Math, Set 7, Rounds 13-17
13) MATH Short Answer A store has been buying a certain type of calculator at $25 and selling them at $40. At this price, they have been selling 50 calculators per month. The owner of the store wishes to increase the price of the calculator and estimates that for each $1 increase in price, 3 fewer calculators will be sold each month. Determine the price, to the nearest dollar, that maximizes profit.
ANSWER: $41
5) MATH Short Answer Find the open intervals over which a function whose derivative is 5x^2(x + 5)(x − 3) [5 x squared times open parenthesis x plus 5 close parenthesis times the quantity x minus 3] is increasing.
ANSWER: (−∞, −5) AND (3, ∞)
15) MATH Short Answer Find an antiderivative for ln(𝑒^(−𝑥)^2 ) [the natural log of the quantity e to the power of negative x squared].
ANSWER: -(1/3)x^3 (ACCEPT: GIVEN ANSWER PLUS ANY CONSTANT)
20) MATH Short Answer Find an antiderivative for 𝑦 = xe^(-x)
ANSWER: -(x + 1) e^(-x) (ACCEPT: -xe^(-x) -e^(-x) , OR EITHER ANSWER +/- ANY CONSTANT)
16) MATH Short Answer Solve for x: 14 − √(3 − 𝑥) = 9 [14 minus the square root of the quantity 3 minus x equals 9].
ANSWER: -22
8) MATH Short Answer If the graph of the complex number z = 3 + 4i is rotated 90° counterclockwise, what complex number is represented by the result?
ANSWER: -4 + 3i
3) MATH Short Answer For the implicit function xy^2 + x^3y = 2, what is the slope of its tangent line at the point (1, 1)?
ANSWER: -4/3
15) MATH Short Answer A curve is defined parametrically by x = cost [x equals cosine of t] and y = tant [y equals tangent of t]. What is dy/dx when t = π/6?
ANSWER: -8/3
21) MATH Short Answer What is the scalar product of the vectors 2i - 3j + 2k and i + 2j + 2k?
ANSWER: 0
10) MATH Short Answer Evaluate ∫∞ 1 (𝑥^2)/((𝑥^3+3)^2) 𝑑𝑑 ∞ 1 [the integral from 1 to infinity of the fraction with numerator x squared and denominator open parenthesis x cubed plus 3 close parenthesis squared dx].
ANSWER: 1/12
23) MATH Short Answer There is a well-shuffled deck of 52 cards containing 13 cards each of spades, clubs, diamonds, and hearts. You draw 2 cards without replacement. What is the probability that both cards are diamonds?
ANSWER: 1/17
13) MATH Short Answer Given 𝑔(𝑥) = 1−cos(𝑥)/𝑥2 [g of x equals the fraction with numerator 1 minus cosine of x and denominator x squared], find lim𝑥→0 𝑔(𝑥) [the limit as x approaches 0 of g of x].
ANSWER: 1/2
19) MATH Short Answer Given 𝑓(𝑥) = ln√(𝑒^𝑥) [f of x equals the natural log of the square root of the quantity e to the x], find its first derivative with respect to x.
ANSWER: 1/2
16) MATH Short Answer What is the area, in square meters, of a triangle whose sides are of length 10, 12, and 14 meters?
ANSWER: 24√6
1) MATH Short Answer Let 𝑦 = 𝑒^(𝑥^2) [e to the power of x squared]. Find dy/dx.
ANSWER: 2𝑥𝑒^𝑥2
1) MATH Short Answer How many distinct permutations can be made from the letters in the word INFINITY?
ANSWER: 3360
23) MATH Short Answer Given the equation: x + 1/x = 3 [x plus 1 over x equals 3], what is the value of x^4 +( 1)/(x^4) [x to the fourth plus 1 over x to the fourth]?
ANSWER: 47
3) MATH Short Answer Given 𝑔(𝑥) = 5 + 𝑥2/sin(𝑥^2) [g of x equals 5 plus the fraction with numerator x squared and denominator sine of the quantity x squared], find lim𝑥→0 𝑔(𝑥) [the limit as x approaches zero of g of x].
ANSWER: 6
7) MATH Short Answer Assuming y^2 + 3x^2y = 7x^2 - 5 implicitly defines y as a function of x, find dy/dx at the point (3, 2).
