2. Shew how to solve two simultaneous equations of the forms

ax2 + by2 + cxy = d

a'x2 + b'y2+ c'xy d'.

Solve the equations

x2 + y2 = a2 + b2

xy=ab.

3. If the terms of a ratio be positive, the ratio is made more nearly equal to unity by adding the same quantity to each of its terms.

4. Define an arithmetical progression, and find the sum of n terms of an arithmetical progression, having given the first term and the common difference.

Five numbers whose sum is 10 are in arithmetical progression, and the sum of their squares is 30; find the numbers.

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

If tan 0 = , and 0 be in the third quadrant, find sin 0 and cos 0.

2. If A and B be positive angles, each less than a right angle, prove that

sin (A

- B) = sin A cos B

cos A sin B.

If A = 60° and b = 2c, find B and C.

4. Shew how to solve a triangle, having given two sides and the angle opposite one of them. If a 1, c√2, A = 30°, find b, B, C.

UPPER MATHEMATICS.

Professor Nanson.

(Candidates must answer satisfactorily in each of the three divisions of this paper.)

I.—1. Prove that the locus of the middle points of any system of parallel chords of a parabola is a straight line parallel to the axis.

2. Prove that each of the two tangents which can be drawn to a conic from any point on its directrix subtends a right angle at the focus.

3. Prove that the sum or difference of the focal distances of any point on a central conic is constant and equal to the transverse axis.

4. Prove that supplemental chords are parallel to conjugate diameters.

II.-1. If a1, a2

...

an be the roots of an equation f(x) =0 of the nth degree which is in its simplest form, shew that

2. If un, vn be the nth terms of two series in each of which every term is positive, and if u, v, be finite and different from zero however large n may be, prove that the series are both convergent or both divergent.

3. Prove the binomial theorem for a negative or fractional exponent.

4. State and prove the rule for forming successive convergents to a given continued fraction.

5. Define a recurring series, and find a formula for the sum of n terms of a recurring series of the second order.

7

III.—1. State and prove De Moivre's theorem.

2. Shew how to find the n nth roots of an expression of the form a + b √ 1.

3. Express cos "0 in terms of cosines of multiples of 0.

4. Find the sum of the sines of a series of angles in arithmetical progression.

5. Prove the rule of proportional differences in the. case of the natural tangent.

the result to find the differential coefficient of x".

2. If y = (z) and z = ↓ (x), prove that

3. Find the nt differential coefficient of ea* cos (bx+c).

4. Shew how to find the value of an expression which

assumes the indeterminate form

0

5. Shew how to find the maxima and minima values

of a function of one variable.

Find the maxima and minima values of

(x-a)2m (x - b)2N,

m and n being positive integers.

6. Find the angle between two straight lines whose equations are given, and deduce the condition that the lines may be (i) parallel, (ii) perpendicular.

7. Find the locus of the middle points of a system of parallel chords of a circle.

Find the locus of the middle points of chords which are at a given distance from the centre.

8. Define a parabola and find its equation.

Shew that the equation

y2 = ax + by + c

represents a parabola.

9. Find the equation of the tangent at any point of an ellipse.

Find the equations of the tangents to

x2 y2

+ =1

a2 b2

which make an angle a with the axis of x.

10. Find the relation between the eccentric angles of the extremities of conjugate diameters in the ellipse.

Prove that the locus of the intersection of the tangents at the extremities of conjugate diameters of an ellipse is a second ellipse.

11. Define a definite integral and an indefinite integral, and state the relation between them.

Prove that f(x) dx = − ƒ ̧¢(x)dx.