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224. Two persons travel at different uniform rates along the same road, in the same direction, starting simultaneously from points at a given distance from each other ; find where they will be together, and explain the result when it is negative.

225. Find, with the help of the tables, what must be the annual payment for the whole life of an individual to begin immediately, in order to secure the reversion of a given sum at his death.

226. Shew how to find the root of x – 1= 0, n being a prime number, without Trigonometry.

227. Express the product of .2727 and 1.166 by a circulating decimal; also obtain the cube root of 74 + 23/11 under the form of a binomial surd.

1835

228. When n is even, find for what value of r the number of combinations of n things taken r together is the greatest possible.

229. Required the amount of £819. 48. in 6 years, allowing £12. 10s. per cent, both simple and compound interest.

230. Investigate the rules for the multiplication and division of algebraical fractions; and reduce x + 22-1

1 x + √x2

1 to its simplest form.

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a

231. If a be prime to b, there is no other fraction equal to

b whose terms are not equimultiples of a and b.

232. When four quantities are proportionals, the sum of the first and second is to their difference as the sum of the third and fourth to their difference.

233. Shew how the logarithm of a number consisting of 6 digits may be found from a table calculated for numbers not exceeding 5 digits, and having given log 33819 = 4.5291608, log 33818 = 4.5291479, find log 338185.

234. Express

a + brī

and log, (a + bv – 1) in the C+ dv

form A + BV1.

235. Shew how to transfer a number from one scale of notation to another, and express 123.45 in a scale whose radix is 5.

236. If the diameter of the earth, supposing it a sphere, be 7916 miles, find the length of a French metre which is one ten-millionth part of a fourth of its circumference.

237. A number is divisible by 9, if the sum of its digits is divisible by 9; and by 11, if the sum of the 1st, 3rd, 5th .... digits is equal to the sum of the 2nd, 4th, 6th, .... digits.

238. Explain the method of solving the indeterminate equation ax + by = c by continued fractions; and shew how the number of solutions may be determined. ad be

bd 239. Supposing

each 6

b a+b+c+d of them

4

ac

a

C + d

a

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3

15.12 +

03375 240. Reduce

to its equivalent simple V 80

✓.01 decimal.

241. The coefficients of the (r + 1)th term of (1 + x)"+1 is equal to the sum of the coefficients of the oth and (r + 1)th terms of (1 + x)"; prove this, and express (1 + x)" in a continued fraction,

242. Every number consisting of n digits will have 2n or 2n – 1 digits in its square. Hence explain the rule for pointing in the extraction of the square root of a number.

243. If a body a inches long weighs m pounds, find the length of a similar body that weighs n pounds.

244. In the expansion of (a + b)" where n is an integer, the coefficients of terms equidistant from the two extremes are equal. Write down the pth term of expansion of (1 – x)

1836

245. If there be a chances of an event's happening in any one trial, 6 of another, and c of a third ; find the probability of the first event's happening p times, the second q times, and the third r times, in p + 9 + r trials.

246. Find the volume of a cube whose edge is 13 feet 8 inches.

247. Determine the present value of p pounds, due n years hence, at a given rate of interest.

Ex. £1000. 108. due 5 years 4 months hence, at 4} per cent. per annum.

248. Define the least common multiple of two quantities ; and

prove that it measures every other common multiple of them. Find the least common multiple of

6.03 11x2 + 5x 3 and 9.03 9.x2 + 5x – 2.
249. Reduce to the simplest form

1
4

9

+ 2x + 2

2 (+ + 3) 250. Investigate the law of formation of the product of a series of binomial factors, (2 + a), (x + b), :.. and deduce the coefficient of x in the product

(x + 2).(x + 6). (+ 10). (x + 14).
251. Shew how to pass from one system of logarithms to
another. What is the advantage of the common system?
Assuming the expansion of ar, calculate the value of the base
of the Napierian system to six places of decimals.
252. Shew, that if n and N be nearly equal,

13
N

I n + N

+
N tn
4

very nearly.
In using this formula, we may sufficiently know the degree of
correctness attained, by observing that if

N

X + 2

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n

N + N

and 1 N tn

.

n

have their p first decimal figures equal, the approximation may be relied upon to 2p decimals at least. Prove this, and apply the formula to find an approximate value of v 30 true to eight decimal places.

253. The corners of a common die are filed away till the faces which before were squares become regular octagons. Compare the respective probabilities, when the die is thrown, of turning up a triangular and an octagonal face. (Considerations of a dynamical nature to be neglected.)

254. Expand loge (1 + x) in a series ascending by powers of x. Shew that no hypothesis, which renders nugatory a preceding step, has been introduced in the demonstration. p

then a'a po' p'q = +1. Prove this, and apply it to find all the solutions, in positive integers, of the equation

33x + 177 743. 256. Find the least number, the product of which by 882 shall be a perfect cube.

257. By selling a given quantity of a certain article for 10s. the seller loses 5 per cent. ; what will be the loss or gain when it is sold for 12s. 6d. ?

258. Explain the popular meaning of the term ratio : shew that it may be represented algebraically by a fraction. On this assumption prove that the geometrical definition of proportion is a consequence of the algebraical.

259. The product of any r consecutive integers is divisible by 1.2.3......

260. Prove that the nth roots of unity form a geometrical progression. Interpret the values of a.(1), when a represents a line.

261. Find the mean proportional between .016 and 2.704. Express 1000 in the scale whose radix is 11.

262. Prove that the number of combinations of n things taken r together is equal to the number when taken n together. What is the nth term of (a? – x2)-3?

-r

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nth

2. If we denote the sum, sum of the squares, cubes, powers of the m quantities a, b, c, d, &c., by S, S,, S.,.. Sn, then the sum of the nth powers of the differences of a, b, c, d,

&c. or

(a - b)" + (b a)" + (Q – c)” + &c.

ms, – ns, S.

- S.1+

n(n-1) s,Su-2 – &c.

1.2

3. Every equation has at least as many changes of sign from + to and from – to +, as it has positive and possible roots; and as many continuations of sign from + to t, and from – to –, as it has negative and possible roots.

4. Give Clairaut's approximation to the solution of a cubic equation in the irreducible case.

1822

5. Transform x3 2.x2 + 2:0 - 4 = 0 into an equation, the roots of which are the squares of the roots of the original equation.

6. Prove that Waring's solution of a biquadratic equation fails when all the roots are impossible, and of the form

a + b = 1.

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