[Ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.] I. Draw a straight line at right angles to a given straight line, from a given point in the same; and show that two straight lines cannot have a common segment. 2. Give Euclid's Axiom on parallel straight lines; and show that the straight lines which join the extremities of two equal and parallel straight lines towards the same parts are also themselves equal and parallel. Two quadrilaterals ABCD, EFGH, have the sides AB, DC respectively equal to the sides EF, HG; and also the angles which the diagonals AC, BD make with AB, CD, namely, CAB, ABD, BDC, DCA respectively equal to the angles which the diagonals EG, FH make with EF, HG, namely, GEF, EFH, FHG, HGE; show that the quadrilaterals are equal in every respect. 3. To a given straight line apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. 4. Enunciate the two propositions represented by (2a+b)2+b2 = 2{a2+(a±b)2}. AB is bisected at C and produced to D; AE, which is at right angles to AB, is divided into two equal parts at F and into two unequal parts at G; CH, BI, DKL are drawn parallel to AE; and FK, GHL, EI parallel to AD. Show that the sum of the squares on GD and IL is double of the sum of the squares on FC and HK. 5. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. 6. From any point in the diameter of a circle which is not the centre there can be drawn to the circumference two straight lines, and only two, which are equal to one another, one on each side of the diameter. If two such pairs of equal straight lines are drawn, prove that the chords joining the extremities of the unequal straight lines meet upon the diameter of the circle (produced if necessary). 7. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it. 8. A quadrilateral ABCD is circumscribed round a circle, touching the circle in E, F, G, H. Find the difference of two opposite angles of ABCD in terms of the difference of two adjacent angles of EFGH. 9. Describe a circle about a given triangle. IO. Inscribe a regular quindecagon in a given circle. II. The sides about the equal angles of equiangular triangles are proportionals. If in two triangles ABC, DEF, AB is to BC as DE is to EF, and the angle A is equal to the angle D, but the angle B unequal to the angle E; show that the angles C and Fare supplementary. 12. In equal circles angles, whether at the centres or at the circumferences, have the same ratio which the arcs on which they stand have to one another. II. ALGEBRA. (Up to and including the Binomial Theorem, the theory and use of Logarithms.) [N.B.--Great importance will be attached to accuracy.] I. Find the value of 1680+(r−7)[1470+(r−8){378+(r−9) (35+r−10)}] when r=-7. 2. Multiply 2x5+4x+6x3 +8x2+10x+5 by x2-2x+1. Find an algebraic function such that when it is divided by a2 – ab+b2, the quotient is 3a2 – 2ab+b2 and the remainder is 2a3(b − a). 3. Write down one factor of x17 − 17x2 + 16. Divide x17 – 17x2 + 16 by x2 − 2x + 1, so far as to show the four terms of highest degree in the quotient. 4. Define the algebraical Greatest Common Measure and Least Common Multiple of two or more functions. Find the Greatest Common Measure of 7x+6x3- 8x2-6x+1 and 11x+15x3 − 2x2 - 5x + 1. 5. Simplify the expression (x-y)+(-y)3(x-2y+2) Examine the meaning of a and ao in the theory of indices. 7. If show that 3c = 3/100+23/10+3, 3(c− 1)3 — 20c = 0. 8. Solve the equations 9. (i.) x2-18x-72 = 0; (ii.) √3√(x+2x2+3)+§x2+ 3 = }ax2. Find the sum of 19 terms of the Arithmetical Progression whose 3rd and 13th terms are 17 and 87 respectively. If x! denote the product of the first x integers, show that 1.1!+2.2! +3.3! + ... + x . x! = (x + 1)! — I, IO. Find the number of permutations of n different things taken 4 together. If there are 4 flags of each of 6 different colours, prove that 4 flags may be hoisted into a vertical line in 64 different ways. I (1 - x)n II. Expand by the binomial theorem, stating the necessary imitations on the value of x. If (1 − x)-1(1 − x2)−o( 1 − x3)−9(1 − x4)−r = 1 +x+2(px2+qx3+rxa)+... show that p = q = fr = 1. 12. If a number has no integral part, show that the characteristic of its logarithm to base 10 is negative, and numerically exceeds by unity the number of zeros following the decimal point. I. III. PLANE TRIGONOMETRY AND MENSURATION. (Including the Solution of Triangles.) [N.B.-Great importance will be attached to accuracy.] (i.) (sec2A+tan2A) (cosec2A + cot2A) = 1+2 sec2A cosec2A. (ii.) sec A+tan A cosec A+ cot A sec Atan A cosec Acot A = 2(sec A - cosec A). 3. Having given sin A = ‡ and sin B =, find the value of tan(A + B). 4. Prove that sin (AB){cos A+ cos B - sin (A+B)} = cos(A+B){ sin (A-B) + cos A cos B}. 5. Find an expression for all the angles which have the same sine as a given angle. Having given the value of sin 2A, find the number of values of tan A. 6. Prove that 7. Find the sine and cosine of 18° and also the sine and cosine of 54°. 8. If R is the radius of the circumscribing circle of a triangle ABC, prove that cos A + c sin B cos B I |