Sat Mathematics Practice Book With Solutions 3: Passport to Advance Math
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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
Table of Contents
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
Quadratic Equations Test 1 ……………………………………………………….
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
Answer Key ……………………………………………………….
74
Solutions ……………………………………………………….
76
1
PASSPORT TO ADVANCE MATH
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
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m+3 -n =0
3 9 27 33
1
CONTINUE
Test 1
1
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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-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
PASSPORT TO ADVANCE MATH
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)
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2
8 2
6
CONTINUE
Test 2
1
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
9
CONTINUE
Test 3
1
PASSPORT TO ADVANCE MATH
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
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Which of the following is equivalent to #$%&'($√*#' *$√&
A)
y
B)
5x y
C)
5 y
D)
5 3y
10
?
CONTINUE
Test 3
1
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
6ø
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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.
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17
CONTINUE
Test 1
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
20
CONTINUE
Test 1
2
PASSPORT TO ADVANCE MATH
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)
Unauthorized copying or reuse of this page is illegal
x-3 4 4x -1 3 x 1 3 4 x 1 4 3
21
CONTINUE
Test 2
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
22
2
CONTINUE
Test 2
2
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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)
Unauthorized copying or reuse of this page is illegal
&'"( '$)
?
6-5 1+ 2 17 6x+2 5 6x+2 6 5 x+2 x+2
25
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
4t ? t-2
4 - 2t
26
CONTINUE
Test 3
2
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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è4ø
A) B) C) D)
3x + 4 3x - 4 4x + 3 4x - 3
28
CONTINUE
Test 3
2
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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)
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2x -1 x +1 2x x +1 2x + 1 x +1 x
32
CONTINUE
3
PASSPORT TO ADVANCE MATH
1
Quadratic Equations
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)
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2 15 15 2 1 4 8
33
CONTINUE
Test 1
3
PASSPORT TO ADVANCE MATH
5
Quadratic Equations
7 What are the solutions to 3x2 - 6 x + 1 = 0 ? A)
1±
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
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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
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
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
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35
CONTINUE
Test 1
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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}
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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
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
5
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
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-3 -2 -1 2
39
CONTINUE
Test 2
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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
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40
3
3
CONTINUE
Test 2
3
PASSPORT TO ADVANCE MATH
1
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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
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44
CONTINUE
Test 3
3
PASSPORT TO ADVANCE MATH
1
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
5
Quadratic Equations
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 -
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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
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
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
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of x1 × x2 ?
47
CONTINUE
Test 4
3
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
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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
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
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49
CONTINUE
Test 5
3
PASSPORT TO ADVANCE MATH
5
Quadratic Equations
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 $320 profit? A) 12 B) 15 C) 20 D) 24
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!'
𝑥+ = 12 # In the equation above, if 𝑥 > 0, what is the solution of the equation? A) 6 B) 9 C) 12 D) 36
50
CONTINUE
Test 5
3
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
11 What are the solutions to 6𝑥 " + 5𝑥 = 4 − 7𝑥 − 3𝑥 " ? A) B) C) D)
"±"√" % "±&"√"
𝑎𝑥 ! + 𝑏𝑥 " + 𝑐𝑥 + 𝑑 = 0 In the equation above, 𝑎, 𝑏, 𝑐, 𝑎𝑛𝑑 𝑑, constant. If the equation has roots −5, −3, 𝑎𝑛𝑑 4 , which of the following is a factor of 𝑎𝑥 ! + 𝑏𝑥 " 𝑐𝑥 + 𝑑 = 0 ? A) 𝑥 − 5 B) 𝑥 − 3 C) 𝑥 + 4 D) 𝑥 − 4
% '&"±"√" &( '"∓"√" %
10
12 !
I𝑡 " + 15 + 𝑥 = 11 In the equation above, if 𝑥 = 7, 𝑎𝑛𝑑 𝑡 > 0, what is the value of 𝑡? A) −7 B) 7 C) 14 D) 49
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What are the solutions to 𝑥 + √𝑥 = 6 ? A) B) C) D)
{−3, 2} {4, 9} {−3, 2, 4, 9} {4}
51
CONTINUE
Test 5
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
15 What are the solutions to A) B) C) D)
# " (#(") # " ("#("$
=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}
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𝑥 " + √2𝑥 − 4 = 0 A) √2 B) -√2 C) 4 D) -4
52
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
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53
CONTINUE
Test 1
4
PASSPORT TO ADVANCE MATH
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
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54
CONTINUE
Test 1
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
(
m
)( x +
m
)
2
55
CONTINUE
Test 1
4
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
-3
B) -2 C) 2 D) 4
57
CONTINUE
Test 2
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
58
CONTINUE
Test 2
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
59
CONTINUE
Test 2
4
PASSPORT TO ADVANCE MATH
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
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g ( x) = 5 - 2 f ( x)
60
CONTINUE
Test 3
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
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
PASSPORT TO ADVANCE MATH
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
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D)
2
(x + 3) + 2 (𝑥 + 6)" + 2
62
CONTINUE
Test 3
4
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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64
CONTINUE
Test 4
4
PASSPORT TO ADVANCE MATH
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
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be
65
CONTINUE
Test 4
4
PASSPORT TO ADVANCE MATH
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
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66
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
67
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
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68
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
69
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
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70
CONTINUE
Test 6
4
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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
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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
PASSPORT TO ADVANCE MATH
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 -
Unauthorized copying or reuse of this page is illegal
1
CONTINUE
Test 1
1
PASSPORT TO ADVANCE MATH
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 →
Unauthorized copying or reuse of this page is illegal
2
CONTINUE
Test 1
1
PASSPORT TO ADVANCE MATH
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
answer
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
Unauthorized copying or reuse of this page is illegal
3
b
;
Fb b 't
'T ( at)↳= ( b )
⇒
ki I -
=
=
b
>o
's
{ tpaokweeraog
Iz both sides
bit
CONTINUE
1
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
6532
t
4
CONTINUE
1
PASSPORT TO ADVANCE MATH
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#
Unauthorized copying or reuse of this page is illegal
solution :
Fx Fx
-
y
-2
=
=
5
;
y
=L
5
(a) 2=175 -
5
X
=
49
CONTINUE
Test 2
1
PASSPORT TO ADVANCE MATH
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-
Unauthorized copying or reuse of this page is illegal
t
-
JE
since the bases are
F.
