II.

1. Shew how to solve two simultaneous equations in two unknowns, one equation being of the first degree and the other of the second degree.

Solve

ax + by = ap + bq

ci + đường = a2 +pq.

2. A ratio is made more nearly equal to unity by adding the same quantity to each of its terms. Arrange in order of magnitude

3. Define an arithmetical progression, and prove the formulæ for the general term and the sum of any number of terms.

Find the sum of all the numbers between 100 and 1,000 which are divisible by 3.

4. State and prove the binomial theorem for a positive integral exponent.

Find the two middle terms in the expansion of

1. Define the trigonometrical ratios of an angle, and express all the other ratios in terms of the

tangent.

a/b and 0 be in the third quadrant,

If tan

find the values of sin 0, cos 0.

E

2. If A, B be positive angles each less than a right

angle, prove that

3. Prove that in any triangle

B C b -c A

tan

2

=

b+c 2*

cot

If A = 30° b = c √3, find B, C.

4. Discuss the ambiguous case in the solution of triangles.

If B, b, c are given, and a1, a2 are the two values of a, prove that

a1 + a2 = 2c cos B

a1 ac2 - b2.

PURE MATHEMATICS.-PART II.

The Board of Examiners.

1. Find the equation of a straight line in terms of the magnitude and direction of its distance from the origin.

Determine the length of the perpendicular from the origin on the join of the points x, y1; X2, y2.

2. Find the general polar equation of a circle.

Determine the locus of the middle points of all chords of a given circle which pass through a given point.

3. Find the equation of any tangent to a parabola in terms of the angle it makes with the axis of the parabola.

Find the equation of the locus of the intersection of two tangents to a parabola which make a given angle with one another.

4. Shew that the sum of the squares of two conjugate diameters of an ellipse is constant.

Find when the product of two conjugate diameters is greatest and when it is least.

5. State and prove the rule for differentiating a function of a function.

Differentiate

sin 1+2 in 1 + sin(1+)"

x

6. Prove that subject to certain conditions

f(a + h) = f (a) + hf' (a) + {h2ƒ” (a + 0h) where is a positive proper fraction, and state the conditions.

Apply this formula to justify the use of the principle of proportional parts in the case of the logarithmic sine of an angle, except when the angle is small or nearly equal to a right angle.

7. Investigate a rule for finding maximum and minimum values of a function of one independent variable.

Find the maxima and minima values of

8. Define the curvature of a curve, and prove the

9. State and prove the rule for integration by substitution.

10. If ƒ (x, y) be a rational algebraic function of x, y, shew how to reduce

ff(x, y)dx,

where y2= ax2 + 2hx + b, to the case of rational fractions.

Find

do

a cos2 0 + 2h cos 0 sin 0 + b sin2 0

11. Shew how to find the area bound by the curve r = f(0) and by two radii vectores.

Find the whole area of the curve

(x2 + y2)2 = a2x2 + b2y2.

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

Find the volume generated by the revolution of the curve

(x − a)y2 = (b − x)3

about its asymptote.

PURE MATHEMATICS.-PART III.

The Board of Examiners.

1. If u is a function of x, y, z, and x, y, z are functions of t, state and prove the rule for finding the differential coefficient of u with respect to t.

If

find

m(x + y")=na"-2mxy",

d2y dx2

2. Determine the conditions for a maximum or minimum value of a function of three variables which are connected by two equations.