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1. Evaluate the integral of xlnx dx. (1/4)x(2lnx + 1) + C (1/4)x²(2lnx – 1) + C (1/4)x²(2lnx + 1) + C (1/4)x(2lnx – 1) + C Solution: 2. In a discrete probability distribution, the sum of all probabilities is always _____. Infinite 1 Undefined 0 Solution: 3. The number of arbitrary constants in the particular solution of a differential equation of third order is: 1 0 3 2 Solution: 4. A given function f(t) can be represented by a Fourier series if it: Is periodic Is single valued All of the choices Has a finite number of maxima and minima in any one period 5. Find the area of the surface of revolution generated by revolving about the y-axis the arc of y² + 4x = 2 ln y from y = 1 to y = 3. 32π/ 34π/ 30π/ 31π/ Solution: 6. Find the dimensions of the right circular cone of minimum volume V that can be circumscribed about a sphere of radius 8 inches. h = 32 in, r = 8/sqrt 2 in h = 32 in, r = 8/sqrt 3 in h = 38 in, r = 8/sqrt 3 in h = 38 in, r = 8/sqrt 2 in Solution: 7. Mrs. Johnston’s class broke into teams of three students each to play a game. The winning team received a 1/ pound jar of jellybeans. How many pounds of jellybeans will each team member get if the 1/2 pound jar is shared equally among the three teammates?
1/3 pound 2/5 pound 1/6 pound 2/9 pound Solution:
8. Find all values of z for which e^3z = 1. kπi (1/8)πi + (1/2)kπi kπi/ 2kπi/ Solution: 9. Describe the locus represented by |z + 2i| + |z – 2i| = 6. Ellipse Parabola Circle Hyperbola 10. Three circles with radii 3, 5 and 9 cm are externally tangent. What is the area of the triangle formed by joining their centers? 46 cm² 44 cm² 48 cm² 42 cm² Solution: 11. A confidence interval is defined as: An interval that has a 95% probability of containing the population parameter A point estimate plus or minus a specific level of confidence. A lower and upper confidence limit associated with a specific level of confidence A lower and upper confidence limit that has a 95% probability o containing the population parameter Solution: 12. Harold used a 3% iodine solution and a 20% iodine solution to make a 95 - ounce solution that was 19% iodine. How many ounces of the 3% iodine solution did he use? 80 60 20 5 Solution: 13. Evaluate the integral of dx/(sqrt of 9 - x²) with limits from 0 to 3. π/ 3π/ π/ 2π/ Solution: 14. If the point (3, k) lies on the line with slope m = −2 passing through the point (2, 5), find k. 3 2 4 1 Solution:
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22. A boy is flying a kite at a height of 150 ft. If the kite moves horizontally away from the boy at 20 ft/sec, how fast is the string being paid out when the kite is 250 ft from him? 14 ft/s 16 ft/s 12 ft/s 18 ft/s Solution: 23. Six married couples are standing in a room. If 2 people are chosen at random, find the probability that they are married. 1/ 6/ 3/ 2/ Solution: 24. Find the area enclosed by the curve y² = x² − x⁴. 4/ 5/ 3/ 3/ Solution: 25. A cylindrical container with circular base is to hold 64 in³. Find its dimensions (in inches) so that the amount (surface area) of metal required is a minimum when the container is an open can. h = r = 4/(square root of π) h = r = 2/(cube root of π) h = r = 4/(cube root of π) h = r = 2/(square root of π) Solution: 26. Let cos z = 2. Find cos 2z. 21 14 7 26 Solution: 27. Find the area S of the surface of revolution generated by revolving about the x axis the arc of the parabola y² = 12x from x = 0 to x = 3. 24(sqrt2 – 1)π 24(2sqrt2 – 1)π 24(sqrt2 + 1)π 24(2sqrt2 + 1)π Solution:
