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

Cracking the SAT Subject Test in Math 2, 2nd Edition CHAPTER 12 MISCELLANEOUS DRILL EXPLANATIONS Drill 1 Logarithms 1 25 = 32, so log2 32 = 5 2 34 = 81, so x = 81 3 103 = 1,000, so log 1,000 = 3 4 43[.]

EXPLANATIONS

Drill 1: Logarithms

1 25 = 32, so log2 32 = 5

2 34 = 81, so x = 81

3 103 = 1,000, so log 1,000 = 3

4 43 = 64, so b = 4

5 Exponents and logs undo each other, so = xlogx y = y

6 70 = 1, so log7 1 = 0

7 logx x = 1

8 x to what power is x12? The 12th power, of course! logx x12 = 12

9 1.5682—use your calculator

10 0.6990—use your calculator

Drill 2: Logarithmic Rules

1 log 20

2 log5 2

3 log 3

4 log4 16 = 2

5 log 75

Drill 3: Logarithms in Exponential Equations

1 Take the log of both sides and use the power rule Your new

equation will be 4 log 2 = x log 3 Now divide both sides by log 3

and you get Using your calculator, you get x =

2.5237

2 Use the Change of Base formula:

3 Use the definition of a logarithm to convert the equation: n =

log 137 = 2.1367

4 Use the Change of Base formula:

multiplying exponents with like bases

6 Use the Change of Base formula:

8 Use the change of base formula for the left side of the equation

So you know that log4 x = 2.2619 Now use the definition of logs to see that 42.2619 = x, and x is about 23.

Make sure not to round too early so that your answer is as close

Drill 4: Natural Logarithms

8 E The graph of ln x is negative at just above x = 0 and then crosses

the x-axis; the given graph never crosses the x-axis Eliminate (A) The graph of −ln x therefore starts positive and then cross the x-axis, so eliminate (B) as well The graphs of e x and e −x will always be in positive territory; eliminate (C) and (D) and choose (E)

You can also Plug In on this problem When x = 0, the function must be negative Choices (A) and (B) are undefined when x =

0, and (C) and (D) equal 1 when x = 0 The only choice that is negative at x = 0 is (E).

18 B The equation e z = 8 converts into a natural logarithm: ln 8 = z.

To find the value of z, just type ln 8 into your calculator and see

what happens You’ll get 2.07944

23 C All you need to know to solve this one is that π ≈ 3.14 and e ≈

2.718 Then it’s easy to put the quantities in order; just remember that you’re supposed to put them in descending order

38 A To solve this equation, start by isolating the e term:

Then, use the definition of a logarithm to change the form of the equation:

Finally, use your calculator to evaluate the logarithm:

n = 3(−0.1823)

n = −0.5469

40 D The equation “ln 1.5x = 1.5” is equivalent to e1.5 = 1.5x, so solve

this second equation:

, which is (D) Another approach is to Plug In The Answers If you start with (C), you find that ln(1.5 × 0.405)

= −0.498, which is way too small, so eliminate (A), (B), and (C) Plugging in (D), however, gives you ln(1.5 × 2.988) = 1.5, which makes (D) the answer

Drill 5: Visual Perception

12 A

like the above As you can see, line l is parallel to line n, but

neither of the other two statements is true

30 D Whenever a circle intersects with a face of cube and pokes

through (as opposed to touching at just one point), it creates a circle

It’s probably easiest to get six intersections by having the sphere intersect with all six faces of the cube

It’s also possible to get five by having nearly the same picture

but moving the sphere up a little so as not to intersect with the base of the cube Seven is too complicated, though; there are only six faces of the cube, so how are there going to be seven circles?

47 D Draw it! Start by drawing circles A and B, with circle B on the

inside and the radius of A twice that of B:

You are looking for the maximum number of regions you can create with your two chords, so draw them one at a time Chord

XY, on its own, can split both circles into two regions:

Chord YZ can also split both circles into two regions Because chord YZ must share point Y with chord XY, it cannot split the regions created by XY:

Drill 6: Arithmetic and Geometric Sequences

14 E An arithmetic sequence is formed by adding a value again and

again to an original term From the second term to a sixth term

is 4 steps Going from 4 to 32 is a change of 28 in four steps, making each step an increase of 7 The fifth term must then be 7 less than the sixth term, or 25

19 E Going from the first to the seventh term of a sequence is 6 steps,

and going from 2 to 16 is a difference of 14 That makes each step an increase of , or 2.33 To find the 33rd term, plug

these numbers into the formula for the nth term of an arithmetic sequence: a33 = 2 + (33 − 1) = 2 + 74 = 76.67

26 D From the second term of a sequence to the fourth is two steps—

that is, 4 is multiplied by the factor r twice to get to 25 That means that 4 × r2 = 25 Solve for r and you’ll find that r = 2.5.

