OCR Mathematics Midterm Exam Answer Key PDF

GEn 422-MATHEMATICS MIDTERM EXAMINATION NAME:____________ DATE:_____________ General Instructions: -Remember that talking is not allowed at any time in the exam area. - You are not permitted to share calculator...

GEn 422-MATHEMATICS MIDTERM EXAMINATION NAME:____________ DATE:_____________ General Instructions: -Remember that talking is not allowed at any time in the exam area. - You are not permitted to share calculator, pen or any other materials during the examination. - Just raise your hand if you have a question. - Keep your eyes on your own paper. - Once you’re done, fold (crosswise) the exam booklet together with the test paper and leave the exam area quickly and quietly. - Stop writing immediately when the teacher says it is the end of the exam. -Corresponding action will be taken if you’re not following the general instructions. TEST 1: Choose the correct answer from the provided choices. *Erasures will be marked as wrong. *Credit point = 1 point each 1. What type of number is √4. 5. Given a square and a circle, which of the A. Real number B. Positive integer following statements is always true? C. Complex number D. A and B A. Exactly one of the square and the circle can 2. Which of the following is/are false. be circumscribed about the other. A. The system of real numbers consists of all B. Exactly one of the square and the circle can rational and irrational numbers. be inscribed in the other. B. Every real number can be expressed as a C. The square can never be circumscribed about decimal, and the decimal representations for the circle. rational numbers are either terminating or D. The circle can never be inscribed in the nonterminating and repeating. square. C. A and B D. None of the above 6. Consider the system of inequalities: 𝑥 + 𝑦 ≤ 1 (unit circle centered at origin) and 3. Which of the following statements about 𝑦 ≥ 𝑥 logarithms and natural logarithms (ln) is false? Which of the following statements about the A. Solving the equation log(x) = 3 is the same as solution set of this system is ALWAYS true? solving ln(x) = 3, but the numerical answers will A. The solution set includes the origin (0, 0). be different. B. The solution set is symmetrical across the line B. ln(x) is equivalent to loge(x), where e is an y = x. irrational number approximately equal to C. Every point in the shaded region below the 2.71828. line y = x is a solution. C. Common logarithms (log) are typically used D. The solution set contains exactly two points. with a base of 10, while natural logarithms always use base e. 7. Triangle ABC has sides with lengths a, b, and D. A and C c, and angles A, B, and C respectively. Given that a = 2b and cos(C) = -1/2, which of the 4. Consider a circle with center O. If line following statements is always true? segment AB intersects the circle at points P and A. Triangle ABC is acute. Q, and O lies on line segment AB, which of the B. Triangle ABC is obtuse. following must be true? C. Angle A is the largest angle in the triangle. A. Segment PQ is always a diameter. D. Triangle ABC cannot exist. B. Length OP is always equal to length OQ. C. Segment AB is always a secant. D. Triangle OPQ is always isosceles. 8. Question: In a quadrilateral ABCD, where AB 18. One endpoint of a line segment has = CD and ∠ 𝐴 = ∠ 𝐶, which of the following coordinates (2,-1), and the midpoint has must be true? coordinates (3,2). Find the coordinates of the A. ∠ 𝐵 = ∠ 𝐷 other endpoint. B. ∠ 𝐵 + ∠𝐷 = 180 A. (2, 3) B. (3, 4) C. (5, 6) D. (4, 5) C. BD bisects AC D. 