6. What do you understand by international bimetallism? Notice some principal objections to such a system.

7. Discuss the "infant industries" argument for protection. 8. What witness do the exports and imports of Great Britain bear (a) to its relations with other countries, (b) to its prosperity? 9. State the chief criticisms passed by Adam Smith on the Mercantile System.

10. State Adam Smith's four maxims of taxation. Criticise the first.

QUESTIONS FOR ORDINARY DEGREE OF M.A., FOR FIRST SCIENCE, AND FOR FIRST M.B., Ch.B. EXAMINATIONS.

MATHEMATICS (FIRST PAPER).

4TH OCTOBER 1900.-9 TO 11 A.M.

(Candidates must satisfy the Examiners in each of the subjects.) I. GEOMETRY.

1. Prove completely that the angle at the centre of a circle is double an angle at the circumference standing on the same arc, and deduce that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

Two circles ABC, ABD, with centres O and Q, intersect at right angles; AC and AD are two chords at right angles. Prove that the angle QOB is equal to the angle ABQ, and that C, B, D, are in one straight line.

2. Describe an isosceles triangle having each of the angles at the base double of the third angle.

The square on a side of a regular pentagon inscribed in a circle is greater than the square on a side of the regular decagon inscribed in the same circle by the square on the radius.

3. If two triangles have one angle of the one equal to one angle of the other, and the sides about these angles proportional, they shall be similar.

ABC and ADE are similar triangles; ABC is fixed but ADE is rotated round A. Show that the locus of the intersections of the straight lines which join the corresponding vertices B and D, C and E, is the circle circumscribed about the triangle ABC.

4. If a transversal cut the sides, or the sides produced, of a

triangle, the product of the three alternate segments is equal to the product of the other three.

If a circle be circumscribed about a triangle, the points in which tangents at the three vertices meet the opposite sides are collinear.

5. If a Harmonic Pencil be cut by a transversal parallel to one of the rays, the part of the transversal between two other rays is bisected by the conjugate ray. Prove this, and show that every transversal is cut harmonically by a Harmonic Pencil.

6. When is (i) a straight line perpendicular to a plane, and (ii) a plane perpendicular to a plane?

If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.

Perpendiculars, AE, BF, are drawn to a plane from the two points A, B above it, and a plane is drawn through A perpendicular to AB. Show that its line of intersection with the given plane is perpendicular to EF.

II. TRIGONOMETRY.

1. State and represent graphically the variations of cote and of sin + cos e, as e increases from 0 to 2 π.

(2) cos e cos, sin • sin ¢= √5,
(3) tan-10+tan-1 20+tan-i 30=T,

giving the general solution in (1).

4. I observe the angle of elevation of the top of a tree to be 15°, and that of a taller tree, 60 feet behind it, to be 30°. I walk towards them until their tops are in a line at a common elevation of 45°. Find the distance I walk, and the height of the smaller tree.

5. Find the expression for the cosine of half an angle of a triangle in terms of the sides, and find the greatest angle in a triangle whose sides are 5, 6, 7 respectively.

Prove that in any triangle a cos

B-C
2

=(b+c) sin

A
2

6. Prove that (cos +i sin e)" = cos ne+i sin ne, when n is any positive or negative integer. Hence find the six values of (-1).

1. State all the steps by which we arrive at the meaning of such expressions as a, a, a-3.

Simplify ((2-)-4 × (21)! × ((§) −1)i × (41) − }},

and √31 +460- √31 −4√60.

3. Prove that a quadratic equation can have at most two roots. If a, B be the roots of x2-px+q=0, show that

(a” +ßn) − p(an−1+ßn−1)+q(an−2+ ßn−2) = 0,

and find the sum of the fourth powers of the roots of x2-x+1=0.

be three finite fractions, then in general each is

5. Define an Arithmetical Progression. If s be the sum of an A. P. whose first term is a and last l, find the number of terms and the common difference. The first term of an A. P. is 10, the sum of 25 terms is 200; find the 15th term and the last. 6. Find the number of permutations of n things taken r together.

