Free SAT Maths Questions with Solutions and Explanations - Sample 4
The detailed solutions and explanations to the maths questions in sample 4, to practice for the SAT maths, are presented.
A) Multiple Choice Questions
Assuming that each small division represents one unit, which of the labeled points in the figure below is at a distance of 10 units from the origin of the rectangular coordinate system of axes?
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Solution
We first write the coordinates of all 5 points.
M(-7 , 7) , P(2 , 4) , N(8 , 7) , Q(-6 , -8) , R(7 , -7)
The distances from the origin to each of the points are
D(O,M) = √ ( (-7 - 0)^{2} + (7 - 0)^{2} ) = 7 √ 2
D(O,P) = √ ( (2 - 0)^{2} + (4 - 0)^{2} ) = 2 √ 5
D(O,N) = √ ( (8 - 0)^{2} + (7 - 0)^{2} ) = √ 113
D(O , Q) = √ ( (-6 - 0)^{2} + (-8 - 0)^{2} ) = 10
The distance from point Q to the origin is equal to 10 units.
If y = kx + 2k, where k is a constant, and if y = -14 when x = 5, what is the value of x when y = -24?
Solution
Let us first find k using y = - 14 when x = 5.
- 14 = 5k + 2k
Solve for k.
k = - 2
Substitute k by -2 in the given equation y = kx + 2k.
y = -2x - 4
We now substitute y by -24 and solve for x.
-24 = -2x - 4
-2x = -20 , x = 10
m and n are integers greater then zero such that 3^{m} = 9^{1/n}. What is the value of m*n?
Solution
We first write the two sides of the given equation to the same base
3^{m} = 9^{1/n}
3^{m} = (3^{2})^{1/n}
3^{m} = 3^{2/n}
Hence
m = 2/n
m * n = 2
If f is a function such that f(x+1) = 2x - 1, then f(2x) =
Solution
Let t = x + 1 and therefore x = t - 1 and substitute x by t - 1 in f(x+1) = 2x - 1 to obtain
f(t) = 2(t - 1) - 1
We now substitute t by 2x to find f(2x)
f(2x) = 2(2x - 1) - 1 = 4x - 3
A large rectangle is made up of 4 congruent smaller rectangles. What is the length of a small rectangle if the area of the larger rectangle is equal to 432 square units?
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Solution
The area of one small rectangle is gievn by
432 / 4 = 108
Let L and W be the length and width of one samll rectangle. From the given diagram, L = 3W. The area of one small rectangle is gievn by
L W = 3W * W = 108
Solve for W
W^{2} = 36 , W = 6
length L is given by
L = 3 W = 3 (6) = 18
The result of 2 + 25% + 1.6 is equivalent to
A) 3.625
B) 3.25
C) 3(1/4) (mixed number)
D) 385%
E) 28.6%
Solution
Convert 25% into decimal and add
2 + 25% + 1.6 = 2 + 0.25 + 1.6 = 3.85
Which is equivalent to 385%. Answer D)
Which of these points is inside the closed curve defined by the equation (x - 2)^{2} + (y + 3)^{2} = 4?
A) (0 , -3)
B) (3 , -2)
C) (4 , 1)
D) (1 , 4)
E) (0 , 0)
Solution
The given equation is that of a circle with center O and radius R = 2. The coordinates of O are
(2 , -3)
The point that is inside the circle has a distance to the center less than the radius which 2. We now find the distances between the center of the circle O(2,-3) and the given points and compare these distances to the radius of the circle.
d(O,A) = √ [ (2 - 0)^{2} + (-3 + 3)^{2} ] = 2
d(OB) = √ [ (2 - 3)^{2} + (-3 + 2)^{2} ] = √ 2
The distance d(OB) is less than 2 the radius and therefore point B is inside the circle.
The perimeter of a rectangular field is 8 times its width. The area of the field is 48 meters squared. What is the perimeter, in meters, of the field?
Solution
Let L and W be the length and width of the rectangular field. Let P be the perimeter. Hence
P = 2L + 2W = 8 W
Solve the equation 2L + 2W = 8 W for L
L = 3 W
The area is 48 meters squared. Hence
48 = L * W = 3 W * W = 3 W^{2}
Solve 3 W^{2} = 48 for W
W^{2} = 48 / 3 = 16
W = 4
Find L and the perimeter P
L = 3 W = 12
P = 2L + 2W = 24 + 8 = 32 meters.
