3rd.prep first term .math

١
Alg.
Sheet (1)
[1] Find the values of a and b in each of the following if :
1) ( a , b ) = ( -5 , 9 )
2) ( a – 2 , b + 1 ) = ( 2 , - 3 )
3) ( 6 , b – 3 ) = ( 2 – a , -1 )
4) ( 3 , b ) = ( 5 a – 1 , 4 a )
[2] If X = {1 , 2} , Y = {3 , 4 , 5} find X × Y and represent it by :
a) An arrow diagram b) Graphical diagram
[3] If X = {a , b} , find X2
and represent it by an arrow diagram
[4] Complete the following :
1) If X = {1 , 2 , 3} , Y = {4} , then X × Y = ………………..
2) If X = {5 , 6} , Y = {a} , then Y × X = ………………
3) If X = {1 , 2} , then X × Ø = ……………
4) {2 , 3} × {4 , 5} = …………….
5) If X2
= {(1 , 1) , (1 , 2) , (2 , 1) , (2 , 2)} , then X = ……………
6) If X × Y = {(2 , 5) , (3 , 5)} , then (3 , 2) ∈ ………..
7) If ( X – 1 , 11 ) = ( 8 , Y + 3 ) , then YX 2+ = ………..
[5] Choose the correct answer from those given :
1) If : ( 5 , x – 8 ) = ( y + 1 , - 5 ) , then x + y = ………………
a) 4 b) 5 c) 6 d)7
2) {3} × {3} = ……………
a) {9} b) {3} c) {(3 , 3)} d) 9
Cairo Governorate
Nozha Directorate of Education
Nozha Language Schools
Ismailia Road
Department : Math
Form : 3rd
prep.
Sheet

٢
3) If n (X) = 3 , n (X × Y) = 12 , then n (Y) = …………..
a) 4 b) 9 c) 15 d) 36
4) If n (X2
) = 4 , n (X × Y) = 8 , then n (Y2
) = …………
a) 1 b) 4 c) 16 d) 64
5) If X = {3 , 4} , then n (X × Ø ) = ……………
a) zero b) 1 c) 2 d) Ø
[6] If X = {2 , -1} , Y = {4 , 0} , Z = {4 , 5 , -2} , find :
a) X × Y b) Y × Z c) X2
d) n (X × Z) e) n (Y2
) f) n (Z2
)
[7] If X = {2 , 3} , Y = {3 , 4 , 5} , find
a) X × Y and represent it by an arrow diagram and a Cartesian diagram .
b) n (X × Y) c) n (Y2
) d) (X × Y) ∩ Y2
[8] If X = {3 , 4} , Y = {4 , 5} and Z = {6 , 5} , then find :
a) X × (Y ∩ Z) b) (X – Y) × Z c) (X – Y) × Y (Y – Z)
[9] If X = {1} , Y = {2 , 3} , Z = {2 , 5 , 6}
Represent each of X , Y and Z by venn diagram , then find :
First : a) X × Y b) Y × Z c) X × Z d) Y2
Second : (X × Y) U (Y × Z)
Third : X × (Y ∩ Z)
Fourth : (X × Y) ∩ (X × Z)
Fifth : (Z – Y) × ( X U Y )

٣
Sheet (2)
1) If the point (X , 2) lies on Y – axis , then X = …………..
a) zero b) 1 c) 2 d) 3
2) If the point (5 , b – 7) is located on the X – axis , then b = …………..
a) 2 b) 5 c) 7 d) 12
3) If the point (-4 , Y) lies on the X – axis , then 2 Y – 1 = …………
a) 1- b) 1 c) -8 d) -9
4) If the point (X – 4 , 2 – X) where X ∈ Z is located on the third quadrant then X
equals …………
a) 2 b) 3 c) 4 d) 6
Sheet (3)
1) If F is a function from the set X to the set Y , then : X is called ……………
a) the range of the function F b) the domain of the function F
c) The codomain of the function F d) the rule of the function F
2) If F is a function from the set X to the set Y , then : Y is called …………..
a) the domain of the function . b) the codomain of the function .
c) the range of the function . d) the rule of the function .
3) The opposite diagram represents
A function on X , its range is ……………
a) {a} b) {a, b , c}
c) {a , b} d) {b , c}
[2] If X = {3 , 4 , 5} , Y = {4 , 6 , 8 , 10} and R is a relation from X to Y where "a R b"
means "a =
2
1
b" for each a ∈ X , b ∈ Y
Write the set of the relation R and show that R is a function , then write its range .
X
a
b
c

