Cracking the SAT Subject Test in Math 2, 2nd Edition PRACTICE TEST 1 EXPLANATIONS 1 E The question asks for the value of c in an equation that is true for all values of y, so plug in a value for y To[.]
PRACTICE TEST 1 EXPLANATIONS E The question asks for the value of c in an equation that is true for all values of y, so plug in a value for y To make the math easy on the right side, plug in y = Substitute y = into the equation to get 2(6) + 6 = (6 + 3) Simplify to get 18 = (9) Cancel the 9s on the right side to get 18 = c The correct answer is (E) B The question asks for the relationship between R and C Since there are variables in the choices, Plug In Let C = 10 If C = 10, then F = (10) + 32 = 50 If F = 50, then R = 50 + 460 = 510 Plug C = 10 and R = 510 into each choice and eliminate any that aren’t true Choice (A) is 510 = (10) − 32 + 460 This is false, so eliminate (A) Choice (B) is 510 = (10) + 32 + 460 This is true, so keep (B) Choice (C) is 510 = (10) + 32 − 460 This is false, so eliminate (C) Choice (D) is 510 = (10) + 860 This is false, so eliminate (D) Choice (E) is 510 = (10) − 828 This is false, so eliminate (E) The correct answer is (B) C To find the slope of a line, use the slope formula: slope = Let (x1, y1) = (1, 13) and (x2, y2) = (−3, 6) The slope of the line is B The correct answer is (C) The question includes three equations and three variables, so find a way to combine the equations Because the second equation provides a value for a + b, substitute this value into the equation for a + b + c to get the value of c When a + b = 4 is substituted into a + b +c = 12, the result is 4 + c = 12 Subtract 4 from both sides to get c = 8 Now, substitute c = 8 into a + c = 7 to get a + 8 = 7 Subtract 8 from both sides to get a = −1 The correct answer is (B) B The question asks for g(h(7)) On questions involving composition of functions, start on the inside and work toward the outside Find h(7) Since h(x) = ln(x), h(7) = ln(7) ≈ 1.94591015 To find g(h(7)), find g(1.94591015) Since g(x) = 2ex − 2, g(1.94591015) = 2e1.94591015 − = 12 The correct answer is (B) C The question asks for which of the three listed figures could be the intersection of a plane and a cylinder Go through one at a time For (I), since the bases of a cylinder are circles, the plane could intersect the cylinder in a way that the plane contains one of the bases and forms a circle Since the intersection could form a circle, (I) must be included Eliminate (B) and (D), which don’t include (I) Try (II) There doesn’t seem to be an obvious way to form a triangle However, don’t eliminate (II) right away in case there is a way that isn’t obvious Try (III) Determine whether a rectangle can be formed If the plane passes through the diameters of each base, then a rectangle is formed Therefore, (III) must be included, so eliminate (A) Now, come back to (II) If the plane is parallel to the bases, a circle is formed rather than a triangle If the plane is perpendicular to the bases, a rectangle is formed If the plane is at any other angle, a curved path is formed, so the result cannot be a triangle Therefore, eliminate (E), which includes (II) The correct answer is (C) A The question asks for the distance between X and Y Use the vertical height to form two right triangles Find the base of each triangle The sum of the two bases will be the distance between X and Y For reference, call the balloon point Z and the point on the ground directly below the balloon point W Look at triangle XWZ Angle X is 72.4° WZ, which is opposite the angle X, is 54 The needed side is XW, which is adjacent to the angle X Therefore, tan Multiply both sides by a to get atan (72.4°) = 52 Divide both sides by tan (72.4°) to get Now, the same for triangle YWZ Angle Y is 50.8° WZ, which is opposite angle Y, is 54 The needed side is YW, which