The area of a triangle with side lengths 2.4, 1.8 and 3.0 is equal to
Solution
Since we are given the three sides of the triangle, the most suitsble formula to find the area is Heron's formula given by
Area = √ [ S(S - a)(S - b)(S - c) ] where a, b and c are the lengths of the sides of the traingle and S = (1/2)(a + b + c). Hence
S = (1/2)(2.4 + 1.8 + 3.0) = 3.6
Area = √ [ 3.6(3.6 - 2.4)(3.6 - 1.8)(3.6 - 3.0) ] = 2.16
The number of books N to be sold at a book fare is given by the function
N(p) = 25000 / (3p + k)
where p is the price per book and k is a constant. If 1000 books are sold at $7 per book, how many books will be sold at the price of $10 per book?
Solution
N = 1000 when p = 7. Hence
1000 = 25000 / (3(7) + k)
Solve the above for k
21000 + 1000 k = 25000
k = 4
What is N, the number of books, when p = 10?
N = 25000 / (3(10) + 4) = 735.29
The number of books cannot be a decimal number. Hence the number of books that can be sold at $10 is 735
For what value(s) of the parameter m does the equation
-2x^{2} + mx = 2
have one solution only?
A) 0
B) -2 , 2
C) -1 , 1
D) 16
E) -4 , 4
The average of the first 4 data values of a data set is equal to 21. The average of the last two data values of the set is equal to 27. What is the average of the 6 data values?
A) 24
B) 22
C) 23
D) 20
E) 21
If the values of the 4th and 7th terms of an arithmetic sequence are 6.5 and 11, then the value of the 20th term is
A) 32
B) 31
C) 31.5
D) 30.5
E) 30
A store is offering a 15% reduction of all regular prices such that the new price of a flat screen TV set is $680. What was the price, in dollars, of the TV set before the reduction?
A) 591
B) 800
C) 665
D) 780
E) 900
(xy)^{4n} - (xy)^{2n} is equivalent to
A) (xy)^{6n}
B) (xy)^{2n}
C) [ (xy)^{2n} - (xy)^{n} ] [(xy)^{2n} + (xy)^{n} ]
D) [ (xy)^{2n} - (xy)^{n} ] [(xy)^{2n} - (xy)^{n} ]
E) [ (xy)^{2n} - (xy)^{n} ] ^{2}
The pie graph below shows the expenses of the Taylors family in a particular month. If the Taylors spent $600 on food, how much did they spend that month on clothing?
.
A) 350
B) 400
C) 500
D) 600
E) 700
Which of these is not correct?
A) sqrt[ x^{2} + 2x + 1 ] = |x + 1|
B) |- x^{2} - 1| = x^{2} + 1
C) sqrt(x^{4}) = x^{2}
D) (-x^{2} - 6x - 9) / (- x - 3) = - x - 3
E) |e - 1/2 - 3| = 3.5 - e
Functions f and g are graphed below. what are the values of x between -1 and 5 (inclusive) for which f(x) > g(x)?
.
A) [-1 , 5]
B) [-1 , 0) U (0 , 4) U (4 , 5]
C) [-1 , 0) U (4 , 5]
D) [2 , 5]
E) [0 , 4]
A circle of center C is shown below. A, B and D are points on the circle such that A, C and D are collinear. If the lengths of the segments AB and BC are equal, then the size of angle BDC is equal to
.
A) 10
B) 30
C) 50
D) 60
E) 90
Twice the difference of the squares of two consecutive positive integers is 4n. What is the sum of the two integers?
A) 2n
B) 2
C) 4
D) 4n
E) 1
B) Student-Produced Response Questions
Points A(2,2), B(x,y) and C(8,6) are collinear and point B is between points A and C. The length of segment AB is equal to 1/5 the length of segment AC. Find the coordinates of point B.
What is the value of x If (x^{2} + 3)(2|x| + 4)(-x + 3) = 0?
n is a positive integer divisible by 3 and 7. What is the reminder when 2(n + 1) + 3 is divided by 42?
x and y are numbers such that (1/2)x + (1/5)y = 1/5 and (1/5)x + (1/2) y = 1/5. What is the value of 2(x + y)?
Pedro drove for one hour and a half at the rate of 60 kilometers per hour and for 3 hours at the rate of 70 kilometers per hour. What was Pedro's average speed, kilometers per hour, for the whole journey?
Find the smallest number that is larger than 1000 and divisible by 3, 11 and 12.
Find a positive integer N such that when N is divided by 7 it gives a reminder equal to 3 and when N is divided by 5 the reminder is equal to 1.
The difference of the squares of two consecutive positive even integers is 20. What is the average of the two numbers?
If x + 2y + 3z = 20 and 3x + 2y - z = 10, what is the value of 5x + 4y?
Each one of the 30 students at the cafeteria bought a sandwich only, a drink only or a sandwich and a drink. If 8 students bought a sandwich only and 12 students bought a drink only, how many students bought a sandwich and a drink?
If 2/3 of a number is equal to 5/3, what is twice the number?