MATH #26
If the squared length of the diagonal of one face of a cube is 162, what is the surface area in square units of the cube?
ANSWER: 486
Find a value for a such that the lim x→∞ (ax⁵ - x³ + x) / (x - x³ + 2x⁵ = 3 [limit as x approaches infinity of the fraction with numerator ax to the fifth minus x cubed plus x and denominator x minus x cubed plus 2 x to the fifth equals 3].
ANSWER: 6
For Valentine's Day, Adam bought several dozen roses and several dozen carnations. The roses cost $15 per dozen, and the carnations cost $8 per dozen. Adam bought a total of 17 dozen flowers and paid a total of $192. How many dozen roses did he buy?
ANSWER: 8
What is the solution to the following system of three equations: equation one: 2x - y + 3z = 8; equation two: x - 6y - z = 0; and equation three: -6x + 3y - 9z = 24?
ANSWER: NO SOLUTION
Expressing the answer in simplest form, solve | 2r-¹³/₄ | < ⁵/₂ [the absolute value of the quantity 2r minus 13 fourths is less than 5 halves].
ANSWER: 3/8 < r < 2 ⁷/₈ (ACCEPT: 3/8 < r < ²³/₈ )
Given f(x) = x + 1 and g(x) = 3x, what is f(g(x)) [f of g of x]?
ANSWER: 3x + 1
Factor completely the expression: 5rs + 25r - 3s - 15.
ANSWER: (s + 5)(5r - 3)
Find the remainder when f(x) = x⁶ + 5x⁵ - x³ + x - 6 is divided by (x + 1).
ANSWER: -10
In terms of a, b, c, and d, what is the slope of the line 3ax + 2by = cd?
ANSWER: -3a/2b
In the three by three matrix, [-2,2,0] [4,1,-5] [8,3,7] [such that row one contains negative 2, 2, and 0; row two contains 4, 1, and negative 5; row three contains 8, 3, and 7], which is the [x sub 2, 3] element?
ANSWER: -5
Differentiate sin(cosx) [sine of cosine of x].
ANSWER: -cos(cos x)sinx [ACCEPT: cos (cos x)(-sin x) or -sinx cos(cos(x)); DO NOT ACCEPT: -cos2x sinx]
What is the value of i⁴?
ANSWER: 1
If g(b) = 6b³ - 4b² + b + 4b⁻¹ [g of b equals 6b cubed minus 4b squared plus b plus 4 b to the negative 1], what is g'(b) [g prime of b]?
ANSWER: 18b² - 8b + 1 - 4b⁻² (ACCEPT: 18b² - 8b + 1 - 4/b²)
Evaluate the following: ∫0,2 1/t × e^ln(t) dt [the integral from 0 to 2 of 1 over t times e to the power of the natural log of t dt].
ANSWER: 2
How many terms does the binomial expansion of (x² + 2y³)²⁰ [open parenthesis x squared plus 2y cubed close parenthesis to the power of 20] contain?
ANSWER: 21
How many points of intersection do the graphs of the functions f(x) = x² and g(x) = 2ˣ have?
ANSWER: 3
What is the multiplicity of the zero 1 in the polynomial p(x) = x⁴ + x³ - 9x² + 11x - 4?
ANSWER: 3
Which of the following functions has an oblique asymptote? W) f(x) = (x⁵ + 1)/(x⁴ + 3x² + 2) [f of x equals open parenthesis x to the fifth plus 1 close parenthesis over open parenthesis x to the fourth plus 3 x squared plus 2 close parenthesis] X) f(x) = (x² + 1)/(x³ - x² - 1) [f of x equals open parenthesis x squared plus 1 close parenthesis over open parenthesis x cubed minus x squared minus 1 close parenthesis] Y) f(x) = (4x² + x + 1)/x² [f of x equals open parenthesis 4 x squared plus x plus 1 close parenthesis over x squared] Z) f(x) = x⁵/(x² - 1) [f of x equals x to the fifth over open parenthesis x squared minus 1 close parenthesis]
ANSWER: W) f(x) = (x⁵ + 1)/(x⁴ + 3x² + 2)
Which of the following points lies in the solution set for the system of two inequalities: 2y - x ≥ -6 and 2y - 3x < -6? W) (-4, -1) X) (3, 1) Y) (0, -3) Z) (4, 3)
ANSWER: X) (3, 1)
Which of the following equations is that of a line perpendicular to 3x - 7y = 5? W) 3x + 7y = 10 X) 7x + 3y = 10 Y) 7x - 3y = 10 Z) 3x - 7y = 10
ANSWER: X) 7x + 3y = 10
The hyperbola with equation y²/25 - x²/36 = 1 possesses which of the following characteristics? W) It opens left and right and has asymptotes y = ⁵/₆x and y = -⁵/₆x X) It opens up and down and has asymptotes y = ⁵/₆x and y = -⁵/₆x Y) It opens left and right and has asymptotes y = ⁶/₅x and y = -⁶/₅x Z) It opens up and down and has asymptotes y = ⁶/₅x and y = -⁶/₅x
ANSWER: X) IT OPENS UP AND DOWN AND HAS ASYMPTOTES y = ⁵/₆x AND y = -⁵/₆x
Which of the following is an identity? W) cos(2x) = 2 cos(x)[cosine of 2x equals 2 cosine of x] X) sin(2x) = 2 sin(x) cos(x) [sine of 2 x equals 2 sine of x cosine of x] Y) cos(x + y) = cos(x) + cos(y) [cosine of the quantity x plus y equals cosine of x plus cosine of y] Z) sin(x - y) = sin(x) - sin(y) [sine of the quantity x minus y equals sine of x minus sine of y]
ANSWER: X) sin(2x) = 2 sin(x) cos(x)
Bacteria in a culture are growing exponentially, such that on day 0, there are 100 bacteria, on day 1, there are 200 bacteria, and on day 2, there are 400 bacteria. Which of the following equations expresses the number of bacteria, y, present at any time t? W) y = 100 + 2ᵗ [y equals 100 plus 2 to the power of t] X) y = 100(2ᵗ) [y equals 100 times the quantity 2 to the power of t] Y) y = 2ᵗ [y equals 2 to the power of t] Z) y = 200(2ᵗ) [y equals 200 times the quantity 2 to the power of t]
ANSWER: X) y = 100(2ᵗ)
In the graph of the function f(x) = ³/₂sin(ˣ/₂) [f of x equals three halves the sine of the quantity x over 2], at which of the following x coordinates would f(x) = 0? W) π/2 X) π Y) 2π Z) 3π/2
ANSWER: Y) 2π
If g(u) = ∫0, u sec²t dt [g of u equals the integral from 0 to u of secant squared of t dt], find g'(u) [g prime of u].
ANSWER: sec²u