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9. Calculate the logarithms of 27 and 3 to base 9, and find the characteristic of log, 2345.

10. Find the sum of 25 terms of an A.P. whose 3rd term is 5 and 14th term 27.

If a be the 1st term, ab the (2m + 1)th term of a G.P., find the middle term.

The sum of 3 numbers in G.P. is 21, and the sum of the products taken two and two is 126 ; find the numbers. 11. Solve:



x + 6a

- За
(i) x2 + 4xy=35; 2xy - 16yo = 1.

(iii) 2 (2x - 3) (2 - 4) - V2.x* — 11x + 16 = 60.
– –

12. If show that

6 d
1 1 1 1 1 h


+ + ma

19 +P 13. Find the coefficient of *4 in the expansion of (1 - 2x + 30% - 403 + )-1

14. Determine the limits between which the expression x2 – 2x – 3

lies for all real values of x.
2x2 + 2x +1
15. Prove the inequalities :

1 1
greater than

(ii) Va% +b Vo? + d greater than ac + bd.


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a? , 1, 18x – 5y.


3. 6.

4. a - 2. a3 - 68

- 2
5. c a+b+c.
6. (i) 2 =

=4 or
Na +4;

(iii) x =

X =3 or-4.

y=5 or 1 7. 3 miles an hour.

8. § ; 9.

9. 4. 10. n (35 4 3n)

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A 1. Express sin in terms of sin A, and explain why there are four values.

Find the sine and cosine of 18°, and thence find those of 9o.

b% +-a 2. In a triangle prove cos A=

2bc If (a + b2) cos 2A= 52 a’, show that the triangle is right angled. 3. Solve completely the equation

cos n 8+ cos (n − 2) Q = cos 0.

4. Two straight rods OA and OB, of equal weight, lengths a and b, are rigidly joined at 0, so that AOB is a right angle. They are hung up by a string from 0; if a be the inclination of 0 A to

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5. Find the relation between P and W in the first system of pullies.

If W=12 lbs., and there are three movable pullies, the lowest weighing 4 lbs., the next 2 lbs., and the upper í lb., find P.

6. If G be the centre of gravity of the triangle ABC, show that 3(GA? + GB? + GC2)=AB? + AC? + BC%.

7. Find the position of equilibrium of a solid cone, floating with axis vertical and vertex upwards in a fluid, whose density is to that of the cone as 27 : 19.

8. A hollow cone, filled with water and closed, is held with its axis horizontal; find the resultant vertical pressure on the upper half of its curved surface.

Higher Mathematics.



1. If the forces represented by OP, OQ, OR, be in equi-. librium, O is the centre of gravity of the triangle PQR.

2. Two spheres are supported by strings attached to a given point, and rest against each other. Find the tensions of the strings.

3. Two forces in the ratio of (1 + n) to 1 act at a point at an angle a ; show that the sine of the angle made by the resultant. with the larger force = (

(1-3) sin a nearly, n being supposed



very small.

4. Define a cycloid, and show how a pendulum can be made to oscillate in one. Find the time of one oscillation of a pendulum of given length,

5. Explain how the times of a star's rising and setting alter during the year. Orion's belt is in the equator, and has R. A: 5 h. 30 m.; during what part of the night is it visible at the vernal. and autumnal equinoxes ?

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1. Solve sin 0 + sin 3 = sin 2 0 + sin 4 0.

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2. A tower 51 feet high has a mark at the height of 25 feet from the ground; find at what distance the two parts subtend equal angles to an eye 5 feet from the ground.

3. Find by De Moivre's Theorem the value of:

(i) (cos 4 A + V-1 sin 4 A)} (ii) (-1)}

4. Given the focus, the position of the axis, and a tangent, construct a parabola.

5. Define the circle of curvature, and show that its chord of intersection with a conic is inclined to the axis at the same angle as the tangent at the point of contact.

6. Two intersecting chords of an ellipse PQ, RS, make equal angles with the axis ; show that the lines PR, QS, and PS, QR also make equal angles with the axis.

7. If from any point of a circle, PC be drawn to the centre C, and the chord PQ parallel to the diameter ACB be bisected in R, show that the locus of intersection of AR and CP is a parabola.

Natural Philosophy.

Junior and Senior.



Junior Students will only be examined in three of the subjects (a), (b), (c), (d),

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

NOTE.—(6) cannot be taken with (a), nor (9), nor (k), without (f).

(a) 1. Ammonia is termed an alkali. Explain this expression.

2. Prove that a molecule of ammonia contains one atom of nitrogen and three of hydrogen.

3. Give a brief account of the oxides of Nitrogen.

(6) 4. Explain how lead occurs in both the silver and copper groups.

5. How would you recognize bismuth in solution ?

6. Show clearly how you would prove the presence of arsenic, and how distinguish it from antimony.

(c) 7. Find the pressure at any depth in a heavy, homogeneous liquid at rest. A cubical box is filled with a cubic foot of water, and closed by a piston which weighs 2 lbs.; find the whole pressure on a side of the box, having given that a cubic foot of water weighs 1000 ounces.

8. Show that a solid immersed in a fluid loses as much of its weight as is equal to the weight of the fluid it displaces. · Compare the densities of two fluids in which a body weighing 10 ounces rests with half its volume immersed when loaded with 2 and 5 ounces respectively.

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