Academic Team Questions: Math
Match each of the following four mathematical terms to its definition. Terms: A. centroid B. modulus C. radian of a triangle D. parameter Definitions: 1. numerical description of a population or model 2. the absolute value of a complex number 3. the point of concurrency of the three medians of a triangle 4. measure of a central angle whose intercepted arc has a length equal to the circle's radius
A-3, B-2, C-4, D-1
Match each of the following four polar equations to its description. Polar Equations: A. r = 3 + 3 cos Θ B. r = 3 cos 4Θ C. r cos Θ = 4 D. Θ = π Descriptions: 1. vertical line 2. horizontal line 3. limaçon 4. rose curve
A-3, B-4, C-1, D-2
To the nearest square centimeter, state which one of the following two shapes is larger and by how much; an equilateral triangle with sides 10 centimeters in length or a circle with a diameter of 10 centimeters.
circle, 35 (square centimeters)
Identify the mathematician whose name is associated with the coordinate plane.
(René) Descartes
Name the individual credited with discovering that the number of possible positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients.
(René) Descartes
Factor completely the following expression: x^6 + 26x^3 - 27
(x + 3)(x^2 - 3x + 9)(x - 1)(x^2 + x + 1) (any order)
Factor completely x^3 + 2x^2 - 9x - 18.
(x - 3)(x + 3)(x + 2) (any order)
Factor completely x^5 - x^3 - 8x^2 + 8
(x-2)(x^2 + 2x + 4)(x - 1)(x + 1)
In vertex form, state the equation of the parabola with directrix x = -3 and focus at (1, 2).
(y - 2)^2 = 8(x + 1)
In simplest form, state the inverse of the function y = log(10x).
(y =) 10^(x - 1)
If a standard six-sided die is rolled twice, state as a reduced fraction the probability that the first roll produces an even number and the second gives a number greater than two.
1/3
Solve for x in the following equation. sqrt(x + 2) - sqrt(x) = 1
1/4
Approximate the area under the curve over the given interval using four right endpoint rectangles of equal width. The function f(x) = - x2 + 4; the interval [ - 2, 2]
10
Calculate in centimeters the perimeter of a square whose diagonal measures 5sqrt(2) centimeters.
20 (centimeters)
In terms of Π, determine the total area in meters squared watered by a sprinkler head which rotates at a radius of 10 meters and covers 240 degrees.
200Π/3 (meters squared)
Find the area of a triangle with vertices (-4, 2), (5, 0), and (1, - 3).
35/2 (Note: |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| / 2 ((-4)(0 - -3) + 5(-3 - 2) + 1(2 - 0))/2 = 35/2)
To the nearest tenth of a degree, find the measurement of the smallest angle for the triangle with side lengths 23, 18, and 30 units.
36.8 (degrees)
To the nearest tenth, state in square inches the surface area of a right triangular prism with a height of 11 inches whose bases are isosceles right triangles with legs 8 inches.
364.5 (square inches)
The number of pepperoni slices that Kim's Pizza Shop puts on a pizza varies directly as the square of the diameter of the pizza. Given that Kim puts 15 slices of pepperoni on a pizza with a diameter of 10 inches, to the nearest whole slice, state the number of pepperoni slices she should put on a pizza with a diameter of 16 inches.
38 (pepperoni slices)
Determine the slope of the tangent to the curve of f(x) at x = 6 given that f(x) = 3x^3 + 6x^2 - 10x + 4.
386 (Note: f'(x) = 9x^2 + 12x - 10 f'(6) = 386)
Given the following set of x-values and their respective y-values, find the function rule. X-values: 0, 1, 2, 3 Y-values: 4, 10, 28, 82
3^(x+1) + 1
Differentiate the following function with respect to x. f(x) = (1/2)x^6 + 3x^(4/3) + 4x
3x^5 + 4x^(1/3) + 4
A rectangular prism has a volume of 27x^3*y^2*z cubic units. The length and width respectively measure 9x and xyz units. State in units the measure of the height.
3xy
In how many ways can four people be selected at random from a committee of twenty people?
4,845 (ways)
State 43 billion, 740 million in scientific notation.
4.374 x 10^10
Given that the diameter of the circle is 21.8, to the nearest tenth, state the length of x.
4.8
For the next question you will need this information. A certain basket contains a combination of basketballs, footballs, and soccer balls totaling 100. The ratio of basketballs to footballs is 2 to 3. The ratio of basketballs to soccer balls is 3 to 5. Determine hte number of soccers balls in the basket.
40
Give all the real solutions to the following equation; the square root of the quantity x + 9 equals x - 3.
7
State the number of students that take statistics and computer science but not calculus.
7
Determine the type of conic represented by the equation r = 15/(3 - 2cos(θ)) and state the exact eccentricity.
ellipse, eccentricity = 2/3
State the inverse of g(f(x)) given that g(x) = 3x3 - 4 and f(x) = x - 2.
cbrt((x + 4)/3) + 2
State respectively the center and the radius of a certain circle given by the equation x^2 + y^2 + 2x = 4y + 11.
center (-1, 2), r = 4
In slope intercept form, state the equation of the line tangent to y = ln x at (e, 1).
y = (1/e)x
Given f(x) = cbrt(x^2 - 4) and g(x) = x^3 - 3x find g(f(sqrt(5)))
-2
Calculate X4 by using Newton's method to approximate the real zero of f(x) = x3 - 2x + 2. Let X1 = - 1.0000 as the initial and round to 4 decimal places after each iteration.
-2.1467
If f(x) = x^2 - px - 4 and f(3) = 7, solve for p.
-2/3
Solve the following absolute value inequality: |3x - 2| ≤ 4.
-2/3 ≤ x ≤ 2
Express as a rationalized fraction the cosine of an angle that has a terminal side that goes through the point (-2, 7).
-2sqrt(53)/53
Find the limit of the quantity x cubed minus 8 divided by the quantity 4 minus x squared as x approaches 2.
-3
Solve for x in the following equation: (53)(3+x) = (252)(x+3)
-3
Determine the measure of line segment SR given that R is the midpoint of line segment ST and SR = 3x + 8 and RT = 5x - 6.
