a b = a 6. IP ==be prove that ау 7. 8 is the sum of a geometrical progression and R is the sum of the reciprocals of the terms of the progression. Prove that the product of the first and last terms of the given progression. 8. How many words can be formed from the 21 consonants and 5 vowels of the alphabet, each of which contains 4 con sonants and 2 vowels? 9. If a, b, c be three consecutive coefficients in the expansion of 2ae+b(a+c) a power of (1+x), prove that the index is 6?-ac 10. Prove that logab x log.c=log.c. Given log9=a, log;5=b, log57=e, find the logarithms to base 10 of 5, 6, 7, 8, 9. GEOMETRY. 1. Describe a parallelogram that shall be equal in area to a given triangle, and have one of its angles equal to a given angle. 2. The side AB of a triangle ABC is greater than the side AC; shew that the median AX makes a smaller angle with AB than it does with AC. 3. Enunciate and prove a proposition of the second book equiva lent to the algebraical equation a2 +22=2ab+(a−b)?? 4. A, B and C are three fixed points in a straight line, and P is any other point in this line; prove that PA.BC+PB.CA+PC.AB=0, due regard being had to the representation of opposite directions by the signs + and —. 5. A straight line is drawn from one angular point of a triangle to bisect the opposite side; prove that the sum of the squares on the sides containing this angle is equal to twice the sum of the squares on this bisector and on half the side bisected. Prove also that in a triangle the shortest median bisects the longest side. а 6. The diameter is the greatest chord in a circle, and of others that which is nearer the centre is greater than one more remote. 7. If iwo chords of a circle cut one another, the rectangle con tained by the segments of the one is equal to the rectangle contained by the segments of the other. 8. The bisector of the vertical angle A of a triangle ABC meets the circumscribing circle in D, and DK is drawn perpendicular to AB; shew that AK is equal to half the sum of the two sides containing the angle A. 9. AB, CD are two parallel straight lines cut by two transversals OAD, OBC; shew that the circles circumscribing OAB, OCD will touch one another at 0. 10. Circumscribe a circle about a given triangle. TRIGONOMETRY. 1. What is meant by a radian? A circular cylinder is found by measurement to be 1.94 inches in diameter and 6:09 inches in circumference. Assuming these estimates to be correct, find in degrees, correct to one decimal place, the value of a radian. 2. Prove the following formulae, cos(A+B)=cos A cos B-sin A sin B, cos A +cos B=2cos }(A+B) cos }(A-B). 3. Assuming the values of the sine and cosine of 30° and 45°, find those of 15°. Also find the value of cos 973°, explaining the reason for the sign. 4. Prove that a’=? +0—2bc cos A. ADEB and AFC are a square and equilateral triangle respectively, described externally on the sides BA, AC, of any triangle ABC. Prove thať DF’=+c+bcí v 3cos A + sin A). 5. In any triangle prove b-C (i.) tan }(B-C)= )= cot A, b+c tan(+A+C)=(c—a)(c+a—b) cta atb If b, c, B are given and a, and an are the two values of a, prove that 8 + qj3+4,3=2c(cocos 3B+36 cos B). 7. If m=cosec 0—sin 0, n=sec 0—cos O show that m3+n}=(mn) – 3. 8. Prove (i.) r=4, .) = 1 1 1 1 (ii.) + P P2 Po where P1, P2, P3 are the perpendiculars from the angular points on the opposite sides. 9. A person standing on the bank of a river observes the eleva a log 1.55823=.192631, L sin 46°=9.856934, L sin 5°=8.940296. |