Cracking the SAT subject test in math 2, 2nd edition

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[.]

aren’t true Choice (A) is 510 = (10) − 32 + 460 This is false,

so eliminate (A) Choice (B) is 510 = (10) + 32 + 460 This istrue, so keep (B) Choice (C) is 510 = (10) + 32 − 460 This isfalse, so eliminate (C) Choice (D) is 510 = (10) + 860 This isfalse, so eliminate (D) Choice (E) is 510 = (10) − 828 This isfalse, so eliminate (E) The correct answer is (B)

3 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 ofthe line is The correct answer is (C)

4 B 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)

5 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) = 2e x − 2, g(1.94591015) = 2e1.94591015 − 2 = 12 The correctanswer is (B)

6 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 atime For (I), since the bases of a cylinder are circles, the planecould intersect the cylinder in a way that the plane contains one

of the bases and forms a circle Since the intersection couldform a circle, (I) must be included Eliminate (B) and (D),which don’t include (I) Try (II) There doesn’t seem to be anobvious 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 planepasses through the diameters of each base, then a rectangle isformed Therefore, (III) must be included, so eliminate (A).Now, come back to (II) If the plane is parallel to the bases, acircle is formed rather than a triangle If the plane isperpendicular 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) Thecorrect answer is (C)

7 A The question asks for the distance between X and Y Use the

vertical height to form two right triangles Find the base of eachtriangle The sum of the two bases will be the distance between

atan (72.4°) = 52 Divide both sides by tan (72.4°) to get

Now, do 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° Multiply both sides by a to get atan (50.8°)

= 54 Divide both sides by tan (50.8°) to get

Add XW to WZ to get 17.13 + 44.04 =

8 B The question asks for the value of y2 Since ,

square both sides to get y2 = 342 − 302 Put the right side of theequation into a calculator to get 342 − 302 = 1,156 − 900 = 256.The correct answer is (B)

9 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 alength 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 = 5 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 (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 answeredusing POE Find points that satisfy this equation Start withpoints (2, 0, 0), (0, 2, 0), and (0, 0, 2) Because there is morethan one point, eliminate (A) These three points do not form aline, 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, soeliminate (D) Since all points in any circle must be on the sameplane, eliminate (C), as well Only one choice remains Thecorrect answer is (E)

13 B The question asks for the x-values at which g has vertical

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

= 4 Since n represents the number of nights, let this be the

denominator Determine the total cost Her stay at the hotelcosts $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 theaverage 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 (B) Choice (C) is Eliminate (C) Choice (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 thetwo 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 = 3 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 Thecorrect answer is (A)

18 B A geometric series is one with nth term ar n−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

This is close to Using either method, the correctanswer 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 ≤

= 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 <n.

Eliminate (B) The correct answer is (E)

21 B The question involves percents, so plug in 100 as the total

number of employees in the office 60% of the employeescommute over an hour each day on average Since 60% of 100 is

60, exactly 60 employees commute over an hour each day onaverage 25% of those 60 employees commute by train Since

employees who commute over an hour by train The questionasks for the probability that a randomly selected employeecommutes over an hour on average by train There are 15employees who do this out of 100 total employees, so theprobability is = 0.15 The correct answer is (B).

22 E The second smallest angle is the one opposite the second

shortest side The second shortest side is the 12 Mark the angle

opposite the 12 as x° The hypotenuse is 13 Therefore, sin

Since sinx = take sin−1 of both sides to get x ≈

67.38 The question asks for the nearest degree, so the correctanswer is (E)

24 D The question asks for the range of g(x), which is a cosine

function The range of cos (x) is −1 ≤ cos (x) ≤ 1 Anything that’s

inside the parentheses does not affect the range The coefficient

3 only affects the period and not the range Similarly, 7π onlyshifts the graph to the left and does not affect the range

Therefore, the range of cos (3x + 7π) is −1 ≤ cos (3x + 7π) ≤ 1.

