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Q. 29. A uniform thin elliptic lamina has its major and minor axes of lengths 2a, 2b, respectively.

Find (1) the centre of gravity of one of the quadrants into which it is divided by the major and minor axes:

(2) The moment of inertia of this quadrant about the minor axis, also the moment of inertia about the parallel axis through its centre of gravity.

Q. 30. Apply the integral calculus to obtain the volume of the solid generated by the revolution about its minor axis of the portion of a given elliptic area included between the minor axis and a latus-rectum.

Find also the whole area of the bounding surface of this solid of revolution.

Q's 27, 29, were usually made out, but in Q. 30, candidates showed much weakness in dealing with the solid of revolution, though some of them, starting from the formulæ

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Q's 31, 32, on differential equations were generally taken, and in many cases were well answered.

The work sent up in this Stage included some excellent papers, and, on the whole, was distinctly satisfactory.

STAGE 7.

Results 1st Class,; 2nd Class, 6; Failed, 2; Total, 8.

Of the 8 candidates, one was very weak; each of the other seven made out three or four questions fairly completely, and did portions of some others, but no one attained the standard of a First Class in this Stage.

HONOURS IN DIVISION II.

Result 1st Class,; 2nd Class,; Failed, 1; Total, 1.

There was only one candidate; his work did not reach the standard of a pass in Honours,

DAY EXAMINATIONS.

STAGE 1.

Results 1st Class, 195; 2nd Class, 287; Failed, 101; Total, 583.

:

The work is of about the same standard as that of the Day Examination of last year.

Q. 1. What is a prime number?

A.

Write down the prime numbers that are greater than 10 and less than 20.

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What is the value of 5'91374 true to the nearest third place of decimals.

A common mistake was to give 06761 instead of 0'6762 for 0'67619.... true to four places.

Q. 2. Assuming that a metre equals 3281 feet, find which is longer, 10 miles or 16 kilometres. How much per cent. of the shorter length must be added to it so that it may equal the longer length?

The area of a hectare being 10,000 square metres, express a hectare as a decimal of a square mile.

Comparatively few found the correct results to all the parts of this question, though the methods were usually right.

Q. 3. Two trains, A and B, are moving uniformly in opposite directions along parallel rails. A's pace is 9-10ths of B's pace; find the rates per hour at which they are moving, if 10 minutes after passing each other they are 144 miles apart.

Was not often attempted, but nearly all who tried it sent up a correct solution.

Q. 5. A man buys eggs at 1s. 3d. a dozen, and sells them at 11s. 8d. per hundred; find his gain per cent.

How many eggs must he sell in order to make a profit of £1. It was a frequent error to say the gain was 15 per cent. when the profit on 100 eggs had been found to be 15 pence.

B.

Q. 7. Draw a straight line AB, and by use of your ruler and compasses find the middle point of AB.

Prove that your method is correct.

How would you proceed to obtain the middle point when AB is longer than the diameter of the greatest circle that can be drawn by your compasses?

Q. 8. In a triangle ABC the angle ABC is greater than the angle ACB; show that the side AC is greater than the side AB.

AC is a diagonal of a parallelogram ABCD, and AC is less than the side AB; show that the angle ABC is less than the angle CAD.

These two questions were often well answered.

Q. 9. State the axiom on which your proof of the following theorem depends :-If a straight line fall on two parallel straight lines it makes the alternate angles equal to one another.

Give the proof of the theorem.

In the quadrilateral ABCD the sides AB and CD are parallel and AB is half as long as CD; each of the diagonals AC and BD is as long as CD. Choosing a length of about 2 inches for AB, construct the figure accurately to scale.

Very many had an erroneous idea of the axiom required, and gave instead the definition of parallel straight lines.

Q. 11. Two given unequal parallelograms are equiangular to each other; show how to draw a straight line which shall divide the larger into two parallelograms, one of which shall be equal to the smaller given parallelogram.

There were a few correct solutions; not many selected this question.

Q. 12. Show that in a right-angled triangle the sum of the squares on the sides containing the right angle is equal to the square on the side opposite the right angle.

Construct a square so that its area may be five times the area of a given square.

Though often attempted, the second part was seldom correctly constructed.

