II.-ALGEBRA AND MENSURATION.

EXTRA PAPER.

(Algebra, up to and including the Binomial Theorem; the Theory and Use of Logarithms; and Mensuration.)

[Great importance will be attached to accuracy.]

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2. Resolve the expression 2(a + b) — ab(a2 + b2) (2ab3a2+362) into 5 simple factors.

3. Prove that the sum or difference of any multiples of A and B is divisible by all the common divisors of A and B.

What numerical value of y will make the expressions

2(μ3 + y2)x3 + (11y2 - 2y)x2 + (y2 + 5y)x+5y1
and 2(y2 + y)x2 + (11y − 2)x + 4

have a common measure other than unity?

6. A wine merchant bought a cask of sherry for £9, and after losing 3 gallons by leakage, sold the rest of the cask at 6s. per gallon above cost price, thereby realising a profit of 333 per cent. on his whole outlay. How many gallons did the cask contain?

7. There are 2m + 1 terms in Arithmetic Progression. The first term is a, and the last is b; what is the middle term?

A series whose 1st, 2nd, and 3rd terms are respectively

8. Determine what value of r will make the number of combinations of 2n things taken together greatest.

+ Cn+3

the symbol "C, being used to denote the number of combinations of n things taken together.

9. Expand (1 - x)-4 to 5 terms by the Binomial Theorem, and write down the r + 4th term in its simplest form.

Apply the Binomial Theorem to prove that the sum of the series

10. The base of the Napierian system of logarithms being defined

as the limiting value of (1+1)* when a is infinitely large, calcu

late its value to 3 decimal places.

Given log10 2 = 30103, log10 3 = '47712, find the values of

log10 (54) and log, (5′4)*.

12. The three conterminous edges of a rectangular block are 93, inches; find the length of its diagonal.

13, and 14

13. If a cubic foot of cast iron weigh 450 lbs., what will be the weight of a cast iron spherical shell whose external diameter is 6 inches and thickness an inch (π = 22)?

III.-EUCLID AND TRIGONOMETRY.

[Ordinary abbreviations may be used in answers to the first six questions, but the method of proof must be geometrical. Great importance will be attached to accuracy.]

1. If a straight line be divided into any two parts, prove that the squares on the whole line and on one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

Prove that the rectangle contained by the diagonals of the squares on the whole line and on one of the parts is equal to twice the rectangle contained by the whole line and that part.

2. Enunciate and prove a proposition which gives the excess of the square on the longest side of an obtuse-angled triangle over the sum of the squares on the other sides.

3. Prove that the sum of either pair of opposite angles of a quadrilateral inscribed in a circle is equal to two right angles.

Two circles intersect in the points P and Q. A straight line MPN is drawn, terminated by the circles in M and N. Through M and N tangents are drawn to the circles intersecting in T. Prove M, N, 2, and Tall lie on the circumference of a circle.

4. Describe an isosceles triangle having each of the angles at the base double of the third angle.

Describe an isosceles triangle having each of the angles at the base one-third part of the remaining angle.

5. If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, prove that the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

6. Draw a circle through two given points (1) to touch a given circle, (2) to touch a straight line given in position.

7. The tangent of an angle is 24. Find the cosecant of the angle, the cosecant of half the angle, and the cosecant of the supplement of double the angle.

11. If A', B', C' are the points in which the sides BC, CA, AB respectively of the triangle ABC are touched by the three escribed circles, and ▲ is the area of the triangle ABC, prove that the area A B C sin sin

of the triangle A'B'C' is 2A sin

12. Two sides of a triangle are 540 yards and 420 yards, and the included angle is 52° 6. Find the remaining angles.

13. The angle of elevation of the top of a tower rising out of a horizontal plane is cot-1 at a point A in the plane.

AB (= 32 feet) is drawn in the plane at right angles to the line joining A to the base of the tower; and the angle of elevation of the top of the latter is observed at B to be cot-1. Find the height of the tower.

HIGHER MATHEMATICS.

IV. CONIC SECTIONS AND DIFFERENTIAL
CALCULUS.

[Full marks may be obtained for about four-fifths of this paper. Great importance will be attached to accuracy.]

1. In the parabola the subnormal is of constant length.

The ordinate PN and normal PG are drawn at the point B of a parabola whose vertex is A and axis ANG. Prove that GP produced through P touches the parabola whose vertex is N and focus at A' such that A bisects NA'.

2. If CN and NP be the coordinates of a point of the hyperbola, with centre C, referred to the asymptotes as axes, prove that CN. NP is constant.

If CV be the perpendicular from C upon the tangent at P, prove that CP. CY is also constant when the hyperbola is rectangular.

3. Find the equation of a straight line (1) in rectangular coordinates, (2) in polar coordinates.

Taking the polar equations of two straight lines, find the condition among the constants that the lines may be at right angles to one another.

4. Through the point (3, 4) two straight lines are drawn each making the angle 45° with the line x - y = 2. Find their equations. Find also the area of the triangle between the two lines and the given line.

5. Find the equation of the circle of radius (a) which cuts the axis of x in the points A and B whose abscissæ are b and + b.

6. Find the equations of the diameter through A, and the tangent at B, and prove that this diameter and tangent intersect on the curve