458
⇒
6
naff
-
-
8D
CONTINUE
1
PASSPORT TO ADVANCE MATH
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
' .
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/ ↳
Unauthorized copying or reuse of this page is illegal
7
CONTINUE
Test 2
1
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
8
CONTINUE
Test 2
1
PASSPORT TO ADVANCE MATH
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_/
Unauthorized copying or reuse of this page is illegal
9
CONTINUE
Test 3
1
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
10
CONTINUE
1
PASSPORT TO ADVANCE MATH
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
=
8¥
-
.
23
2213×-2 GX -4 921
)
=
-
=
=
2-3
-
3 X
CX
-
Cxtsy
g
)
-15
15
11
x-/ a
Unauthorized copying or reuse of this page is illegal
11
CONTINUE
Test 3
1
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
12
CONTINUE
1
PASSPORT TO ADVANCE MATH
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/ ✓
Unauthorized copying or reuse of this page is illegal
13
CONTINUE
2
1
PASSPORT TO ADVANCE MATH
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/
Unauthorized copying or reuse of this page is illegal
14
CONTINUE
1
PASSPORT TO ADVANCE MATH
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 : .
6ø
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 -
Unauthorized copying or reuse of this page is illegal
64
3×2=75
I
6
=
15
CONTINUE
Test 4
1
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
12
5 ? 108
HII
-
16
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
¥3
17
=I4-¥I
CONTINUE
Test 1
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
18
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
=/¥-xyT Unauthorized copying or reuse of this page is illegal
-
19
CONTINUE
Test 1
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
20
CONTINUE
Test 1
2
PASSPORT TO ADVANCE MATH
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
3¥
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
-
-
Unauthorized copying or reuse of this page is illegal
21
CONTINUE
2
PASSPORT TO ADVANCE MATH
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-
Unauthorized copying or reuse of this page is illegal
22
CONTINUE
Test 2
2
PASSPORT TO ADVANCE MATH
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
=¥-2xy+4y# Unauthorized copying or reuse of this page is illegal
23
CONTINUE
Test 2
2
PASSPORT TO ADVANCE MATH
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 -
Unauthorized copying or reuse of this page is illegal
24
CONTINUE
2
PASSPORT TO ADVANCE MATH
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 ?
⑧
A) -1 B) 1 C) 5 D) 5x stfu
:
2/25×2)
-
5/4×2)
-6×2=50%-40×2
solution :
¥ 'T,⇒""IxT
7¥
-
,
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
Unauthorized copying or reuse of this page is illegal
25
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
26
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
b
-
Ky
=
X
-
=
6
m=3T -
27
CONTINUE
-
-
t
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
28
a
also
factor of
be
can
factor
a
PIX ) CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
-
A
ly
Atbtctd
-
29
-
F
;
B
-_
IT
=I÷=÷ CONTINUE
2
PASSPORT TO ADVANCE MATH
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
=
-
Unauthorized copying or reuse of this page is illegal
30
CONTINUE
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
31
CONTINUE
3b
2
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
=x4axII 32
⇒
xu
-
=
CONTINUE
atta
3
PASSPORT TO ADVANCE MATH
1
Test 1
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
33
CONTINUE
2
3
PASSPORT TO ADVANCE MATH
5
Quadratic Equations
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)
6±
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
}
Unauthorized copying or reuse of this page is illegal
34
CONTINUE
Test 1
3
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
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 -
Unauthorized copying or reuse of this page is illegal
3=5
35
and
2b
-
3=-5
26=-2
b L
CONTINUE
Test 1
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
36
CONTINUE
Test 1
3
PASSPORT TO ADVANCE MATH
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
'
f-
x
)
'
XZ 6=0 x
-127=0 t
X
=
-2
FEI
-
-
C-2)
2=2 True
CONTINUE
Test 2
3
PASSPORT TO ADVANCE MATH
5
Test 2
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
check
:
52.2=2-4 2=-2 False
38
-
and
T X check
.F=8 -4 ra 4=4
time CONTINUE
3
PASSPORT TO ADVANCE MATH
Test 2
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
39
CONTINUE
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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
t=-2zandt=Z# ✓ Unauthorized copying or reuse of this page is illegal
2
Xia
=
The
40
273T
larger
root
is
12-531
CONTINUE
3
PASSPORT TO ADVANCE MATH
1
Quadratic Equations
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 ✓
