Citation preview

SAT

PRACTICE BOOK WITH SOLUTIONS

PREFACE

"The struggle is real." That is how I describe my many years of teaching Mathematics! From finding reference materials that contain almost everything I need to preparing students for international assessments like SAT, PSAT, CIE and IB have been quite a huge challenge because textbooks and other resources vary accordingly. This practice book contains more exercises with solutions based on blended SAT Math curriculum. I have compiled as many questions as possible with corresponding solution for each type. SAT and PSAT exam preparation exercises are the central components and main focus of this workbook. I want users of this book to have an all-in-one go-to material where they can find everything they need to enhance their Math skills and eventually earn mastery of the topics. Preparing students for SAT was my inspiration on why I came up with designing this practice book. My objective is to be able to help students achieve 600 points and above. This book is special as it contains questions with handwritten solutions. This practice book is a product of my many years of teaching experience, and it took me a year to finally complete it. I am sure this will be a big help for students. Though this has a simple structure, I still suggest that students first review the topics from different resources, then take hold of this book and solve all practice questions to attain the target score.

Arif Yetik, MA

CHAPTERS Exponents and Square Roots Test 1 ……………………………………………………….

1

Test 2 ……………………………………………………….

5

Test 3 ……………………………………………………….

9

Test 4 ……………………………………………………….

13

Test 1 ……………………………………………………….

17

Test 2 ……………………………………………………….

21

Test 3 ……………………………………………………….

25

Test 4 ……………………………………………………….

29

Polynomials

33

Test 2 ……………………………………………………….

37

Test 3 ……………………………………………………….

41

Test 4 ……………………………………………………….

45

Test 5 ……………………………………………………….

49

Functions and Quadratic Functions Test 1 ……………………………………………………….

53

Test 2 ……………………………………………………….

57

Test 3 ……………………………………………………….

60

Test 4 ……………………………………………………….

63

Test 5 ……………………………………………………….

66

Test 6 ……………………………………………………….

70

74

Solutions ……………………………………………………….

76

1

1

Exponents & Square Roots

3 2

If a = 4 2 and 2a = 4b , what is the value of b ? A) B) C) D)

8 16 32 64

2

Which of the following is equal to b 5 ? 2

A)

b5

B)

b2

5

C)

2

b5

D)

5

b2

4 2

x If a 2 = a 54 , a > 1 and x - y = 6 , what is the ay value of x + y ?

In the equation above, if n = 6 , what is the value of m?

A) B) C) D)

A) B) C) D)

3 9 24 27

m+3 -n =0

3 9 27 33

1

CONTINUE

Test 1

1

5

Exponents & Square Roots

7 3

Which of the following is equal to 25 4 ? A)

3

25

B)

5 5

C)

4

D)

5

1

-3 2 Which of the following is equal to a × b ? -4 a ×b a A) 2 b

a b

B)

5

1

C)

a

-7

a ×b

D)

b 2

b

8

6 If

x + 16 = 25 , what is the value of x ?

A) 1 B)

9

C) 6 D) 9

If a A)

-

1 3

= m, where a > 0, , what is the value of m ? 3

1 a

B)

3

a

C)

3

a2

D)

2

a -3

2

CONTINUE

Test 1

1

9

Exponents & Square Roots

11 25a 2

If a>0, which of the following is equivalent to the given expression? A) 50a4 B) 10a C) 5a2 D) 5a

10

1

Which of the following is equivalent to ( 4t ) 3 ? A)

3

B)

23 t

C)

1 12t 3

D)

( 4t )

4t

3

12 y2 = y Which of the following values of y is NOT solution to the equation above?

A) 2 B) 1 C) 0 D) -2

If a3 = b , where b > 0, what is the value of a in terms of b? 1

A)

b7

B)

b

5 12

C)

b12

D)

b4

1

3

3

CONTINUE

Test 1

1

13

Exponents & Square Roots

15 Which of the following is equal to A)

1 2 +1

?

2 1 2 2

B)

2 -1

C)

2 1 + 2 2

D)

2 +1

14

If y is positive integer and x = 2 y +3 + 2 y , what is 2 y + 2 in terms of x? A)

x 9

B)

4x 9

C) D)

13x 36x

16 If x3 = 30, and y2 = 10 , what is the value of ( x × y )6 ? A) B) C)

9 ´104

D)

9 ´105

3 ´104 3 ´105

What is the value of 5 8 + 6 32 ? A)

10 2

B)

11 40

C)

34 2

D)

36 2

4

CONTINUE

Test 1

1

1

Exponents & Square Roots

3 What is the value of 4 50 - 3 32 ?

What is the value of 9 24 ? 4 27

A)

8 2

B)

12 40

C)

20 2

D)

32 2

2

A)

3 2

B)

3 2 2

C)

2 2

D)

2 2 3

4 If

16b = 2a +5 , what is the value of b ?

A)

32 × 2

B)

2a+ 2

C)

2 × 2a+ 2

D)

2a +1

a

-x - y = 5 In the equation above, if y = 2 , what is the value of x? A) -49 B) -27 C) 27 D) 49

5

CONTINUE

Test 2

1

Exponents & Square Roots

7

5 4x Which of the following is equal to 25 ? 1253 x

A)

1 5

B) C)

1 5x 5

D)

5x

6

Which of the following is equivalent to (a + a + a + a) × (a + a) ? 4-1 A) 16 × a B) 32 × a C)

16 × a2

D)

32 × a 2

8 If 8 = 4 , what is the value of x ? 12

A) B) C) D)

x

9 12 18 36

Which of the following is equivalent to 32 + 4 18 ? 8

A)

B) 2 2 C) 8 D)

2

8 2

6

CONTINUE

Test 2

1

Exponents & Square Roots

11

9 If a 2 = 9, b2 = 16, and ( a + 3) × ( b - 4 ) ¹ 0 what is the value of a + b ?

If 1 = 3 3 what is the value x? x

A) -2 B) -1 C) 1 D) 2

A)

3

B)

10

3 2 2 3 -

C)

2 3

D)

3 2

12 If

1 = x-2

2 what is the value x?

A)

2+4 2

B)

2 -4 2

C)

2 -2 2

D)

2+2 2

If ( 3,000 ) × ( 4,000 ) = 1.2 ´10m what is the value m? A) 6 B) 7 C) 8 D) 9

7

CONTINUE

Test 2

1

Exponents & Square Roots

15

13 If 2a = 5 what is the value 5 × 4a + 2a ? A) B) C) D)

If

x 4+ 7

55 80 125 130

A) B) C) D)

=

4 - 7 , what is the value of y

3

( xy ) 2 ?

9 27 54 81

16

14

(

If 3 + 2 2

)

2

= a + b 2, where a and b are

integers, what is the value of A) B) C) D)

a +b?

17 23 29 37

If 4x = 25 , what is the value of 2 x- 2 ? A)

5 4

5 2 C) 5 D) 50

B)

8

CONTINUE

Test 2

1

1

Exponents & Square Roots

3 If f ( a, b ) = 2a2 , where a and b are positive b

integers, what is the value of f ( 2a, 4b ) ? f ( a,3b )

A) B) C) D)

9 16 !

What is the smallest integer value of x such that 10- x < 0.000036 ? A) B) C) D)

4 5 6 7

" 9 4 9 2

4

2

2 = 4 8 , what is the value of m ? 2- m

If a is positive integer, and 3a = b , which of the following express 3b2 in terms of a?

If

A) 3 × 3a

A)

B)

5 2

B)

3

C)

7 2

D)

9 2

C) D)

32 a a+ 2

3

2 a+1

3

9

CONTINUE

Test 3

1

5

Exponents & Square Roots

7 If 4 50 - 3 8 = n 2 , what is the value of n ? A) B) C) D)

6 12 14 20

6

Which of the following is equivalent to 2 ? 3-1 × ( a + a + a ) × ( a + a )

A)

1 a

B)

1 a2

C)

a

D)

a2

8 If 812 = 4x , what is the value of x ? A) B) C) D)

9 12 15 18

Which of the following is equivalent to #\$%&'(\$√*#' *\$√&

A)

y

B)

5x y

C)

5 y

D)

5 3y

10

?

CONTINUE

Test 3

1

Exponents & Square Roots

11

9 x If 2 x - y = 5 what is the value of 4 ? 2y A) 8 B) 16 C) 32 D) 64

10

If m = 11 + 13 + 15 + ... + 199 what is the value m? A) B) C) D)

25 49 50 99

12 If x

-

1 3

= 2 what is the value x?

A)

-8

B)

1 8

C) - 1 8

D)

8

If 43 x - 2 = A) B) C)

1 8

x +5

what is the value x?

-11 -9

11 9

D) - 11 9

11

CONTINUE

Test 3

1

13

Exponents & Square Roots

15 If

3y+1 = 10 what is the value 3

y- 2

?

If

x x x = x m , what is the value of m ?

A)

3 10

A)

B)

10 3

7 4

B)

3 4

C)

10 27

C)

3 2

D)

27 10

D)

1

14

16 If a b = 64, value a × b ? 2 3

A) B) C) D)

and a3b2 = 16 what is the

2 4 5 10

If 5 x + 3 = 4 5 , what is the value of x ? A)

-

B)

1 5

C) D)

5

1 5

10

12

CONTINUE

Test 3

1

1

Exponents & Square Roots

3

2n +10 32 m × 2n 32 m -10

Assuming n is nonzero, which of the following expressions is equivalent to the expression above? A)

210

B)

310

C)

610

D)

620

If

3x 3 ´ 3 = 81 , and y = 9m -1, 3 x

y

m

what is

the value of x in terms of m? A) B) C) D)

6m - 3 6m - 2 3m - 2 3m - 1

4

2 If

(x )

a + b a -b

the value of A) B) C) D)

= x64 , and a + b = 16 , what is

a -b?

4 8 12 16

If 6 3 x + 13 = 43, what is the value of x ? A) B) C) D)

5 25 100 125

13

CONTINUE

Test 4

1

5

Exponents & Square Roots

7 If a = 3 2 and 2a= 3x , what is the value of x? A) B) C) D)

6 12 24 36

Which of the following is equivalent to (4 × x 2 × y 3 )3 (8 × x3 × y ) ? × (2 × x × y 2 ) 4 (2 × x × y ) 2

A)

x2 2

B)

x3 4

C)

2x2 D) 8𝑥 !

6

8 If A) B) C) D)

3

y 2 + 4 = 5 , what is the value of

y?

-9 -11 -12 -121

Which of the following is equivalent to (-1)9 × (-1)10 + (-1) 21 ? (-1)11 × (-1)8

A) B) C) D)

-2 -1 1 2

14

CONTINUE

Test 4

1

9

Exponents & Square Roots

11 If 2t of t?

× 43 = 8s what is the value of s in terms

A)

t+6

B)

t+6 3

C)

t -6

D)

t -6 3

What is the value æç 1 + 1 ö÷ × 8 ? ç ÷ è

A)

3

1

B)

2

C) 2 D) 4

12

10 If A) B) C) D)

4x +3 = 162 x what is the value x? 1 2 4 8

If A) B) C) D)

3

3 x 2 - 11 = 4 what is the value x? 3 5 8 16

15

CONTINUE

Test 4

1

13

Exponents & Square Roots

15 What is the value A) B) C) D)

21 + 13 + 6 + 9 ?

4 5 6 7

What is the value A)

10

B)

2

C)

10 10

D)

0.04 + 0.09 + 0.25 ?

1

16

14 What is the value A) B) C) D)

13 + 2 ×

13 - 2 ?

3 9 13 15

What is the value 5 3 × 45 × 27

A)

7 9

B)

20 3

C)

25 36

D)

18 25

16

12

5 ? 108

CONTINUE

Test 4

2

1

Polynomials

3 Which of the following is equivalent to 1 ? 1 1 x-2 x-3 A) B)

x2 - 5x + 6 x +1 2 - x + 5x - 6

C)

x +1 x2 - 5x + 6

D)

x -1 x - 5x + 6

Which of the following is equivalent to

9𝑥 ! − 12𝑥 " 𝑦 " + 4𝑦 ! ?

A)

( 3x - 2 y )

B)

( 3x

2

- 2 y2 )

C)

(9x

2

- 4 y2 )

D)

(9x - 4 y )

4 2

2

4

2

4

2 For a polynomial P( x) , the value of P(-2) is 3. Which of the following must be true about P( x) ?

Which of the following is equivalent to 4 x - 3 ? x +1 A)

1

A)

x - 2 is a factor of P( x)

B)

4-

B)

x + 2 is a factor of P( x)

1 x +1

C)

x + 3 is a factor of P( x)

C)

4-

7 x +1

D)

4 - 7 ( x + 1)

D) The remainder when P( x) is divided by x + 2 is 3.

17

CONTINUE

Test 1

2

5

Polynomials

7

( -4 x

2

+ 7 x - 3) - 3 ( -2 x 2 + 3x - 1)

If the expression above is rewritten in the form of polynomial ax2 + bx + c , where a, b, and c are constant, what is the value of a? A) -10 B) -2 C) 2 D) 3

4x + 5

( x - 2)

2

-

4 x-2

If the expression above is rewritten in the form of polynomial

t

( x - 2)

2

, where t is positive

constant and x ¹ 2 . What is the value of t? A) -13 B) -3 C) 3 D) 13

8

6 If x2 + y 2 = m, and xy = n , which of the following is equivalent to 4m - 8n ? A) ( 4 x + 4 y )2 B) ( 4 x - 4 y )2 C) ( 2 x + 2 y )2 D) ( 2 x - 2 y )2

Which of the following is equivalent to 2 (1.2x - 1.5) - ( 4.44 x2 + 1.25) ? A) B) C) D)

−3𝑥 " − 3.6𝑥 + 1 -3x2 + 3.6 x 3x2 - 3.6 x + 1 -3x2 + 3.6 x + 1

18

CONTINUE

Test 1

2

9

Polynomials

11 Which of the following is equivalent to 3x 2 + 4 x ? x-2 A) 3x - 10 x x-2

B)

3x + 10 +

C) D)

3x

20 x-2

Which of the following is equivalent to 4 ( x - 3) + 11 ? A) B) C) D)

x -1 12 x + 11 -8 x + 11 4x -1

x-2

12

10 Which of the following is equivalent to 2

æx ö ç - y÷ ? è2 ø

A) B)

x2 + y2 4 2

x + y2 2

C)

x2 - xy + y 2 4

D)

x2 + xy + y 2 4

Which of the following is equivalent to x2 + 10 x + 17 ? A)

( x + 5)

B)

( x + 5)

C)

( x + 5)

D)

( x + 5)

19

2

+ 17

2

- 17

2 2

-8 +8

CONTINUE

Test 1

2

13

Polynomials

15 Which of the following is equivalent to x2 + 4 x + 3 ? x2 - x - 2 A)

x+3 x+2

B)

x+3 x-2

C)

x +1 x-2

D)

x -1 x+2

( 5019 - 163x ) + 23( 7 x 2

2

- 217 )

The expression can be rewritten as mx 2 + n , where m and n are positive constant. What is the value of m + n ? A) B) C) D)

-30 -28 26 30

16

14 The expression 1 x 2 - 3 can be rewritten as 5 + (𝑥 − 𝑚)(𝑥 + 𝑚 ), ,

where m is positive

constant. What is the value of m?

A) -3 B) 10 C) 15 D)

(5 - x ) - 2 (7 - x 2

2

+ 3x ) + 6 x

Which of the following is equivalent to expression above? A) 𝑥 " − 9 B) - x2 - 9 C) - x2 + 12 x - 9 D) - x2 - 12 x - 9

15

20

CONTINUE

Test 1

2

1

Polynomials

3 Which of the following is equivalent to 5 + 2? 3x - 1

A)

6x - 2 3x - 1

B)

6x + 1

C) D)

If x + 4 is a factor of the expression x2 + mx - 12 , what is the value of m? A) -3 B) 1 C) 3 D) 4

6x + 3 3x - 1 6x + 7 3x - 1

4

2

( 3x - 2 )( ax + 5) - 6 x

2

+ 7 = bx - 3

In the expression above, a and b are constant. What is value of a? A) B) C) D)

2 3 9 11

Which of the following is equivalent to 4 x - 3 ? 12 A) B) C) D)

x-3 4 4x -1 3 x 1 3 4 x 1 4 3

21

CONTINUE

Test 2

2

Polynomials

7

5 If the polynomial

P ( x ) , has factors of

4x 12 ÷ x - 3 3x - 9

6, ( x - 4 ) , and ( x + 5) , Which of the following must also be a factor of P ( x ) ? A) 6 x2 - 24 B) 6 x2 + 30 C) x2 - 8x + 16 D) 3x2 + 3x - 60

Which of the following is equivalent to the expression above, given that x ¹ 3 ? A) x B) 2x C) x - 3 D) x

3

6

8 Which of the following is a factor of x3 - 8 ?

Which of the following is NOT equivalent to 8a 2 ?

A) x 2 - 2 B) 𝑥 " + 2𝑥 + 4 C) 𝑥 " − 2𝑥 + 4

A)

4a 2 + 4a 2

B)

( 2a 2 )

C)

8a ( a )

D)

( 4a ) × ( 4a )

D) x + 2

22

2

CONTINUE

Test 2

2

9

Polynomials

11 What

is the coefficient of x 2 when 1 1 21x 2 - x + 3 is multiplied by the 10 x - ? 5 7 A) -5 B) -1 C) 1 D) 5

4 3 Which of the following is equivalent to x + x ? x3 + x 2

A) B) C) D)

x7 x5 2x

x 1

12

10 Which of the following is equivalent to "

#

. − 2𝑦0 ? " 2

A)

x + y2 4

B)

x2 + y2 2

C)

x2 - xy + y 2 4

D)

x2 + xy + y 2 4

Which of the following is factor of polynomial of 𝑎" − 12𝑎 + 9 ? A) B) C) D)

(𝑎 − 6)" − 9 (𝑎 − 6)" − 27 (𝑎 + 6)" − 9 (𝑎 + 6)" + 27

23

CONTINUE

Test 2

2

13

Test 2

Polynomials

15 Which of the following is equivalent to 16𝑎\$ + 8𝑎" 𝑏" + 𝑏\$ ? A)

( 2a - b )

B)

( 2a - b ) ( 2a + b )

C)

( 4a + b ) ( a - b )

D)

( 4a

4 2

2

2

+ b2 )

2

2

2

14

Which of the following is equivalent to ab - a ? ab + a 2 A) - a b B) - b a !"# C) #\$! !\$# D) #"% 2

16 1 x? Which of the following is equivalent to 1 2+ x 2 x 1 A) 2x + 1 2-

B)

2x + 1 2x -1

C)

4x2 - 1 x2

D)

1

Which of the following is equivalent to 4 x 4 - 4 ? A)

4 ( x 2 - 1)

B)

4 ( x 2 + 1)

C)

4 ( x 3 - 1) ( x + 1)

D)

4 ( x 2 + 1) ( x - 1)( x + 1)

24

2

2

CONTINUE

2

Test 3

Polynomials

3

1 Which of the following is equivalent to 2 + 3 ? x

A) B) C) D)

4

5 4x 5 4+ x 8 3x 8 + 3x 4x

2 (5x ) - 5( 2 x ) 2

If x ¹ 0 , what is the value of

2

( 2 x ) × ( 3x )

? A) -1 B) 1 C) 5 D) 5x

4

2 3x 2 + 4 7 =m+ x -1 x -1 In the expression above, m is a constant. What is the value of m?

A)

3x - 5

B) C)

3x - 4

D)

3x + 4

3x + 3

Which of the following is equivalent to A) B) C) D)

&'"( '\$)

?

6-5 1+ 2 17 6x+2 5 6x+2 6 5 x+2 x+2

25

CONTINUE

2

5

Polynomials

7 5a 2 - 4a + 1 n = 5a + 6 + a-2 a-2 In the expression above, n is a constant. What is the value of n?