٤
[3] If X = {4 , 6 , 8 , 10} , Y = { 2 , 3 , 4 , 5} and R is a relation from X to Y , where
“ a R b ” means “ a = 2b ” for each a ∈ X , b ∈ Y
Write R and represent it by an arrow diagram .
[4] If X = {1 , 3 , 4 , 5} , Y = {1 , 2 , 3 , 4 , 5 , 6} and R is a relation from X to Y , where
“ a R b ” means “ a + b = 7 ” for each of a ∈ X , b ∈ Y
Write R and represent it by an arrow diagram and also by a Cartesian diagram .
[5] If X = {1 , 2 , 3} , Y = {2 , 3 , 7} and R is a relation from X to Y , where “ a R b ”
means “ a + b = a prime number ” for each a ∈ X , b ∈ Y
Write R and represent it by an arrow diagram . is R a function ?
[6] If X = {-2 , -1 , 1 , 2} , Y = {
8
1
,
3
1
, 1 , 3 , 8 } and R is a relation from X to Y ,
where “ a R b ” means “ a3
= b ” for each a ∈ X , b ∈ Y
Write R and represent it by an arrow diagram and also Cartesian diagram .
[7] If X = {2 , 5 , 8} and Y = {10 , 16 , 24 , 30} and R is a relation from X to Y where
“ a R b ” means “ a is a factor of b ” for each a ∈ X , b ∈ Y
Write R and represent it by an arrow diagram and by Cartesian diagram . is R a function ?
and why ?
[8] If X = {2 , 3 , 4} , Y = {6 , 8 , 10 , 11 , 15} and R is a relation from X to Y , where
“ a R b ” means “ a is a factor of b ” for each a ∈ X , b ∈ Y
write the relation R .
[9] If X = {6 , 4 , 2 , 0 , -2 , -4 , -6 } , and R is a relation on X where “ a R b ” means
“ a is the additive inverse of b ” for each a ∈ X , b ∈ X
Write R and represent it by an arrow diagram and show with reason if R is a function or
not ? and if R is a function , mention its range .

٥
[10] If X = {0 , 1 , 2 ,
2
1
} and R is a relation on X where “ a R b ” means “ a is the
multiplicative inverse of b ” for each a ∈ X , b ∈ X . write R and represent it by
an arrow diagram and show if R is a function or not .
[11] If X = {1 , 2 , 4 , 6 , 10} and R is a relation on X where “ a R b ” means “ a is a
multiple of b ” for each a ∈ X , b ∈ X .
Write R and represent it by an arrow diagram and also by a Cartesian diagram.
is R is a function ? and why ?

٦
Sheet (4)
1) The function F where (X) = 2 X – 3 X4
+ 1 is a polynomial function of ……….degree
a) first b) second c) third d) fourth
2) The function F : F (X) = (X – 5 )3
is a polynomial function of ………..degree .
a) zero b) second c) third d) fourth
3) The function F : F (X) = X (X- 2 X2
) is a polynomial of the ……….degree .
4) The function F : F (X) = X2
(X-3)2
is a polynomial of the ………….degree .
5) If : F (X) = X2
– X + 3 , then : F (3) = ………
a) 3 b) 6 c) 9 d) 12
6) If : F (X) = a X + 6 , F (2) = 2 , then a = …………..
a) 2 b) -2 c) 4 d) 6
7) If : F (X) = X – 5 and
2
1
F (a) = 3 , then a = ……….
a) 2 b) 8 c) 11 d) 16
1) If (3 , y) ∈ the set of the function F where F (X) = X + 2 , then y = …………..
2) If (a , a) ∈ the set of the function F where F (X) = 2 X + 3 , then a = …………
[3] If : F (X) = 2 X2
– 5 X + 2
a) Mention the degree of F b) Prove that : F (2) = F (
2
1
)

٧
Sheet (5)
1) The function F : R R where F (X) = 5 is represented by a straight line parallel to
………….and intersects y-axis at the point …………….
2) If F (X) = 3 , then F (5) + F (-5) = ……………
3) If F (X) = 5 , then
)10(
)5(
F
F
= ……………
4) The liner function given by the rule = 2 X – 1 is represented graphically by a straight
line intersecting the X-axis at the point …………..
5) The liner function given by the rule Y = 3 X + 6 is represent graphically by a straight
line intersecting the X-axis at the point ……………
6) The point of the vertex of the curve of the function F : F (X) = 2 X2
– 4 X + 5 is ………
7) If ( -2 , y) belongs to the curve of the function F : F (X) = X2
+ 1 , then : Y = ……..….

