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take the whole of this paper; those for the first year's partial examination,

the first part of it; those for the second year's partial examination, the

second part.

Note 2. â€” State at the head of your pai^er what text-book you have

studied on the subject, and to what extent.

I.

1. Resolve x^ + Sy'^ into two factors.

. Cxiveu = ex, to nnd the value of x.

a+b aâ€”h

3. There are three workmen, A, B, C. A and B together

can perform a certain piece of work in a days ; A and C to-

gether, in h days ; and B and C togetlier, in c days. In

what time could each singly perform it, and in what time

could they finish it if all worked togetlier?

4. Prove that if

", (I., (^-5 ., Ui + a., + u^

â€” = â€”=â€” = i\ then â€” 'â€”^ â€” - = r.

b, b, b, u, + b,-\-b,

5. (a) Write equivalent expressions for the following:

138 ALGEBRA.

(b) Reduce the following expressions to simpler forms

( 1 ) V4.") f' â€” VSOc-' ^- Voa-c ;

(3) 4a/|x2V|.

II.

6. Given 3a;- -\-x â€” 7, to find x.

7. Given ^ ,_ â€” = \ -, to find x.

^x ^a;'"+\

^"^ af-\-2bx + 2ax-\-bf' ^ ^ a^bx-h^a?

2. Given â€” - ^^^^^^ -bx = ae-S bx, to find x.

d a

140 ALGEBHA.

3. A sum of money, at simple interest, amounted in m

years to a dollars, and in n years to b dollars. Find the sum

and the rate of interest.

4. Prove that if ^~^ = 450, and A =

27Â° 50', to first determine whetlier the triangle admits of

more Hum one solution, and tlien solve it.

June, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. "Write tlie values of the different trigonometrie func-

tions for angles of 0Â°, 90Â°, 180Â°, 270Â°, 300Â°.

2. "Write the simplest equivalent expressions for

sin I -â€”a), cos( - (-(tj, tan(-|7râ€” a), sin(27râ€” a).

3. Write the fundamental formulae connecting two or

more of the trigonometric functions of the same angle, and

deduce any two of them.

4. Given tan a = â€” , to find sin a and cos a.

n

5. Deduce the formulae,

cos 2 a =1â€”2 siu^a ; sin J-a = Vy(1â€” cos a).

6 . Prove that tan ^ |^ + tan ^ ^ = -.

4

7. In a plane triaugle the side a is 4575, the side b is

5G56.7, and the angle B (opposite b) is 125Â° 25'; find the

remaining parts of the triangle.

September, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. Find the angle 1.8 in degrees.

2. Calculate the different functions of - and V-tt.

6 '

3. Represent the different trigonometric functions of an

angle in the second quadrant by lines.

144 TRIGONOMETRY.

4. Deduce the formula,

cos a â€” cos h = â€” 2 am -\ {a + b) sin .', (a â€” b) ,

a and ?/ being any angles.

m

5. Given tan^= â€” . to lind the value of

u

1)0 COS 20 + n sin 2^.

G. In a triangle ABC, given the sides a, 6, and c, respec-

tively, opposite the angles A, B, and C, as follows :

a = 7GG.3, b = G72, c = 565. U5, to lind the angle A.

Juuc, 1882.

[State what text-book you have studied on the subject, aud to what extent.]

1. Define the different trigonometric functions of an

angle.

2. Give the algebraic sign of each function of an angle

terminating in each of the four quadrants.

3. Show that sin (tt + a) = â€” sin a ; and that cos (2 tt â€” a)

= cos a.

4. Find the formula for tan (a -f^) in terms of tan a and

tan /S. Find also the formula for tan 2 a in terms of tan a.

5. Given 2 a tan ^ sin ^ â€” 2 a cos ^ -f- /j = 0, to find 6.

6. Given two sides of a plane triangle IHO and 100, and

the angle opposite the former 127Â° 33', to find the other parts.

September, 1882.

[State what text-book you have studied, and to M'liat extent.]

1. Find tlie luniibor of degrees in the angle subtended at

the centre of a circle whose radius is lU ft. by an arc 9 in.

loner.

TRIGONOMETRY. 145

2. Determine all the funetious of the followiug tingles :

30Â°5 7, f 7I-, 585Â°.

4

3. Given the functions of an}' angle to find the functions

of half that angle.

4. Given tan ^ = â€” , to find the value of sin 2^.

n

5. Show that 2tau"'^a = tan~'

l-a^

6. In a plane trianglo, given a =100, c = 200, and B =

112Â° 30', to find the remaining parts.

