SAT II Mathmatics level 2: Designed to get a perfect score on the exam.

406 Pages·2018·5.667 MB·English

by  John Chung

Dr. John Chung’s

  SAT II

  Mathematics

Level 2

Good Luck!

Dr. John Chung's SAT II Math Level 2 1
Dear Beloved Students, 
With this SAT II Subject Test Math Level 2 Third Edition, I like to thank all students who sent me 
email to encourage me to revise my books. As I said, while creating this series of math tests has 
brought great pleasure to my career, my only wish is that these books will help the many students 
who are preparing for college entrance. I have had the honor and the pleasure of working with 
numerous students and realized the need for prep books that can simply explain the fundamentals 
of mathematics.  Most importantly, the questions in these books focus on building a solid 
understanding of basic mathematical concepts. Without understanding these solid foundations, it 
will be difficult to score well on the exams. These book emphasize that any difficult math question 
can be completely solved with a solid understanding of basic concepts. 
As the old proverb says, “Where there is a will, there is a way.” I still remember vividly on fifth‐
grader who was last in his class who eventually ended up at Harvard University seven years later. I 
cannot stress enough how such perseverance of the endless quest to master mathematical 
concepts and problems will yield fruitful results. 
You may sometimes find that the explanations in these books might not be sufficient. In such a 
case, you can email me at [email protected]  and I will do my best to provide a more detailed 
explanation. Additionally, as you work on these books, please notify me if you encounter any 
grammatical or typographical errors so that I can provide an update version. 
It is my great wish that all students who work on these books can reach their ultimate goals and 
enter the college of their dreams. 
Thank you. 
Sincerely, 

Dr. John Chung 

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Contents

61 Tips

TIP 01  Identical Equation  Page 06 
Tip 02  Remainder Theorem  Page 07 
Tip 03  Factor Theorem  Page 08 
Tip 04  Sum and Product of the roots  Page 09 
Tip 05  Complex Number  Page 11 
Tip 06  Conjugate Roots  Page 12 
Tip 07  Linear Function  Page 13 
Tip 08  Distance from a Point to a line  Page 14 
Tip 09  Distance from a Point to a Plane  Page 15 
Tip 10  Quadratic Function   Page 16 
Tip 11  Discriminant  Page 17 
Tip 12  Circle  Page 18 
Tip 13  Ellipse  Page 19 
Tip 14  Parabola  Page 21 
Tip 15  Hyperbola  Page 23 
Tip 16  Function  Page 25 
Tip 17  Domain and Range of a Composite Function  Page 26 
Tip 18  Piecewise‐Defined Function  Page 28 
Tip 19  Odd and Even Function  Page 29 
Tip 20  Combinations of functions  Page 30 
Tip 21  Periodic Functions  Page 31 
Tip 22  Inverse Functions  Page 32 
Tip 23  The Existence of an Inverse Function   Page 33 
Tip 24  Leading Coefficient Test (Behavior of Graph)  Page 34 
Dr. John Chung's SAT II Math Level 2 3
Tip 25  Arithmetic Sequence  Page 35 
Tip 26  Geometric Sequence  Page 36 
Tip 27  Exponential Functions  Page 37 
Tip 28  Logarithmic Functions  Page 39 
Tip 29  Basic Trigonometric Identities  Page 41 
Tip 30  Circle(Trigonometry)  Page 43 
TIP 31  Reference Angles and Cofunctions  Page 44 
Tip 32  Trigonometric Graphs  Page 45 
Tip 33  Inverse Trigonometric Functions  Page 46 
Tip 34  Sum and difference of angles  Page 48 
Tip 35  Double Angle Formula  Page 49 
Tip 36  Half Angle Formula  Page 50 
Tip 37  Trigonometric Equation  Page 51 
Tip 38  The Law of Sines  Page 58 
Tip 39  The Law of Cosines  Page 54 
Tip 40  Permutation  Page 55 
Tip 41  Combination  Page 56 
Tip 42  Dividing Group  Page 57 
Tip 43  Binomial Expansion Theorem  Page 58 
Tip 44  Sum of Coefficients of a Binomial Expansion  Page 59 
Tip 45  Binomial Probability  Page 60 
Tip 46  Probability with Combinations  Page 61 
Tip 47  Heron’s Formula  Page 62 
Tip 48  Vectors in the Plane  Page 63 
Tip 49  Interchange of Inputs  Page 65 
Tip 50  Polynomial Inequalities  Page 67 
Tip 51  Rational Inequalities  Page 69 
Tip 52  Limits  Page 71 
Tip 53  Rational Function and Asymptote  Page 73 
Tip 54  Parametric Equations  Page 76 
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Tip 55  Polar Coordinates  Page 77 
Tip 56  Matrix  Page 79 
Tip 57  Inclination Angle  Page 81 
Tip 58  Angle between Two Lines  Page 82 
Tip 59  Intermediate Value Theorem  Page 83 
Tip 60  Rational Zero Test  Page84 
Tip 61  Descartes Rule of Sign  Page 85 

Practice Tests 

Test 01    Page 087 
Test 02    Page 113 
Test 03    Page 141 
Test 04    Page 167 
Test 05    Page 195 
Test 06    Page 223 
Test 07    Page 249 
Test 08    Page 277 
Test 09    Page 303 
Test 10    Page 329 
Test 11    Page 355 
Test 12    Page 381 

