ALGEBRA 1

PROPERTIES OF REAL NUMBERS

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

LAWS OF EXPONENTS

RADICALS

QUADRATIC EQUATION

CUBIC EQUATION

LOGARITHM

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 

Term with yr

rth term

SEQUENCE AND SERIES

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

Ans. A.

What is the discriminant of the equation 4x2=8x-5?

a.8b.-16c.16d.-8

Ans. B. -16

7. Given: log3 ( x2-8x)=2. Find x.

a. -1b.9c.-1 and 9d. 1 and 9

Ans. c. -1 and 9

A. C.

B. D.

Ans. A.

Practice:

Find x, if

a.√2b.√3c.√5d.√7

Ans. B.

Find the value of k so that 4x2+6x+k is a perfect square.

a. 36 c. 9

b. 2.5 d. 2.25

Ans. D.

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

Ans. c. 218

ECE Board April 1993Solve for the value of x in the ffequation: x3logx=100x

a. 12 c. 30

b. 8 d. 10

Ans. D. 10

Find the term involving y5 in the expansion of (2x2+y)10

a. 8064x10y5

b.8046x5y5

c. 8046x10y5

d. 4680x5y5

Ans. A.

4. what is the 4th term of the of the expansion of (x+x2)100?

a. 1650x103

b.161700x103

c.167100x103

d.167100x103

Ans. B.

5. what is the coefficient of the term free of x of the expansion of ( 2x-5y)4

a.256b.526c.265d.625

Ans. D. 625

6. Whatis the sum of the coefficients of the expansion of ( 2x-1)20

a. 0b.1c.2d.3

Ans. A. 0

21. The sum of three arithmetic means between 34 and 42 is?

a.114b.124c.134d.144

Ans. A. 114

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

Ans. D. 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

Ans. D.

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

Ans. c. 432 ft

Arithmetic ProgressionFind the 14th term in the arithmetic sequence: 1,3,5,7

a. 25 c. 29

b. 27 d. 31

Ans. B. 27

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

Ans. A. 1030

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

Ans. B. 17496

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

Ans. B. 2046

What is the sum of the 6+9+12…………..+171?

a. 4956 c. 5198

b. 4389 d. 5462

Ans. A. 4956