𝛥𝐴𝐵𝐶 𝑖𝑠 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 𝑡𝑜 𝛥 𝐶𝐷𝐴 19. Find the equation of the line through (1,-1) and the point of intersection of lines 𝑥 + 2𝑦 = 9. Find the number of edges and the number of 8 𝑎𝑛𝑑 3𝑥 − 𝑦 = 3. vertices of a regular dodecahedron. A. 4x + y = 5 B. x + 4y = 5 A. e =20, v = 20 B. e =30, v = 10 C. 4x − y = 5 D. x − 4y = 5 C. e =30, v = 20 D. e =30, v = 30 20. Find the equation of the line that passes 10. A portion of the surface of a sphere included through the origin and the point (-2, 3). between two parallel planes is a A. 3x − 2y = 0 B. x + 2y = 0 A. lune B. spherical sector C. 3x + 2y = 0 D. 2x + y = 0 C. zone D. spherical segment 21. Find the equation of the line perpendicular 11. The lateral face of a frustum of a regular to 6𝑥 − 2𝑦 − 1 = 0 and passing through (1,5). pyramid is in the shape of a A. y + 3x = 16 B. y + 2x = 6 A. rectangle B. square C. triangle D. trapezoid C. 3y + x = 16 D. 2y + x = 5 22. Which of the following is equivalent to 12. A portion of the surface of a sphere included 𝑎+ ? between two parallel planes is a A. 𝑎 + B. 𝑎 + A. lune B. zone C. spherical sector D. spherical segment C. 𝑎 + + D. 𝑎 + 𝑎𝑏 + 13. Consider the expression (𝑥 ). Which of the 23. Trapezoid ABCD is similar to trapezoid PQRS. following statements accurately represents the The length of each side of trapezoid PQRS is 3 simplified form of this expression? times the length of its corresponding side of trapezoid ABCD. The area of trapezoid ABCD is 6 A. |𝑥| B. 𝑥 C. 𝑥 D. 𝑥 square centimeters. What is the area, in square centimeters, of trapezoid PQRS? 14. A _____ is an extension of a chord in a circle A. 9 B. 18 C. 216 D. 54 which is a straight line segment of which the endpoints lie on the circle. 24. A rectangle was altered by increasing its A. diameter B. chord C. tangent D. secant length by 10 percent and decreasing its width by p percent. If these alterations decreased the 15. What is the name of a polygon with 40 sides? area of the rectangle by 12 percent, what is the A. tetracontagon B. pentagon value of p ? C. nonagon D. hexacontagon A.20 A. 12 B. 15 D. 22 16. Solve 𝐴𝑟𝑐𝑡𝑎𝑛 𝑥 + 𝐴𝑟𝑐𝑡𝑎𝑛 (1 − 𝑥) = 𝐴𝑟𝑐𝑡𝑎𝑛 4/3. A. 𝑥 = 1/2 B. 𝑥 = 1/3 C. 𝑥 = 0 D. 𝑥 = 0.4567 17. Simplify + A. 3 cos 𝐴 B. sin 𝐴 cos 𝐴 C. 2sec 𝐴 D. 2 sin 𝐴 tan 𝐴 25. The three candidates who ran for a city 31. 8. Find the general form of the equation of a council position in Davao City in 2015 received a circle with center at (3,1) and tangent to the line total of 14,705 votes. The ratio of votes for the 4x+3y+5=0. first candidate to votes for the second candidate A. 𝑥 + 𝑦 + 6𝑥 − 2𝑦 − 6 = 0 was approximately 4 to 1. The ratio of votes for B. 𝑥 + 𝑦 − 6𝑥 − 2𝑦 + 6 = 0 the first candidate to votes for the third C. 𝑥 + 𝑦 − 6𝑥 − 2𝑦 − 6 = 0 candidate was approximately 100 to 13. Based D. 𝑥 + 𝑦 + 6𝑥 + 2𝑦 + 6 = 0 on this information, which of the following is closest to the number of votes received by the 32. A regular triangular pyramid has an altitude first candidate? of 9 m and a volume of 187.06 cu. m. What is the A. 14,601 B. 10,656 C. 3,676 D. 1,911 base edge in meters? A. 13 m B. 12 m C. 14 m D. 12.45 m 26. According to a 2008 study, there were five known subspecies of tigers, including the Amur 33. A swimming pool is rectangular in shape and Bengal, living in the wild. Scientists length 12 m and width 5.5 m. It has sloping