28. 8 nails and 12 nuts are present in a box. Among them, half of the nails and half of the nuts are rusted. If an item is chosen at random, what is the probability that it is rusted or is a nail? 4/ 7/ 4/ 3/ Solution: 29. Find the limit of sin(πcos²x)/x² as x → 0. 1 0 π - 1 Solution: 30. Determine the Laplace transform of cos(ωt + θ). (scosθ – ωsinθ)/(s² + ω²) (scosθ + ωsinθ)/(s² - ω²) (scosθ + ωsinθ)/(s² + ω²) (scosθ – ωsinθ)/(s² - ω²) Solution: 31. The earth’s orbit is an ellipse with the sun at one of the foci. If the farthest distance of the sun from the earth is 105.50 million km and the nearest distance of the sun from the earth is 78. million km, find the eccentricity of the ellipse. 0. 0. 0. 0. Solution: 32. If z = 2 – 3i, then find z². - 5 – 12i 5 – 12i 5 + 12i - 5 + 12i Solution: 33. What is the fourth term of the expansion of (x + x²)¹⁰⁰? 161700 x¹⁰³ 167100 x¹⁰³ 1650 x¹⁰³ 167100 x¹⁰³ Solution: 34. A guy wire 78 ft long runs from the top of a telephone pole 56 ft high to the ground and pulls on the pole with a force of 290 lb. What is the horizontal pull on top of the pole? 221 lb 201 lb 241 lb 261 lb Solution: 35. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axis. 4 5
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42. What is the equation of the line joining the points (3,-2) and (-7, 6)? 2x + 3y = 0 4x – 5y = 22 4x + 5y = 2 5x + 4y = 7 Solution: 43. An ellipse with an eccentricity of 0.65 and has one of its foci 2 units from the center. The length of the latus rectum is nearest to _____. 3.2 units 3.8 units 4.2 units 3.5 units Solution: 44. Find the area of the region bounded by the parabolas y = 6x − x² and y = x² − 2x. 65/ 64/ 63/ 62/ Solution: 45. Find the volume of the solid obtained by revolving about the y axis the region in the first quadrant bounded above by the parabola y = 2 − x² and below by the parabola y = x². 3π 2π/ π 2π Solution: 46. Suppose f(z) = (2z + 1)/(3z – 2), z ≠ 2/3. Find f{f(z)}. (2 + z)/(3 – 2z) (2 – z)/(3 – 2z) (2 + z)/(3 + 2z) z Solution:
47. Two trains leave the same city at the same time, one going east and the other going west. If one train is traveling at 65 mph and the other at 72 mph, how many hours will it take for them to be 822 miles apart? 7 9 6 8 Solution: 48. Describe the locus represented by |z – i| = 2. Circle Hyperbola Ellipse Line 49. The temperature in Hillsville was 20° Celsius. What is the equivalent of this temperature in degrees Fahrenheit? 68° 132°
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50. A ladder 20 ft long leans against a house. If the foot of the ladder is moving away from the house at the rate of 2 ft/sec, find how fast the top of the ladder is moving downward when the foot of the ladder is 12 ft from the house. 1/2 ft/s 3/2 ft/s 1/3 ft/s 2/3 ft/s Solution: 51. Suppose z = (2 – i)² + [(7 – 4i)/(2 + i)] – 8, express z in the form of x + iy such that x and y are real numbers. 3 – 7i 3 + 7i - 3 – 7i - 3 + 7i Solution: 52. A given rectangular area is to be fenced off in a field that lies along a straight river. If no fencing is needed along the river, what is the relation between the length and width of the field if the least amount of fencing will be required? The length of the field is three times its width. The length of the field is equal to its width. The length of the field is twice its width. The length of the field is four times its width. Solution: 53. Find L-^1 {(k²)/[s(s² + k²)]. 1 + coskt 1 + sinkt 1 – coskt 1 – sinkt
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60. A solid has a base in the form of an ellipse with major axis 10 and minor axis 8. Find its volume if every section perpendicular to the major axis is an isosceles triangle with altitude 6. 70π 60π 50π 40π Solution: 61. A coin is tossed and a die is rolled. What is the probability that the coin shows the head and the die shows 3? 1/ 1/
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62. A calibration study needs to be conducted to see if 20 scales are giving the same weights. How many ways may be selected to perform a preliminary study? 14, 13, 16, 15, Solution: 63. The integrating factor of the differential equation (xlogx)dy/dx + y = 2logx is: x log(log x) eˣ log x Solution: 64. If A and B are two events such that P (A) = 0.4 , P (A + B) = 0. and P (AB) = 0.2, then P (B) = 0. 0. 0. 0. Solution: 65. In how many ways can a teacher choose one or more students from six eligible students? 68 58 63 53 Solution: 66. The solution of the differential equation (x² + 1)dy/dx + (y² + 1) = 0 is: y = 2 + x² y = (1 – x)/(1 + x) y = (1 + x)/(1 – x)
