Algebra
Intermediate Algebra
Geometry
Probability
Trigonometry
100
Add 8x to 2x and then subtract 5 from the sum. If x is a positive integer, the result must be an integer multiple of?
a) 2 b) 5 c) 8 d) 10 e) 15
5
The mathematical expression given in the questions is 8x+2x-5, which is equivalent to 10x-5. The expression 10x-5 can be factored as 5(2x-1). For every positive integer x, 5(2x-1) must be a multiple of 5. If x=1, then 5(2x-1)=5 which is not an integer multiple of 2, 8, 10, or 15.
100
One side of square ABCD has a length of 12 meters. A certain rectangle whose area is equal to the area of ABCD has a width of 9 meters. What is the length in meters of the rectangle?
a) 12 b) 16 c) 18 d) 20 e) 24
18
You need to find the area of the square first since you know the side length and the area of the rectangle is the same as the area of the square. A=s^2 you get A=12^2 which gives you 144. Area of a rectangle is A=wl since we know the area and width we can find teh length now. 144=8l divide both sides by 8 and you get 18.
100
A point at (2, 5) in the standard (x, y) coordinate plane is shifted down 2 units and left 7 units. What are the new coordinates of the point?
a) (-5, 3) b) (-2, 2) c) (0, -2) d) (4, 12) e) (9, 7)
a) (-5, 3)
A downward shift happens on the y axis and a left shift happens on the x axis. So our y unit starts at 5 and moves down 2 to become 3. Our x unit starts at 2 and moves 7 units to the left to become -5.
100
If two regular six-sided dice are rolled at the same time, what is the probability that the sum of the their numbers will be prime?
a) 7/18 b) 5/12 c) 1/2 d) 1/4
5/12
Explanation:
Thenumber of possible outcomes is equal to six times six, or thirty-six. The sum of the two dice must be either 2, 3, 5, 7, or 11. There are 15 out of the 36 outcomes that would result in a sum that is a prime number:
[1,1], [1,2], [1,4], [1,6], [2,1], [2,3], [2,5], [3,2], [3,4], [4,1], [4,3], [5,2], [5,6], [6,1], [6,5]
100
What is the value of Θ in the right triangle above? Round to the nearest hundredth of a degree.
a) 67.33 b) 59.04 c) 36.87 d) 26.57 e) 53.13
59.04
Recall that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side of the triangle. You can solve for the angle by using an inverse tangent function:
200
When x=3 and y=5, by how much does the value of 3x2-2y exceed the value of 2x2-3y?
a) 4 b) 14 c) 16 d) 20 e) 50
14
When you use x=3 and y=5 in the given expressions, you get 3(3)2-2(5) which gives you 27-10 which gives you 17. And you get 2(3)2-3(5) which gives you 18-15 which gives you 3. Then you subtract 3 from 17 and get 14.
200
In the standard (x,y) coordinate plane, the point (2,1) is the midpoint of line CD. Point C has coordinates of (6,8). What are teh coordinates of point D?
a) (-2. -7/2) b) (-2, -6) c) (4, 9/2)
d) (10, 10) e) (10, 15)
(-2, -6)
Since the midpoint formula is (x1+x2)/2 and (y1+y2)/2 we know that 6+x/2 needs to equal 2 and 8+x/2 needs to equal 1.
200
The figure on the board is a quadrilateral CDEF. What is the measure of <D?
a) 95 degrees b) 105 degrees c) 115 degrees
d) 135 degrees e) 295 degrees
c) 115 degrees
The sum of the interior angles of any quadrilateral is 360 degrees. So we add up our angles and subtract the total from 360. 100+70+75=245 and 360-245=115
200
Maria is planning the seating for the head table at a college gala. There are eight speakers that will be seated along one side of the table. Richard wants to sit beside Hang, and Maria knows that Thomas and Lily should not be seated together. In how many ways can Maria make up the seating plan?
