This equation when cleared of fractions is of the second degree in 0, and is satisfied by the three values 0=X, 0=μ, 0=v, in virtue of the given equations; hence it must be an identity. [Art. 310.]

To find the value of x, multiply up by a +0, and then put a +0=0;

8. Prove that the real root of the equation 3+12x-12=0 is 23/2-$4.

17. x1+9x3+12x2 – 80x – 192=0, which has equal roots.

18. Find the relation between q and r in order that the equation 23+qx+r=0 may be put into the form x=(x2+ax+b)2.

Hence solve the equation

8x3-36x+27=0.

19. If

x3+3px2+3qx+r and x2+2px + q

have a common factor, shew that

4 (p2 − q) (q2 — pr) − ( pq − r)2=0.

If they have two common factors, shew that

p2-q=0, q2-pr=0.

20. If the equation ax3+3b.x2+3cx+d=0 has two equal roots, bc-ad shew that each of them is equal to

2 (ac-b2)*

21. Shew that the equation 4+px3+qx2+rx+8=0 may be solved as a quadratic if r2=p2s.

22. Solve the equation

26-18x4+16x3+28x2 - 32x+8=0,

one of whose roots is 6-2.

23. If a, ß, y, d are the roots of the equation

x+qx2+rx+8=0,

find the equation whose roots are ẞ+y+d+(Byd)−1, &c.

24. In the equation x-pr3+qx2 − rx+8=0, prove that if the sum of two of the roots is equal to the sum of the other two p3 - 4pq+8r=0; and that if the product of two of the roots is equal to the product of the other two r2=p2 s.

25. The equation x5 - 209x+56=0 has two roots whose product is unity determine them.

26. Find the two roots of x5-409x+285-0 whose sum is 5.

(1+a2) (1+b2)......(1+k2)=(1 − P2+P4-...)2+(P1−P3+P6 - ...)2.

28. The sum of two roots of the equation

x4-8x3+21x2 - 20x+5=0

is 4; explain why on attempting to solve the equation from the knowledge of this fact the method fails.

MISCELLANEOUS EXAMPLES.

1. If 81, 82, 8, are the sums of n, 2n, 3n terms respectively of an arithmetical progression, shew that 83=3 (82-81).

2. Find two numbers such that their difference, sum and product, are to one another as 1, 7, 24.

3. In what scale of notation is 25 doubled by reversing the digits? 4. Solve the equations:

(1) (x+2)(x+3)(x-4) (x-5)=44.

(2) x(y+z)+2=0, y(z-2x)+21=0, z(2x-y)=5.

In an A. P., of which a is the first term, if the sum of the
terms =0, shew that the sum of the next
q terms

7. Find an arithmetical progression whose first term is unity such that the second, tenth and thirty-fourth terms form a geometric

series.

8. If a, ẞ are the roots of x2+px+q=0, find the values of

a2+aẞ+B2, a3+ß3, a1+a2ß2+84.

9. If 2x=a+a-1 and 2y=b+b-1, find the value of

11. If a and ẞ are the imaginary cube roots of unity, shew that

a1+ß1+a ̄1ß ̄1=0.

12. Shew that in any scale, whose radix is greater than 4, the number 12432 is divisible by 111 and also by 112.

13. 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?

14. Eliminate x, y, z between the equations:

x2-yz=a2, y2 - zx= b2, 22 — xy=c2, x+y+z=0.

-

15. Solve the equations:

[R. M. A. WOOLWICH.]

ax2+bxy+cy2=bx2+cxy+ay2= d.

[MATH. TRIPOS.]

16. A waterman rows to a place 48 miles distant and back in 14 hours: he finds that he can row 4 miles with the stream in the same time as 3 miles against the stream: find the rate of the stream.

17. Extract the square root of

(1) (a2+ab+be+ca) (bc+ca+ab+b2) (bc+ca+ab+c2).

(2) 1−x+√√/22x – 15 – 8x2.

10

18. Find the coefficient of 6 in the expansion of (1 – 3x)3, and the 4 3 9 term independent of x in

19. Solve the equations:

2x

(2) x2 — y2=xy —ab, (x+y)(ax+by)=2ab (a+b).

[TRIN. COLL. CAMB.]

20. Shew that if a (b-c) x2+b (c− a) xy+c (a - b) y2 is a perfect square, the quantities a, b, c are in harmonical progression.

21. If

[ST CATH. COLL. CAMB.]

(y-2)2+(2-x)2+(x − y)2=(y + z −2x)2 + (z+x-2y)2+(x+y-22)2,

and x, y, 2 are real, shew that x=y=z.

ST CATH. COLL. CAMB.]

22. Extract the square root of 3e58261 in the scale of twelve, and find in what scale the fraction would be represented by ⚫17.

1 5

...

23. Find the sum of the products of the integers 1, 2, 3, n taken two at a time, and shew that it is equal to half the excess of the sum of the cubes of the given integers over the sum of their squares.

24. A man and his family consume 20 loaves of bread in a week. If his wages were raised 5 per cent., and the price of bread were raised 2 per cent., he would gain 6d. a week. But if his wages were lowered 7 per cent., and bread fell 10 per cent., then he would lose 1d. a week: find his weekly wages and the price of a loaf.

25. The sum of four numbers in arithmetical progression is 48 and the product of the extremes is to the product of the means as 27 to 35: find the numbers.

(a+b+c+3x) (a+b+c-x)=4(bc+ca+ab);

and if a+b+c=0, shew that (a+b+c)3=27abc.

28. 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.

29. Solve the equations:

2x+y=2z, 9z-7x=6y, x3+y3+23=216.

[R. M. A. WOOLWICH.]

30. Six papers are set in examination, two of them in mathematics: in how many different orders can the papers be given, provided only that the two mathematical papers are not successive?

31. In how many ways can £5. 4s. 2d. be paid in exactly 60 coins, consisting of half-crowns, shillings and fourpenny-pieces?

32. Find a and b so that x3+ax2+11x+6 and 3+ bx2+14x+8 may have a common factor of the form x2+px+q.

[LONDON UNIVERSITY.]

33. In what time would A, B, C together do a work if A alone could do it in six hours more, B alone in one hour more, and C alone in twice the time?