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8. If through a fixed point, within or without a circle, chords be drawn, the rectangle of their segments is invariable.
9. DB is the diameter of a senii-circle, and BF, DG, chords intersecting in P. Shew that the circles passing through B, G, P, and P,F,D, cut the base in the same point : also that
BF. BP+ DG. DP=BD. 10. Two half-dollar coins, lying flat in the corner of a rectangular box, are so moved about, as to touch each other and also the sides of the box. Find the locus of the point where they touch each other.
ARITHMETIC AND ALGEBRA.
(EXHIBITIONS AND BURSARIES.)
TIME: THREE HOURS.
1. If 3 A=5 B, and 7 B=9 C, and 13 C=14 D, and 35 D=52 E: how many A = 100 E?
find m in terms
(ac- bd)? + (ad + bc)2 – az, and 3. Reduce to simplest form,
c2 + d2 2/*) % (*- 4x^2 + 4x4
4. Shew that, (a + b)2 + (a + c)2 + (a +d)2+(6+c)2 +(6+d)2+(c+d)2 = (a+b+c+d;2+2 (a2 +62 + c2 + do).
5x + -3 7 x2 – 3x – 9 5. Solve the equations,
52 - 4 7x - 10
√x 6. Solve the simultaneous equations, xy=a (x+y), w z=b (x+z),
yz=c(y+z). 7. A warehouse will hold 24 boxes and 20 bales; and 6 boxes and 14 bales will half fill it. Find how many of each kind it will hold.
8. Shew that, in the common process for finding “Greatest Common Measure or Highest Common Factor,” the measure so found is the greatest.
9. Shew that, if a2 + ax +b and x4 + a,x+b, have a Common Measure, x+p, their Least Common Multiple is
(2 + p) (x + a-p) (x + a, - p).
ELEMENTARY MECHANICS OF SOLIDS AND FLUIDS.
..J. G. MacGREGOR, D.Sc.
(Higher Matriculation Examination, Fnculty of Medicine.)
TIME: THREE HOURS.
N. B.-Answer only ten questions.
1. Enunciate and prove the proposition called the Polygon of Velocities.
2. A stone is thrown vertically upwards with a velocity of 90 feet per second.
Find its velocity at the end of the 4th second, and its positions at the end of the 3rd and 12th seconds respectively.
3. When is force said to do work on a body? How is the work done measured ? Define the foot-pound, and find the work done by a man who lifts 120 lbs. through a height of 20 yards.
4. Two masses, m and M, are acted upon for the same time, by forces of 20 and 25 units respectively, and the same velocities are produced. Find m if M is 10 lbs.
5. Find the resultant of two forces acting on a particle, whose magnitudes are 10 and 12 respectively and whose directions are inclined at an angle of 60°.
6. A shell, initially at rest, bursts into two fragments whose masses are 12 and 20 pounds. The first moves off with a velocity of 25 ft. per second. Find the velocity of the second.
7. Shew that if three forces acting on a particle are in equilibrium, and if their directions are represented by the sides of a triangle taken in order, their magnitudes also are represented by those sides respectively.
8. Find the centre of Inertia (or of gravity) of a uniform, thin triangular plate.
9. Shew that the algebraic sum of the moments of two parallel forces about any point in their plane is equal to the moment of their resultant about the same point.
10. Describe the common Balance ; and show that the mass of a body may be determined by means of it.
11. The height of a smooth inclined plane is 3 ft. and its length 5 ft. Find what force acting up the plane will keep a body of 20 lbs. mass, in equilibrium on it.
12. Shew that the pressures, at all points in a horizontal plane, of a heavy fluid, are the same.
13. The pressure on the surface of a lake is equal to the weight of 14.5 lbs. per square inch. Find the pressure at a point 20 ft. below the surface. Find also the pressure on a horizontal surface of 2 sq. ft. area at that depth. (A cubic foot of water weighs 1000 oz.)
14. Find the specific gravity of a body weighing 15 oz. in air and 11.5 oz. in water.
15. Describe the structure of the Mercury Barometer; and shew how to use it to determine which of two hills is the higher.
.J. G. MACGREGOR, D.Sc.
(Higher Matriculation Examination, Faculty of Medicine.)
TIME: THREE HOURS.
N. B.—Answer only ten questions.
1. Describe the ordinary Air Pump; and shew how by means of it the pressure of the air in a vessel may be reduced.
2. State and explain the Law of the Conservation of Energy. Illustrate it by reference to the motion of the Pendulum, and the phenomena of the galvanic circuit.