a) 2880 b) 10,080 c) 3975 d) 8100 e) 7200
Correct answer: 7200
Explanation:
The simplest way to solve this is to find the number of seating arrangements in which Richard and Hang are seated together and then subtract those in which Thomas and Lily are also seated together. Consider Richard and Hang as a unit. This pair can be arranged with the other six speakers in 7P7 ways. For each of these ways, Hang could be either on Richard’s left or his right. Thus, there are 7P7 × 2 = 10 080 arrangements in which Richard and Hang are seated together. Now also consider Lily and Thomas as a unit. The two pairs can be arranged with the remaining four speakers in 6P6 ways, and the total number of arrangements with each of the pairs together is 6P6 × 2 × 2 = 2880.
Therefore, the number of seating arrangements in which Richard and Hang are adjacent but Thomas and Lily are not is 10 080 − 2880 = 7 200.
200
What’s tangent of C in the given right triangle?
a) CB/AB b) AB/AC c) AC/BC d) AB/BC e) CB/AC
AB/BC
Tangent= opposite over adjacent
300
What is the value of x when 2x+3=3x-4?
a) -7 b) -1/5 c) 1 d) 1/5 e) 7
7
You can solve this problem by first subtracting 2x from each side of the equation to get 3=x-4. Then add 4 to each side, so x=7
300
Given the matrices x=[-1 0] and y= [-2]
[-1]
Which of the following matricies is xy?
a) [-4] b) [-3] c) [-2] d) [2] e) [3]
[2]
-1 x -2 + 0 x -1
you get 2+0 which gives you 2
300
A farmer’s rectangular field is 600 ft by 80 ft and is all grazing land for a cow. The farmer decides to build a circular pig sty with a radius of 8ft in the field. How much land in square feet is left for grazing?
a) 2800+8(pi) b) 48000-64(pi) c) 60000-100(pi)
d) 40000 -64(pi) e) 48000
b) 48000 -64(pi)
The area of a circle is (3.14)r^2 so we find that first. We end up with 64(pi) The area of the rectangle is length x width so we get 600 x 80 which gives us 48,000
So the land left over is 48,000-64(pi)
300
{10, 12, 24, 50, 60, 100, 260, 480, 606, 1000}
What is the probability that a number selected randomly from the set will be divisible by both 4 and 6?
a) 3/10 b) 1/10 c) 1/2 d) 2/5 e) 3/5
2/5
Explanation:
First, find the numbers that in the set that are divisible by 4. 12, 24, 60, 100, 260, 480, and 1000 are all divisible by 4. Now find the numbers that are divisible by 6. 12, 24, 60, 480, 606 are all divisible by 6. The numbers that are divisible by both 4 and 6 are 12, 24, 60, and 480, or 4 total numbers from the set. So 4 out of the 10 numbers are divisible by 4 and 6. The probablility is 4/10, which reduces to 2/5. The correct answer is 2/5.
300
Consider a right triangle with an inner angle x(x<90degrees). If
cosx=3/5 and Sinx=4/5 what is tanx?
a) 5 b) 4/3 c) 3/4 d) 1/5 e) 1
b) 4/3
tangent is opposite over adjacent cosine is adjacent over hypotenuse and sin is opposite over hypotenuse so opposite side length is 4 and adjacent side length is 3.
400
Katrerina runs 15 miles in 2.5 hours. What is the average number of minutes it takes her to run 1 mile?
a) 6 b) 10 c) 12.5 d) 16 2/3 e) 17.5
10
multiply your 2.5 hours by 60 since there are 60 minutes in an hour. You get 150 minutes. Then set up a proportion and solve for x. You get 150/15=x/1 you end up with x=10
400
x=-2 and x=4 are two solutions to which of the following quadratic equations?