٨
Represent Graphically
[1] Represent the following function graphically , where X ∈ R :
a) F (X) = 5 b) F (X) = - 4
[2] Represent each of the following linear function graphically and find the point of
intersection of the straight line which represents each of them with the coordinate axes ,
where X ∈ R :
a) F : F (X) = X + 2 b) F : F (X) = -2 X + 3
[3] Represent each of the following function graphically and from the graph , deduce the
coordinates of the vertex of the curve and the equation of the line of symmetry and the
maximum or minimum value of the function , where X ∈ R :
a) F : F (X) = X2
+ 2 X + 1 taking X ∈ [ - 4 , 2 ] .
1) If : X = {1 , 3 , 5} , F : X R and F (X) = 2 X + 1 , then the range of F = ………….
2) The liner function F : F (X) = X + 7 is represented by a straight line cuts X-axis at the
point ……………
3) The liner function F : F (X) = 2 X – 1 is represented by a straight line cuts y-axis at the
point ………….

٩
Unit (2)
Sheet (6)
1) The proportion is ………………….
2) If a , b , c and d are proportional quantities , then c is called ………………
3) If the quantities a , b , c and d are proportional , then :
b
a
= ………….
4) The fourth proportional for the numbers 4 , 12 and 16 is ……………
5) The second proportional for the numbers 2 , 4 and 6 is ……………
6) The third proportional for the numbers 8 , 6 and 12 is …………..
7) The first proportional for the numbers 5 , 27 and 45 is ………….
8) If 3 , 4 , X and 11 are proportional , then : X = ………..
9) If 7 X = 3 Y , then :
Y
X
= …………
10) If 5 a – 4 b = 0 , then :
b
a
= …………..
11) If
ba
ba
118
75
+
−
= 0 then :
a
b
= ……………
12) If 9 a2
– 25 b2
= 0 where a ∈ R+
and b ∈ R+
, then :
b
a
= ……….
13) If
Y
X
=
5
2
, then :
Y
X
2
2
= …………..
14) If
2
a
=
3
b
, then :
b
a
3
2
= ………….
1) If
b
a
5
3
=
2
1
, then :
b
a
= ……….
a)
5
6
b)
6
5
c)
3
2
d)
2
3
2) If : 5 a , 2 , 3 b , 7 are four proportional quantities , then :
b
a
= …………..
a)
7
3
b)
35
6
c)
5
3
d)
2
3

١٠
3) If
ba
ba
−
+ 2
=
3
2
, then :
a
b
= …………
a)
8
1
b) 8 c) -
8
1
d) – 8
[3] Find the value of X in each of the following , If :
1) ( 2 X – 3) : (X – 5) = 1 : 4
2) (X2
– 8) : (2 X2
+ 1) = 1 : 3
3) If
YX
YX
−
+
2
3
=
3
4
, find the ratio X : Y
4) If X2
– 4 Y2
= 3 X Y , find X : Y
5) If
b
a
=
4
3
, then find the value of :
a) b)
6) If
Y
X
=
3
2
, find the value of the ratio :
XY
YX
−
+
6
23
7) find the number that if it is added to each of the numbers 3 , 5 , 8 and 12 , it becomes
proportional .
8) Prove that : a , b , c and d are proportional quantities if :
a)
b
ba +
=
d
dc +
b)
ba
a
−
=
dc
c
−
9) If a : b : c = 5 : 7 : 3 and a + b = 27.6 , find the value of each of : a , b and c .
10) If 2 a = 3 b = 4 c , find a : b : c
[4] Answer the following :
1) Find the number which if it is added to the two terms of the ratio 7 : 11 it will be 2 : 3
2) Find the number that if we subtract thrice of it from each of the two terms of the ratio
69
49
, the ratio becomes
3
2
3) Find the number which if its square is added to each of the two terms of ratio 7 : 11 it
becomes 4 : 5
4) Find the positive number which if we add its square to each of the two terms of ratio
5 : 11 it becomes 3 : 5
b2
– a2
a2
– b2
4 a + b
2 a – b