A) B) C) D)

2a - 1 11

13 11x + 13

6

6 x2 + 5x + 2 1 = +m 2x + 1 2x + 1

In the expression above, m is a constant. What is the value of m? A) B) C) D)

3x + 1 3x - 1

3x 2 + x 2 x2 + x + 2

8 P ( x ) = 2 x3 - kx 2 + 4 x - 5

In the expression above, k is a constant. If P(x) is divisible by 𝑥 + 1, what is the value of k? A) B) C) D)

Which of the following is equivalent to

-13

A) B)

2

-11

C)

4-

8 t+2

D)

4+

8 t-2

11

13

4t ? t-2

4 - 2t

26

CONTINUE

Test 3

2

9

Polynomials

11

3x2 + 5x - 6 = ( ax + b )( x + 2) + c

P ( x ) = 2 x3 - kyx 2 + 5xy + 2 y - 2 In the expression above, k is a constant. If P(x) is divisible by 𝑥 − 1, what is the value of 𝑘?

In the expression above, where a, b and c are constant, what is value the of a+b+c?

A)

-7

A)

-5

B)

-7 y

-2

C) D)

7

B) C) D)

7y

10

0 8

12

2 x 2 - 4 x - 3 = A ( x - 1) + B In the expression above, A and B are constant, what is the value of A in terms of x? A) B) C) D)

2𝑥 − 2 2𝑥 + 2 -2𝑥 + 2 -5

P ( x ) = mx 4 + nx3 - 3x 2 + 5x In the expression above, m and n are constant. If P(x) is divisible by 𝑥 − 1, and 𝑥 + 1, what is the value of m?

A) B) C) D)

-2 1

3 5

27

CONTINUE

Test 3

2

13

Polynomials

15 For a polynomial P ( x ) , P ( -2 ) = 0 . Which of the following must be true about P ( x ) ?

P ( x)

A)

-2x is a factor of

B)

-2 x + 2 is a factor of

C)

x - 2 is a factor of

P ( x)

D)

x + 2 is a factor of

P ( x)

P ( x)

14

3 3 Which of the following is equivalent to x2 y - xy 2 ?

x y - xy

A)

x+ y

B)

x- y

C)

xy

D)

x+ y x- y

16 If the polynomial P ( x ) is divisible by 𝑥 − 5,

For a polynomial P ( x ) , P æç 3 ö÷ = 0 . Which of

which of the following could be P ( x ) ?

the following must be a factor of P ( x ) ?

A) 2𝑥 " − 7𝑥 − 15 B) 2𝑥 " + 7𝑥 − 15 C) 2𝑥 " − 7𝑥 + 15 D) 2𝑥 " + 7𝑥 + 15

è4ø

A) B) C) D)

3x + 4 3x - 4 4x + 3 4x - 3

28

CONTINUE

Test 3

2

1

Test 4

Polynomials

3 1 A B = + x + 3x + 2 x + 2 x + 1

Which of the following is equivalent to

6 xy ( 4 x - 5 ) - 3x ( 8 y - 7 xy ) 3xy

A) B) C) D)

2

?

-18 15𝑥 − 18 18 15𝑥 + 18

2

In the expression above, A, and B are constant. What is the value of A × B ?

-2 B) -1 C) 1 D) 2 A)

4

2 x2 + 3x + ax + b = ( c - 2 ) x3 + dx 2 + c

3x - 2 A B = + ( x - 3)( x + 1) x - 3 x + 1

In the expression above, a, b, c, and d are constant. What is the value of a + b + c + d ? A) B) C) D)

3 4 6 8

In the expression above, A, and B are constant. What is the value of A ? B 5 A) 4 B) 7 5 C) 7 4 D) 1 4

29

CONTINUE

2

Test 4

Polynomials

7

5 If a - b = 4, and a × b = 3 , where a and b are real numbers. What is the value of a 2 + b2 ? A) B) C) D)

10 16

22 28

6

If b - a = 8, and a - c = 10 , where a, b and c are real numbers. What is the value of ab + bc - ac - b2 ? A) 144 B) 108 C) 88 D) -144

8 If a + b = 4, and a × b = 3 , where a and b are real numbers. What is the value of a3 + b3 ? A) B) C)

52

D)

56

28 48

Which of the following is equivalent to t -1 ÷ t - 1? 1 -1 t A) B)

-2

0

C)

t 1- t

D)

t

30

CONTINUE

2

9

Test 4

Polynomials

11 P ( x ) = x3 + 4 x 2 + 2mx - 5 In the expression above, m is a constant. If P(x) is divisible by 𝑥 − 1, what is the value of m?

A) B) C) D)

-2 -1

0 1

10

4𝑥 " + 10𝑥 + 2 = (𝑎𝑥 − 𝑏)(𝑐𝑥 + 3) +8 In the expression above, where a, b and c are constant, what is value the of a+b+c? A) 3 B) 5 C) 6 D) 9

12

7 x2 + x + 5 = A ( x + 2) + B In the expression above, A and B are constant, what is the value of B? A) B) C) D)

7𝑥 − 13 7𝑥 + 13 13 31

P ( x ) = 2 x3 + ax2 + 5x + 2 In the expression above, a is a constant. If P(x) is divisible by x + 2 , what is the value of a? A) B) C) D)

-6 -4

4 6

31

CONTINUE

2

Test 4

Polynomials

15

13 Which of the following is equivalent to a 2 - b2 a 2 - b2 + =? a -b a+b

Which of the following is equivalent to x - 1 x 2 + 3x + 2 ? × x+2 x2 - 1

A) B)

A) 1

C) D)

a 2a b 2

B) C) D)

14

1 x -1 1 x+2 x+2

16 Which of the following is equivalent to x2 - x 1 - x2 ? ÷ x 2 - x + 1 x3 + 1

Which of the following is equivalent to x2 - 1 x2 + 1 ? x -1 x +1

A)

1- x

B)

A)

x +1

C) −𝑥 D) x

B) C) D)

2x -1 x +1 2x x +1 2x + 1 x +1 x

32

CONTINUE

3

1

3 If m > 0 and m 2 -9 = 0, what is the value of m?

What is the sum of all values of x that satisfy 12 x2 - 13x + 1 = 0?

A) B)

A)

C) D)

-9 -3 3

B)

9

C) D)

2

12 13 13 12 1 13 1 12

4

2t 2 + 46 - x = 0 If t > 0 and x = 4, what is the value of t ? 3

A) 1 B) 3 C) 6 D) 9

What is the product of all values of x that satisfy 2 x2 - 15x - 8 = 0? A) B) C) D)

2 15 15 2 1 4 8

33

CONTINUE

Test 1

3

5

7 What are the solutions to 3x2 - 6 x + 1 = 0 ? A)

6 3

B)

6 6± 3

C)

-1 ±

6 3

D)

-6 ±

6 3

6

2 x3 + 3x2 - 7 x - 8 = 0 For what values of x , is the equation above

true? A) -2 B) -1 C) 1 D) 2

8 x3 ( x 2 - 10 ) = -9 x

If x > 0 what is the one possible solution to the equation above? A) 2 B) 3 C) 6 D) 9

What are the solutions of the quadratic equation 2 x2 - 10 x - 12 = 0 ? A) B) C) D)

x = -6 and x = -1 x = -6 and x = 1 x = -1 and x = 6 x = 1 and x = 6

34

CONTINUE

Test 1

3

9

11 y = 2 x2 2 y + 4 = 2 ( x + 3)

If ( x, y ) is a solution of the system of equations above and x > 0 what is the value of xy ? A) B) C) D)

-2 -1 1 2

10

What is the sum of solutions to ( x + 5.2 )( x - 1.2 ) = 0? A) B) C) D)

-6.4 -4 4

6.4

12

2 x 2 - 5x = m In the equation above, m is a constant. If equation has no real solutions, which of the following could be the value m ? A) -4 B) -3 C) 3

In the equation ( bx - 3)2 = 25, b is a constant. If x = 2 is one solution to the equation, what is a possible value of b ? A) B) C) D)

-4 -2 -1 2

D) 4

35

CONTINUE

Test 1

3

13

15 𝑦 = 3𝑥 " − 𝑥 − 3 𝑦 = −2𝑥 + 7 If ( x, y ) is a solution of the system of equations above, what is the one possible value of y ?

How many solution are there to the system of equations above?

A) B) C) D)

A) B) C) D)

-2 -1 1 11

y = x2 - x + 3 y - 4x + 7 = 0

There are no solutions. There is exactly one solution. There are exactly two solutions. There are exactly four solution.

16

14 3x + 10 + 5 = x + 7

What is the solution set of equation above? A) {-3, -2} B) {-3, 2} C) {-2,3} D) {2}

ax3 + bx2 + cx + d = 0 In the equation above, a, b, c, and d are constant. If the equation has roots, -2, 3, and 7 , which of the following is a factor of ax3 + bx2 + cx + d = 0? A) B) C) D)

x-7 x-2 x+3 x+7

36

CONTINUE

Test 1

3

3

1 𝑚" − 𝑚 − 20 = 0 If m is a solution of the equation above and m < 0 , what is the value of m ? A) B) C) D)

x 3x = x-2 3

Which of the following are a possible value of x ? A) B)

-5 -4

C) D)

-3 -1

2

0 and 2 0 and 3 0 and 5 2 and 5

4 9 = x-2 x-2

In the equation above, which of the following is a possible value of 𝑥 − 2? A) −9 B) −1 C) 3 D) 9

x + 6 = -x

What is the solution set of equation above? A) {−2} B) {3} C)

{-2,3}

D) There are no solutions.

37

CONTINUE

Test 2

3

5

7

x2 = x Which of the following is the solution of the equation above? A) B) C) D)

-4

x2 - 4 =4 x-2

What are the solutions to the equation above? A) There are no solutions. B) x = 2

-2 2

There are no solutions.

C) D)

x = 2 and x = 4 x = -2 and x = 4

8

6 x2 - 4 = -2 x+2

What are the solutions to the equation above? A) B)

x = -2 and x = 0 x = -2 and x = 2

C) x = 0 and x = 2 D) 𝑥 = 0

2x = x - 4

What are the solutions to the equation above? A)

{-8, -2}

B) C)

{8}

{2,8}

D) There are no solutions.

38

CONTINUE

Test 2

3

11

9

2 x2 = 5x + c

In the equation above, c is a constant. If x = 2 is a solution of this equation, what other value of x satisfies the equation? A) -1 2 B) 0 C)

1 2

D)

3 4

Which of the following are solutions to the quadratic equation ( x + 2 )2 = 4 ? 9

A) B) C) D)

10

8 4 x = - and x = 3 3 8 -4 x = - and x = 3 3 2 2 x = - and x = 3 3 x = -2 and x = 2

12 25 = 10 x In the equation above, if x > 0 , what is the solution of the equation?

In the equation ( ax + 1)2 = 9, a is a constant. If x = 2 is one solution to the equation, what is a possible value of a ?

A) B) C) D)

A) B) C) D)

x+

-5 1 2

5

-3 -2 -1 2

39

CONTINUE

Test 2

3

13

15 x2 - x - 6 = 3x + 10 x+3 In the equation above can be solved for two solutions, one of which is extraneous. What is the value of the extraneous solution?

A) B) C) D)

-3

2 y = 3 y 12

What are the solutions to the equation above? A) B)

-2

C)

2

D)

3

{-2, 2} {-4, 4} {2,8}

{-2

2, 2 2

}

16

14 If 6𝑡 " − 5𝑡 − 6 = 0, what is the possible value of t?

The equation x2 - 4 x = -1 has two solutions for x . What is the larger value of x ?

A) - 3

A)

2-

B)

3

C)

2+

D)

5

2

B) - 2 3

C)

2 3

D)

3 4

40

3

3

CONTINUE

Test 2

3

1

3 3x 2 -4 x - 2 = 0

Which of the following is solution to the quadratic equation shown above? A)

4 ± 10 6

B)

2 ± 10 6

C)

2 ± 10 4

D)

2 ± 10 3

2

If 3x2 = 63 - 12 x and x < 0 , what is the value of x 2 ? A) B) C)

36

D)

9

64 49

4 2 x2 = 7 x - m

If one of the solutions of the equation above x=-

3 , which of the following is the other 2

possible value of x ? A) −15 B) −5 C) 2 D) 5

If x2 = 2 x + 15 and x > 0, what is the value of x3 ? A) B) C)

27

D)

125

-125 -27

41

CONTINUE

Test 3

3

7

5 Which of the following equation has NO real solutions? A) x2 - 5x - 2 = 0 B) x2 - 3x - 3 = 0 C) x - 6 x + 11 = 0 D) x2 - 2 x - 15 = 0 . 2

3x 2 = 10 x - 8

If m and n are the two solutions to the equation above and m > n , what is the value of n 2 ? A)

9

B)

4 9 *&

C) D)

+ 4

8

6 2 x2 + 8x + 4 = 0

What are the solutions to the equation above? A) −2 ± √2 B)

-4 ± 4 2

C)

4±4 2

D) 2 ± √2

What is the sum of the solutions of the equation 2 ( 2x - 3) = 4 x - 3? A) B) C)

-4 -3 3

D) 4

42

CONTINUE

Test 3

3

9

11 x2 + 2 x - 7 = 0

What are the solutions to the equation above? A)

-1 ± 2 2

B)

1± 2 2

-1 ± 4 D) 1 ± 4 C)

Which of the following are solutions to the 2 quadratic equation x - 3x + 2 = 0 ? x2 + 5x - 6 A) {1, 2} B) C) D)

{2} {-6,1} {-1}

12

10 x4 - 81 = 0 In the equation above, if x < 0 , what is the

solution of the equation? A) -9 B) -6 C) -3 D) -2

Which of the following are solutions to the quadratic equation x3 + 10x2 + 24x = 0 ? A) {0} B) C) D)

{-6, -4,0} {-6, -4} {-2, 2}

43

CONTINUE

Test 3

3

13

15 x + y = 5ü ý x× y = 6 þ

Which of the following are solutions to the quadratic equation 6 x2 + 5x = 4 - 7 x - 3x2 ? A) - 2 ! 2 2 3 B)

12 ! 2 2 18

C)

2 ! 12 2 3

D) There is no solution.

14

What are the solutions to the system of equation above? A) B) C) D)

{(1,3)( 2,1)} {( 2,3)(3,3)} {( 2,3)(3, 2)} {( 3, 2 )}

16 If the equation of mx 2 + ( 2m + 1) x + m - 1 = 0 , has double root, what is the possible value of m?

What is the product of the roots of the equation of x 2 + 2 x - 4 = 0 ?

A) 8 B) 4

A)

2

B) C) D)

- -2

C)

1 8

D) - 1

4 -4

8

44

CONTINUE

Test 3

3

1

3 ax 2 - 2 ( a - 1) x + a = 0

If x1 and x2 are solution of the equation above and x1 = x2 , what is the value of a ? A)

-

B)

1 2 0

C) D)

1 2

3x2 + 2x - 1 = 0

Which of the following is a possible value of x ? 1 3

A)

-

B) C)

-1

D)

1

2 3

1

4

2

2 x2 + ax + 2 = 0 In the equation above, one of the roots of equation is -2, which of the following is a possible value of 𝑎? A) B) C) D)

1 2

3 5

2 x2 + mx + m - 2 = 0

If x1 and x2 are solution of the equation above and x1 = x2 , what is the value of m ? A)

1 2

B)

2

C)

8 3

D)

4

45

CONTINUE

Test 4

3

5

7 Which of the following is the sum of the roots of the equation of ( x - 1)( x - 4 ) = 2 - 2 x ? A) 2 B) 3 C) 4 D) 5

6

x2 - 2tx + t 2 = 0

If x1 and x2 are solution of the equation above and x1 + x2 = x1 × x2 , what is the value of t ? A) B) C) D)

-2 -1 1 2

8 𝑥 " − 13𝑥 − 36 = 0 If x1 and x2 are solution of the equation above, x + x2 what is the value of 1 ? x1 × x2 A) B) C) D)

13 12 1 3 13 36 0 -

2 + 2x + 11 = x What are the solutions to the equation above? A) B) C) D)

{7} {1, 7} {-7, -1} {-1,7}

46

CONTINUE

Test 4

3

9

11 x 2 + ( 3m - 1) x + 24 = 0 In the equation above, m is a constant. If x = 3 is a solution of this equation, what other value of x satisfies the equation?

Which of the following are solutions to the quadratic equation 2 x 2 - 14 x + 21 + 4 = x ?

A) B) C) D)

-8

B)

3

C)

6 8

A)

D)

{-1,1} {5} {4,5} {1, 2}

12

10 x-3 1 = x + 2 2x - 2 In the equation above, if 𝑥 > 1, what is the solution of the equation?

In the equation mx 2 + 3m2 x - 6 = 0, m is a constant. If x1 + x2 = 6 , what is a possible value

A) 1 B) 2 C) 3 D) 4

A) -3 B) -2 C) 3 D) 6

of x1 × x2 ?

47

CONTINUE

Test 4

3

15

13 x2 + 7 x - 8 =0 x 2 - 3x + 2 What are the solutions to the equation above?

A) B) C) D)

{1, 2} {-8,1} {-8,1, 2} {-8}

14

y +1 3 = 2 y+2

What are the solutions to the equation above? A) B) C) D)

{-1, 4} {-1, -4} {1, -4} {1, 4}

16 If 3m2 + 15m + 12 = 0 , what is the possible value of m? A) - 4 3

B) -1 C)

4 3

D) 4

x 2 - 3x + 2 =0 x2 + 5x - 6 What are the solutions to the equation above?

A) {2} B) C) D)

{-6,1} {-1} {6}

48

CONTINUE

Test 4

3

3

1 What are the solutions to 2𝑥 " + 5𝑥 + 3 = 0 ? %

A) A− , 3C

If −1 is one of the zeros of the equation (𝑚 + 1) 𝑥 " − 7𝑥 + 3𝑚 − 4 = 0, what is the value of the 𝑚?

"

%

B)

A" , 3C

C)

A−1, !C

%

A) −2 B) −1 C) 0 D) 1

"

D) A− ! , −1C

4

2 If the quadratic equation 𝑎𝑥 " − (3𝑎 − 1)𝑥 + 1 = 0 has two equal roots, what could be the value of 𝑎? A)

% & %

B) ' %

C) \$

A rectangle has length 4 cm greater than width. If the area is 96 cm2, what is the perimeter of rectangle? A) 20 B) 24 C) 40 D) 48

D) 1

49

CONTINUE

Test 5

3

5

7 When a tennis ball is hit directly upwards, its height ℎ above the ground is given by ℎ = 2 + 6𝑡 −𝑡2 meters, where 𝑡 is the time in seconds after the ball is hit. When is the ball height of 11 meters? A) 2 B) 3 C) 6 D) 11

What are the solutions to 𝑥 \$ − 17𝑥 " + 16 = 0 ? A) {−4, −1, 1, 4} B) {−4, 4} C) {1, 16, } D) {−1, 1, 16}

8

6 A baker baking cakes finds that his profit per hour, \$𝑃, 𝑃 = 36𝑑 −𝑑2, where d is the number of cakes made per hour. How many cakes must the company make per hour in order to make \$320 profit? A) 12 B) 15 C) 20 D) 24

!'

𝑥+ = 12 # In the equation above, if 𝑥 > 0, what is the solution of the equation? A) 6 B) 9 C) 12 D) 36

50

CONTINUE

Test 5

3

9

11 What are the solutions to 6𝑥 " + 5𝑥 = 4 − 7𝑥 − 3𝑥 " ? A) B) C) D)

"±"√" % "±&"√"

𝑎𝑥 ! + 𝑏𝑥 " + 𝑐𝑥 + 𝑑 = 0 In the equation above, 𝑎, 𝑏, 𝑐, 𝑎𝑛𝑑 𝑑, constant. If the equation has roots −5, −3, 𝑎𝑛𝑑 4 , which of the following is a factor of 𝑎𝑥 ! + 𝑏𝑥 " 𝑐𝑥 + 𝑑 = 0 ? A) 𝑥 − 5 B) 𝑥 − 3 C) 𝑥 + 4 D) 𝑥 − 4

% '&"±"√" &( '"∓"√" %

10

12 !

