8. £3,000 is to be divided between A, B, and C; if each received £1,000 more than he actually does, the sums received would be proportional to 4, 3, and 2. Find the share of each.

9. (i) In the two series 2, 5, 8 . . . . and 3, 7, 11 each continued to 100 terms, how many are identical?

=

=

(ii) The (p + q)th term of a G.P. P, the (p-q)th term =Q; find the pth and qth terms.

10. Find the number of combinations of n things taken r at a time, without assuming the formula for permutations.

The ratio of the number of combinations of 4n things taken 2n together, to that of 2n things taken n together is

11. Prove the Binomial. Theorem for a positive integral exponent. Write down the term of the expansion of (+ 2) " 1 involving a, n being an even number.

12. Find the co-efficient of x5 in (1 + x + x2

)2.

not

2. (2x-3y) (2x+3y) (4x2+9y2); (x+1) (x+2) (x+3) (x+4).

1. Trace the changes in sign and magnitude of cos as changes from 0 to π.

sec 0,

2. Find the values of sin {w++(-1)*a}, n being any

integer, positive or negative.

3. Express sin a + siny and cos x + cos y as products of sines and cosines.

4. Define a logarithm, and show that loga

Find log1 01; log23125; log•

=

Of what use are logarithms in Trigonometry?

1

logab®

5. Two forces sustain each other by means of a string passing over a tack. Prove that either force is to the pressure on the tack as is to the cosine of half the angle between the directions of the forces.

6. The sum of the moments of any two forces is equal to the moment of their resultant, about whatever point they are taken.

7. From a given rectangle ABCD show how to cut off a triangle CDO (O being a point in AD), so that when the figure ABCO is suspended from O the sides AO, BC may be horizontal.

8. If a lighter fluid rest upon a heavier, and their specific gravities be a and b, and a body, specific gravity c, rest with one part P in the upper fluid, and the other part Q in the lower, then

9. Show how the relations are found by experiment (i) between pressure and volume of a gas, temperature being constant ; (ii) between temperature and volume, pressure being constant.

Higher Mathematics.

Women.

STATICS, DYNAMICS, ASTRONOMY.

1. Define the terms Statics, force. How are forces (i) measured, (ii) represented?

2. State and illustrate the principle of the transmission of force.

3. Enunciate and prove the parallelogram of forces. If three forces are in equilibrium, the sum of any two of them is greater than the third.

4. If O be a point within a triangle ABC, and D, E, F, the middle points of the sides, the forces represented by OA, OB, OC, are equivalent to those represented by OD, OE, OF.

5. How are physical quantities measured? An acceleration is denoted by ƒ when t seconds and s feet are units of time and length. What is its measure when T seconds and S feet are units?

6. Enunciate and prove the parallelogram of velocities. A ship is sailing due N. with a velocity of 10 knots an hour, another is steaming S.W. at the rate of 15 knots an hour. Find the velocity of the second relative to the first.

7. State Newton's three laws of motion.

8. What do we infer from the general changes in the appearance of the heavens during a night?

9. What observations lead us to believe that the earth revolves about the sun?

10. Define axis, poles, equator, meridian, zenith.

Find

of 2 rt. angles, express the angles A° in circular measure. the number of degrees in the angle whose circular measure is 1.

2. Find all the trigonometrical ratios of the angle whose cosine is

3

5

3. Prove that tan (90+ A) = cot (180-A). Find all the trigonometrical ratios for angles of 45°, 30°, and 510°.

4. Prove geometrically sin (A + B) = sin A cos B + cos Asin B, tan Atan B

and tan (A+B) =

1-tan A tan B'

5. Find an expression for all the angles which have a given tangent. Find all the values of which satisfy the equation

(1-tan 0) (1 + sin ) = 1 + tan 0.

6. Define Parabola, Ellipse, and Hyperbola. Of which class of curves are (i) a circle, (ii) a straight line, (iii) two straight lines, particular cases?

7. Prove that no straight line can meet a conic in more than two points.

8. Show that a single equation between the co-ordinates represents a locus. What is this locus when the equation is of the first degree?

9. Show that the three lines joining the pairs of points (1,-2), (2,−3); (0, 1), (2, 3) ; (3,—2), (−3, 1) meet in a point, and represent them in a figure.

Natural Philosophy.

Junior and Senior.

(a) CHEMISTRY; (b) PRACTICAL CHEMISTRY; (c) STATICS, DYNAMICS, AND HYDROSTATICS EXPERIMENTALLY TREATED; (d) THE EXPERIMENTAL LAWS OF HEAT; (e) ELECTRICITY AND MAGNETISM; (f) ELEMENTARY BIOLOGY; (g) ZOOLOGY; (h) BOTANY; (k) PHYSICAL GEOGRAPHY.

Junior Students will only be examined in three of the subjects (a), (b), (c), (d), (g), (h). Senior Students will only be examined in three of the subjects (a), (b), (c), (d), (e), (ƒ), (g), (k).

NOTE. (b) cannot be taken with (a), nor (g), nor (k), without (ƒ).

(a)

1. How is phosphorus obtained, and what are its chief forms?

2. How many phosphates of hydrogen exist? How are they prepared?

3. Give an account of the chief compounds of arsenic.

(b)

4. Given a piece of iron, how can I dissolve it, and subsequently test it?

5. Why are phosphates precipitated with the metals of the third group?

6. How is chromium recognised in solution?

(c)

7. State the laws of friction, and show how the co-efficient of friction may be practically determined.

8. If μ1 in the case of a rough plane, and a body resting upon it, what inclination may be given to the plane before the body will begin to slide down?

9. Describe methods by which reciprocating motion can be made to produce motion of rotation in machinery.