Х 2 10 15 21 5/14 226. Simplify ; and find the value of 327 4715 748 1 given that 5=2.236. 315-6? 227. By the Binomial Theorem find the cube root of 128 to six places of decimals. 228. There are 9 books of which 4 are Greek, 3 are Latin, and 2 are English ; in how many ways could a selection be made so as to include at least one of each language ? 229. Simplify 745.23 – 780x3 + 15a_x (1) a - 2 230. Form the quadratic equation whose roots are 5 + /6. If the roots of x2 - px+1=0 are two consecutive integers prove that p2-47-1=0. 231. Solve 28+1=81(y2 +y); x2+x=9(78+1). 232. Find log1& 128, log4 V128, log2 ; and having given log 2=-3010300 and log 3=.4771213, find the logarithm of .00001728. 233. A and B start from the same point, B five days after A ; A travels 1 mile the first day, 2 miles the second, 3 miles the third and so on; B travels 12 miles a day. When will they be together? Explain the double answer, 234. Solve the equations : (1) 22=8v+1, 9y=32-9; (2) 2=y24, 22=2 x 47, x+y+z=16. 235. The sum of the first 10 terms of an arithmetical series is to the sum of the first 5 terms as 13 is to 4. Find the ratio of the first term to the common difference. 236. Find the greatest term in the expansion of (1 - x) 3 when 12 237. Five gentlemen and one lady wish to enter an omnibus in which there are only three vacant places; in how many ways can these places be occupied (1) when there is no restriction, (2) when one of the places is to be occupied by the lady? 238. Given log 2=301030, log 3=.477121, and log 7=845098, .0056 216 Find x from the equation 188–4x=(54 12)34– 1 239. If P and Q vary respectively as y2 and y3 when z is con stant, and as z2 and 23 when y is constant, and if x=P+Q, find the equation between x, y, z; it being known that when y=z=64, x= =12; and that when y=42=16, x=2. 240. Simplify 133 13 143 77 +log 171. 241. If the number of permutations of n things 4 at a time is to the number of combinations of 2n things 3 at a time as 22 to 3, find n. log log 90 -C 1 1 1 1 242. If + + , prove that 2b is either the aritha с 26 26 metic mean between 2a and 2c, or the harmonic mean between a and c. 243. If nCr denote the number of combinations of n things taken r together, prove that *+2Cr +1 =*Cr+1+"Cr-1+(2 x "Cr). 244. Find (1) the characteristic of log 54 to base 3; (2) logo (-0125)}; (3) the number of digits in 345. Given log10 2=30103, log10 3=.47712. 245. Write down the (r+1) th term of (2ax2 – x3), and express it in its simplest form. 246. At a meeting of a Debating Society there were 9 speakers ; 5 spoke for the affirmative, and 4 for the negative. In how many ways could the speeches have been made, if a member of the affirmative always spoke first, and the speeches were alternately for the affirmative and the negative ? 247. Form the quadratic equation whose roots are 2ab a+b+ va? +62 and a+b+ Va? +62 248. A point moves with a speed which is different in differ ent miles, but invariable in the same mile, and its speed in any mile varies inversely as the number of miles travelled before it commences this mile. If the second mile be described in 2 hours, find the time taken to describe the nth mile. 249. Solve the equations : (1) x2(6-c)+ ax(c-a) +a'(ab)=0, (2) (zo-px+p%)(+p9+p) =223+pq+p4. 250. Prove by the Binomial Theorem that 3 3.5 3.5.7 + ... ad inf. = V8. 4.8 4.8.12 251. A and B run a mile race. In the first heat A gives B a start of 11 yards and beats him by 57 seconds; in the second heat A gives B a start of 81 seconds and is beaten by 88 yards : in what time could each run a mile? 1+ 252. A train, an hour after starting, meets with an accident which detains it an hour, after which it proceeds at three-fifths of its former rate and arrives 3 hours after time: but had the accident happened 50 miles farther on the line, it would have arrived 1} hrs. sooner: find the length of the journey. 253. Expand for 4 terms by the method of Undetermined 2 Coefficients 3x2–2x3 254. A body of men were formed into a hollow square, three deep, when it was observed, that with the addition of 25 to their number a solid square might be formed, of which the number of men in each side would be greater by 22 than the square root of the number of men in each side of the hollow square: required the number of men. 255. Expand into a series Va?+62. 256. Solve the equation V2x–1+ V3x2=V42–3+ V5x – 4. 7x2 +22x+5 257. Separate into partial fractions. (x+3)(x2-1) 258. Solve the equation 24 – 5x2 - 6x — 5=0. 259. Find the generating function of 1+5x+7x2 + 1733 +31x4 + ... 3x – 22–4 260. Separate into partial fractions. (x2+1) (22-2-2) 261. Solve the equation 24+3x2=16x+60. 763 262. Express as a continued fraction and find the fourth 396 convergent. 263. What is the sum of n terms of the series 1, 8, 27, 64, ... ? 264. The sum of 6 terms of the series 1–xv-1- x2 + ... is equal to 65 times the sum to infinity : find x. 265. Convert 2 v5 into a continued fraction. 916 266. Find limits of the error when is taken for V23. 191 267. Sum to infinity the series 3-x-2x2–16x3 – 28x4 - 676x5 + 1 1 1 1 1 1 268. Find value of + 3+ 2+ 1+ 3+ 2 1+ 269. Solve, by Cardan's Method, the equation 23 — 30x +133=0. 270. Solve the equation 23-13x2 +15x+189=0, having given that one root exceeds another root by 2. ...... 271. Solve the equation 24 – 4x2 +8x+35=0, having given that one root is 2+ V3. 272. Sum to infinity the series 4-9x+16x2 – 25x3 +36x4 - 49.25 +. 273. Solve the equation 24 – 12x8 +4722 — 72x+36=0. 274. Solve the equation 4x 6x +2 8x + 1 =0. 6x +2 9x +3 12x 8x +1 12x 16x +2 275. Solve the equation 2.25 +24+x+2=1268 +12x2. |