Post on 26-Dec-2015
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The commutative and associative laws do not hold for subtraction or division:
a – b is not equal to b – aa ÷ b is not equal to b ÷ aa – (b – c) is not equal to (a – b) – ca ÷ (b ÷ c) is not equal to (a ÷ b) ÷ c
PROPERTIES OF EQUALITY Reflexive property: x = x
Example: 2 = 2 or I am equal to myself
Symmetric property: If x = y, then y = x
Example: Suppose fish = tuna, then tuna = fish
Transitive property: If x = y and y = z, then x = z
Example: Suppose John's height = Mary's height and Mary's height = Peter's height, then John's height = Peter's height
Addition property: If x = y, then x + z = y + z
Example: Suppose John's height = Mary's height, then John's height + 2 = Mary's height + 2
Or suppose 5 = 5, then 5 + 3 = 5 + 3
Subtraction property: If x = y, then x − z = y − z
Example: Suppose John's height = Mary's height, then John's height − 5 = Mary's height − 5
Or suppose 8 = 8, then 8 − 3 = 8 − 3
Multiplication property: If x = y, then x × z = y × z
Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight × 4 = Darline's weight × 4
Or suppose 10 = 10, then 10 × 10 = 10 × 10
Division property: If x = y, then x ÷ z = y ÷ z
Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight ÷ 4 = Darline's weight ÷ 4
Or suppose 20 = 20, then 20 ÷ 10 = 20 ÷ 10
Substitution property: If x = y, then y can be substituted for x in any expression
Example: x = 2 and x + 5 = 7, then 2 can be substituted in x + 5 = 7 to obtain 2 + 5 = 7
BINOMIAL THEOREM
(x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x 2 +2xy + y 2 (x + y)3 = x 3 +3x 2 y + 3xy 2 + y 3 (x + y)4 = x 4 +4x 3 y + 6x 2 y 2 +4xy 3 + y 4 (x + y)5 = x 5 +5x 4 y +10x 3 y 2 +10x 2
y 3 + 5xy 4 + y 5
PROGRESSION
A sequence of values in which each term is obtained from the preceding term in the same way.
Ex. 5,7,9,11,13,15….. 2,4,8,16,32,64…..
ARITHMETIC PROGRESSION(A.P)
Is a sequence in which thereis a common difference “d” between any two consecutive terms.
Ex. 2,4,6,8,10….
FOR AN A.P.
Where: an= nth term or last term AM=arithmetic mean
am= any term before an d=common difference
S= sum of the first n terms n= number of terms
GEOMETRIC PROGRESSION(G.P)
Is a sequence in which there is a common ratio of each term to its preceding term.
Ex. 2,4,8,16,32….
FOR A G.P
Where: an= nth term or last term GM=geometric mean
am= any term before an r=common ratio
S= sum of the first n terms n= number of terms
HARMONIC PROGRESSION(H.P)
Is a sequence of terms in which each term is the reciprocal of the corresponding term of a series in arithmetic progression.
Ex. ¼, 1/6, 1/8, 1/10…..Harmonic Mean
SAMPLE PROBLEMS
The equation whose roots are the reciprocal of the roots of 2x2-3x-5=0 is
a.5x2+3x-2=0b.2x2+3x-5=0c. 3x2-3x+2=0d.2x2+5x-0=0
EE Board April 1996, March 1998The polynomial x3+4x2-3x+8 is divided by x-5, then the remainder is
a. 175 c. 218
b. 140 d. 200
Find the term involving y5 in the expansion of (2x2+y)10
a. 8064x10y5
b.8046x5y5
c. 8046x10y5
d. 4680x5y5
4. what is the 4th term of the of the expansion of (x+x2)100?
a. 1650x103
b.161700x103
c.167100x103
d.167100x103
22. in a pile of logs, each layer contains one more log than the layer above and the top contains just one log. If there are 105 logs in the pile, how many layers are there?
a. 11b.12c. 13d. 14
23. determine x so that :x, 2x + 7, 10x-7 will be a geometric progression.
a. 7, -7/12b. 7, -5/6c.7,14/5d.7,-7/6
In a potato race, 8 potatoes are placed 6 feet apart on a straight line, the first being 6 ft from the basket. A contestant starts from the basket and puts one potato at a time into the basket. Find the total distance he must run in order to finish the race.
a.222 ftb.342 ftc. 432 ftd.532 fft
Arithmetic ProgressionFind the 14th term in the arithmetic sequence: 1,3,5,7
a. 25 c. 29
b. 27 d. 31
CE Board May 1995What is the sum of the progression 4,9,14,19 up to the 20th term?
a. 1030 c. 1040
b. 1035 d. 1045
EE Board Oct 1991The Fourth term of G.P. is 216 and 6th term is 1944. Find the 8th term ?
a. 17649 c. 16749
b. 17496 d. 17964
ECE Board April 1998
The sum of the first 10 terms of a geometric progression 2,4,8 is?
a. 1023 c. 225
b. 2046 d. 1596