Mathematics Tests 295 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 3. The correct answer is (B). Square both sides: (does not check) Thus x = is the only root. 4. The correct answer is (D). 3 3(6 – r) = 3 2(r – 1) If the bases are equal, the exponents are equal. 5. The correct answer is (B). The first digit can be filled in 3 ways (not zero). The second digit can be filled in 3 ways; the third digit in 2 ways, and the fourth digit in 1 remaining way. Hence the number of possible intergers is 3 ⋅ 3 ⋅ 2 ⋅ 1 = 18. 6. The correct answer is (E). Part VI296 www.petersons.com/arco ARCO ■ SAT II Math 7. The correct answer is (E). 8. The correct answer is (C). 9. The correct answer is (D). Let each root = r, then (x – r) 3 = x 3 – 3rx 2 + 3r 2 x – r 3 10. The correct answer is (B). One real root is 2. By De Moivre’s Theorem, the complex roots are separated by Hence, in the second quadrant the root is 2(cos 144° + i sin 144°). 11. The correct answer is (C). x 2 – 3x – 10 < 0 (x – 5)(x + 2) < 0 then x – 5 > 0 and x + 2 > 0 x > 5 and x< – 2 Impossible or x – 5 < 0 and x + 2 < 0 x < 5 and x > – 2 thus –2 < x < 5. Mathematics Tests 297 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 12. The correct answer is (E). The length of O P is . The amplitude of O P is 225° or radians. Hence, the polar form is . 13. The correct answer is (C). Since log 10 1 = 0 and the log N decreases as it approaches zero, it follows that log 10 x < 0. 14. The correct answer is (D). The inverse of an implication is obtained by negating the hypothesis and the conclusion. Thus the inverse of the given proposition is p → q. 15. The correct answer is (A). 16. The correct answer is (D). For f to be a function, there must be a unique value of y for any given value of x. This is apparently true for all the sets above except where y ≥ x + 1; in this case, for any given value of x, there is an infinity of values of y. Hence the nonfunction is (D). 17. The correct answer is (D). Eliminate fractions in both equations. Part VI298 www.petersons.com/arco ARCO ■ SAT II Math 18. The correct answer is (A). 19. The correct answer is (A). 20. The correct answer is (B). Mathematics Tests 299 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 21. The correct answer is (C). 22. The correct answer is (D). The graph of y = tan x indicates that the function increases in all four quadrants. 23. The correct answer is (B). Let the area of the section be A Part VI300 www.petersons.com/arco ARCO ■ SAT II Math 24. The correct answer is (C). Since sin x and cos x are positive, x is a first quadrant angle. Method 1: Method 2: 25. The correct answer is (B). From the following diagram, we see that we can get to P by (–5, 150º). 26. The correct answer is (D). ~ p means two heads do not show; therefore ~ p ^ q means three heads show. The probability of 3 heads is . 27. The correct answer is (D). 28. The correct answer is (D). cos x (cos x – 2) = 0 cos x = 0, cos x = 2 (impossible) x = 90º. Mathematics Tests 301 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 29. The correct answer is (C). 30. The correct answer is (C). The maximum value of the function is determined by the coefficient of the function. Hence maximum value is 2m. 31. The correct answer is (A). Method 1: or Method 2: 32. The correct answer is (D). or y = 2x 33. The correct answer is (A). Let then or thus However, the range of f(x) consists of all real numbers > 0 so that f –1 (x) is defined for x > 0. Part VI302 www.petersons.com/arco ARCO ■ SAT II Math 34. The correct answer is (B). By similar triangles. 35. The correct answer is (B). The x-intercepts, also called roots or zeros, are found by setting f(x) = 0 36. The correct answer is (B). Multiply by e x : e 2x + e x – 1 = 0 By quadratic formula: Since e x can only be a positive number, reject the minus sign in the formula. Thus only 1 root Mathematics Tests 303 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 37. The correct answer is (C). When x ≥ 0 and y ≥ 0, the equation is x + y = 4. When x ≤ 0 and y ≤ 0, the equation is – x – y = 4. When x ≤ 0 and y ≥ 0, the equation is – x + y = 4. When x ≥ 0 and y≤ 0, the equation is x – y = 4. The graph thus becomes the following: The graph consists of the sides of a square. 38. The correct answer is (C). From the right triangle, By equating areas, we get Part VI304 www.petersons.com/arco ARCO ■ SAT II Math 39. The correct answer is (D). Let the original radius = 1 40. The correct answer is (E). The two numbers must both be even or both be odd. The probability of choosing two even numbers is : the same for two odd numbers, hence the probability of one or the other is . 41. The correct answer is (D). Hence no value of x. 42. The correct answer is (C). Let the series be a, ar, ar 2 … ar 12 . [...]