256. Solve the equations: 257. (1) ax+by+2=zx+ay+b=yz+bx+a=0. (2) x + y + zu = 12,) If P agree with unity as far as the 7th decimal place, to how many places will this approximation in general be correct? [MATH. TRIPOS.] 258. A lady bought 54 lbs. of tea and coffee; if she had bought five-sixths of the quantity of tea and four-fifths of the quantity of coffee she would have spent nine-elevenths of what she had actually spent; and if she had bought as much tea as she did coffee and viceversa, she would have spent 5s. more than she did. Tea is more expensive than coffee, and the price of 6 lbs. of coffee exceeds that of 2 lbs. of tea by 58.; find the price of each. 259. If s, represent the sum of the products of the first n natural numbers taken two at a time, then 260. If Q = pa2+2qab+rb2 pac+q(bc-a2)-rab pc2-2qca+ra2 R prove that P, p; Q, q; and R, r may be interchanged without altering the equalities. 261. If a+ẞ+y=0, shew that [MATH. TRIPOS.] a”+3+ßn+3+yn+3=aßy (a”+ß"+y") + 1 (a2+B2+y3)(a®+1+ßm+1+yn+1). [CAIUS COLL. CAMB.] 262. If α, B, y, & be the roots of the equation x2+px3+qx2+rx+8=0, find in terms of the coefficients the value of (a – ẞ)2 (y − 8)2. [LONDON UNIVERSITY.] 263. A farmer bought a certain number of turkeys, geese, and ducks, giving for each bird as many shillings as there were birds of that kind; altogether he bought 23 birds and spent £10. 11s.; find the number of each kind that he bought. 264. Prove that the equation (y + z −8x)3 + (z+x − 8y)3 + (x + y − 8z)3 =0, is equivalent to the equation (1) x+y+z=ab, x−1+y−1+z-1=a-1b, xyz=a3. (2) ayz+by+cz=bzx+cz+ax=cxy+ax+by=a+b+c. 268. In a company of Clergymen, Doctors, and Lawyers it is found that the sum of the ages of all present is 2160; their average age is 36; the average age of the Clergymen and Doctors is 39; of the Doctors and Lawyers 32; of the Clergymen and Lawyers 363. If each Clergyman had been 1 year, each Lawyer 7 years, and each Doctor 6 years older, their average age would have been greater by years: find the number of each profession present and their average ages. 269. Find the condition, among its coefficients, that the expression ɑx2+4a1x3y+6a„x2y2+4a ̧xy3+α ̧ya should be reducible to the sum of the fourth powers of two linear expressions in x and y. [LONDON UNIVERSITY.] 271. It is a rule in Gaelic that no consonant or group of consonants can stand immediately between a strong and a weak vowel; the strong vowels being a, o, u; and the weak vowels e and i. Shew that the whole number of Gaelic words of n+3 letters each, which can be formed of n consonants and the vowels aeo is peated in the same word. 2n+3 where no letter is re [CAIUS COLL. CAMB.] 272. Shew that if x2+y2=2z2, where x, y, z are integers, then 2x=r(12+2lk-k2), 2y=r(k2+2lk-12), 2z-r (12+k2) (2) 3ux-2vy=vx+uy=3u2+2v2=14; xy=10uv. 277. If a, b, c,... are the roots of the equation when n is an integer, and the series stops at the first term that vanishes. [MATH. TRIPOS.] 279. Two sportsmen A and B went out shooting and brought home 10 birds. The sum of the squares of the number of shots was 2880, and the product of the numbers of shots fired by each was 48 times the product of the numbers of birds killed by each. If A had fired as often as B and B as often as A, then B would have killed 5 more birds than A: find the number of birds killed by each. 280. Prove that 8(a3+b3+c3)2>9 (a2+bc)(b2+ca) (c2+ab). What is the limit of this when n is infinite? [KING'S COLL. Camb.] 282. If ?" is the nth convergent to the continued fraction In 1 1 1 1 1 1 a+b+c+ a+b+c+ shew that P3n+3=bp3n+(bc+1) 9зn⋅ ... [QUEENS' COLL. CAMB.] ... 283. Out of n straight lines whose lengths are 1, 2, 3, n inches respectively, the number of ways in which four may be chosen which will form a quadrilateral in which a circle may be inscribed is 1 48 {2n (n-2) (2n −5)-3+3(-1)"}. [MATH. TRIPOS.] 284. If u, u are respectively the arithmetic means of the squares and cubes of all numbers less than n and prime to it, prove that n3 - 6nu2+4u3=0, unity being counted as a prime. [ST JOHN'S COLL. CAMB.] 285. If n is of the form 6m-1 shew that (y-z)+(z-x)n+(x − y)n is divisible by x2 + y2+z2 − yz − zx - xy; and if n is of the form 6m+1, shew that it is divisible by (x2+ y2+z2-yz — zx — xy)2. 286. If S is the sum of the mth powers, P the sum of the products m together of the n quantities a1, α2, az, ... an, shew that [n-1.S>\n- m.\m. P. 287. Prove that if the equations [CAIUS COLL. CAMB.] x3+qx-r=0 and rx3-2q2x2 - 5qrx – 2q3 — r2=0 have a common root, the first equation will have a pair of equal roots; and if each of these is a, find all the roots of the second equation. 288. If [INDIA CIVIL SERVICE.] x√2a2 · 3x2 + y √√/2a2 − 3y2+z √√/2a2 − 3z2=0, where a2 stands for x2+ y2+22, prove that (x+y+2)(−x+y+z) (x−y + z) (x+y−z)=0. ก [TRIN. COLL. CAMB.] 289. Find the values of X1, X2, xn which satisfy the following system of simultaneous equations: zx-y2 xy-z2 yz-x2 xy-z2 yz - x2 zx - y2 where 72x2+y2+z2, and u2=yz+zx+xy. |