estimated that there were a total of 4,000 tigers bottom and is 1 m deep at one end and 3.6 m in the wild. Of these, 450 were Amur tigers and x deep at the other end. The water from a full were Bengal tigers. Which inequality represents cylindrical reservoir 3.6 m in diameter and 10 m all possible numbers of tigers in the wild in 2008 deep is emptied to the pool. Find the depth of that belonged to the Bengal tiger subspecies? water at the deep end. A. 450 > 𝑥 ≥ 4000 B. 5 ≤ 𝑥 ≤ 3547 A. 2.695 m B. 2.842 m C. 2.912 m D. 2.754 m C. 3547 < 𝑥 ≤ 4000 D. 1 ≤ 𝑥 ≤ 3547 34. A solid spherical steel ball 20 cm in diameter 27. An airplane flew from Davao whose latitude is placed into a tall vertical cylinder containing is 14o N and longitude of 121o30’ E on a course S water, causing the water level to rise by 10 cm. 30o W and maintaining a uniform altitude. At What is the radius of the cylinder? what point will the plane cross the equator? A. 11.55 cm B. 9.08 cm C. 10.28 cm D. 12.14 cm A.113 33 E B. 160 59 E C. 7 57 E D. 121 30 E 35. The volume generated by the circles 𝑥 + 𝑦 + 4𝑥 − 6𝑦 − 12 = 0 revolved about the line 2𝑥 − 3𝑦 − 12 = 0 is: 28. The area of a rhombus is 143 m2. If the longer A. 3242 cubic units B. 3422 cubic units diagonal is 26 m, find the angle opposite the C. 3452 cubic units D. 3342 cubic units shorter diagonal. A. 45.86o B. 40.86o C. 25.86o D. 15.86o 36. Which of the following is perpendicular to the line + = 1? 29. The sum of the interior angles of a regular A. 3𝑥 − 4𝑦 − 5 = 0 B. 4𝑥 − 3𝑦 − 6 = 0 polygon is 1260o. Find the area of the polygon if C. 𝑥 − 4𝑦 − 8 = 0 D. 4𝑥 + 3𝑦 − 11 = 0 its perimeter is 45 centimeters. A. 134.681 cm2 B. 100.6 cm2 37. Find the shortest distance from the point C. 104.6 cm2 D. 154.6 cm2 (1,2) to a point on the circumference of the circle defined by the equation 𝑥 + 𝑦 + 10𝑥 + 6𝑦 + 30. A lawn in the form of a 30o right triangle 30 = 0. whose leg is 67 m on the long leg and 15 m from A. 5.61 B. 5.71 C. 5.81 D. 5.91 the short leg, is a peg to which a goat is tied such 38. Find the equation of the ellipse whose center that the farthest distance its mouth can reach is is at (-3,-1), vertex at (2,-1) and focus at (1,-1). 30 m from the peg. Find the area on the lawn A. 9𝑥 + 25𝑦 + 54𝑥 + 50𝑦 − 119 = 0 from which it can graze. B.9𝑥 + 36𝑦 − 54𝑥 + 50𝑦 − 116 = 0 A. 1056 sq. m B. 1023 sq. m C. 9𝑥 + 25𝑦 + 50𝑥 + 50𝑦 + 109 = 0 C. 1000.45 sq. m D. 23.78 sq. m D. 4𝑥 + 25𝑦 + 54𝑥 − 50𝑦 − 112 = 0 39. Two vertices of a triangle are (6,-1) and (7,- 48. Find the equation of hyperbola satisfying the 3). Find the ordinate of the third vertex such that following conditions: Center at the origin, one the centroid of the triangle will lie at the origin. focus at (4,0), one vertex at (3,0) A. -4 B. 13 C. 4 D. -13 A. − =1 B. − =1 40. 4. A wave has a wavelength of 3.00 m. C. − =1 D. − =1 Calculate the frequency of the wave if it is a sound wave. 49. Find the standard equation of the circle with A. 174 Hz B. 110 Hz C. 114 Hz D. 14 Hz center (3,5) that is tangent to the line 12𝑥 − 41. A ball is thrown from the top of a building 5𝑦 + 2 = 0. with an initial velocity of 20.0 m/s straight A. (𝑥 − 7) + (𝑦 + 6) = 1 upward, at an initial height of 50.0 m above the B. (𝑥 − 1) + (𝑦 − 4) = 1 ground. The ball just misses the edge of the roof C.