Free Full-Length Adaptive SAT Practice Test Online with Complete Analysis - Module 1
MATH DIGITAL SAT PRACTICE-MODULE-1 -TEST -1
| YOUR TOTAL SCORE IS : 15 | YOUR PERCENTAGE IS : 68 |
SCORING SUMMARY
TOTAL QUESTIONS :22
TOTAL TIME TAKEN : 0
TOTAL CORRECT QUESTION :15
TOTAL INCORRECT QUESTION:7
TOTAL QUESTION ATTEMPTED :22
QUESTION AND ANSWER REVIEW
1. Module -1
If triangle ABC is a right triangle and B is 90° and the longest side of the triangle is 61 and the shortest side is 11, what is the length of the third side of the triangle?
If triangle ABC is a right triangle and B is 90° and the longest side of the triangle is 61 and the shortest side is 11, what is the length of the third side of the triangle?
40606259
Key Explanation: Choice B is correct. The Pythagorean theorem states that the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse of the triangle, or a²+ b² = c. Plugging in 11 for a and 6 1 for c yields 11² +b²=61² Therefore, b = 60.
2. Module -1
In the equation 3 = |x − 1|, how many solutions are there?
In the equation 3 = |x − 1|, how many solutions are there?
Exactly 1Exactly 2NoneInfinite
*(B) For an absolute value equation 3 = |x − 1|, set up two separate
equations to represent both the possible positive and negative values of
what is inside the absolute value symbol:
Equation 1:
3=x-1->x=4
Equation 2:
-3 =x-1 ->x=2
3. Module -1
If a/3 = 10-7b and a≠ 0 , which of the following correctly expresses b
in terms of a?
If a/3 = 10-7b and a≠ 0 , which of the following correctly expresses b
in terms of a?
\( b = \frac{a – 21}{30} \)\( b = \frac{30 – a}{21} \)\( b = 10 + \frac{a}{3} \)\( b = 10 + \frac{3}{a} \)
\( b = \frac{30 – a}{21} \)
4. Module -1
If \( \frac{x^2 – 6x + 10}{x + 2} = A + \frac{B}{x + 2} \) what is the value of B?
__________
If \( \frac{x^2 – 6x + 10}{x + 2} = A + \frac{B}{x + 2} \) what is the value of B?
__________
5-3A(1) 26
*Key Explanati on: 8 is the remainder when x² – 6x + IO is divided by x +2. When the divisor is equated to 0, we find x = – 2, we can then find the remainder by substitute – 2 in place of x . (-2² -6(-2 )+ IO )which yields 26, this is the value of B.
5. Module -1
Assuming Baahir’s heartbeats per minute and breaths per minute have a
linear relationship with one another, what would Baahir’s pulse (heartbeats
per minute) be if he is breathing at a rate of 24 breaths per minute?
__________
| Measurement | First | Second | Third |
|---|---|---|---|
| Heartbeats Per Minute | 60 | 70 | 80 |
| Breaths Per Minute | 12 | 15 | 18 |
Assuming Baahir’s heartbeats per minute and breaths per minute have a
linear relationship with one another, what would Baahir’s pulse (heartbeats
per minute) be if he is breathing at a rate of 24 breaths per minute?
__________
24(1) 100
*(100) Since the heartbeats and breaths have a linear relationship, you
can solve for the number of heartbeats per minute by noticing that the
breaths go up by 3 for every increase of 10 heartbeats per minute. So, if
Baahir breathes 24 breaths per minute, add 10 + 10 to 80 to get 100
heartbeats for minute as his pulse.
6. Module -1
If $$ \frac{x^{\frac{2}{3}}}{x^{\frac{1}{6}}} = \frac{x^a}{\sqrt{x}} $$, what is the value of a?
__________
If $$ \frac{x^{\frac{2}{3}}}{x^{\frac{1}{6}}} = \frac{x^a}{\sqrt{x}} $$, what is the value of a?