How many different "words" may be formed of the first 8 letters of the alphabet, each containing 6 letters? In how many of these will the letters a, b, be adjacent?

IV. CO-ORDINATE GEOMETRY.

1. Find the perpendicular distance of a given point from a given straight line.

Hence find the equations of the straight lines bisecting the angles between the straight lines.

4x+3y-24=0 and 3x+4y-12=0.

2. Draw the loci represented by the following equations(1) 5x-3y=10; (2) x2-4y2=0; (3) x2-y2=9;

π

(4) r cos
cos (-)=3; (5) x cos 10° + y sin 10° = 4.

3. Find the equation to a straight line through a given point and making a given angle with a given straight line.

The points (3, 4), (− 1, − 2), are the extremities of one side of a square. Find the equations to two other sides and a diagonal. 4. What two straight lines are represented by the equation 2y2 — 3xy — 2x2+10x+5y-12=0? Show that these lines are at right angles to one another.

5. Find the equation of a circle referred to rectangular axes. Find the radius of the circle 1+b2(x2+ y2) − 2ax – 2aby=0. Prove that the circle

x2+y2-2ax cos 0-2by sin e-b2 cos2 0=0 intercepts a length 2b on the axis of y.

6. Find the condition that the straight line y=mx+c may touch the circle x2+y2=a2.

Find the equations of the two tangents to the circle x2+y2=9, which make an angle of 30° with the axis of x.

MATHEMATICS-(FIRST PAPER).

26TH MARCH 1901-9 TO 11 A.M.

(Candidates must satisfy the Examiners in each of the subjects.)

I.-GEOMETRY.

1. The line joining the mid-points of the sides of a triangle is parallel to its base and equal to half of it.

D is the mid-point of the base BC of a triangle ABC, BM and CN are drawn perpendicular to the bisector of the angle A. Prove that M and N lie on the circumference of a circle which has D for centre.

2. The opposite angles of a quadrilateral inscribed in a circle are supplementary.

Perpendiculars are drawn to the sides of å triangle from any point on its circumcircle. Show that the feet of the perpendiculars are collinear.

3. Find a mean proportional between two given straight lines. Two circles touch one another and also touch a given straight line. Show that the part of the line intercepted between the points of contact is a mean proportional between the diameters of the circles.

4. If three planes intersect two and two, show that their lines of intersection are parallel or concurrent.

Prove that in all cases two straight lines which are parallel to a third straight line are parallel to one another.

5. If the points of section of a pencil of four rays by any one transversal form a harmonic range, and if through one point of section a line be drawn parallel to the conjugate ray and terminated by the other rays, it is bisected at the point of section ; prove this and the converse, and also that the points of section of the pencil by every transversal will form a harmonic range. Hence show that the arms of an angle and its internal and external bisector form a harmonic pencil.

6. OP, OQ are tangents to a parabola whose focus is S, at the points P, Q; show that the triangles SPO, SOQ, are similar. Three tangents to a parabola intersect and form a triangle; prove that its circumcentre passes through the focus.

7. Prove that an angle at the centre of a circle of any size is correctly measured by the ratio of the arc on which it stands to the radius of the circle.

Transform, approximately, to sexagesimal measure the angle which in circular measure is 2.

Assuming that an object which subtends less than a minute of angle at the eye ceases to be visible, find in feet how far off a man six feet high can be seen.

8. Prove

(i) cos (A+B)=cos A cos B-sin A sin B, when A and B are both acute and their sum obtuse;

(ii) cos 10a+cos 8a+ 3 (cos 4a+cos 2a)=8 cos a cos3 3a; (iii) cos 4a cos a-cos 3a cos 2a=sin 4a sin a- sin 3a sin 2a. 9. Find all the values of x which satisfy tan x=tan a, and verify your result by tracing the graph of y=tan x.

Show by a general solution of the two simultaneous equations, tan (20+30)=cot (30+24) and tan (20-3)=cot (30 - 2p), that and must both be multiples of 18°.

10. Show that if in a triangle a, b, A be given b>a>b sin A], then there are two possible values of the third side c, c, and c2 say, such that c1+c=2b cos A, and c1c2= b2 — a2.