(𝑥 − 3) + (𝑦 − 5) = 1 on its way down. Determine the time needed for D. (𝑥 − 1) + (𝑦 − 1) = 1 the ball to reach its maximum height. A. 2.04 sec B. 1.2450 sec C. 5 sec D. 3.095 sec 50. Find the equation of the line that passes through the point of intersection of the lines 42. Find the term involving 𝑥 in the expansion 3𝑥 − 2𝑦 = 6 and 𝑥 + 3𝑦 = 13 and whose distance from the origin is 5. of 𝑥 +. A. 4𝑥 + 3𝑦 = 25 B. 𝑥 + 𝑦 = 15 A. 120𝑎 𝑥 B. 130𝑎 𝑥 C. 2𝑥 − 𝑦 = 5 D. 𝑥 + 3𝑦 = 45 C. 140𝑎 𝑥 D. 150𝑎 𝑥 51. Find the point on the line 3x + y = - 4 that is 43. A train is moving at the rate of 8 mi/h along equidistant from ( - 5 , 6 ) and (3,2). a piece of circular track of radius 2500 ft. A. ( - 1 , 3 ) B. ( - 1 , 1 ) C. ( 1 , 3 ) D. ( - 2 , 2 ) Through what angle does it turn in 1 min? A. 34.13 B. 20.13 C. 16.13 D. 11.1312 52. From the top of a lighthouse, 175 ft above the water, the angle of depression of a boat due 44. Express 𝑠𝑒𝑐 455 in terms of function of a south is 18 50′. Calculate the speed of the boat positive acute angle. if, after it moves due west for 2 min, the angle of A. −𝑠𝑒𝑐 85 B. cos 85 C. 𝑠𝑖𝑛 20 D. 𝑐𝑜𝑠 45 depression is 14 20′. A. 427.88 ft/min B. 226.84 ft/min 45. The graph 𝑦 = sin (𝑥 + ) is shifted ______. C. 256.50 ft/min D. 342.50 ft/min A. 𝑟𝑖𝑔ℎ𝑡 units B. 𝑙𝑒𝑓𝑡 units C. left 1 unit D. 𝑙𝑒𝑓𝑡 𝜋 units 53. Determine the pair of perpendicular lines. A. y = 3x + 2 and y = 3x − 2 46. Find the range of 𝑓(𝑥) = 2𝑥 − 6𝑥 + 7. B. 3x − 2y = 5 and 2x + 3y = 4 C. 6x + 3y = 1 and 4x + 2y = 3 A.𝑓(𝑥) ≥ B.𝑓(𝑥) ≤ D. x = −2 and x = 7. C.𝑓(𝑥) ≥ 7 D.𝑓(𝑥) ≤ 7 54. Form the equation with integral coefficients 47. Find the equation of an ellipse satisfying the of ±3𝑖, ± √2. following conditions: Center at the origin, one A. 2𝑥 − 17𝑥 + 9 = 0 focus at (0,5), length of semimajor axis is 13 B. 2𝑥 + 17𝑥 + 9 = 0 A. + =1 B. + =1 C. 2𝑥 + 17𝑥 − 9 = 0 C. + =1 D. + =1 D. −2𝑥 − 17𝑥 − 9 = 0 55. The triangle has an area of 𝑥 − 𝑥 − 2 and 63. A sphere is inscribed in a rectangular solid 5 a base of 𝑥 + 1. What is the height of the cm in length, 4 cm in width, and 3 cm in height. triangle? Another sphere of the same volume is also A. (𝑥 − 2) B. (𝑥 − 2) inscribed in a cube. Find the volume of the cube. A. 15 cm3 B. 27 cm3 C. 42 cm3 D. 67.43 cm3 C. 2 (𝑥 − 2) D. (𝑥 − 2)(𝑥 + 1) 64. Two ships have radio equipment with a range 56. If a sphere is inscribed in a cylinder, the of 200 km. One is 155 km N42 40′E and the volume of the sphere is that of the cylinder, other is 165 km N45 10′W of a shore station. A. one-half B. two times Can the two ships communicate directly? C. one-third D. two-thirds A. Yes, they are 190.0234 km apart B. No, they are 222.07 km apart 57. The volume of water in a spherical tank C. Yes, they are 192.98 km apart having a diameter of 4 m is 5.236 m3. D. No, they are 215.0347 km apart Determine the depth of the water in the tank. A. 0.8 m B. 1.0 m C. 1.2 m D. 1.4 m 65. A sphere of radius 3 m is circumscribed about a regular triangular prism. If the base edge of the 58. A right circular cone of altitude h is cut into prism is 2 m, find the volume of the prism. two parts of equal weight. How far from the A. 9.59 m3 B. 2.51 m3 C. 21.01 m3 D. 7.07 m3 vertex should the cut, parallel to the base, be made? 66. The radii of parallel circular sections of a A. 0.794h B. 0.595h C. 0.691h D. 0.29413h sphere are 4 in. and 4.8 in., while the radius of the sphere is 5 in. Find the volume of the portion 59. Find the total surface area of a right circular of the sphere included between these sections. cone whose base radius is 10 cm and whose A. 314.4 in3 B. 113.1 in3 C. 215.5 in3 D. 111.7 in3 altitude is 15 cm. A. 384.52 cm2 