SABIS School Network

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1 EOY Revision Packet Mathematics, IM3 Page 1 of 14 SABIS School Network EOY Revision Packet Ch.1 Ch.8 Highlights: Understand the geometric approach to square a binomial. Apply the formula for the square of a binomial. Expand powers of polynomials and apply Pascal s triangle. Know and apply the binomial theorem. Use the binomial theorem to estimate decimals. Use the binomial theorem to evaluate powers of complex numbers. Mathematics IM3 Divide a polynomial by monomials, binomials, and other polynomials. Divide using synthetic division. Know and apply the formula for the sum of geometric series. Know and apply the remainder theorem. Know and apply the factor theorem. Solve a quadratic equation with complex roots. Know and apply the properties of conjugates. Work on quadratic equations with complex roots. Apply the fundamental theorem of algebra. Work out questions on rational roots of polynomial equations. Graph a cubic and a quartic polynomial function. Operate with rational expressions, work out complex rational expressions, and find the zeros of rational expressions. Graph rational functions. Solve rational equations. Solve rational inequalities algebraically and using a calculator. Solve a radical equation and a literal equation. Find the inverse relation for different types of expressions. Solve a nonlinear equation graphically and numerically using linear interpolation, interval bisection, and rearranging.

2 EOY Revision Packet Mathematics, IM3 Page of 14 Express a logarithmic expression in exponential form and an exponential expression in logarithmic form. Evaluate logarithmic expressions from the definition, and estimate the value of a logarithmic expression. Write a log expression in terms of ln. Solve a literal equation involving exponentials. Solve for the time the formula of amortization. Solve for the time the formula of compound interest. Work out questions on radioactive material. Apply Newton s law of cooling. Graph logarithmic functions and know the inverse property of the logarithms and exponentials. Apply the equality and the inequality properties of logarithms. Apply the basic properties of the logarithmic function. Apply the product rule and quotient rule of logarithms. Solve simple logarithmic equations and simplify logarithmic expressions. Apply the change of base formula. Solve exponential equations and inequalities. Solve continuous compounding problems, exponential growth and decay problems, loudness of sound problems, acidity problems, and earthquake intensity problems. Identify data that is modeled using a cubic polynomial and build a cubic model to represent data. Use a simple rational function to model data given in a table of values. Identify data that is modeled using an exponential function. Use a logarithmic function or a power function to model a set of data. Given one algebraic measure of an angle, in degrees or in radians, give other positive/negative ones. Draw a directed angle in a coordinate plane and know the six trigonometric ratios of a directed angle. Know the signs of the trigonometric ratios of directed angles. Find the reference angle of a directed angle. Know the relations between special trigonometric ratios. Express a trigonometric ratio in terms of its reference angle. Solve many types of trigonometric equations. Draw the graph of the sine function and identify its period, amplitude, vertical shift, and phase shift. Express a sine function as a cosine function and sketch their graphs. Sketch graphs of tangent functions. Work out different problems on the trigonometric functions.

3 EOY Revision Packet Mathematics, IM3 Page 3 of 14 Vocabulary: Binomial, Square of a binomial, Pythagorean triplet, Pascal s triangle, Binomial theorem, Estimate, Expansion Divide a polynomial, Factor, Remainder, Synthetic division, Quotient, Divisor Geometric series, Deposit, Interest, Compounded, Rate, Financed Remainder theorem, Factor theorem, Linear factors, Multiplicity Complex roots, Conjugate, Root of a polynomial, Fundamental theorem of algebra Zeros of a polynomial, Roots of odd multiplicity, Roots of even multiplicities End behavior, Cubic function, Quartic polynomial Rational expression, Domain, Simplest form, Least common multiple, Least common denominator, Simplest form, Complex rational expressions, Asymptotes, Large values Rational equation, Extraneous solution, Rational inequality, Graphing calculator Non-linear equation, Radical equation, Literal equation, Inverse of a relation, Graphically, Numerically, Linear interpolation, Interval bisection, Rearranging Change of base, Logarithmic equation, Exponential equation, Continuous compounding, Annual percentage rate, Decay, Growth, Relative intensity, Sound loudness, Acidity of a substance, Concentration, Magnitude of an earthquake Modelling, Polynomial function, Rational function, Logarithmic function, Power function, Exact math model Trigonometric ratios, Directed angles, Algebraic measure, Remarkable angle, Reference angle, Related angle, Special angles, Cofunctions, Trigonometric equation, Periodic functions, Amplitude, Period, Phase shift, Vertical shift Ferris wheel, Electricity supply, Seasonal sale, Arterial blood flow, Temperature fluctuation