١١
5) What is the number which is subtracted from the antecedent of the ratio 15 : 13 and
added to its consequent to become 3 : 4
6) Two integers , the ratio between them is 3 : 7 and if we subtracted 5 from each term , the
ratio between each of them becomes 1 : 3 , find the two umbers .
7) The ratio between two integers is
4
3
, if we add 4 to the small number and subtract 3
form the great number , the ratio will become 8 : 9 find the two numbers .
8) Two integers , the ratio between them is 2 : 3 , if you add to the first 7 and subtract from
the second 12 , the ratio between them becomes 5 : 3 find the two numbers .
Sheet (7)
1) If
b
a
=
d
c
=
5
3
, then :
db
ca
+
+
= ………………
2) If
b
a
=
d
c
=
f
e
=
5
3
, then :
fdb
eca
+−
+−
2
2
= ……………..
3) If
X
4
=
Y
7
=
XY
a
−
, then : a = …………
4) If
b
a
=
d
c
=
f
e
, then :
fb
ca
4.......5
.......35
++
++
=
b
a
[2] If a , b , c and d are proportional quantities , prove that :
1)
db
ca
35
35
+
+
=
bb
ca
23
23
−
−
2)
ca
ca
35
23
+
−
=
db
db
35
23
+
−
3)
bd
ac
= (
db
ca
−
−
)2
4)
db
ca
53
53
−
−
=
b
a
where a , b , c and d are positive quantities .
5) 3
35
35
db
ca
−
−
=
db
ca
+
+
2 2
2 2
3 3
3 3

١٢
[3] If
b
a
=
d
c
=
f
e
prove that :
1)
db
ca
5
5
+
+
=
fd
ec
3
3
−
−
2)
fdb
eca
472
472
−+
−+
=
fb
ea
8
8
−
−
[4] If
YX
a
+4
=
YX
b
4−
, prove that :
YX
ba
35 −
+
=
YX
ba
53 +
−
[5] If
19
yx +
=
7
zy +
, prove that :
13
2 zyx ++
=
6
zx −
[6] If
cba
X
+−
=
acb
y
+−
=
bac
z
+−
, prove that :
a
yX +
=
b
zy +
[7] If
ba
X
+2
=
cb
y
−2
=
ac
z
−2
, then prove that :
cba
yX
−+
+
44
2
=
ba
zyX
63
22
+
++
[8] If
yX
a
−2
=
Xy
b
−2
, prove that :
ba
ba
2
2
+
+
=
y
X
[9] If
yX
a
+2
=
Xy
b
−3
=
yX
c
54 +
, prove that :
cb
ba
+
+
4
2
=
17
7
[10] If
2
a
=
7
b
=
3
c
, find the value of :
cb
ba
−
+ 2
[11] If
7
yX +
=
5
zy +
=
8
Xz +
, prove that :
zX
zyX
−
++
= 5
[12] If
y
X
=
4
3
,
z
X
=
5
2
and 3 X + 2y + z = 49 , find the value of each of : X , y and z

١٣
Sheet (8)
[1] Find the middle proportion between :
1) 3 , 27
2) 2 a , 8 ab2
[2] If b is the middle proportion between a and c , prove that :
1)
cb
ba
−
−
=
bc
ba
+
+
3
3
2) (
ba
cb
−
−
)2
=
a
c
3)
cb
ba
+
+
=
c
a
[3] If a , b , c and d are in continued proportion , prove that :
1)
cb
ba
2
2
−
−
=
dc
cb
43
43
+
+
2)
db
ca
53
53
+
+
=
db
ca
4
4
−
−
3)
cb
cdab
−
−
=
b
ca +
4)
dcb
cba
++
++
=
bd
ac
2 2
2 2
2 2
2 2 2
2 2 2

١٤
Sheet (9)
1) If X α y then : X = ……………….
2) If z =
X
m
where m is a constant , then : z α …………….
3) If y α X , then :
X
X
=
.......
.......
4) If X varies inversely as y , then
y
y
=
.......
.......
5) If y =
5
3
X , then : y α ……………
6) If y α
X
5
, then : y varies inversely as ……………
7) If X – 2 y = 0 , then : X α ……………..
8) If 2 X y = 5 , then : X α ………………..
9) If y α X and y = 2 as X = 8 , then : y = ……… when X = 12
10) If y α
X
1
and y = 3 as X = 20 , then : y = ……….. when X = 12
11) If y α X and y = 2 as X = 4 , then : y = …………X
12) If α X and y = 6 as X = 4 , then :
X
y
= …….. ( in simples form )
[2] If y varies directly as X and y = 20 as X = 7
Find : X when y = 40
[3] If a varies inversely as b and a = 12 as b = 8 , find :
a) The value of a as b = 1.5
b) The value of b as a = 2
[4] If y α X and y = 14 when X = 42 , find :
a) The relation between X and Y
b) The value of y when X = 60
2
2
1
2
1

