(ac + a)(a + b)(a + c) 7. Show how to find a factor which will rationalise the surd quantity

1 1
ap + bo, where (p) and (9) are any whole numbers.

51 - 7
Express by an equivalent fraction with a rational

7.
denominator.
8. Solve the following equations :-

(1.) (x – a) (x – b) = (x – c) (x – d);
3 5

8
+
Y

15

Зху 9y – 22.

53

+ 79

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(2.)

25

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= 108

22 1 43x 8ac2
(3.)
6
7

42
aca (2C+y+z)
(4.) y(+ y + x) = 192

za (ac +y+z) 300) 9. What are the roots of a quadratic equation? If (a) and (8) are

the roots of the equation x? – px + 9 = 0, and (a) (B4) the roots of 22 - P: + Q = 0; express P and Q in terms of (p) and (9)

a

a
=
с C

10. Two workmen (A) and (B) of unequal efficiency, working together

are capable of making a drain, which they undertake to do, in 60 days. After working for 20 days A falls ill, and his place is supplied by another workman (C), whose efficiency is the same as that of B, and it is found that the whole time consumed in making the drain is 80 days. In how many days would A

or B working alone make the drain? 11. A cubical tank contains 512 cubic feet of water. It was required

to enlarge the tank, the depth remaining the same, so that it should contain seven times as much water as before, subject to the condition that the length added to one side of the base should be four times the length added to the other side. Find

the sides of the new rectangular base. 12. Define the three progressions, Arithmetical, Geometrical, Har

monical; and show that the three equations following severally
determine for the three quantities (a) (b) () the conditions of
each progression :
b
6

b
(1.)
; (2.)

; (3.)
b

b
b

b

1
Insert four harmonic means between and 2.

3
A watch, which is set right at noon, gains two minutes the
first hour afterwards, three the second, four the third, and so
on; after what interval will the watch be an hour and a half

fast, and what time will it then indicate ? 13. Assuming the expression for the number of permutations of (n)

things taken (r) together, find the expression for the number of combinations of (n) things taken (r) together.

A dealer has for sale 8 bay, 7 grey, and 5 black horses. A purchaser requests that 12 horses, four of each colour, may be sent him; in how many different ways can the dealer execute

the order? 14. Assuming the form of the expansion of a binomial when the index is a positive integer, and that f(m) denotes the series

m (m – 1) 1+ mx +

ac? + &c. for any value of (m); show how

1.2 to obtain a proof of the theorem when the index is a negative integer.

Since (1. - )" (1 2) ". 1, it would follow that if the two expanded series for (1 - )" and (1 - 2)-" be multiplied together all the coefficients of the different powers of (a) should vanish separately; show that this is the case for the whole coefficient of x*.

PLANE TRIGONOMETRY.

cos 135o.

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1. What in plane geometry is the limit of the magnitude of an angle?

In what sense in analytical Trigonometry is the magnitude of
an angle considered unlimited, and either positive or negative?
Which of the ordinary trigonometrical functions are limited in
magnitude and which unlimited ?

If sin A = a, express by a general formula all the angles

which have (a) for their sine ; if cos A b, prove a + b2 = 1. 2. Find sin A and cos A in terms respectively of the sines and

cosines of their supplements. Through what portions of the
first four quadrants is the sine greater than the cosine in
magnitude, and the tangent greater than the cotangent; when
are they respectively equal and of the same sign?
Find sin 120°, tan 60°,

tan A
Prove sin A

1+ (tan A) 3. Prove cos (A + B) = cos A cos B sin A sin B, and deduce

from the formula the corresponding expressions for sin (A+B),

sin (A – B), and cos 2A. 4. Prove

Sin A + sin B A +B
(1.)
Cos A + cos B

2

15-1
(2.) Cos 72° =

4
(3.) Sin A. sin (A + 2C) + sin B. sin (B + 2A) + sin C.

sin (Ć + 2B) = 0, if A +B+C = 180°.
5. Explain the notation sin-8, tan-'t.
Prove-
3

4 (1.) Sin- + sin

5 (8-6)(8-0)

- 8c

- a)(8-5)
+ tan-
t tan-

= 90°
8(8—a)
8(8-6)

8(8-)
a+b+c
when 8 =

2

= tan

AB.

5

= 90°.

(2.) Tan-/18-1

arc 6. Show that

be properly taken for the measure of an
radius

may
angle; of what angle is 1.5708 the circular measure.

If (0) be the circular measure of an angle less than a right
angle.

1
Prove sin o less than 09, and verify this when a =

6

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