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5. Shew that in the case when m and n are positive integers and m is greater

a" ; a" = a"-".

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7. Solve the equations:

(a) 5x + 2y— 1 = 3x – y + 14 = x + 197 +6,
(B)

(25x® – 9y* = 675,

(3y + 5x = 45. 8. A train travelling from A to C direct at a uniform rate of 54 miles an hour accomplishes the distance in the same time as a 'train which travels from A to a station B between A and C at the uniform rate of 60 miles an hour and without stopping at B proceeds from B to C at the uniform rate of 50 miles an hour. If the distance between B and C be 3 miles greater than that between A and B, find the number of miles between each pair of stations.

9. Find the sides of a rectangle whose area is unaltered if its length be increased by 4 feet while its breadth is diminished by 3 feet, and which loses one third of its area if its length be increased by 16 feet while its breadth is diminished by 10 feet.

10. Shew that a ratio of greater inequality is diminished by adding the same quantity to its antecedent and its consequent. If

X – 2 : Y-2=x* :y", shew that

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11. Shew how to insert two geometrical means between 2com and yon.

If xy, , be in arithmetical progression, shew that y, 2, 2y « are in geometrical progression. 12. Sum to 10 terms each of the series :

(a) 64 -34 +14

– -
(B) 9-51-193 -

ANSWERS.

SECOND GENERAL EXAMINATION.

WEDNESDAY, November 26, 1884.

STATICS. (A.) (Page 140.) 1. 7 ft. 4 in. and 5 ft. 4 in.

3. 45 in. from the top of the cross-bar. 7. 1 ft. 5 in. and 1 ft. 4 in.

10. 3 ft.

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HYDROSTATICS AND HEAT. (B.) (Page 143.) 3. 2 and 4. 4. 311 lbs.

5. 9 inches. 9. 200 cubic inches.

10. 79°C.

7. 8R.

FRIDAY, November 28, 1884.

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ALGEBRA. (A.) (Page 149.) 1. (i) x = 1; ) (ii) x= 4; (iii) x = 15, y=6;

= (iv) x = 2 or - . 2. 9 and 10. 3. 10 of each kind.

a? 12 4. (i) x = }, y=1; (ii) 2

(iii) x =

b

y 6. £9900; £90 and £110.

9. 12s.

10. 12 and 11: 11. (i) 27; ) (ii) - 1705; (ii) 9.

12. 5 and 45.

3 or

};

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ALGEBRA. (B.) (Page 150.)

1. (i) x=21; ) (ii) x= 3; (iii) x=6,

y=15; (iv) x = } or – 2. 2. 8 and 9. 3. 10 of each kind.

a?

62 4. (i) x = 1, y=$; (ii) x=4 or (iii) x =

or b, y

or a

b 6. £9975; £95 and £105.

9. 8s.

10. 16. 11. (i) 29; (ii) - 1023;

-
(iii) 14.

12. .3 and 75.

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1;

a

PREVIOUS EXAMINATION.

FRIDAY, December 5, 1884. 121–3.

ALGEBRA. ADDITIONAL. (A) (Page 154.) 3. (1) 480; (2) n; (3) 17955; (4) 32.

4.

5, 6, 7. 5. 5, 20, 80, or -9, - 41, -21.

6. 40 men; 80 days. 7. 4; 398. 8. a + (r – 1) (B –a).

9. 2:5105450.

ALGEBRA. ADDITIONAL. (B.) (Page 155.) 3. (1) 483; (2) m; (3) 17955; (4) 27.'

4. 4, 5, 6.

6 5. 7, 21, 63. 6. 40 men ; 80 days.

7. 4; 3858 B-a 8. at

9. 2.8115750.

r-1.

FRIDAY, December 12, 1884. 9–111.

.

ARITHMETIC (A) · (Page 190.)

. ( 1. 660539. 2. (1) 362 p. 26 gal. 2 qts. 1 pt. (2) 15 a. 3 r. 10} p.

3. 850. 4. (@) 14. (B) 1-259525. 5. £1. 12s. 10% d. 7. £82. 0s. 337d. 8. 10 months. 9. 93 yds. 2 ft. 6,3 in. 10. £630. 11. 1663

ARITHMETIC. (B) (Page 191.) 1. 582216. 2. (1) 1843 p. 5 yds. 1 ft. 3 in. (2) 36 a. 2 r. 17 p. 4. (a) , (B) +50381. 5. 16s. 35d. 7. £184. 10s. 7491d. 9. 83 yds. 1 ft. 3 in. 10. £1155.

11. 9.

3. 1700. 8. £30.

2 24

FRIDAY, December 12, 1884. 121-3.

For candidates under the old regulations.

ELEMENTARY ALGEBRA, (A.) (Page 192.) 1. a+c; 216. atc 2. 20% – 4y + 5y*.

