4. Factor the expression completely. Begin by factoring out the lowest power of each common factor: 20x 1/2 + 9x 1/2 + x 3/2

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1 M180 Final Exam practice 1.Simplify each expression, and eliminate any negative exponents. st s t. Simplify the expression. Assume that x, y, and z denote any positive real numbers. 3. Rationalize the denominator 5 x y x y z 4. Factor the expression completely. Begin by factoring out the lowest power of each common factor: 0x 1/ + 9x 1/ + x 3/ 5. Rationalize the denominator Perform the multiplication or division and simplify. 7. Find all solutions of the equation 3x3 1 x 4y 9 y y 3 y 7 y 15 y 4y 5 9. What amount of a 60% acid solution must be mixed with a 30% solution to produce 300 ml of a 55% solution? 10. It took a crew h 15 min to row 3 km upstream and back again. If the rate of flow of the stream was1 km/h, what was the rowing speed of the crew in still water? x Solve the nonlinear inequality. Express the solution using interval notation: 3 x 3 1. Solve the absolute value inequality. Express the answer using interval notation: 3x Determine whether the function f is even, odd, or neither. f(x) = x 3 + 4x f x x x 14. Suppose f(x) = 5x 4 and g(x) = 3 x. (f g)( )= (g f)(x)= 15. Find the inverse for f x x. x Find a function f whose graph is a parabola with the given vertex and that passes through the given point, vertex (8, 3); point (9, )

2 17. If a ball is thrown directly upward with a velocity of 50 ft/s, its height (in feet) after t seconds is given by y = 50t 16t. What is the maximum height attained by the ball? (Round your answer to the nearest whole number.) 18. Use either synthetic or long division to divide P(x) by D(x), where P(x) = 3x 3 + 9x 6x 1, D(x) = x + 4 Express P in the form P(x) = D(x) Q(x) + R(x). 19. Let P(x) = x 3 3x (a) Find the zeros (b) Write the polynomial in factored form. 0. Find a polynomial with integer coefficients that satisfies the given conditions: degree 3 and zeros 4 and 1 + i. Solve the logarithmic/exponential equations. 1. log(x) + log(x ) = 8. e 3x = 10, 3. 5 x = 4 x A woman invests $6300 in an account that pays 6% interest per year. How long will it take for the amount to be $11,000? (Round your answer to two decimal.) If the interest compounded monthly? 5. A sum of $1000 was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to $ in the given time, what was the interest rate? (Round your answer to two decimal places.) 6. The half-life of radium-6 is 1600 years. Suppose we have a 4-mg sample. (a) Find a function m(t) = m 0 t/h that models the mass remaining after t years. (b) Find a function m(t) = m 0e rt that models the mass remaining after t years. (Round your rvalue to six decimal places.) (c) How much of the sample will remain after 5000 years? (Round your answer to one decimal place.) (d) After how many years will only 15 mg of the sample remain? (Round your answer to one decimal place.) 7. Find the terminal point P(x, y) on the unit circle determined by the given value of t = Suppose that the terminal point determined by t is the point on the unit circle which is terminal point determined by π t. 1, 5 5. Find the

3 Write the first expression in terms of the second if the terminal point determined by t is in the given quadrant. 9. tan(t), cos(t); quadrant II 30. csc(t), cot(t); quadrant III 31. sin(t), sec(t); quadrant IV 3. If cos(t) = -4/5 and terminal point of t is in quadrant III, find cot(t). 33. If tan(t) = 1/5 and terminal point of t is in quadrant III, find csc(t). 34. If sec(t) = 5 and sin(t) < 0, find tan(t). 35. If tan(t) = -5 and csc(t) > 0, find sec(t). 36. If cos t = 1/5 and t, compute the following: sin( t) + cos( t) Find the amplitude, period, and phase shift of the function, y-intercept and graph one complete period. 37. y cos x 38. y = sin( πx + π) The graph of one complete period of a sine or cosine curve is given. Write an equation that represents the curve in the form y = a sin(k(x - b)) or y = a cos(k(x - b)). 39.

4 Find an angle between 0 and π that is coterminal with 7 /5. Give your answer in terms of π. 4. Find the radius of each circle if the area of the sector is 80. (Round your answer to two decimal places.) 43. Find x.write exact length. Assume A = Find the cosθ from the information given: cot θ = 1/6, sin θ < Find the area of the shaded region in the figure where α = π/6, b = 0. Write exact length. No calculator. 46. Use the Law of Sines to solve all possible triangles that satisfy the given conditions if possible. Round to one decimal. a 1, b 16, A For the triangle shown, find the length AD. Assume u = 0, v = 0, x = 35, y = 35.(Round your answer to one decimal places.) 48. A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 0 mi apart, to be 3 and 48, as shown in the figure. Round your answers to the nearest tenth mile.

5 Find the elevation of the plane. 49. Find the area of the shaded figure. (Round your answer to two decimal places.) a = 1, b = 1, y = Airport B is 300 mi from airport A at a bearing N 50 E (see the figure). A pilot wishing to fly from A to B mistakenly flies due east at 00 mi/h for 30 minutes, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? Round your answer to one decimal. (b) What bearing should he head his plane in order to arrive at airport B? Round your answer to one decimal. 51. A 500 ft tower is located on the side of a mountain that is inclined 3 to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed. 5.Find the solution of the system using Gauss-Jordan elimination. x + y + z = -7 x y + z = 3 4x + y + z = - 5. Solve the system of equations by converting to a matrix equation (AX=B) and using the inverse of the 1 coefficient matrix. First find A. -X+Y+Z = -10 X+Y+3Z = - X+Y+Z = Find the partial fraction decomposition of the rational function x1 x 55. An equation of an ellipse is given: 9x 36x + 4y = 0 4x

6 (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. 56. An equation of a parabola is given: x + 6x + 1y + 9 = 0 (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph with intercept if exist. 57. An equation of a hyperbola is given: x 16 4 y x (a)find the center, vertices, foci, and asymptotes of the hyperbola (b)sketch a graph 58. A partial sum of an arithmetic sequence is given. Find the sum: The 0th term of an arithmetic sequence is 101, and the common difference is 3. Find a formula for the nth term The fourth term of geometric sequence is 1 and the seventh term is. Find the first and nth term Express the repeating decimal as a fraction: Prove by mathematical induction: n 3 = 1 n n 65. Use mathematical induction to determine 5 n is divisible by 4 for all natural numbers n. 66. Find the indicated term in the expansion of the given binomial. The term that does not contain x in the expansion of x 3x Determine the coefficient of ab in the expansion of (a + b) 8 by using the Binomial Theorem.