multiplying by 2.5 until you’ve counted up to the ninth term Or

figure out that the first term in the sequence is , or 1.6 Then

just plug those values into the formula for the nth term of a geometric sequence: a9 = 1.6 × 2.58 = 2441.41

34 C Because no end term is given, this is an infinite geometric

sequence It’s decreasing, not increasing, so its sum is finite Its first term is 3, and each successive term is multiplied by a factor

of Plug those values into the formula for the sum of an infinite geometric sequence, Of course, you can also approximate If you add the first four terms on your calculator, you get 4.4444 So you can eliminate (A) and (B) All the numbers you add after this are really tiny, so you’ll never reach 5.0 or an infinite size Therefore you can eliminate (D) and (E)

38 D Use the formula for the sum of a geometric sequence:

, where a1 is the first term in the sequence, r is the factor between terms, and n is the number of terms 1.5 is the second term and 4.5 is the third term, so r will be If 1.5

is the second term, the first term is You want the first

10 terms, so your equation is , (D).

Drill 7: Limits

30 B This expression factors into The binomial 4x − 5

cancels out, leaving you with This expression is no

longer undefined when x equals 1.25, so just plug 1.25 into the

expression—the result is the limit Alternatively, plug something very close to 1.25 (say, 1.24999) into the expression and use your calculator to evaluate it It should be very close to (B)

38 A This expression factors into The binomial x − 3

cancels out, leaving you with This expression is no longer

undefined when x = 3, so just plug x = 3 into the expression to

obtain the limit Alternatively, plug in a number very close to 3 (say, 2.99) into the expression and use your calculator to evaluate it The result should be very close to (A)

40 E This expression factors into The binomial x + 7

factors out, leaving you with Notice, however, that the

expression is still undefined when x = −3 The limit remains

undefined and does not exist Alternatively, plug in a number very close to −3 (say, −2.99) and evaluate the expression Notice that it doesn’t appear to be close to any of the numbers in the answer choices If you want to verify that it’s not going anywhere, try numbers even closer to −3 (say, −2.999 or

−2.9999) until you’re convinced that the limit does not exist

Drill 8: Vectors

36 D You can see from the graph that the components of are (−2,

4), and the components of are (−2, −1) So the components of must be the result of adding the components, (−4, 3) If you draw this, you can see that the magnitude of is the length of the hypotenuse of a 3:4:5 triangle

41 B Subtracting the components of a and b, you get (5, 12) So the

magnitude of c is the length of the hypotenuse of a 5:12:13

triangle

44 E

Either move u so that its tip points to the tail of v or vice-versa;

you’ll get the same result either way In any event, the angle between the vectors is now 140° Draw in the resulting vector

(we’ll label it with a length of c) from tail to head, closing off the triangle The Law of Cosines then gives us c2 = 92 + 72− 2(9)(7)

cos140° So c = 15.05 You could also approximate c, once you

draw in the resulting vector

Drill 9: Logic

28 D The basic statement here is: sophomore → not failing The

contrapositive of this statement would be: failing → not sophomore Choice (D) is the contrapositive

33 B The basic statement here is: arson → building burns The

contrapositive would be: no building burns → no arson Choice (B) directly contradicts the contrapositive

35 C The basic statement here is: not genuine → ruby The

contrapositive would be: not ruby → genuine Choice (C) paraphrases the contrapositive

40 A You are looking for the answer choice which is ALWAYS false

Reading carefully is important throughout the SAT Subject Test

in Math 2, but especially on logic questions such as this If all dogs are smelly, it doesn’t matter whether or not a certain dog is

a pet; it will always be smelly Therefore (A) is always false; it is impossible according to the information to have a dog that is not smelly Choice (B) must be true, because you know some pets are dogs, and all dogs are smelly, so eliminate it Choice (C) could be true, because you do not know about the smelliness of any non-dog pet Because it could be true, (C) is not MUST be false, so eliminate it Similarly, (D) could be true because maybe pets other than dogs are smelly, or maybe all of Ralph’s pets are dogs; eliminate (D) Choice (E) is tricky; it is certainly not the case that it is necessarily true However, it is possible with the given information that all smelly pets are dogs (though not necessary) So (E) could be true with the given information; eliminate (E)

11 C Remember that the powers of i repeat in a cycle of four Divide

51 by 4, and you’ll find that the remainder is 3 i51 will be equal

to i3, which is −i.

23 D Only (D) contains two values that do not cancel each other out

i11 = i3(i8) = i3(1) = i3 = −i, and i9 = i(i8) = i(1) = i So −i − i =

−2i, not 0.