I𝑡 " + 15 + 𝑥 = 11 In the equation above, if 𝑥 = 7, 𝑎𝑛𝑑 𝑡 > 0, what is the value of 𝑡? A) −7 B) 7 C) 14 D) 49

What are the solutions to 𝑥 + √𝑥 = 6 ? A) B) C) D)

{−3, 2} {4, 9} {−3, 2, 4, 9} {4}

51

CONTINUE

Test 5

3

13

15 What are the solutions to A) B) C) D)

# " (#(") # " ("#("\$

=0?

{−4, 5} {−4, 5, 6} {5} {5, 6}

14

If −2 is one of the roots of the equation 3𝑥 " − (𝑎 − 𝑏)𝑥 − 6 = 0, what is the value of the other root of the equation? A) 3 B) 1 C) −1 D) −3

16 What are the solutions to 𝑥 ! + 𝑥 " − 4𝑥 − 4 = 0 ? A) B) C) D)

{−2, 2} {−2, −1, 2} {−1, 1, 4} {−1, 1}

𝑥 " + √2𝑥 − 4 = 0 A) √2 B) -√2 C) 4 D) -4

52

CONTINUE

Test 5

4

1

Functions, and Parabola

3 f ( x ) = mx 2 + 24 For the function f defined above, m is a constant

and f ( -4 ) = 8 . What is the value of f ( 4 ) ? A)

-8

B) C) D)

-1

A function f satisfies f ( 3) = 11 and f ( 5) = 6 . A function g satisfies g ( 3) = 4 and g ( 6 ) = 3 What is the value of f ( g ( f ( 5) ) ) ? A)

3

B) 4 C) 6 D) 11

4 8

4

2 g ( x) =

1

( x - 3)

For what value of undefined?

2

+ 4 ( x - 3) + 4

x is the function g above

A) -2 B) -1 C) 1 D) 2

y = a ( x - 3)( x + 5) In the quadratic equation above, a is nonzero constant. The graph of the equation in the xy plane is parabola with vertex (ℎ, 𝑘). Which of the following is equal to k ?

A) B) C)

-16a -12a 12a

D) 16a

53

CONTINUE

Test 1

4

Functions, and Parabola

7

5 In the xy - plane is parabola with equation

y = f ( x ) = ( x - 8) intersects the line with the equation y = 9 at two points, M and N . What is the length of MN ? 2

A) 5 B) 6 C) 11 D) 16

6

2 x+n 3 In the function above, n is a constant. If f ( 6 ) = 9 , what values of f ( -3) ? f ( x) =

A) B)

-2

3

C) 5 D) 7

8 If f ( x ) = -3x + 7 , what values of f ( -2 x ) ? The table above shows some values of the linear function f . Which of the following defines f ? A) B) C) D)

2x - 3 2x -1

A)

f ( x ) = -6 x + 7

B)

f ( x ) = 6x + 7

C)

f ( x ) = -6 x + 3

D)

f ( x ) = 6x + 3

2x + 1 2x + 3

54

CONTINUE

Test 1

4

9

Functions, and Parabola

11 In the xy - plane, the graph of function f has x - intercepts at −2, 1, 𝑎𝑛𝑑 4. Which of the following could define f ? A)

f ( x ) = ( x - 2 )( x + 1)( x + 4 )

B)

f ( x ) = ( x + 2 )( x + 1)( x - 4 )

C)

f ( x ) = ( x - 2 )( x - 1)( x + 4 )

D)

f ( x ) = ( x + 2 )( x - 1)( x - 4 )

10

The table above shows some values of the linear function f . Which of the following defines f ? A) 2 x - 5 B) 2 x + 5 C) 3 x - 5 D) 3 x + 5

12 h ( x ) = ( x - 6 )( x + 4 ) Which of the following is an equivalent form of the function above in which the minimum value of h appears as a constant?

In the equation y = x2 - m and m is a positive constant the graph of the equation is a parabola. Which of the followingt is an equivalent form of the equation?

A) h ( x ) = ( x - 1) + 25

A)

y = ( x - m )( x + m )

B) h ( x ) = ( x - 1) - 23

B)

y= x-

C) h ( x ) = ( x - 1) - 25

C)

y = ( x - m) - m

D) h ( x ) = x 2 - 23

D)

m öæ mö æ y = ç x - ÷ç x + ÷ 2 øè 2ø è

2

2 2

(

m

)( x +

m

)

2

55

CONTINUE

Test 1

4

13

Functions, and Parabola

Test 1

15 In the xy - plane, the point ( 2,3) lies on the graph of the function g . If g ( x ) = - x 2 + t , where t is a constant, what is the value of t ? A) -4 B) -1 C) 3 D)

7

14

The table above shows some values of the linear function h . Which of the following defines h ( 3) ? A) 5 B) 8 C) 9 D) 12

16

The table above shows some values of the functions g and h and g. For which values of x is g ( x ) = h ( x ) ?

A) -2 B) -1 C) 3 D) 6

In the xy-plane, the graph f ( x ) = 2 x 2 - 9 x intersects the graph of the y = - x at the points (0,0) and (m,-m). What is the value of m ? A) B) C) D)

4 6 8 9

56

CONTINUE

4

1

Functions, and Parabola

3 h( x) =

For what value of undefined? A) B)

4 x + x - 12 2

x is the function g above

-4

-3

C) 1 D) 4 Which of the following could be equation of the graph above? A)

y = ( x - 1)( x + 1)( x - 2 )

B)

y = x 2 ( x + 1)( x - 2 )

C)

y = x ( x - 1)( x + 2 )

D)

y = x 2 ( x - 1)( x + 2 )

2

4 f ( x ) = 3x - 7 g ( x) =

1 2 ( x + 2) - 6 2 g ( x) = -x - 8 f ( x) =

4 - f ( x) 3

The functions f and g are defined above. What value of g(3) ?

The functions f and g are defined above. If f ( a ) = g ( a ) , what is possible value of a ?

A)

A)

-2

B) - 2 3 C) 2 3 D) 3

-3

B) -2 C) 2 D) 4

57

CONTINUE

Test 2

4

5

Functions, and Parabola

7 In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, the graph of the polynomial function 𝑔 crosses the 𝑥 − 𝑎𝑥𝑖𝑠 at exactly two points, ( m,0 ) and ( n,0 ) , where m and n are both negative. Which of the following could define 𝑔 ? A) g ( x ) = ( x + m )( x + n ) B) g ( x ) = ( x - m )( x - n ) C) g ( x ) = ( x - m )( x + n ) D) g ( x ) = ( x + m )( x - n )

The graph above shows 𝑓 in the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒𝑠. Which of the following defines 𝑓 ? A) B) C) D)

1 2 x + 4x + 7 3 2 y = - x2 + 4x + 7 3 2 y = x2 - 4x + 7 3 2 y = x2 + 4x + 7 3 y=

6

8 If f ( x ) = 3x 2 + 3 and f ( x - a ) = 3x 2 - 12 x + 15 , what is the value of a ? A) B) C) D)

-3 -1

1 2

If f ( x ) = 5x 2 + 7 x + 2 is graphed in the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation? A)

y-

intercept

B) x - intercept(s) C) y - coordinate of the vertex D) x - intercept of the line of symmetry

58

CONTINUE

Test 2

4

Functions, and Parabola

11

9 In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, the graph of function h ( x ) = x 2 + 6 x + 8 has two x - intercepts. What is the distance between the x - intercepts? A) B) C) D)

4

3 2

If f ( x ) = 3x - 5 , what is the value of f ( 0 ) ? x+5 A) -10 B) -5 C) -1 D) 1

1

12

10

If f ( x ) = 3x - 2 x , what is the value of f ( 3) - f ( 2 ) ? A) B) C) D)

19

14 13 11

In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, the graph of 2 g ( x ) = x + 7 x + 11 is shown above. If the function crosses the 𝑦 − 𝑎𝑥𝑖𝑠 at point ( 0, k ) , what is the value of k ? A) B) C) D)

4

7

10 11

59

CONTINUE

Test 2

4

1

Functions, and Parabola

3 2 If f ( x ) = 3x + 4 x - 1 , what is f ( -2 ) ? 2x + 3

A) B)

-6 -3

C) 3 D) 6

Which of the following could be the equation of the graph above? A)

y = ( x - 2 )( x + 2 )

B)

y = x 2 ( x - 2 )( x + 2 )

C)

y = x ( x - 2 )( x + 2 )

D)

y = x ( x - 2) ( x + 2) 2

4

2 g ( x ) = x2 + 6 x + 8

f ( x ) = 2 x 2 - 3x

The graph of the g in the xy - plane is a parabola. Which of the following intervals contains the x - coordinate of the vertex of the graph of g ?

The functions f and g are defined above. What is value of g ( -1) ?

A) -6 < x < -5 B) -4 < x < -2 C) -2 < x < 0 D) 2 < x < 4

A) −5 B) 1 C) 5 D) 15

g ( x) = 5 - 2 f ( x)

60

CONTINUE

Test 3

4

Functions, and Parabola

7

5

In

the

xy - plane,

the graph of g ( x ) = 3x + 11x + 7 is shown. If the graph 2

The graph above shows f in the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒𝑠. Which of the following defines f ? A)

y = 3( x + 1) + 2

B)

y = 2 ( x + 1) + 3

C)

y = ( x + 1) + 2

D)

y = ( x + 2) + 3

2

2

crosses the y - axis at point ( 0,n ) , what is the value of n ? A) - 11 6 B) 3 C) 6 D) 7

2

2

6

8 If h ( x ) = x + 2 and h ( x + k ) = x - 6 x + 11 , what is the value of k ? 2

A) B) C) D)

2

-3 -1

0 3

In the

xy - plane,

the graph of function g ( x ) = x - x - 6 has two x - intercepts. What is the distance between the x - intercepts? 2

A) 6 B) 5 C) 2 D) 1

61

CONTINUE

Test 3

4

Functions, and Parabola

11

9

If f ( x ) =

3x + 2 , what is the value of f ( 2 ) ? x +1

A) 9 B) 18 C) 27 D) 81

In the xy - plane, the graph of f is shown above. Which of the following defines f ( 0 ) ? A) B) C) D)

-1

0 1 2

10

12 If f ( x ) = 6 x - 5 , what is the value of f -1 ( 2 ) ?

If f ( x - 3) = x 2 + 6 x + 11 for all real values of x which of the following is equal to f (x)?

A)

7 6

A)

x2 - 2

B)

B)

6 7

x2 - 2

C)

C)

1 7

D)

7

D)

2

(x + 3) + 2 (𝑥 + 6)" + 2

62

CONTINUE

Test 3

4

1

Functions, and Parabola

3 If f (x)= 5 x + k , where k is a constant, and 𝑓(−2) = −4. What is value of k ? A) −6 B) −4 C) 4 D) 6

Which of the following could be the equation of the graph above? A)

f (x)= x 2 + 2 x + 3

B)

f (x)= x 2 - 4 x + 3

C)

f (x)= x 2 + 4 x + 3

D)

f (x)= x 2 - 5x + 3

4

2 If

f (x)= 1- x 2 , and m is a positive real !

number, then 𝑓 " # ? "

A)

1+

1 m2

(m + 1)(m + 1) B) 2 m

C) D)

(m - 1)(m + 1) m2

1- m2 m2

The functions

f is defined by the equation

f (x)= x - x . 2

Which of the following represents a quadratic equation with no real zeros? A)

f (x)- 1 #

B) 𝑓 " # C) 𝑓 " D)

\$ \$#%! \$

#

f (x)+ 3

63

CONTINUE

Test 4

4

Functions, and Parabola

6

5

In the xy - plane, the graph of y = f (x) is shown. If the graph crosses the y - axis at The graph above shows f in the xy - planes. Which of the following defines f ? A) 𝑦 = 𝑥 " − 4𝑥 + 1 B) C) D)

1 2 (x - 2) - 3 2 3 2 y = 2(x - 2) 2 y=

point (0, k ), what is the value of k ? A) B)

5 2 3

C) 4 D) 5

2

y = 4(x - 2) - 3

64

CONTINUE

Test 4

4

Functions, and Parabola

9

7 If

f (x)= - x + 5 and g ( f (x))= 4 x - 7 , what

is value of g (3)? A)

- 7

- 1 C) 1 B)

D) 7

In the xy - plane, the graph of f is shown above. Which of the following defines f ( x ) ? A)

y = x3 - 7 x + 6

B)

y = (x - 3)(x + 1)(x + 2)

C)

y = x2 - 2 x + 1

D)

y = x3 - 6x + 7

10

8 In the xy - plane, the graph of function with

Let

the

function

f

defined

by

f (x)= 3 - x - 5 for all real values of x . What

zeros at x = 3 and x = 5 and vertex at (4, - 1). What is the y - intercept of the graph?

is the greatest value of f ?

A) - 15 B) - 8 C) 8 D) 15

A) - 2 B) 3 C) 5 D) 8

be

65

CONTINUE

Test 4

4

Functions, and Parabola

3

1 If 𝑓(𝑥) = 𝑥 " − 𝑥, what is the value of 𝑓(−1) + 𝑓(1) + 𝑓(2) ?

If 𝑓(𝑥) = 3"#(% , and 𝑓(2) = 5𝑚 + 2, what is the value of 𝑚 ?

A) 7 B) 6 C) 5 D) 4

A) 1 B) 3 C) 4 D) 5

4

2 If 𝑔(𝑥) = 3𝑥 + 2, and 𝑓(𝑥) = 𝑥 " + 2, what is the value of 𝑔(𝑓(2)) ?

If 𝑓(2𝑥 − 5) = 𝑥 " + 2𝑥 − 1, what is the value of 𝑓(5)?

A) 8 B) 12 C) 20 D) 24

A) 36 B) 34 C) 25 D) −1

66

CONTINUE

Test 5

4

5

Functions, and Parabola

7 𝑓(𝑥) = 3𝑚𝑥 − 7 For the function 𝑓 defined above, m is a constant and 𝑓(2) = −13. What is the value of 𝑓(−2)? A) −1 B) 1 C) 13 D) 17

A function 𝑓 satisfies 𝑓(11) = 5, 𝑎𝑛𝑑 𝑓(3) = 1. A function 𝑔 satisfies 𝑔(5) = 11, 𝑎𝑛𝑑 𝑔(1) = 11. What is the value of 𝑓(𝑔(𝑓(3)))? A) 11 B) 5 C) 3 D) 1

8

6 ℎ(𝑥) =

% # # (%!# " *!'

For what value of 𝑥 is the function ℎ above undefined? A) −9 B) −4 C) 3 D) 4 The graph of function 𝑓(𝑥) is shown above, +((%)*+(") what is the value of ? +((")*+(%)

A) −1 B) −1/2 C) 1/2 D) 1

67

CONTINUE

Test 5

4

9

Functions, and Parabola

11

The graph of function ℎ(𝑥) is shown above, .())*.(!) what is the value of ? .(%)*.(\$)

A) −1 B) 0 C) 1 D) 3/2

10

The graph of function 𝑓(𝑥), and g(𝑥) are shown above, what is the value of 𝑔(f(0)) + 𝑓(3)? A) 2 B) 3 C) 5 D) 8

12

The graph of function 𝑓(𝑥) is shown above, what is the value of 𝑓(𝑓(0)) + 𝑓(−3)?

The graph of function 𝑓(𝑥) is shown above, what is the value of 𝑓(5) − 𝑓(−1)?

A) −3 B) 0 C) 3 D) 4

A) −5 B) −1 C) 1 D) 5

68

CONTINUE

Test 5

4

13

Functions, and Parabola

15 If 𝑓(𝑥) = √𝑥 + 1, what is the value of 𝑓(3) + 𝑓(15) − 𝑓(24) ? A) −1 B) 1 C) 6 D) 11 The graph of function 𝑓(𝑥) is shown above, and 𝑔(𝑥) = 2𝑓(𝑥 − 2) − 1, what is the value of 𝑔(𝑓(8))? A) 19 B) 20 C) 21 D) 25

16

14

The graph of function 𝑓(𝑥) is shown above, and 𝑓(𝑓(−5)) + 𝑓(𝑚) = 2, what is the value of 𝑚?

The graph of function 𝑓(𝑥) is shown above, and 𝑓(4 − 𝑛) + 6 = 𝑓(𝑓(8)), what is the value of 𝑛?

A) 0 B) 3 C) 4 D) 8

A) −2 B) −1 C) 0 D) 4

69

CONTINUE

Test 5

4

Functions, and Parabola

3

1 In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, 𝑓(𝑥) = (𝑥 − 3)" intersects the line with the equation 𝑦 = 16 at points A, and B. What is the length of |𝐴𝐵|? A) 6 B) 7 C) 8 D) 11

𝑓(𝑥) = (𝑥 − 2)(𝑥 + 4) Which of the following is an equivalent form of the function above in which the minimum value of 𝑓 appears as a constant? A) 𝑓(𝑥) = (𝑥 + 1)" + 9 B) 𝑓(𝑥) = (𝑥 + 1)" − 9 C) 𝑓(𝑥) = (𝑥 − 1)" − 9 D) 𝑓(𝑥) = (𝑥 − 1)" + 9

4

2 In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, graph of function 𝑓 has 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 at −1, 0, 𝑎𝑛𝑑 2. Which of the following could define 𝑓?

In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, point (1, 4) lies on the graph of function 𝑓. If 𝑓(𝑥) = −2𝑥 " + 𝑚𝑥, where 𝑚 is constant. What is the value of 𝑚?

A) 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥 + 2) B) 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥 − 2) C) 𝑓(𝑥) = 𝑥(𝑥 + 1)" (𝑥 + 2) D) 𝑓(𝑥) = 𝑥(𝑥 − 1)(𝑥 − 2)

A) 1 B) 3 C) 4 D) 6

70

CONTINUE

Test 6

4

Functions, and Parabola

7

5 In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, the graph of function 𝑓(𝑥) = −𝑥 " + 𝑥 + 12, has two 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠. What is the distance between the x− 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠?

In the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒, the graph 𝑓(𝑥) = −𝑥 " + 4𝑥 intersects the graph of the y=x at the points (0, 0), 𝑎𝑛𝑑 (𝑛, 𝑛). What is the value of n?

A) −7 B) 1 C) 4 D) 7

A) −4 B) −3 C) 3 D) 5

6

8 𝑓(𝑥) = −𝑥 " − 2𝑥 + 15 The graph of the 𝑓 in the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒 is a parabola. Which of the following intervals contains the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the vertex of the graph of 𝑓? A) −5 < 𝑥 < 3 B) −3 < 𝑥 < 5 C) −1 < 𝑥 < 16 D) −3 < 𝑥 < −1

ℎ(𝑥) = 𝑥 " − 𝑥 + 1 𝑔(𝑥) = −30 + 2ℎ(𝑥) The functions ℎ and g are defined above. What is value of g(−3)? A) −4 B) 4 C) 9 D) 13

71

CONTINUE

Test 6

4

9

Functions, and Parabola

11 If ℎ(𝑥) = 𝑥 " − 𝑥, 𝑎𝑛𝑑 ℎ(𝑥 − 𝑡) = 𝑥 " + 3𝑥 + 2, what is the value of 𝑡 ?

If 𝑓(𝑥) =

A) −2 B) 1 C) 2 D) 4

A) −16 B) −2 C) 2 D) 4

10

# " (% #*%

, what is the value of

+((!)*+(!) +())

?

12 Let the function 𝑓 be defined by 𝑓(𝑥) = 7 − 2|𝑥 − 3|, for all real values of 𝑥. What is the greatest value of 𝑓? A) 0 B) 4 C) 7 D) 13

If 𝑓(𝑥) = 𝑓(1) ?