is adjacent to angle Y Therefore, tan50.8° = 54 Multiply both sides by a to get atan (50.8°) Divide both sides by tan (50.8°) to get Add XW to WZ to get 17.13 + 44.04 = 61.17 The correct answer is (A) B The question asks for the value of y2 Since , square both sides to get y2 = 342 − 302 Put the right side of the equation into a calculator to get 342 − 302 = 1,156 − 900 = 256 The correct answer is (B) E Because there are variables in the choices, plug in Pick coordinates for point A Because the question involves distance, choose a point that can be used to make a Pythagorean triple Let A be (x, y) = (3, 4) Point A’ is (3x, 3y) = (9, 12) The distance between A and the origin is c Draw a segment vertically from A to the x-axis, forming a right triangle The distance to the x-axis is 4, and the distance along the x-axis is 3 Therefore, this is a 3:4:5 right triangle, and c = 5 Do the same for A’ Draw a vertical line from (9, 12), forming a right triangle The horizontal side has a length of 9, and the vertical side has a length of 12 Therefore, this is a 9:12:15 right triangle, and the distance from A’ to the origin is 15, which is the target (Alternatively, use the Pythagorean Theorem to determine the hypotenuse of both triangles.) Plug c = into each choice and eliminate any that aren’t 15 Choice (A) is , so eliminate (A) Choice (B) is , so eliminate (B) Choice (C) is 5, soeliminate (C) Choice (D) is 5 , so eliminate (D) Choice (E) is 3(5) = 15, so keep (E) The correct answer is (E) 10 D There are variables in the choices, so plug in Let x = 3 Then, ≈ 15.389 This is the target Go through the choices, one at a time and determine for which expression for q(x) does p(q(3)) ≈ 15.389 For (A), if q(3) = 32 − = 7, then Eliminate (A) For (B), if q(3) = 32 = 9, then Eliminate (B) For (C), if q(3) = 3, then Eliminate (C) For (D), if , then , so keep (D) For (E), if q(3) = then so eliminate (E) The correct answer is (D) 11 C The question asks for sin (90° − x) There are two possible approaches to this problem One is to find the value of x by using the inverse cosine function If cosx = 0.6, then take the inverse cosine of both sides to get x = cos−1(0.6) ≈ 53.13 Therefore, sin (90° − x) ≈ sin (90° − 53.13°) ≈ 0.6 Alternatively, use the identity cosx = sin (90° − x) Therefore, if cosx = 0.6, then sin (90° − x) = 0.6 Using either method, the correct answer is (C) 12 E In xyz-coordinates an equation with the graph x2 + y2 + z2 = r2 is a sphere with radius r and center at the origin However, if this equation is not familiar, the question can still be answered using POE Find points that satisfy this equation Start with points (2, 0, 0), (0, 2, 0), and (0, 0, 2) Because there is more than one point, eliminate (A) These three points do not form a line, so eliminate (B) These points could make a circle, plane, or sphere, so Plug In more points Try (−2, 0, 0), (0, −2, 0), and (0, 0, −2) These six points are not on the same plane, so eliminate (D) Since all points in any circle must be on the same plane, eliminate (C), as well Only one choice remains The correct answer is (E) 13 B The question asks for the x-values at which g has vertical asymptotes A function has a vertical asymptote for x-values at which the denominator is and the factor that makes the denominator equal to cannot be canceled out with the numerator Since the numerator of g cannot be factored, only worry about where the denominator is Set x2 − 6x + = Factor the left side, finding two factors of with a sum of −6 These are −3 and −3 Therefore, the factored form of the equation is (x − 3)(x − 3) = 0 Set both factors to 0 and solve In both cases, the equation is x − 3 = 0, so add to both sides to get x = 3 The correct answer is (B) 14 C The question asks for the value of k, which is the constant term in the polynomial The constant