Mathematics Placement Test Part D - Sample Questions Calculators are not permitted (An answer key is included) #1. For the parabola x = 16y + 4y + 13, for what value of y does x have a minimum? #. If sin 4 ω cos 4 ( ω) = 0, and π ω π, then ω = (?) #3. If the distance between (a, ab) and (ab + a, ab + b) is b (b 0), then what is the value(s) of a? #4. If 1 3 = log 7k 30, then what is the value of k? #5. If tan θ = 3 and sin(θ + π) > 0, what is the exact value of cos θ? #6. Find the solution set for the system of equations: y = 3x ; y = x + 1 #7. If a 0, a ±b, and b < 0, which of the following is FALSE? a 1. a + b a a + b ( ) 1 1. b + = 0 b a ab + b 3. b = a b b 4. 3 b 5 b6 = 15 b 3 5. a (a 1 b ) 1 = 1 ab #8. If g(x) = 1, simplify completely (g(x + h) g(x)) x #9. Given below is a cube. The length of an edge of the cube is x. If the surface area of each face of the cube is halved, determine the length of the inner diagonal of the resulting smaller cube in terms of x
Mathematics Placement Test: Part D Sample Questions #10. If there are 150 grams of carbon-14 in an organism that dies today, in how many years will there be less than 50 grams left in the organism? The half-life of carbon-14 is 5730 years. (Note: if A 0 is the present amount of a material with half-life H, then the amount remaining H time units later is A(H) = A 0 e Hk = 1 A 0, where k is the constant of decay.) 1 #11. Express 1 + cos x + 1 in terms of csc(x). 1 cos x #1. If x and y are real numbers, what is the domain of the function defined by y = log x 7x 15? #13. The graph of g(x) = ax x c passes through ( 1, ) and (c, 0). If 3 is not in the range of g, determine a and c. #14. A quadrilateral whose sides are the x axis, the y axis and the lines y = 4 + x and x = c, lies in Quadrant I. If the area of the quadrilateral is 1 square units, determine the value of c. #15. Given the system of equations, 79 b/(a +3) = 1 3, and 41 b = 1, determine a and b. 64 #16. In the figure above are two concentric circles and an equilateral triangle. The equilateral triangle is inscribed in the larger circle and the smaller circle is inscribed in the equilateral triangle. If the length of a side of the equilateral triangle is 5 units, determine the exact area of the annulus (i.e. the difference of the areas of both circles).
Mathematics Placement Test: Part D Sample Questions #17 Determine the equations of function graphed below. 1 x 1 0 0 1 1 1. f(x) = cos(x π). f(x) = sin(x π ) 1 3. f(x) = sin(4x π ) 4. f(x) = sin(4x) 5. f(x) = cos(4x π) 1 #18. In the figure below, T A and T B are tangents to a circle with center O that has radius 3 cm. What must be the measure of the angle AOB and the length of AB if the triangle ABT is equilateral? #19. If acute angles P and Q are complementary, (i.e. P +Q = π ), and cos(q P ) cos Q cos P < 0, then is the expression sin Q < tan Q TRUE or FALSE?
Mathematics Placement Test: Part D Sample Questions #0. Given below is the graph of the polynomial p(x) = a(x + 5)(x b) c. If c = b and p(0) = b, determine b and a. 30 y 0 10 0 5.0.5 0.0 x 10.5 5.0 #1. If f(x) = x sin x, the which of the following is FALSE? 1. The domain of f is the set of all real numbers. f is a periodic function 3. f is not a one-to-one function 4. f is an even function (the graph of f is symmetric with respect to the y axis) 5. The graph of f crossed the x axis infinitely many times (f has infinitely many x intercepts) # A right circular cone of radius 6m and height 10m is initially filled with water. An empty right circular cylinder of radius 8m and height 7m is set beneath the cone. If the water in the cone begins to flow out and into the cylinder below, express the height H of the water in the cylinder as a function of the height h of water remaining in the cone.
Answers and Hints to Sample MPT - Part D Questions 1. y = x = (y + ) + 3. ω = ± π (sin ω cos ω)(sin ω + cos ω) = (sin ω cos ω)(sin ω + cos ω) = 0 4 3. a = ± 3 a b + b = 4b 4. k = 1000 5. tan θ = 3 θ QI or QIII sin(θ + π) > 0 θ QIII 13 6. (1, 1), ( 43 ), 6 7. answer 5 is FALSE h 8. ( x h) ( x) 3 9. x Length of an edge of the smaller cube must be x resulting length of an ( ) x inner diagonal of the smaller cube is + x. 10. 908 years 11. csc x 1. (, 3 ) (5, ) Solve the polynomial inequality x 7x 15 > 0 13. a = c = 3 a + c = a = c. c c c c = 0 c(c 3) = 0. As 3 is not in the range of g and 0 and are in the range of g, g is not a linear function (i.e. a 0) a = c = 3 < 0, and the graph of g must be a parabola which opens downward, thereby having a maximum value (less than 3 - check!) 14. c = 3 4c + 1 c(c + 4 4) = 4c + c = 1 (c + 7)(c 3) = 0 c = 3 > 0 15. b =, a = ±3 16. 75π 1 Ratios of corresponding sides of similar triangles are equivalent. Use the fact that the right triangle shown the the diagram has acute angles π 6 and π 3. 17. Answer 3 is correct. Period is π, phase shift is π, amplitude is, y-intercept is (0. ). 8 18. AOB = π 3, AB = 3 3
Answers and Hints to Sample MPT - Part D Questions 19. TRUE cos(q P ) = cos(p Q) > 0. P < Q and cos Q cos P < 0 must be true. π > Q > π tan Q > 1 > sin Q 4 0. b =, a = 1 p(0) = a 5 4 = 80a = a = 1 40 40 1. answer is FALSE. f is not periodic. H(h) = 15 8 3 1600 h3 for h [0, 10]