Mathematics Tests 195 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 4. The correct answer is (A). The slope of a line perpendicular to has a slope of . This is the negative reciprocal of . 5. The correct answer is (D). 6. The correct answer is (E). When r = 60°, When r = 45°, If 60 > r > 45, Part V196 www.petersons.com/arco ARCO ■ SAT II Math 7. The correct answer is (B). The locus of points at distance d from one plane consists of two planes parallel to the given plane and distance d from it. The locus of points at distance e from the second plane consists of two planes parallel to the second plane and at distance e from it. Since the two original planes are perpendicular to each other, the first pair of planes is perpendicular to the second pair of planes, resulting in 4 lines of intersection. 8. The correct answer is (D). The graph of y = –2x + 3 has a slope of –2 and a y-intercept of 3. By setting y = 0, we see that the x-intercept is . Thus, the graph lies in quadrants I, II, and IV only. 9. The correct answer is (D). Perimeter of square = 4x Mathematics Tests 197 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 10. The correct answer is (C). Let x and y be two odd integers and represent x = 2a + 1 and y = 2b + 1. Where a and b are any integers, need not be an integer at all. There is closure of the odd integers only under multiplication. 11. The correct answer is (E). If If 2x + 3 < 0, 2x < –3 and x < – 3 2 Therefore the solution set is represented by 12. The correct answer is (B). Rationalize the denominator of 13. The correct answer is (C). If the roots of 9x 2 – 4Kx + 4 = 0 are equal, the discriminant must be zero. Part V198 www.petersons.com/arco ARCO ■ SAT II Math 14. The correct answer is (E). Make a sketch of the graphs of the two equations. The graph of x 2 + y 2 = 7 is a circle with center at (0, 0) and radius = . The graph of x 2 – y 2 = 1 is a hyperbola with intercepts at (1, 0) and (– 1, 0). Thus there are 4 points of intersection. 15. The correct answer is (C). As K varies from 1 to n, (2K + 1) becomes 3, 5, 7, 9 … (2n + 1). This is an arithmetic progression with first term a = 3, last term l = 2n + 1. The sum S is given by the formula: 16. The correct answer is (D). If we tried to divide a number n by 0, and defined the quotient as a number K, and K · 0 equals n. But K · 0 is always 0, and we could not get any number n as the product. Mathematics Tests 199 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 17. The correct answer is (D). Solve the equation for y. Multiply by 225. y is imaginary if Thus there are no points in the vertical strip between x = 5 and x = –5. 18. The correct answer is (C). P = EI, E = IR Substitute E = IR in first equation. P = IR · I = I 2 R (1) Also, solve second equation for I and substitute in first equation . (4) Only (1) and (4) are possible. Part V200 www.petersons.com/arco ARCO ■ SAT II Math 19. The correct answer is (C). || || Also, Add these two equalities. 20. The correct answer is (B). Mathematics Tests 201 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 21. The correct answer is (A). 22. The correct answer is (D). Multiply both sides by r + y. Factor the right-hand member. pr = y(r – p) Divide both sides by (r – p). 23. The correct answer is (C). Part V202 www.petersons.com/arco ARCO ■ SAT II Math 24. The correct answer is (C). Construct a right ∆ with one acute angle = x, and designate the opposite leg as p and the adjacent leg as 1, so that By the Pythagorean Theorem, the hypotenuse is equal to and cos 25. The correct answer is (A). Total amount of solution = (p + q) lb 26. The correct answer is (C). 131, 132, 135 … 258, 259 Divide by 3. 44, 45 … 86 There are as many numbers divisible by 3 as there are numbers between 44 and 86 inclusive. 86 – 44 + 1 = 43 27. The correct answer is (E). Mathematics Tests 203 ARCO ■ SAT II Math www.petersons.com/arco GO ON TO THE NEXT PAGE 28. The correct answer is (D). Multiple by mn. 29. The correct answer is (B). The equation x 2 + 4y 2 = 25 is of the form which are the standard forms of an ellipse with center at the origin. The graph of the equation is therefore an ellipse. 30. The correct answer is (D). 31. The correct answer is (B). Let the radius of the circle = x. Then, the area of the circle = πx 2 . Since the area of the square is a 2 , the equation expressing the given conditions of the problem is πx 2 = 2a 2 32. The correct answer is (E). The set of points to the left of the line x = 2 is designated by the inequality x < 2. The set of points above the x-axis is designated by the inequality y > 0. The set of points under the line y = 2x is designated by the inequality y < 2x. Hence, the sketched region is designated by the intersection of the sets described by the inequalities y < 2x, x < 2, y > 0 33. The correct answer is (C). (63) 21 = 6.11155 × 10 37 The decimal point is moved 37 places to the right. This yields a 38-digit number. 