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Q. 51. (a) The sum of n terms of the series 1, 2, 3, 4, 500500; find n.

(b) Show that the sum
progression

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of 100 terms of the geometrical

8, 12, 18, 27, . .

exceeds the sum of 100 terms of the arithmetical progression

by

8, 12, 16, 20, . . . .

81 (15)96-20616.

(c) Find the harmonical progression whose third term is 5, and whose fifth term is 9.

These two questions were frequently answered in very good style.

Q. 52. Adopting the usual notation, assume that

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C' n + 1an + 1

State carefully the reasoning by which it can be shown hence that the Binomial Theorem is true for all positive integral values of n.

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in ascending powers of x, as far as x4 (inclusive).

Was not often well done. A clear statement of the reasoning was rarely made, and in the example numerical mistakes often occurred. More attention ought to be paid to the Binomial Theorem.

C.

Q. 53. (a) Show, from a geometrical construction, that

sin 24 2 sin A cos A,

where 24 is less than a right angle.

(b) If A is an angle between one and two right angles, show, without a new diagram, that the formula just given holds good.

(c) Express cos2 24 sin 24 in terms of sin 64 and sin 24.

(a) and (c) were well answered, though few made out (b) properly. Theorems on angles greater than a right angle are very difficult to most students, and require much careful teaching before they are clearly understood.

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where x denotes the measure, in radians, of an acute angle.

These two questions were attempted freely; Q. 54 was often fully answered, and the first part of Q. 55 was nearly always done correctly, but there were not many answers to parts (i) and (ii) of the question.

Q. 56. In a triangle ABC a line is drawn to bisect the angle A and to meet BC in D ; find the length of AD in terms of a, b, and

Show that twice the area of ABC equals
BC.AD cos(B − C).

Was answered a fair number of times.

A.

Q. 57. (a) Show that the ratio of the radii of the inscribed and circumscribed circles of the triangle ABC is

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(b) If the inscribed circle passes through the orthocentre, show

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The first part (a) was often made out; (b) was established now and then.

Q. 58. Find the number of angles in two regular polygons, such that an angle of the one may be six-sevenths of an angle of the other.

Of all pairs of polygons which satisfy this condition find the pair with the largest number of angles.

A good many answered the first part, and several were also successful with the second part.

STAGE 4.

Results 1st Class, 5; 2nd Class, 4; Failed, 5; Total, 14.

The number of candidates was small, but the work as a whole was decidedly satisfactory.

The questions in Solid Geometry were generally answered rightly. In Descriptive Geometry the diagrams were carefully drawn, and, as a rule, full explanations were given of the methods of construction.

Good work was done by the candidates who took up Geometrical Conics, but only in three or four cases were the questions on Spherical Trigonometry really well answered.

HONOURS IN DIVISION I.

Results 1st Class, 5; 2nd Class, 48; Failed, 52; Total, 105.

There were 105 Candidates who took the paper, i.e. thirteen more than there were last year.

All the questions were answered once at least, except Q. 84 (6), which was several times attempted in vain.

Q. 84. (a) Show that the sum of the squares of the coefficients in the expansion of (1 + x)" is

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where n is a positive integer.

(b) Find the coefficient of x" +4 in the expansion of
(1 + x)"log (1+x)

where <1, and n is a positive integer.

(c) Show that the coefficient of x" in the expansion of

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Q. 89. (a) If a straight line drawn through S, one of the centres of similitude of two circles whose centres are C and C', cut one circle in P, Q, and the other in P', Q', show that the rectangles SP. SQ and SQ. SP' are each equal to the rectangle contained by the tangents from S to the circles.

(b) Show how to describe a circle to pass through a given point and touch two given circles.

Q. 90. Given two sides of a quadrilateral inscribed in a circle and the ratio of the other two sides; show how to construct the quadrilateral, (a) when the given sides are adjacent, (b) when the given sides are opposite.

In Q's. 89 and 90 the first parts were answered fairly often; but there were very few correct answers to the second parts.

Q. 92. Two given circles have their centres at a distance d apart, one of them with centre I and radius r being entirely within the other, whose centre is 0 and radius R; MN is a chord of the outer circle drawn so as to touch the inner circle, and C is the centre of the circle which passes through M, N, and I. Show that the locus of C is a circle of radius

R2d2
2r

having its centre at 0.

Was answered far less often than might have been expected.

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and if a, 8, 7, 8, are different positive angles, each less than 2, show that

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(a) was well answered six times but (b) only once.

Nos. 71400 and 71202 sent up very good answers. It was No. 71400 who answered Q. 95 (b), and a very good answer it was.

As usual the candidates showed a distinct tendency to pay more attention to questions in Algebra (81–86) than to the other questions.

STAGE 5.

Results 1st Class, 79; 2nd Class, 198; Failed, 112; Total, 389.

The general results are good, but not quite as good as those of last year. The questions in Descriptive Geometry were taken fairly often, and the drawing was usually good, but the necessary explanations of the methods adopted were in many cases not clearly stated.

The work in the more elementary portions of Analytical Geometry showed a sound knowledge of the principles and methods of the subject, but want of accuracy in the arithmetical calculations frequently marred the results. In the more advanced part of the subject weakness was shown in dealing with the general equation of the second degree. Thus in the first of the two examples in

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represents a parabola having its vertex at (3, 4). Give a diagram in each case.

Very few candidates discovered that the left hand side of the former equation was the product of two linear factors, and even they were frequently unable to interpret the result.

In the Differential and Integral Calculus most of the work was very satisfactory.

Q. 8. Explain fully what is meant by a differential coefficient, and give an illustration taken from the motion of a point along a straight line.

Find, from first principles, the differential coefficient of x, and of cos x.

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Q. 9. Find the equation of the tangent to the curve

y = f(x) at the point (x'y').

In the catenary

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show that the length of the perpendicular let fall from N, the foot of the ordinate PN, upon the tangent at the point P, is of constant length.

Also, if the normal at P meet the axis of x at G, show that PG varies at PN2.

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The methods required in these three questions were well understood, and the candidates seemed to have had considerable practice in obtaining numerical results.

Q. 11. The base of an uniform solid cone is a circle of radius a, and the altitude is h.

Apply the integral calculus to obtain

(1) the volume;

(2) the position of the centre of gravity;

(3) the moment of inertia about the axis.

Q. 12. The axes of an ellipse are of lengths 6a, 12a. Find the area of the segment cut off by a line through one extremity of the major axis, and inclined 45° to that axis.

Show also that the volume generated by the rotation of this ellipse about the major axis is half the volume generated by its rotation about the minor axis.

These two questions, though not quite so often taken, were answered correctly a good many times.

STAGE 6.

Results 1st Class, 25; 2nd Class. 24; Failed, 12; Total, 61.

The results in this stage are very good. Several excellent papers were sent in, and quite a large proportion of the candidates deserve a First Class.

Q. 21. Obtain the equation of a plane in the form

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in the line PQ. A plane is drawn through the point (21, 4), perpendicular to PQ.

Show that this new plane contains the

point (2, 3, 5).

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