4092704 res mathematics

MATHEMATICS

2. Identity – an equation which is true for all permissible values of the literals.

Conditional equation – satisfied by some but not all values of the variable.

Defective – has fewer roots than the original.

Singular – general term for basic equations usually containing one variable.

4. Axiom – result if equals are added to equals; statement of truth admitted without proof.

Theorem – statement of truth of which must be established by proof.

Postulate – (in geometry) construction / drawing of lines and figures admitted.

Corollary – statement of truth which follows little or no proof from the theorem.

8. Lemma – proved proposition used mainly as a preliminary to the proof of theorem.

9. twice as long – timetwice as fast - rate

12. The speed of the plane is 120 mi/hr in calm. With the wind it can cover a certain distance in 4 hrs, but against the wind it can cover only 3/4 of that distance in the same time. Find the velocity of the wind. a. 10 mi/hr c. 20 mi/hrb. 30 mi/hr* d. 40 mi/hr

Solution:

d = rtdopp = dwith (3/5)(120 – W)(4) = (120 + W)(4)(3/5)W = 30 mi/hr

25. Find the term containing x^2 in the expression of [x^3+(a/x)]^10.

Solution:

nCr ( ) ( )= 10C7 (x3)3 (ax-1)7

= 10C7 x9 a7 x-7

= 120 a 7 x 2 28. Find the values of x in degrees.

sin x cos x + sin 2x = 1

Solution:

2 [sin x cos x + sin 2x = 1] 22 sin x cos x + 2 sin 2x = 2sin 2x + 2 sin 2x = 23 sin 2x = 2x = 20.9°

29. The number between 1 and 1000 (inclusive) is randomly selected. What is the P that it will be divisible by 4 and 5?

Solution:

Ps = S / total4 x 5 = 20n = 1000 / 20n = 50 successPs = 50 / 1000Ps = 0.005

35. Find the value of A for which the equation A(2x+3)-(x-4) = 3x+10 is identity.

Solution:

2Ax+3A-x+A = 3x+10(2A-1)x+3A+4 = 3x+10

2A-1 = 3A = 2

3A+4 = 10A = 2

38. In how many ways can 9 different books be arranged on a shelf so that 3 of the books are never all 3 together?

Solution:

Ntogether = 7(3!)(6!)

Nnot together = total - Ntogether = 9! - 7(3!)(6!)

B = 332,640

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Vopp = Vc – WVwith = Vc + W

3 2 1 6 5 4 3 2 1

RES-M1 (Algebra)

MATHEMATICS

39. Compute the sum of 4-digit numbers which can be formed with the 4 digits 2, 3, 5, 8 if each digit is used only once in each arrangement.

Solution:

4P4 = 242+3+5+8 = 1824 / 4 = 6

1(6x18)+10(6x18)+100(6x18)+1000(6x18) = 119,988

40. There are 3 copies each of 4 different books. In how many different ways can they be arranged on shelf?

Solution:

12! / (3!3!3!3!) = 369600

42. 1 bag contains 4 white balls and 2 black. Another bag contains 3 white and 5 black. If one ball is drawn from each bag, determine the probability that the balls drawn are of different colors?

Solution:

Black = 2/6White = 3/8

Black = 5/8White = 4/6

mutually exclusive (add)

Answer:13/24

43. A probability that A can solve a given problem is 4/5, that B can solve it is 2/3 and that C can solve it is 3/7. If all 3 try to solve the problem, compute the probability that the problem will be solved.

Solution:

Ps = 1 – Pnot solved

Pnot solved = (A)(B)(C)Pnot solved = (1/5)(1/3)(4/7)Ps = 0.96

44. Three friends A, B, and C are in a race. The odds that A will win are 7 to 5, and

the odds that B will win are 1 to 3. What are the odds in favor of C?

Solution:

7:5failure = 7success = 5total = 7+5 = 12

A: Pf = 7 / 12 = 7 / (7+5) = 7 / 5B: Pf = 1 / 4 = 1 / (1+3) = 1 / 3C: Pf = 1 – Pfail(A) – Pfail(B)

= 1 / (1+5) = 1 / 5

45. Find the probability of getting 9 exactly once in 3 throws with a pair od dice?

Solution:

failure = 32 / 36success = 4 / 36

nCr (F) (S)= 3C1 (32 / 36)2 (4 / 36)1

= 0.263

46. Tree broken over by the wind forms a right triangle with the ground. If the broken part makes an angle of 50° with the ground and the top of the tree is now 20 ft from its base, how tall was the tree?

Solution:

tan 50 = x / 20x = 23.83

cos 50 = 20 / hh = 31.17

47. Find the perimeter of an isosceles triangle whose base is 40 and whose vertical angle is 40°.

Solution:

of 440°

40 cm

50°

20 ft

hxh

MATHEMATICS

402 = x2+x2-2(x)(x)cos 40x = 58.5P = 2x+40P = 157

48. A wheel, 5 ft in diameter, rolls up an incline of 18° 20’, what is the height of the center of the wheel above the base of the incline when the wheel has rolled up 5 ft up the incline?

