Page 27 - Schaum's Outline of Theory and Problems of Advanced Calculus

18 NUMBERS [CHAP. 1

1.39. Show that a fraction with denominator 17 and with numerator 1; 2; 3; .. . ; 16 has 16 digits in the repeating
portion of its decimal expansion. Is there any relation between the orders of the digits in these expansions?
p ffiffiffi p ffiffiffi
1.40. Prove that (a) 3, (b) 3 2 are irrational numbers.
p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi
1.41. Prove that (a) 3 5 4 3, (b) 2 þ 3 þ 5 are irrational numbers.
1.42. Determine a positive rational number whose square differs from 7 by less than .000001.

1.43. Prove that every rational number can be expressed as a repeating decimal.

1.44. Find the values of x for which
2
3
3
2
4
2
(a)2x 5x 9x þ 18 ¼ 0, (b)3x þ 4x 35x þ 8 ¼ 0, (c) x 21x þ 4 ¼ 0.
p ffiffiffi p ffiffiffiffiffi p ffiffiffiffiffi
Ans. 5 (c) 1 1
2 ð5 17Þ; ð 5 17Þ
2
(a)3; 2; 3=2(b)8=3; 2
ffiffiffiffi
p
p
ffiffiffiffi
1.45. If a, b, c, d are rational and m is not a perfect square, prove that a þ b m ¼ c þ d m if and only if a ¼ c
and b ¼ d.
p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffiffiffi p ffiffiffi
5 12 5 2 15 þ 14 3 7
1.46. Prove that 1 þ p 3 þ p ffiffiffi ¼ :
ffiffiffi
5 11
1 3 þ
INEQUALITIES
1.47. Find the set of values of x for which each of the following inequalities holds:
1 3 x x þ 3
A 5; xðx þ 2Þ @ 24; jx þ 2j < jx 5j; > :
ðaÞ þ ðbÞ ðcÞ ðdÞ
x 2x x þ 2 3x þ 1
1
1
Ans. (a)0 < x @ , (b) 6 @ x @ 4, (c) x < 3=2, (d) x > 3; 1 < x < ,or x < 2
2
3
1.48. Prove (a) jx þ yj @ jxjþjyj, (b) jx þ y þ zj @ jxjþjyjþjzj, (c) jx yj A jxj jyj.
2
2
2
1.49. Prove that for all real x; y; z, x þ y þ z A xy þ yz þ zx:
2
2
2
2
1.50. If a þ b ¼ 1 and c þ d ¼ 1, prove that ac þ bd @ 1.
1 1
n
1.51. If x > 0, prove that x nþ1 þ > x þ where n is any positive integer.
x nþ1 x n
1.52. Prove that for all real a 6¼ 0, ja þ 1=aj A 2:
1.53. Show that in Schwarz’s inequality (Problem 13) the equality holds if and only if a p ¼ kb p , p ¼ 1; 2; 3; ... ; n
where k is any constant.
1
1.54. If a 1 ; a 2 ; a 3 are positive, prove that ða 1 þ a 2 þ a 3 Þ A p a 1 a 2 a 3 .
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
3
EXPONENTS, ROOTS, AND LOGARITHMS
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1.55. Evaluate (a)4 log 2 8 , (b) 3 log 1 Þ, (c) ð0:00004Þð25,000Þ , (d)3 2log 3 5 , 1 4=3 2=3
4 1=8 128 5 (e) ð Þ ð 27Þ
8
ð
ð0:02Þ ð0:125Þ
Ans. (a) 64, (b)7/4, (c) 50,000, (d)1/25, (e) 7=144
r
1.56. Prove (a)log MN ¼ log M þ log N, (b)log M ¼ r log M indicating restrictions, if any.
a
a
a
a
a
1.57. Prove b log b a ¼ a giving restrictions, if any.