25.18 (a) How many 5-digit numbers can be formed from the 10 digits 0,1,2,3,. . . . . In (2) the transverse axis lies on the y axis, the vertices are at V(0,a ) and V ’ ( 0 ,- a ) , and the foci are at F(0, c) and F‘(0, - c ) (see Fig. 29.31 In the expansion of determine the signs associated with the terms d2b4c3al and b3c2a4dl. . . . . . A=P(l+i)" or A 2000 P=---- $1102.52 using tables. . . . = u2 b2, we get c = 5 . Operations with Matrices . . . . . . Since the specified 3 books are always together, they may be considered as 1 thing. If the objective is a linear function and the constraints are linear inequalities, the values, if any, that maximize o r minimize the objective occur at the corners of the region determined by the constraints. . . ways, and the three groups in 3! If the rate of decay of carbon-14 is 0.0124% per year, how much carbon-14 will be left in the wood after 200 years? Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. . The distance, d , between two points ( x l , y l ) and (x2,yz) is d = d x2 - x1)2;Q2 - y1)2.Thus, the distance from the center C ( - 1 , 3 ) to (5,4) is 1 - = d ( 5 - ( - 1 ) ) * + ( : - 3 ) ~ = 62+l2=*. Multiplying both sides of x + y > 1 by the positive number x - y, (x + y ) ( x - y ) > (x - y ) or ( 6 ) Since x < y, x - y < 0. . . Another method. If ( a + l/a) L 2, then a2 + 1 L 2a, a2 - 2a + 1 I0, and ( a - 1)2 2 0 which is true. . . . . . . . . Missed Lectures? - -Y* 6 - 2 - 6 --6 2.Y 3 6 - y-12 3.~42 [APPENDIX C This will always eliminate the denominators. . . . . . (3! INDEX . . 17-15(a)). D) If equals are divided by equals, t h e results are equal provided t h e r e is no division by zero. . 25.20 From 11 novels and 3 dictionaries, 4 novels and 1 dictionary are to be selected and arranged on a shelf so that the dictionary is always in the middle. In how many ways can these three positions be filled? When lines are drawn through the points R and C and the points S and C, we have the asymptotes of the hyperbola. 5 ~ + 0 . . SOLUTION Number of lines formed if no 3 of the 10 points were in a straight line = Number of lines formed by 4 points, no 3 of which are collinear = 4C2 = 4.3 7 = 6. . . . 25.45 In how many ways can a person choose 1 or more of 4 electrical appliances? . . . . . 24 Chapter 25 Permutations and Combinations 25.1 FUNDAMENTAL COUNTING PRINCIPLE If one thing can be done in m different ways and, when it is done in any one of these ways, a second thing can be done in n different ways, then the two things in succession can be done in mn different ways. . . . SYSTEMS OF EQUATIONS INVOLVING QUADRATICS 192 18.9 Solve the system: (1) 2 + y3 = 35, (2)x [CHAP. 5! . . . . 3 . -: 1 -1 (b) A=[: (c) A = F [-:I 2 3 5 41 3 -2 (d) A = gd 363 MATRICES 2 1 and B:[- and 1 2 1 11 1 -3 2 B=[: -6 and B=[-S 1 and B=[10 :] 3 121 30.11 Solve each system of equations using a matrix equation of the form AX = B. . EXAMPLE 18.2. . . System of Real Numbers . Thus 4(7.6) = (4-7)6. . . Thus O ~ x < 4and - 4 < x s O , or - 4 < x < 4 . SOLUTION There are 8! and y - - I - y--5 4 Simplify each APPENDIX C] SAMPLE SCREENS 393 Check Using Study Works: Remember extraneous roots may be introduced when taking t h e root of both sides. . . arrangements of the beads on the bracelet, but half of these can be obtained from the other half simply by turning the bracelet over. -:I=[ -4 -12 I -6 -8 0 2 -2A = -2 The product AB where A is an m x p matrix and B is a p X It matrix is C,an m X n matrix. 18.16 The hypotenuse of a right triangle is 41 ft long and the area of the triangle is 180 ft2. . . . 0,- 3)2 x2 The equation of the ellipse is -+ - = 1 36 27 EXAMPLE 17.10. . Solve the system (1) x 2 + x y = 6 + ( 2 ) x2 5xy - 4y2 = 10 Method 1. . (c) The number of arrangements of the remaining 8 letters, of which 3 0’s are alike, = 8!