a) (x-4)(x-2)=0 b) (x-4)(x+2)=0 c) (x+4)(x-2)=0
d) (x+4)(x+2)=0 e) (x+4)^2=0
b) (x-4)(x+2)=0
x+2=0 gives you x=-2 and x-4=0 gives you x=4
400
A circle has a radius of 5. What is the measure of an arc on that circle that is interrupted by a central angle of 144 degrees?
a) 4(pi) b) 10 c) 10(pi) d) 25 e) 25(pi)
a) 4(pi)
we need the circumference of the circle first which is 2(pi)r which gives us 10(pi). To find the arc length we take (central angle)/360 x circumference which gives us 4(pi)
400
Jacob was 27 years old when his son Mike was born. Mike was 23 years old when his son Sam was born. Sam will celebrate his seventh birthday in 2014. What year was Jacob born?
a) 2007 b) 1987 c) 1984 d) 1957 e)1964
1957
Explanation:
If Sam will celebrate his seventh birthday in 2014, then he was born in 2007. So Mike was born 23 years before in 1984 and Jacob was born 27 years before that in 1957.
If you answered 1964, then you did not factor that Sam was 7 years old at the time of calculation.
If you answered 1984, then you found the year that Mike was born, not Jacob.
If you answered 2007, then you found the year that Sam was born, not Jacob.
If you answered 1987, then you just subtracted 27 years old from the day of the party in 2014.
400
Triangle ABC is a right triangle. If tangent of angle c is 3/7, what is the length of BC?
a) √21 b)3 c)3.5 d)7 e)√58
7
Explanation:
Use the definition of the tangent and plug in the values given:
tangent C = Opposite / Adjacent = AB / BC = 3 / 7
Therefore, BC = 7.
500
The solid rectangular prism with a length of 7 cubes, a height of 6 cubes and a width of 5 cubes, was built by alternating congruent black and white cubes such that 2 cubes with the same color have at most 1 edge touching. What is the total number of white cubes that were used to build this prism?
a) 45 b) 102 c) 105 d) 140 e)210
105
Find the volume of the prism first. Since the cube is set up like a checkerboard, you will divide that in half to get the final number. v= lwh so v=(5x6x7) which gives you 210. Divide that by 2 and you get 105.
500
For a certain quadratic equation, the factors are (4x+1) and (3x+4). What are the two solutions to this quadratic equation?
a) -1/4 and -4/3 b) -1/3 and -3/4
c) -1/3 and 3/4 c) 1/3 and -3/4
e) 1/4 and 4/3
a) -1/4 and -4/3
4x+1=0 subtract 1 and you get 4x=-1 divide by 4 and you get x=-1/4
3x+4=0 subtract 4 and you get 3x=-4 divide by 3 and you get x=-4/3
500
A triangle has sides with lengths 4x, 5x, and 7x. If x=4, what is the perimeter of the triangle?
a) 14 b) 54 c) 64 d) 128 e) 172
c) 64
4(4)=16 5(4)=20 7(4)=28 16+20+28=64
500
What is the probability of choosing two consecutive red cards from a standard deck of cards, if replacement is not allowed?
a) 61/204 b) 23/25 c)51/100 d) 17/25 e) 25/102
25/102
Probability = what you want ÷ total number
A standard deck of playing cards has 52 cards, with 4 suits and 13 cards in each suit
Choosing two red cards = 26 * 25 = 650
Choosing two cards = 52 * 51 = 2652
So the probabiulity of choosing 2 red cards is 650/2652 = 25/102
If replacement is allowed, then the probability of choosing 2 red cards becomes 676/2704 = 1/4
500
For the above triangle, o=21 and a=8. Find θ.
a) 22.4 degrees b) 67.6 degrees c) 20.9 degrees d) 69.1 degrees
With right triangles, we can use SOH CAH TOA to solve for unknown side lengths and angles. For this problem, we are given the opposite and adjacent sides of the triangle with relation to the angle. With this information, we can use the tangent function to find the angle.
tan=opposite/adjacent
tan(θ)=21/8=2.63
arctan(2.63)=θ
69.1∘=θ