7. Given log1, 2, and log,, 3, find the integral values be

10

10

tween which x must lie in order that the integral part of (1·08)* may contain four digits.

8. Given log10 6.10257855078,

logo 6*10267855149,

MISCELLANEOUS PROBLEMS.

1. Shew that y"-1 is divisible by either of the quantities y" — 1 or y" - 1 without remainder, m and n being positive integers.

2. Of the two quantities a+ab2+a2b*+b® and (a3 + b3)3, determine which is the greater.

3. Prove that

2

2

(a1+a2+ ... + a1)2= n (a‚2+a,2 + ... + a‚3) — (α, — a„)2 — (α, — ag)2 ... -(α-a)2.

4. If a be greater than b, a"-b" is greater than nb"-1 (a - b) and less than na"-1 (a - b).

5. If a be an approximate value of the square root of n, and n-a2±b, then will

6. If n be greater than 3, then will √n be greater than in +1.

7. The number of different combinations of n things taken 1, 2, 3,... n at a time, of which there are p of one sort, q of another, and r of another,

= (p+1) (q+1) (r + 1) − 1.

8. If (b-a) (y-ma) = (nb — ma) (x — a),

:

and (b'a') (y-mb') = (mb' — na') (x — b'),

9. Prove that out of the combinations of n things the number of combinations involving an odd number of things exceeds the number of those involving an even by one.

14. Prove that the value of -2x+3 corresponding to r = 1 is smaller than for any other value of x.

16. Prove that x+y is never less than 2xy.

19. Shew that (a2+b2)* (c2 + d2)* is greater than ac+bd.

20. Shew that abc is greater than (a+b−c) (a + c − b) (b+c-a) unless a = b = c.

21. Shew that a +1 is greater than a2+a unless a = 1.

24. Prove that x2+ px3+ qx2+rx+s is a perfect square,

25. If ax2+ bx2+ cx + d is a perfect cube, prove that

bc9ad; ac3=b3d.

26. Can x, y, and z be obtained from the following equations?

3x-2y+5z14,

2x+y-8210,

8x-3y+2z=38.

27. If x, x, are the roots of the equation ax + bx + c = 0, prove that

28. The quantity x2+ax+b is always positive, whatever be the value of x, provided a2 is less than 46.

29. The same value of x satisfies the equations

and

ax2 + bx + c = 0,

a'x2+ bx + c' = 0;

prove that (ac' a'c)3 — (ab' — a'b) (bc' — b'c).

30. The same values of x and y will satisfy the three

if (a'b" — a"b') c + (a′′b — ab") c' + (ab' — a′b) c" = 0.

31. If a, b, c be any quantities in geometrical progression, a2+b2+c2 is greater than (a−b+c)2.

12

32. If a2+b2+c2= 1, and a22+b22+ c22= 1, then aa' + bb'+ cc' is never greater than 1.

(a+b+c)*−(b+c)* — (c+a)* − (a+b)*+a*+b*+c*=12abc (a+b+c).

36. If a and B be the roots of the equation

37. Prove that the sum of the products of the first n natural numbers taken two and two together, is

(n − 1) n (n + 1) (3n+2)

24

TRIGONOMETRY.

GENERAL QUESTIONS CONCERNING TRIGONOMETRICAL FUNCTIONS.

1. GIVE the proper signs to sin {(2n + 1) 180° +A} and cos {(2n+1) 180° + A}, supposing A to be less than 90°.

2. Trace the sign of the quantity sin + cos 0, while changes from 0 to 2π.

π

3. If be between and T, what are the proper signs of

2

sin 20, cos 20, and tan 20?

4. Write down a formula for all angles the tangent of which is - tan 0.

5. Write down a formula for all angles the cosine of which is = 1.

6. Express all the trigonometrical functions in terms of the sine.

7. Express the same in terms of the tangent.

8. The same in terms of the versedsine.

9. The same in terms of the suversedsine, (or versedsine of the supplement).

10. If tan + cot 0 = m, express all the trigonometrical functions in terms of m.

11. Find sin from the equation sin ✪ cos 0 = m.

12. If the right angle were divided into 100° instead of 90°, what would be the value of an angle of 36° 7'?