34. The correct answer is (B). For the roots of 2x 2 – 3x+c = 0 to be real and irrational, the discriminant must be positive but not a perfect square. Hence, . This applies to all listed values except 2. However, C = –2 makes the discriminant 25; C = –1 makes it 17; C = 0 makes it 9; and C = 1 makes it 1. All are perfect squares except 17. Hence, the only possible value of C is –1. Part V204 www.petersons.com/arco ARCO ■ SAT II Math 35. The correct answer is (C). Slopes are negative reciprocals. 36. The correct answer is (D). OB = r. Draw and extend to point D on . Then and m∠ COB = 60º Thus m∠ B = 30º. In right triangle OBC, it follows that BC = 4" and . By the Pythagorean Theorem, [...]... x + 1 (B) y = |x – 1| (C) y = x2 + 1 (D) y = |x + 1| (E) y = |x| Fig 15 GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 21 4 Part V 16 Select the correct order for defining the following terms: I—natural number II imaginary number III—rational number IV—integer (A) I, IV, III, II (B) I, II, III, IV (C) I, III, II, IV (D) IV, I, III, II (E) I, IV, II, III 17 If the reciprocal of y –... (A) 2. 67 (B) 2. 71 (C) 2. 78 (D) 2. 81 (E) 2. 84 10 In figure 10, FG HK , FH ⊥GH , and GK ⊥ HK If FG = 5 and m∠ F = r, HK = (A) (B) (C) (D) (E) 5 sin2 r 5 cos2 r 10 sin2 r 5 sin r none of these Fig 10 11 If the graph of the equation y = 2x2 – 6x + C is tangent to the x-axis, the value of C is (A) 3 (B) (C) 4 (D) (E) 5 www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 12 If (A) (B) (C) (D) (E) 21 3... same time divided by 4, V remains the same 50 The correct answer is (E) Let RS = x, QS = 21 –x, and PS = h By the Pythagorean Theorem h2 + x2 = 100 and h2 + (21 – x )2 = 28 9 Subtract the first equation from the second Thus, www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 20 9 PRACTICE TEST 3 Answer Sheet Math Level IC Directions: For each question in the sample test, select the best of the... of (sin 17° )2 + (cos 17° )2? (A) 03 (B) 18 (C) 72 (D) 1.00 (E) 1 .27 6 In figure 6, m∠ P = 2 m∠ Q and m∠ QRS = 108° Triangle PQR is (A) isosceles (B) right (C) obtuse (D) scalene (E) equilateral Fig 6 7 For what value or values of y is the equation (A) (B) (C) (D) (E) satisfied? ±3 + 3 only – 3 only ±9 + 9 only GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 21 2 Part V 8 In figure 8, m∠... increases 2 units, so that the slope of the line must be two It can be seen also from the table that the slope is constant, for when x increase 3 units, y increases 6 units, again yielding a slope of 2 The equation of the line is y = 2x + 1 41 The correct answer is (A) Take the product of 3, 4, 5, and 7, which is divisible by 3, 4, 5, 6, and 7 Now add 2 to this product 3 ⋅ 4 ⋅ 5 ⋅ 7 + 2 = 420 + 2 = 422 42. .. 422 42 The correct answer is (E) 43 The correct answer is (A) MATH - 175 SCIENCE - 1 42 TOTAL NUMBER OF STUDENTS = 25 0 x = No of students taking both math and science www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 20 7 44 The correct answer is (C) Since m∠ P is 60° and PS = 5, PR = 10 and Therefore, the coordinates of R are (0, 2 + ) 45 The correct answer is (B) r–s>r+s Add – r to both sides... (E) 5 2 In triangle NJL the measure of angle N is 90° and the measure of angle L is 24 ° If NL = 10, what is the approximate length of JL ? (A) 10.75 (B) 10.85 (C) 10.95 (D) 11.05 (E) 11.15 3 If 4x – 3 > x + 9, then (A) x > 2 (B) x > 3 (C) x > 4 (D) 8 > x > 4 (E) x > 0 www.petersons.com/arco ARCO ■ SAT II Math Mathematics Tests 21 1 4 (A) (B) (C) (D) (E) 5 What is the approximate value of (sin 17° )2 +... angle bisector divides the opposite leg? (A) 2: 3 (B) 3:4 (C) 1 :2 (D) 3:5 (E) 2: 5 19 The equation 4y2 – 3y + C = 0 has real roots The value of C for which the product of the roots is a maximum is (A) (B) (C) (D) (E) 20 The sum of all the even numbers between 1 and 51 is (A) 1300 (B) 650 (C) 325 (D) 675 (E) none of these www.petersons.com/arco ARCO ■ SAT II Math ... below as needed GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 21 0 Part V SOLID VOLUME REFERENCE DATA OTHER Right circular cone L = cl V = volume r = radius h = height Sphere S = 4 πr2 V = volume r = radius S = surface area Pyramid L = lateral area c = circumference of base l = slant height V = volume B = area of base h = height PRACTICE TEST 3 MATH LEVEL IC 50 Questions • Time—60 Minutes...Mathematics Tests 20 5 Or, since we have a 30°–60°–90° triangle, 37 The correct answer is (E) If Divide the numerator and denominator by K and simplify 38 The correct answer is (C) Let F = C = x, and substitute in Multiply through by 5 39 The correct answer is (E) GO ON TO THE NEXT PAGE ARCO ■ SAT II Math www.petersons.com/arco 20 6 Part V 40 The correct answer is (B) . terms: I—natural number II imaginary number III—rational number IV—integer (A) I, IV, III, II (B) I, II, III, IV (C) I, III, II, IV (D) IV, I, III, II (E) I, IV, II, III 17. If the reciprocal of y – 1. h. By the Pythagorean Theorem h 2 + x 2 = 100 and h 2 + (21 – x) 2 = 28 9 Subtract the first equation from the second. Thus, Mathematics Tests 20 9 ARCO ■ SAT II Math www.petersons.com/arco GO. + 2 = 420 + 2 = 422 42. The correct answer is (E). 43. The correct answer is (A). MATH - 175 SCIENCE - 1 42 TOTAL NUMBER OF STUDENTS = 25 0 x = No. of students taking both math and science. Mathematics