40 B Use FOIL on the top of the fraction, and you get , or

43 D Plot the point on the complex plane; it will have a real

coordinate of 5 and an imaginary coordinate of −12 The Pythagorean Theorem will give you the point’s distance from the origin, 13

48 B Remember that |a + bi| is the distance from the origin in the

complex plane Therefore, this question is asking for the point which is closest to the origin

Drill 11: Polynomial Division

21 C Divide both sides by (x + 2) Now Plug In x = 2 g(2) = = 4,

your target number Plug In 2 for x in the answer choices, to see

which one turns into 4 Only (C) works

27 A Plug In x = 10, and use your calculator When 970 is divided by

7, you get 138.571 Well, 7 × 138 = 966, so the remainder is 4, your target number Choice (B) is wrong Choices (C), (D), and (E) are nowhere near 4, so only (A) can be correct Choice (E) is

41 B Plug In! Because you want remainders, you want to choose a

larger value of x Make x = 10 Putting x into both statements,

you find that you want the remainder when 119,716 is divided

by 120 A trick to find the remainder at this point is to do the division in your calculator: you find that 119,716 ÷ 120 = 997.633 Subtract what comes before the decimal, so your calculator reads 0.633 Then multiply by what you divided by:

0.633 × 120 = 76, which is your remainder Finally, plug x = 10

into each answer choice, looking for that one that equals 76 The only choice that works is (B)

Drill 12: Matrices

30 D In matrix multiplication, the two inner dimensions (the number

of columns in the first matrix and the number of rows in the second matrix) must be equal; the resulting matrix will have as many rows as the first matrix and as many columns as the second matrix

40 E The determinant is (1)(0) − (2)(−1) = 2

45 C You can write the matrix next to itself; then the six parts of the

formula form straight (diagonal) lines So we get 0 + (−4) + (−3) − (−2) − 0 − (−6) = 1 Alternatively, if you have a calculator that can do matrices, use your calculator

46 A Make a matrix out of the coefficients on the left-hand side of the

equations

Alternatively, if you have a calculator that can do matrices, use your calculator

Comprehensive Miscellaneous Drill

7 E To find − , find +(− ) If =(−3, −5), then, − = (−3, −5), so

+ (− )+(5, 3) + (−3, −5) = (2, −2), (E)

10 D To find the absolute value of a complex number, use the

Pythagorean Theorem: 102+(−242)=c2, 676 = c2, c = 26 Choice

(D) is correct

You can also use your calculator on this question On the TI-84, press MATH -> NUM -> abs to find absolute value, then input

the expression (i is 2ND -> “.” on the TI-84), close the

parenthesis, and press ENTER

17 C

Draw it! The graph of f(x) = (x − 2)2 + 3 is a parabola 2 units to

the right of the y-axis and 3 units above the x-axis, while g(x) =

−(x − 2)2− 4 is a downward-opening parabola 2 units to the

change value at equal rates, so the line equidistant between the two will be a horizontal line; eliminate (A), (D), and (E) The

midpoint between the two parabolas will be below the x-axis, so you need a line for which y is negative; choose (C).

out the (x − 2) terms and you are left with = Plug

in x = 2 and you find , (A)

26 E Plug In! Make x = 2, y = 3, and z = 5 The expression becomes

log The only choice that matches is (E)

Alternatively, use logarithm rules: multiplication of terms after log is the same as adding the logarithms, and division is the same as subtraction, so log The power rule lets you pull the exponent out in front of the log as

multiplication: logx3 + log y − log z = 3 log x + log y − log z.

29 E The equation log7 x = 14 tells you the same information as x =

714 Therefore, you are looking for log14714 Use the change of

31 A Plug In! Make x = 3 The long expression then becomes 2(3)5 −

3(3)4 + 6(3)3 + 23(3)2 − 25(3) + 6 = 219 You can divide that by

each answer choice Only (A) equals 73

32 D Bethany says that if the customer ordered a double-breasted

suit, then the suit won’t be ready until after Monday The contrapositive of Bethany’s statement is that if the suit is ready

on Monday or earlier, then the customer did not order a double-breasted suit Because Diana states that the suit will be ready on Monday, then we know from the contrapositive of Bethany’s statement that the customer did not order a double-breasted suit

39 D Because you need to test Statement III, plug in numbers for x

and z that are opposite, like 3 and −3, and set y equal to 0, as in

Statement II If you now multiply these matrices on your

calculator, you will see that this works Because x wasn’t set to

zero, Statement I doesn’t have to be true, and (A), (C), and (E) can be eliminated Both (B) and (D) say that Statement II

works, so test Statement III again If you make x = 3, y = 0, and

z = −4, the product matrix does not work Statement III must

always be true, so choose (D)

You can also approach this question using matrix multiplication The product of the two matrices (ignoring the given product for the moment) would be:

The first row will obviously equal the given zeroes, so you only have to make sure

that 2x + 2z = 0 and that 2y = 0 Solving each equation, you find that x = −z and y = 0, so Statements II and III are true.

44 C All the weird symbols break down into asking for the sum of an