2. Find the highest common factor of 4x4-9x2+6x-1 and 6x3–7x2+1. When of two algebraical expressions the H. C. F. has been found, what is the rule for determining their L. C. M.? 1 (1) (21a+3b) (3Ẫa – 21b) — 6a (a2 – b2) + 23ab. (2) (35√10+77 √2+63 √3+28 √15) × (√10−√2−√3). 7. Divide 11 into three parts so that the 1st may be to 1 1 1 the 2nd as to, and the 2nd to the 3rd as to 8. Shew that p and q represent respectively the sum and product of the roots of the equation x2-px+q=0. The area of an oblong room is 328 square feet, and its perimeter is 73 feet; write down and solve the quadratic equation which gives the lengths of the sides of the room. 9. If a b c d, prove that 4a6+5b6: 4c+5d6 :: a3b3 : c3d3. 10. Prove the rule for expressing a given integer in any proposed scale. 758 t83 cubic feet being the volume (expressed in the duodenary scale) of a cubic; find the number of feet and inches in an edge of the cube. 11. If the first and last terms of a geometrical progression of 8 terms be respectively a7 and 87; find the sum of the series. Prove the rule for finding the value of a recurring decimal in which all the figures recur. 12. Find the number of permutations (p,) of n things taken r at a time. Shew that (P3 −P2) (P4 −P3) ..... (Pn−1−Pn−2) = P2P3Pn-1Pn P3Pn-1 13. Assuming the Binomial Theorem when the exponent is a positive integer, prove it when the exponent is any positive quantity. Find the 13th term in the expansion of 11 (28+26x) 2. XXXIII. Royal Military College, Sandhurst. Further Examination. 1. Shew that (ax+by+cz)3+(cx-by+az)3 is divisible by (a+c)(x+2). Find the H. C. D. of 25+11x-12 and 5+11+54. 2. Find an expression containing no higher power of x than the first which added to xa+6x3 + 13x2+6x+1 will make it a complete square. 4. If a and B are the roots of the equation a2+px+q=0, express a2+ẞ2 and a3+ß3 in terms of p and q. 6. Three numbers are as 1, 2, 3; the sum of their squares is sixty-three times the sum of the numbers. Find them. 7. If the first term of an A. P. is a and the common difference d, find an expression for the sum of n terms. If the common difference is -d, and the sum of n terms (2a+d)2 9d find n. 8. Write down the general term (the 7th) in the expansion of (1 − x)−”, and find the condition that the sum of the coefficients of the first three terms may equal that of the fourth. Generalise this result. 9. Prove that if m, n, and a are any numbers, Given that log102=30103, find log10 500 and log10 *0008. 10. Assuming the expansion for log,(1+x), prove that XXXIV. Army Preliminary Examination. February, 1886. 1. Find the value of {15α-5 (b÷c)} - {17b — 7 (a–c)} when a=5, b=3, c=2. 2. Multiply by 3. 6x3+13x2-4x - 15 6x313x2-4x+15. Divide 35xa – 22x3+58x -35 by 5x2-6x+7. --- 4. Reduce {x2- (x − a)2} × {(x+a)2 − x2} × {(2x2+a2)2 – 4x1} to its equivalent form without brackets. 5. Find the Greatest Common Measure of x1- 2x3+x2 −1 and x4 -3x2+1. 6. Find the Least Common Multiple of 15a3x (a+x)3, 20αx3 (a−x)3, 36a2x2 (a2 — x2)2. 10. A roll of cloth was bought at 5s. 6d. a yard; and another roll, twenty-five yards longer, at 5s. a yard; the two together cost £100. 158. How many yards were there in each roll? XXXV. Army Preliminary Examination. March, 1886. 1. Subtract 4x3-3x2 - x+2 from 7x3-6x2+2x-1; and find the value of the answer when x=3. 2. Divide îa — 9x2 – 6xy – y2 by x2+3x+y. 5. Find the Highest Common Divisor of x2+4x-5 and x3-6x+5. 6. Find the Least Common Multiple of x2-y2; x3-y3; and 23+y3. 7. Find the square root of x2+8x3-26x2 - 168x+441. 8. Shew that (ax+by)2 + (bx − ay)2=(a2+b2) (x2+y2). 10. Divide £1120 between A and B, so that for every halfcrown which A receives B may receive a shilling. |