/3! . If world population decline was exponential, what was the annual rate of decline? . . . (2) Associative property for addition The terms of a sum may be grouped in any manner without affecting the result. For example, if z = 6, then x = 3,y = 3; if z = -4, then x = -2,y = -2; etc. 25 25.14 Six different biology books, 5 different chemistry books and 2 different physics books are to be arranged on a shelf so that the biology books stand together, the chemistry books stand together, and the physics books stand together. . 3! Since (1) and (3) are symmetric in x and y , substitute x = U + U, - U in ( 1 ) and (3) and obtain y =U (4) U u2+u2=90 and (5) 824-u2+u2 = O . Hence the required number of orders = 5 . . X ~ ' X ~ = S 3.10.5 X ~ = 150 outfits 25.2 PERMUTATIONS A permutation is an arrangement of all or part of a number of things in a definite order. . 201 INEQUALITIES CHAP. . . Common terms and phrases. This paper. . 25.22 Compute the sum of the 4-digit numbers which can be formed with the four digits 2,5,3,8 if each digit is used only once in each arrangement. . . . Total number of ways = 3P3-7P7 = 3!7!= 6.5040= 30 240 ways. . . * * k2d kd k2d + kd The right-hand side of this equation = ka +-2+a+kd= - kd(k + 1) + 2a(k + 1 ) --k + 1 - q L L + 2ka + 2u 2 (h+kd) which is the value of (n/2)[2a+ ( n - 1)dl when n is replaced by ( k + 1). 24.33 Find the Richter number of an earthquake if its intensity is 3 16OOOO times as great as the reference intensity . is )(: .~ + d, [2a + (n - l)(il, that is a + (a +d) + ( a +2d) + - + [a + (n - l)d] = -n2[ 2 u + (n - l)d]. . [ ;-2 [ ;]=[(-3/2)+1 3 + 0 (3/2) o + o(3/2) -1 -4 -4 andX=-$A+fB. Each wind chime requires 3 hours of work from Jean and 1 hour of work from Wesley. . . . . . . . A = $20000(1.005)36 A = $23933.61 use the power key to compute (1.005)% To the nearest cent, the amount has been increased by $33.61 from the amount found in Example 24.3. The vertices are at (a, 0) and ( - U , 0), so V(5,O) and V ( - 5 , O ) . . . 29.3 Write the expansion of the determinant a1 61 c1 a2 62 63 43 c2 c3 by use of inversions. 17-18 cf) ellipse, Fig. a 24.31 Find the amount that will result if $90oO is invested for 2 years at 12% compounded continuously. . . It is interesting to note that these coefficients may be arranged as follows. . Missed Lectures? . 29.24 Determine whether the system + 4~ - 2y 62 = 8 2x - y + 32 = 5 2x - y 32 = 4 + is consistent. . 2 - 1 3 2 3 1 2 4 1 -3 -1 3 -1 2 -2 -3 3 - 1 2 1 4 2 0 - 3 -2 1 -3 2 1 3 - 1 4 - (d) 3 2 - 1 3 2 -2 0 3 4 3 1 -3 -2 1 0 2 4 1 0 1 -1 -1 2 1 0 29.39 Factor each determinant: (a) I f2 a3 :2 (6) b3 c3 1 1 1 1 1 x 1 1 y 2 x2 y2 z2 x3 y3 z3 29.40 Solve each system: x - 2y + z - 3 w = 4 2 x + y - 3 z = -5 3y + 42 + w = 5 22 - w - 4x = 0 w+3x-y=4 2 x + 3 y - z - 2 w = -4 3x-4y+22-4w = 12 2 x - y - 3 z + 2 w = -2 29.41 Find il and i4 for the system I 2il - 3i3 - i4 = -4 3il + i2 - 2i3 2i4 + 24 = 0 -il - 3i2 + 2i4 + 3is = 2 il 2i3 - is = 9 2i1+i2=5 + + 1 2 - 1 1 -2 3 2 -1 3 -1 1 -4 -1 4 -3 2 DETERMINANTS OF ORDER n 352 29.42 Determine whether each system is consistent. . il - 2i2 + i3 = 3 i2 3i4 - i5 = -5 il + i2 + i3 - is = 1 2i2 + i3 - 2i4 - 2is = 0 il i3 2i4 + i5 = 3 + + + SOLUTION D3= 1 - 2 3 0 0 0 1 -5 3 -1 1 1 1 0 -1 ~ 3 8 , 0 2 0 -2 -2 1 0 3 2 1 29.23 Determine whether the system x D= 1 - 2 1 0 0 0 1 0 3 - 1 1 1 1 0 -1 ~ 1 9 , 0 2 1 -2 -2 1 0 1 2 1 - 3y + 22 = 4 2x+y-3~=-2 4X - 5y =5 + is consistent. . . . . 25.76 In how many ways can 4 men and 4 women sit at a round table so that no two men are adjacent? . . . . Hence nothing can be concluded from these facts. (c) 4 x + 9 y 2 = 3 6 , y 2 = - (9-4, 3 9 Note that if x is greater than 9, y is imaginary. Since the circle goes through ( 2 , 6 ) , we substitute x = 2 and y = 6 to determine 3. . SOLUTION I R = log 10 I 6.90 = log 10 I -= antilog6.90 antilog0.9000 = 7.94 10 I = (antilog 6.90) I. I = (7.943 x 106)10 I = 7 943 OOOI, - 0.8998 + 0.9000 (0.01) 0.9004 - 0.8998 0002 + 0O.OOO6 -(O.ol) = 7.94 + 0.003 = 7.94 antilog0.9000 = 7.943 antilog6.9000 = 7.943 x 106 24.18 A sound that