MATHEMATICS AND Osama Rashad

MECHANICS
URT EXAM
1. Two equal forces intersects at a point the magnitude of their resultant equals 8 N, if one
them is reversed, then the magnitude of their resultant equals 6 N, then the magnitude of
each forces equal

a) 16
b) 5
c) 9
d) 3

2. If the forces 𝐹⃗ 1 = 5𝑖̂ – 4𝑗̂, 𝐹⃗ 2 = -6𝑖̂ + a𝑗̂ and 𝐹⃗ 3 = b𝑖̂ + 7𝑗̂ are meeting at a point and are in
equilibrium. find the value of a+b

a) -3
b) -2
c) 1
d) 4

3. A weight of 200 gm.wt is suspended by two strings of lengths 90 cm and 120 cm fixed in
two horizontal points , the distance between them 150 cm find the value of the tension in
each of the two strings

a) 120,150
b) 100,120
c) 120,160
d) 100,110

4. A metallic sphere of weight 15 kg.wt is put such that it touches two smooth planes , one
of them is vertical and the other inclines to the vertical by an angle of measure 30° find
the reaction of the vertical plane

a) 15
b) 5√3
c) 30
d) 30√3

5. Two trains moving in opposite direction pass a man at rest on a platform in 27 sec and 17
sec respectively and pass each other completely in 23 sec then the ratio between their
speed = ……..
 a) 1:3
b) 1:2
c) 3:2
d) 3:4

6. A body moves from rest with a uniform acceleration for 20 sec , if it covers distance S1
in the first 10 seconds and s2 in the next 10 seconds then

a) S1= S2
b) S2= 2S1
c) S2= 3S1
d) S2= 4S1

7. If a body loses half its speed for penetrating 3 cm inside a wooden barrier of thickness 10
cm, then the distance of the body can move after that to come to rest = ……. cm

a) 1
b) 12
c) 3
d) 4

8. A particle is projected vertically upward, then the acceleration of the particle at its
maximum height equal = ……..

a) zero
b) 9.8m /sec2 downward
c) 9.8m /sec2 upward
d) depends on the initial speed

9. A boy stands on abridge and projects a stone downward, assume the positive direction is
upward then which of the following graphs represents the acceleration – time graph of the
stone mention?
 a) b)

c) d)

10. If A and B are two events of the sample space of a random experiment
4 1
, P(A\ B\ ) =
U
P(A - B) = 0.3, P(B - A) = then P(A ⋂ 𝐵) =
15 5

1
a)
2
1
b)
5
c) 0.6
7
d)
30

11. If the sample space of random experiment is S ={ A,B,C,D } and P({B}) = 0.33 ,
P({B,C}) = 0.45 , P({ C,D }) = 0.44 then P({A}) =

a) 0.77
b) 0.23
c) 0.11
d) 0.01
12. a fair die is thrown three consecutive time , then the probability of getting three similar
numbers
1
a)
36
1
b)
6
5
c)
6
1
d)
216

13. The number of solutions for the equation x =∟x in z equals

a) zero
b) 1
c) 2
d) infinite solutions

14. If
n x
P + Pn = 1440 then
n+4
C x – 5 = …….
x
a) 6
b) 9
c) 10
d)12
n m
15. If m= C2 then C2 = ……..
n
a) 3× C3
b) 4× n+1C3
n+1
c) 4× C4
n
d) 3× C4

16. if we want to form a committee from 4 persons chosen from 9 men and 3 women such
that the committee includes one woman at least the number of ways is …….

a) 495
b) 11880
c) 369
d) 252
 𝑓(ℎ+3)−𝑓(ℎ−2)+𝑓(−2)−𝑓(3)
17. lim
ℎ→0 ℎ

a) f \ (3) + f \ (-2)
b) f \ (3)
c) f \ (-2)
d) f \ (3) - f \ (-2)
1 𝑑𝑦
18. If 5 f ( x ) + 3f ( ) = x + 2 and y = x f ( x ), then = ………. At X=1
𝑥 𝑑𝑥
a) 1
7
b)
8
c) 10
d) 14

