7. In a plane triangle of which A, B, C are the angles, and (a), (b), (c) the sides subtending them.

Prove

b2+c2 - a2

Deduce from these expressions the value cos A = 2bc

Prove a+b2 + c2 = 2ab cos C+2ac cos B + 2bc cos A.

8. Find the area of a triangle in terms of any two sides and their included angle, and also in terms of the three sides. Find the area of the greatest triangle that has for two of its sides 50 and 60 feet.

9. Given the difference between the angles at the base of a triangle 17° 48′, and the sides subtending those angles 105 25 feet and 76.75 feet: find the angle included by the given sides. 10. An observer in a balloon when it is one mile high observes the angle of depression of a conspicuous object on the horizontal ground to be 35° 20', then after ascending vertically and uniformly for 20 minutes, he observes the angle of depression of the same object to be 55° 40′: find the rate of ascent of the balloon in miles per hour.

11. A tower which stands on a horizontal plane is 200 feet high, and there is a small loophole in the tower at a certain height above the ground; an observer is at a horizontal distance from the tower of 300 feet, but stands on a mound so that his eye is 12 feet above the ground on which the tower stands, and in that position, the angles subtended at his eye by the portions of the tower above and below the loophole are equal: find the height of the loophole from the ground.

12. In a circle which has a radius of 10 feet two chords AB, CD are drawn at right angles to each other, and intersecting in O, AO and CO are three and four feet respectively: find the sides and angles of the quadrilateral ACBD formed by joining the extremities of the chords.

PURE MATHEMATICS. (1.)

1. If two straight lines be at right angles to the same plane, they shall be parallel to one another.

2. If two straight lines be cut by parallel planes, they shall be cut in the same ratio.

3. If two straight lines do not intersect, and along one of them AB, BC, CD, &c. be taken all equal to one another, and along the other ab, bc, cd, &c. all equal to one another, the straight lines Aa, Bb, Cc, &c. shall be all parallel to a certain plane.

4. Define a logarithm, and prove that

5. Find the expansion of log. (1+x) in ascending powers of x. Prove that the coefficient of x" in the expansion of

COS

1

( + ON − 1) = 2 √2 {(e® + e −o) − √ = 1 (e° − e−0) }

4

7. If a be the circular measure of an angle, prove that

8. Find an expression for cos"e in terms of cosines of descending multiples of when n is an even integer.

9. Prove that the foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex.

A long rectangular slip of paper is folded so that one of the corners always lies on the opposite side. Prove that the crease always touches a parabola of which the aforesaid side is the directrix.

10. In the ellipse prove that if CP and CD be conjugate semi

diameters

CP”? + CD = CA* +CB”.
CA2

(9)

11. In the hyperbola the foot of the perpendicular from the focus on the tangent lies on the circle whose diameter is the transverse axis.

What portion of this circle corresponds to either branch of the hyperbola?

12. Show how to pass from one set of rectangular axes to another without changing the origin.

Find the angle through which the axes must be turned in order that the equation

Ax+By+ C = 0

may be reduced to the form x =

constant, and find the value of this constant. What is the geometrical meaning of this result?

13. Find any form of the equation of the ellipse, and interpret the

constants.

A number of ellipses are described upon the same major axis, prove that the locus of the extremities of their latera recta is a parabola.

14. Find the equation of the tangent at any point of an hyperbola and deduce that of the normal.

If the hyperbola be equilateral, prove that the portion of the normal intercepted between the axes is bisected by the curve.

PURE MATHEMATICS. (2.)

1. ACB, ADB are. two segments of circles on the same base AB; if through any point C in the arc ACB, two straight lines ACD, BCE be drawn to meet the arc ADB in D and E, prove that the arc DE is of constant length.

2. If two circles intersect each other, prove that their common chord bisects their common tangent.

3. Having given the base of a triangle, and the ratio of the sides, prove that the locus of the vertex is a circle.

6. If the sides of a triangle be in Arithmetical progression, the cotangents of its semi-angles will also be in Arithmetical progression.

7. A man walking towards a tower on the edge of which a flagstaff is fixed, observes that when his distance from the tower is c feet, the flagstaff subtends the greatest angle at his eye, and he finds that this angle is a; prove that the height of the flagstaff is 2c tan a.

8. Find the present value of an annuity of £A per annum, to continue for n years, allowing compound interest.

What is the present value of a perpetual annuity of £100 a year, to commence at the end of two years from the present time, allowing compound interest at 4 per cent. per annum?

9. Prove that the convergents to a continued fraction, taken in order, are alternately less and greater than the continued fraction. Find the continued fraction equivalent to √10 and its first four convergents.

2

Prove that the difference between any two consecutive odd convergents to a2 + 1 is a fraction whose numerator is 2a. 10. If a, b, and c be positive integers, prove that one solution of the equation ax by = c can always be found such that x and y are positive integers, and hence find the general solution in positive integers. Prove also that the successive values of x are in an arithmetical progression of which b is the common difference.

Find the least solution in positive integers of the equation 19y = 10.

15x

11. Find all the solutions, in positive integers, of the equation xy- 2x - y = 8.

13. Find the number of cannon balls in a pyramidal pile: 1st. When the base is an equilateral triangle having n balls in a side: 2nd. When the base is a square having n balls in a side. Hence show that a square pile is equivalent to two triangular piles.

x, find the coefficient of x2 and of x*.

15. A sum of money is distributed amongst a certain number of persons. The second receives one shilling more than the first, the third two shillings more than the second, the fourth three shillings more than the third, and so on. If the first person receives one shilling and the last £3. 78., what is the number of persons, and the sum distributed?

16. If an event may happen in different independent ways, prove that the probability of its happening is the sum of the probabilities of its happening in the different ways.

Find the chance of throwing 6 with two dice. Find also the odds against throwing 6 twice at least in three throws with two dice.