8. Prove that if a number of two digits is equal to four times the sum of the digits, then the number with the same digits but in opposite order is equal to seven times the sum of the digits.

9. The united ages of a man and his wife are 24 times the united ages of their children. Two years ago their united ages were three times the united ages, at that time, of the same children and five years hence the united ages of the same children, if living, will be of the united ages of their parents How many children are there?

10. Find two numbers whose difference, sum, and product are as the numbers 1, 2, 3.

For Admission as Teacher of the Second Class.

(Three hours allowed.)

b:a

bcded.

1. (a) If a:b::e:d, show that a (b) What number must be subtracted from each of the numbers 9, 12, 13, 22, in order that the remainders may be proportional?

2. (a) If a + vb = c + vd, neither nor d being a perfect square, prove that a = c, and b = d.

(b) Find the square root of 174 +36 6, and show where the foregoing theorem is applied in the course of your work.

3. (a) Form the quadratic equation which has 13 for the sum of the squares of its roots, and 6 for the product of its roots.

(b) Show that the difference of the roots of the equation x2 + ax + b = 0 is equal to the difference of the roots of the equation x2 + 3ax + 2a2 + b = 0.

4. (a) If a + b, c + a, be, are in Harmonical Progression, show that a2, b2, c2 are in Arithmetical Progression.

(b) The sum of the first four terms of a Geometrical series is to the sum of the first eight terms as 81 is to 97. Find the common ratio.

5. How many different numbers greater than a million can be made with the digits 0, 1, 2, 2, 2, 3, 3:

6. Solve

(a) 3x-57x3 - 648 = 0.

(b) xy2 + x3y = 84; x3 + y3 = 91.

7. Give the third term, reduced and simplified, in (1 - x) ̄*· 8. If the whole number of persons born in any month be of the whole population at the beginning of the month, and the number of persons who die, find the number of months in which the population will be doubled. Given, log 2 = 3010300, log 34771213, log 7 = 8450980.

9. Two boys start from the right angle of a triangular field, and run along the sides with velocities in the ratio of 13:11. They meet first in the middle of the hypotenuse, and again 64 yards from the starting point. Find the length of each side of the field.

H

GEOMETRY.

18th December-Afternoon, 2 to 5.

DIRECTIONS TO EXAMINEES.-1. Draw figures where necessary to explain the answers. 2. When an answer is shown on more than one page, the figure must be repeated on each page.

NEW SCHEDULE.

For Admission as Pupil-Teacher of the Second Class.

(One hour and a-half allowed.)

1. What is a parallelogram? Name and define three special kinds of parallelogram.

2. A cuboid is 3 feet long, 1 ft. 9 ins. wide, and 1 ft. 3 ins. deep. Draw its net, scale I foot to an inch, and mark the lengths.

3. A circle is said to be symmetrical with r gard to a diameter." Explain this statement. In such circumstances what name may be applied to the diameter ?

4. (a) Construct a triangle ABC, having BC= 4 inches, angle B = 90°, angle C = 30°.

(6) Construct a triangle A'B'C', having B'C' = 2 inches, angle B' = 90°, angle C' 30".

(c) Measure and compare the other sides of the two

triangles.

5. P, Q, R are three villages P lies 4 miles to the N. E. of Q, and Qlies 11 miles to the N.W. of R. Find the distance and bearing of P from R. (Scale, 2 miles to an inch.)

6. A man standing on the bank of a river sees a tree on the far bank in a direction 20 W. of N. He walks 200 yards along the bank, and finds that the direction of the tree is now N.E. If the river flows east and west, find its breadth. (Scale, 50 yds. to an inch.)

7. A man observes the angle of elevation of the top of a spire to be 23; he walks 40 yds. towards it, and then finds the angle to be 29. What is the height of the spire? (Scale, 40 yds. to an inch.)

NEW SCHEDULE,

For Admission as Pupil-Teacher of the Third Class. (One hour and a-half allowed.)

1. Specify all the conditions under which two triangles are congruent.

2. How many triangles can be formed, two of whose sides are 3 inches and 4 inches long, and the third side an exact number of inches? Explain clearly how your answer is arrived at. 3. If a straight line cuts two parallel straight lines, then (a) the alternate angles are equal, () the corresponding angles are equal, (c) the interior angles on the same side of the cutting line are together equal to two right angles.