term represents the yintercept According to the graph, the curve crosses the y-axis between and 40 Only one choice is between these The correct answer is (C) 15 C The question asks for cscx, which is equivalent to Substitute the value of sinx given by the question to get csc The correct answer is (C) 16 D The question asks for the average cost for each night, which is There are variables in the choices, so plug in Let n = Since n represents the number of nights, let this be the denominator Determine the total cost Her stay at the hotel costs $80 per night for four nights for a total of 4 × $80 = $320 Furthermore, the three-night hotel stay for her friend costs 3 × $80 = $240 She must also pay airfare, which is $170 Therefore, the total cost is $320 + $240 + $170 = $730, and the average cost is This is the target number Plug n = 4 into each of the choices, and eliminate any that aren’t 182.5 Choice (A) is Eliminate (A) Choice (B) is Eliminate Eliminate (B) (C) Choice Choice (C) is (D) is Keep (D) Choice (E) is Eliminate (E) The correct answer is (D) 17 A The set of points equidistant between two points is the perpendicular bisector of the segment whose endpoints are the two points The segment with endpoints (0, 0) and (6, 0) lies on the line y = 0 and has midpoint (3, 0) Since a line in the form y = c, where c is a constant, is a horizontal line, the perpendicular line must be a perpendicular line the form x = k, where k is a constant To be a bisector, the line must go through the midpoint, which is (3, 0), so the line is x = Alternatively, sketch the two points and sketch each of the choices Choices (B), (C), (D), and (E) all have points that are clearly closer to (0, 0) to (6, 0) and vice versa, so they can be eliminated The correct answer is (A) 18 B A geometric series is one with nth term arn−1, where a represents the first term, and r represents the common ratio, i.e the number by which each term must be multiplied to get the next term If 0 < r < 1, then the sum of an infinite series can be found using the formula The first term is , fill in this for a To find the common ratio, set up the equation Multiply both sides by 9 to get Therefore, the sum is Alternatively, find the sum of the known terms on a calculator This is close to Using either method, the correct answer is (B) 19 D Simplify the inequality by combining like terms Subtract a from both sides to get −b ≥ b Add b to both sides to get 0 ≥ 2b Divide both sides by 2 to get 0 ≥ b This can be rewritten as b ≤ The correct answer is (D) 20 E The question says must be true, so eliminate any choice that can be false It is unknown whether g is increasing, decreasing, or neither, so attempt as many cases as is needed to eliminate four choices Let g be an increasing function Try g(x) = x In this case, if g(m) >g(n), then m > n Eliminate (C) and (D) Furthermore, it could be that m = 2 and n = 0 In this case, mn = 0, so eliminate (A) as well Now try g(x) = −x In this case, if g(m) > g(n), −m > −n Divide both sides by −1 to get m 1 Therefore, T2 = T1 + 2 = 1 + 2 = 3 Continue to T11 T3 = T2 + 3 = 3 + 3 = 6 T4 = T3 + 4 = 6 + 4 = 10 T5 = T4 + 5 = 10 + 5 = 15 T6 = T5 + 6 = 15 + 6 = 21 T7 = T6 + 7 = 21 + 7 = 28 T8 = T7 + 8 = 28 + 8 = 36 T9 = T7 + 9 = 36 + 9 = 45 T10 = T9 + 10 = 45 + 10 = 55 T11 = T10 + 11 = 55 + 11 = 66 The correct answer is (C) 40 E The question asks which statements are true about the function f(x) = x3 + x2 − 16x + 12 are true If a graphing calculator is available, graph the function The graph crosses the x-axis three times, so statement I is true Also, the graph has a relative minimum at (2, −8) and has no points for which y < −8 on the positive side of the x-axis Therefore, II and III are also true If no graphing calculator is available, then factor to determine the number of solutions One method is