!#(/ # " (!#

, what is the value of 𝑓(−1) +

A) −6 B) 0 C) 3 D) 6

72

CONTINUE

Test 6

4

13

Functions, and Parabola

15

The graph of function 𝑓(𝑥) is shown above, 𝑓(𝑥) = 𝑥 " − 2𝑥 + 𝑚 intercepts at the points A, and B, and |𝐴𝐵| = 4, what is the value of 𝑚? A) −3 B) −1 C) 1 D) 3

14

What is the equation of function shown above? A) ℎ(𝑥) = −𝑥 " − 4𝑥 − 5 B) ℎ(𝑥) = 𝑥 " − 6𝑥 + 5 C) ℎ(𝑥) = (𝑥 − 1)(𝑥 + 5) D) ℎ(𝑥) = 𝑥 " − 3𝑥 + 6

16

What is the equation of function shown above? A) g(𝑥) = (𝑥 − 1)(𝑥 − 4) B) g(𝑥) = (𝑥 + 1)(𝑥 − 4) C) g(𝑥) = 𝑥 " − 4𝑥 − 5 D) g(𝑥) = 2𝑥 " + 4𝑥 + 5

What is the equation of function shown above? A) g(𝑥) = (𝑥 − 1)(𝑥 − 4) B) g(𝑥) = −𝑥 " − 4𝑥 − 5 C) g(𝑥) = −𝑥 " − 4𝑥 + 3 D) g(𝑥) = −𝑥 " − 4𝑥 + 5

73

CONTINUE

Test 6

SOLUTIONS

1

1

Exponents & Square Roots

3 2

If a = 4 2 and 2a = 4b , what is the value of b ? A) B) C) D)

8 16 32 64

Solution

:

Which of the following is equal to b 5 ? 2

A)

b5

B)

b2

5

C)

2

b5

D)

5

b2

solution a

452

-_

La

=

:

Fib

(2) 452

Xmf

n

Vfb

=

( 8525=15455

b

=

's

XIN

using

,

this

formula

ib

64.2=4 b 128

=4b

|b=32I

2

4 2

x If a 2 = a 54 , a > 1 and x - y = 6 , what is the ay value of x + y ?

In the equation above, if n = 6 , what is the value of m?

A) B) C) D)

A) B) C) D)

3 9 24 27

Solution x.

m+3 -n =0

3 9 27 33

solution

:

Fz

2

a

"

54

q x'

-

=

a.

y 2=54

(X y ) ( xty ) -

-

x

-

a

-

y

=

Y

:

'

-

=D

N

" =

a

mtf

( Tmf)

6

=

N

=

( 6)

Z

M -13=36

54

M€324

61×+43=54

Xty=gT -

1

CONTINUE

Test 1

1

5

Exponents & Square Roots

7 3

Which of the following is equal to 25 4 ? A)

3

25

B)

5 5

C)

4

D)

5

1

-3 2 Which of the following is equal to a × b ? -4 a ×b a A) 2 b

a b

B)

5

Solution

1

C)

:

25¥

't

( 54¥

=

b-

g

=

I

;

-3*2

FEM

=

a

-7

a ×b

D)

b

21%

solution

so ;

b 2

:

-3

53

a

I

=

525.5

=

I

b

-3

a-

3+4

b

-

I

a

=

t 4) -

.

b

's

-1

b

Ab'12

=

=

-

a

=

y,

5125

=

.

NET

÷

=

-

FI

8

6 If

x + 16 = 25 , what is the value of x ?

A) 1 B)

9

If a m? A)

C) 6 D) 9 solution

Fx Tx

:

+

t

The 4

(FIFA )

=

=

525

5

-

1 3

= m, where a > 0, , what is the value of

3

1 a

B)

3

a

C)

3

a2

D)

2

a -3

solution

'

:

ai

xT

's

m

=

;

a

> o

'

at

=

m

/÷ra=mT →

2

CONTINUE

Test 1

1

9

Test 1

Exponents & Square Roots

11 25a 2

If a>0, which of the following is equivalent to the given expression? A) 50a4 B) 10a C) 5a2 D) 5a Solution :

52552

1

Which of the following is equivalent to ( 4t ) 3 ? A)

3

B)

23 t

C)

1 12t 3

D)

( 4t )

4t

3

solution : =

( at )÷

feat

=

1501

=

the

,

since

is

sat

a >0

59

.

10

12 y2 = y Which of the following values of y is NOT solution to the equation above?

A) 2 B) 1 C) 0 D) -2 Solution

If a3 = b , where b > 0, what is the value of a in terms of b? 1

A)

b7

B)

b

5 12

C)

b12

D)

b4

'

1

.

✓y2

=

y

3

solution

. .

t

ly if so

ly

t

I -

-2

=

y is

to the

a'

y then Not

yzo a

solution

equation

.

ME

=

af

of

=

=

q3 its a

3

b

;

Fb b 't

'T ( at)↳= ( b )

ki I -

=

=

b

>o

's

{ tpaokweeraog

Iz both sides

bit

CONTINUE

1

13

Exponents & Square Roots

Test 1

15 1

Which of the following is equal to

2 +1

?

2 1 2 2

A) B)

2 -1

C)

2 1 + 2 2

D)

2 +1

If y is positive integer and x = 2 y +3 + 2 y , what is 2 y + 2 in terms of x? A)

x 9

B)

4x 9

C) D)

Solution :

13x 36x

solution :

÷

1(F

=

( 52+17152

-

X

D

X

=r÷

+

29

23.29+29

÷÷

I ⇒

=

291-3

."

REI

=

=

)

"

"÷÷÷ .

=4gx⑦

reel

14

16 If x3 = 30, and y2 = 10 , what is the value of

( x × y)

6

?

A) B) C)

9 ´104

D)

9 ´105

3 ´104 3 ´105

3=30

ya =

=

=

=

,

yo

}

(2/3) 900

9

A)

10 2

B)

11 40

C)

34 2

D)

36 2

solution :

solution : X

What is the value of 5 8 + 6 32 ?

.

"

( XY )

-

.

-

X

" -

y

558

'

( y 2) 3=302.103

103

102.103

=

=

=

6156-2

y-ff.IT g.

252

Iofz

t

t

6-452

24T

3452

=

9×10547

6532

t

4

CONTINUE

1

1

Exponents & Square Roots

3 What is the value of 4 50 - 3 32 ?

What is the value of 9 24 ? 4 27

A)

8 2

B)

12 40

C)

20 2

D)

32 2

A)

3 2

B)

3 2 2

C)

2 2

D)

2 2 3

Solution :

4550 =

=

=

3532

-

4525.2

-

4.552

solution :

94M¥ %÷=9

3-452

-

.

T2

20

3516T

-

1252

NEI

÷ '

.

4.353

E-

Era

-

¥7

-

4

2 If

16b = 2a +5 , what is the value of b ?

A)

32 × 2a

B)

2a+ 2

C)

2×2

D)

2a+1

A) -49 B) -27 C) 27 D) 49

a+ 2

Solution : 16 b

La

=

246=2

-15

a

-15

at 5

b

2 =

-x - y = 5

In the equation above, if y = 2 , what is the value of x?

-24

b=2#

solution :

Fx Fx

-

y

-2

=

=

5

;

y

=L

5

(a) 2=175 -

5

X

=

49

CONTINUE

Test 2

1

Test 2

Exponents & Square Roots

7

5 4x Which of the following is equal to 25 ? 1253 x

A)

1 5

B)

1 5x 5

C) D)

5

x

Which of the following is equivalent to

(a + a + a + a) × (a + a) ? 4-1

A) B)

16 × a

C)

16 × a2

D)

32 × a 2

32 × a

Solution : solution :

4A

=f:3=5I

2544

I

=

594

892

=

=

58×-9

"

¥1 '

X

-

5

=

-

La

-

IT

-

4

=

-

892

32/ '

=

8

6 If 812 = 4x , what is the value of x ? A) B) C) D)

Which of the following is equivalent to 32 + 4 18 ? 8

9 12 18 36

Solution

:

812=44

A)

2

B) C)

2 2 8

D)

8 2 Solution :

( 23112--127

" =

JE

t

-

236

=

22

"

,

ya ⇒

equal

4592

54.2 ⇒

4f2tl2B 252

36--221

1x-

t

-

JE

since the bases are

F.

458

6

naff

-

-

8D

CONTINUE

1

Exponents & Square Roots

11

9 If a 2 = 9, b2 = 16, and ( a + 3) × ( b - 4 ) ¹ 0 what is the value of a + b ?

If 1 = 3 3 what is the value x? x

A) -2 B) -1 C) 1 D) 2

A)

3

B) C)

Solution :

a. =3

a =-3

or

( A -13 ) (

b

I at -3

and

d. =3

and

so

Atb

62=16

/

aka

=3

b.

-

-4

=

4)

or

D)

b=4

÷

f- O

tf 4) -

353=73-7-3

=

3-4=3

b -44 =

3 2

Solution :

I

b

3 2 2 3 2 3 -

" -

3

'T

=3

' .

?

I

-4

-

x

-

312

21=-37 ↳

=-If 12

10 1 = x-2

If

2 what is the value x?

A)

2+4 2

B)

2 -4 2

C)

2 -2 2

D)

2+2 2

If ( 3,000 ) × ( 4,000 ) = 1.2 ´10m what is the value m? A) 6 B) 7 C) 8 D) 9 Solution : 3000

12×106

¥2 -

2

X

x

=

-

-

4000

2

=

1. 2X

( 3×103/(4×103)=1.2

Solution :

x

-

=

1.2

X

ION

x

10M

10M

-

Ira -

F-

1.2×1007=1.2 XlO⑨

T2

=

-

|m=I

TI

-

L

I 2

+

a

/r2a/ ↳

7

CONTINUE

Test 2

1

Exponents & Square Roots

15

13 If 2a = 5 what is the value 5 × 4a + 2a ? A) B) C) D)

4+ 7

Solution :

La

-

x

If

55 80 125 130

5

5.49+20

A) B) C) D)

=

4 - 7 , what is the value of y

3

( xy ) 2 ?

9 27 54 81

Solution :

5.220+20

=

51292+200

=

x.

5155+5

=

X

1251-5

=

1131T

=

(

14

-

y y

=

=

×yy%

( 4 -1ft)

-

( 4 ft) -

42-7--9 ⇒

QI

3£ Zz -

'

=3

=27

16

(

If 3 + 2 2

)

2

= a + b 2, where a and b are

integers, what is the value of A) B) C) D)

a +b?

17 23 29 37

Solution

A)

5 4

B)

5 2

C) D)

:

( 3+25212 a

If 4x = 25 , what is the value of 2 x- 2 ?

at

=

-11252 -18 171-12 ra a- It

=

=

bfd

at

at

and

b

5 50

solution :

44--25

bra

224=52

bra -

12

22×-2

=

24=5

2X -

Atb

22=290 ⇒

17-112

YEI

8

CONTINUE

Test 2

1

1

Exponents & Square Roots

3 If f ( a, b ) = 2a2 , where a and b are positive b

integers, what is the value of f ( 2a, 4b ) ?

A) B) C) D)

f ( a,3b )

A) B)

9 16

Solution :

9 ••

! "

a. b) f (

C)

9 4

D)

9 2

f ( =

2-291442

-2A

4 5 6 7

solution :

62

10

,4b)

-

X

L

31

=

36.10-6

106

-

la

f

=

La

=

What is the smallest integer value of x such that 10- x < 0.000036 ?

,

3b

)

1X/

YE '¥* :* IIT .

.

=

÷

a

.

2

4 If a is positive integer, and 3a = b , which of the following express 3b2 in terms of a?

If 2 = 4 8 , what is the value of m ? -m

A) 3 × 3a

A)

5 2

B)

3

C)

7 2

D)

9 2

B) C) D)

32 a a+ 2

3

2 a+1

3

solution :

30=6

3. b'

=

?

362=3 ( 392 =3

( 329)

=/3T

2

solution : z

Tm gum

=

=

48

=

22.2%

4

Fas

=

2%

I Itm

-

I

a-

/m=I_/

9

CONTINUE

Test 3

1

5

Test 3

Exponents & Square Roots

7 If 4 50 - 3 8 = n 2 , what is the value of n ? A) B) C) D)

6 12 14 20

Which of the following is equivalent to 2 ? 3-1 × ( a + a + a ) × ( a + a )

Solution :

4550

-

358

4525T 4.

-

552

Lora

-

-

=

nT2

354.2 3-252 652

=

A)

1 a

B)

1 a2

C)

a

D)

a2

Nfd

=

solution :

#

nfd

-

=

q2

IT -3/9.2/9

nrz

1-

n8

6 If 812 = 4x , what is the value of x ? A) B) C) D)

9 12 15 18

Which of the following is equivalent to #\$%!&'\$√)#& )\$√!

Solution

:

812=44

A)

y

B)

5x y

C)

5 y

D)

5 3y solution :

(23/2=44 236

=22X

Miry

7xtTytX2y ⇒

-

=

=

7XFy+3Xr3y -

X=T

2×53

+

x

-

2×53

✓ 36=221

?

2×53

=

10x

By

-

2×53

=5

10

CONTINUE

1

Exponents & Square Roots

11

9 x If 2 x - y = 5 what is the value of 4 ? 2y A) 8 B) 16 C) 32 D) 64

If m = 11 + 13 + 15 + ... + 199 what is the value m? A) B) C) D)

Solution :

solution : 2x

-

y

5

=

yy=2g÷ 4

25 49 50 99

m

X

=

22×-9

It 1+1-1

=

.

.

+1

-

25

times

50

m=50T ✓

3247

10

.

.

12 If x

-

1 3

= 2 what is the value x?

A)

-8

B)

1 8

If 43 x - 2 = A) B)

11 9

D)

D) - 11 9

Solution :

(

x

's

solution :

) 4253 -

X

what is the value x?

-9

C)

8

8

-11

C) - 1 8

1 x +5

=

L

43×-2

L

3 =

-

1

=

-

.

23

2213×-2 GX -4 921

)

=

-

=

=

2-3

-

3 X

CX

-

Cxtsy

g

)

-15

15

11

x-/ a

11

CONTINUE

Test 3

1

13

Test 3

Exponents & Square Roots

15

3y+1 = 10 what is the value 3

If

y- 2

?

x x x = x m , what is the value of m ?

If

A)

3 10

A)

7 4

B)

10 3

B)

3 4

C)

10 27

C)

3 2

D)

27 10

D)

1

Solution :

solution :

34.3=10

3

'

39=1

X

3

TF

=

Xm

xTxrx=rxFE=rE×

-2--37=3%457 -

=

4fxT= X

74

=

-

E

Xm

Xm

m=Y4T ✓ 14

16 If a b = 64, value a × b ? 2 3

A) B) C) D)

and a3b2 = 16 what is the

2 4 5 10

Solution : >

a

+

b'

=

64=26

a3b2= 16=24

-

a5b5 ab

=

26.24

If 5 x + 3 = 4 5 , what is the value of x ? A)

-

B)

1 5

C) D)

5

1 5

10

solution :

( 55*55=(4557 251×+3)

=

16.5

25×+75=80

5=210--645

ab

=

2521

=

5

/X=#

22

-

=④

12

CONTINUE

1

1

Test 4

Exponents & Square Roots

3

2n +10 32 m × 2n 32 m -10

If

Assuming n is nonzero, which of the following expressions is equivalent to the expression above? A)

210

B)

310

C)

610

D)

620

3x 3 ´ 3 = 81 , and y = 9m -1, 3 x

m

what is

the value of x in terms of m? A) B) C) D)

6m - 3 6m - 2 3m - 2 3m - 1

Solution :

3×3

solution : n t

y

10

2M

2£ 33¥ .

" "

-

n

y

=

.

"

2

=

=

.

3

32M

-

'

3z÷

2Mt "

=

=

"

81M ⇒ 3×+4=3

gm

Xt

y

3×7=3

'

-

=

"m

⇒ x + -m

-

2⇒

x

-

y

y

4M

=

=

2M

-

4M

tX-y=2m#

16T)

2x

-

Gm -2

-

X

-

=6m

!X=3m#

4

2

(x )

a + b a -b

If

the value of A) B) C) D)

= x64 , and a + b = 16 , what is

a -b?

If 6 3 x + 13 = 43, what is the value of x ? A) B) C) D)

4 8 12 16

5 25 100 125

Solution :

solution :

( xatb )

a

-

x

=

ya!!!!! I 16

( a b) -

:)

6

64

b

=

FX

=

=

"

"

"

43

-

13

30

res

21=1257 Ly

64

a-b.LT/ ✓

13

CONTINUE

2

1

5

Test 4

Exponents & Square Roots

7 If a = 3 2 and 2a= 3x , what is the value of x? A) B) C) D)

6 12 24 36 Solution : A.

=

352

2A

652

=

Which of the following is equivalent to (4 × x 2 × y 3 )3 (8 × x3 × y ) ? × (2 × x × y 2 ) 4 (2 × x × y ) 2

A)

x2 2

B)

x3 4

C)

2x2 D) 8" !

=f3X

La

solution :

(65212--1535)

'

V

/

i:'÷i¥:s 43×6 y

*

'

=

213.23

=I8

6

8 If A) B) C) D)

3

y 2 + 4 = 5 , what is the value of y ?

Which of the following is equivalent to (-1)9 × (-1)10 + (-1) 21 ? (-1)11 × (-1)8

-9 -11 -12 -121

A) B) C) D)

Solution :

( 3¥43 y

-

-

(5)

-2 -1 1 2

3

2+4=125

solution :

=

I

42=121

-

1) (

t

)

t

L

-

t) =

IT

-_

247

y=I/

14

CONTINUE

1

9

Exponents & Square Roots

11 If 2t of t?

× 43 = 8s what is the value of s in terms

A)

t+6

B)

t+6 3

C)

t -6

D)

t -6 3

What is the value æç 1 + 1 ö÷ × 8 ? ç ÷ 3

è

A)

Lt

1

B)

2

C) 2 D) 4 Solution :

ftp.f

solution : .

26

235

=

¥-1

=

get -16=2,35

TF

=

t -16=35

/s=tfI/ -

Ira

=

-

V8

-

-

B

=fg⇒ 10

TF

54--11

12

4x +3 = 162 x what is the value x?

If A) B) C) D)

1 2 4 8

3

3 x 2 - 11 = 4 what is the value x?

A) B) C)

8

D)

16

3 5

solution :

solution :

( 221×+3=124 )2× 22×+6

If

=

2

fgfzxz.pt/3=f4)3 3×2-11

Ex

2×+6=8 X =

x

GX

2=25

x

=

I

5

21=57 L

121=17 -

64

3×2=75

I

6

=

15

CONTINUE

Test 4

1

13

Test 4

Exponents & Square Roots

15 What is the value A) B) C) D)

21 + 13 + 6 + 9 ?

What is the value

4

A)

10

5

B)

2

C)

10 10

6 7

D)

Solution :

0.04 + 0.09 + 0.25 ?

1

solution :

#tFE

O

6t}=g

-

2

0.3

t

t

0.5

=D

t

13+37=165

EY

-255=547

-

14

16 What is the value A) B) C) D)

13 + 2 ×

13 - 2 ?

3 9 13 15

Solution :

EYE

=

M¥25

=

What is the value 5 3 × 45 × 27

A)

7 9

B)

20 3

C)

25 36

D)

18 25

solution :

53¥ :# Ear .

=E -

-

=

'

-

-

ID

÷÷

,

,

ra

12

5 ? 108

HII

-

16

CONTINUE

2

1

Polynomials

3 Which of the following is equivalent to 1 ? 1 1 x-2 x-3

Which of the following is equivalent to

9" ! − 12" " & " + 4& ! ?