١٥
[5] If X α
X
1
and y = 3 when X = 2 , find :
a) The relation between X and y B) The value of y when X = 1.5
[6] If y α X3
and y = 46 as X = 2 , find the relation between X and find the value of y
as X =
2
1
[7] If y2
α X3
, find the relation between X and y where y = 3 as X = 2
[8] If y2
α 3
1
X
and X = 8 as y = 3 , find X as y = 1.5
[9] If y α ( X+ 1) , find the relation between X and y if X = 3 when y = 2
[10] If
zX
yX
−
−
7
21
=
z
y
, prove that : y α z
[11] If : 4 a2 + 9 b2
= 12 a b , prove that : a varies as b
[12] Connecting with physics :
A car moves with a uniform velocity where the distance varies directly with the time
(t) . If the car covered a distance of 150 km. in 6 hours , find the distance covered by
that car in 10 hours ?
[13] Connecting with astronomy ;
If the weight of a body on the moon (W) is directly proportional with its weight on the
ground ( R ) . If the body weight 84 kg. , on the ground and its weight on the moon is
14 kg. . What will its weight be on the moon if its weight on the ground is 144 kg. ?

١٦
Sheet (10)
Important Rules :
1) The standard deviation of set of values .
σ =
2) The standard deviation of frequency distribution .
σ =
3) The standard deviation of frequency distribution of sets .
σ =
A) Complete the following :
1- The resources of collecting data are …………… and …………………. .
2- The personal interview is a ………………. resource of collecting data .
3- Central agency for public mobilization and statistics is a ……………… resource of
collecting data .
4- The suitable method for checking the production of a factory is …………………
5- ………………. Is secondary resource of collecting data .
6- Choosing a sample from the society’s layers in statistics is called a ………… sample .
7- Dispersion measurements are …………….. and ………………..
8- The simplest measure of the dispersion is ……………………….
9- The difference between the greatest value and the smallest value in a set of values is
called ………………………
10- The positive square root of the average of squares of deviation of the values from their
mean is called …………………..
11- If the standard deviation equals zero , then …………………….. .
12- The dispersion to any set equally values equals ………………… .
13- The mean of the set of the values : 7 , 5 , 9 , 11 and 3 is ………………….
14- The range of the set of the values : 6 , 5 , 9 , 4 and 12 is ………………….
15- The most repeated value in a set of values represents …………………….
Σ (x –x )2
n
Σ (x –x )2
×k
Σ K
Σ (x –x )2
×k
Σ K

١٧
16- If the mean of numbers : 3k – 3 , 3k – 1 , 2k + 1 , 2k + 3 and 2k + 5 is 13 , then
k = ……….
17- If Σ ( x – x )2
= 36 of a set of values and the number of these values = 9 , then the
standard deviation = ……………………..
B) Calculate the standard deviation of the values : 8 , 9 , 7 , 6 and 5 .
C) The following tables shows the distribution of ages of 20 persons in years :
The age 15 20 22 23 25 30 Total
Number
of persons
2 3 5 5 1 4 20
Find the standard deviation of the ages .
D)The following is the frequency distribution of weekly incentives of 100 workers in a
factory :
Incentives in pounds 35- 45- 55- 65- 75- 85-
Number of workers 10 14 20 28 20 8
Find the standard deviation of this distribution .

١٩
Geom.
Sheet (1)
[1] In the opposite figure :
If ABC is a right-angled triangle at B ,
then : sin A = ………..
∆ ABC is a right-angled triangle at B ,
AB = 3 cm , AC = 5 cm ,
Then : sin C × cos C = ………
[3] If the ratio between the measures of two supplementary angles is 3 : 5 , find the
measure of each one by degree measure .
ABC is a right-angled triangle at B in which :
AB = 8 cm , AC = 17 cm.
Find each of :
Sin C , tan A , cos A , cos C , tan C , sin A
[5] XYZ is a right-angled triangle at Z where XZ = 7 cm. and XY = 25 cm.
Find the value of each of the following :
1) tan X × tan Y 2) sin2
X + sin2
Y
[6] XYZ is a right-angled triangle at Y , if YZ = 2 XY
Find the value of each of : tan Z , tan X , cos Z , cos X
[7] ABC is a right-angled triangle at B , if 2 AB = 3 AC
Find : the main trigonometrical of the angle C .
A
C B
12 cm.
5cm.
A
C B
5cm.
3cm.
A
C B
17cm
8cm

