SAT II Math Episode 2 Part 8 pdf

Mathematics Tests 315 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 37. In figure 37, what is the approximate area of triangle NJL? (A) 11.79 (B) 13.85 (C) 15.31 (D) 17.10 (E) 17.82 38. A right circular cylinder is circumscribed about a sphere. If S represents the surface area of the sphere and T represents the total area of the cylinder, then (A) (B) (C) (D) (E) 39. If |x – 2| < 5, what are the possible values of x? (A) 0 < x < 5 (B) 0 < x < 2 (C) –3 < x ≤ 7 (D) –3 ≤ x < 7 (E) –3 < x < 7 40. How many even numbers greater than 40,000 may be formed using the digits 3, 4, 5, 6, and 9 if each digit must be used exactly once in each number? (A) 36 (B) 48 (C) 64 (D) 96 (E) 112 Fig. 37 Part VI316 www.petersons.com/arco ARCO ■ SAT II Math 41. A regular octagon is formed by cutting off each corner of a square whose side is 6. Find the length of one side of the octagon. (A) 2 (B) (C) (D) (E) 42. Approximately, what is ? (A) 1.42 (B) 3.18 (C) 5.38 (D) 7.00 (E) 8.67 43. All triangles in the set of triangles having a given side and a given angle opposite that side (A) are congruent (B) are similar (C) are equivalent (D) have the same inscribed circle (E) have the same circumscribed circle 44. Find the value of (A) (B) (C) (D) (E) 45. For what positive value of m will the line y = mx + 5 be tangent to the circle x 2 + y 2 = 9? (A) 1 (B) 2 (C) (D) (E) Mathematics Tests 317 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 46. The graph of the curve whose parametric equations are x = a sin t and y = b cos t is a(n): (A) ellipse (B) circle (C) parabola (D) hyperbola (E) straight line 47. The graph of y = |x – 2| + 2 consists of (A) one straight line (B) a pair of straight line rays (C) the sides of square (D) a circle (E) a parabola 48. The converse of ~ p → q is equivalent to (A) p → ~ q (B) p → q (C) ~ q → p (D) q → p (E) ~ p → ~ q 49. Write an equation of lowest degree, with real coefficients, if two of its roots are –1 and 1 + i. (A) x 3 + x 2 + 2 = 0 (B) x 3 – x 2 – 2 = 0 (C) x 3 – x + 2 = 0 (D) x 3 – x 2 + 2 = 0 (E) none of these 50. If log r 6 = S and log r 3 = T, then log r is equal to (A) log 2 r for any r (B) 1 – S + T (C) 1 – S – T (D) log r 2 – 1 (E) zero, if r = 4 STOP IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON THIS TEST ONLY. DO NOT WORK ON ANY OTHER TEST IN THIS BOOK. Part VI318 www.petersons.com/arco ARCO ■ SAT II Math PRACTICE TEST 3 Answer Key Math Level IIC SOLUTIONS 1. The correct answer is (B). For y to be a function of x, there must be a unique value of y for any given value of x. This would be true only for y = x 2 + 1. In A, C, D, and E, y may take on 2 or more values for a given x. 2. The correct answer is (A). The set P ∩ Q includes as elements all rectangles that are also rhombi. These elements make up the set of squares. 3. The correct answer is (B). 4. The correct answer is (C). 8 2.5 = p and 2 5 = q then (2 3 ) 2.5 = p or p = 2 7.5 = (2 5 ) 1.5 thus p = (2 5 ) 3/2 = q 3/2 5. The correct answer is (E). R is internally tangent to S and its diameter is half that of S. Hence S has an area 4 times that of r, or 16 square inches. 1. B 2. A 3. B 4. C 5. E 6. D 7. C 8. D 9. D 10. A 11. E 12. D 13. D 14. B 15. C 16. D 17. E 18. B 19. C 20. A 21. E 22. C 23. D 24. B 25. A 26. B 27. C 28. B 29. D 30. D 31. B 32. A 33. E 34. A 35. B 36. C 37. B 38. A 39. E 40. A 41. D 42. B 43. E 44. B 45. E 46. A 47. B 48. A 49. D 50. B Mathematics Tests 319 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 6. The correct answer is (D). Consider a cube of edge 1. Then the longest diagonal = The diagonal of a base = Hence the ratio = 7. The correct answer is (C). x 2 – 6x + 9 + y 2 + 4y + 4 = 12 + 13 (x – 3) 2 + (y + 2) 2 = 25 Center is (3, –2) 8. The correct answer is (D). sin 135º cos x + cos 135º sin x + sin 135º cos x – cos 135º sin x 9. The correct answer is (D). 