29
Solve for x in the following logarithmic equation. log(x + 21) + log x = 2
4
State the exact value of the integral of e raised to the x from 0 to π.
e^π - 1
State the solution set of the equation (2x)/(x^2-4) + 8/(x-2) = 2/(x+2)
x = -2.5
State the vertical asymptotes of the following function: f(x) = (x^2 - 81)/(4x^2 + 4x - 24).
x = -3 and x = 2
Solve for x in the following equation. sqrt(2x + 3) = x + 2
-1
State exactly the definite integral of (x - 4)^3 with respect to x from 1 to 4.
-81/4 (accept: -20.25)
Solve for x in the following equation. 3(x + 1)^(2/3) = 12
-9, 7
Simplify arcsin(cos(2π/3)).
-π/6
To the nearest hundredth, solve for x in the following equation 5^(3x) = 37
.75
State in radians the exact solutions of the equation from 0 to 2π inclusive.
0, π, 2π, π/6, 7π/6
If five fair, six-sided dice are rolled simultaneously, to the nearest thousandth, state the probability of rolling no threes.
0.402
John passes 80% of all his SMAD tests. State the exact probability John will pass three out of the next four tests.
0.4096 (accept: 256/625)
There are four men and six women who all want to be chosen for the Hospitality Committee at work. Three will be selected at random. State the probability that all three will be the same gender.
1/5 (accept: .2)
Find the projection of vector u = [3, 5] onto vector v [6, 3].
11/15 [6, 3]
A boat is anchored to the sea floor. The anchor is attached to a 150 foot rope. The rope makes a 50° angle with the surface of the sea and is pulled tight. To the nearest foot, state the depth of the water.
115 (feet)
Mickey Mouse is building a pyramid. It has a square base with sides of 254 meters. The faces of the pyramid form an angle of 43.19° with the base. To the nearest meter, calculate the height of the pyramid.
119 (meters)
A coin purse contains equal amounts of pennies, nickels, and dimes. The sum of the coins is $2.08. State the number of each coin present.
12
Given that f(x) = sqrt(x - 3) and g(x) = x^2 + 3, evaluate f of g of 12.
12
To the nearest tenth, determine the sum of the infinite geometric series that begins with 10 + 2 + 2/5 + 2/25.
12.5
In simplest form, state the exact area of a triangle with sides 7, 8, and 9.
12sqrt(5)
In terms of π, state the area of sector AOB from the circle with center O whose diameter is 24 inches and the measure of angle AOB = π/6.
12π
To the nearest tenth of a degree, for the interval from 0° to 360°, solve for x in the following equation: 3sin^2(x) + 4cos(x) = -1.
131.8 (degrees) and 228.2 (degrees)
State the remainder when 2x^4 + 19x^3 - 8x^2 - 11x +4 is divided by x - 2.
134
A trapezoid has an area of 112 square centimeters and a height of 7 centimeters. Given that one base is 18 centimeters long, state in centimeters, the length of the other base.
14 (centimeters)
The measures of the angles of a triangle are in the ratio. In degrees, state the measure of the smallest angle.
15 (degrees)
State in degrees the interior angle of a regular polygon with 12 sides.
150 (degrees)
Find all of the values for x at which the graph of x^2 - 1 and x + 1 intersect.
2, -1
Find the x-values where the tangent lines to the graph of y = 3x3 - 4x + 1 have a slope of
2, -2
Solve for a in the following equation: 3a + 2 = a (x - 3).
2/(x-6)
Determine the fifth term in the binomial expansion (x - 3y)^8
5670x^4 * y^4
In simplest standard form, state the equation of the line that passes through (3, - 3) and (1, 2).
5x + 2y = 9
In French or Spanish state and spell the shape suggested by the following description: This convex polygon has interior angles whose measures total 1,080 degrees.
French: Octogone Spanish: Octágono (Note: Sum of interior angles is (n - 2) * 180, where n is the number of sides. (n - 2) * 180 = 1080; n = 8. Octagon.)
This is a two part question regarding the polynomial 4x3 - 7x2 - 7x + 3 Part 1: x - 7 is a factor of the polynomial. True or false? Part 2: State the remainder when the polynomial is divided by x - 7.
Part 1: false Part 2: 983
Given that g(x) = 2x + 3, what is the inverse of g(x)?
½ (x - 3)
A cylinder has a diameter that is one half as long as its height. The volume is 16,717 cubic units. To the nearest whole number, determine respectively the height and diameter of the cylinder.
height: 44 (units) diameter: 22 (units)
Find the radius of convergence to the series negative one raised to the n multiplied by the quantity x plus 1 raised to the n divided by 4 raised to the n from n equals zero to infinity.
4
On the unit circle, which one of the following four positions does not lie in the third quadrant? 1. 225 degrees 2. negative three pi over four 3. negative seventeen pi over six 4. one thousand four hundred twenty degrees
4
Which one of the following four is not a function? 1. y = x^2 2. y = x^3 3. sqrt(x) 4. x = y^2
4
State as a complex number the product of (i^3 - i^2)(3i^4 + i)
4 - 2i
One half of the sum of two numbers is negative three. One half of the difference is seven. Identify the two numbers.
4 and -10
State the dimensions of a rectangle with an area of 12 square inches given that the length is 5 inches less than 3 times the width.
4 and 3
Given that 3x - y = 12, state the value of g^x/2^y.
4096 (accept: 2^12)
Solve for x in the following equation: -2,493 = -4(66 - x)^(4/2) + 7
41
An equilateral triangle is inscribed in a circle of radius 6 centimeters. To the nearest whole number, give the percent of the circle's area covered by the triangle.