The coefficient 5 changes the amplitude, which is defined ashalf the distance from the maximum and minimum values of

the cosine graph Therefore, the range of 5cos (3x + 7π) is −5 ≤ 5cos (3x + 7π) ≤ 5 The constant −2 represents a downward

y Divide both sides by 100 to get 6 = e 0.06y Take the natural

log of both sides to get ln (6) = 0.06y and 1.791759 = 0.06y Divide both sides by 0.06 to get 29.86 = y, which is closest to

III Eliminate (A) and (D) The question also states that

If a fraction is positive, then the numerator and

denominator must be of the same sign Since cosθ is negative,

of whether it can be inferred The given statement is a

conditional statement in the form if p then q In the original statement, p is a multiple of 10 and q is a multiple of 5 A

conditional statement is logically equivalent to its

contrapositive, which is in the form if not q, then not p Therefore, the contrapositive of the given statement If a is a

multiple of 10, then a is a multiple of 5 is If a is not a multiple

of 5, then a is not a multiple of 10 Go through each choice one

at a time Choice (A) is in the form if q then p A reversal of the

order of the original condition statement cannot be assumed to

be equivalent to the original (For example, consider a = 15.)

Mark this choice as N Choice (B) is the contrapositive Thecontrapositive is logically equivalent, so mark this choice with

Y Choice (C) rephrases an if p then q statements as an p implies

q This is logically equivalent, so mark this choice with Y Choice

(D) discusses a necessary condition For a statement in the form

if p then q, p is referred to as the sufficient condition and q is

referred to as the necessary condition Since (D) refers to q as

the necessary condition, mark (D) with Y Similarly, (E) refers

to p as the sufficient condition, so mark (E) with Y Four choices

are marked as Y and one choice is marked with N, so select the

30 C The question asks for how many different arrangements of 8

flowers in a line Draw 8 spaces in a line for the 8 flowers.Consider the number of possibilities for each space For the firstspace, any of the 8 flowers can be chosen, so put an 8 in the firstspace Once that flower is chosen, there are 7 possible flowersremaining for the second choice, so put a 7 in the second space.Similarly, there are 6 flowers remaining for the 3rd space, 5 forthe 4th space, 4 for the 5th space, 3 for the 6th space, 2 for the7th space, and 1 for the 8th space Because the question asks for

different orders, the order matters, so multiply the numbers in

the spaces without doing any division to get

The correct answer is(C)

31 D The question asks for what value an expression approaches as x

approaches 0 Plug in a value close to 0 Let x = 0.1 In a

calculator, compute to get 2.098 Eliminate (A), (B),and (C) as they are not close to 2.098 To determine whether

the answer could be (E), plug in a value for x that is slightly less than 0 Let x = −0.1 In a calculator, compute to get1.898 This is still close to 2, so the function does approach 2 as

x approaches 0 As an alternative, graph on a graphingcalculator The graph is a mostly smooth graph with a hole at(0, 2) Using either method, the correct answer is (D)

32 E The question asks for f(1) Replace x with 1 in f(x) to get f(1) = |7

of a tangent graph in the form y = Atan (Bx + C), the period is

equal to the expression In this equation B = 2π, so theperiod is Alternatively, graph the equationand find the distance between zeroes This distance is equal tothe period The correct answer is (B)

34 D The question asks for how long it would take Katherine to get

from Point X to Point Y along Maple Street and Elm Street

Because this is a rate question, use the formula d = rt, with d representing distance, r representing rate, and t representing

time According to the question, the rate is 15 kilometers per

hour, so substitute r = 15 to get d = 15t To get the time, find the

distance According to the question, the straight-line distancefrom Point X to Point Y is 40 kilometers However, the questionalso says that she bikes along Maple Street and Elm Street, so

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 To find the distance from intersection to

Point Y, use the Pythagorean Theorem: a2 + b2 = c2 Draw thestraight-line distance from Point X to Point Y and label it 40

Since 40 is the hypotenuse, plug in a = 10 and c = 40 to get 102+ b2 = 402 Simplify to get 100 +b2 = 1600 Subtract 100 from