June, 1883.

1. Explain the circular measure of an angle. Express an

angle of 1.5 in degrees.

2. Give the algebraic signs of the trigonometric functions

for angles terminating in each of the four quadrants, and the

q

values of the functions of 0, -, rr, â€” .

3. State some of the fundamental relations existing be-

tween the different functions of the same angle.

4. Find an angle whose tangent is double its sine.

5. "Write the different formulae which serve to solve right-

angled triangles, and explain their application.

6. Find a formula for expressing sin ^a in terms of cos a.

7. If a, b, and c are the sides of a triangle, and a the angle

opposite the side a, show that

a- = &- + c^ â€” 2bc cos a.

Also state any application you have made of this formula.

8. In a triangle given A = 32Â°, a = 40, 6 = 50, to find

one solution of the triangle. Is there more than one? If so,

how do you recognize the fact?

146 TRIGONOMETRY.

September, 1883.

1. Express an angle of 75Â° in radians.

2. Express each of the functions of ( - + a j by a function

of a. ^^ ^

3. Form a general expression for all angles which have

the same cosine.

4. What must the value of the cosine be in order that the

tangent may be n times the sine?

5. To find a formula for the tangent of the sum of two

angles in terms of the tangents of the angles ; and from this

formula to derive an expression for the tangent of twice an

angle in terms of the tangent of the angle.

6. In a triangle ABC, given the angle A= 118Â° 04', and

the adjacent sides, &= 9G0, and c = 1686, to find the remain-

ing angles and side of the triangle.

LATIN. 147

LATIN.

Julj-, 18S0.

I. Translate as literally as possible : â€”

a. Qnum oa ita siut, tamcn, si obsides ab iis sibi den-

tin-, uti ea, quae polliceantnr, factnros intelligat, et si

Aeduis de injuriis. qnas ipsis sociisque eorum intulerint,

item si Allobrogibus satisfaciant, sese cum iis pacem esse

5 factm-nm. Divico respondit : Ita Helvetios a majoribus

suis iustitutos esse, iiti obsides accipere, non dare, con-

sueriut ; ejus rei populum Romanum esse tesfem.

h. Ibi vadis repertis partem suarura copiarum trans-

ducerc conati simt, co cousilio, uti, si posseat, eastellum,

10 cui praeerat Quiutus Titurius legatus, expuguarent, pon-

temque interscindereut ; si minus potuissent, agros Re-

morum popularentur, qui magno nobis tisni ad l)ellura

gereudum erant, commeatuque nostros prohibereut.

c. Impeditis hostibus propter ea, quae ferebant, ouera,

15 subito quabus portis eruptionem fieri jubet. Factum est

opportuuitate loci, hostium inscientia ac defatigatione,

virtute militum, et superiorem pugnarum exercitatione, ut

ne uuum quidem nostrorum irapetum ferreat ac statim

terga verterent.

20 d. Dum in his locis Caesar navimn parandarum causa

moratur, ex magna parte Moriuorum ad eum legati vene-

runt, qui se de superioris temporis consilio exousarent,

quod homines barbari et nostrao consuetadmis imperiti

bellura populo Romano fecissent, seque ea, quae impc-

25 rasset, facturos pollicereutur.

e. Britanniae pars interior ab iis incolitur, quos natos

in insula ipsi memoria prodititm dicunt ; maritima pars

ab iis, qui praedae ac belli inferendi causa ex Belgis

transieraut ; qui oumes fere iis nominibus civitatum ap-

14y LATIN'.

30 pellantnr, quilnis orti ex civitatibus eo pcrvcucrunl, et,

bello illato, ibi porinauserunt atcpie agros colore coepc-

runt.

/. Aiiibiorix copias suas Jadlcione iiou conduxerit,

quod proclio diraicandura non cxistimarit, an tempore

35 cxclusus ct rcpeutino equitum advcntu prohibitum., quuia

r.'liquum exercltmn subsequi crederet, dubium est ; sed

certe dimissis per agros uuntiis sibi quemquc cousulerc

jussit.

II. 1. Decline obskles (1), majoribus (5), loci (IG).

2. Compare imj^eriti (23), interior (26), certe (37).

3. "Write the synopsis, active and passive, of intelligat (2).

4. State the mood and tense of the following : consuerint

(G), praeerat (10), ferrent (18), imjjerasset (24), existi-

marit (34).

5. Give the principal parts of the following : 2^^^^ic

the first part of it; those for the second year's partial examination, the

second part.

