11. In obtuse angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and that part of the straight line without the triangle intercepted between the obtuse angle and the foot of the perpendicular.

If a triangle be isosceles, and one of its angles equals twice the sum of the other two, the square on the longest side will be three times as great as the square on either of the others.

12. The straight lines drawn at right angles to the sides of a triangle from the points of bisection of the sides meet in a point.

ALGEBRA AND MENSURATION.

Candidates are not permitted to answer more than nine questions in Algebra nor more than three in Mensuration.

2. Multiply the sum of a2+ 6ab + 962 and a2-6ab + 9b2 by the difference of a-2ab+b2 and a'+2ab-176.

3. Divide (x+1)+4(x+1)® +6 (x+1)*+4(@+ 1)2 + 1 by x+2x+2.

4. Prove that ama" = am¬n.

7. Show that (x - y)3 + (y — 2)3 + (≈ − x)3

=3(x − y) (y—2) (z — x).

8. Show that (— a3) = (x —— a) (x2 + ax +a2)

is not really an equation.

Solve the equation-

9. Solve the equation √ +8 + √x + 19=11.

10. Solve the equations

9x2 + y2 + 12xy + 11 = 0.

5x+2y-1=0.5

11. Prove the rule for the extraction of the square root of a number.

Apply your method to the extraction of the square root of 165649.

12. Find an expression for the sum of n terms of a geometric series whose first term is a and common ratio r.

Apply your method to the conversion of 1.3459 to a vulgar fraction.

13. A's share is of B's and C's together; A's and C's shares together are more than twice B's share by £217 and the whole amounts to £736, find their shares.

14. Three numbers are in A. P., the sum of the the first and second exceeds the third by 1: and the difference of the first and third is less than their sum by the same quantity. Find the numbers.

15. Two trains A and B start together to meet each other from stations 200 miles apart in opposite directions: the intermediate stations are 5 miles apart; A travels 30 miles an hour and stops 2 minutes at every fourth station, B stops 5 minutes at every station; find B's rate of travelling if the two trains meet each other when A has just travelled 120 miles.

MENSURATION.

[The answers need not be carried beyond two places of decimals.]

1. A triangular space which is isosceles and rightangled, is traversed by a fence parallel to the hypotenuse which divides the triangle into two equal parts, the length of the fence is 30 yards; find the sides of the triangle.

2. How many sleepers are required for a railway 22 ft. wide and occupying 8 acres of ground, the sleepers being laid at intervals of 24 yards?

3. If a cubic yard of stone weighs 73.5 times as muchas acubic foot of water, find in pounds the weight of a block of stone 9 ft. long, 3 ft. broad, and 1 ft. deep, a cubic inch of water weighing 252.5 grains.

4. Find the side of an equilateral triangle equal in area to a rhombus whose shorter diagonal is 42 ft. and side 35 feet.

MALE CANDIDATES-SECOND YEAR.

ALGEBRA AND MENSURATION.

Candidates are not permitted to answer more than TWELVE of these questions.

[NOTE. In all problems, where required, the circumference of a circle may be assumed to be (2)ths of the diameter. Not more than 2 decimal places are required in the answers.]

1. Prove the rule for finding the cube root of a

number.

Apply your method to finding the cube root of

67419143.

2. Explain and prove the method for proving a multiplication sum by casting out the nines.

Give examples to show that the method will not detect all cases of error.

3. Extract the square root of 7-4/3

and show under what conditions

a b c are positive integers.

4. Solve the equations

ã+ √b = √ē if

}

(1) x2-3xy + y2+190
x + y =24

(2) 22 +2+2 16896.

5. If abb: c

=

6. Assuming the Binomial Theorem to be true for (1 + x)", show that it will be also true for the first three terms of (1 + x)”+1.

Write down the middle term of (a + x) and the 6th term of (1 — π)—'.

7. Assuming the number of permutations of n things taken (1) together to be n (n - 1).. (n-r+2), find their number taken together.

r

If out of 32 men 8 firing parties are to be made, in how many of such parties might the same two men be found together?

8. Transfer 1,000,000 from the denary to the quinary scale.

In what scale will a number be expressed by 304 if it be expressed in the senary by 211?

given log10 6·7781513 log10 13 1139434

=

=

m log. a - n log

find, log10 216, log10 169, log10 2, log10 78,

10. Prove the formula M = P (1 + r)” where P and M are the number of pounds in the principal and amount of a given sum, r the interest of one pound for one year, compound interest being given.

Show by expansion of the above function that a sum of money will more than quadruple itself at 10 per cent. compound interest in 20 years.

11. The rent of a house and field is £129 10s. if the rent of each had been £4 15s. less, the rent of the house would have been 5 times as great as that of the field; find their rent.

12. A man walks from A to B at a certain rate : if he had walked half a mile an hour faster he would have walked the distance in 9 hours less, if a mile per hour faster in 16 hours less; find the distance from A to B.

13. Find the difference in feet of the perimeters of a circle and an oblong having the same area, viz., 496947 square feet, one of the sides of the oblong being three times as long as the other,

14. A grain of gold is beaten so thin as to cover 80 square inches; find the cost of the gold required to cover the curved surface of a cylinder 10 feet long and whose end has a radius of 4 inches, if an ounce of gold be worth £4.