Factorizing
If (2X + 1) is a factor of the expression 6 x2 + 5x + 1 then the other factor is ……
The quotient of dividing x2 – 3x – 4 by ( x + 1) is ............
The length of a rectangle is 2x + 3, if its area is 2 x2 – 7 x – 15 then its width is ……....
5 x2 – 2 x – 7 = (x + ……) (…….– 1)
If a + b = 3 and x + y = 5 then a(x + y) + b (x + y) = ………
If x2 + 5 x y + 6 y2 = 21, x + 3 y = 7 then x + 2 y = ……..
If x2 – y2 = 10, x + y = 5, then x – y = ……
If a + b = 7, a – b = 5 then a2 – b2 = ……..
If x2 + k x + 9 is a perfect square then k = ……..
If a x2 – 20 x + 4 is a perfect square then a = …….
X2 – 14 x + 40 + ……. is a perfect square.
4 x2 + 2 x y + 9 y2 + ……. or ……. is a perfect square.
If x2 – 8x y + 16 y2 = 25 then x – 4y = …….
If 2x + 3y = 3 then 4x2 + 10 x y + 9 y2 = …….
X2 + y2 = (x + y)2 – ……., X2 + y2 = (x + y)2 + ……. .
x2 + y2 = ( x – y)2 + ……, x2 + y2 = ( x – y)2 – …… .
If x2 + y2 = 11, x y = 3 then: (x + y)2 = …., x + y = …. ,(x – y)2 = …., x – y = …..
If (x + y)2 = 14, x2 + y2 = 9, then x y = ……, if (x – y)2 = 13, x2 + y2 = 8 then x y = ....
If (x + y)2 = 15, x y = 2 then x2 + y2 = ……, if (x – y)2 = 10, x y = 3 then x2 + y2 = … .
If x + y = 5, x y = 6 then x2 + x y + y2 = …., If x – y = 1, x y = 6 then x2 + x y + y2 =
Let x3 – y3 = 28 and x - y = 2, Find the value of the expression x2 + x y + y2
20
20
 75 
 25
7
7
( 73 )2 + 2 ( 27 )(73 ) + ( 27 )2 = (….………….)2 = ……2 = …….
32 2  2(32)( 68)  68 2 = …….
(7.3)2 + 2 × 7.3 × 2.7 + (2.7)2
(103)(97) = (…… + …….) (……. – …….) = …… × …….. = …….
31 × 29 = ………………
(763)2 – (237)2 = (…… + …….) (……. – …….) = …… × …….. = ……. .
(999)2 – 1 = ( …… + …… )( ……. – ……) = …… × …….. = ……. .
s.s. of the equation 3x ( x2 – 4)is ….……, s.s. of the equation 4x ( x2 + 9) is ………
s.s. of the equation (x – 1) (x + 3) is ….….., s.s. of the equation (2x + 5) (3x – 1) is ……..
s.s. of the equation 4 x 2 = 9 is ………, s.s. of the equation 9x2 + 25 = 0 is ……….
Factor each of the following:
 x2 +11 x + 10 …….…..…….………………, x2 - 7x + 10 …..………...…………………,
x2 - 3x – 10 ……………..………..…………, x2 - 7x + 12 ……………..…………...…….
 3 a2 + 7a + 2 ………………………..………, 3m2 - 19 m + 6 ……………………………,
10a2 + 11ab - 18 b2 ………...……..………, 8x3 - 27x2 - 20x ………...………………….
 9 a2 + 6 a b + b2……………………..………, 25 b2 - 10 b +1….………..………………,
4 b2c + b c 2 + 4 b3 …………………………, 20 ay2 - 60ay + 45a……………………….
 x2 – 4 ………………..…………….….……, -9x2 + 25 ……….……..……………….……,
(x + 1)2 – (x - 1)2……………………...……, (2x + 1)2 – 4 a2 …...…………………….… .
 x3 + 8 …………………………….………, A 512 x3 – y3 …..………….…………..………,
1 3
a – 8 b3 ………………………………, (m
8
– 2 n)3 – 8 n3 ………..……………………,
(x + y)3 – (x – y)3 …………….…………………………………………………………..… ,
.................................................................................................................................…,
5 x3 – 40x …………………………….……………………………………...……………….
 a x + b x + a y + by……………………………….…………………………………………,
a3 + a2 + a + 1 …………………………………….…………………………………………,
9x2 - 4 a 2 + y2 + 6 x y…………………………….…...……………………………………,
x3 + 2x2 – x – 2 ……………………………………………………………………………….
 4x4 + y4 ……………………………………………………...……………..…………………,
………………………………………………………………………………………………… ,
8x4y2 + 162 z4y2 ………………………………………………………..………..……………
……………………………………………………………………………..………….……..,
x4 + x2 y2 + 25 y4 ……………………………………………………………...………………
……………………………………………………………………………….……..……..…,
m4 - 11 m2 n2 + n4 ……………………………………………………………………………
………………………………………………………………………………….…………....,
m4 - 7 m2 n2 + n4 ………………………………………………………..……………………
………………………………………………………………………………...…………..…,
 Find s.s. of each of the following: x R,