Note 2. â€” State at the head of your pai^er what text-book you have

studied on the subject, and to what extent.

I.

1. Resolve x^ + Sy'^ into two factors.

. Cxiveu = ex, to nnd the value of x.

a+b aâ€”h

3. There are three workmen, A, B, C. A and B together

can perform a certain piece of work in a days ; A and C to-

gether, in h days ; and B and C togetlier, in c days. In

what time could each singly perform it, and in what time

could they finish it if all worked togetlier?

4. Prove that if

", (I., (^-5 ., Ui + a., + u^

â€” = â€”=â€” = i\ then â€” 'â€”^ â€” - = r.

b, b, b, u, + b,-\-b,

5. (a) Write equivalent expressions for the following:

138 ALGEBRA.

(b) Reduce the following expressions to simpler forms

( 1 ) V4.") f' â€” VSOc-' ^- Voa-c ;

(3) 4a/|x2V|.

II.

6. Given 3a;- -\-x â€” 7, to find x.

7. Given ^ ,_ â€” = \ -, to find x.

^x ^a;'"+\

^"^ af-\-2bx + 2ax-\-bf' ^ ^ a^bx-h^a?

2. Given â€” - ^^^^^^ -bx = ae-S bx, to find x.

d a

140 ALGEBHA.

3. A sum of money, at simple interest, amounted in m

years to a dollars, and in n years to b dollars. Find the sum

and the rate of interest.

4. Prove that if ^~^ = 450, and A =

27Â° 50', to first determine whetlier the triangle admits of

more Hum one solution, and tlien solve it.

June, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. "Write tlie values of the different trigonometrie func-

tions for angles of 0Â°, 90Â°, 180Â°, 270Â°, 300Â°.

2. "Write the simplest equivalent expressions for

sin I -â€”a), cos( - (-(tj, tan(-|7râ€” a), sin(27râ€” a).

3. Write the fundamental formulae connecting two or

more of the trigonometric functions of the same angle, and

deduce any two of them.

4. Given tan a = â€” , to find sin a and cos a.

n

5. Deduce the formulae,

cos 2 a =1â€”2 siu^a ; sin J-a = Vy(1â€” cos a).

6 . Prove that tan ^ |^ + tan ^ ^ = -.

4

7. In a plane triaugle the side a is 4575, the side b is

5G56.7, and the angle B (opposite b) is 125Â° 25'; find the

remaining parts of the triangle.

September, 1881.

[State what text-book you have studied on the subject, and to what extent.]

1. Find the angle 1.8 in degrees.

2. Calculate the different functions of - and V-tt.

6 '

3. Represent the different trigonometric functions of an

angle in the second quadrant by lines.

144 TRIGONOMETRY.

4. Deduce the formula,

cos a â€” cos h = â€” 2 am -\ {a + b) sin .', (a â€” b) ,

a and ?/ being any angles.

m

5. Given tan^= â€” . to lind the value of

u

1)0 COS 20 + n sin 2^.

G. In a triangle ABC, given the sides a, 6, and c, respec-

tively, opposite the angles A, B, and C, as follows :

a = 7GG.3, b = G72, c = 565. U5, to lind the angle A.

Juuc, 1882.

[State what text-book you have studied on the subject, aud to what extent.]

1. Define the different trigonometric functions of an

angle.

2. Give the algebraic sign of each function of an angle

terminating in each of the four quadrants.

3. Show that sin (tt + a) = â€” sin a ; and that cos (2 tt â€” a)

= cos a.

4. Find the formula for tan (a -f^) in terms of tan a and

tan /S. Find also the formula for tan 2 a in terms of tan a.

5. Given 2 a tan ^ sin ^ â€” 2 a cos ^ -f- /j = 0, to find 6.

6. Given two sides of a plane triangle IHO and 100, and

the angle opposite the former 127Â° 33', to find the other parts.

September, 1882.

[State what text-book you have studied, and to M'liat extent.]

1. Find tlie luniibor of degrees in the angle subtended at

the centre of a circle whose radius is lU ft. by an arc 9 in.

loner.

TRIGONOMETRY. 145

2. Determine all the funetious of the followiug tingles :

30Â°5 7, f 7I-, 585Â°.

4

3. Given the functions of an}' angle to find the functions

of half that angle.

4. Given tan ^ = â€” , to find the value of sin 2^.

n

5. Show that 2tau"'^a = tan~'

l-a^

6. In a plane trianglo, given a =100, c = 200, and B =

112Â° 30', to find the remaining parts.

