MATH #19

What are the coordinates of the hole in the graph of the function, g(x) = (2x²-8)/(x-2)

ANSWER: (2, 8) (Solution: simplifies to 2(x + 2); graph looks like a line except at the indicated hole)

Short Answer Factor the following polynomial completely over the real numbers: x³ + 3x² - 5x - 15.

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

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)?

ANSWER: 1

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

ANSWER: 1 + 0i (ACCEPT: 1)

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?

ANSWER: 1 OR 3 (MUST GIVE BOTH ANSWERS)

To what value does the geometric series that starts 1/3 + 1/9 + 1/27 + ... converge?

ANSWER: 1/2

Cards numbered 1 through 10 lie on a table. If two cards are picked at random, without replacement, what is the probability, as a fraction in lowest terms, that the two cards will have values within one of each other?

ANSWER: 1/5 (Solution: 9/(10 nCr 2) = 9/45 = 1/5)

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

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

ANSWER: 10

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

ANSWER: 10π

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

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

ANSWER: 2

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

ANSWER: 2a² - 1

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

ANSWER: 30°

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

ANSWER: 3π/5

How many solutions are there for the equation, cosine 4x = 21, for the interval 0 ≤ x < 2π?

ANSWER: 8 (Solution: 4 cycles, 2 solutions per cycle)

For which real values of x does the following the series converge: ∑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

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

ANSWER: LEIBNIZ AND NEWTON

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

Which of the following is the equation for the tangent line for y = x⁴ at the point (1, 1): W) y = 4x - 3 X) y = 4x - 1 Y) y = 4x + 1 Z) y = 4x + 3

ANSWER: W) y = 4x - 3 (Solution: using the derivative, y' = 4x³, the slope of the tangent line at x = 1 is 4. The line must pass through (1, 1) so y - 1 = 4(x - 1))

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)

ANSWER: X) [0 1] [−1 0]

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!

ANSWER: X) x − x³/3! + x⁵/5!

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

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