__________
1/2(1) 1
*When dividing exponential expression that have the same base,
subtract the exponents from one another:
The square root of x is equivalent to x^1/2. Substitute this in to the
equation and simplify:
\( x^{\frac{1}{2}} = \frac{x^a}{\sqrt{x}} \implies x^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = x^a \implies x^1 = x^a \implies x = 1 \)
7. Module -1
*A school has received a donation of $20,000 for the purchase of new laptops. If each laptop costs $149, no tax is charged, and the laptop manufacturer offers a 7.5% discount on orders of at least 100 laptops, what is the maximum number of laptops the school can purchase with
the donation?
*A school has received a donation of $20,000 for the purchase of new laptops. If each laptop costs $149, no tax is charged, and the laptop manufacturer offers a 7.5% discount on orders of at least 100 laptops, what is the maximum number of laptops the school can purchase with
the donation?
124134145146
*C
The question asks for a maximum value given a specific situation.
Since the question asks for a specific value and the answers contain
numbers in increasing order, plug in the answers. Rewrite the answer
choices on the scratch paper and label them “number of laptops.”
Next, pick a value to start with. Since the question asks for the
maximum, start with the largest number, 146. The question states that
each laptop costs $149, so multiply that by the number of laptops to
get ($149)(146) = $21,754. The question also states that there is a
7.5% discount on orders of at least 100 laptops. Since 146 is more
than 100, the discount applies. Take 7.5% of the cost and subtract the
result from the cost to get . This
is greater than the donation of $20,000, so eliminate (D). The result
was close, so plug in the next largest value, 145, for the number of
laptops. The initial cost becomes ($149)(145) = $21,605. Apply the
7.5% discount to get . This is
less than the donation of $20,000, so the school can purchase 145
laptops. The correct answer is (C).
8. Module -1
In the xy-plane, the equation (x – 7)²+ (y + 7)²= 64 defines circle O, and the equation (x – 7)² + (y + 7)² = c defines circle P. If the two circles have the same center, and the radius of circle P is three less than the radius of circle O, what is the value of constant c?
__________
In the xy-plane, the equation (x – 7)²+ (y + 7)²= 64 defines circle O, and the equation (x – 7)² + (y + 7)² = c defines circle P. If the two circles have the same center, and the radius of circle P is three less than the radius of circle O, what is the value of constant c?
__________
25
*25
The question asks for the value of a constant, given information about
circles in the coordinate plane. The equation of a circle in standard
form is (x – h)
2
+ (y – k)
2
= r
2
, where (h, k) is the center and r is the
radius. In the equation given for circle O, r
2
= 64. Take the square root
of both sides of the equation to get r = 8. The question states that the
radius of circle P is three less than the radius of circle O, so the radius
of circle P is 8 – 3 = 5. Plug r = 5 into the equation of circle P to get (x
– 7)2
+ (y + 7)2
= 52
, or (x – 7)2
+ (y + 7)2
= 25. Thus, c = 25. The
correct answer is 25.
9. Module -1
\( \frac{2x – 6}{2} = x + k \)
If the equation above has an infinite number of solutions, what is the value of
the constant k?
\( \frac{2x – 6}{2} = x + k \)
If the equation above has an infinite number of solutions, what is the value of
the constant k?
k = −7k = −3k = 0k = 2
*(B) For the equation to have infinitely many solutions, the two sides of
the equation should be equivalent. Simplify the left-hand side by
dividing everything on that side by 2:
x-3 =x+k
So, for the equation to have equivalent sides, the constant k must equal
−3.
10. Module -1
In normal atmospheric conditions, the speed of sound through dry air can be
approximated with the following equation, in which V represents the velocity
in meters per second and T represents the air temperature in degrees Celsius.
V = 331.4 + 0.6T
What is the speed of sound in air that is 15 degrees Celsius?
In normal atmospheric conditions, the speed of sound through dry air can be
approximated with the following equation, in which V represents the velocity
in meters per second and T represents the air temperature in degrees Celsius.
V = 331.4 + 0.6T
What is the speed of sound in air that is 15 degrees Celsius?
−537.3 meters per second14.8 meters per second340.4 meters per second423.5 meters per second
(C) Plug 15 degrees in for T to solve for the corresponding speed:
V= 331.4 +0.6T
V= 331.4 +0.6(15) = 340.4
11. Module -1
Nyla is driving X miles per hour. Given that there are 5,280 feet in a mile,
what is her speed in feet per second?