ANSWER: 6/31
3) MATH Multiple Choice What is dy/dx for x^2 + y^2 = 2 at the point with coordinates (1, 1)? W) -1 X) -1/2 Y) 0 Z) 3/2
ANSWER: W) -1
9) MATH Multiple Choice What is lim𝑥→𝜋 (sin𝑥)/(𝑥− 𝜋) [the limit as x approaches pi of the fraction with numerator sine of x and denominator x minus pi]? W) -1 X) 0 Y) 1 Z) Undefined
ANSWER: W) -1
9) MATH Multiple Choice The line determined by points (0, 2b) and (2a, 0) intersects the line determined by points (0, b) and (a, b). What represents the x-value of the point where the two lines intersect? W) a X) 2a Y) a + b Z) b/a
ANSWER: W) a
8) MATH Multiple Choice Which of the following is a solution for x of the equation 2^(𝑥−2)= 𝜋^𝑥 [2 to the quantity x minus 2 equals pi to the x]? W) log(4)/log(2)−log(𝜋) [the fraction with numerator log of 4 and denominator log of 2 minus log of pi] X) log(4)/log(𝜋)−log(2) [the fraction with numerator log of 4 and denominator log of pi minus log of 2] Y) log(𝜋)−log(2)/log(4) [the fraction with numerator log of pi minus log of 2 and denominator log of 4] Z) log(2)−log(𝜋)/log(4) [the fraction with numerator log of 2 minus log of pi and denominator log of 4]
ANSWER: W) log(4)/log(2)−log(𝜋)
21) MATH Multiple Choice What is the largest possible real domain of the function h(x) = ln(x^2 - 9) [h of x equals the natural log of the quantity x squared minus 9]? W) (−∞, ∞) X) (−∞, −3) ∪ (3, ∞) [the open interval from negative infinity to negative 3 union the open interval from 3 to positive infinity] Y) (−3, 3) Z) (3, ∞)
ANSWER: X) (−∞, −3) ∪ (3, ∞)
3) MATH Multiple Choice The points on a scatter plot are extremely close to their least-squares regression line, which is expressed by the equation y = -2x + 3. The correlation coefficient r between x and y is closest to which of the following? W) -2 X) -1 Y) 1/2 Z) 1
ANSWER: X) -1
14) MATH Multiple Choice A jar contains 500 jellybeans, 100 each of red, white, blue, orange, and green. If you reach in without looking, how many jellybeans must you grab to be certain that at least five are of the same color? W) 6 X) 21 Y) 25 Z) 26
ANSWER: X) 21
9) MATH Multiple Choice What is the principal value in radians for cos^−1(− √3/2 ) [inverse cosine of negative square root of 3 over 2]? W) − 𝜋/6 X) − 𝜋/3 Y) 5𝜋/6 Z) 11𝜋/6
ANSWER: Y) 5𝜋/6
20) MATH Multiple Choice Box A contains 4 chips numbered 1 through 4 and Box B contains 5 chips numbered 1 through 5. A chip is drawn from each box and the product of the two values drawn is denoted by x. What is the probability that x is even? W) 1/2 X) 2/3 Y) 7/10 Z) 3/4
ANSWER: Y) 7/10
2) MATH Multiple Choice Given the two sequences s1 with nth term 2/𝑛+6 and s2 with nth term 𝑛/5+𝑛^2 , which of the following statements is correct? W) s1 and s2 are both increasing X) s1 is increasing and s2 is decreasing Y) s1 is decreasing and s2 is increasing Z) s1 and s2 are both decreasing
ANSWER: Y) s1 IS DECREASING AND s2 IS INCREASING
5) MATH Multiple Choice Which of the following equations represents two intersecting lines? W) x^2 + y^2 = 9 X) x^2 - y^2 = 9 Y) x^2 + 4y^2 = 1 Z) x^2 - y^2 = 0
ANSWER: Z) x^2 - y^2 = 0
16) MATH Multiple Choice Which of the following is a first order non-linear ordinary differential equation? W) y' + y = 0 [y prime plus y equals zero] X) y'' - 2y^2 - 2 = 0 [y double prime minus 2y squared minus 2 equals zero] Y) y' - xy = 0 [y prime minus xy equals zero] Z) y' - 2y^2 - 2 = 0 [y prime minus 2y squared minus two equals zero]
ANSWER: Z) y' - 2y^2 - 2 = 0
7) MATH Short Answer Solve for x: ln(x + 1) - 1 = 0 [the natural log of open parenthesis x plus 1 close parenthesis minus 1 equals 0].
ANSWER: e - 1
14) MATH Short Answer Find h(t) if 2h'(t) - h(t) = 0 [2h prime of t minus h of t equals zero] and h(0) = 10.
ANSWER: h(t) = 10 e^(1/2)t (ACCEPT: 10 e^(1/2)t )
21) MATH Short Answer What are the equations of all asymptotes -- vertical, horizontal, or oblique -- for the graph of the function 𝑓(𝑥) = (𝑥^2−3𝑥)/(𝑥−1) [f of x equals the fraction with numerator x squared minus 3x and denominator x minus 1]?
ANSWER: x = 1, y = x - 2
2) MATH Short Answer What is the slope-intercept form of the equation of the line that is tangent to the circle with equation x^2 + y^2 = 25 at the point (-3, 4)?
ANSWER: y = (¾)x + 25/4
19) MATH Short Answer Providing your answer in slope intercept form, what is the equation of the line tangent to the graph of 𝑦 = 𝑒^2𝑥/𝑥^2 [y equals the fraction with numerator e to the power of 2x and denominator x squared] where x = 1?
ANSWER: y = e^2 (ACCEPT: y = 0x + e^2)
21) MATH Short Answer Solve the following equation for x over the interval 0 ≤ x < 2π: 2sin^2x − sinx - 1 = 0 [2 sine squared of x minus sine of x minus 1 equals 0].
ANSWER: π/2 , 7π/6, AND 11π/6
16) MATH Short Answer A circle of radius 2 is centered at the point (2, 3). What is the slope of the line tangent to the circle at the point where x = 3 and y is greater than 3?
ANSWER: −√3/3 (DO NOT ACCEPT: −1/√3 )
10) MATH Short Answer Evaluate tan(sin^-1(-1/2)) [tangent of the inverse sine of negative one half].
ANSWER: −√3/3 (DO NOT ACCEPT: −1/√3)
9) MATH Short Answer Find tan(arccos (2x)) [the tangent of the arc-cosine of 2x] in terms of x.
ANSWER: √(1 − 4𝑥^2)/2x