A)

( 3x - 2 y )

x2 - 5x + 6 x +1 2 - x + 5x - 6

B) C)

( 3x (9x

C)

x +1 x2 - 5x + 6

D)

(9x - 4 y )

D)

x -1 x2 - 5x + 6

A) B)

2

- 2 y2 )

2

- 4 y2 )

2

4

9×4

^

X

-

=

21-3-1×-22

' -

5×+6

(

-

X -3

-

X -12

-

12x2y2t4y4

34×42-2.3×2 .2y2t¥y'T /-1 ( 3×2 Ly 2)

( x -371×-2)

=

2

Solution :

Solution :

=

4

2

-

212-55+2

-xt5T

-

2

4 For a polynomial P( x) , the value of P(-2) is 3. Which of the following must be true about P( x) ? A)

x - 2 is a factor of P( x)

B)

x + 2 is a factor of P( x)

C)

x + 3 is a factor of P( x)

D) The remainder when P( x) is divided by x + 2 is 3. solution : Pf 2) -

This

when

Which of the following is equivalent to 4 x - 3 ? x +1 A)

1

B)

4-

1 x +1

C)

4-

7 x +1

D)

4 - 7 ( x + 1)

solution : 4

=3

means

Xt I

the remainder

pcx)

divide

by

3

is

X -12

-

(4×+4)

I

.

so

¥3

17

=I4-¥I

CONTINUE

Test 1

2

5

Test 1

Polynomials

7

( -4 x

2

+ 7 x - 3) - 3 ( -2 x 2 + 3x - 1)

If the expression above is rewritten in the form of polynomial ax2 + bx + c , where a, b, and c are constant, what is the value of a? A) -10 B) -2 C) 2 D) 3

( x - 2)

2

-

4 x-2

If the expression above is rewritten in the form t

of polynomial

( x - 2)

2

, where t is positive

constant and x ¹ 2 . What is the value of t? A) -13 B) -3 C) 3 D) 13

Solution :

-

4x + 5

4×7+7×-3 -16×2-921+3 2×2

-

2x

ax 't

=

b. Xtc

c

-

-

-

4×+5

late b=

solution :

-

¥⇒=4"Y;x÷

=e¥*=÷ 4×+5-4×+8

2

=

O

,

1t=I 8

6 If x2 + y 2 = m, and xy = n , which of the following is equivalent to 4m - 8n ? A) ( 4 x + 4 y )2

A) B)

B) ( 4 x - 4 y )2

C) D)

C) ( 2 x + 2 y )2 D) ( 2 x - 2 y )2

X2ty2=m -

4M

−3" " − 3.6" + 1 -3x2 + 3.6 x 3x2 - 3.6 x + 1 -3x2 + 3.6 x + 1

Solution :

solution :

xy

Which of the following is equivalent to 2 (1.2x - 1.5) - ( 4.44 x2 + 1.25) ?

N

4×2+442=4 me -

(

1.44×2

8xy=8n)

8h

3.6×-12.25

-

4.44212

-

1.25

-3×2-3.621+17 ✓

-

-

-

-1442=112×-2912-1

=

4×2-8 Xy

18

CONTINUE

2

9

Polynomials

11 Which of the following is equivalent to 3x 2 + 4 x ? x-2 A) 3 x - 10 x x-2

B)

3x + 10 +

C) D)

3x

20 x-2

Which of the following is equivalent to 4 ( x - 3) + 11 ? A) B) C) D)

x -1 12 x + 11 -8 x + 11 4x -1

Solution :

x-2

4( X 3) -

solution :

3×-110 X -2

4×-121-11

3X2t4X

1421-1-1

-

-13×2-6×-7 O

10X

1-

-

+11

dox

)

-20

-

O -120

so

.si?aXL--f3xtiotF#

-

10

12 Which of the following is equivalent to 2

æx ö ç - y÷ ? è2 ø

A)

x2 + y2 4

B)

x2 + y2 2

x2 - xy + y 2 4

C)

Which of the following is equivalent to x2 + 10 x + 17 ? A)

( x + 5)

B)

( x + 5)

C)

( x + 5)

D)

( x + 5)

2

+ 17

2

- 17

2

-8

2

+8

Solution :

2

x + xy + y 2 4

D)

XZ -110×-117

Solution :

( 21

-

y

)2=2¥

X 't

2.45×+52-52+17

-

2

-

Ig

-

y

-192

Cxt55

251-17

llxtT

-

19

CONTINUE

Test 1

2

13

Polynomials

15 Which of the following is equivalent to x2 + 4 x + 3 ? x2 - x - 2 A)

x+3 x+2

B)

x+3 x-2

C)

x +1 x-2

D)

x -1 x+2

( 5019 - 163x ) + 23( 7 x 2

2

- 217 )

The expression can be rewritten as mx 2 + n , where m and n are positive constant. What is the value of m + n ? A) B) C) D)

-30 -28 26 30

Solution : 5019

-

163×2+161×2

solution : -

YETI

=

2×2+28 a-

-

a

→ Mtn

=

-

-

4991

=

=

mxrdtnx

MXZTN

;n=ao 21-28

|mtN=26T

-

14

16 The expression 1 x 2 - 3 can be rewritten as 5 * +

(" − ))(" + ) ),

where m is positive

constant. What is the value of m?

15

f-

x'

F-

CXZ

2

+ 3x ) + 6 x

Which of the following is equivalent to expression above?

solution :

solution : -

2

A) " " − 9 B) - x2 - 9 C) - x2 + 12 x - 9 D) - x2 - 12 x - 9

A) -3 B) 10 C) 15 D)

(5 - x ) - 2 (7 - x

3=15 -

15)

-

15

=

(

Xtm

¥ -

=

) (

x

-

my

5

-

X

2-

14

-12×2-16×+64

1×2-9=7

( x' m2 ) -

m2

1m=f

20

CONTINUE

Test 1

2

1

Test 2

Polynomials

3 Which of the following is equivalent to 5 + 2? 3x - 1

A)

6x - 2 3x - 1

B)

6x + 1

C) D)

A) -3 B) 1 C) 3 D) 4

6x + 3 3x - 1 6x + 7 3x - 1

Solution : X

Solution

If x + 4 is a factor of the expression x2 + mx - 12 , what is the value of m?

Xta )

( Xt 4) (

:

3. X

,

=

means

XZTMX

-

12

-12=75+6×-22 -

factor

is

-14

4a=

1

-

12

a =-3

't

12=135×77 so ,

(

X -14

) (

X

-

3)

=

x

X

=x4mx

-

-

b

n

-

2

4

( 3x - 2 )( ax + 5) - 6 x

2

+ 7 = bx - 3

In the expression above, a and b are constant. What is value of a? A) B) C) D)

Which of the following is equivalent to 4 x - 3 ? 12 A)

2 3 9 11

B) C)

Solution :

3aX2tl5X

X' (

3A

-

6)

-

+

Lax X

(

-

15

-

6=0

IAI)

-

-

2A

)

3

-

=

bx -3

b

15

-

D)

solution :

o

3A

-6×47=6×-3

to

x-3 4 4x -1 3 x 1 3 4 x 1 4 3

4×-3

-12=4,7--7-2

2a=b

15-212)

-

b

=If

lb-Ft

-

-

21

CONTINUE

2

Polynomials

7

5

P ( x ) , has factors of

If the polynomial

4x 12 ÷ x - 3 3x - 9

6, ( x - 4 ) , and ( x + 5) , Which of the following must also be a factor of P ( x ) ?

Which of the following is equivalent to the expression above, given that x ¹ 3 ?

A) 6 x2 - 24 B) 6 x2 + 30

A) x B) 2x C) x - 3

C) x2 - 8x + 16 D) 3x2 + 3x - 60

D) x

Solution : P

( x)

3

=

6

(

x

=

6

(

x

=

\

-

4) (

't X

)

solution :

X -15

-

20

)

4x

tx ✓)

or

Tat '

'

II. YEI

12×312×2 ) =

12

-12×-40

=D

8

6 Which of the following is a factor of x3 - 8 ?

Which of the following is NOT equivalent to 8a 2 ?

A) x 2 - 2 B) " " + 2" + 4 C) " " − 2" + 4

A)

4a 2 + 4a 2

B)

( 2a 2 )

C)

8a ( a )

D)

( 4a ) × ( 4a )

D) x + 2 Solution

X

3-

8

one

:

=

X

=

(

of

3-

23

X

2) ( XZ -12×-14 )

892

=

2 =

=

=

492

-

2

=

892

892 16

a

2

-

the

factors

is

X2t2x-

22

CONTINUE

Test 2

2

9

Polynomials

11 What

is the coefficient of x 2 when 1 1 21x 2 - x + 3 is multiplied by the 10 x - ? 5 7 A) -5 B) -1 C) 1 D) 5 Solution

(

21×2

21×2

:

-

Fx

) (

-13

tox

-

T)

4 3 Which of the following is equivalent to x + x ? x3 + x 2

A) B) C) D)

x 1

solution :

xT¥ 2131×+1 )

x4 -1×3

X

-

x7 x5 2x

¥2

=

-1

.

7-

f-

-

3×2 -

x

-

10X

XLT

=

2×2=-5×2

⑤ 12

10 Which of the following is equivalent to "

#

. − 2/0 ? "

A) B) C) D)

2

A)

x + y2 4

B)

x2 + y2 2

C)

x2 - xy + y 2 4

D)

x2 + xy + y 2 4

-2g )

?

(1 − 6)" − 9 (1 − 6)" − 27 (1 + 6)" − 9 (1 + 6)" + 27

Solution :

AZ q2

-

-

12A

2.6

-

a

+9

+62

-

62+9

#

Solution :

(I

Which of the following is factor of polynomial of 1" − 121 + 9 ?

II

-

221.2g -1492

Ca (

a

-65-36+9 -612

-

27

23

CONTINUE

Test 2

2

13

Test 2

Polynomials

15 Which of the following is equivalent to 161\$ + 81" 5" + 5\$ ? A)

( 2a - b )

B)

( 2a - b ) ( 2a + b )

C)

( 4a + b ) ( a - b )

D)

( 4a

4 2

2

2

+ b2 )

2

2

2

Solution :

watt

8a2b 't b

-

4a2b

"

2

solution :

-1%72

I

( 4992+2

't

Which of the following is equivalent to ab - a ? ab + a 2 A) - a b B) - b a !"# C) #\$! !\$# D) #"%

ab

-

AZ

(497-627) ✓

acb =

a

-

)

Fa

y:⇒ 14

16 1 x? Which of the following is equivalent to 1 2+ x 2-

2x -1 2x + 1 2x + 1 2x -1

A) B) C)

4x2 - 1 x2

D)

1

Which of the following is equivalent to 4 x 4 - 4 ? A)

4 ( x 2 - 1)

B)

4 ( x 2 + 1)

C)

4 ( x 3 - 1) ( x + 1)

D)

4 ( x 2 + 1) ( x - 1)( x + 1)

2

2

Solution :

4x4

-

4=41×4-1 )

solution : =

2-

III

Lx

-

21-4 2x =

-

=

-

=

2x

-11

41×2-171×41 ) 4 ( 1) ( -1111×41) T ✓ x

-

x

-

X

I

×-

.

I 2×+1

4:I -

24

CONTINUE

2

Test 3

Polynomials

3

1 Which of the following is equivalent to 2 + 3 ? x

A) B) C) D)

4

5 4x 5 4+ x 8 3x 8 + 3x 4x

2 (5x ) - 5( 2 x ) 2

If x ¹ 0 , what is the value of

2

( 2 x ) × ( 3x )

HE ?

A) -1 B) 1 C) 5 D) 5x stfu

:

2/25×2)

-

5/4×2)

-6×2=50%-40×2

solution :

¥ 'T,⇒""IxT

-

,

HIFI

=D

2

4 3x 2 + 4 7 =m+ x -1 x -1 In the expression above, m is a constant. What is the value of m?

A) B) C) D)

3x - 5 3x + 3

C)

3x + 4

Solution :

-

i

A) B)

3x - 4

x

Which of the following is equivalent to

D)

¥¥3. -

'\$)

?

6-5 1+ 2 17 x+2 5 6x+2 6 5 x+2 x+2 6-

solution :

xt2T6¥

(3212-34)

(-6×-112)-3%32 ¥ 6Is=a-'f/

¥4

-

'

&'"(

17

,

m=

3×+3

25

CONTINUE

2

5

Test 3

Polynomials

7 5a 2 - 4a + 1 n = 5a + 6 + a-2 a-2 In the expression above, n is a constant. What is the value of n?

A) B) C) D)

2a - 1

6 x2 + 5x + 2 1 = +m 2x + 1 2x + 1

In the expression above, m is a constant. What is the value of m? A) B)

11

13

C) D)

11x + 13

Solution :

3x + 1 3x - 1

3x 2 + x 2 x2 + x + 2

Solution :

b- a -16

5a€at

a -2

( 5aZ

-

-

3×+1

÷÷t:÷÷i÷

Ioa )

2×+1675-125

-69+1

(6×2-1321)

.

-

' ÷

OTB

5at6+a!÷→ln6

8 P ( x ) = 2 x3 - kx 2 + 4 x - 5

In the expression above, k is a constant. If P(x) is divisible by " + 1, what is the value of k? A) B) C) D)

Which of the following is equivalent to

-13

A) B)

2

-11

C)

4-

8 t+2

D)

4+

8 t-2

4 - 2t

11

13

Solution :

divisible

21-113

4t ? t-2

solution :

by -

Xtl

KI 1)

'

-

PC 17=0 -

means

+4ft)

-

5=0

-1 72=4-1*24 4T

4 -

2

-

K

-

4-5=0

t

k/

-

2

Tft -

14T

-

8)

-

1- 8

26

CONTINUE

2

9

Test 3

Polynomials

11

3x2 + 5x - 6 = ( ax + b )( x + 2) + c

P ( x ) = 2 x3 - kyx 2 + 5xy + 2 y - 2 In the expression above, k is a constant. If P(x) is divisible by " − 1, what is the value of 6?

In the expression above, where a, b and c are constant, what is value the of a+b+c?

A)

-7

A)

-5

B)

-7 y

-2

C) D)

7

B) C) D)

7y

8

solution :

solution :

divisible

21113

0

by

X

-

I

Pcl )

O

3×2+5×-6=(3×-171×-2)

3×-1

Xt2M3x¥J

Ky @ 12+5

-

-

-

I

-

-2=0 y -12g

-

-4

w

to

dxtb

13×46×3

-

12

-12=0

Ky t5yt2y

-

My

O

-

6

C=

(x 2)

-

4

a =3 ;

;

-

atbtc =3

-

o

-

4

-

I -4

=f2J

1k=I 10

12

2 x 2 - 4 x - 3 = A ( x - 1) + B In the expression above, A and B are constant, what is the value of A in terms of x? A) B) C) D)

2" − 2 2" + 2 -2" + 2 -5

Solution x

-

I

P ( x ) = mx 4 + nx3 - 3x 2 + 5x In the expression above, m and n are constant. If P(x) is divisible by " − 1, and " + 1, what is the value of m?

A) B) C) D)

:

¥745.3 -

12×2

-

-2 1

3 5

Solution :

)

2x

divisible -

( 2x

-12 )

by

t

PCD

X -11

PC -11=0

X

-

-

o

-

o

2×2-4×-3

=

-

(x

5

-

1) ( 2x

-

2)

-

+

5

To

Mtn m

-

n

A

-

-

3-15=0 3-5=0

}

/ mm 't :

-2

8

-

2M

1A=2x-2J

b

-

Ky

=

X

-

=

6

m=3T -

27

CONTINUE

-

-

t

2

13

Test 3

Polynomials

15 For a polynomial P ( x ) , P ( -2 ) = 0 . Which of the following must be true about P ( x ) ?

P ( x)

A)

-2x is a factor of

B)

-2 x + 2 is a factor of

P ( x)

Which of the following is equivalent to x3 y - xy 3 ? x 2 y - xy 2

A)

x+ y

B)

x- y

C)

x - 2 is a factor of

P ( x)

C)

xy

D)

x + 2 is a factor of

P ( x)

D)

x+ y x- y

Solution :

PC

-

solution :

27=0

x3y

-

Xy

's

-

(

x -12

)

factor

is

XZY

P Cx)

of

-

=

My

( XZ

-

y

)

'

-

XYZ

Xy

(X

-

y)

lxfylcxty )

=

-

( xfy )

x-D

=

16

14 a

If the polynomial P ( x ) is divisible by " − 5, which of the following could be P ( x ) ? A) 2" " − 7" − 15

PC

"

p (

B) 2" + 7" − 15 C) 2" " − 7" + 15 D) 2" " + 7" + 15

5) 15)

PC 5)

PC 5)

=

=

O

to

=

=

30

100

Solution :

divisible PCs)

-

-

O

For a polynomial P ( x ) , P æç 3 ö÷ = 0 . Which of è4ø the following must be a factor of P ( x ) ? A) B) C)

3x - 4

D)

4x - 3

3x + 4 4x + 3

solution :

by or

-

5

x

-

x

means

5

is

a

factor

P

(F)

=

o

X

-

F

is

factor

a

in 4×-3

g4 X

-

3

is

28

a

also

factor of

be

can

factor

a

PIX ) CONTINUE

2

1

Test 4

Polynomials

3 1 A B = + x + 3x + 2 x + 2 x + 1

Which of the following is equivalent to

6 xy ( 4 x - 5 ) - 3x ( 8 y - 7 xy )

A) B) C) D)

In the expression above, A, and B are constant. What is the value of A × B ?

?

3xy

2

-2 B) -1 C) 1 D) 2

-18 15" − 18 18 15" + 18

A)

solution :

solution :

=

24×29

30

-

Xy

-

×÷×*=

2421g -121×24

-

3.

45x2y

=

XY

A -113=0

54xy

-

at

=

-

3.

3Xy (

=

,

}

AX-xA.tl?I+-z2B-=CAtBHtAt2BX2t3X

BB

'

,

a.

=

,

Xy

15×-18 )

-12

}

AB

C- DID

-

-

ID

=I5xJ

⇒ 2

4 ( xtl)

2 x + 3x + ax + b = ( c - 2 ) x + dx + c 2

3

In the expression above, a, b, c, and d are constant. What is the value of a + b + c + d ? A) B) C) D)

3 4 6 8

x

-

3)

In the expression above, A, and B are constant. What is the value of A ? B 5 A) Solution 4 321-2 XCATB ) + A 313 B) 7 5 3A -1313=9 31A -113=3 7 ⇒ C) A 313=-2 A -313=-2 4 ta D) 1 4 :

solution :

=

2×2-13×-1 axtb =

(

3x - 2 A B = + ( x - 3)( x + 1) x - 3 x + 1

2

-

Cc 2) -

X3tdX2tc

l÷n*⇐J

V

=

Ty

b=c=zLT

=

-

F-

3+2+2+2

=③

-

A

ly

Atbtctd

-

29

-

F

;

B

-_

IT

=I÷=÷ CONTINUE

2

Test 4

Polynomials

7

5 If a - b = 4, and a × b = 3 , where a and b are real numbers. What is the value of a 2 + b2 ? A) B) C) D)

10

If b - a = 8, and a - c = 10 , where a, b and c are real numbers. What is the value of ab + bc - ac - b2 ? A) 144 B) 108 C) 88 D) -144

16

22 28

Solution :

solution : a

-

=4

b

ab

=3

(

AZ a

} a

" -

=

?

abtbc

ac

-

b

-

'

w

-

Lab

-

94 b

-

Clb a)

b) 2=42

-

c.

-162=16

( b

-

a

b( a

t

)

-

b

)

b( b- a)

-

=

2.3+62=16 b

a4be+6y 1a

b

-

b) ( b

-

a)

-

t

t

-18

8

=

_

-

-

C

-10=8

go ¢ c t

't 62=22

(

-1444T

0=18

C

6

-

b=

-

18

8 If a + b = 4, and a × b = 3 , where a and b are real numbers. What is the value of a3 + b3 ? A) B) C)

52

D)

56

28 48

Which of the following is equivalent to t -1 ÷ t - 1? 1 -1 t A) B)

Solution :

Atb ab

-

4

=3

}

FIT

-2

0

C)

t 1- t

D)

t

solution :

t' .

-

a3tb3

I

-

t

#

Hht I

-

I -

t -1

t

F

fy.tt#/-128J 33+13

=

-

30

CONTINUE

2

9

Test 4

Polynomials

11 P ( x ) = x3 + 4 x 2 + 2mx - 5 In the expression above, m is a constant. If P(x) is divisible by " − 1, what is the value of m?