Nyla is driving X miles per hour. Given that there are 5,280 feet in a mile,
what is her speed in feet per second?
\( \left(\frac{60}{5{,}280}\right) X \)\( \left(\frac{3600}{5,280}\right) X \)\( \left(\frac{5,280}{50}\right) X \)\( \left(\frac{5,280}{3600}\right) X \)
Convert the miles per hour to feet per second by cancelling out
terms: =>X * 5,280/3,600
This corresponds to choice (D)
12. Module -1
The psychology department of a school conducted a study on 20 random students in a third grade class of 58 students. 20 of the students were then offered a supplement. The study found that 15 of these students did better in their end- term exams compared to those who did not take the supplements. Which of the following statements can best be concluded from the above study?
The psychology department of a school conducted a study on 20 random students in a third grade class of 58 students. 20 of the students were then offered a supplement. The study found that 15 of these students did better in their end- term exams compared to those who did not take the supplements. Which of the following statements can best be concluded from the above study?
Students who take supplements do better on exams.Students who do not take supplements do not do well on their exams.Supplements improve students’ performance in their exams.No conclusion can be drawn about the causeand- effect relationship between test taking and supplement taking.
*Key Explanation: Choic.e D is correct. This statement is true because it is not certain that taking supplements would directly equate to an improvement in student performance. It is not known if all other variables were kept constant. Therefore, a direct cause-and- effect relationship cannot be determined. Also. the sample size is too small to generalize to a larger population
13. Module -1
A study conducted by a school’s medical board found that 23 out of the 48 students surveyed practice sanitary routines such as washing their hands before meals. If there are 2,280 students in the school) approximately how many students in the school do not practice sanitary routines (rounded up to the nearest whole number)
A study conducted by a school’s medical board found that 23 out of the 48 students surveyed practice sanitary routines such as washing their hands before meals. If there are 2,280 students in the school) approximately how many students in the school do not practice sanitary routines (rounded up to the nearest whole number)
1,1881,1891,1871,186
*Key Explanation: The students surveyed who do not practice sanitary routines are (48 – 23) =25. It can therefore be extrapolated that 25/48 of all the students in the school donot practice sanitary routines. Therefore (25/48)x 2,280 = 1187.5 students. This rounds up to 1,188 students.
14. Module -1
Which of the following is the equation of g(x) = 2x when it’s moved 1 unit to the left and 1 unit up?
Which of the following is the equation of g(x) = 2x when it’s moved 1 unit to the left and 1 unit up?
y=2x-1y=2x+1y=2x+5y=2x + 3
Key Explanation: Choice D is correct. To move one unit up) l is added to the equation. To move 1 unit to the left, 1 is added to the x term in the
equation as follows
y=2(x+1)+1
y=2x+2+ 1
y=2x+3
15. Module -1
What is the x coordinate of the vertex for the parabola represented by the equation
y=2x² +8x+12
What is the x coordinate of the vertex for the parabola represented by the equation
y=2x² +8x+12
6-42-2
Explanation; Choice A is incorrect as this is the product of the solutions to the quadratic equation. Choice B is incorrect as this is the sum of the solutions to the quadratic equation. Choice C is incorrect and is found by incorrectly using (b/a)
16. Module -1
How many values for y satisfy the equation –6(4y + 2) = 3(4 – 8y)?
How many values for y satisfy the equation –6(4y + 2) = 3(4 – 8y)?
012Infinitely many
**A
The question asks for the number of solutions to an equation.
Distribute on both sides of the equation to get –24y – 12 = 12 – 24y.
Add 24y to both sides of the equation to get –12 = 12. This is not true,
so the equation has no solutions. The correct answer is (A).
17. Module -1
*For all positive values of y, the expression \( \frac{3}{y+c} \)
is equivalent to \( \frac{15}{5y+30} \) . What is the value of constant c?
*For all positive values of y, the expression \( \frac{3}{y+c} \)
is equivalent to \( \frac{15}{5y+30} \) . What is the value of constant c?