Solution:

sin 18° 20’ = x / sx = 1.57 + radius x = 4

1. If the straight lines ax+by+c=0 and bx+cy+a=0 are parallel, then which of the following is correct?

2. The capacities of 2 hemispherical tanks in the ratio 64:125. If 4.8 kg of paint is required to paint the outer surface of the smaller tank, then how many kgs of paint would be needed to paint the outer surface of the layer tank?

3. Find the equation of perpendicular bisector of the segment joining the points (2,6) and (-4,3).

3. An advertisement claim that a certain 1200 kg car can accelerate from rest to a speed of 25 m/s in a time of 8 s. What is

the average power? Ignore friction losses. 62.8 hP

4. A 0.25 hP motor is used to lift a load at the rate of 5 cm/s. How great a load can it lift at constant speed? 381 kg

5. An 8 g bullet is fired horizontally into a 9 kg block of wood and sticks in it. The block which is free to move, has a velocity of 40 cm/s after impact. Find the velocity of the bullet. 450 m/s

6. A 2 kg is moving at a speed of 6 m/s. How large a force F is needed to stop the brick in a time of 7 x 10 ^ -4 s. -1.71 x 10 ^ 4 N

7. A car has wheels of radius r=30 com. it starts from rest and accelerates uniformly to a speed of 15 m/s in a time of 8 sec. Find the number of rotations one wheel makes in this time. 32

8. What is the maximum speed at which a car can round a curve of 25 m radius on a level road if the coefficient of static friction between the tires and road is 0.80? 14 m/s

9. A uniform solid sphere rolls on a horizontal surface at 20 m/s and then rolls up the incline. If the friction losses are negligible, what will be the value of h where the ball stops? 28.54 m

10. An object traveling in a circular path makes 1200 rev in 1 hr. If the radius of the path is 10 m, calculate the speed of the object. 21 m/s

11. A 1200 N box is pulled along a horizontal surface at a uniform speed by means of a rope that makes an angle of 30º above the horizontal. If the tension in the rope is 100 N. What is the coefficient of friction? 0.0721

12. When a mass is hung on a spring, the spring stretches 6 cm. Determine its period of vibration if it is then pulled down a little and released. 0.49

13. Atmospheric pressure is about 1.01 x 10 ^ 5 Pa. How large a force does the atmosphere exert on a 2 square cm area on top of your head? 20 N

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18° 20’

5 ft

5 ft

RES-M2 (Trigonometry and Geometry)

RES-M4 (Physics)

MATHEMATICS14. The gauge pressure in a car tire is 305

kPa when its temperature is 15ºC. After running at high speed, the tire has heated up and its pressure is 360 kPa. What is then the temperature of gas in the tire? Assume atmospheric pressure to be 101 kPa? 54ºC

15. A running fork makes 284 vib/sec in air. Compute the wavelength of the tone emitted in 25ºC. 1.22 m

16. Elastic Potential – energy stored in a stretched or compressed elastic material such as a spring.

17. A ball is dropped from a height h above a tile floor and rebounds to a height of 0.65 h. Find the coefficient of restitution between the ball and floor. 0.81

18. A 30 m long aluminum bar is subjected to a tensile stress of 172 MPa. Find the elongation. 0.0746 m

19. - 40 ºC = - 40 ºF

20. Inelastic collision is a collision in which the total kinetic energy after collision is less that before collision.

21. Specific Gravity – ratio of the density of a substance and density of water.

22. Newton’s First Law – Inertia

23. Newton’s Second Law – Acceleration

24. Newton’s Third Law – Action and Reaction

25. A train is traveling in a speed of 60 mi/hr is brought to an emergency stop in 2000 ft. What is the time required for the train to stop? 46

26. By use of a pulley a man raises a load 120 lb to a height of 40 ft in 65 sec. Find Power in hP. 0.134 hP

27. What average force is necessary to stop a bullet of mass 20 gm and speed of 250 m/s as it penetrates wood to a distance of 12 m? 52.08 N

28. How much heat energy is needed to raise the temperature of 20 kg of liquid water from 5ºC to 20ºC? 1257 kJ

29. A projectile is launched a 45º to the horizontal on a level ground at a speed of 60 m/s. Neglecting air resistance, What is the range?

30. A 0.5 kg ball falls past a window that is 1.5 m in vertical length. How much did the kinetic energy of the ball increase as it fall past the window? 7.35 J

31. A car wheel 30 cm in radius is turning a rate of 8 rev/sec when the car begins to slow uniformly to rest in a time of 14 sec. Find the number of revolutions made by the wheel in this time. 56

32. An automobile moving at 30 m/s is approaching a factory whistle that has a frequency of 500 Hz. If the speed of sound in air is 340 m/s, what is the apparent frequency of the whistle as heard by the driver? 544 Hz

33. A tank continuing water has an orifice

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4092704 res mathematics

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