41 (%)
Find the inverse of f(x) given that f(x) = (3x - 2)/4
f^(-1)(x) = (4x + 2)/3
Find the inverse of the logarithmic function f(x) = log_4(x - 2)
f^(-1)(x) = 4^x + 2 (accept: y = 4^x + 2 or 4^x + 2)
If a sphere and a cylinder both have the same volume and the same radius, express the height of the cylinder in terms of the radius of either figure.
h = (4/3) * r
Expressing your answer as a logarithm, state the solution of 2^(x+5) = 2 + 2^x
log_2(2/31)
State the following expression as a single logarithm: 6log_2(m) + log_2(x)/3
log_2(m^6 * cbrt(x))
Simplify the following expression to a single logarithm. 3 log_b(X) + 2 log_b(Y) - ½ log_b(Z)
log_b((x^3 * y^2)/sqrt(z))
Simplify the quotient using only positive exponents: (x^n * y^(2n))/(n^(-1) * y^n)
n * x^n * y^n
Solve for x in the following equation. 1/(x - 4) + 1/3 = 3/(3x - 12)
no solution
Solve for x in the following equation: 81^(x + 2) = 9^(2x).
no solution
State respectively both the oblique and vertical asymptotes for the following equation. y = (3x^2 - 5x + 2)/(x - 5)
oblique: y = 3x + 10 vertical: x = 5
Identify the period in terms of π radians and the amplitude for the sinusoid y = -3 + 2cos(4x - 20).
period = π/2 (accept: 2π/4) amplitude = 2
Identify the trigonometric function represented by 1/cos(x)
sec(x)
Rationalize the denominator of sqrt(3)/(2sqrt(7) - 4).
sqrt(21)/6 + sqrt(3)/3 (accept: (sqrt(21) + 2sqrt(3))/6)
Give the absolute value of the complex number 3 - 5i.
sqrt(34)
Solve for x in the following absolute value inequality. |2x + 7| ≤ 0
x = -7/2 (accept: x = -3 1/2)
Solve for x in the following equation x - 3 = 6sqrt((2/3)x - 2).
x = 27 and x = 3
Solve for x in the following absolute value equation. |2x + 1| = |3x - 4|
x = 5 or x = 3/5
The volume in cubic feet of a CD holder can be expressed as V(𝑥) = −𝑥3 − 𝑥2 + 6𝑥, or, when factored, as the product of its three dimensions. The depth is expressed as 2−𝑥. In terms of 𝑥, state the dimensions of the other two sides.
x, x +3
State whether the equation has symmetry with respect to the x-axis or the y- axis.
x-axis
Simplify the following. (2x^(-2) * y^3)^(-2)/(2xzy^4 * x^(2)y^(-2) * y^4)
x/(8y^12 * z)
Two of the roots of a cubic equation are and . Given that the coefficients of the cubic equation are real numbers, state the equation in standard form.
x^3 - 6x^2 + x + 34 = 0
In point-slope form, state the equation of the tangent line to the curve y = 5x2 - 10x + 10 at x = 3.
y - 25 = 20(x - 3)
State in slope-intercept form the equation of a line that runs through the point (-18, 5) and is perpendicular to the line y = 3x - 6.
y = (-1/3)x - 1
Find the inverse of the function y = (x - 4)/(x + 3)
y = (3x + 4)/(1 - x) (accept: y = (-3x - 4)/(x-1))
In slope intercept form, state the equation of the line that passes through the point (- 1, - 1) and is perpendicular to 3x - 2y =8.
y = -(2/3)x - 5/3
State the equation of the horizontal asymptote of the quantity one fourth x plus 13 divided by the quantity two-fifths x cubed plus 2 x minus 7.
y = 0
Write an equation for the following transformations of y = x^2: up 7, left 6, vertical stretch by a factor of 2, down 3.
y = 2(x + 6)^2 + 4 (accept: y = 2x^2 + 24x + 76)
In slope-intercept form, state the slant asymptote of the function (2x^3 + 4x^2 + 1)/(x^2 + x + 4).
y = 2x + 2
In standard form, state the polynomial equation with the following zeros: 0, ½, and 3 with a multiplicity of 2.
y = 2x^4 - 13x^3 + 24x^2 - 9x (accept: 0 = 2x^4 - 13x^3 + 24x^2 - 9x)
State the equation of a line perpendicular to the line x = 8, containing the point (3, 4).
y = 4
Find the inverse of the following function: y = (x - 2)^(3/2).
y = cbrt(x^2) + 2
State in terms of π radians all angles from 0 to 2π having a cosine of sqrt(3)/2.
π/6, 11π/6
For the next question, you will need this information. A student wants to take a trip around South America after his college graduation in 4 years. The student believes he will need $3,000. His bank is offering a four-year certificate with 4% interest compounded monthly. To the nearest whole dollar, how much must the student initially invest with the certificate to have enough money for his trip?
$2,557
The value of an industrial machine has a decay rate of 0.25 per year. After six years, the machine is worth $7,500. To the nearest cent, state the original value of the machine.
$42,139.92
State the simplest radical form of sqrt(75/100)
sqrt(3)/2
State the inverse of the function y = x^2 - 1.
(y =) ±sqrt(x + 1)
Evaluate cos(3π/2) - csc(π/2)
-1
Find the limit as x approaches 3 of the function f(x) = x - 4
-1
In reduced fraction form, find the value of the derivative of 4x2 + 2y3 - 3x = 8 when x = 2 and y = - 1.
-13/6
Solve for x in the following logarithmic equation. ln(x^2 + 5x + 9) = ln 3
-3 and 2
Evaluate: limit t --> 5 ((t - sqrt(4t + 5))/(5 - t))
-3/5
In most reduced form, find the x-value where the tangent to the graph of y = 3x2 - 4x + 1 has a slope of -12.
-4/3
Simplify the following sqrt(-25) * sqrt(-81)
-45
State the remainder of (x^3 + 5x^2 + 2x - 8) divided by (x^2 + 2x).
-4x - 8
Solve for x in the following equation. cbrt(2 - 3x) - cbrt(-x + 12) = 0
-5
Solve for x in the following equation. 3^(2x - 1) = 3 * 9^(2x + 6)
-7
In a nearby county, 40% of students pass their driving test on the first try. 80% pass on all subsequent attempts. State the probability that a random student would need exactly three tries to pass his test.
.096 (accept 9.6%)
In a particular orange grove, the function h(x) = -10x2 + 280x + 17,400 represents the number of oranges that will be harvested with x number of trees being added to the existing trees in the grove. Determine the maximum number of oranges that can be harvested.