368150
*The question asks for the value of a constant given two equivalent
expressions. Start by rewriting the expressions with an equal sign
between them to get . Next, start to solve by crossmultiplying. The equation becomes (y + c)(15) = (3)(5y + 30).
Distribute on both sides of the equation to get 15y + 15c = 15y + 90.
Subtract 15y from both sides of the equation to get 15c = 90. Divide
both sides of the equation by 15 to get c = 6. The correct answer is
(B)
18. Module -1
If x+3y=9 and 2x+2y=14, what is the value of y-x?
If x+3y=9 and 2x+2y=14, what is the value of y-x?
-5516
Key Explanation: Choke A is correct. The most efficient way to answer this question is to not solve the system of equations for x and y individually, but rather to subtract the second equation from the first, yielding – x + y = – 5. Therefore y – x = – 5
19. Module -1
Given that the length of a rectangle is 3 meters more than its width, what is the perimeter of the rectangle given that the area is 28 square meters?
Given that the length of a rectangle is 3 meters more than its width, what is the perimeter of the rectangle given that the area is 28 square meters?
11221432
* Explanation: Choice A is incorrect as this value is the sum of the length and the width and not two times the sum. Choice C is incorrect as this is half the area. Choice D is incorrect and may be due to conceptual error.
20. Module -1
3x²– y – 26 = 0
y = –3x + 10
The point (a, b) is an intersection of the system of equations above
when graphed in the xy-plane. What is a possible value of a?
3x²– y – 26 = 0
y = –3x + 10
The point (a, b) is an intersection of the system of equations above
when graphed in the xy-plane. What is a possible value of a?
-462026
*A
The question asks for the value of the x-coordinate of the solution to a
system of equations. The quickest method is to enter both equations
into a graphing calculator, then scroll and zoom as needed to find the
points of intersection. The graph shows two points of intersection: (3,
1) and (–4, 22), so the x-coordinate is either 3 or –4. Only –4 is in an
answer choice, so choose (A). To solve the system for the x-coordinate
algebraically, substitute –3x + 10 for y in the first equation to get 3x ^2
(–3x + 10) – 26 = 0. Distribute the negative sign to get 3x^2+ 3x – 10 –
26 = 0, then combine like terms to get 3x^2+ 3x – 36 = 0. Factor out 3
to get 3(x^2+ x – 12) = 0. Factor the quadratic to get 3(x + 4)(x – 3) =
0. Set each factor equal to 0 and solve to get x = –4 and x = 3. Using
either method, the correct answer is (A).
21. Module -1
A parabola represents the graph of the function f in the xy-plane, where y = f (x). If the vertex of the parabola is (5, –4) and one of the xintercepts is (–1.5, 0), what is the other x-intercept?
A parabola represents the graph of the function f in the xy-plane, where y = f (x). If the vertex of the parabola is (5, –4) and one of the xintercepts is (–1.5, 0), what is the other x-intercept?
(–6.5, 0)(1.5, 0)(3.5, 0)(11.5, 0)
*The question asks for an x-intercept of a parabola. Sketch a graph
using the given points, and label those points. The vertex of a parabola
is on the axis of symmetry, so the axis of symmetry of this parabola is
the line x = 5;
The two x-intercepts are an equal distance from the line of symmetry.
The x-coordinate of the given x-intercept is –1.5, so the distance from
the line of symmetry is 5 – (–1.5) = 6.5. The x-coordinate of the other
x-intercept is thus 5 + 6.5 = 11.5. The correct answer is (D).
22. Module -1
For the line l, y = ax + n, where a and n are distinct nonzero constants.
Which of these lines must be perpendicular to line l?
For the line l, y = ax + n, where a and n are distinct nonzero constants.
Which of these lines must be perpendicular to line l?
y = nx + ay = ax − ny= -1/ax – ny = −ax + n
(C) Since this line is written in slope-intercept form, y = mx + b,
recognize that the slope is going to be a. A line that is perpendicular to
this one will have a slope that is the negative reciprocal of a: -1/a . The
only option that has this as its slope is choice (C).