A) B) C) D)

4" " + 10" + 2 = (1" − 5)(:" + 3) +8 In the expression above, where a, b and c are constant, what is value the of a+b+c? A) 3 B) 5 C) 6 D) 9

-2 -1

0 1

Solution :

solution :

divisible

by

x

13+41112 I

-

I

PCI )

-12M t

(1)

-

4×2-1

O

10X -12

( ax b) ( -

Cx -13

)

-18

-

ACXZ

5=0

2m=

=

at

5=0

-

4 t 2M

-

ac=4

O

3A

mI

-

box

3421

} II

bc= to

b+s= ,

# at

10

-

btc

=

7

12

7 x2 + x + 5 = A ( x + 2) + B In the expression above, A and B are constant, what is the value of B? A) B) C) D)

7" − 13 7" + 13 13 31

P ( x ) = 2 x3 + ax2 + 5x + 2 In the expression above, a is a constant. If P(x) is divisible by x + 2 , what is the value of a? A) B) C) D)

Solution :

-6 -4

4 6

solution : TX

-

B

divisible

TEA

X -12 -

Ll 2) 3-1 -

(7×4140)

-

-

O

-

-

16

13×-15

(13×-26) f-

t

a

49

x -12

PC -21=0

(-212+51-2) -

-12=0

101-2=0

49=24

L)

AT

-

O

by

31

7×4×+5=(7×-13)

(

x -12

)

-13,1 B

31

CONTINUE

3b

2

Test 4

Polynomials

15

13 Which of the following is equivalent to a 2 - b2 a 2 - b2 + =? a -b a+b

Which of the following is equivalent to x - 1 x 2 + 3x + 2 ? × x+2 x2 - 1

A) B)

A) 1

C) D)

a 2a

1 x -1 1 x+2 x+2

B)

b 2

C)

Solution :

D) +

solution :

III 'Y#¥F¥i④ '

atxtaix

-42J

14

16 Which of the following is equivalent to x2 - x 1 - x2 ? ÷ x 2 - x + 1 x3 + 1

Which of the following is equivalent to x2 - 1 x2 + 1 ? x -1 x +1

A)

1- x

B)

A)

x +1

C) −" D) x

B)

Solution :

C)

!¥÷¥"⇒ I T * ¥ I

D)

2x -1 x +1 2x x +1 2x + 1 x +1 x

solution :

cx-nf.x.tn xI -

-

KD

=x4axII 32

xu

-

=

CONTINUE

atta

3

1

Test 1

3 If m > 0 and m 2 -9 = 0, what is the value of m?

What is the sum of all values of x that satisfy 12 x2 - 13x + 1 = 0?

A) B)

A)

C) D)

-9 -3 3

B)

9

C)

Solution :

m2

-

9=0

m2 M

-_

=

D)

9

Solution :

73

> o

m

12 13 13 12 1 13 1 12

M=3

i

12×2-13×+1=0 A

Xp

=

t

I

;

Xz

b= -13

=

-

by

C=1

;

-

=

-

13

-

12

tied 2

4

2t 2 + 46 - x = 0 If t > 0 and x = 4, what is the value of t ? 3

A) 1 B) 3 C) 6 D) 9

What is the product of all values of x that satisfy 2 x2 - 15x - 8 = 0? A) B)

solution :

C)

(TETH XP

D)

-

2 15 15 2 1 4 8

solution :

2+2+46--713=721--4 21-2+46=43--64

2×2-15×-8=0 e

×

242=18

2X

#

-

t

g 't

-

f- 2--9 t

>

o

;

;

t

=

I 3

-

next

X

=

}

(

)(

Xj 2×417=0

tf L

x=

-

I

-1521

t=3

33

CONTINUE

2

3

5

7 What are the solutions to 3x2 - 6 x + 1 = 0 ? 6 1± 3

A)

2 x3 + 3x2 - 7 x - 8 = 0 For what values of x , is the equation above

true?

6 3

B)

C)

-1 ±

6 3

D)

-6 ±

6 3

A) -2 B) -1 C) 1 D) 2 Solution :

Solution :

3×2-6×+1=0 a. =3

D=

b

;

=

-

6 ;

C

-

I

62-490=(-672-411113)--24 FITT

6

X ,z=

L

-

=

roots

roots

can

be

Let 's

plugin

-

X

-

-

4

-

-

,

-21=4

-2

,

-1,21214

I

7C t ) -

-

8=0

2+31-7-8=0

-

61

3

0=0

/1FI -

=

is

21-113+31-112

6/7524

-

product of

-1 :.x8

6 x3 ( x 2 - 10 ) = -9 x

If x > 0 what is the one possible solution to the equation above? A) 2 B) 3 C) 6 D) 9

What are the solutions of the quadratic equation 2 x2 - 10 x - 12 = 0 ? A)

x = -6 and x = -1

B) C) D)

x = -6 and x = 1 x = -1 and x = 6 x = 1 and x = 6

Solution :

solution :

2×2-10×-12=0×5

215

10×3

-

(

X

't

-

ax

XZ

10×3+921=0

-

X

=

-

10×2-191=0

-0×410×2+9=0

( X

-

-

5×-6=0

6) (

X -11 )

=

O

x

-

X2=1

-

3

'

-

X2=g

and

iX=6AX=# ✓

X=I3

X==, y

{

( X 1) ( x'-97=0

,

-1

,

O

,

1,3

}

34

CONTINUE

Test 1

3

9

11 What is the sum of solutions to ( x + 5.2 )( x - 1.2 ) = 0?

y = 2 x2 2 y + 4 = 2 ( x + 3)

If ( x, y ) is a solution of the system of equations above and x > 0 what is the value of xy ? A) B) C) D)

-2 -1 1 2

A) B) C) D)

-6.4 -4 4

6.4

Solution :

( X -15.2

solution : 212×2) -14=21×+3)

X,

=

-

) (

5.2

X

2×2

a

-

-

2x X

-

1=0

-

:*

X,

2=0

}

'

,

'""

21=1

1.21=0

Xz

;

4×2+4=2×-16 4×2

-

-12/2=-5.2

=

1.2

-11.2=1-447

12×+17=0 ;

X=

-

I

t

y

-

2

;

y

-

L

2 1.4--1-2=24 X. y -3.12=-4 -

10

12 2 x 2 - 5x = m In the equation above, m is a constant. If

equation has no real solutions, which of the following could be the value m ? A) -4 B) -3 C) 3

A) B) C) D)

D) 4

No

-

-4 -2 -1 2

solution :

solution :

2×2

In the equation ( bx - 3)2 = 25, b is a constant. If x = 2 is one solution to the equation, what is a possible value of b ?

5X

-

real

m

-

roots

O mean

in

plug

21=2

DLO

in

( bx

-

312=25

( 2b 3) 2=25 -

D= C- 5)

2

ML

-

4121km )

-

2b

-

2b

25

8-

-_

8

b=4T L

m=Ij -

3=5

35

and

2b

-

3=-5

26=-2

b L

CONTINUE

Test 1

3

13

15 / = 3" " − " − 3 / = −2" + 7 If ( x, y ) is a solution of the system of equations above, what is the one possible value of y ?

How many solution are there to the system of equations above?

A) B) C) D)

A) B) C) D)

-2 -1 1 11

X

-

3×2

X

t

y - 4x + 7 = 0

There are no solutions. There is exactly one solution. There are exactly two solutions. There are exactly four solution.

solution :

solution :

3×2

y = x2 - x + 3

-

-

X

# 3 X

XZ

3=-2×+7 10=0 2 -

5

y

=

-

1×+2 )

)

X= -2

X

-

I

=

t

D=

3

D=

3

4×-7

=

f

-

5)

25

2 -

-

4

-

48

I

12

-

-2320

=

-

21-2 ) -17 There

Y y

X

212-5×+12--0

(3×-5)=0 ;

-

-

solution

no

are

( Es ) -17

2

y= 16

14

ax3 + bx2 + cx + d = 0

3x + 10 + 5 = x + 7

What is the solution set of equation above? A) {-3, -2} B) {-3, 2} C) {-2,3}

In the equation above, a, b, c, and d are constant. If the equation has roots, -2, 3, and 7 , which of the following is a factor of ax3 + bx2 + cx + d = 0? A) B) C) D)

D) {2} Solution :

jzxtt ZX -110

X 't

X

( x -13 )

X -12

=

-

(

6=0 X

-

x-2 x+3 x+7

solution :

212+4×1-4

=

x-7

Roots

so

27=0 =

X Check

=-3

;

21=2

are

(

x -12 )

(

X

-

7

and

-2,3

3) (

X

-

7

)

gax

't

b x

-

t

ex

+

d

t.se/ai:5n :

31-37-1107=-3-12 1=-1

-110=2-12

4=4

fTmeJ

36

CONTINUE

Test 1

3

3

1 )" − ) − 20 = 0 If m is a solution of the equation above and m < 0 , what is the value of m ? A) B) C) D)

-5 -4

-3 -1

x 3x = x-2 3

Which of the following are a possible value of x ? A)

0 and 2

B) C) D)

0 and 3

(m

m

-

m

-

321=3×(21-2)

20--0

321--3×2-621

5) ( m -147=0

-

M

2 and 5

solution :

solution : m2

0 and 5

-

-

5

-4

}

3×2

Meo

921=0

3×(21-3)=0

MIT -

2

-

-1 121--0,21=32

4 9 = x-2 x-2

x + 6 = -x

In the equation above, which of the following is a possible value of " − 2? A) −9 B) −1 C) 3 D) 9

A) {−2} B) {3} C)

{-2,3}

D) There are no solutions. solution :

solution : 9-

What is the solution set of equation above?

( x 2) ( -

(X

-

x

-

2

( Fte

)

Xt 6

2) 2=9

XZ

73 =/ ✓ X

-

2

)

X

-

=

(X

-

t

35+6=-3 z

=3

False

37

-

=

-

3) (

1=3

'

f-

x

)

'

XZ 6=0 x

-127=0 t

X

=

-2

FEI

-

-

C-2)

2=2 True

CONTINUE

Test 2

3

5

Test 2

7

x2 = x Which of the following is the solution of the equation above? A) B) C) D)

-4

x2 - 4 =4 x-2

What are the solutions to the equation above? A) There are no solutions. B) x = 2

-2 2

C) D)

There are no solutions.

Solution :

x = -2 and x = 4

solution :

xf

Ex

x = 2 and x = 4

X

-

III a=o⇒*4jIa=o -

-

X

XZO

x'-4×+4

=

-

X

121=220-1

=o

2

-

212-4×+4

-

and

( X -232=0 X

X -21=0

X # 2

2

-

¥saqlict So

,

.

solution

no

.

8

6 x2 - 4 = -2 x+2

2x = x - 4

What are the solutions to the equation above?

What are the solutions to the equation above? A) B)

x = -2 and x = 0 x = -2 and x = 2

A)

{-8, -2}

B) C)

{8}

{2,8}

D) There are no solutions.

C) x = 0 and x = 2 D) " = 0

Solution :

solution :

XIII

+2--0=7%-4+22×+4

X42X -

X

(F) Efx -412 ax=x2

-0

(

XCX -127=0 and

x

-

2) ( X 81=0

€-21

X -12

and

xtzto xt -2

X

-

-

8×+16

-12×2-10×-116=0

=D

1/42×-0

-

2✓ conflict

check

:

52.2=2-4 2=-2 False

38

-

and

T X check

.F=8 -4 ra 4=4

time CONTINUE

3

Test 2

11

9

2 x2 = 5x + c

In the equation above, c is a constant. If x = 2 is a solution of this equation, what other value of x satisfies the equation? -1 2 B) 0

A)

Which of the following are solutions to the quadratic equation ( x + 2 )2 = 4 ? 9

A) B) C)

C)

1 2

D)

3 4

D)

8 4 x = - and x = 3 3 8 -4 x = - and x = 3 3 2 2 x = - and x = 3 3 x = -2 and x = 2

Solution :

solution :

-232×2-5×+2=0

X

2.22=5.2

211=2

,

Xz

=

f-

=

f-

-1C

=

c=

-

1+2=23

2

and

X -12

=

x=-÷and#) 3€

I

22/2=1

212--1-1 a 12

10 25 = 10 x In the equation above, if x > 0 , what is the solution of the equation?

In the equation ( ax + 1)2 = 9, a is a constant. If x = 2 is one solution to the equation, what is a possible value of a ?

A) B) C) D)

A)

-3

B) C) D)

-2

x+

-5 1 2

5

-1 2

solution :

solution :

-1172--9×2=10 '

X

X=2⇒ ( za

-125

La -11=3 20+1=-3×2-10×+25 20=-4 and

×_=0

2a=2

9=17 ↳

2/2-10×+25=0

9=-27 ✓

(X 572=0 -

21=547

and

X

#

O

39

CONTINUE

3

13

Test 2

15 x2 - x - 6 = 3x + 10 x+3 In the equation above can be solved for two solutions, one of which is extraneous. What is the value of the extraneous solution?

A) B) C) D)

2 y = 3 y 12

What are the solutions to the equation above? A) B)

-3 -2

C)

2

D)

3

{-2, 2} {-4, 4} {2,8}

{-2

2, 2 2

}

solution :

solution :

Ty

1×-3447522=73×+10

3.

+

Xt-3T extraneous

-

' =

y

y

X -13 to

-

I 24

2--8

T y=F2€ root

16

14 If 6= " − 5= − 6 = 0, what is the possible value of t?

The equation x2 - 4 x = -1 has two solutions for x . What is the larger value of x ?

A) - 3

A)

2-

B)

3

C)

2+

D)

5

2

B) - 2 3

C)

2 3

D)

3 4

3

3

solution :

XZ

Solution : let 3T

2-

5T

-

XZ

6=0

#

f

D=

72

-

-

=

-

1

4×+1=0

-

4)

4X

2 -

4117117--16-4=12

4725313ft g2) Lt

-3

-

4

x, ,z=

FITZ

=

( Lt -33=0

2

Xia

=

The

40

273T

larger

root

is

12-531

CONTINUE

3

1

3 3x 2 -4 x - 2 = 0

Which of the following is solution to the quadratic equation shown above? A)

4 ± 10 6

B)

2 ± 10 6

C)

2 ± 10 4

D)

2 ± 10 3

solution

If 3x2 = 63 - 12 x and x < 0 , what is the value of x 2 ? A) B) C)

36

D)

9

64 49

Solution :

3×2+12×-63=0 x

(

X

3×2-421-2=0

x ✓

4)

µ

=L

I

Xp ,2=

-

2 -

41371-21--16+24=40

XO

4×-21=0

't

(Xt7 )

:

;

-

37=0 and

21=37

X

4j.FI#=4F2I 6

×

"

C

-

/xs=2;/

712=1497

-

2

4 2 x2 = 7 x - m

If one of the solutions of the equation above 3 x = - , which of the following is the other 2

possible value of x ? A) −15 B) −5 C) 2 D) 5

27

D)

125

-125 -27

'

X

(

2×2-7×-1 ,

A) B) C)

solution :

solution :

X

If x2 = 2 x + 15 and x > 0, what is the value of x3 ?

m=o

X

;

=

,

-

I

Xz

=

X

2X -

-

5) (

15=0

X

-131=0

1¥51

-1×2=-21 =3

-3--1

-

¥-3

I

X

It

> O

x3= 53

1254T

212=57 ✓

41

CONTINUE

Test 3

3

Test 3

7

5 Which of the following equation has NO real solutions? A) x2 - 5x - 2 = 0 B) x2 - 3x - 3 = 0

D

>

D

>

C) x - 6 x + 11 = 0 D) x2 - 2 x - 15 = 0 . 2

O

0

A

0

Solution :

3x2 = 10 x - 8

If m and n are the two solutions to the equation above and m > n , what is the value of n 2 ? A)

9

B)

4 9 *&

C) real

No

solution

D LO

solution :

2/2-5×-2=0 ; A =L

D)

3×2

5) 2-4-1.1-21=33

-

+ 4

10 X

-

t

8=0

4×2-3×-2=0 X 3 X

;

-

x

D= f- 3) 2-4-1.1-27=17

"

XZ A

-

15=0

2

-

X

III. III. nuts 2X

-

II

-

in ! ;

man

=L

X

;

;

}n=f

(-2/2-4.11-15)=64

=

6

8 2 x2 + 8x + 4 = 0

What are the solutions to the equation above? A) −2 ± √2

What is the sum of the solutions of the equation 2 ( 2x - 3) = 4 x - 3 ?

-4

B)

-4 ± 4 2

A) B)

C)

-3

4±4 2

C)

3

D) 2 ± √2 solution

D) 4

:

solution :

4×2-12×+9 4×-3×2+4 =

X

D= x

,

,a=

t

2

0

=

42-4.2

.

I

=

-472€

4×2

8

=

-41=25-2 2

X

( x

-

16 X t 12

-

Z -

0

4×+3=0

3) ( X

X. =3

=

;

-

X

1) -

=O

I

XatXJ -

42

CONTINUE

3

9

Test 3

11 x2 + 2 x - 7 = 0

What are the solutions to the equation above? A)

-1 ± 2 2

B)

1± 2 2

B)

-1 ± 4 D) 1 ± 4 C)

C) D)

Solution : X

't 2X

D=

X.

Which of the following are solutions to the 2 quadratic equation x - 3x + 2 = 0 ? x2 + 5x - 6 A) {1, 2}

22 =

4

-

Solution :

7=0

-

{2} {-6,1} {-1}

11-77=32

-

(

X

-

2) ( x

t

)

=O

-2'r (x

12711 ) -

=

2

F4f2

-

2) (

x

21=2

-

;

-

2

=

-

17=0

(

and

X -16

) ( X DFO

Xt

21=1

-

-

6

;

conflict

-1-+271 ✓

XII L

12

10 x4 - 81 = 0 In the equation above, if x < 0 , what is the

solution of the equation? A) -9 B) -6 C) -3 D) -2

Which of the following are solutions to the quadratic equation x3 + 10x2 + 24x = 0 ? A) {0} B) C) D)

solution :

X

X

=

{-6, -4,0} {-6, -4} {-2, 2} solution :

214=81

X

20

;

(212+10×+24)=0

XCX -1471×+61=0

I3

21=-37 -

X=O;X=-4;X=# -

43

X# I

CONTINUE

3

13

Test 3

15 x + y = 5ü ý x× y = 6 þ

Which of the following are solutions to the quadratic equation 6 x2 + 5x = 4 - 7 x - 3x2 ?

What are the solutions to the system of equation above?

A) - 2 ! 2 2 3 B)

12 ! 2 2 18

C)

2 ! 12 2 3

B) C)

D) There is no solution.

D)

solution :

solution :

3212

=⇒÷=⇒÷÷¥÷÷f÷÷÷÷n

6×2+521=4

-

7 X

-

X

9×2+12×-4=0

122-41911-47=288

D=

x.

{(1,3)( 2,1)} {( 2,3)(3,3)} {( 2,3)(3, 2)} {( 3, 2 )}

A)

5-

=

Xy

if

6

=

(g y ) ( y )

=

-

7=2

6

1-273252-1

=

-

14

16 If the equation of mx 2 + ( 2m + 1) x + m - 1 = 0 , has double root, what is the possible value of m?

What is the product of the roots of the equation of x 2 + 2 x - 4 = 0 ?