Unauthorized copying or reuse of this page is illegal
41
CONTINUE
Test 3
3
PASSPORT TO ADVANCE MATH
Quadratic Equations
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 -
Unauthorized copying or reuse of this page is illegal
42
CONTINUE
3
PASSPORT TO ADVANCE MATH
9
Test 3
Quadratic Equations
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 -
Unauthorized copying or reuse of this page is illegal
X=O;X=-4;X=# -
43
X# I
CONTINUE
3
PASSPORT TO ADVANCE MATH
13
Test 3
Quadratic Equations
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= -
Unauthorized copying or reuse of this page is illegal
44
CONTINUE
3
PASSPORT TO ADVANCE MATH
1
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
45
CONTINUE
Test 4
3
PASSPORT TO ADVANCE MATH
5
Test 4
Quadratic Equations
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) -
T Unauthorized copying or reuse of this page is illegal
12
50
and
X # O
CONTINUE
Test 5
3
PASSPORT TO ADVANCE MATH
9
Quadratic Equations
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
Unauthorized copying or reuse of this page is illegal
t
( t -13 ) (t
+2=49 t
Fx -6=0
51
=-3
solution
-
;
27=0 t
-
2
fI=2
x=4a CONTINUE
3
PASSPORT TO ADVANCE MATH
13
Quadratic Equations
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/ -
Unauthorized copying or reuse of this page is illegal
A-
1
=
,
X' Xa
52
Ig b
-
-
T2
c=
-
4
,
=
-447
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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/ ✓
Unauthorized copying or reuse of this page is illegal
53
CONTINUE
4
PASSPORT TO ADVANCE MATH
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
=
-
Unauthorized copying or reuse of this page is illegal
54
CONTINUE
Test 1
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
55
CONTINUE
MZ
-
m
4
PASSPORT TO ADVANCE MATH
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
-
-
⑧
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
X=⑦ Unauthorized copying or reuse of this page is illegal
56
CONTINUE
Cmn)
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
g
=
21 Ila -125-6
3
=
(a)
a=
57
(
-4
a
-127=0
;a=1
CONTINUE
Test 2
4
PASSPORT TO ADVANCE MATH
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
÷÷÷÷÷÷÷÷÷÷÷÷÷
:
Unauthorized copying or reuse of this page is illegal
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
PASSPORT TO ADVANCE MATH
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
KH L Unauthorized copying or reuse of this page is illegal
59
CONTINUE
Test 2
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
60
CONTINUE
Test 3
4
PASSPORT TO ADVANCE MATH
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 -
K€31 Unauthorized copying or reuse of this page is illegal
61
CONTINUE
4
PASSPORT TO ADVANCE MATH
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
m=%/✓ Unauthorized copying or reuse of this page is illegal
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
PASSPORT TO ADVANCE MATH
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 )
T Unauthorized copying or reuse of this page is illegal
63
is
CONTINUE
roots
4
PASSPORT TO ADVANCE MATH
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 -
Unauthorized copying or reuse of this page is illegal
64
CONTINUE
Test 4
4
PASSPORT TO ADVANCE MATH
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
Y-15-7y-intercep.IT) Unauthorized copying or reuse of this page is illegal
65
.
CONTINUE
3
.
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
f Cs )
=
→
f- 1215)
-
5)
X
42×-1
21=5
I
52+215)
-1
fl5)=34T -
66
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
-
Unauthorized copying or reuse of this page is illegal
-
7¥
fl -27=0 f 111=2
.
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
4-10=40 Unauthorized copying or reuse of this page is illegal
68
CONTINUE
Test 5
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
-
=
-
⇒
6
O
4
-
n
-
4
T n CONTINUE
I
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
lies
70
CONTINUE
Test 6
4
PASSPORT TO ADVANCE MATH
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 -
Unauthorized copying or reuse of this page is illegal
71
CONTINUE
Test 6
4
PASSPORT TO ADVANCE MATH
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 -
Unauthorized copying or reuse of this page is illegal
72
=-3
t
3=00
CONTINUE
Test 6
4
PASSPORT TO ADVANCE MATH
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
Unauthorized copying or reuse of this page is illegal
al -
(x
-
t)
) (-2-1)
⇒
-2-15
tix -15) (
x
-
I -2,9 )
;
a
--
-
y
D
y=-X2-4xT
73
CONTINUE
Test 6
Answer Key
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
Unauthorized copying or reuse of this page is illegal
74
CONTINUE
Answer Key
Quadratic Equations TEST 1
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
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75
CONTINUE