June, 1883.

1. Explain the circular measure of an angle. Express an

angle of 1.5 in degrees.

2. Give the algebraic signs of the trigonometric functions

for angles terminating in each of the four quadrants, and the

q

values of the functions of 0, -, rr, â€” .

3. State some of the fundamental relations existing be-

tween the different functions of the same angle.

4. Find an angle whose tangent is double its sine.

5. "Write the different formulae which serve to solve right-

angled triangles, and explain their application.

6. Find a formula for expressing sin ^a in terms of cos a.

7. If a, b, and c are the sides of a triangle, and a the angle

opposite the side a, show that

a- = &- + c^ â€” 2bc cos a.

Also state any application you have made of this formula.

8. In a triangle given A = 32Â°, a = 40, 6 = 50, to find

one solution of the triangle. Is there more than one? If so,

how do you recognize the fact?

146 TRIGONOMETRY.

September, 1883.

1. Express an angle of 75Â° in radians.

2. Express each of the functions of ( - + a j by a function

of a. ^^ ^

3. Form a general expression for all angles which have

the same cosine.

4. What must the value of the cosine be in order that the

tangent may be n times the sine?

5. To find a formula for the tangent of the sum of two

angles in terms of the tangents of the angles ; and from this

formula to derive an expression for the tangent of twice an

angle in terms of the tangent of the angle.

6. In a triangle ABC, given the angle A= 118Â° 04', and

the adjacent sides, &= 9G0, and c = 1686, to find the remain-

ing angles and side of the triangle.

LATIN. 147

LATIN.

Julj-, 18S0.

I. Translate as literally as possible : â€”

a. Qnum oa ita siut, tamcn, si obsides ab iis sibi den-

tin-, uti ea, quae polliceantnr, factnros intelligat, et si

Aeduis de injuriis. qnas ipsis sociisque eorum intulerint,

item si Allobrogibus satisfaciant, sese cum iis pacem esse

5 factm-nm. Divico respondit : Ita Helvetios a majoribus

suis iustitutos esse, iiti obsides accipere, non dare, con-

sueriut ; ejus rei populum Romanum esse tesfem.

h. Ibi vadis repertis partem suarura copiarum trans-

ducerc conati simt, co cousilio, uti, si posseat, eastellum,

10 cui praeerat Quiutus Titurius legatus, expuguarent, pon-

temque interscindereut ; si minus potuissent, agros Re-

morum popularentur, qui magno nobis tisni ad l)ellura

gereudum erant, commeatuque nostros prohibereut.

c. Impeditis hostibus propter ea, quae ferebant, ouera,

15 subito quabus portis eruptionem fieri jubet. Factum est

opportuuitate loci, hostium inscientia ac defatigatione,

virtute militum, et superiorem pugnarum exercitatione, ut

ne uuum quidem nostrorum irapetum ferreat ac statim

terga verterent.

20 d. Dum in his locis Caesar navimn parandarum causa

moratur, ex magna parte Moriuorum ad eum legati vene-

runt, qui se de superioris temporis consilio exousarent,

quod homines barbari et nostrao consuetadmis imperiti

bellura populo Romano fecissent, seque ea, quae impc-

25 rasset, facturos pollicereutur.

e. Britanniae pars interior ab iis incolitur, quos natos

in insula ipsi memoria prodititm dicunt ; maritima pars

ab iis, qui praedae ac belli inferendi causa ex Belgis

transieraut ; qui oumes fere iis nominibus civitatum ap-

14y LATIN'.

30 pellantnr, quilnis orti ex civitatibus eo pcrvcucrunl, et,

bello illato, ibi porinauserunt atcpie agros colore coepc-

runt.

/. Aiiibiorix copias suas Jadlcione iiou conduxerit,

quod proclio diraicandura non cxistimarit, an tempore

35 cxclusus ct rcpeutino equitum advcntu prohibitum., quuia

r.'liquum exercltmn subsequi crederet, dubium est ; sed

certe dimissis per agros uuntiis sibi quemquc cousulerc

jussit.

II. 1. Decline obskles (1), majoribus (5), loci (IG).

2. Compare imj^eriti (23), interior (26), certe (37).

3. "Write the synopsis, active and passive, of intelligat (2).

4. State the mood and tense of the following : consuerint

(G), praeerat (10), ferrent (18), imjjerasset (24), existi-

marit (34).

5. Give the principal parts of the following : 2^^^^ic