10. The correct answer is (A). 11. The correct answer is (E). Consider y = sin x cos x sin 2x has a period of 180º and reaches its maximum at x = 45 ° or . 12. The correct answer is (D). There are 4! = 24 ways of lining up the 4 men. Consider the certain two as one unit and determine the number of ways of lining up three; 3! = 6. However the two men can be next to each other in twice as many ways as indicated by merely switching places. Hence, there are 12 ways of the four men lining up so that a certain two are always next to each other. Probabilty . Part VI320 www.petersons.com/arco ARCO ■ SAT II Math 13. The correct answer is (D). 14. The correct answer is (B). The converse, inverse, and negative are not equivalent statements. The contrapositive, ~ Q → ~ P, is equivalent and this is the same as saying that Q is a necessary condition for P. 15. The correct answer is (C). Hence, 16. The correct answer is (D). log x ≥ log 2 + log x 1/2 log x – log x 1/2 ≥ log 2 17. The correct answer is (E). 2 cos 3 A sin A + 2 sin 3 A cos A = 2 sin A cos A (cos 2 A + sin 2 A) = 2 sin A cos A = sin 2 A 18. The correct answer is (B). Mathematics Tests 321 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 19. The correct answer is (C). Let the roots be r, – r, and s 20. The correct answer is (A). Since log 10 x = y, 10 y = x Substituting, we get 21. The correct answer is (E). 22. The correct answer is (C). Writing this to two significant figures, we get 8.6 ✕ 10 8 23. The correct answer is (D). The exponent is the sum of an infinite geometric series S a r = − = − = 1 1 1 1 2 1 2 Thus 3 1 = 3. Part VI322 www.petersons.com/arco ARCO ■ SAT II Math 24. The correct answer is (B). 25. The correct answer is (A). 26. The correct answer is (B). He saves . Percent saved = − ⋅ = − ⋅=− () ≈− () ≈⋅ ≈ ≈ 22 2 100 22 2 100 2 2 50 50 2 1 414 50 586 29 3 29 xx x . . . Mathematics Tests 323 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 27. The correct answer is (C). Longest dimension is along the diagonal d 2 = 24 2 + 8 2 + 6 2 d 2 = 576 + 64 +35 = 676 28. The correct answer is (B). x 2 – x – 6 < 0 (x – 3)(x + 2) < 0 Either x – 3 < 0 and x + 2 > 0 or x – 3 > 0 and x + 2 > 0 x < 3 and x > –2 or x > 3 and x < –2 –2 < x < 3 this is impossible 29. The correct answer is (D). 30. The correct answer is (D). or 1– K + 2 + 1 = 0 K = 4 31. The correct answer is (B). Part VI324 www.petersons.com/arco ARCO ■ SAT II Math 32. The correct answer is (A). By De Moivre’s Theorem 33. The correct answer is (E). Method 1: 17º ∆ 33º Method 2: 34. The correct answer is (A). 35. The correct answer is (B). 36. The correct answer is (C). Since the period of cos x is 360°, the period of cos 2x is 180°. 37. The correct answer is (B). [...]...Mathematics Tests 325 38 The correct answer is (A) 39 The correct answer is (E) If x – 2 > 0 or x > 2, then x – 2 < 5 x –5 x>–3 Thus, – 3 < x < 7 or |x – 2| < 5 –5 . = − ⋅ = − ⋅=− () ≈− () ≈⋅ ≈ ≈ 22 2 100 22 2 100 2 2 50 50 2 1 414 50 586 29 3 29 xx x . . . Mathematics Tests 323 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 27 . The correct answer. inches. 1. B 2. A 3. B 4. C 5. E 6. D 7. C 8. D 9. D 10. A 11. E 12. D 13. D 14. B 15. C 16. D 17. E 18. B 19. C 20 . A 21 . E 22 . C 23 . D 24 . B 25 . A 26 . B 27 . C 28 . B 29 . D 30. D 31. B 32. A 33 diagonal d 2 = 24 2 + 8 2 + 6 2 d 2 = 576 + 64 +35 = 676 28 . The correct answer is (B). x 2 – x – 6 < 0 (x – 3)(x + 2) < 0 Either x – 3 < 0 and x + 2 > 0 or x – 3 > 0 and x + 2 >