to test factors of 12 for a solution The factors of 12 are 1, 2, 3, 4, 6, and 12 Try x = Since f(1) = 13 + 12 − 16(1) + 12 = −2, x = 1 is not a solution Try x = Since f(2) = 23 + 22 − 16(2) + 12 = −8, x = is not a solution Since f(3) = 33 + 32 − 16(3) + 12 = 0, x = 3 is a solution and (x − 3) is one factor Rewrite x3 +x2 − 16x + 12 in a way that makes it easy to factor (x − 3) First, rewrite it as x3 − 3x2 + 3x2 + x2 − 16x + 12, and factor the first two terms to get x2(x − 3) + 4x2 − 16x + 12 Now, rewrite it as x2(x − 3) + 4x2 − 12x + 12x − 16x + 12, and factor 4x2 − 12x to get x2(x − 3) + 4x(x − 3) − 4x + 12 Now factor −4x + 12 to get x2(x − 3) + 4x(x − 3) − 4(x − 3) Factor (x − 3) to get (x − 3)(x2 + 4x − 4) Now determine the number of factors of x2 − 4x + 4 To determine the number of factors of a quadratic in the form ax2 + bx + c, use the discriminant: b2 − 4ac If the discriminant is positive, there are two real solutions If the discriminant is 0, there is one real solution If the discriminant is negative, there are no real solutions In the quadratic x2 + 4x − 4, a = 1, b = 4, and c = −4, so b2 − 4ac = 42 − 4(1)(−4) = 32 > 0 Since the discriminant is positive, x2 + 4x − 4 has two solutions and (x − 3)(x2 + 4x − 4) has three solutions Thus, (I) is true Eliminate (B) and (D) Test (II), which says that f(x) ≥ −8, for all x ≥ 0 Set up x3 + x2 − 16x + 12 ≥ −8 Get one side equal to 0 Add 8 to both sides to get x3 + x2 − 16x + 20 ≥ Similarly, factor the polynomial on the right by testing the factors of 20: 1, 2, 4, 5, 10 and 20 Since 13 + 12 − 16(1) + 20 = 6, x = 1 is not a solution Since 23 + 22 − 16(2) + 20 = 0, x = 2 is a solution, so (x − 2) is a factor Rewrite x3 + x2 − 16x + 20 as x3 − 2x2 + 2x2 + x2 − 16x + 20 = x2(x − 2) + 3x2 − 16x + 20 = x2(x − 2) + 3x2 − 6x + 6x − 16x + 20 = x2(x − 2) + 3x(x − 2) − 10x + 20 = x2(x − 2) + 3x(x − 2) − 10(x − 2) = (x − 2) (x2 + 3x − 10) Factor to get (x − 2)(x − 2)(x + 5) = (x − 2)2(x + 5) Therefore, the expression x3 + x2 − 16x + 20 = 0 when x = 2 and x = −5 The statement only refers to what happens when x ≥ 0, so ignore x = −5 Since x3 + x2− 16x + 20 = 0, when x = 2, x3 + x2 − 16x + 12 = f(x) = −8, when x = Determine what happens to the left and right of x = 2 If x = 1, then f(1) = 13 + 12 − 16(1) + 12 = −2 ≥ −8 If x = 3, then f(3) = 33 + 32 − 16(3) + 12 = 0 ≥ −8 Therefore f(x) ≥ −8, whenever x ≥ 0, so (II) is true Eliminate (A) and (C) Only one choice remains The correct answer is (E) 41 D The question asks which of the following could be a portion of the graph of gh, the product of the graphs of g and h Look at the graphs in pieces When x is negative, the graphs of g and h are both negative Therefore, the product must be positive Eliminate (A) and (C), which are positive when x is negative If x is positive, g is positive but h is negative, so the product must be negative Eliminate (B), which is positive when x is positive If x = 0, the f and g are both 0, so the product is 0, and the graph of the product must go through the origin Eliminate (E), which does not go through the origin The correct answer is (D) ... The list with the greatest standard deviation is the list in which the numbers are farthest apart The numbers in (D) have the greatest separation The correct answer is (D) 26 E First, plug in the known values into the equation The initial investment is $100, so plug in I = 100... don’t use the straight-line distance for d Instead use the sum of the distances from Point X to the intersection and from the intersection to Point Y for d The distance from the Point X to the intersection is 10... Because it is between 3 and 4 hours, eliminate (A), (B), and (C) The remaining two choices only differ by the remainder in minutes To get the remainder in minutes, set up the proportion Cross multiply to get x ≈ 15 minutes Therefore,