Dr. John Chung's SAT II Math Level 2 5
Tips

TIP 01 Identical Equation

An identical equation is an equation which is true for all values of the variable.
10x5x15x is an identical equation because it is always true for all real x.
10x515 is an algebraic equation because it is true for x1only.
 Identical equation has infinitely many solutions.
 In an identical equation, the expressions of both sides are exactly same.
Example 1: 5x55x5
Example 2: axb0 for all real values of x  axb0x0  a0 and b0

PRACTICE

1. If axb3x2 is always true for all realx, what are the values of a and b?

2. If a(x1)b(x1)x9 is true for all real values of x, what are the values of aand b?

3. If x2 2x6a(x1)2 b(x1)c is true for all real x, where a,b,and care constants, what are the values
of a,b,andc?

EXPLANATION

1. The coefficients must be equal.

2. axabxb(ab)xab, (ab)x(ab)x9
Coefficients must be equal.
ab1 and ab9
Therefore, a5, b4.

3. Since x2 2x6ax2 (b2a)xabc, then a1, b2a2,and abc6.
Therefore, a1, b4, and c1.

Answer: 1. a3,b2 2. a5,b4 3. a1,b4,c1
6
Tips

TIP 02 Remainder Theorem

When polynomial P(x)is divided by (xa), the remainder R is equal to P(a).
Polynomial P(x)can be expressed as follows.

P(x)(xa)Q(x)R
The identical equation is true for any value of x, especially xa. Therefore,

P(a)R
Example: If P(2)5, then you can say that “When polynomial P(x) is divided by (x2),
the remainder is 5.

PRACTICE

1. If a polynomial f(x)2x2 3x5is divided by (x1), what is the remainder?

2. If a polynomial g(x)x3 2x2 2x3is divided by (x1)(x2), then what is the remainder?

EXPLANATION

1. R f(1)2(1)2 3(1)54

2. g(x)(x1)(x2)Q(x)axb When divided by degree 2 polynomial, the
remainder is represented by axb
At x1 g(1)8ab
At x2 g(2)232ab Therefore, a15 and b7.

The remainder is 15x7

Also the remainder can be obtained using long-division or synthetic division.
Dr. John Chung's SAT II Math Level 2 7
Tips

TIP 03 Factor Theorem

If p(a)0, then p(x) has a factor of (xa).
Polynomial p(x) can be expressed with a factor of (xa) as follows.
p(x)(xa)Q(x), where Q(x) is quotient.
If (xa) is a factor of p(x), then the remainder after division should be 0.
Example: If p(5)0, then p(x) has a factor of (x5).

PRACTICE

1. If a polynomial P(x)x2 kx8has a factor of (x2), then what is the value of constant k?

2. If a polynomial f(x)x3ax2 bx1has a factor of (x2 1), what are the values of aand b?

EXPLANATION

1. Using the factor theorem,
P(2)0  22 2k80  k 2
2. Sincex3ax2 bx1(x1)(x1)Q(x),
f(1)ab20 and f(1)ab0
Therefore, a1 and b1.

Answer: 1. k 2 2. a1 and b1

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Tips

TIP 04 Sum and Product of the Roots

For a polynomial P(x)a xn a xn1a xn2 a xa 0
n n1 n2 1 o
a
Sum of the roots  n1
a
n
a
Product of the roots  o 1n, where n is the degree of the polynomial
a
n
Example 1: P(x)ax2 bxc0, r and s are the roots of the quadratic equation.
b
Sum of the roots: r s 
a
Difference of the roots: (rs)2 (rs)2 4rs
If r and s are real and r  s, then
b2 4c b2 4ac
rs (rs)2 4rs  rs  
a2 a a
c c
Product of the roots: rs  (1)2 
a a
Example 2: P(x) Ax3Bx2 CxD0
B
Sum of the three roots 
A
D D
Product of the three roots =  13 
A A

PRACTICE

1 1
1. If the roots of a quadratic equation 2x2 5x40are and , what is the value of  ?
 

2. What is the sum of all zeros of a polynomial function P(x)2x7 3x35x2 4?

3. What is the product of all zeros of g(x)3x7 5x33x2 x2?

4. If one of the roots of a quadratic equation is 2i, what is the equation?

Dr. John Chung's SAT II Math Level 2 9
Tips

EXPLANATION

5 4 1 1  5
1. Since   and  2, therefore    .
2 2    4

a 0
2. Because the coefficient of x6is 0, the sum of the roots  n1  0
a 2
n

a 2 2
3. Because the product of all zeros is o (1)n  17 
a 3 3
n

b c
4. The quadratic equation can be defined by x2  x 0.
a a
b b
The sum of the roots (2i)(2i)4  4
a a
c c
The product of the roots (2i)(2i)5  5
a a
Therefore, the equation is x2 4x50.

Proof
Sum of the two roots:

b b2 4ac b b2 4ac 2b b
  
2a 2a 2a a

Product of the two roots:

 2
b b2 4acb b2 4ac b2  b2 4ac
  
 2a  2a  4a2
4ac c
 
4a2 a

Answer: 1. 5 2. 0 3. 2 4. x2 4x50
4 3

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