4 EOY Revision Packet Mathematics, IM3 Page 4 of 14 Revision Exercises: 1. Verify the given identity. a) b) 3 6 x x x 1 x 1. The length of the hypotenuse of a right triangle is 34 units. Find the lengths of the two legs knowing that they are integers with no common factors. 3. Construct Pascal s triangle up to row 5 to answer the following questions. a) What is the coefficient of x 3 in the expansion of ( + x) 4? b) What are the last two terms when (1 + x ) 5 is expanded in an ascending order of powers of x? c) What is the middle term in the expansion of ( + x) 4? d) Expand and simplify 4 1. e) Expand and simplify 5 1 t t. 4. Use the Binomial Theorem to find the indicated terms. a) The 7th term in the expansion of (u + ) 11 b) The 5th term in the expansion of (4 x) 8 6 a x 5. The middle term in the expansion of x is 1. Find the value of a. 6. Simplify (1 + i) 5 (1 i) The total cost function for a product is modeled in dollars by C(x) = 0.0x + 8x + 00, 0 x 00. a) Derive a simplified expression for the average cost function. b) What is the change in the average cost per unit when the production increases from 10 units to 0 units? c) What is the change in the average cost per unit when the production increases from 0 units to 30 units? d) What conclusion can be drawn from the above results? 8. Find the quotient and the remainder using long division and verify your answers. a) (x +5x + ) (x + 4) b) (x 3 + 3x 6x + ) (x + 1) 9. When the polynomial P(x) is divided by (x + 3), the quotient of the division is x + and the remainder is 3. Find P(x).

5 EOY Revision Packet Mathematics, IM3 Page 5 of Determine P(x) in the synthetic division: 3 Px ( ) When x + ax + is divided by (x + ), the remainder is 6. Find the value of a. 1. Consider the division a) Find the value of a. b) Write the result of the division. 3 3 k x x x x R x a x a. 13. The floor of a rectangular hall is to be covered with rectangular tiles each having a length of x + 5 cm and a width of x + 5 cm. Derive a simplified expression for the minimum number of tiles needed to cover the whole room given that its area is given by A = 40x x + 650x The marketing department expects the 00t 0 monthly t4 sales of a new product during the first year to be Nt () modeled by the equation t 1, where N is the number of units sold in thousands and t is the time in months. 00t 0t4 a) Perform the division t 1. b) Using the result of the division, estimate the maximum monthly sales. c) Verify your conclusion by calculating N(3), N(4), and N(6). 15. Find the sum of a geometric series given its first term a, its last term an, and the common ratio r. Express your answer in a simple rational form. 1 a) a = 1, an = 6,561, r = 3 b) a = 1,04, an = 1, r = A young worker s goal is to have $ 50,000 by the time of his retirement. He plans to set aside a fixed amount at the end of each month in an investment that earns an annual rate of 8% compounded monthly. If the worker s intent is to retire after 30 years, what is the amount of money he should set aside each month? 17. Apply the Remainder Theorem to find P(a). a) P(x) = 3x 3 1x 6x + 6, a = b) P(x) = 18 10x 3 1x 0x, a = The graph of the equation y = 3x 3 + kx x 4 intersects the x-axis at x =. Find k. 19. The remainder of the division of a polynomial P(x) by (4x 1) is 3. Find P(0.5).