6x2 - 7x -3 = 0.
5x2 + 12x = 44.
(x - 3) (x + 1) = 5
(x + 3)2 + 3 (x +3) - 10 = 0
(x + 3)2 - 49 = 0
What is the real number if added to its square results 12?
Find two real numbers whose product is 45 and one of them is 4 more than
the.
What is the real number if added to its square results 12?
The length of a rectangular piece of land is more than its width by 5 meters
and if its area 500m square then find its dimensions.
Hatem is 4 years older than Hanan now, and the sum of squares of their ages
now is 26. Find their ages now?
Integer powers
 How many numbers can be formed from five even digits?
 Find the value of x in each of the following
If a 5  3 , then  a 5 
, if  a 11  4 , then a11 
, (-1)101 =
if x 395  x 396 = zero, then x =
, (-1)3576 =
 2
3x
 If

4x
3
x
 3
5 x 1
= 26 ,
= 3, then 4
3
= 5, then
+ 37 + 37 =
33 x  6
x 1
x2
=
,
 5
2 x 1
3
x2
, the third of
3 x 1
3
3 x is
=
x 3
= 7, then
, if 5
=1, then x =
2 x 5
n
x = , a +1=a (
+
= 1 then x =
, if  x  4 
, If
.......
zero
9
, if x =
 if x =


5
2 3 , y =
 if a = 7 , b =
3x  3x  3x

=
3x  3x  3x

 If 5
x2
3
=
1

2 3
5
then x y =
then find the value of a101b100

 5 x 1  5 x =
x
x 1
 If 2  2 
3
find the value of x
2
 2 x  1  10 find the value of x
2 x 1  2 x
x–2
 If x
find the value of x
x 1 = 2
2 2
 If 2
x 1
 7

10
, 28 + (-2)8 =
=
zero
= 1, then x 
= 1 then x 
,
.
then x – 1 = ………, If x3 = 4, then
)
 27
2
, 37
x  5zero is undefined then x =
1
3
if
, 215 + 215 =
=
,If  x  5
2
2
.
x 1
, 25 +
 1 
zero
–1
 =
 2 + (2) + 
 2
 1 
zero
 +
 3 +  
3


2 x 3
2 x1 =
x  5zero = 1 then x 
zero
if 2 x  6  = 1 then x 
 xm  …….. = 1, 0.002 × 0.05 = 10
=7
, the sixth of 25  35 is
x
,if 2
n
 7
,
3


4
,if  3  = , then  
2
 2 9
, if
-3
=5
, if 3x + 3x + 3x = 1 then x =
, 220 + 221 =
 if
x 1
=3
3 x1  3 x  2
 Prove that
= 12
3x
2 3x  3  2 3x  1
 Prove that
=6
4  2 3x  6  2 3x 1


2 3

5
 if 3x = 5,

5
2 3 =
1
= 7 then 3x + y =
y
3
 312 × 214 =
9
then x =
4
 if 2x – 1× 31 – x =
2 2 n 1  5 2 n 1

=
10 2 n
8x 9x
 If
= 64 then find the value of x and calculate the value of (4) – x
x
18
1
2 x 3x
 If
=
then x =
2
12 x
 Put each of the following in the simplest form
 3   3
 3
15   5   3
9  5 
7

8
6
2


3
3
3
10 2 10  7
0.12  0.001
4 x 1  9 2  x

and find the numerical value when x = 1
62x
49 n  25 2 n  3 4 n
 If
= 343 then calculate the value of 62n
7  n 15 4 n
 Prove that
27 x  1  8 x
2 2   3 3 
2x
2x
=
1
27
 Find the value of x in each of the following
 2x = 32
 2x – 1 = 1

5x
2
4
 3
x–2
=
74 x
2
1
=
9
2
  
5
2 x 1
2
  
3
x4
125
=
8
= 2
1
4
 If 3x = 27, 4x + y = 1 calculate the value of x and y
Probability
Probability is the ratio between number of times event occur and number of all
possible outcomes.
It takes a value from zero to 1 (100%) P (A) ∈ [0, 1]
The probability of the impossible event is zero and the probability of certain event is 1
If the probability of occurring an event is x then the probability of not occurring
it is 1 – x
To calculate the probability we divide number of occurring this event over
number of all possible outcomes and to calculate the number of occurring the
event we multiply the probability of its occurring by the total number.
A numbered card is selected randomly from a set of similar cards numbered
from 1 to 24 Find the probability of getting a card carries:
A) a multiple of 4
B) a multiple of 6
C) a multiple of 4 and 6 together.
D) a multiple of 4 or 6
E) a number divisible by 25
F) a positive integer less than 25
During the training of a soccer-team for the final-match of world-cup, a player
scored 12 goals out of 15 kicks and another player scored 9 goals out of 12 kicks.
Whom will the coach choose to kick the penalties during the match? Why?
2 A calculator manufacturing company examined randomly electronic-circuits
in a sample of 200 units. The defective production was 6%.
A) How many units are out of order in this sample?
B) If the total production in one month was 1500 units. How many units are
functional units for marketing?
Good Luck :)