A) 8 B) 4

A)

2

B) C) D)

- -2

C) 1 8

D) - 1 8

D=

-4

Solution :

solution D=

4

:

o

(2

me

1) 2-

4

Mlm

412 -14M -11 -14mF 8M

=

X

root

double

-

-

17=0

't fax

q=1

b

;

4m=O

X

l

,

Xz

-

=

4=0

=D

8-

=

c

;

-4T

=

=

-

4

-447

1m= -

44

CONTINUE

3

1

3 ax 2 - 2 ( a - 1) x + a = 0

3x2 + 2x - 1 = 0

If x1 and x2 are solution of the equation above and x1 = x2 , what is the value of a ? A)

-

B)

1 2 0

C) D)

1 2

Which of the following is a possible value of x ? -

B) C)

-1 1

2 3

D)

Solution :

1

solution :

X

=

,

D=

1 3

A)

3×2+2×-1=0

212

f- 21A

-1772

-

492-89+4

XX

1=0

4A

-

ZX -

a

1 -

I

-

( Xt 1) (3×-1)--0

402=0

=/

80--4

a=

or

x=J

4

2

2 x2 + ax + 2 = 0 In the equation above, one of the roots of equation is -2, which of the following is a possible value of 1? A) B) C) D)

1 2

3 5

Solution : 2×4

ax

21-212-1

+2=0 ;

X ,

=

-2

2 x2 + mx + m - 2 = 0

If x1 and x2 are solution of the equation above and x1 = x2 , what is the value of m ? A)

1 2

B)

2

C)

8 3

D)

4

Solution :

at -21+2=0

2×2-1

29=10

a=5

x

,

=X

MX t

(

,

D= m2 m

m

-

4=0

4.2cm

-

2=0

-

2)

m2

-

8M -116=0

4) 2=0

-

MII L

45

CONTINUE

Test 4

3

5

Test 4

7 Which of the following is the sum of the roots of the equation of ( x - 1)( x - 4 ) = 2 - 2 x ? A) 2 B) 3 C) 4 D) 5

x2 - 2tx + t 2 = 0

If x1 and x2 are solution of the equation above and x1 + x2 = x1 × x2 , what is the value of t ? A) B) C) D)

Solution :

-2 -1 1 2

solution : '

X

-

5×+4=2

x'

-

X

,

-

2X

3×-12=0

-1212

=

-aI

-

=

=

t2

-

Xz

'

,

tf

=

1

Lt

3D

X

=

-2ft

-

II

Xz

f-

X,

=

t2

27=0

+ It -27=0

_g t=2

f- =O

6

8 " " − 13" − 36 = 0 If x1 and x2 are solution of the equation above, x + x2 what is the value of 1 ? x1 × x2 13 12 1 3 13 36 0

A)

C) D)

solution : ,

A) B)

D)

{7} {1, 7} {-7, -1} {-1,7}

Solution :

TF

=

X -2

2×+11=212-4×+4 212

± a

¥,=z=-bT= X

What are the solutions to the equation above?

C)

-

B)

2 + 2x + 11 = x

-1212

=

-

¥6

1-3%7 -

(x

-

7) (

X

-117=0

and

check

TET

=

a

5=5

46

#

-2

0, what is the solution of the equation? A) 6 B) 9 C) 12 D) 36 Solution :

solution :

320=36 d d

(d

' -

-

la

36 d

) (

d- 16

d

-

t

-

d2 320=0

201=0

or

212+36 =

y

XZ

-

12×+36

-

=

O

X

d=2I

2=0 (X 6) -

12

50

and

X # O

CONTINUE

Test 5

3

9

Test 5

11 1" ! + 5" " + :" + G = 0

What are the solutions to 6" " + 5" = 4 − 7" − 3" " ? A) B) C)

In the equation above, 1, 5, :, 1JG G, constant. If the equation has roots −5, −3, 1JG 4 , which of the following is a factor of 1" ! + 5" " :" + G = 0 ?

"±"√" !

A) " − 5 B) " − 3 C) " + 4 D) " − 4

"±%"√" ! *%"±"√" %&

Solution :

D)

*"∓"√" !

roots

5

-

,

Solution :

txt 3) ( x

( X -15 )

9×2+12×-4=0 144-41911-47=288 A

-

-3

and

4

4) =ax3tb2x+cx+d

--

71.2=-12 FF - =

2.9

=

-

1271252

-

18

-2727J ⇐

10

12 !

I= " + 15 + " = 11 In the equation above, if " = 7, 1JG = > 0, what is the value of =? A) −7 B) 7 C) 14 D) 49

What are the solutions to " + √" = 6 ? A) B) C) D)

{−3, 2} {4, 9} {−3, 2, 4, 9} {4}

Solution : Xt

solution :

FEE -17=11 3¥15 THI

-_

4

5=64

Tx=t⇒X=t2 t2_

t

>

0

=

;

-

6=0

f- =-3

Fx

I 7

No

t=If

t

( t -13 ) (t

+2=49 t

Fx -6=0

51

=-3

solution

-

;

27=0 t

-

2

fI=2

x=4a CONTINUE

3

13

15 What are the solutions to A) B) C) D)

# " *#*", # " *"#*"\$

=0?

If −2 is one of the roots of the equation 3" " − (1 − 5)" − 6 = 0, what is the value of the other root of the equation?

{−4, 5} {−4, 5, 6} {5} {5, 6}

A) 3 B) 1 C) −1 D) −3

Solution :

(x

-

5) ( Xt 't )

solution : =D

-6321=5

-

(

X

-

6) ( x -14 )

( X 5) -

(

X

;

-141=0

( Xt 6) ( X -41=0 Xt -6

21=-4

Xi

-

X

-2 's

,

212

=

g-

=

X , 212=-2

:X -44

-

conflict

2×2=-2

Xz7

XT

14

16 " " + √2" − 4 = 0

What are the solutions to " ! + " " − 4" − 4 = 0 ? A) B) C) D)

A) √2 B) -√2 C) 4 D) -4

{−2, 2} {−2, −1, 2} {−1, 1, 4} {−1, 1}

solution :

solution :

X' ( Xtl )

1×2-4 )

-

41×-117=0

Xi Xz

( X -111=0

212-4=0 21--1=2

X -11=0 and

x=

-

y

{-2,-iT/ -

A-

1

=

,

X' Xa

52

Ig b

-

-

T2

c=

-

4

,

=

-447

CONTINUE

Test 5

4

1

Functions, and Parabola

Test 1

3 f ( x ) = mx 2 + 24 For the function f defined above, m is a constant

and f ( -4 ) = 8 . What is the value of f ( 4 ) ? A)

-8

B) C) D)

-1

8

Solution :

) f- ( x

)

solution : MX 't 24

-

-47=8

them

-

=

fix ) f-

(

What is the value of f g ( f ( 5) ) ? A) 3 B) 4 C) 6 D) 11

4

ft

A function f satisfies f ( 3) = 11 and f ( 5) = 6 . A function g satisfies g ( 3) = 4 and g ( 6 ) = 3

16

=

14 )

=

-

-

ml ⇒

X

'

-

f-

4) 2+24=8

m=

-

1

(

f (5) =g

g 16 ) )

so

f

so

fl3#) ✓

-124

42+24=-161-24

( 91ft 5 ) ) )

9161=3

fl4 L 2

4

g ( x) =

1

( x - 3)

For what value of undefined?

2

+ 4 ( x - 3) + 4

x is the function g above

A) -2 B) -1 C) 1 D) 2

y = a ( x - 3)( x + 5) In the quadratic equation above, a is nonzero constant. The graph of the equation in the xy plane is parabola with vertex (ℎ, 6). Which of the following is equal to k ?

A) -16a B) -12a C) 12a D) 16a

solution :

undefined is

denominator

means Zero

solution :

3) ( X -15 ) 9=9 ( X

.

-

(

XZ

X

-

-

3) 2+41×-31+4=0

2=-5×2

6×+9+421 -

2X

(x

-

+

-

X , =3

12+4=0

1=0

15=0

×T

ah

-

k=

X

,

t

X

,

Xz

=

a-

f

(h )

-

32-52=-1

fl

ft

-

-

1)

t

) =

-

-

al

-

I

-

3) ft -15)

16A

k=-I6T/ ✓

53

CONTINUE

4

Functions, and Parabola

7

5 In the xy - plane is parabola with equation

y = f ( x ) = ( x - 8) intersects the line with the equation y = 9 at two points, M and N . What is the length of MN ? 2

A) 5 B) 6 C) 11 D) 16

y

A) B)

-2

3

C) 5 D) 7 Solution :

solution -

2 x+n 3 In the function above, n is a constant. If f ( 6 ) = 9 , what values of f ( -3) ? f ( x) =

:

(x

-

812 (X

X

-

,

-

3-

8) 2=9

8=73

( 61--9

+

y=9

I

6th

-

g

n=5

×=yy

f-

×=g

1×7=2-3211-5

f C-

#

¥!e

37=3-1-3)

-15

tt3T

N

M

IMNI

-

6

8

6

If f ( x ) = -3x + 7 , what values of f ( -2 x ) ? The table above shows some values of the linear function f . Which of the following defines f ?

A)

f ( x ) = -6 x + 7

B)

f ( x ) = 6x + 7

A) B) C) D)

C)

f ( x ) = -6 x + 3

D)

f ( x ) = 6x + 3

2x - 3 2x -1 2x + 1

Solution :

2x + 3

solution :

fox)

f- ( x )

fly )= Mtn

-

-

-

f-

121--1

2mtn=1

-1

-3×-17

fl 2×1=-31-2×7-17

mxtn

I

-1

=

ft2X)=6X ✓

I

-mImm÷÷]Ii¥ax Mtn

=

-

54

CONTINUE

Test 1

4

9

Functions, and Parabola

Test 1

11 In the xy - plane, the graph of function f has x - intercepts at −2, 1, 1JG 4. Which of the following could define f ? A)

f ( x ) = ( x - 2 )( x + 1)( x + 4 )

B)

f ( x ) = ( x + 2 )( x + 1)( x - 4 )

C)

f ( x ) = ( x - 2 )( x - 1)( x + 4 )

D)

f ( x ) = ( x + 2 )( x - 1)( x - 4 )

A) 2 x - 5 B) 2 x + 5 C) 3 x - 5 D) 3 x + 5

Solution : X

-

at

intercept zeros

Are

-2

,

The table above shows some values of the linear function f . Which of the following defines f ?

I and

off function

4

solution : f (2)

11 ;

=

f

(4)

=

It

f- ( 61--23

;

.

f- ( x )

so

flxt2xx4 #

=

mxtn

2Mt

( 4mi

-

-

n n

-

-

M

17

)

-

-

2M m

h

10

=

-

6

f- 1×7=3×-15 \ } -

3

=

5

=

12 h ( x ) = ( x - 6 )( x + 4 ) Which of the following is an equivalent form of the function above in which the minimum value of h appears as a constant?

In the equation y = x2 - m and m is a positive constant the graph of the equation is a parabola. Which of the followingt is an equivalent form of the equation? XZ m2 → A) y = x - m x + m

(

A) h ( x ) = ( x - 1) + 25 2

)(

(

)( x +

)

B) h ( x ) = ( x - 1) - 23

B)

y= x- m

C) h ( x ) = ( x - 1) - 25

C)

y = ( x - m) - m

D) h ( x ) = x 2 - 23

D)

m öæ mö æ y = ç x - ÷ç x + ÷ 2 øè 2ø è

2 2

solution : h (X )

=

=

XZ XZ

-

-

2x 2X

-

t

2

m

)

x

XZ

-

X2

-

2-

m

2M Xt

MI 4

24 t

-

I

-

24

hlx)=(x-D2-25f

55

CONTINUE

MZ

-

m

4

13

Functions, and Parabola

Test 1

15 In the xy - plane, the point ( 2,3) lies on the graph of the function g . If g ( x ) = - x 2 + t , where t is a constant, what is the value of t ? The table above shows some values of the linear function h . Which of the following defines h ( 3) ? A) 5 B) 8 C) 9 D) 12

A) -4 B) -1 C) 3 D)

7

Solution :

y=

91×1=-212

it

-

@ i3 )

i

3=-22 it

solution : hlx )

-

t=T/ ✓

hut

mxtn

=

-2

=

men -

=

( 4mtn

h 141=13

; -

2

)

13

=

-

-

3M

=

15

-

}hlXk5X

FEI

h( 31=5-3-7

,

14

7

-

-

16

The table above shows some values of the functions g and h and g. For which values of x is g ( x ) = h ( x ) ?

A) -2 B) -1 C) 3 D) 6

-

=

A) B) C) D)

4 6 8 9

Solution :

1×1=2×2

f( o.o)

solution : glx ) gu )

In the xy-plane, the graph f ( x ) = 2 x 2 - 9 x intersects the graph of the y = - x at the points (0,0) and (m,-m). What is the value of m ?

mxtn 12

;

9141=27

.

inn -7 ,

}glxl=5xt7

:¥ HIM

-

-

-

18

h

;

141=24

-

,

m

)

-

X

at Coco)

and

intersection

are

of

f- ( m )

18

9=2

b -46

}h

-

"" " " '

=

functions

.

-

(

m=o

5×+7=2×-116

M

msn.em.int: am

-

hlx )=gCx)

Im

y=

hlxtaxtb Atb

¥¥÷÷±

and

ax

-

m

-

4) or

=D

MIT ✓

321=9

56

CONTINUE

Cmn)

4

1

Functions, and Parabola

3 4 x + x - 12

h( x) =

2

For what value of undefined? A) B)

x is the function g above

-4

-3

C) 1 D) 4 Which of the following could be equation of the graph above? A)

y = ( x - 1)( x + 1)( x - 2 )

B)

y = x 2 ( x + 1)( x - 2 )

C)

y = x ( x - 1)( x + 2 )

D)

y = x 2 ( x - 1)( x + 2 )

Solution :

XZ

X

+

y

-

=

intercepts

(

( X -14 ) X= -4

X

-

37=0 7=3

;

t Makes

-1,0

are

( x -11 ) ( x ) ( x 2) -

,

undefined

function

g

solution : x

12=0

-

and 2

X

2

4 f ( x ) = 3x - 7 g ( x) =

1 2 ( x + 2) - 6 2 g ( x) = -x - 8 f ( x) =

4 - f ( x) 3

The functions f and g are defined above. What value of g(3) ?

The functions f and g are defined above. If f ( a ) = g ( a ) , what is possible value of a ?

A)

A)

-2

B) - 2 3 C) 2 3 D) 3

solution : f-

solution :

gt3 )

=4-z

-3

B) -2 C) 2 D) 4

4 =

-

13.3

-

7)

-

3

=

(a

+

2)

2 -

92+491-4

4-22

12

-

ca )

=

=

a

-

-

2A

-

-

8 16

12+201-16--0

a2t6a -18=0

( at 4)

}Ef

g

=

21 Ila -125-6

3

=

(a)

a=

57

(

-4

a

-127=0

;a=1

CONTINUE

Test 2

4

5

Functions, and Parabola

7 In the "/ − LM1JN, the graph of the polynomial function P crosses the " − 1"QO at exactly two points, ( m,0 ) and ( n,0 ) , where m and n are both negative. Which of the following could define P ? A) g ( x ) = ( x + m )( x + n ) B) g ( x ) = ( x - m )( x - n ) C) g ( x ) = ( x - m )( x + n ) D) g ( x ) = ( x + m )( x - n )

solution

=

a

ex

-

m

) Cx

-

n

)

The graph above shows K in the "/ − LM1JNO. Which of the following defines K ? A)

1 2 x + 4x + 7 3 2 y = - x2 + 4x + 7 3 2 y = x2 - 4x + 7 3 2 y = x2 + 4x + 7 3 y=

.BG⇐→ B) C)

D)

.

¥÷÷÷÷÷¥¥x⇒

. .

6

8 If f ( x ) = 3x 2 + 3 and f ( x - a ) = 3x 2 - 12 x + 15 , what is the value of a ? A) B) C) D)

-3 -1

If f ( x ) = 5x 2 + 7 x + 2 is graphed in the "/ − LM1JN, which of the following characteristics of the graph is displayed as a constant or coefficient in the equation? A)

1 2

÷÷÷÷÷÷÷÷÷÷÷÷÷

:

y-

intercept

B) x - intercept(s) C) y - coordinate of the vertex D) x - intercept of the line of symmetry

:÷÷÷÷÷÷

so

58

.im

is

.

.

CONTINUE

Test 2

4

Functions, and Parabola

11

9 In the "/ − LM1JN, the graph of function h ( x ) = x 2 + 6 x + 8 has two x - intercepts. What is the distance between the x - intercepts? A) B) C) D)

4

3 2

If f ( x ) = 3x - 5 , what is the value of f ( 0 ) ? x+5 A) -10 B) -5 C) -1 D) 1 Solution :

1

solution : h (x)

f

X 't 6×48

=

101=3101-54

=

-

l

o -15

712+6×+8--0

(X -14) (

X -121=0

21=-4

X

;

=

-2

✓ 2

distance is the

of

the

points

10

12 If f ( x ) = 3x - 2 x , what is the value of f ( 3) - f ( 2 ) ? A) B) C) D)

19

14 13 11

solution :

In the "/ − LM1JN, the graph of 2 g ( x ) = x + 7 x + 11 is shown above. If the function crosses the / − 1"QO at point ( 0, k ) , what is the value of k ? A) B) C) D)

) f- 13

-

f-

(2)

=

=

( 33

-

23)

127-8 )

-

-

(32-22) (9-4)

19-5--140

=

4

7

10 11

Solution :

(

O

,

K)

-7

9107=02-1710 ) g lo)

=

-111

11

59

CONTINUE

Test 2

4

1

Functions, and Parabola

3 2 If f ( x ) = 3x + 4 x - 1 , what is f ( -2 ) ? 2x + 3

A) B)

-6 -3

C) 3 D) 6 Solution : C- 2)

f

31-212-141-2 ) =

-

t

-

C

2

12

=

8

-

2)

-

-

-13

t

-

-4-13 =

=

-347

Which of the following could be the equation of the graph above? A)

y = ( x - 2 )( x + 2 )

B)

y = x 2 ( x - 2 )( x + 2 )

C)

y = x ( x - 2 )( x + 2 )

D)

y = x ( x - 2) ( x + 2) 2

Solution : X

-

intercepts

are

-

2

,

0.2

y=Cxe2)(x-2/ 4

2 g ( x ) = x2 + 6 x + 8

f ( x ) = 2 x 2 - 3x

The graph of the g in the xy - plane is a parabola. Which of the following intervals contains the x - coordinate of the vertex of the graph of g ?

The functions f and g are defined above. What is value of g ( -1) ?

A) -6 < x < -5 B) -4 < x < -2 C) -2 < x < 0 D) 2 < x < 4

A) −5 B) 1 C) 5 D) 15

solution : XZ

(X -14 ) X=

x

-

-

solution :

text

+

4

g ( x) = 5 - 2 f ( x)

8=0

gf

( X -121--0 and

X

intercepts

-

1)

=

=

=

-

2

Are =

so

vertex

5-

2ft

5-2 5-

-

t)

( 21-172-31-1 ))

212-13)

gtD=-5I

4 -

L

X

L

-

2

60

CONTINUE

Test 3

4

Functions, and Parabola

Test 3

7

5

In

the

xy - plane,

the graph of g ( x ) = 3x + 11x + 7 is shown. If the graph 2

The graph above shows f in the "/ − LM1JNO. Which of the following defines f ? A)

y = 3( x + 1) + 2

B)

y = 2 ( x + 1) + 3

C)

y = ( x + 1) + 2

D)

y = ( x + 2) + 3

solution

2

:

VI 112 ) -

2

) alex -1112-12

( 0,3

2

y

2

-_

crosses the y - axis at point ( 0,n ) , what is the value of n ? Solution :

A) - 11 6 B) 3

glo )

C) 6 D) 7

3

=

n

(012-11110)-17

-_

n

NIT ↳

3=91172-12 A.

=

1

y=Cxt1)2tI 6

8 If h ( x ) = x + 2 and h ( x + k ) = x - 6 x + 11 , what is the value of k ? 2

A) B) C) D)

2

-3 0 3

Solution : =

=

=

(

xy - plane,

the graph of function g ( x ) = x - x - 6 has two x - intercepts. What is the distance between the x - intercepts? 2

A) 6 B) 5 C) 2 D) 1

-1

h ( Xtk )

In the

( Xtk )

'

-12

212-6×-111 (

Xtk ) 't

X

solution :

XZ

-372-12

2

=

Ex

( X 3) 42 -

X=3

-

-

X

3) (

-

X

and

6=0

-121=0 X

=

-

2

Idistancebetweenxinterceptsaresf -

61

CONTINUE

4

Functions, and Parabola

11

9

If f ( x ) =

3x + 2 , what is the value of f ( 2 ) ? x +1

A) 9 B) 18 C) 27 D) 81 •

solution :

32-12

In the xy - plane, the graph of f is shown above. Which of the following defines f ( 0 ) ? A) B) C) D)

f- (2)

2+7

=

-1

4=33

0 1 2

=

solution :

fCo) 12

10 If f ( x ) = 6 x - 5 , what is the value of f -1 ( 2 ) ? A)

7 6

B)

6 7

C)

1 7

D)

7

x which of the following is equal to f (x)?