٢٠
ABC is a right-angled triangle at B ,
AB = 6 cm , tan C =
4
3
, find :
1) The length of each of BC and AC
2) Sin A + cos A
A
C B
6 cm

٢١
Sheet (2)
1) sin 45° = …………….
2) cos 60° + sin 30° = …………
3) sin 30° + cos 60° - tan 45° = …………..
4) sin 60° + cos 30° + tan 60° = ………….
5) sin2
45° + cos2
45° = …………
6) tan2
60° + cos 60° - tan 45° = ………….
7) tan 45° × sin 30° = ………….
8) 4 cos 30° tan 60° = …………
[2] Without using the calculator , prove each of the following :
1) sin 60° = 2 sin 30° cos 30°
2) cos 60° = 2 cos2
30° - 1
3) 2 cos2
45° - 1 = 1 – 2 sin2
45°
4) cos 60° = cos2
30° - sin2
30°
5) tan 60° =
1) If cos C =
2
1
where C is an acute angle , then : m (∠ C) = …………..
a) 30° b) 60° c) 45° d) 90°
2) If sin X =
2
1
where X is an acute angle , then : m (∠ X) = ………..
a) 30° b) 60° c) 45° d) 90°
3) If tan X =
3
1
where X is an acute angle , then : tan 2 X = …………
a)
3
2
b) 2 3 c) 3 d) 3
4) If X is the measure of an acute angle and sin X =
2
1
, then : sin 2 X = ………..
a) 1 b)
4
1
c)
2
3
d)
2
1
2 tan 30°
1 – tan2
30°

٢٢
5) If 2 sin X = tan 60° where X is an acute angle , then : m (∠ X) = ………..
a) 30° b) 45° c) 60° d) 40°
6) If tan 2 X =
3
3
where 2 X is an acute angle , then : m (∠ X) = ………..
a) 15° b) 30° c) 60° d) 45°
7) If sin 2 X =
2
3
, then : X = ………… (where 2 X is an acute angle ) .
a) 20° b) 30° c) 45° d) 60°
8) If cos
2
X
=
2
1
where
2
X
is an acute angle , then : m (∠ X) = ………..
a) 30° b) 45° c) 60° d) 120°
9) If cos ( X + 10° ) =
2
1
where ( X + 10° ) is an acute angle , then X = ……….
a) 30° b) 40° c) 50° d) 70°
10) If tan ( X - 5° ) =
3
1
where ( X - 5° ) is an acute angle , then : X = ……….
a) 35° b)65° c) 60° d) 30°
11) If sin ( X + 5° ) =
2
1
where (X + 5°) is the measure of an acute angle ,
then : tan ( X + 20° ) = ………….
a)
2
2
b)
2
1
c)
2
3
d) 1
12) tan 75° = …………
a) b) c) 3 tan 25° d) 3 sin 25° cos 25°
[4] Find the value of X in each of the following :
1) tan X = 4 sin 30° cos 60° where X is an acute angle .
2) sin X = sin 60° cos 30° - cos 60° sin 30° where X is an acute angle .
3) 2 sin X = sin 30° cos 60° + cos 30° sin 60° where X is an acute angle .
cos 75°
Sin 75°
sin75°
cos 75°

٢٣
[5] ABCD is trapezium in which : AD // BC and m (∠ ABC) = 90°
If AB = 12 cm , AD = 16 cm , and BC = 25 cm .
Find : 1) The length of DC 2) m (∠ C)
3) sin (∠ DCB) – tan (∠ ACB)