6 EOY Revision Packet Mathematics, IM3 Page 6 of Use synthetic division to calculate the value of k that makes the given binomial a factor of the corresponding polynomial. a) x 3 5x + kx 15; (x 3) b) 18x 3 + kx x + 1; (x 1) 1. Given that (x ) and (x + 3) are factors of the polynomial x 4 + x 3 3x + ax + b, find a and b using synthetic division.. Show that the polynomial P(x) = x 4 3x 3 + 8x 6x + 8 can be expressed as P(x) = (x +) Q(x). Determine Q(x). 3. Determine the range of values of k for which the roots of the equation are complex. a) 7x x + k = 0 b) kx + x + = 0 c) 3kx 6kx + 3k 1 = 0 z z, z z,and z z 4. Evaluate. a) z = i b) z 3 i c) 1 i z 5. Determine the value of k that satisfies the given condition. a) One root of x + kx + = 0 is 1 + i. b) One root of x + kx + 3 = 0 is 1 i. 6. Find a quadratic equation with real coefficients that has the given complex number as a root. a) i b) 1 i 7. Use the given condition to determine the zeros of the polynomial. a) x 3 + 9x + 6x + 4 = 0; P( ) = 0 b) x 3 3x 5x + 75 = 0; P(5) = 0 8. Find the cubic polynomial of real coefficients P for which P() = P(1 i) = 0 given that P(0) = Given 1 i is a zero of the polynomial x 3 + ax + b, a and b are real numbers. Find a and b and the other zeros of the polynomial 30. The height of a projectile launched from an elevated hill is given in meters by the equation h(t) = t 3 + 3t + 10t + 40, where t is the time in seconds after its launch. a) What is the height of the hill of the launch? b) On its way down, when was the projectile back to its original height (height of the launch)? c) When was the projectile at a height of 70 meters? d) For how long was the projectile in the air before hitting the ground?

7 EOY Revision Packet Mathematics, IM3 Page 7 of Find the equation of a quartic polynomial P of real coefficients that touches the x-axis at 1 and 3, and that has a y-intercept of 7. Use a calculator to verify that the graph of the polynomial satisfies the given conditions. 3. Find the zeros of the quartic polynomial and then sketch its graph. Whenever applicable, estimate the local maxima and the local minima, and verify the answers using a calculator. a) P(x) = x 4 9x b) P(x) = x 4 x 3 x + x 33. The water reserve in a lake during the last year is given in millions of liters by the equation V(t) = t t 338t , where t is the number of months from the beginning of the year. a) What was the volume of the water reserve at the beginning of the year? b) What was the volume of the water reserve at the end of the year? c) Sketch the graph of the function V. d) Estimate the minimum reserve reached during the year. e) The state of drought is declared in the region when the volume of the water reserve drops below 00 million liters. Was the state of drought declared last year? If so, during which months was it declared? f) Based on the water reserve at the beginning of this year and assuming the pattern of water consumption is the same as last year, derive an equation for the expected water reserve this year. 34. State whether the expression is given in its simplest form. If not, simplify the expression and state the values excluded from the answer. a) ( x3)( x 4) ( x )( x 4x 3) b) 3 x 7x 1x 3 x x 15x 35. Perform the indicated operation and express your answer in simplest form. 3x x x x 3 a) x4 x 4 b) x 4 x 4 c) x x 3 x x Perform the division and simplify. a) x 3x x 9 x 3x x 1 b) x x x x 5x x 3x c) y3 4y6 y y 8 y 5y Simplify x the complex rational expression. x 5 4x a) x 5 b) 1 1 a b a b

8 EOY Revision Packet Mathematics, IM3 Page 8 of Solve for x. a) 4x 0 x b) x 3 0 x 4x The area of a rectangle is 64 cm. If the length of the rectangle is x cm, derive a simple rational expression in terms of x to represent its perimeter. 40. A square piece of cardboard of side a feet is used to make a rectangular box open at the top. After removing a small square of side 1 a from each corner, the board is folded along the dotted lines shown in the adjacent figure. a) Find the area of the base of the box in terms of a. b) Find the total surface area of the box in terms of a. c) Derive an expression for the ratio of the area of the base to the total surface area and express it in simplest form. d) Calculate the ratio in c) for each of the cases where the cardboard has a side of length feet, 3 feet, and 4 feet. Describe the behavior of this ratio as the length of the side increases. 41. The graph of the function ax b f( x) cx d asymptote at x =. Find the values of a, b, and c. has an x-intercept of 8, a y-intercept of, and a vertical 4. A company charges its customers a total cost of x + 5 dollars for printing x DVDs. a) Express the average cost C per DVD as a function in x. b) Find the equation of the horizontal asymptote of the graph of C. c) What is the interpretation of this asymptote within the context? d) What is the average cost per disk when printing 8 DVDs? e) When does the average cost per DVD drop to $.50? 43. Solve the equation and verify the answer. 1 x a) x 5 x 5 b) x x1 x x 44. Find the values of A and B that satisfy the equation A B x 5 x x1 x x. 45. Solve the given inequality and use the graphing utility to verify your answer. a) x 0 x 3 b) x 4 x 1 a a