A)

x2 - 2

B)

x2 - 2

C) D)

2

(x + 3) + 2 (" + 6)" + 2

Solution :

solution :

f

If f ( x - 3) = x 2 + 6 x + 11 for all real values of

-

3) f (x -

112 )

m ⇒

=

6M

-

f (m )

=L

5=2

f (x f (x

f- ( X -13

-

z)

-

-

3)

3)

X 't 6 X -111

=

X2t6X -19+2

=

=

=

(

(

Xt

X

3)

2+2

-13+312+2

fcxi-CX-65.IT 62

CONTINUE

Test 3

4

1

Test 4

Functions, and Parabola

3 If f (x)= 5 x + k , where k is a constant, and K(−2) = −4. What is value of k ? A) −6 B) −4 C) 4 D) 6 Solution : f l 2) -

5

I 2)

tk

-

-

k

IO t

=

-

=

-

=

-

4 4

4

Which of the following could be the equation of the graph above?

KII L

A)

f (x)= x 2 + 2 x + 3

B)

f (x)= x 2 - 4 x + 3

C)

f (x)= x 2 + 4 x + 3

D)

f (x)= x 2 - 5x + 3

solution :

Xp

1

=

y

X

,

3

y

2=3

f IX )

=

=

=

=

(x

-

a a

t

,

(x C

-

(

O

t

) CX 3)

-

,

3) -

1) I 3) -

a

=

1

) CX -13)

y=X2-4X# 4

2 If

f (x)= 1- x 2 , and m is a positive real !

f (x)= x - x .

Which of the following represents a quadratic equation with no real solution zeros?

"

B) C) D)

1+

1 m2

:

A)

(m + 1)(m + 1)

#

(m - 1)(m + 1) m

f (x)- 1

B) ! " #

m2

1- m m2

f is defined by the equation

2

number, then ! " # ? A)

The functions

C) ! "

2

D)

2

\$

#

f (x)+ 3

Z

YI

\$ \$#%!

X

X

-

-

If

(2×2-1)

XZ

-

-

=

"

-

1=0 0

;

there

there

is

2×2-1=0 ;

X t 3=0

A

;

;

is

roots

roots

there

No real roots

< 0

Solution :

f

Hm )

=

I

-

tmz

( m 1) ( -

=

=

MIS m

2

mtl )

63

is

CONTINUE

roots

4

Functions, and Parabola

6

5

In the xy - plane, the graph of y = f (x) is shown. If the graph crosses the y - axis at The graph above shows f in the xy - planes. Which of the following defines f ? A) / = " " − 4" + 1

1 2 (x - 2) - 3 2 3 2 y = 2(x - 2) 2

C)

Solution :

2

y=

solution : V

y 1

(

2

,

=

a

=

a

y=

5 2 3

C) 4 D) 5

y = 4(x - 2) - 3

D)

A) B)

y=

B)

point (0, k ), what is the value of k ?

k=

Ix C

(x

-

-

coil )

and

-3 )

-

25

212 2)

-

X

'

02

-

-

k€7 ,

-

z

3

5×-14

;

(o

,

K

)

510 ) -14 →

y

-

intercept

a

2 -

3

y=X2-4XtIJ -

64

CONTINUE

Test 4

4

Test 4

Functions, and Parabola

9

7

f (x)= - x + 5 and g ( f (x))= 4x - 7 , what

If

is value of g (3)? A)

- 7

- 1 C) 1 B)

D) 7 Solution :

g

(t (

l

g

-

x)

)

Xt

4×-7

=

5)

=

4 X

7

-

✓ 3

g

plug

21=2

(-2+5)

412 )

=

In the xy - plane, the graph of f is shown above. Which of the following defines f ( x ) ?

in

-

7

9131=1-1 ✓

A)

y = x3 - 7 x + 6

B)

y = (x - 3)(x + 1)(x + 2)

C)

y = x2 - 2 x + 1

D)

y = x3 - 6x + 7

X

-

intercepts y

=

( X -13 ) (

solution : or

X

-

-

1

3,1

,

and

2

) ( x 2) =X3 -

-

7×+6

10

8 In the xy - plane, the graph of function with

Let

the

function

is the greatest value of f ?

A) - 15 B) - 8 C) 8 D) 15

A) - 2 B) 3 C) 5 D) 8

y -

I

=

y

a

X

-

14

3) ( -

X

-

-

-

-

-

-

O

;

5) ; 14

3) 14 5)

( X 3) ( x

=

by

solution :

(x

a

=

defined

f (x)= 3 - x - 5 for all real values of x . What

zeros at x = 3 and x = 5 and vertex at (4, - 1). What is the y - intercept of the graph?

solution :

be

f

y

=

l

-

-

i

D

aT

5) 3) C- 5)

) f ( x =3

l

-

X

-

51

-

Absolute For so

value

greatest the

is

value

greatest

always positive of f

value

,

of

Ix f

-

51=0 is

65

.

CONTINUE

3

.

4

Functions, and Parabola

3

1 If K(") = " " − ", what is the value of K(−1) + K(1) + K(2) ?

If K(") = 3"#*% , and K(2) = 5) + 2, what is the value of ) ?

A) 7 B) 6 C) 5 D) 4

A) 1 B) 3 C) 4 D) 5 solution :

solution :

fl

-

+ (

1)

1)

+ (2)

f- 1)

=

=

=

2 -

C 1) =L

f (2)

-

12-1--0

=

5Mt 2

322-1=5 m -12

22-2=2

ff¥¥2

27

)

=

5M -12

m=5I

4

2 If P(") = 3" + 2, and K(") = " " + 2, what is the value of P(K(2)) ?

If K(2" − 5) = " " + 2" − 1, what is the value of K(5)?

A) 8 B) 12 C) 20 D) 24

A) 36 B) 34 C) 25 D) −1

solution :

g

(f

solution :

( 2) )

=

=

g

f ( 2x -5)

( 242)

9161=3.6+2 =

200

f Cs )

=

f- 1215)

-

5)

X

42×-1

21=5

I

52+215)

-1

fl5)=34T -

66

CONTINUE

Test 5

4

5

Functions, and Parabola

7 K(") = 3)" − 7 For the function K defined above, m is a constant and K(2) = −13. What is the value of K(−2)? A) −1 B) 1 C) 13 D) 17

A function K satisfies K(11) = 5, 1JG K(3) = 1. A function P satisfies P(5) = 11, 1JG P(1) = 11. What is the value of K(P(K(3)))? A) 11 B) 5 C) 3 D) 1 Solution :

solution

:

f- (2) 3M

-

2

=

7

-

am

=

f- ( g ( f-

13

=

=

m

-

( gli ) ) f

13

-

)

f 137=1

gli )

-

-

11

fUD=5T ↳

-6 -

( 3) )

I

)

fl -21=-31-2

-

7--6-7=-10 8

6 ℎ(") =

% # # *%!# " -!'

For what value of " is the function ℎ above undefined?

A) −9 B) −4 C) 3 D) 4

Solution :

X4

-

13×2-136=0

(212-411×2-9)=0 2/2=4 21=72

X

X

? 9 =

I3

-

The graph of function K(") is shown above, .(*%)-.(") what is the value of ? .(*")-.(%)

A) −1 B) −1/2 C) 1/2 D) 1

solution : f C D= -

-

67

t

f- ( 27=2

-

-

fl -27=0 f 111=2

.

CONTINUE

Test 5

4

9

Functions, and Parabola

11

! \

'T ht

hl37=

'

htt

't

I

••→hlo)=

-1

The graph of function ℎ(") is shown above, 1(,)-1(!) what is the value of ? 1(%)-1(\$)

The graph of function K("), and g(") are shown above, what is the value of P(f(0)) + K(3)? Solution :

A) 2 B) 3 C) 5 D) 8

A) −1 B) 0 C) 1 D) 3/2

gfflo) ) tf 13) =

g

10

-12

=D

hits:3 :# =D

it's

2

t

+ o

=

Solution :

(5)

12

• •

E

fl 37--0 -

The graph of function K(") is shown above, what is the value of K(K(0)) + K(−3)?

The graph of function K(") is shown above, what is the value of K(5) − K(−1)?

A) −3 B) 0 C) 3 D) 4

A) −5 B) −1 C) 1 D) 5

solution :

f- Cfto

f-

))

-1ft

-3 )

V

3

o

13 )

solution :

fC5 ) t 3

-

fl

-

t)

t

I -2 )

31-2

to

-

=⑤

68

CONTINUE

Test 5

4

13

Test 5

Functions, and Parabola

15 If K(") = √" + 1, what is the value of K(3) + K(15) − K(24) ? A) −1 B) 1 C) 6 D) 11 Solution :

f- (3)

Ftl

=

The graph of function K(") is shown above, and P(") = 2K(" − 2) − 1, what is the value of P(K(8))?

=L

It:'t

solution :

A) 19 B) 20 C) 21 D) 25

g

(f

18 ) )

=

g

( 101=2 f 110-2)

-

-

Lf (8)

=

=

2

( 107

-

-

I I

=④

16

14 f lol

'll

f- (3) =D

A

To

f 181=9

fl 7) =D

@

V

f l

-

5)

=

O

If 141=-4 The graph of function K(") is shown above, and K(4 − J) + 6 = K(K(8)), what is the value of J?

The graph of function K(") is shown above, and K(K(−5)) + K()) = 2, what is the value of )? A) 0 B) 3 C) 4 D) 8

A) −2 B) −1 C) 0 D) 4

solution :

5) f- ( fl -

t

f ( m))

=

2

if

f 14

m) f lo) tf (

=L

let f

Solution :

( m)

=L ⇒

f

m f ( )

-

-4

14

n

-

)

6

t

)

-

Y

m-F)

L

69

)

n

f I -4 )

=

t 6

n

f 14

t

-

=

-

6

O

4

-

n

-

4

T n CONTINUE

I

4

Functions, and Parabola

3

1 In the "/ − LM1JN, K(") = (" − 3)" intersects the line with the equation / = 16 at points A, and B. What is the length of |XY|? A) 6 B) 7 C) 8 D) 11

-

X

3) 2=16 -

Which of the following is an equivalent form of the function above in which the minimum value of K appears as a constant? A) K(") = (" + 1)" + 9 B) K(") = (" + 1)" − 9 C) K(") = (" − 1)" − 9 D) K(") = (" − 1)" + 9

Solution :

(X

K(") = (" − 2)(" + 4)

solution :

3=7457,1 ! }

f- Cx )

-

=

IABI

-

7-1-11--80

X

' -

2x

-

8

212-2×+1

-

I -8

fhd=lX

4

2 In the "/ − LM1JN, graph of function K has " − QJ=NZ:NL=O at −1, 0, 1JG 2. Which of the following could define K?

In the "/ − LM1JN, point (1, 4) lies on the graph of function K. If K(") = −2" " + )", where ) is constant. What is the value of )?

A) K(") = "(" − 1)(" + 2) B) K(") = "(" + 1)(" − 2) C) K(") = "(" + 1)" (" + 2) D) K(") = "(" − 1)(" − 2)

A) 1 B) 3 C) 4 D) 6 solution :

solution :

f-

1×7=1×-11 ) ( X ) ( X -2)

Iv X= -1

Iv

Iv ;

X=o

;

X

( 114 ) -

on

2412rem

.

text

-

-

2x 't

mx

1=4

m/

-2

lies

70

CONTINUE

Test 6

4

Functions, and Parabola

7

5 In the "/ − LM1JN, the graph of function K(") = −" " + " + 12, has two " − QJ=NZ:NL=O. What is the distance between the x− QJ=NZ:NL=O?

In the "/ − LM1JN, the graph K(") = −" " + 4" intersects the graph of the y=x at the points (0, 0), 1JG (J, J). What is the value of n?

A) −7 B) 1 C) 4 D) 7

A) −4 B) −3 C) 3 D) 5 solution :

solution :

-

2/4×+12=0 212

X

x -4

) (

-

(

f- CX )

1=4

-

X

-131=0

n=

-

NZ

X =-3

n

#

n

-

o

-

-

n

,

)

,

n

-

-

x

)

f- CX)

in

't 4N

n

3h

O

=

In -31=0

distance.be/-weeT

x-interceptsi.SI) 6

(n

in

plug

y

;

(n

and

( 0,0 )

12=0

and

-212+421

=

nT

8 K(") = −" " − 2" + 15 The graph of the K in the "/ − LM1JN is a parabola. Which of the following intervals contains the " − :[[ZGQJ1=N of the vertex of the graph of K? A) −5 < " < 3 B) −3 < " < 5 C) −1 < " < 16 D) −3 < " < −1

Solution : -

X

2-

2×+15=0

X 't 2X

(

15=0

-

( X 37=0

X -15)

-

X= -5

;

vertex

is

X

so ,

-

X =3

between

ℎ(") = " " − " + 1 P(") = −30 + 2ℎ(") The functions ℎ and g are defined above. What is value of g(−3)? A) −4 B) 4 C) 9 D) 13 solution :

gl

-

3)

=

-

30 -12hL -3)

30+24-35 l 3) t D -

=

intercepts

-52×23-1 ✓

=

=

-

-30-12 -

-

191-3-11)

30426

91-37=-4-1 -

71

CONTINUE

Test 6

4

9

Functions, and Parabola

11 If ℎ(") = " " − ", 1JG ℎ(" − =) = " " + 3" + 2, what is the value of = ?

If K(") =

A) −2 B) 1 C) 2 D) 4

A) −16 B) −2 C) 2 D) 4

solution : hlx

-

t

)

212 x

' -

-

Ex

-

tf

-

2txtt2

txt )

-

Xt

t

-

=

X 't 3×+2 X

I3! 3;

III

-4

=

-

, what is the value of

+

2+3×-12

""

t

x

-

#-%

.(*!)-.(!) .(,)

?

solution :

-

-

# " *%

12ft 1) 2T -

-

I

=3

t =

-

-

y

=D

=

=3

2t=4

t I ✓

10

12 Let the function K be defined by K(") = 7 − 2|" − 3|, for all real values of ". What is the greatest value of K?

!#*2

If K(") = K(1) ?

# " *!#

, what is the value of K(−1) +

A) −6 B) 0 C) 3 D) 6

A) 0 B) 4 C) 7 D) 13

Solution :

solution :

fix) since we

7

=

21×-3)

absolute

can

so ,

-

value is

consider

greatest

fl always possible

of

f

is

1)

3t"t9)-

=

C- 1) 2- 31 D

=

1-13

-

f (1) 7

=

,

21×-31=0

value

-

=

3,175%

=3

=

.

1) tf It ) f l -

72

=-3

t

3=00

CONTINUE

Test 6

4

13

Functions, and Parabola

15

The graph of function K(") is shown above, K(") = " " − 2" + ) intercepts at the points A, and B, and |XY| = 4, what is the value of )? A) −3 B) −1 C) 1 D) 3

Solution :

-Ia=2yHtXz=2 X

,

Xz

-1212

-

=

Xp -4 -

A-

}

X

-

2×2=6

-

Xi -_

,

m=X

X3-

m=

,

-

A) ℎ(") = −" " − 4" − 5 B) ℎ(") = " " − 6" + 5 C) ℎ(") = (" − 1)(" + 5) D) ℎ(") = " " − 3" + 6 Solution :

13=4

Xz

What is the equation of function shown above?

-

4

X,

Vlh

,

-9 ,

h=Xyt# -52-1=-2

I

=

212

y=

-113 )

14

212=1

-5 ;

=

-

g

=

a

1×+511×-1)

a

1-2-1571-2

VC-2

-

,

a)

I -2 , -9)

;

-

)

D 's

9=1

y=#txH⇒y=x44x -5 16

What is the equation of function shown above? A) g(") = (" − 1)(" − 4) B) g(") = (" + 1)(" − 4) C) g(") = " " − 4" − 5 D) g(") = 2" " + 4" + 5

A) g(") = (" − 1)(" − 4) B) g(") = −" " − 4" − 5 C) g(") = −" " − 4" + 3 D) g(") = −" " − 4" + 5

Solution :

y

-_

a

( 0,4 )

IX 1) ( x -4) ;

4=91

-

-

1) I 4) -

a

-4

solution :

y=alxt5) g

y=Cx-t)Cx# ✓ y= 212-5×-14

What is the equation of function shown above?

-

y=

V

al -

(x

-

t)

) (-2-1)

-2-15

tix -15) (

x

-

I -2,9 )

;

a

--

-

y

D

y=-X2-4xT

73

CONTINUE

Test 6

Exponents & Square Roots TEST 1

TEST 2

TEST 3

TEST 4

1

C

1

A

1

B

1

C

2

B

2

D

2

D

2

A

3

D D

3

B A

3

B A

3

D D

6

C D

5

6

B C

5

6

B A

5

6

C B

7

B

7

D

7

B

7

D

8

A

8

C

8

C

8

D

9

D

9

B

9

C

9

B

10

D

10

A

10

B

10

A

11

11

12

C D

11

12

A B

11

12

A C

C B

13

B

13

D

13

C

13

B

14

D

14

C

14

B

14

A

15

B C

15

B A

15

A B

15

D C

4 5

16

4

16

4

16

4

12

16

`

Polynomials TEST 1

TEST 2

TEST 3

TEST 4

1

B

1

C

1

D

1

B

2

D

2

A

2

C

2

A

3

B

3

B

3

C

3

B

4

C

4

C

4

B

4

B

5

C

5

D

5

C

5

C

6

D

6

B

6

B

6

A

7

D

7

A

7

A

7

D

8

A

8

D

8

D

8

C

9

B

9

A

9

C

9

C

10

C

10

C

10

A

10

D

11

D

11

C

11

B

11

B

12

C

12

B

12

C

12

D

13

B

13

D

13

D

13

B

14

D

14

A

14

A

14

C

15

C

15

C

15

A

15

A

16

A

16

D

16

D

16

B

74

CONTINUE

TEST 2

TEST 3

TEST 4

TEST 5

1

C

1

B

1

D

1

B

1

B

2

B

2

C

2

D

2

D

2

D

3

B

3

B

3

B

3

B

3

B

4

D

4

A

4

D

4

D

4

C

5

A

5

C

5

C

5

B

5

B

6

B

6

D

6

A

6

C

6

C

7

B

7

A

7

C

7

D

7

A

8

C

8

B

8

D

8

A

8

A

9

D

9

C

9

A

9

D

9

D

10

A

10

D

10

C

10

D

10

B

11

B

11

B

11

B

11

B

11

D

12

C

12

B

12

B

12

C

12

D

13

D

13

A

13

A

13

D

13

C

14

D

14

B

14

D

14

B

14

B

15

A

15

D

15

C

15

C

15

B

16

A

16

C

16

D

16

A

16

D

Functions and Quadratic Functions TEST 1

TEST 2

TEST 3

TEST 4

TEST 5

TEST 6

1

D

1

B

1

B

1

D

1

D

1

C

2

C

2

C

2

B

2

C

2

C

2

B

3

D

3

A

3

C

3

B

3

D

3

C

4

A

4

B

4

A

4

D

4

B

4

D

5

B

5

C

5

C

5

A

5

A

5

D

6

A

6

D

6

A

6

C

6

C

6

A

7

B

7

B

7

D

7

C

7

B

7

C

8

B

8

A

8

B

8

D

8

C

8

A

9

D

9

C

9

A

9

A

9

B

9

A

10

C

10

D

10

A

10

B

10

D

10

C

11

D

11

C

11

C

11

A

11

C

12

B

12

B

12

D

12

D

12

B

13

D

13

A

13

A

14

C

14

C

14

A

15

B

15

A

15

C

16

A

16

C

16

D