11. Shew that in any triangle

c2 = a2 + b2 — 2ab cos C.

12. Shew how to solve a triangle, having given two sides and an angle opposite one of them.

If A = 60°, a = √3, b

triangle.

=

√2, solve the

PURE MATHEMATICS.-PART II.

The Board of Examiners.

1. Find the equation of a straight line in terms of the intercepts which it makes on the axes.

If the sum of the reciprocals of the intercepts be constant, shew that the line passes through a fixed point.

2. Find the general polar equation of a circle.

If from a fixed point a straight line be drawn. to cut a given circle, the rectangle contained by the segments is constant.

3. Find the equation of the straight line passing through two given points on a parabola.

If the sum of the reciprocals of the ordinates of the extremities of a chord of a parabola be constant, the chord passes through a fixed point.

4. The sum of the squares of two conjugate semidiameters of an ellipse is constant.

Find the magnitude and position of the equiconjugate diameters of an ellipse.

5. Define a differential coefficient, and find that of ". Differentiate

6. Prove that under certain conditions

f(a + h) =ƒ(a) + hf'(a + 0h),

and state the conditions.

Expand a cot x in ascending powers of x as far as the term in 4.

7. Shew how to find the value of an expression which takes the indeterminate form co

Find the value when x = 0 of cot x

cosec x.

8. State and prove the rule for integration by substi

tution.

10. Find a formula for the area of a curve referred to polar coordinates.

Find the area of a quadrant of the ellipse

11. Find a formula for the volume of a solid of revolution.

Find the volume generated by the revolution about the axis of x of any portion of the ellipse

y2

x2
+

= 1.

b2

PURE MATHEMATICS.-PART III.

The Board of Examiners.

Write essays on three of the following subjects:(i) Change of variables.

(ii) Tangents and normals to plane curves.

(iii) Definite integrals.

(iv) Straight lines in space.

(v) Generating lines of a conicoid.

(vi) Ordinary differential equations of the first order but not of the first degree.

(vii) Linear partial differential equations of the first

order.

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

1. Define power, and express a horsepower in c.g.s. units, taking a kilogramme as 2-2 pounds, and a foot as 30.5 centimetres.

2. Two particles of masses mi, m, are connected by an inextensible string of length 7. Being placed close together they are acted on by forces F, Fa in opposite directions. Find their velocity immediately after the string becomes tight, neglecting gravity.

3. Investigate the acceleration of a particle moving uniformly in a circle.

4. Two equal particles of mass m are attached to two points B, C of an endless string of length 31 forming an equilateral triangle ABC in a vertical plane. The whole rotates under gravity about a vertical axis through A, with BC horizontal, making n revolutions a second. Find the tensions of the string.

5. Prove that two couples on a rigid body in the same plane and of equal and opposite moments, will balance.

6. A uniform stick of weight W is supported partly by a string attached at one end, and partly by a rough plane, inclined at an angle a, with which the other end is in contact, the string and stick lying in a vertical plane through a line of

greatest slope with the string making an angle with the inclined plane, and with the stick. Find the tension of the string and the least possible coefficient of friction.

7. Investigate the formula for finding the distance of the centre of mass of a system of particles from a plane.

8. A simple pulley sustains a weight W, and the rope which supports the pulley leaves the wheel at an angle a with the vertical. Find the tension of the rope, and examine separately the equilibrium of the wheel and the case.

9. Assuming the normality of fluid pressure, deduce its equality in all directions at the same point.

10. Find the least force required to open outwards a rectangular door on a vertical hinge to an exhausted vessel, when the top of the door is at a depth h below the surface of water, its breadth being b, and height 1.

11. Volumes v1, v, of gas at pressures P1, P2, and equal temperatures are introduced into a vessel of volume V. Find the pressure of the mixture.

12. The top of a barometer tube contains air, so that when the atmospheric pressure is 30 inches it reads 29 inches, and 4 inches is unoccupied by the mercury. Find the reading when the pressure is 29.5 inches, assuming the level of the mercury in the cistern and also the temperature

constant.