SAT Preparation Diagnostic
On this page, you will find a very helpful SAT Practice Test shown below, which will give you valuable insights on how well you would do on an actual SAT Practice Test. Our math tutors are experts at crafting SAT Practice Tests. The SAT is divided up in to two parts, the first part is the Mathematics portion and the second part is on reading/writing. Each section has 20 questions, for a total of 40 questions. With not only correct solutions to the test, but with detailed explanations to each problem as well. This can be found at the end of each section of the test. It should take you a total of 70 minutes to complete all 40 questions. It is customized SAT prep for math success Good luck!
Part 1.SAT Math Practice Test
Instructions:

Calculator use is not permitted for this practice test.

You have 30 minutes to complete these 20 questions.
Multiple Choice Questions:

What is the value of 5/2−3/4?

A) 5/8

B) 11/8

C) 7/4

D) 13/8


If x2−4x−5=0, what is the value of x?

A) −1

B) 1

C) 5

D) −5


Which of the following is equivalent to sqrt49x4?

A) 49x2

B) 7x2

C) 7x

D) 49x


Solve for x in the equation 3x−4=2x+1.

A) −5

B) 5

C) 0

D) 1


What is the sum of angles in a triangle?

A) 90

B) 120∘

C) 180∘

D) 360∘


The probability of picking a red ball from a bag of 4 red, 3 blue, and 2 green balls is?

A) 1/9

B) 4/9

C) 1/4

D) 2/3


If f(x)=x2+2x+1, then f(−1)=?

A) 0

B) 1

C) 2

D) −1


The value of sin(30∘) is?

A) 1/2

B) Sqrt2/2

C) Sqrt3/2

D) 1


The slope of the line 2x−3y=6 is?

A) 2/3

B) 2/3

C) 3/2

D) −3/2


If 2^x=8, what is the value of x?

A) 2

B) 3

C) 4

D) 1

FreeResponse Questions:

Find the roots of x2−3x+2.

What is the area of a triangle with base 10 and height 5?

Solve the inequality 3x−4>5.

Calculate limx→3(x2−9)/(x−3).

Find the derivative of f(x)=x3−4x2+2x−7.

Determine the equation of the line passing through the points (3,4) and (1,2).

Calculate the volume of a cylinder with radius 4 and height 3.

If A and B are matrices, and AB=BA, prove that: A−B and A+B are commutative.

Evaluate the integral ∫(3x2−4x+1)dx.

If log2(8)=x, find the value of x.
Answers:

Multiple Choice: 1C, 2 AC, 3B, 4B, 5C, 6B, 7A, 8A, 9A, 10B

FreeResponse: 1. x=1,2

25

x>3

6

f′(x)=3x2−8x+2

y=x+1

48π

(A−B)(A+B)=A2−B2

x3−2x2+x+C

x=3

Explanations:
Multiple Choice Questions:
1. What is the value of 5/2−3/4?
To find the value, we need a common denominator:
5/2−3/4=10/4−3/4=7/4
Answer: C
2. If x2−4x−5=0, what is the value of x?
We can solve this quadratic equation by factoring:
(x−5)(x+1)=0
So, x=5 x=−1
Answer: A) −1 and C) 5 (Note: Multiple correct answers)
3. Which of the following is equivalent to SQRT49x4?
SQRT49×x4=SQRT49xSQRTx2=7x2
Answer: B) 7x2
4. Solve for x in the equation 3x−4=2x+1.
3x−2x=4+1=5
Answer: B) 5
5. What is the sum of angles in a triangle?
The sum of angles in a triangle is always 180.
Answer: C) 180∘
6. The probability of picking a red ball from a bag of 4 red, 3 blue, and 2 green balls?
Total balls = 4+3+2=9
Probability of picking a red ball = 4/9
Answer: B) 4/9
7. If f(x)=x2+2x+1, then what is f(−1)?
f(−1)=(−1)2+2(−1)+1=0
Answer: A) 0
8. The value of sin(30∘)sin(30∘) is?
sin(30∘)=1/2
Answer: A) 1/2
9. The slope of the line 2x−3y=6 is?
We can rewrite the equation in slopeintercept form y=mx+b:
2x−6=3y
dive both sides by 3, to isolate for y.
y=2/3x2
The slope m=2/3
Answer: A) 2/3
10. If 2^x=8, what is the value of x?
log both sides of the equation: log(2^x)=log(8). Then apply one of the
logarithmic properties, xlog2=log(8). Next divide both sides by log2 in order
to isolate for x. Hence, x=log(8)/log(2)
Answer: B) 3
FreeResponse Questions:
1. Find the roots of x2−3x+2?
To find the roots, we factor the quadratic equation:
((x−2)(x−1)=0
So, the roots are: x=2 and x=1.
2. What is the area of a triangle with base 10 and height 5?
The area A of a triangle is given by:
(base× height/2)=(5x10/2)=25
3. Solve the inequality: 3x−4>5.
solve for x, let 3x>4+5=3x>9=x>3
4. Calculate limx→3 (x29)/x3
We can simplify the expression by factoring:
limx−3(x−3)(x+3)/x3 =lim(x+3)=3+3=6
5. Find the derivative of f(x)=x3−4x2+2x−7?
f′(x)=3x2−8x+2
6. Determine the equation of the line passing through the points (3,4) and (1,2).
The slope of m is:
m=42/31=2/2=1
Using the pointslope form of a line y−y1=m(x−x1), with(x1,y1)=(3,4):
Substituting the coordinates into the point slope equation, we get: y−4=1(x−3)
solving for y, we get: y=4+x3. In other words, y=x+1.
7. Calculate the volume of a cylinder with radius 4 and height 3.
The volume of V is then:
V=π×r2×h=π×16×3=V=48π
8. If A and B are matrices, and AB=BA, prove that: A−B and A+B are commutative.
(A−B)(A+B)=A2−B2=(A+B)(A−B)
Since AB=BA, A2−B2=A2−B2, proving A−B and A+B are commutative.
9. Evaluate the integral ∫(3x2−4x+1)dx.
∫(3x2−4x+1)dx=x3−2x2+x+C
where C is the constant of integration.
10. If log28=x, find the value of x?
3log28=x=3
Part 2.
SAT Reading/Writing Test:
Time Limit: 40 minutes
Instructions:
Read the passage carefully and answer the questions that follow. You will have 10 multiplechoice questions and 10 shortanswer questions.
Passage:
(Excerpt from Plato's "Allegory of the Cave")
Imagine a cave where prisoners have been held since birth. These prisoners are chained so they can only see a blank wall in front of them. Behind them is a fire, and between the fire and the prisoners is a walkway where objects are moved. The prisoners can only see the shadows of these objects and hear the echoes of people talking. They believe these shadows and echoes to be the only reality. One day, a prisoner is freed and sees the outside world. He realizes the truth but finds it difficult to explain to the other prisoners.
Multiple Choice Questions:

What is the main idea of the passage?

A) Discovery of a new cave

B) The limitations of human perception

C) The joy of freedom

D) The power of fire


What do the prisoners represent in the allegory?

A) Philosophers

B) Ignorant people

C) Children

D) Animals


Why are the prisoners chained?

A) To protect them

B) To limit their perception

C) As a form of punishment

D) To keep them focused


What does the fire symbolize?

A) Wisdom

B) Illusion

C) Fear

D) Enlightenment


What is the significance of the shadows?

A) They are distractions

B) They represent reality for the prisoners

C) They are just a game

D) They are irrelevant


How does the freed prisoner feel about his discovery?

A) Happy

B) Confused

C) Enlightened

D) Indifferent


Why is it difficult for the freed prisoner to explain the truth to the others?

A) He is not articulate

B) The others are not willing to listen

C) He lacks credibility

D) The others have limited perception


What is the purpose of the walkway?

A) For people to exercise

B) To carry objects that cast shadows

C) To escape the cave

D) To separate the prisoners


Who are the people talking and moving objects?

A) Prison guards

B) Philosophers

C) Reality manipulators

D) Fellow prisoners


What is the allegory mainly critiquing?

A) The educational system

B) Human perception and ignorance

C) Prison conditions

D) The concept of freedom

Short Answer Questions:

In your own words, explain the main idea of the passage.

How does the allegory relate to realworld situations?

What does the freed prisoner learn from his experience?

Why do the prisoners believe the shadows to be real?

What moral or lesson can be drawn from the allegory?

How might the prisoners react if they were told the shadows were not real?

What role does the fire play in the prisoners' understanding of the world?

Why does the freed prisoner find it difficult to communicate the truth?

What might the prisoners' chains symbolize in a broader context?

How does the allegory explore the concept of enlightenment?
Answer Key
Multiple Choice:

B

B

B

B

B

C

D

B

C

B
Short Answer:

The main idea is about the limitations of human perception and how one's reality is shaped by their experiences and environment.

The allegory can relate to how people are often limited by their own perspectives or societal norms, unaware of greater truths.

The freed prisoner learns about the broader reality outside the cave and becomes enlightened.

The prisoners believe the shadows to be real because that is all they have ever known.

The moral is that enlightenment comes from questioning and breaking free from one's limited perspectives.

They might react with disbelief, anger, or confusion.

The fire creates the illusions (shadows) that the prisoners take as reality.

The freed prisoner finds it difficult because the other prisoners have never experienced anything beyond the cave.

The chains could symbolize ignorance, societal norms, or limitations imposed by lack of education.

The allegory explores enlightenment as a difficult journey that requires breaking free from one's limited perspective.
Below are very useful resources on the SAT, to help students get used to the structure and the types of questions asked, the links are posted for each of the websites. I hope you find this helpful.
Resources:
here are the links to resources and tips for the SAT:

Understanding the SAT Math Content:

Comprehensive guide on SAT Math content areas: PrepScholar Guide1.


Employing Effective Study Materials:

Kaplan’s SAT Math prep book: Amazon Link (Note: You may need to search for the exact title as the URL was not provided).

Official SAT practice tests: PrepScholar Resource1.


Identifying and Bridging Knowledge Gaps:

Tips on identifying knowledge gaps: Winward Academy Tips2.


Mastering Key Strategies:

Essential math formulas and application: PrepScholar Resource1.

Tips for tackling the SAT Math section: CollegeVine Tips3.


Regular Practice and Review:

Free SAT Math practice problems: PrepScholar Resource1.


Engaging a Competent Math Tutor:

For personalized tutoring services, you can explore www.STEM1Online.com

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