... (approximately)? (A) 22 (B) 29 (C) 33 (D) 20 (E) 24 www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 313 27 What is the largest rod that can just fit into a box 24 ″ × 8″ × 6″ (in inches)? (A) 24 (B) 25 (C) 26 (D) 28 (E) 30 28 Find the solution set of the inequality x2 – x – 6 < 0 (A) x > 2 (B) 2 < x < 3 (C) x > 3 and x < 2 (D) x > 3 or x < 2 (E) x < 3 29 If f(x) = x2 + 2 and , write the... 10 – 3 + 5 = –13 47 The correct answer is (B) r1 + r2 = p and r1r2 = q Then (r1 + r2 )2 = p2 or r 12 + r 22 + 2r1r2 = p2 Thus r 12 + r 22 = p2 – 2q 48 The correct answer is (E) 49 The correct answer is (D) 50 The correct answer is (B) The maximum value for 3 – cos 2x is attained when cos 2x is at a minimum value Since the least value for the cosine of an angle is –1, let cos 2x=–1 then 2x=π www.petersons.com/arco... 62 × 108 23 Find the product of an infinite number of terms: 31 /2 × 31/4 × 31/8 × 31/16 × … (A) 31 /2 (B) (C) 32 (D) 3 (E) 1 24 Which of the following is an (x,y) coordinate pair located on the ellipse 4x2 + 9y2 = 100? (A) (1, 3.5) (B) (1.4, 3 .2) (C) (1.9, 2. 9) (D) (2. 3, 3.1) (E) (2. 7, 2. 6) 25 If log4 44 = x, what is the approximate value of x? (A) 2. 73 (B) 2. 70 (C) 1. 67 (D) 1.64 (E) 1.61 26 A boy walks... ■ SAT II Math www.petersons.com/arco 3 12 Part VI 21 Find the angle between π and 2 that satisfies the equation: 2 sin2x + 5 cos x + 1 = 0 (A) (B) (C) (D) (E) 22 Two heavenly bodies are respectively 8.9 × 108 miles and 2. 7 × 1 07 miles distant from the earth Approximately how much farther way from the earth is the first than the second? (A) 6 .2 × 1 07 (B) 8.6 × 1 07 (C) 8.6 × 108 (D) 86 × 106 (E) 62 ×... www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 311 15 T varies directly as the square of r and inversely as the cube of s If r is tripled and s is doubled, the value of T is (A) multiplied by 3 /2 (B) multiplied by 6 (C) multiplied by 9/8 (D) multiplied by 2 (E) divided by 2 1 16 If log x ≥ log 2 + log x, then 2 (A) x ≥ 2 (B) x ≤ 2 (C) x ≤ 4 (D) x ≥ 4 (E) x ≥ 1 17 2 cos3 A sin A + 2 sin3 A cos A equals... (B) (C) (D) (E) 2: 1 7 Which ordered number pair represents the center of the circle x2 + y2 – 6x + 4y – 12 = 0? (A) (9,4) (B) (–3 ,2) (C) (3, 2) (D) (–6,4) (E) (6,4) 8 sin (135° + x) + sin (135° – x) equals (A) (B) –1 (C) (D) (E) 9 If 37 = 7x, what is the approximate value of x? (A) 95 (B) 1.95 (C) 2. 95 (D) 3.95 (E) 4.15 GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 310 Part VI 10 In... equals which one of the following? (A) cos 2A (B) 2 sin A (C) 2 cos A (D) cos2 A (E) sin 2A 18 An isosceles triangle with base 24 and legs of 15 is inscribed in a circle Find the radius (A) 7 (B) (C) 25 (D) (E) cannot be determined 19 Two roots of 4x3 + 8x2 + Kx – 18 = 0 are equal numerically but opposite in sign Find the value of K (A) 2 (B) +2 (C) –9 (D) +9 (E) 20 If log10 x = y and , then (A) (B) (C)...Mathematics Tests 305 43 The correct answer is (E) m m 44 The correct answer is (C) 45 The correct answer is (C) At the y–intercept, x = 0 GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 306 Part VI 46 The correct answer is (E) The remainder is 5(–1)2n+1 – 10(–1)2n + 3(–1)2n–1 + 5 Since 2n + 1 and 2n – 1 are odd and 2n is even, the remainder = 5(–1)... (D) (E) www.petersons.com/arco ARCO ■ is SAT II Math Mathematics Tests 309 4 If log8 p = 2. 5 and log2 q = 5, p expressed in terms of q is (A) p = q2 (B) p = q8 (C) p = q3 /2 (D) p = q5 /2 (E) p = q3 5 If circle R, of area 4 square inches, passes through the center of and is tangent to circle S, then the area of circle S, in square inches, is (A) 8 (B) (C) (D) 12 (E) 16 6 In a cube, the ratio of the longest... value of K if (x + 1) is a factor of x8 + Kx3 – 2x + 1 (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 31 The portion of the plane, whose equation is 8x + 20 y + 15z = 120 , that lies in the first octant forms a pyramid with the coordinate planes Find its volume (A) 90 (B) 120 (C) 140 (D) 150 (E) 160 GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 314 Part VI 32 Write in the form a + bi (A) (B) (C) (D) (E) . 10 – 3 + 5 = –13 47. The correct answer is (B) r 1 + r 2 = p and r 1 r 2 = q Then (r 1 + r 2 ) 2 = p 2 or r 1 2 + r 2 2 + 2r 1 r 2 = p 2 Thus r 1 2 + r 2 2 = p 2 – 2q 48. The correct. ellipse 4x 2 + 9y 2 = 100? (A) (1, 3.5) (B) (1.4, 3 .2) (C) (1.9, 2. 9) (D) (2. 3, 3.1) (E) (2. 7, 2. 6) 25 . If log 4 44 = x, what is the approximate value of x? (A) 2. 73 (B) 2. 70 (C) 1. 67 (D) 1.64 (E). of these Part VI3 12 www.petersons.com/arco ARCO ■ SAT II Math 21 . Find the angle between π and 2 that satisfies the equation: 2 sin 2 x + 5 cos x + 1 = 0 (A) (B) (C) (D) (E) 22 . Two heavenly