3. (a) a (x - 12y) (x – 7y), (B) 9 (x 4y) (x + 4y) (x 2y) (x + 2y), (w) (1 – 4xy°) (1 + 4xy’ + 16x+y). x2 + 2y

1 4. (a) (B) 5. ** – 5xy + 7y.

6. 2x? – 7xy + 1ly. a + 8y2'

1-a' 7. a.

8. (a) x = 7, (B) x = 9, or – 63.
()

9. (a) x=1, y=6,

1 (B) x = 10, y = 9.

10. X:y :: 5 : 4.

x

.

=

12. y=+3

=

ELEMENTARY ALGEBRA. (B.) (Page 193.) 1. a + 6c; -76. 2. 2x2 + 9xy 4y. 3. (a) x (x – 12yo) (x - 5y), (B) (x 3y) (x + 3y) (5x y) (5x + y), (W) (1 – 3xy') (1 + 3xy* + 9x+y). y

* 202 + 2y

2.cc + 1 4. (a) (B) 5 5. – 7xy +9y.

6. 2* — 9xy + 7yo. ma + 9ya' 7. 8. (a) x=9, (B) x = 2, or – 111.

9. (a) x = 6,

1 (B) x=6, y = 5.

10.
X : y :: 7 : 2.

12.

y =

=t

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+1.

a.

y=1;

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FRIDAY, December 12, 1884. 121-3.

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For candidates under the new regulations.
ELEMENTARY ALGEBRA. (A.) (Page 195.)

2cm + 2y 1. a +c; 216. 2. 2x® – 4xy + 5y".

3. (a)

a + 8ye? 1 (B) 4. x – 5xy + 7y.

5. a.

6. (a) x = 7 1 - x (B) x = 9, or -63.

7. (a) x= 1, y=6, (B) x = 10, y = 9. 8. The distance between A and B is 131 miles.

9. The length and breadth of the rectangle are 9 inches and 8 inches respectively.

12. (a) 8.27 (B) – 24971.

256)

ELEMENTARY ALGEBRA. (B.) (Page 196.)

una + 2y* 1. a + 6c; -76. 2. 2.x2 + 9xy 4y?.

3. (a)

x +9y' 2x + 1 (B) 4. X* 7 xy +9y?.

5. 6. (a) x = 9, (B) x = 2, or -14.

7. (a) x=6, y=1, (B) x= 6, 8. The distance between A and B is 12 miles,

9. The length and breadth of the rectangle are 16 feet and 15 fect respectively.

12. (a) 4162 (B) – 5511.

a.

XC +1

=

y = 5.

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MATHEMATICAL TRIPOS.

PART III.

MONDAY, January 5, 1885. 9 to 12.

GROUP B.

n

n.n

1-2

1. PROVE that the Zonal Surface Harmonic P, satisfies the difference equation

nP (2n – 1) uPm-1 - (n − 1) Pn-z. Shew how the Solid Zonal Harmonics are associated with the Surface Harmonics.

X', is the Solid Zonal Harmonic of positive order n having the axis of , for its axis and the origin of coordinates for its origin; Xm is the Solid Zonal Harmonic of positive order m having the same axis and a point distant a from the origin for its origin; prove that

1
X', = X, + na X-+ a’X -- + ... + na*-! X, +a".

1.2 The corresponding Solid Zonal Harmonic of negative order being denoted by Y', prove that, for points included within any sphere whose radius is less than å and whose centre is the new origin,

1

n +1! X, n + 2! X n + 3! X Y'.

+ n!

2!n! a 3!n! a Obtain the expression for Y', for points outside any sphere whose radius is greater than a and whose centre is the new origin in the form

n+1!
n+ 2!

n+3!
+
a? Y

a Y.., t.
n!
2! n!

3! n!

1

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2

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anti

a

n

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a Y

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n+1

n+2

+3

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2. Explain Kirchhoff's kinetic analogue to the statics of a bent wire.

A uniform wire in the shape of a helix is pushed into a uniform tube in the shape of a different helix and of the same length, which the wire just fits; investigate the shape of the new helix formed by the combination, and point out the kinetic analogue. 3. Establish the equations in the Planetary Theory

da 2 dR do 6 dR
dt

dt

na' de Prove that, if the disturbance is due only to a resisting medium producing retardation kum/pl and if u denote the excentric anomaly, then da

- (m-1) - P
2ka' (1 + e cos u)(m+1) (1 – e cos u)
du
de
du

(1 – e®) cos u (1 +e cos ujtim-1) (1 – e cos u) (m-1)=p
do
(1 –e)sin u (1 + e cos u)
s u)t(m1) (1 – e cosu)

-}(m-1)-p
du

na de'

m-p

m - 2 n

-1)-9,

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