19. In the opposite figure

The straight line L is a tangent to the curve of the function F at the point C and cuts X-
axis at the point A(-4,0) and B (4,0) , F(4) + F\ (4) = 9 , then the area of triangle abc =
……. Square unit

a) 30
b) 32
c) 36
d) 42

1−2𝑥
20. ∫|𝑥|√ dx = ……. +c
𝑥2

2
a) (1-2x)3/2
3
1
b) - (1-2x)3/2
3
2 3
c)
3
√(1 − 2𝑥)2
1
d) − 3√(1 − 2𝑥)2
3

21. If x , y are two positive real numbers where x + y = k then x y is maximum when :

a) X = ky b) y = kx c) y = x d) xy = 1
 𝜋 𝜋
22. If f(x) = tan X , then f ( x + ) × f(x- )
4 4

a) -1
b) 1
c) 2 tanx
d) cot x

23. Abc is an acute – angled triangle tan A = 0.75 , tan b = 2.4

then a:b:c =

a) 31:21:20
b) 13:20:21
c) 20:13:21
d) 12:13:20

24. The solution set of the equation cos 3x – cos 2x + cosx = 0 where x is an acute angle

a) {30,60}
b) {30,45}
c) {45,60}
d) {15,75}

25. In triangle XYZ if x = y then cos x

2𝑦
a)
𝑧
𝑧
b)
2𝑦
𝑧
c)
4𝑥
𝑦
d)
2𝑥

26. The strongest correlation coefficient of the following is ……

a) -0.8
b) -0.5
c) 0.4
d) 0.7
27. For the complex z1 and z2 arg(z1z2) is ………

a) arg(z1) - arg(z2)
b) arg(z1) + arg(z2)
c) arg(z1) × arg(z2)
d) arg(z1) ÷ arg(z2)

28. The roots of the functions f(x) = 6x2 + x - 2

1 −2
a) { , }
2 3
−1 2
b) { , }
2 3
1
c) { , 1}
3
−1
d) { , 1}
3

29. The total area of the right cone equals

a) π𝑟𝑙 L
π h
b) 𝑟𝑙
3
c) π𝑟(𝑟 + 𝑙)
π r
d) 𝑟(𝑟ℎ + 𝑙)
3

30. Which of the following nets doesn’t regular quadrilateral pyramid when if folded
31. The curve of g(x) = |𝑥 + 3| is the same curve of the f(x)= |𝑥| by translation of magnitude
3 units in the direction of

a) 𝑜𝑥
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗\
b) 𝑜𝑥
c) 𝑜𝑦
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗\
d) 𝑜𝑦

1
32. A point of symmetry of the function f where 𝑓 (𝑥 ) = + 4 is
𝑥−3

a) (3,-4)
b) (3,5)
c) (-3,-4)
d) (-3,4)

33. 2x+1 =8 then x= …….

a) 1
b) 2
c) 3
d) 4

34. log 𝑎 16 = 4 then a € …….

a) {16}
b) {2,-2}
c) {2}
d) {1}

𝑎𝑥+2 +𝑎𝑥+1 +𝑎𝑥

35. = ……..
𝑎𝑥+1 +𝑎𝑥 +𝑎𝑥−1

a) a
b) a-1
c) ax
d) 1
36. The ninth term in the sequence ( 2,4,8,……) is …..

a) 256
b) 512
c) 1024
d) 128

37. Which of the following is a factor of the function x2+3x-4

a) x+1
b) x+4
c) x-4
d) 2x+1

1 𝑤 2 𝑖 1
38. (| |) + | |=
1 𝑤2 2 𝑖

a) -6
b) 0
c) -3
d) -5

39. Mr Osama made the opposite stem-and-leaf plot to show the test scores in mathematics
class, what is the range of the test score?

Stem leaf

a) 49 2 0 1 2 2
b) 85 3 0 0 1 4 4 5 8
c) 29 4 0 2 3 4 6 6 9
d) 38 5 1 1 1 5 8

40. For what values of k does the equation kx2-4x+x=0 has real roots?

a) -4 ≤ k ≤4
b) k ≥ -2
c) -2 ≤ k ≤2
d) k ≤ 2
41. The solution set of the equation x + 5y = 16 and x + 17 y = -20

a) {( -31,3 )}
b) {(31,-3)}
c) {(3,31)}
d) {(-3,31)}

42. The scatter diagram representing inverse correlation is:

43. Find the greatest possible value of the function P = 3x + 6y under the following
restrictions: x ≥ 0 , y ≥ 0 , x + y ≤ 5 , 2x + y ≤ 6 , x + 2y ≤ 8
a) 24
b) 28
c) 6
d) 9

−8 2
−3 2 5
44. ( ) ×( 1 5) =
7 0 1 0 −3
−1 25 10 −13 52 5
a) ( ) b) ( )
29 1 18 27 0 −1
−3 2 5 3 − 22 15
c) ( ) d) ( )
7 0 1 14 −5 −1

45. The of the triangle which its vertices (-1 , -3 ) , ( 2 , 4 ) , ( -3 , 2 ) is :

a) 18 square units
b) 17 square units
c) 19 square units
d) 20 square units
46. -4K + 5 ≤ 21 then :

47. Y = | x + 4 |
48.

49.
50.