19,360
Identify respectively the mathematical property illustrated in each of the following four statements. 1. 7c + (a + 5b) = 7c + (5b + a) 2. 1 x A = A 3. 4(x + 3) = 4x + 12 4. 2 + (a + b) = (2 + a) + b
1. Commutative (Property of Addition); 2. Identity (Property of Multiplication) 3. Distributive (Property); 4. Associative (Property of Addition)
Classify each coordinate point by its quadrant number. 1. (- 2, - 3) 2. (5, 4) 3. (- 2, 4) 4. (5, - 3)
1. III; 2. I; 3. II; 4. IV
At a charter high school in Las Vegas, hot lunch costs $3. Students can roll a die to try to win a free lunch. If they roll a number greater than 4, lunch is free. If they roll a number less than 3, they pay full price. However, lunch costs $5 for all other results. To the nearest penny, in dollars and cents, state the expected cost of lunch for students who choose to roll the die.
($) 2.67
If the equations y = -2x + 1 and y = x^2 + x + 3 were both graphed on the same coordinate plane, specify the coordinates of the point or points of intersection.
(-2, 5), (-1, 3)
In simplest form, state the derivative of the function f(x) = (3x - 8)/e^x.
(-3x + 11)/e^x
Simplify the following. (3x^(-1) - 3y^(-1))/(y^(-2) - x^(-2))
(-3xy)/(x + y)
Find the midpoint between the coordinate points (- 6, 7) and (- 2, - 3).
(-4, 2)
State the coordinates of the point where the perpendicular bisector of the line segment joining points A (2, -4) and B (-6, -2) hits the x-axis.
(-5/4, 0) (accept: (-1.25, 0))
In simplest fractional form, find by implicit differentiation of the following equation. 8x^4 + 3y^4 = 32.
(-8x^3)/(3y^3)
Determine the interval for which f (x) is concave down given that f (x) = 3x^3 - 4x^2 + 1.
(-∞, 4/9) (accept: x < 4/9)
Solve for x in the following absolute value inequality. |x + 8| > -2
(-∞,∞) (accept: all real numbers)
Solve the following system of equations: x + y = 5 y + 1 = 3x^2 + 2x
(1, 4) and (-2, 7)
Differentiate the function f(x) = x^2/sin(x)
(2x sin x - x^2 cos x)/sin^2(x) (accept: 1/2 tan^2(x/2))
Find the solution set for the equation sqrt(3x + 7) = x.
(3 + sqrt(37))/2 (accept: x = 3/2 + sqrt(37)/2)
Factor the quadratic expression 6x^2 + x - 15.
(3x + 5)(2x - 3)
Given that x represents a positive real number, reduce the following expression: (3x^2 + 10x - 8)/(5x^2 + 19x - 4)
(3x - 2)/(5x - 1)
Factor 3y^2 + 7y + 4 completely over the real numbers
(3y + 4)(y + 1)
For the next question you will need this information. Slugging percentage in baseball is calculated by taking the sum of the number of singles plus two times the number of doubles plus three times the number of triples plus four times the number of home runs a player hits. This sum is divided by the number of at bats and is reported to three decimal places. In 2008, Matt has 107 at bats. He hit 17 homeruns, 3 triples, 11 doubles and 2 singles. To three decimal places, state Matt's slugging percentage.
.944
Find the sum of all real roots of the following equation: 3x^4 + 10x^2 - 8 = 0
0
Given f of x = log x and g of x = what is g of f of 1000?
0
Solve for x in the following equation: (2^3)^x * (4^4) * (8)^x = 16 * 4^2
0
To three decimal places, solve for x in the following equation: 9^(3x) = 45.
0.577
State the probability of A and B, given that the probability of A or B = 0.58, the probability of A = 0.6, and the probability of B = 0.75.
0.77
If a fair die is rolled 15 times, to the nearest thousandth, calculate the probability of rolling a 5 at least once.
0.935
Find the limit as x approaches zero to the function f(x) = sin(x)/x
1
Which one or ones of the following four quadrilaterals would not necessarily have perpendicular diagonals? 1. rectangle 2. parallelogram 3. kite 4. square
1, 2
Which one or ones of the following four statements are true concerning the function f(x) = x^3 + 2x^2 ? 1. There is one point of inflection. 2. There are two critical points. 3. At x = 1, the graph is increasing at a decreasing rate. 4. The graph is concave down for all real numbers.
1, 2
Captain Jack's sword blade is 90 centimeters in total length with a base of 4 centimeters. The pointy tip of the blade is 10 centimeters in length shaped like a right circular cone while the base of the blade is shaped like a cylinder. Rounding to the nearest whole number, in cubic centimeters, give the volume of the entire blade.
1,047 (cubic centimeters)
Using the letters in the word BASHFUL, state the number of arrangements that can be made using all of the letters given that the first letter must be a vowel.
1,440
Identify respectively the following four mathematical terms beginning with the letter M. 1. arc that is smaller than a semicircle 2. theorem that relates the average rate of change to the derivative of a function 3. line segment drawn from one vertex of a triangle to the midpoint of the opposite side of the triangle 4. the lowest point in a particular section of a graph
1. minor (arc); 2. mean value (theorem); 3. median; 4. minimum
Identify respectively the following three mathematical terms beginning with the letter T. 1. line that intersects two coplanar lines at two distinct points 2. quadrilateral with exactly one pair of parallel sides 3. line in the plane of a circle that intersects the circle in exactly one point
1. transversal; 2. trapezoid; 3. tangent
To the nearest tenth, find the sixth term of the sequence a_n = 2^n/n^2
1.8
For the next question you will need this information. According to the 1990 census, the population of Florida was 12.9 million. In 2010, it was 18.8 million. Assume the population grew exponentially over the entire time period. To the nearest tenth, give the rate of growth as a percent.
1.9 (%)
You can rent a car from Risky Rental for $40 per day plus $0.08 per mile. Comfy Car Rental charges $35 per day plus $0.13 per mile. Find the first whole number of miles for which it is cheaper to rent from Risky Rental if you rent a car for 1 day.
101 (miles)
In terms of pi, state the difference between the circumference of a circle that circumscribes a square of side length 10 and the circumference of a circle inscribed in a square of side length 10.
10πsqrt(2) - 10π
Solve for x in the following equation. log_2(x - 5) + log_7(1) = log_3(27)
13
To the nearest hundredth, state in square units the area of a triangle that has a side length of 4, a side length of 15, and an included angle of 26º.
13.15 (square units)
Calculate the percent of change on a $24 shirt that is on sale for $20.
16 2/3 (%) decrease (Note: (24-20)/24 = 0.1666...)
Given that the half-life of carbon-14 is approximately 5,700 years, calculate in whole years the time it will take a 50 gram sample to deteriorate to a 6.25 gram sample.
17,100 (years)
Determine the sine of a certain angle given that the cosine of the angle is 7/25.
24/25 (Note: sin(arccos(7/25)) )
Determine the sum of the infinite geometric series 3 - 3/2 + 3/4 - 3/8 + ...
2
Identify respectively each of the following four equations as a function or a relation. 1. x = 2y + 3 2. x^2 + y^2 = 16 3. the absolute value of y equals the absolute value of x 4. x^2 - 2(x) = y
2
Solve the following equation. 3sqrt(4x^3) * (4th root of x^6) = 48
2
Solve the following logarithmic equation: log_7(x) + log_7(x + 1) = log_7(6).
2
Which one of the following four is the logical conclusion of the following true statements? If M is yellow, then Q is green. Q is not green. 1. Q is yellow 2. M is not yellow 3. M is yellow 4. M is green
2
Which one of the following four logarithms results in a negative number? 1. log base 2 of 8 2. log base one half of 4 3. log base 2 of negative 8 4. log base 8 of 2
2
Zach and Jen each cut grass. Zach charges $50 annually and $20 every time he cuts a lawn. Jen charges $30 annually and $30 for every lawn cut. How many lawns must be cut for both Zach and Jen to earn the same amount of money?
2
Arrange the following four in order from least to greatest. 1. the square root of 441 2. the greatest common factor of 60 and 75 3. the least common multiple of 6 and 14. 4. the quotient of 132 and 6
2, 1, 4, 3 (Note: 21, 15, 42, 22)
To the nearest whole number, calculate in square centimeters the area of a regular hexagon inscribed in a circle with a radius of 30 centimeters.
2,338 (cm^2)
State the exact derivative of secant x evaluated at x = π/6 radians.
2/3
Stating your answer in reduced fraction form, evaluate: lim x -> 3 ((x - sqrt(15 - 2x))/(2x - 6))
2/3
What is the exact area bounded by the parabola y = 2x^2, the y-axis and the line that is tangent to y = 2x^2 at the point (1, 2)?
2/3
Solve for x in the following absolute value equation. |(2/5)x - 1| = 7
20, -15
State both the number of terms and the sum of this number of terms for the sequence that starts with 4, 7, 10 and ends with 64.
21 (terms), 714 (sum)
Two sides and the included angle of oblique triangle ABC measure 17 units, 68º, and 27 units. To the nearest square unit, find the area of triangle ABC.
213 (square units)
Determine the number of full years John would need to wait to triple his investment given that he opened an account for $2,000 paying at a rate of 5% compounded continuously.
22 (years)
An athletic field is a rectangle measuring 120 yards by 40 yards with a semicircle at each of the short sides. There is a 10-yard-wide running track that surrounds the field. In square yards, state the exact area of the track in terms of π.
2400 + 500π
State the exact value to the solution log (x - 21) + log x = 2.
25
Give the equation in general from of the conic section centered at the origin such that the sum of the distances from any point on this curve to its foci is 20 units and one focus is located at (0,8).
25x^2 + 9y^2 = 900 (accept: 25x^2 + 9y^2 - 900 = 0)
A number is greater than a second number by 23. The difference of four times the smaller and half the larger is 90. Identify the two numbers.
29, 52
To the nearest tenth of an inch, determine the length of side AB in triangle ABC given that side BC measures 15 inches, side AC measures 20 inches, and angle ACB measures 112 degrees.
29.2 (inches)
State the following linear equation in standard form: y - 4 = (2/3)(x + 9)
2x - 3y = -30
The sum of two quadratic expressions is three times the difference of the expressions. The first expression is 4x2 - 6x + 10. Give the second expression.
2x^2 - 3x + 5
Integrate the following function y = 4x ln x.
2x^2 ln x - x^2 + C (accept: any variable for C other than x)
State the exact value of: 4 * integral from -1 to 1 (dx/(1_x^2))
2π
Chord AB and Chord CD of circle O intersect at point E. Determine the length of ED given that AB = 11, AE = 9, and CE = 6.
3
Find the product of the roots of the equation x^2 - 4x + 3 = 0.
3
Solve for x in the following equation: x + 5 = sqrt(5x + 49)
3
In simplest radical form, state the product of cbrt(9) * sqrt(3)
3*(6th root of 3)
A cargo ship is carrying shipping containers that have a total volume of 374,970 cubic meters. Shipping containers are 14.5 meters long, 3 meters tall and 2.5 meters wide. State the number of shipping containers that the cargo ship is carrying.
3,448 (containers)
To the nearest tenth, state in square inches the area of a parallelogram with two acute angles each measuring 37°, a side length of 18 inches, and a base of 28 inches.
303.3 (square inches)
Find the exact value of sum from k=2 to 4, (k^(-2) * (1/k)).
307/1728
State as an improper fraction in simplest form the summation of k going from 1 to 3 of k!/k^k
31/18
To the nearest hundredth, state in square units the area of a triangle that has a side length of 7, a side length of 10, and an included angle of 95º.
34.87 (square units)
An ant can support 5,000 times its body mass. The average mass of a human is 62 kilograms and the average mass of an African elephant is 7,000 kilograms. If an average human could support a proportionate mass to the ant, to the nearest whole number, state the number of African elephants a human could support.
44 (elephants)
Find f of 4 given that f(x) = x^5 - x^4 - 10x^3 - 5x^2 + 1.
49
Find the length in inches of the smaller base of a trapezoid with area of 128 square inches, a height of 16 inches, and one base of 11 inches.
5 (inches)
Factor completely 40x^3 - 60x^2 - 10x + 15
5(2x-3)(2x-1)(2x+1)
The volume of a certain rectangular prism is 200 cubic centimeters. A new rectangular prism is formed by tripling its dimensions. In cubic centimeters, state the volume of the new prism.
5,400 (cubic centimeters)
For the next question, you will need this information. The mean lifespan of a certain brand of battery is 200 hours with a standard deviation of 7.6 hours. Assume the lifespans are approximately normally distributed. To the nearest tenth of a percent, state the probability that a randomly selected battery will have a lifespan of at least 212 hours.
5.7 (%)
In simplest, fractional form, state the probability of rolling a sum equal to a prime number when rolling a pair of dice.
5/12
In terms of π, find the area of the sector of the circle with a radius of 12 centimeters and a central angle of 130 degrees.
52π
For the next question you will need this information. Nacho wants a new trademark for his wrestling blouse. He wants his trademark to be an isosceles red triangle inscribed in a blue circle whose radius is 5 centimeters. The triangle's base cuts through the center of the circle. To the nearest tenth, determine in square centimeters how much of his trademark is blue.
53.5 (square centimeters)
Simplify: 5/(x/y +3)
5y/(x + 3y)
In a certain triangle ABC, the midpoints of the three sides are D, E, and F. Determine the number of lines that can be drawn that contain exactly two of the six points.
6
Solve for x in the following logarithmic equation: 2log_2(x - 4) = log_3(4)
6
An 8 foot ladder leans against a wall and makes an angle of 55° with the ground. To the nearest hundredth of a foot, calculate how far up the wall the ladder reaches.
6.55 (feet)
Determine the measure in degrees of the larger acute angle in a right triangle given that the ratio of the acute angles is 1 to 2.
60 (degrees)
Simplify the following expression: (4y^4)^3
64y^12
Twelve people in a family attend the circus together. Adult tickets are $23 and tickets for children are $21. The family spends a total of $266. State respectively the number of adults and the number of children attending.
7 adults, 5 children
To the nearest square centimeter, calculate the difference between the area of a circle and the area of a square given that the square is inscribed in the circle and the circle has a diameter of 16 centimeters.
73 (square centimeters)
State the product of the largest single digit prime number and the smallest two-digit prime number.
77
Find the sum of the first 20 terms for the series 1, 5, 9, 13, 17....
780
A jar contains 10 blue marbles, 8 green marbles, and 4 red marbles. Given that two marbles are selected at random from the jar, state in lowest terms the probability that both marbles are the same color.
79/231
State the area of the triangle bounded by the equations y = 7 and y = 3 + 2|x - 4|.
8
State the difference between the 5th prime number and the 8th prime number.
8
To the nearest tenth, give the exact distance between (- 3, 1, 2) and (4, 3, - 2).
8.3 (Note: sqrt((4 - -3)^2 + (3 - 1)^2 + (-2 - 2)^2) = 8.30662)
The measures of the angles of a triangle are (3x - 5), (2x + 3), and (x + 2). In degrees, state the measure of the largest angle.
85 (degrees)
Determine the remainder when 3 raised to the power of 102 is divided by 10.
9 (Note: on a calculator, type remain(3^102, 10) )
To the nearest tenth, find the distance between the coordinate points (2, 2/3) and (-7, 4).
9.6
A 13 foot ladder is leaning up against a vertical wall. The foot of the ladder is being drawn away from the wall at a rate of 4 feet per second. To the nearest tenth, in feet per second, how fast is the top of the ladder sliding down the wall at the instant when the foot of the ladder is 12 feet from the wall?
9.6 (feet per second) (accept: -9.6)
The Galactic Empire placed two orders with Kuat. The first order was for 11 AT-STs and 2 AT-ATs which totaled 22,184 credits, the second order was for 23 AT-STs and 4 AT-ATs and totaled 45,294 credits. Rebel spies stole the bill and the Empire wants to know the price they paid for each vessel. State respectively in credits, the amount the Empire paid for each AT-ST and each AT-AT?
926 (credits) AT-ST, 5,999 (credits) AT-AT
To the nearest hundredth of a cubic inch, determine the volume of a cone that has a radius of 8 inches and a base angle of 60 degrees.
928.67 (cubic inches)
For the next question you will need this information. A new tennis company has spotted tennis balls. For their cylindrical container they want a label that shows off their tennis balls. The label will be ¾ the height of the cylinder, it will go around the container completely and overlap 1 inch. In order to expose the spots, there will be a circle with a radius of 1.5 inches cut out of the label. To the nearest whole square inch, determine the area of the label given that the cylinder has a height of 8 inches and a radius of 2.5 inches.
93 (square inches)
Match each of the following four mathematicians to his accomplishment. Mathematicians: A. Euclid B. Euler C. Fibonacci D. Descartes Accomplishments: 1. introduced decimal system to Europe 2. recognized as "Father of Geometry" 3. proved e to be irrational 4. established the Cartesian coordinate plane
A-2, B-3, C-1, D-4
Match each of the following four polar graph descriptions with its polar equation. Descriptions: A. circle B. rose curve C. limacon D. line Equations: 1. r = 4sin(2 * theta) 2. r sin(theta) = 4 3. r = 2 + sin(theta) 4. r = 4
A-4, B-1, C-3, D-2
For the next question, you will need this information. In a factory, there are three machines, A, B, and C. When all three machines are working, they produce 287 cell phone cases per hour. When only machines A and C are working, they produce 197 cell phone cases per hour. When only machines A and B are working, they produce 202 cell phone cases per hour. State respectively the number of cell phone cases each machine can produce per hour.
A. 112; B. 90; C. 85
For the next question, you will need this information. Aaron, Beth, and Carol are painting signs for the student council election. When all three students are working, they paint 17 signs in one hour. If just Aaron and Carol are working, they paint 12 signs in one hour. If just Aaron and Beth are working, they paint eight signs in one hour. State respectively the number of signs that each student paints per hour.
Aaron 3 Beth 5 Carol 9
Name the Italian mathematician who introduced a sequence of numbers in which each number is the sum of the previous two numbers, starting with zero and one.
Fibonacci (accept: Leonardo of Pisa)
This is a two-part question. Part 1: Identify the French mathematician who saw a fly on the ceiling and wondered how he could describe its location. He later developed the coordinate plane. Part 2: Identify the Swiss mathematician who discovered a relationship among the numbers of faces, vertices, and edges of any polyhedron. He also developed a first-order numerical procedure for solving differential equations with a given initial value.
Part 1: (Rene) Descartes Part 2: (Leonhard) Euler
This is a two-part question related to a scalene right triangle with the smallest angle measuring 30 degrees where s represents the length of the smallest side and h represents the length of the hypotenuse. Part 1: State a function for h in terms of s. Part 2: State the exact length of the largest leg given that the hypotenuse is 10.
Part 1: h = 2s Part 2: 5cbrt(3)
This is a two part question related to an isosceles right triangle. Part 1: State a function in terms of h that gives the length of the hypotenuse (h) of an isosceles right triangle with side lengths in terms of s. Part 2: State the length of a side given that the hypotenuse is 6.
Part 1: h = s * sqrt(2) Part 2: 3sqrt(2)
For the next question you will need the following information. A company that manufactures women's glasses has a historic 3% defect rate. Management wants to give bonuses to their employees if they can improve upon this rate. They were going to base their decision on collecting data for a month and creating a 98% confidence interval. This is a three-part question. Part 1: If management performs a hypothesis test, is this one or two tails?Part 2: What significance level did they use? Part 3: If they extend data collection to two months, what effect will it have on the power of the test?
Part 1: one tail Part 2: (α =) .02 Part 3: increase
For the next question you will need the following information. A consumer's organization is testing a new battery that claims to have a longer charge time than the previous model. The results from a sample of 500 batteries produced a z-score of 1.72. This is a three-part question. Part 1: Was the organization's test a one-tail or two-tail? Part 2: At a 5% level of significance, would the null hypothesis have been rejected Part 3: If the level of significance was decreased to 1%, would the power of the test increase or decrease?
Part 1: one-tail Part 2: yes Part 3: increase
Who am I? I was born in Samos and then immigrated to southern Italy. None of my writings have survived; however, I am credited with several important discoveries. My most famous writing was a theorem about right triangles.
Pythagoras
Which one of the following four sets of numbers is not considered a Pythagorean triple? Set A. 6, 8, 10 Set B. 5, 12, 13 Set C. 7, 24, 25 Set D. 8, 15, 16
Set D (Note: plug in to the Pythagorean theorem, a^2 + b^2 = c^2)
State the domain of the function given by f(x) = sqrt(9 - x^2)/(x + 2).
[-3, -2) U (-2, 3] (accept: x - 3 ≤ x < -2 or -2 < x ≤ 3)
State whether a certain triangle with side lengths of 4 inches, 7 inches, and 8 inches is right, acute, or obtuse.
acute
For the next question you will need this information. The seniors are at it again. They need to raise money to fund their senior trip. They convince 1,500 underclassmen to each play a $10 game of chance. The game consists of drawing one card at random from a well-shuffled 52-card deck. If the card is black, the underclassman wins $11. If the card is a red ace, the underclassman wins $50. The underclassman loses if he or she draws any other card. To the nearest hundred dollars, give the expected earnings for the senior class.
$3,900
If a triangle in the x, y plane with vertices A (2, 3), B (- 1, 2) and C (1, - 3) is rotated counter-clockwise 90 degrees about the origin, what are the new coordinates for vertex B?
(-2, -1)
Differentiate the following function with respect to x: 6/(3x^4 + 5)
(-72x^3)/(9x^8 + 30x^4 + 25) or (-72x^3)/(3x^4 + 5)^2
Find the solution set of the following inequality. (2x - 5)/(x - 1) ≤ 1
(1, 4] (accept 1 < x ≤ 4)
Simplify the following expression (3 + i)/(4 - i).
(11 + 7i)/17 (accept: 11/17 + 7i/17)
Give the solution to the following system of equations: x + 2y + 3z = 16 x + 2y + 2z = 10 5x + 3y = 4
(2, -2, 6) (accept: x = 2, y = -2, z = 6)
Give the name of the mathematician associated with the following sequence: 1, 1, 2, 3, 5, 8...
(Leonardo) Fibonacci
State the formula used to determine of the sum of the interior angles of a convex polygon.
(n-2)*180
In simplest fractional form, state the sixth term of the following geometric progression: 27, - 3, 1/3.
-1/2187
Find the exact value of 4 times the sine of 15 degrees times the cosine of 15 degrees.
1
Solve for x in the following logarithmic equation. log_6(x) + log_6(x + 5) = 1
1
Which one or ones of the following four statements are false? 1. An integer divided by an integer is always an integer. 2. An irrational number times an irrational number is always irrational. 3. An integer divided by an integer is sometimes undefined. 4. When you square a rational number, the result is always greater than or equal tothe original.
1, 2, 4
Which one or ones of the following four represent a one-to-one function? 1. y = x 2. y = x^2 3. y = log x 4. y = x^3
1, 3, 4
To three decimal places, solve for x in the following equation. 4^(2x) = 60
1,477
For the next question you will need this information. Normal human cells can duplicate twice every 24 hours. Wolverine's mutant cells can duplicate 3,000 times per 24 hours. To the nearest whole number, state how many more times Wolverine's cells duplicate per hour versus normal human cells.
1,500
Identify respectively each of the following four sequences as being geometric, arithmetic or neither. Choice 1. 25, 31, 37, 43 Choice 2. 8, 24, 72, 144 Choice 3. 3, 12, 48, 192 Choice 4. 4, 10, 16, 21
1. arithmetic; 2, neither; 3. geometric; 4. neither
In simplest form, rationalize the numerator and reduce the resulting fraction of the following expression: (sqrt(x+13) - 3)/(x + 4)
1/(sqrt(x + 13) + 3)
Evaluate the limit as x approaches infinity of the square root of the quantity x squared minus 3 divided by the quantity 2x + 1.
1/2
Fill in the blank. If 0° < x° < 90° and cos x° = sqrt(3)/2, then sin x° = (blank).
1/2
To the nearest tenth, determine in square centimeters the area of a certain parallelogram given that its two acute angles measure 24.2 degrees, its side length is 11 centimeters and its base is 23.2 centimeters.
104.6 (square centimeters)
On her vacation, Julie purchased items totaling $29.98. Her total bill was $32.77. To the nearest tenth, determine the percent of sales tax where Julie purchased her items.
9.3 (%)
A baseball player hits a baseball. The height in feet, y of the ball x seconds after it is hit is given by the equation: y = - 14x2 + 84x + 4. State in feet the height the ball reaches.
130 (feet)
Which one or ones of the following three mathematical properties would not be used to solve the equation (x + 3)/2 = (x - 4)/5? 1. distributive property 2. zero product property 3. addition property of equality
2
Nate scored 37 points. He hit 17 shots. Some shots were worth 2 points and the rest were worth 3 points. State respectively the number of each type of shot Nate hit.
2 point shots hit: 14 3 point shots hit: 3
A vehicle's radiator is filled with 4 liters of a 30% antifreeze solution. To the nearest tenth, calculate the number of liters that should be drained and replaced with an 80% antifreeze solution to leave the radiator filled with a 60% antifreeze solution.
2.4 (liters)
State the exact value of the seventh term of the following geometric progression: 686, - 98, 14...
2/343
State the discriminate of the following function: f(x) = 5x2 + 36x + 54
216
Two circles are concentric. The radius of the large circle is 15 centimeters and that of the small circle is 9 centimeters. Given line segment MN is a chord of the larger circle and is tangent to the smaller circle, state in centimeters the length of chord MN.
24 (centimeters)
If y = x^2/ln(x), then dy/dx = (blank).
2x/ln(x) - x/ln^2(x) (accept: (2x ln x - x)/ln^x(x)
Given y = 2x^4 * e^x, evaluate dy/dx. State your answer in simplest form.
2x^3 * e^x * (x + 4)
A sharp corner at a point on a graph indicates which one of the following three? 1. a point of concavity 2. the slope equals zero 3. the derivative is undefined
3
State respectively in centimeters and square centimeters the perimeter and the area of a square with a diagonal that measures eight times the square root of two centimeters.
32 (centimeters), 64 (square centimeters)
Three tennis balls fit tightly in their cylindrical tube. Calculate in cubic centimeters the exact amount of water required to fill the tube with the balls still in place given that the radius of one ball is 3 centimeters.
54π (cubic centimeters)
Assume there are two billion children on the earth under the age of 19 and that Santa will visit each one. On average there are 3.5 children per household. By visiting different time zones from east to west, Santa has exactly 24 hours to complete his job. To the nearest whole number, state the number of houses he must visit per second to reach them all.
6,614 (houses)
For the next question you will need this information. In right triangle ABC where C is the right angle, the length of AB is 8 inches and the measure of angle B is 37°. To the nearest tenth, state in inches the length of BC.
6.4 (inches)
A piece of pie with a radius of 14 centimeters was cut out and eaten. The empty space has a central angle of 59°. To the nearest whole number, state in centimeters the arc length of the outer edge of the remaining pie.
74 (centimeters)
Solve for x in the following equation (x - 2) = 2sqrt(x+1)
8
In triangle ABC, side AB is congruent to side BC. The measure of angle A is 47°. What is the measure of angle B?
86 (°)
Let T(x) = represent the average cost per TIE fighter, in credits per day, when the Imperial Navy builds x TIE fighters. To the nearest whole number, state the minimum number of fighters the Imperial Navy must build in a day to ensure the average cost per fighter is below 9,487 credits.
9
S.H.I.E.L.D. recruited five members of the Avengers in their first decade. Within the last sixty years, forty superheroes have called themselves Avengers alumni. Given that the amount of new additions has decreased by one every decade, how many new additions were added the second decade?
9
This is a three-part question relating to the function f of x is equal to the quantity x + 2 divided by the quantity x - 2. Part 1: Identify the inverse of the function. Part 2: Identify the domain of the inverse. Part 3: Identify the range of the inverse.
Part 1: (2x + 2)/(x - 1) Part 2: all real numbers except x = 1 Part 3: all real numbers except y = 2
This is a two part question regarding a study suggesting that approximately 12% of all people have coulrophobia, a fear of clowns. Part 1: To the nearest hundredth, state the probability that at least one person in a random sample of ten people has coulrophobia. Part 2: To the nearest hundredth, state the probability that the first person you randomly encounter with coulrophobia is the third person you meet.
Part 1: 0.72 Part 2: 0.09
For the next question you will need this information. Mike is trying to decide which box will hold more of his Legos. Box A is a rectangular box with a width of 6 inches, length of 12 inches and height 6 inches. Box B is a cube with a side length of 7.5 inches. This is a two-part question. Part 1: Identify which box will hold more Legos. Part 2: To the nearest whole number, state in cubic inches the difference between the two volumes.
Part 1: Box A Part 2: 10 (cubic inches)
For the next question you will need this information. Two casino games are based on the outcome observed when a player rolls an ordinary 6-sided die. Both games cost $1 to play. Game 1 pays $2.50 for a roll that results in a prime number and nothing for all other results. Game 2 pays $4 for a result that is a multiple of 3 and nothing otherwise. This is a two-part question. Part 1: Which game has greater expected earnings? Part 2: To the nearest cent, how much more does it pay per play?
Part 1: Game 2 Part 2: $.08
This is a two-part question relating to the function f(x) = (-12x - 7)/(x + 3) Part 1: State the inverse of the function. Part 2: State respectively the domain and range of the inverse.
Part 1: f^(-1)(x) = (-3x - 7)/(x + 12) (accept: y = (-3x - 7)/(x + 12) or (-3x-7)/(x+12)) Part 2: domain: all real numbers except x = - 12 range: all real numbers except y = - 3
Which one or ones of the following four sets of numbers are possible lengths for three sides of a triangle? Set A. 6, 7, 8 Set B. 3, 4, 5 Set C. 2, 2, 4 Set D. 2, 11, 12
Set A, Set B, Set D
State respectively the horizontal and vertical asymptotes of the following equation y = (x + 3)/(x^2 + 3x - 4)
horizontal: y = 0 vertical: x = 1 and x = -4
Determine respectively the horizontal and vertical asymptotes of the following equation. y = (x^2 + 2x + 1)/(x^2 + 4x + 3)
horizontal: y = 1 vertical: x = -3
Find the horizontal and vertical asymptotes of the function f(x) = (3x^2 + 2x + 13)/(6x^2 - 5x - 6)
horizontal: y = 1/2 vertical: x = (-2/3)x = 3/2 *
State the domain of the function y is equal to the natural logarithm of the quantity negative x minus 2.
x < -2
Solve for x in the following equation: |x + 1| = |2x - 1|
x = 0 and 2
Identify all asymptotes of the rational function y equals the quantity x squared minus two x plus one all divided by the quantity x minus four.
x = 4; y = x + 2