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In the result you obtain, substitute a2 + b2 for c2, and show that the product then reduces to 4a2b2.

The multiplication was often wrong. Very few made out the final result.

Q. 14. (a) Divide x2 —'y2 - z2 + 2yz by x + y

2.

(b) Divide (3 a + 2b + c)2· (u+2b+3c) by a + b + c.

Q. 15. (a) Find the following expressions in factors:

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(iii) a1 + b1 c1 2a2b2.

(b) Write down the expression a2

C.

9x+14 in factors, and find

for what values of x the expression will have a positive value.

Questions 14 and 15 (a) were well done. In Question 15 (6) not many got further than writing down the given expression in factors.

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Q. 18. Divide £150 between A, B, and C, so that B and C together may receive £20 more than 4, and that B's share may exceed C's by 24 times as much as A's exceeds B's.

There were many very good answers to these two questions.

STAGE 2.

Results 1st Class, 70; 2nd Class, 401; Failed, 133; Total, 604.

The results are not quite as good as those of the Day Examination of last year.

A.

Q. 21. State what is meant by the projection of a finite line upon another line.

Show that 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 other two sides, together with twice the rectangle contained by either of them and the projection of the other upon it.

The base BC of an acute-angled triangle ABC is produced through B to D, and through C to E, so that DB = BC = CE, and AD, AE are joined; show that

AD + AE AB2 + AC2 +、DC2.

=

22. From one end of a diameter of a circle a straight line is drawn at right angles to the diameter. Show that the line so drawn does not cut the circle.

Define a tangent to a circle.

P is a given point outside a circle, whose centre is A. Show how to produce AP to Q, so that the tangent of the circle drawn from Q may be twice as long as the tangent drawn from P.

Q. 23. If two opposite angles of a quadrilateral are together equal to two right angles, show that a circle can be described through the four corners of the quadrilateral.

ABCD is a quadrilateral inscribed in a circle; the sides AB, DC are produced to meet at E, and the sides BC, AD are produced to meet at F; show that the bisectors of the angles at E and F are at right angles to one another.

24. Show how to inscribe a circle in a given triangle.

A circle inscribed in a triangle ABC touches the sides at points D, E, F; find the angles of the triangle DEF in terms of the angles of ABC.

The bookwork propositions in these questions were on the whole well written out, the chief defect being in Q. 24, where the candidates frequently omitted the proof that the circle touches the sides of the triangle. The deductions in the latter part of these questions were done correctly fairly often,

Q. 25. Let AB be the straight line joining A and B, the centres of two intersecting circles; let the common chord of the circles cut AB in M. From any point N in AB draw a line NP at right angle to AB, and from any point in NP let tangents be drawn to the circles; show that the difference between the squares on the tangents equals twice the rectangle AB. MN.

Q. 26. Find the locus of a point which moves so that the sum of the squares of its distances from two fixed points is constant.

Construct the locus to scale when the fixed points are two inches apart and the sum of the squares is 10 square inches.

These two questions were not often selected, but were rightly made out a few times.

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(c) Find the square root of (x − 9) (x − 1)3 + 64x.

(a) was not often made out ; (c) was well done a good many times.

Q. 28. (a) If a2 5ab6b2 = 0, show that b must be either the half or

the third of a.

(b) Show that the square of any odd number increased by 3 is divisible by 4.

(c) Explain whether there be such an odd number that its square, diminished by 3, is divisible by 4.

Often answered; the reasoning was usually well given.

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+ 16, +

(c) (2a − b — x)2 + 9(a

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(a) was frequently solved; (b) less often; (c) a few times.

Q. 30. (a) If a, ẞ are the roots of the equation

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= a and aß =b,

without solving the equation. Also find the value of a2 + ß

in terms of a and b.

(b) Without solving the equations, find two relations between the
roots of x2
O and the roots of bx2
ax + b

The answers, especially those to (6), were very feeble.

2αx+4=0.

Q. 31. Find the ratio of two numbers, which are such that their product is a mean proportional to the sum of their squares and the difference of their squares.

Also express the ratio as a decimal true to three places.

N.B.

√5=2*236068.

9897.

B

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