٢٤
Sheet (3)
1) The distance between the two points (15 , 0) , (6 , 0) equals ……………
2) The distance between the two points A (6 , 0) , B (0 , 8) = …………
3) The distance between the point (-3 , 4) and the point of origin = …………
4) If A ( 2 , -3 ) , B (-1 , 1) , then AB = ………..
5) If the distance between the two points (a , 0 ) , (0 , 1) is unit length , then a = …………
6) The radius length of the circle whose centre is (7 , 4) and passes through (3 , 1)
equals …………
7) In the square ABCD if A (3 , 5) and B (4 , 2) , then the area of the square equals
………..area unit .
8) In the rhombus ABCD where A ( -1 , 7) , B (-3 , 1) , then the perimeter of the rhombus
equals ………..length unit .
[2] Prove that :
1) The points A ( 4 , 3) , B (1 , 1) and C (-5 , -3) are collinear .
2) Prove that the triangle with vertices of points : A (5 , -5) , B (-1 , 7) and C (145 , 15)
is a right-angled triangle at B , then calculate its area .
3) The points A (0 , 1) , B (4 , 5) , C (1 , 8) and D (-3 , 4) are vertices of a rectangle and
find its diagonal length .
4) ABCD is a quadrilateral where A ( 5 , 3) , B (6,-2) , C (1 ,-1) and D (0 , 4)
Prove that : ABCD is a rhombus , then find its area .

٢٥
5) The points A ( -2 , 5) , B (3 , 3) and C (-4 , 2) are non-collinear and if D (-9 , 4) ,
Prove that : The figure ABCD is a parallelogram .
6) ABCD is a quadrilateral where A ( 2 , 4) , B (-3 , 0) , C (-7 , 5) and D ( -2 , 9)
Prove that : The figure ABCD is a square .
7) The points A (3 , -1) , B (-4 , 6) and C (2 , -2) lie on the same circle whose centre is
M (-1 , 2) , then find the circumference of the circle where π = 3.14
[3] If the distance between the two points A (0 , K) and B (4 , 0) is 5 length units .
Find : The value of K .
[4] Find the value of a in each of the following cases :
1) If the distance between the two points (a , 7) , (-2 , 3) equals 5 length units .
2) If the distance between the two points (a , 7) , (3 a – 1 , - 5 ) equals 13 length units .

٢٦
Sheet (4)
[1] Find the coordinates of the midpoint of AB in each of the following cases :
1) A (3 , 5) , B (7 , 1)
2) A (5 , -3 ) , B (-1 , 3)
3) A (-5 , 4) , B (5 , -4)
4) A (0 , 4 ) , B (8 , 0)
[2] If the point (X , 0) is the midpoint of the line segment whose ends are (1 , -5) and
(2 , 5 ) , find the value of X
[3] If the point (5 , 3) is the midpoint of AB where its terminals are A ( 15 , y ) and
B (-5 , -2) , find the value of y .
[4] If the point (5 , 3) is the midpoint of AB where its terminals are A (15 , y) and
B (-5 , -2) , find the value of y
[5] Find the value of each of X and y if the point (3 , -2) is the midpoint of the line segment
drawn between the two points (X , 2) , (3 , y)
[6] Prove that the points A (3 , -2) , B (-5 , 0) , C (0 , -7) and D (8 , -9 ) are the vertices of
a parallelogram .

٢٧
[7] If the points A (3 , 2) , B (4 ,-3) , C (-1 , -2) and D (-2 , 3) are vertices of the rhombus .
Find :
1) The coordinates of the point of intersection of the two diagonals .
2) The area of the rhombus ABCD .
[8] ABCD is a square whose vertices are A ( 0 , 5) , B ( 3 , 2) , C (0 , -1) and D (X , y )
respectively .
Find the coordinates of the point D .
[9] Prove that : The points A (6 , 0) , B (2 , -4) , C (-4 , 2) are the vertices of a right-angled
triangle at B , then find the coordinates of D that make the figure ABCD a rectangle .

٢٨
Sheet (5)
1) In the opposite figure :
The slope of the straight line L equals …………..
2) The condition of parallelism of two straight lines whose slopes are m1 , and m2
is …………. While the condition of their perpendicularity is …………
3) The slope of the straight line parallel to X-axis = ………..
4) The slop of the straight line parallel to y-axis = ………..
5) The slope of the straight line which makes with the positive direction of X-axis
a positive angle of measure 45° equals ………….
6) If AB // CD and the slope of AB =
3
2
, then : the slope of CD equals …………..
7) If AB ⊥ CD and the slope of AB =
2
1
, then the slope of CD equals ……….
8) The slope of the straight line which is parallel to the straight line passing through the
two points (2 , 3) and (-2 , 3) equals ……….
9) If ABCD is a square whose diagonals AC and BD where A (3 , 5) and C (5 , -1) , then
the slope of BD = ………….
10) If the straight line AB is parallel to the X-axis where A (8 , 3) and B (2 , K) ,
then K = …………
11) If the straight line CD is parallel to the y-axis where C ( M , 4) and D (-5 , 7) ,
then M = ……….
[2] Prove that : The straight line which passes through the two points (4 , 2) and (5 , 6) is
parallel to the straight line which passes through the two points (0 , 5) and (-1 , 1) .
[3] Prove that : The straight line passing through the two points A (-3 , 4) and C (-3 , -2) is
perpendicular to the straight line passing through the two points B ( 1 , 2) and D (-3 , 2)
y
L
X´
y´
Xθ