B. 185.52 cm2 67. How many iron rivets can be made from two C. 582.52 cm2 D. 880.52 cm2 kilograms of metal if each rivet consists of a head in the form of a spherical segment, with an 60. A closed cylindrical tank 10 ft. in height and 4 altitude of 4 mm and a base radius of 8 mm, and ft. in diameter contains water with depth of 3 ft. a cylindrical body with length 15 mm and radius and 5 in. What would be the height of the water of 8 mm? The density of iron is 7.8 × 103 kg/m3 when the tank is lying in a horizontal position? A. 31 B. 57 C. 25 D. 74 A. 2.87 ft B. 3.5 ft C. 2.0167 ft D. 1.5 ft 61. A flower vase, in the form of a hexagonal 68. Given ℎ(𝑥) = find ℎ (𝑥). prism, is to be filled with 512 cubic inches of A. ℎ (𝑥) = B. ℎ (𝑥) = water. Find the height of the water if the wet C. ℎ (𝑥) = D. ℎ (𝑥) = portion of the flower vase and its volume are numerically equal. A. 34.88 in B. 23.88 in C. 45.88 in D. 21.88 in 69. Find the values of p for which (𝑥 − 𝑝 𝑥 + 62. A right section of an oblique prism is a 3 − 𝑝) ÷ (𝑥 − 3) has a remainder of 4. hexagon in which the sides are 3 cm, 4 cm, 4 cm, A. 5, 1 B. -5, 16/3 5 cm, 6 cm, and 7 cm. The measure of one lateral C. -5, -1 D. 5, -16/3 edge is 7 cm. Find the lateral area. A. 203 cm3 B. 313 cm3 C. 428 cm3 D. 710 cm3 70. Find a positive and a negative angle coterminal with a 55° angle. A. −55° 𝑎𝑛𝑑 415° B. −95° 𝑎𝑛𝑑 15° C. −300° 𝑎𝑛𝑑 400° D. −305° 𝑎𝑛𝑑 415° 71. Two seaports are on the same parallel of 77. Find the standard equation of the circle latitude 42◦ 27’ N. Their difference in longitude having as a diameter the segment joining (2, -3) is 137◦ 35’. Ship A and ship B sail at 20 knots from and (6,5). one port to the other. Ship A sails along the A.(𝑥 − 2) + (𝑦 − 4) = 20 parallel of latitude; ship B sails the great circle B. (𝑥 − 4) + (𝑦 + 1) = 80 route connecting the two ports. Calculate the C. (𝑥 + 4) + (𝑦 + 1) = 80 time difference in their arrival times if they leave D. (𝑥 − 4) + (𝑦 − 1) = 20 port together. A. 2 days 10.52 hours 78. Find the diameter of a pulley which is driven B. 1 day 19.77 hours at 360 r/min by a belt moving at 40 ft/s. C. 1 day 5.25 hours A. 1.54 ft B. 2.12 ft C. 1.3289 ft D. 4.32 ft D. 2 days 5 hours 30 minutes 79. Find the term involving independent of x in 72. Assuming the Earth to be a sphere of radius the expansion of 𝑥 −. 6378 km calculate the great circle distance in A. 23 B. 84 C. 121 D. 5 kilometres between London (51◦ 30’ N, 0◦) and New York (40◦ 45’ N, 74◦ W). Find also the most 80. Find the perimeter of a regular octagon northerly latitude reached on the great circle arc. inscribed in a circle of radius 150 cm. A. 53o48’ N B. 43o28’ N o A. 918.44 cm B. 138.58 cm C. 75 08’ N D. 83o21.2’ N C. 57.40 cm D. 459.22 cm 73. Two radio astronomy observatories wish to set up a radio link. Their longitudes and latitudes are respectively +2◦ 18.4’, +53◦ 14.2’ (Jodrell Bank, England) and +79◦ 50.2’, +38◦ 26.3’ (Green Bank, West Virginia, USA). In what angular direction to the north must the antenna at the first station be set to achieve the best transmission (i.e. the great circle direction)? A. 76o45’ B. 72o49’ C. 43o17’ D. 65o04’ 74. If a + 12ab = 679 and 9a b + 12b = 978, find a − 4ab + 4b. A. 1 B. 9 C. 25 D. 49 75. Find the length of the latus rectum of 4𝑦 = 𝑥 + 4𝑥 + 8. A. 8 B. 4 C. 2 D. 16 76. Find the equation of the line midway between the lines 3x + 4 y = 2 and 6 x + 8 y = 1. A. 12x + 16y = 5 B. 7x + 6y = 2 C. 8x + 15y = 10 D. 2x + 3y = 6