3. Show how a sound is propagated through the air ?
4. Describe the motion of a plate or of a string, which is giving out a musical note, explaining nodal lines or nodal points, respectively.
5. What are the “fixed points” of the Mercurial Theomometer, and how are they determined?
6. What is the linear co-efficient of expansion of a substance ? Find the length at 20° C of a bar of brass, whose length at 0° C is 2 ft., the linear co-efficient of expansion being 0.00018.
7. Two lbs. of water at 25° C are mixed with 3 lbs. of mercury at 120° C. Find the temperature of the mixtur (The specific heat of mercury is 0.033.)
8. Explain mechanical equivalent of heat ; and describe one method of determining it.
9. Give the laws of the reflection of light; and show how an image of an object is formed by a plane niirror.
10. Describe the effect of passing a ray of white light through a prism ; and shew how the prism may be used to analyse any ray of light.
11. How would you shew by experiment that “like electricities repel and unlike attract."
12. Describe the Torsion Balance and show how to use it to prove that the attraction of two electrified bodies is inversely proportional to the square of the distance between them.
13. Describe any form of electrical machine, explaining its action.
14. Describe the effect of an electric current on a magnet hanging in its neighbourhood, and the structure of the Galvanometer.
15. Describe fully the apparatus by which you would decompose water by the aid of the electrical current.
SENIOR MUNRO EXHIBITIONS AND BURSARIES.
....JOHN JOHNSON, M. A.
HORACE: ODES, Book IV. CICERO: SELECT LETTERS.
TIME: THREE HOURS.
Ille non inclusus equo Minervae
Falleret aulam ;
Matris in alvo,
Vatis Horati. 1. Note any Greek forms of words or Greek constructions in the extract. 2. (a) “Doctor Argivae fidicen Thaliae." (b) “ Dauniae defende decus Camoenae,
Pollicis ictum.” (d) “ Saeculo festas referente luces." Write explanatory notes. Mark the quantity of syllables in fidicen and compare it with similar compounds.
3. Quote other passages from this book referring to Horace as a poet. 4. (a) Janum Quirini clusit. (6) Major Neronum mox grave proelium
Testis Metaurum flumen.
(6) Decline in the pl. : nix, pinus, vice, Scythes.
spargier. 6. Scan the following lines and name the system to which each belongs :
(a) Ostendet Capitolio.-
II. B. Translate :
Mirabar, quid esset, quod tu mihi litteras mittere intermisisses : indicavit mihi Pansa meus Epicureum te esse factum. O castra praeclara ! quid tu fecisses, si te Tarentum et non Samarobrivam misissem? jam tum mihi non placebas, cum idem tuebare, quod Zeius familiaris meus. Sed quonam modo jus civile defendes, cum omnia tua causa facias, non civium? Ubi porro illa erit formula fiduciae UT INTER BONOS BENE AGIER OPORTET? Quis enim est, qui facit nihil nisi sua causa ? Quod jus statues COMMUNI DIVIDUNDO, cum commune nihil possit esse apud eos, qui omnia voluptate sua metiuntur ? Quo modo autem tibi placebit JOVEM LAPIDEM jrare, cum scias Jovem iratum esse nemini posse ?
Quid fiet porro poupulo Ulubrano, si tu statueris, TOALTEVEĞbai non oportere ? Qua re si plane a nobis deficis, moleste fero ; sin Pansae adsentari commodum est, ignosco. Modo scribe aliquando ad nos, quid agas et a nobis quid fieri aut curari velis.
1. “O castra praeclara !" Quote other examples of military terms thus used.
2. “Quid tu fecisses, si te Tarentum et non Samarobrivam misissein ?" Explain the meaning. Give the modern name of Samarobriva.
3. Quis enim est, qui facit nihil risi sua causa ?” Show that you understand the use of the mood of facit.
4. (a) “JOVEM LAPIDEM jurare ”; (b) Quid fiet porro populo Ulubrano”. Write explanatory notes and account for cases. 5. Ad eum postridie mane vadebam, cum haec scripsi. — Mancipia venibant Saturnalibus tertiis. Cum haec scribebam,
in tribunali res erat ad HS. CXX. (a) Translate these sentences, changing the Latin epistolary
forms into the usual English forms. (6) Give the date in our calendar nominally corresponding to Saturnalibus tertiis. (c) Write in full HS. cxx. and give the value
in English money. Explain the symbol HS. 6. Quote some proverbs from these letters.