y = x(x – 1) Solution:
67. It is known that 60% of a class is female, 50% is majoring in chemical or electrical engineering, and 25% is female and majoring in chemical or electrical engineering. What percentage of class is female or majoring in chemical or electrical engineering? 82 % 85 % 78 % 75 % Solution: 68. Given: x + xy + y = 2. Find y’’. 2(1 + y)/(1 + x)² (1 + y)/(1 + x)² (2 + y)/(1 + x)² 2(1 - y)/(1 + x)² Solution: 69. There are 3 bags which are known to contain 2 white and 3 black balls; 4 white and 1 black balls and 3 white and 7 black balls respectively. A ball is drawn at random from one of the bags and found to be a black ball. Then the probability that it was drawn from the bag containing the most black balls is: 6/ 9/ 7/ 8/ Solution: 70. Find the remainder if we divide 4y³ + 18y² + 8y – 4 by (2y + 3). 10 11 12 13 Solution: 71. Evaluate the integral of x²lnx dx. (x³/3)lnx - x³/9 + C (x³/3)lnx + x³/9 + C x³lnx - x³/9 + C (x³/3)lnx - x³/3 + C Solution:
3/2π deg/s Solution:
77. Find the limit of (sinx)/x as x →∞. - 1 1 ∞ 0 Solution: 78. Gas is escaping from a spherical balloon at the rate of 2 ft³/min. How fast (in ft²/s) is the surface area shrinking when the radius is 12 ft? 2/ 1/ 3/ 1/ Solution: 79. What is the differential equation of the family of circles touching the y-axis at the origin? **2xyy+ x**^2 **= y**^2 xyy + x2 = y 2xyy`` + x= y 2xyy – x2 = y Solution: 80. The solution of differential equation xdy – ydx = Q represents: parabola whose vertex is at the origin a circle whose center is at the origin straight line passing through the origin a rectangular hyperbola 81. Differentiate y = (3 – 2x)/(3 + 2x). - 12/(3 + 2x)² 12/(3 + 2x)² 12/(3 - 2x)²
- 12/(3 - 2x)² Solution: 82. A colony of bacteria is growing exponentially. At time t = 0 it has 10 bacteria in it, and at time t = 4 it has 2000. At what time will it have 100,000 bacteria?
6.
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83. If the average value of a periodic function over one period is zero and it consists of only odd harmonics then it must be possessing _________ symmetry. Odd quarter-wave Even quarter-wave Half-wave Odd 84. Given z₁ = 3 – 4i and z₂ = - 4 + 3i. Find |z₁ x z₂|. - 7 - 24 24 7 Solution: 85. Find the order of the differential equation 2x³ d²y/dx² - 3 dy/dx + y = 0. 2 Undefined 0 1 Solution: 86. What is the length of the line with a slope of 4/3 from a point (6, 4) to the y-axis? 10 75
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87. What refers to a locus of a moving point in a plane so that the difference of its distances from two fixed points (foci) is constant? Parabola Ellipse Circle Hyperbola 88. Find the inverse Laplace transform of (2s – 18)/(s² + 9) 2cos3t + 6sin3t 2 cos 3t – 6 sin 3t 6cos3t – 2sin3t 6cos3t – 2sin3t Solution: 89. The section of a certain solid cut by any plane perpendicular to the x axis is a square with the ends of a diagonal lying on the parabolas y² = 4x and x² = 4y. Find its volume. 144/ 144/ 144/ 154/ 90. What is 19% of 26? 19 21. 5 4. Solution: 91. What is the polar form of the complex number z = 3 + 4i? (3)(cos 36.87° + i sin 36.87°) (3)(cos 53.15° + i sin 36.87°) (4)(cos 53.15° + i sin 53.15°) (5)(cos 53.13° + i sin 53.13°) Solution: 92. The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is a/an:
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99. Which of the following lines is parallel to the line 3x – 2y + 6 = 0? 15x – 10y – 9 = 0 12x + 18y = 15 3x + 2y – 12 = 0 4x – 9y = 6 Solution: 100. How far from the y-axis is the center of the curve 2x² + 2y² + 10x – 6y – 55 = 0? 3.25 units 2.5 units 2.75 units 3.0 units Solution:
