MATH-5
Short Answer A triangle has two sides with lengths of 4 feet and 5 feet, and a 60- degree included angle. What is the length, in feet and in simplest form, of the third side of the triangle?
21
Short Answer Considering the function f(x) = x2, use a left Riemann sum with 4 equal subintervals to estimate the integral from 2 to 10 to the nearest integer.
240
Short Answer Find the sum of the first 10 terms of the arithmetic sequence whose first three terms are 7, 11, and 15:
250 ANSWER: Y) MIXED ANSWER: Y) HIGH LATITUDES
Short Answer By number, identify which one of the following statements is TRUE regarding the set of means of samples of size n from a given population: 1) the mean of the set of sample means varies inversely as the square root of n; 2) the variance of the set of sample means varies directly as n; or 3) the standard deviation of the set of sample means varies inversely as the square root of n.
3
Short Answer What is the limit of (3x2 + 6x)/(5 + 3x + x2) (read as: open parenthesis 3x squared plus 6x close parenthesis over open parenthesis 5 plus 3x plus x squared close parenthesis) as x goes to infinity?
3
Short Answer What is the measure in radians of an interior angle of a regular pentagon?
3 π/5
Short Answer In degrees, what is the smallest angle x > 0 such that sin(x) + cos(2x) = 1?
30??
Short Answer Consider the polar equation r = sin(2??) (read as: r equals the sine of 2 theta). Its graph looks like a rose. How many petals would there be on the ???rose??? in this graph?
4
Short Answer What is the minimum possible value of 2x+ 2y (read as: 2 to the power of x plus 2 to the power of y) if x + y = 2?
4
Short Answer Convert the complex number -2 - 2i 3 to trigonometric form.
4 cos 4?? + i sin 4?? 33
Short Answer Assuming that log 2 = 0.7 and log 3 = 1.1, find the value of log 72 to the nearest tenth.
4.3
Short Answer Each side of a cube is white. Two sides are selected at random and painted green. What is the probability that two selected sides will have a common edge?
4/5
Short Answer What is the coefficient corresponding to x3 in the binomial expansion of (x + 2)5 (read as: the quantity x plus 2 to the power of 5)?
40
Short Answer The four lateral faces of a square pyramid are equilateral triangles. If the area of the base is 49 square centimeters, what is the total surface area, in square centimeters, in simplest radical form?
49 + 49 3 (ACCEPT: 49(1 + 3))
Short Answer If the volume of oil leaked into the ocean from a damaged drillingplatform is given by the equation V = 0.05??r 2 , where r is the radius of the spill, use differentials to estimate the error in cubic meters of the volume if there is a possible error of 0.5 meters in the radius measurement of a spill of radius 100/?? meters.
5
Short Answer If a family has 5 children, what is the probability that there are 4 girls and 1 boy?
5/32
Short Answer Let n be the number of faces of a regular polyhedron. What is the sum of all possible values of n?
50
Short Answer Suppose the average score on a national test is 500, with a standard deviation of 100. If each score is increased by 25, what are the new mean and standard deviation, respectively?
525 AND 100
Short Answer What is the limit as x goes to 0 of xsin(3x) (read as: the quantity x 1 - cos(x) sine of 3x over the quantity 1 minus cosine of x)?
6
Short Answer What is the sum of the smallest prime number and the smallest composite number?
6
Short Answer How many permutations consisting of four letters each, starting with the letter E, can be formed from the letters in the word ENERGY?
60
Short Answer If the area of a regular pentagon is 280 square centimeters, what is the length, in centimeters, of each side if the apothem is 16 centimeters?
7 ANSWER: W) WINDS ANSWER: Y) ALPINE TUNDRA HAS A LOWER SPECIES DIVERSITY THAN ARCTIC TUNDRA
Short Answer Providing your answer in decimal form, what is the limit of(read as: the summation of n equals 0 to infinity of [pause] open parenthesis 3 to the n plus 4 to the n close parenthesis over 5 to the n)?
7.5
Short Answer In how many different ways can a president, vice president, and secretary be elected in a club of 10 members if no one may hold more than one office?
720
Short Answer A population has a mean of 73 and a standard deviation of 8. If the distribution is standardized to create a new distribution with a mean of 100 and a standard deviation of 20, what is the new value of a score of 65 from the original distribution?
80
Short Answer The linear function y = f(x) contains the two points (2, 1) and (5, 8). In simplest form, what is f(13)?
80/3 (ACCEPT: 26 2/3)
Short Answer A circle has a diameter of 32 meters. Find the degree measure of the central angle of a sector of the circle if its arc length measures 8?? meters:
90
Short Answer Evaluate the integral: 0 36 x2 dx (read as: the integral from -6 to 6 0 of the square root of open parenthesis 36 minus x squared close parenthesis d x).
9??
Short Answer What are, respectively, the amplitude, period, horizontal shift, and vertical shift of y = -5 + 7cos [??/6 (x + ??/12)] (read as: y equals negative 5 plus 7 cosine of open bracket pi over 6 open parenthesis x plus pi over 12 close parenthesis close bracket)?
AMPLITUDE = 7, PERIOD = 12, HORIZONTAL SHIFT = - ??/12 (ACCEPT: ??/12 LEFT), VERTICAL SHIFT = -5 (ACCEPT: 5 DOWN)
Short Answer Which two mathematicians are generally credited with the invention of calculus?
LEIBNIZ AND NEWTON
Short Answer For which real values of x does the following the series converge:??? xn (read as the summation from n equals 0 to infinity of [pause] x to the power of n over nn=0 n! factorial)?
ALL REAL NUMBERS
Short Answer If f(x) = x2 ??? 3 and g(x) = 6 ??? x, what is the value of f(g(16)) (read as: f of g of 16)?
1
Short Answer What is the multiplicative identity of the complex numbers in standard a + bi form?
1 + 0i (ACCEPT: 1)
Short Answer Providing your answer in simplest form, give the volume in terms of ?? of a solid generated by revolving the region bounded by the graph of y = ln(x) (read as: y equals the natural log of x), the x-axis, the lines x = 1, and x = e, about the y-axis.
1 ??(e2 +1)[ACCEPT:??(e2 +1 )] 222
Short Answer According to Descartes???s Rule of Signs, what are the possible numbers of positive real roots that the following equation might have: x5 ??? 7x3 + 15x2 + 3 = 10?
1 OR 3 (MUST GIVE BOTH ANSWERS)
Short Answer To what value does the geometric series that starts 1/3 + 1/9 + 1/27 + ... converge?
1/2
Short Answer What is the circumference of the circle with equation (x+3)2 +(y???5)2 =25?
10??
Short Answer Providing your answer in decimal form, what is the area under the curve y = x between 0 and 3?
4.5
Short Answer If the probability of event A is 0.25, the probability of event B is 0.5, and the probability of the union of A and B is 0.55, expressing your answer as a decimal to the nearest tenth, what is the probability of the intersection of A and B?
0.2
Short Answer Assuming that the probability of A is 0.3, the probability of B is 0.3, and the probability of A union B is 0.5, providing your answers as decimals to the nearest tenth, what are the probabilities, respectively, of A intersection B and the complement of the quantity A union B?
0.1 AND 0.5
Short Answer What is the set of all real numbers on which the function ln (x + 2) 1+ 2x(read as: the natural log of the quantity x plus two over open parenthesis 1 plus the square root of the quantity 2 minus x close parenthesis) is well defined?
(-2, 2] (ACCEPT: ALL REAL NUMBERS GREATER THAN NEGATIVE 2 AND LESS THAN OR EQUAL TO 2)
Short Answer The three vertices of a triangle have coordinates (4, 8), (???2, 5), and (3, 1). The longest side of the triangle is opposite which of these three vertices?
(???2, 5)
Short Answer Factor the following polynomial completely over the real numbers: x3 +3x2 ???5x???15.
(x + 3)(x + 5)(x - 5) (IN ANY ORDER)
Short Answer Given f(x) = tanx/secx, find f???(2??/3) (read as: Given f of x equals tangent of x over secant of x, find f prime of 2 pi over 3).
-1/2
Short Answer Suppose that g is the inverse function of f, f(-1) = 2 (read as: f of negative 1 equals 2), and f???(-1) = -2 (read as: f prime of negative 1 equals negative 2). Find g???(2) (read as: g prime of 2).
-1/2
Short Answer Find all values of c such that the given function is continuous from negative infinity to infinity: f(x) = c2x when x ??? 2, and f(x) = (5/4)cx2 + 3 when x > 2.
-1???2, 3
Short Answer If f(x) = x 2 (read as: f of x equals open parenthesis x minus 2 close x2 4parenthesis over open parenthesis x squared minus 4 close parenthesis) at which value or values of x does the graph of f(x) have a vertical asymptote?
-2
Short Answer Considering the function f x = 2x3 + 5x2+ 4, what is the instantaneous rate of change at the point (-1, 7)?
-4
Short Answer Providing your answer as an improper fraction in simplest form, what is the slope of the line tangent to the graph of 3 x2 (read as: the cube root of x squared) at the point with coordinates of (-1/8, 1/4)?
-4/3
Short Answer If f(x) = 2x ??? 1 and g(x) = 6 + 2x ??? x2, what is the standard polynomial form for g(f(x)) (read as: g of f of x)?
-4x2 + 8x + 3
Short Answer Let f(x) be a function with antiderivative g(x) satisfying g(1) = 2 and g(3)=4.Calculate 132f x dx- 135f x dx(readas:theintegralfrom1to3of2fofxdxminus the integral from 1 to 3 of 5 f of x dx).
-6
Short Answer Evaluate limx?????/3 x ??2 (read as: the limit as x approaches pi over3three of open parenthesis sine of x [pause] minus the square root of 3 over 2 close parenthesis over open parenthesis x minus pi over 3 close parenthesis).
1/2
Short Answer At what value of x is the tangent to the curve f(x) = 3x2 ??? 2x + 1 a horizontal line?
1/3
Short Answer Find the slope of the line tangent to the parametric curve at the given point: x = et, and y = lnt (read as: y equals the natural log of t) at t = 1.
1/e
Short Answer An isosceles triangle has a base of 30 feet and height of 15 feet. In feet, how far apart are the legs 10 feet above the base?
10
Short Answer Given: u(4) = -2 and u???(4) = 3 (read as: u of 4 equals negative 2, and u prime of 4 equals 3). If f(x) = xu(x) (read as: f of x equals x u of x), then evaluate f???(4) (read as: f prime of 4).
10
Short Answer A right triangle has sides with lengths of 8, 15, and 17 meters. As a fraction in lowest terms and in meters, what is the length of the altitude to the hypotenuse?
120/17
Short Answer Providing your answer as an improper fraction in simplest form in terms of ??, what is the volume of the solid generated when the area bounded by y = x3 , x = 2, and the x axis is revolved around the x-axis?
128/7 ??
Short Answer Considering the function f x = 3x2- 4x + 1, what is the average rate of change from x = 2 to x = 5?
17
Short Answer Approximate the area under f(x) = x2 + 1 from x = 0 to x = 8 using a mid-point Riemann sum with four equal subintervals.
176
Short Answer What is the name of the type of surface generated by the equation z=x2+y2?
ELLIPTIC PARABOLOID
Short Answer Find the radius and the center, respectively, of the circle with the equation:x2 +y2???16x+12y+75=0?
RADIUS = 5; CENTER = (8, -6)
Short Answer A triangular prism has a base that is an equilateral triangle with sides of length x. If the height of the prism is x, express the volume of the prism in terms of x.
V= x3 3 4
Short Answer What is the name for the test that can be used to visually determine whether or not a relation defined by a graph represents a function?
VERTICAL LINE TEST (ACCEPT: VERTICAL LINE)
Multiple Choice Which of the following can be constructed using only a compass and a straightedge?W) A line segment with one-third the length of a given line segment X) A cube with twice the volume of a given cubeY) An angle with one-third the measure of a given angleZ) A square with the same area as that of a given circle
W) A LINE SEGMENT WITH ONE-THIRD THE LENGTH OF A GIVEN LINE SEGMENT
Multiple Choice The line with equation y = ???3x ??? 5 contains points in each quadrant EXCEPT which of the following?W) I X) II Y) III Z) IV
W) I
Multiple Choice A polyhedron with seven faces, exactly five of which are rectangles, could be which of the following?W) Pentagonal prism X) Pentagonal pyramid Y) Hexagonal prismZ) Hexagonal pyramid
W) PENTAGONAL PRISM
Multiple Choice If the derivative of f(x) is positive on the closed interval [a, b], then which of the following statements is TRUE?W) f(x) is increasing on [a, b]X) f(x) has critical points at a and bY) f(x) is constant on [a, b]Z) f(x) cannot have an inflection point on [a, b]
W) f(x) IS INCREASING ON [a, b] BONUS
Multiple Choice The last two digits, in order, of a positive integer n are 1 and 8. Which of the following statements about n can be TRUE?W) n is divisible by 7 X) n is divisible by 5 Y) n is divisible by 4Z) n is a perfect square
W) n IS DIVISIBLE BY 7
Multiple Choice Which of the following is a function for which f(-x) = -f(x) (read as: f of negative x equals negative f of x)?W) sin(x) X) x2Y) cos(x) Z) ex
W) sin(x)
Multiple Choice Which of the following 2 by 2 matrices is equivalent to a clockwise rotation of 90 degrees?W) 0 ???1 (readas:row1:0,-1;row2:1,0) 10X) 0 1 (readas:row1:1,1;row2:-1,0) ???1 0Y) 0 ???1 (readas:row1:0,-1;row2:-1,0) ???1 0Z) 0 1 (readas:row1:0,1;row2:1,0) 10
X) 0 1 ???1 0
Multiple Choice Which of the following is an even function?W) SineX) CosineY) Tangent Z) Cotangent
X) COSINE
Multiple Choice Which of the following is the sum of the first three nonzero terms of the Taylor series of sin(x) around 0?W) x + x3/3! + x5/5! (read as: x plus x cubed over 3 factorial plus x to the fifth over 5 factorial) X) x ??? x3/3! + x5/5!Y) 1 ??? x2/2! + x4/4!Z) 1 + x2/2! + x4/4!
X) x ??? x3/3! + x5/5!
Short Answer Find the x and y intercepts of the line passing through the point (4, 8) that is perpendicular to the line 3x ??? 6y ??? 12 = 0:
X-INTERCEPT = (8, 0) (ACCEPT: 8); Y-INTERCEPT = (0, 16) (ACCEPT: 16) ANSWER: Z) GABBRO ANSWER: Z) VESICULAR ANSWER: X) EYELID
Multiple Choice For a function, f(x) (read as: f of x), suppose that f is continuous on the closed interval [-1, 3], differentiable on the open interval (-1, 3), and that f(-1) = -2 andf(3) = 2. Which of the following is the value of the derivative for at least one value, c, on the open interval (-1, 3)?W) -1 X) 0 Y) 1 Z) 2
Y) 1
Multiple Choice If e 5 = 30, which of the following is the value of x, assuming that thenatural log of 2 = 0.7, and the natural log of 15 = 2.7:W) 15 X) 16 Y) 17 Z) 18
Y) 17
Multiple Choice Which of the following statements is TRUE?W) A correlation of +0.8 is stronger than a correlation of -0.8 X) A correlation of +0.8 is weaker than a correlation of -0.8Y) A correlation of +0.8 is just as strong as a correlation of -0.8 Z) It is impossible to tell which correlation is stronger
Y) A CORRELATION OF +0.8 IS JUST AS STRONG AS A CORRELATION OF -0.8
Multiple Choice Which of the following statements is NOT true about Taylor series?W) Taylor series are defined for infinitely differentiable functionsX) All Taylor series are power seriesY) All Taylor series are convergentZ) The first n terms of a Taylor series approximate a function with a polynomial
Y) ALL TAYLOR SERIES ARE CONVERGENT
Multiple Choice Which of the following is NOT true about the function y = |x| (read as: y equals the absolute value of x)?W) It is symmetric around the y-axisX) The derivative of the function does not exist at x = 0Y) It is monotonicZ) The integral of the function over the real line is unbounded
Y) IT IS MONOTONIC
Multiple Choice Which of the following decreases when 5 is subtracted from each item of a data set?W) Standard deviation X) RangeY) MeanZ) Variance
Y) MEAN
Multiple Choice A line with the equation y = cx + c, where c is a non-zero real number, must intersect which of the following?W) Positive x-axis X) Positive y-axis Y) Negative x-axis Z) Negative y-axis
Y) NEGATIVE x-AXIS
Multiple Choice Suppose, in a certain part of the world, in any 50-year period, the probability of a major plague is 39%, the probability of a major famine is 52%, and the probability of both a plague and a famine is 15%. Given that there is a plague, which of the following is the probability of a famine?W) 0.240 X) 0.288 Y) 0.370 Z) 0.385
Z) 0.385
Multiple Choice Which of the following statements is NOT an advantage of a repeated-measures design compared to an independent-measures design to compare two treatment conditions?W) It requires fewer subjectsX) It is better suited to study development that takes place over time Y) It reduces problems caused by individual differencesZ) It reduces the confounding effects of other factors
Z) IT REDUCES THE CONFOUNDING EFFECTS OF OTHER FACTORS
Multiple Choice Which of the following is NOT a property of the normal p.d.f.?W) The p.d.f. is symmetricX) The p.d.f. has two inflection pointsY) The variance has to be positiveZ) The domain of the p.d.f. is non-negative real numbers
Z) THE DOMAIN OF THE P.D.F. IS NON-NEGATIVE REAL NUMBERS
Short Answer If f(x) = x (read as: f of x equals the square root of x) over the domain of all nonnegative real numbers and if g(x) = cosx over the domain of closed interval [-??, ??], then what is the domain of f composed with g?
[-??/2, ??/2]
Short Answer Evaluate the following limit: limn?????? 1+ n3 n (read as: the limit as n approaches infinity of open parenthesis 1 plus 3 over n close parenthesis to the power of n).
e3
Short Answer What is the inverse function of f(x) = x5 + 1 (read as: f of x equals x to the fifth plus 1)?
f ???1(x) = 5 x 1
Short Answer If f x, y = xy + 3xy2 (read as: f of x, y equals xy plus 3xy squared), what is the partial derivative of f with respect to y?
x + 6xy
Short Answer What is magnitude of the phase shift of the function y = sinx toy = 2sin(3x -π)- 4 (read as: 2 sine of open parenthesis 3x minus pi close parenthesis minus 4)?
π/3