4. If two sides of a triangle are unequ-1, the greater side has the greater angle opposite to it.

5. The locus of a point which is equidistant from two fixed points is the perpendicular bisector of the straight line joining the two fixed points.

6. The straight line joining the mid-points of the sides of a triangle is parallel to the base.

NEW SCHEDULE.

For Admission as Pupil-Teacher of the Fourth Class. (One hour and a half allowed.)

1. Show how to construct a triangle equal to a given quadrilateral. (Construction and proof required.)

2. If two triangles have two sides of the one equal to two sides of the other each to each, and the included angles unequal, the triangle which has the greater included angle has the greater third side.

3. (a) Give a geometrical illustration of the algebraical identity (a+b) (c + d) = ae bead Ed.

(b) Express as an algebraical identity-If a straight line is divided into any two parts, the rectangle contained by the whole line and one of the parts is equal to the square on that part together with the rectangle contained by the two parts.

4. Construct a straight line whose length shall be exactly 5 inches. (Give the necessary explanation.)

5. In an obtuse-angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle plus twice the rectangle contained by one of those sides and the projection on it of the

State in words the geometrical theorem corresponding to the above algebraic formula, and prove the theorem.

3. Through a point 14 inch outside a circle of radius 14 inch draw a line to pass at a distance of 1 inch from the centre. What is the length of the part of the line inside the circle?

6. In equal circles equal chords are equidistant from the centre. (1) Prove the theorem, and (ii.) enunciate the converse theorem.

7. Construct a rhombus given that the distance between the parallel sides is half the length of a side,

& Construct an interior common tangent to two circles.

9 Briefly describe the method of doing the following

For Admission as Teacher of the Second Class.
Three hours allowed. "

1 The internal bisector of an angle of a triangle divides the appose te side internal y in the ratio of the series containing the angle.

2. In a machine power as being transmitted from one shaft to another parachel shaft by means of a belt passing over two wheels. one 6 feet and the other 2 feet in diameter. If the centres of the wheels are & feet apart, and assuming the belt to he taut ard not mossed between the wheels find the length of Felting from the last point of contact on, the one wheel to the first pint of dulcisc† on the other.

$ Find the mean proportional between two given straight

4. There are twe fixed concer tricaroles AB is a variable diameter of the one, and P a variable point on the other, Pove that AP - BP remains constant.

6. If the line joining two points subtends equal angles at two other points on the same side of it, the four points lie on a circle.

7. Each of two equal circles passes through the centre of the other. AB is their common chord. Through A is drawn a straight line cutting the two circles again in CD. Prove that triangle BCD is equilateral.

8. Transform an equilateral triangle of side 2 inches into an equivalent triangle with a side of 3 inches and an angle of 60* adjacent to that side.

9. The ratio of the areas of similar triangles is equal to the ratio of the squares on corresponding sides.

10. ABC is any triangle and AD is the perpendicular drawn from A to the base BC. Prove that AB - AC = BD2 – DC2. If AB is 4 inches long, AC 2 inches, and BC 5 inches, find the lengths of BD and DC.

EUCLID-OLD SCHEDULE.

For Admission as Teacher of the Third Class.
Three hours allowed.)

1. Detine polygon, rhombus quadrilateral, median, orthe centre, segment of a circle, and gnomon. What is meant by the converse of a proposition? Illustrate by reference to Book II, Prop. 5.

2. Prove that the angles at the base of an isosceles triangle are equal.

3. Construct a triangle, the sides of which shall be respectively equal to three given straight lines, any two of which are together greater than the third.

4. Prove that parallelograms on the same base and between the same parallels are equal.

5. Prove that if a straight line be divided into two parts. the squares on the whole line and on one part are together equal! to twice the rectangle contained by the whole and that part. with the square on the other part.

6. Divide a straight hne into two pirts, so that the rectangle contained by the whole and one part may be equal to the square on the other part.

7. Prove that the ag é a paraleeran bisect one another.

8. Prove that the 'uedians of a triangle are concurrent.

9 Prve that of two sides of a tranche unequal, and if from their point of intersect) & three strieht lines be drawnnamely, the besector of the angle at that point, the median, and the slime-the first shall be internnolate, both in magnitude and position, tɔ the ot`er twu,

10. Construct a track, bacing given the base, the vertical angle, and tire diference of the s des

ARITHMETIC.

15th December-Morning, 9:30 to 12.

Set down the working of the sums, and draw diagrams where necessary, so that the process by which each answer is obtained may be clearly seen. Full marks will not be given if the working is not set down in a neat and orderly way.

(Questions 1 to 6 were the same as set for Admission as PupilTeacher of the First Class.)

7. A floor 20 feet 4 inches long and 16 feet 8 inches wide is laid with square tiles of the largest size possible without cutting; what is the size of the tiles, and how many are needed to cover the floor?

8. A district is 460 miles long and 360 miles wide; what size sheet will be needed to make a map on the scale of inch to the mile, with a border of 2 inches?

9. The gable of a house is 50 feet wide and 14 feet high; what is the length of the rafters? [Make no allowance for projection over the side of the house.]

10. Draw an equilateral triangle ABC and AD perpendicular to the base BC. The length of AB is 1 foot. Find the area of the square on AD, and the height and the area of the equilateral triangle.

11. Find the cost of putting a curbstone 15 inches broad, round a well 7 feet in diameter, at a cost of 7d. per square foot.

MISCELLANEOUS.

17th December-Morning, 9:30 to 12.

Only ten questions to be answered.

1. (a) Construct a scale of one inch and three-quarters to a yard, showing feet, and long enough to measure three yards and a-half.

(b) Using this scale, construct the plan of the top of a table, two yards one foot long, and one yard two feet wide.

2. Construct an isosceles triangle having an altitude of one inch and three-quarters, and angles at the base equal to the given angle.

3. Write an account of the successive steps by which the will of a majority of the electors of our State becomes the law of the land. Your answer should deal with the election of members of Parliament, with the constitution of Parliament, and with the steps by which a Bill becomes an Act of Parliament.

4. Write brief but clear explanatory notes on the following:-The Suffragettes, Old Age Pensions, The Franco-British Exhibition, The American Fleet, The Braddon Clause.

5. Enumerate the most important items of news that you have read during the present year, relating to India, China, Persia, and the United States; and write a concise note on each item.

6. (a) What is meant in needlework by the selvedge way of calico? What is the other way called?

(b) If a piece of calico supplied to you had no selvedge, how would you find out which was the selvedge way? Why is it important to be able to do this?

7. (a) In what direction is button-hole stitch worked-right to left or left to right? Mention two other stitches that are worked in the same direction.

(b) Why should a button-hole at the end of a band have one end made round and the other square or barred? Where should the round end of the button-hole be worked-near the end of the band or away from it? (c) Mention a purpose for which button-hole stitch is often used, other than for making button-holes.

8. (a) Explain briefly the meaning of the signs, words, and marks of expression that occur in the following piece of music.

(b) What is the key in which the piece is written? What is a key signature?

(c) Write a great stave and place the G, C, and F clefs in position.

9. (a) Explain the time signature .

(b) Write the first two bars of the piece in

time.

(c) Write out the scale in which the piece is written, marking the semitones where they occur.

State in words the geometrical theorem corresponding to the above algebraic formula, and prove the theorem.

5. Through a point 1 inch outside a circle of radius 11⁄2 inch draw a line to pass at a distance of 1 inch from the centre. What is the length of the part of the line inside the circle?

6. In equal circles equal chords are equidistant from t centre. (i.) Prove the theorem, and (ii.) enunciate the conv theorem.

7. Construct a rhombus given that the distance bet

parallel sides is half the length of a side.

8. Construct an interior common tangent to tw

9. Briefly describe the method of doing

exercises :

(i.) Describing a circle equal to the

circles.

(ii.) Describing an ellipse mechani (iii.) Trisecting an arc of 90° wit'

10. Two equal chords of a cir segments of the one chord are res of the other.

6. If the line joining two points two other points on the same side of circle.

7. Each of two equal circles other. AB is their commor straight line cutting the tw triangle BCD is equilatera

8. Transform an eq' equivalent triangle w adjacent to that side

9. The ratio ratio of the squ»

10. ABCi from A to t If AB is lengths

For Admission

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