٢٩
[4] Find the slope of the straight line which is perpendicular to the straight line which
passes through the two points A ( 2 , -3 ) , B (3 , 5) .
[5] Prove that : The straight line passing through the two points (2 . -1 ) and (6 , 3) is
parallel to the straight line that makes an angle of measure 45° with the positive
direction of the X-axis .
[6] The triangle whose vertices are A ( 3 , -1) , B (X , 3) and C (5 , 3) is a right-angled
triangle at A , find the value of X .
[7] If the straight line AB // the y-axis , where A ( X , 7) and B (3 , 5) , then find the value
of X .
[8] If the straight line CD // the X-axis where C (4 , 2) and D (-5 , y) , find the value of y
[9] If A (-1 , -1 ) , B (2 , 3) and C (6 , 0) , prove that triangle ABC is a right-angled triangle
at B .
[10] Prove that : The point A (-1 , 1) , B (0 , 5) , C (4 , 2 ) and D (5 , 6) are the vertices of
the parallelogram ABDC .
[11] Prove that : The point A (5 , 1) , B (1 , 5) , C (-1 , 3) and D (3 , -1) are vertices of the
rectangle ABCD .

٣٠
[12] Prove that : The point A ( 1 , 3) , B (6 , 4) , C (7 , 9) and D (2 , 8) are vertices of the
rhombus ABCD .
[13] Prove that : The points A (-1 , -1) , B (2 , 3) , C (6 , 0) and D (3 , -4 ) are vertices of
a square .

٣١
Sheet (6)
[1] Find the slope and the intercepted part of y-axis by each of the following straight
lines :
1) y = 5 X – 3
2) 2 y + 3 X = 8
[2] Find the equation of the straight line if :
1) Its slope = 2 and intercepts from the positive part of y-axis 7 units .
2) Its slope = 1- and intercepts from the positive part of y-axis 3 units .
[3] Find the equation of the straight line if :
1) Which passes through the point and makes with the positive direction of X-axis a
positive angle of measure 135° .
2) Which cuts a part of length 3 units from the negative part of y-axis and is parallel to the
line whose equation : 2 X – 3 y = 6 .
3) Which is perpendicular to the straight line : 3X – 4 y + 7 = 0 and intercepts from the
positive part of y-axis a part of length 6 units .
4) Which passes through the point (2 , -1) and its slope equals 2 .
5) Passing through the point (-2 , 3) and perpendicular to the straight line whose equation :
y =
2
1
X – 5
6) Passing through the point (3 , -5) and it is parallel to the straight line : X + 2y – 7 = 0
7) Which passes through the point (3 , 2) and parallel to the straight line passing through
the two points ( 5 ,6) and (-1 , 2) .
8) Which passes through the two points ( 2 , -1 ) and (1 , 1 )
9) The perpendicular to AB from its midpoint where A (1 , 3) and B (3 , 5) .

٣٢
A particle moves with a constant speed (v)
where the distance (d) is measured by meter
and time (t) by second .
find the following :
1) The distance at the beginning of moving .
2) The velocity of the particle.
3) The equation of the straight line which represent the movement of the particle .
4) The time in which the particle covers a distance of 5 meters from the beginning of
the movement .
[5] The opposite graph :
Represents the motion of a particle moving with uniform
velocity (v) where the distance (d) is measured in meter
and the time (t) in seconds .
Find :
1) The distance at the beginning of the motion .
2) The velocity of the particle .
3) The equation of the straight line representing the motion of the particle .
4) The covered distance after 4 seconds from the beginning of the motion .
5) The time in which the particle covers a distance of 3.5 meters from the beginning of
the motion .
3
2
1
0
1 2 3 4 5 6
D (m.)
T (sec.)
5
4
3
2
1
0
1 2 3 4 5
D (meter)
T (second

3rd.prep first term .math

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to 3rd.prep first term .math (20)

More from أمنية وجدى (20)

Recently uploaded (20)

3rd.prep first term .math