NOBCChE Science Bowl: MATH Set 5

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?

ANSWER: 4/5

Short Answer Factor the following polynomial completely over the real numbers: x^3 + 3x^2 - 5x - 15.

ANSWER (x + 3)(x + √5)(x - √5) (IN ANY ORDER)

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?

ANSWER 10

Short Answer What is the measure in radians of an interior angle of a regular pentagon?

ANSWER 3π/5

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

ANSWER Z) 0.385

What is magnitude of the phase shift of the function y = sinx to y = 2sin(3x - π) - 4 (read as: 2 sine of open parenthesis 3x minus pi close parenthesis minus 4)?

ANSWER π/3

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?

ANSWER: (−2, 5)

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

ANSWER: W) I

Short Answer If f(x) = x^2 - 3 and g(x) = 6 − √x, what is the value of f(g(16)) (read as: f of g of 16)?

ANSWER: 1

Short Answer What is the multiplicative identity of the complex numbers in standard a + bi form?

ANSWER: 1 + 0i (ACCEPT: 1)

Short Answer According to Descartes's Rule of Signs, what are the possible numbers of positive real roots that the following equation might have: x^5 - 7x^3 + 15x^2 + 3 = 10?

ANSWER: 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?

ANSWER: 1/2

Short Answer If the median of the five numbers 1, 7, 14, 28, and x is 10, what is x?

ANSWER: 10

Short Answer What is the circumference of the circle with equation (x + 3)^2 + (y − 5)^2 = 25 ?

ANSWER: 10π

Short Answer Providing your answer as an improper fraction in simplest form in terms of π, 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?

ANSWER: 128/7 π

Short Answer What is the exponent of 5 in the prime factorization of 325?

ANSWER: 2

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.

ANSWER: 240

Short Answer If a = cos(x), what is cos(2x) in terms of a?

ANSWER: 2a^2 - 1

Short Answer In degrees, what is the smallest angle x > 0 such that sin(x) + cos(2x) = 1?

ANSWER: 30°

Short Answer Let n be the number of faces of a regular polyhedron. What is the sum of all possible values of n?

ANSWER: 50

Short Answer For which real values of x does the following the series converge: Σ∞ n=0 xn/n! (read as the summation from n equals 0 to infinity of [pause] x to the power of n over n factorial)?

ANSWER: ALL REAL NUMBERS

Multiple Choice Which of the following is an even function? W) Sine X) Cosine Y) Tangent Z) Cotangent

ANSWER: X) COSINE

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?

ANSWER: √21

Short Answer Which two mathematicians are generally credited with the invention of calculus?

ANSWER: LEIBNIZ AND NEWTON

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.

ANSWER: V= (x3√ 3)/4

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 + x^3/3! + x^5/5! (read as: x plus x cubed over 3 factorial plus x to the fifth over 5 factorial) X) x − x^3/3! + x^5/5! Y) 1 − x^2/2! + x^4/4! Z) 1 + x^2/2! + x^4/4!

ANSWER: X) x − x^3/3! + x^5/5!

Multiple Choice Which of the following decreases when 5 is subtracted from each item of a data set? W) Standard deviation X) Range Y) Mean Z) Variance

ANSWER: Y) MEAN

Multiple Choice Which of the following 2 by 2 matrices is equivalent to a clockwise rotation of 90 degrees? W) 0 −1 1 0 (read as: row 1: 0, -1; row 2: 1, 0) X) 0 1 −1 0 (read as: row 1: 1, 1; row 2: -1, 0) Y) 0 −1 −1 0 (read as: row 1: 0, -1; row 2: -1, 0) Z) 0 1 1 0 (read as: row 1: 0, 1; row 2: 1, 0)

X) 0 1 −1 0