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9x + 1 = 0
56. If two roots of the cubic x3
(b − a)3.
27 57. Required the conditions to be satisfied, that in the division of x4 + 9x2 + rx + s by x2 + ax + b, the remainder may be 0 independently of the value of a, after three terms of the quotient are obtained ; and when these conditions are satisfied, obtain an equation for finding b from q, r, s, given, and shew how to solve it.
58. Prove that in any equation, the greatest negative coefficient increased by unity is greater than the greatest root. Find also a limit less than the least positive root.
59. Find the sum of the mth powers of the roots of an equation of n dimensions in terms of the sums of inferior powers.
60. A recurring equation of an even number of dimensions may be solved by means of an equation of half the number of dimensions; prove this, and solve the equation
204 + 4x3 5x2 + 4x + 1 = 0. 61. Shew that
nxn-1 - (n − 1) pan-2 + (n – 2) (c"-3.... - Q is the sum of the products of every (n − 1) simple factors of хп panel + qxn-2
Qr + R. 62. Solve the equation x* – 2.r3 + 3x2 – 2x + 1 = 0,
1 1 and shew that the roots are of the form a, b,
a 63. Shew that the rational roots of any numerical equation can always be found.
Mention any ways of shortening the operation; and apply the method to the equation
3.03 2:x2 6x + 4 0.
64. If e be the sum of two roots of the equation
X+ + 9x2 + rx + s = shew (without consideration of the reducing cubic) that ez has only three different values.
65. Every equation of an even number of dimensions, of which the last term is negative, must have at least two possible roots, one positive, the other negative.
66. The coefficient of the second term of an equation, with 1833 its proper sign, is the sum of the roots with their signs changed; the coefficient of the third term is the sum of the products of every two roots with their signs changed, &c., and the last term is the product of all the roots with their signs changed.
67. Find a number next greater than the greatest positive root, and next less than the least negative root of the equation
23 - 4.02 4x + 20 = 0.
68. Determine the relation which exists between the coefficients of the equation 23 – px? + 9x - r = 0, when the roots are in arithmetic progression.
69. Jf p be the coefficient of the second term of an equation of n dimensions, shew that the sum of the roots of it and all its successive limiting equations is equal to -- p.
70. No equation can have more positive roots than it has changes of sign, nor more negative roots than it has continuations of the same sign. If one term be wanting, what inference may be drawn respecting the number of possible roots ?
71. Solve the equation 23 + 6x = 2, by Cardan's method, and shew that the possible root is 32748, having given
log 2 = .30103, log 1.58740 = .200686,
log 1.25992 = .100343. 72. Impossible roots enter equations by pairs; make this 1834 appear also from geometrical considerations.
73. Shew how to transform an equation into one which shall want its second or third term : under what circumstances may both be made to disappear by one operation ? Solve the equation
X3 -- 3x2 - x + 3 = 0.
74. Every equation whose roots are possible has as many changes of sign as it has positive roots, and as many continuations of the same sign as it has negative roots. State the proposition when the roots are not all possible. Hence shew that all the real roots of
x3 + x + x2 25x – 36 = 0 lie between 10 and + 10.
75. Solve the recurring equation 2.24 5.x3 + 6x2 + 2 = 0; and having given that 2} is an approximation to a root of x3 5x – 3 0, find its exact value to four places of decimals.
76. Each of a series of numbers is the sum of two roots of the equation XN pxn-1 + qan-2
rich-3 + .. prove that the symmetrical function, formed by combining these numbers in products, three taken together, is equal to (n-1) (n − 2)(n-3)
+ p.q. (n − 2)2 + r. (n – 4). 1.2.3
77. Express the sum of any powers of the roots of an equation
79. Shew how the solution of a biquadratic equation may be made to depend upon that of a cubic; and exhibit the roots of the biquadratic in terms of the three roots of the cubic.
80. Shew that all the rational roots of an equation can always be found. Apply the general method to solve the equation
2x3 3x2 + 23 3 = 0. 81. Every biquadratic equation of the form 24 + px3 + 2x2 + rx + s = 0, can be solved by means of a quadratic equation, if p3 - 4p9 + 8r = 0.
82. Form an equation, whose roots differ by a given quantity 1836 from those of x" + pxn-1 +...+Q=0. Explain the use of this process to determine a superior limit of the roots of an equation.
Ex. x4 2.03 3x2 15x 3
83. Every equation of an even number of dimensions, and whose last term is negative, has at least two real roots. 84. Take away the third term from the equation
24 18.23 – 60x2 + 3 2 = 0. Find the equation whose roots are the squares of the differences of the roots of the equation x3 + qx += = 0. 85. The equation x3 – qx2 + 2 = 0 has two impossible
q3 roots if 4
27 Also, in general, an equation cannot have all its roots possible if the coefficient of any term be less than a mean proportional between those of the adjacent terms.
86. Shew what conclusions may be drawn as to the number of positive and negative roots of an equation, from observing the signs of its terms. Apply them to determine the nature of the roots of the equation
2.24 4x + 8 = 0.
QUESTIONS IN PLANE TRIGONOMETRY.
1. Prove Demoivre's formula
(cos A +V-1.sin Am = cos mA + V-1.sin mA. 2. Why is Cardan's formula for the solution of a cubic equation inapplicable when all the roots are possible? solve the equation in this case by trigonometrical formulæ, and reduce the results for logarithmic computation.
3. Given two sides and the included angle of a plane triangle: find the remaining parts, and reduce the results to logarithmic computation.
4. The sides of a plane triangle are 3, 5, 6: compare the radii of the inscribed and circumscribed circles. 5. If 1, p,p';. ....p-1 are the roots of the equation x" — 1 0; find the value of 1m.pk + p.1" +....tp.p2r + p2m.p" +....
6. Prove that
7. In an isosceles plane triangle, prove geometrically that the versed sine of the vertical angle : radius :: the square of the base : twice the square of either side. .
8. Prove that cos (A + B) sin (A - B) + cos (B + C) sin (B - C) + cos (C + D) sin (C – D) + cos (D + A) sin (D – A)