causes pain has an intensity 1014times the threshold intensity. . A partial list is given in AppendixI. . Find the numbers. How should the company place its order so that costs are minimized? . . . . . . + 19.15 Determine graphically the range of values of x defined by (a) x 2 + 2 x - 3 = 0 (b) X2+2x-3>0 (c) x2 + 2x - 3 CO. The product of two numbers a and b is a number c such that a x b = c. The operation of multiplication may be indicated by a cross, a dot or parentheses. Assume that the theorem is true for n = k. Then a2k - 62k is divisible by a We must show that a2k+2- b2k+2 is divisible by a a2k+2 - b 2 k + 2 = a2(a2k + b. . . . . . O! SOLUTION The first person may take any one of 5 seats, and after the first person is seated, the second person may take any one of the remaining 4 seats, etc. a . (c) 125 - (38 + 27) = 125 - 65 = 60 ( d ) 6 . . . . 29.28 For what values of k will the system x + 2y + kz = 0 2x + k y + 22 = 0 3x+y+z=O have non-trivial solutions? . . The second student can select 4 of the remaining 8 books in g c 4 ways. SOLUTION We are asked for the present value P which will amount to A = $2000 in 10 years. . . units respectively to the left of the origin. Exchange the entries on the main diagonal, swap all and a22. The required expansion is: al b 2 ~ 3- al b3c2 - a2b1c3 + a2b3c1- a3b2c1+ a3bl c2. . . (::) ( :;:) 170 10! 2 - 1 Number of arrangements = -= 302400. . . 17-16 -5 t Fig. . 4) To add or subtract fractions t h a t do n o t have a common denominator, first rewrite each fraction as equivalent fractions with a common denominator t h e n add or subtract t h e numerators t h e n write t h a t sum or difference over t h e common denominator. Q 3x2 - 1 is a rational fraction. . . = 1036 800. . . . The sum of the principal and the interest is called the amount. He was interested in most branches of mathematics, especially those which involved applications to physics and engineering problems. . EXAMPLES 1.10. . (1) Form the partitioned matrix [AII], where A is the given n X n matrix and I is the n x n identity matrix. + n =SOLUTION Step I. I. Moyer. . The symbols for “greater than” and “less than” are > and < respectively. Hence the required number of ways = 10- 10 = 100 ways. . (a) x2+y2= 9 XY= 4x2+y2=16 (g) x 2 + 3 x y + y 2 = 1 6 (4 x 2 - 4 y 2 = 36 (h) x 2 + 4 y = 4 -4 xz+3y2-1=o (f) xy -y2- 7x - 2y +3 =0 goes through ( O , O ) , (-4,0), (d) goes through (2,3), (-1,7), (c) and (0,6) and (1,5) Write the equation of the circle in standard form and state the center and radius. Substituting y = -2r in (1) or (2), 2 = 1 and x = +1. 26.5 (X - y2)6 = x6 + 62(-y2) + 6.5 -x4( 1.2 -y2)2 + 6.5.4 -x3( 1-2.3 + 6.5.4.3-2 x( -y2)5 + (-y2)6 1-2.3-4.5 -y2)3 + = x6 - 6xsy2+ 15x4y4- 20x3y6 15x2y8- 6xy'' + 6.5-4.3 -x2( 1.2-3.4 -y2)4 + yI2 In the expansion of a binomial of the form ( a - b ) " , where n is a positive integer, the terms are alternately + and -. . . How many such arrangements are possible? . . 26.6 + 4.3 - ( 3 ~ ~ ) ~-2b)2 ( + 4.3.2 -(3a3)( 1.2 1-2.3 = 8 1 ~-' 2~ 1 6 ~ + ~ 216a6b2 6 - 96a3b3+ 16b4 + (3a3 - 26)4 = ( 3 ~ ~4 )( 3~~ ~-26) )~( = x7 - 7x6 + 2 1 2 - 3sX4+ 35x3 - 21x2 + 7~ - 1 -26)3 + (-26)4 306 26.8 THE BINOMIAL THEOREM (1 - 2 ~ =)1 +~5(-2X) + 5.4 + -(-zX)2 1.2 = 1 - 1 0 ~40x2 - 80x3 26.9 (z+ 3 Y = (:y + 4( :r( ); [CHAP. . . . . . . . 22.1 Sequences . . . SOLUTION Multiply (2) by 3 and subtract from (1) to obtain 2r2 - 6xy + 4y2 = 0, x2 - 3xy + 2y2 = 0, ( x - y)(x - 2y) = 0 and x = y, x = 2y. . . . . 17-19 186 CONIC SECTIONS [CHAP. I ( a )Ellipse ( b )Hyperbola (c) Parabola Fig. . Number of arrangements = -- -- 369 600. Combinations of different things taken any number at a time The total number of combinations C of n different things taken 1 , 2 , 3 . . Chapter 31 31.1 Principle of Mathematical Induction 31.2 Proof by Mathematical Induction . Not display the result y - 1 4! = 120 - 24 = 2880 1.. 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