2. If be+ca + ab = 0, prove that

b2+c2-a2-2bcc2+a2-b2-2ca a2+b2-c2-2ab

bc

ca

ab

+

+

6abc

= (b + c − a) (c + a − b) (a + b − c) *

3. Solve the equation

(x−b) (x −c) + (x − e) (x − a) + (x − a)(x — b) = 0,

and shew that if a, b, c are in descending order of magnitude one of the roots lies between a and b and the other between b and c.

4. Explain what is meant by a" where m is fractional or negative, and prove that

Express "a" × √ as the root of a rational function of a and b.

5. If p, q be two quadratic surds, prove that the sum of any number of terms, each of which is the product of two integral powers of these surds, can be expressed in the form

A+B √p + C√ q + D √ pq

where A, B, C, D are rational.

If x=√q—√r, y=√r−√p, z=√p−√9,

=2(q− r) √p+2(r− p) √q+2(p − q) √T.

6. Define a homogeneous expression.

x У Z

If = = 5

prove that each of these expressions is equal to

ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy

aε2 + bn2 + c52 + Qƒn< + Qg55 + Qhên If a, a,....a, be in continued proportion, prove that

7. If z be a quantity depending upon n other quantities in such a manner that when n any 1 of them are constant z varies as the remaining one, prove that when all the quantities vary z varies as their product.

The expenses of running a train are partly constant and partly vary as the square of the velocity. The receipts are partly constant and partly vary as the velocity. Prove that there are two velocities for which the receipts are equal to the expenses, and that the profits will be greatest when the velocity is the arithmetic mean of these two velocities.

8. Define a geometrical progression, and find when possible the sum of an infinite number of terms of a given geometrical progression.

If a, b be the first and second terms of a harmonical progression where b a is small compared with a, shew that the sum of n terms of the harmonical progression is approximately

na + } n (n − 1) (b − a).

9. Find the number of combinations of n things r at a time.

At a certain examination there are subjects. In each subject two papers are set, one pass and one honour. In order to pass a candidate must satisfy the examiners in m subjects, but he may take either the pass or the honour paper in any subject. In how many ways can a candidate select m papers for the examination?

10. Find the numerically greatest term in the expansion of (a + x)" where n is a positive integer. If a, be the coefficient of a" in the expansion of (1 + x)", prove that

GEOMETRY AND TRIGONOMETRY.

The Board of Examiners.

In the first six questions the symbol - must not be used; and the only abbreviation admitted for "the square described on the straight line AB" is "sq. on AB," and for "the rectangle contained by the straight lines AB, CD" is "rect. AB, CD.” 1. If a straight line terminated by two intersecting straight lines be bisected, no other straight line drawn through the point of bisection and terminated by the same intersecting straight lines is bisected.

2. Prove that the bisectors of the angles of a triangle meet in a point.

The bisectors of the angles of a triangle ABC meet in D. BD produced meets AC in E, and DF is perpendicular to AC. Prove that the angles ADE, CDF are equal.

3. A straight line AB is bisected in C and divided unequally in D. Prove that the difference of the squares on the straight lines AD, DB is equal to twice the rectangle contained by the straight lines AB, CD.

4. P is any point on the arc of a given segment whose chord is AB. AG, BH are drawn perpendicular to BP, AP produced. Prove that the straight line GH touches a fixed circle.

5. Triangles of equal altitudes are as their bases.

Draw a straight line in a given direction to bisect a given triangle.

6. The sides about the equal angles of equiangular triangles are proportionals, and those sides are homologous which are opposite to equal angles. Prove this, and define the terms in italics.

ABC is a right-angled triangle having the angle A a right angle. ABEK, ACDH are squares falling outside the triangle. CE cuts BA in G, and BD cuts AC in F. Prove that AG and AF are equal, and that each of them is a mean proportional to BG and CF.

7. The circumferences of circles are as their radii. Prove this, and write down the ratio of the circumference of a circle to its diameter correct to five places of decimals.

A strip of paper 2 miles long and ·003 inch in thickness is rolled up into a solid cylinder. Find approximately the radius of the cylinder.

8. Define the sine and the cosine of an angle,

and prove from the definitions that when B >

and < and A > π and <

If

3п

2'

cos A sin B.

y′′) (x — x'),

π

x"=r" cos 0", y′′ = r" sin 0′′,

rr' sin (0 — 0′) + r'r" sin (0′ — 0′′) + r′′r sin (0′′ — 0) = 0.

9. Find an expression for all angles having a given tangent. Find an expression for all the angles which satisfy the equation tan 0 . tan n0 = 1.

D