9 EOY Revision Packet Mathematics, IM3 Page 9 of The sum of a number and six times its reciprocal is 5. Find the possible values of this number. 47. A company introduces a new toy for the holiday season. They expect sales to reach a maximum before starting to decrease gradually. The sales s expected in a certain city, in thousands of units sold, are modeled by the equation s(t) = introduced to the market. 00t t 100 a) Use a graphing utility to graph the function. b) Describe the behavior of the graph., where t is the number of days since the toy is c) When are the sales expected to reach its maximum and what is the maximum number of toys to be sold on that day? d) During which period of time are the sales expected to exceed 6,000 toys? e) When would the sales drop below,000 toys? f) What happens to the number of toys expected to be sold as t gets very large in value? 48. The two sides of the equation have to be squared twice to remove the radicals. Solve the equation and check for extreneous solutions. a) x1 x 14 b) a 9 a A farmer uses a reservoir of recycled water to irrigate his garden. The rate of water consumption h( t) 6 t 3 during an irrigation cycle is modeled by the equation, where h is the height in feet of the water in the reservoir, and t is the time in hours since the begining of the cycle. a) What is the height of the water in the tank at the beginning of the irrigation cycle? b) How long would it take to empty the reservoir? c) If the farmer stops the irrigation when the height of the water in the reservoir drops below 50 feet, what would be the length of the irrigation cycle? d) Use the graphing utility to sketch the graph of h and verify your answers graphically. e) What are the practical domain and the practical range of the function h? 50. Find the inverse relation E that defines x = E(y). a) R( x) 1 x b) 3 x Rx ( ) 3x Find the point(s) of intersection of the graphs of f and g by solving the equation f(x) = g(x). Verify your answer. a) f(x) = x +, g(x) = x + 3x b) f(x) = 3 x + 1, g(x) = 1 7

10 EOY Revision Packet Mathematics, IM3 Page 10 of Show that the equation has a root over the given interval. Use two steps of the linear interpolation to find an approximate value for that root. a) 3 x x x 5 0; 3 b) x x 0; 1 x Show that the equation has a root over the given interval. Use two steps of the bisection method to find an approximate value for that root. 3 x x x 3 0; 1 3 x x x 1 0; 1 a) b) 54. The cost of buying a laptop that has a cash price of $1,00 is to be paid in weekly installments at an annual interest rate of 10% compounded weekly. a) How long would it take to pay off the cost of the laptop and the interest at equal weekly payments of $135 each? b) Calculate the finance charges paid. 55. a) The doubling time of a bacteria is 15 minutes. How long would it take this bacteria to triple in count? b) Find the doubling time of a bacteria that triples in count every 1.5 hours. 56. Use the graph of y = log x to deduce the graph of y = log (x 1) +. Identify the vertical asymptote of the graph and state the domain and the range of the function. 57. Use the equality property to solve the equation and check for extraneous solutions. a) log 3 (m 0) = log m b) ln ( 6x) = log (x + 9) 58. Solve the inequality and verify the answer graphically using the calculator by graphing the expressions on each side of the inequality. a) ln x > ln (6 x) b) log ( x) log (x + 6) 59. Simplify the expression and express the answer as a single logarithm. a) log 4 (x 4) log 4 (x + ) b) log (z 3 z ) log z z 60. The count of bacteria in a lab grown culture is modeled t hours after its preparation by the equation N = t. a) What is the initial count of the bacteria? b) How long does it take the culture to double in count? c) When does the count N exceed,456? 61. A principal P is invested in an account at an annual interest rate compounded continuously. What is the corresponding annual interest rate if the investment doubles in 7 years?

11 EOY Revision Packet Mathematics, IM3 Page 11 of The current share price of a certain commodity is $7 and is expected to increase at a weekly rate of 3%. The share price of a second commodity is $9 and is expected to drop by % each week. After how many weeks will the two commodities have the same share price? Calculate this price. 63. How many times more powerful is a sound of 3 decibels than a sound of 30 decibels? 64. Orange juice has a concentration of hydrogen ions that is 10,000 times the concentration in pure water. What is the ph level of orange juie? 65. The average life expectancy at birth increased noticeably among Americans during the 70 s. It is still on the rise, but at a slower pace. The equation E = ln (t + 1.7) is used to estimate the average life expectancy at birth t years after a) What was the average life expectancy in 1970? b) What was the increase in the average life expectancy at birth from 1971 to 1980? c) What is the expected increase in the average life expectancy at birth from 011 to 00? d) How are the results in a) and b) justified by the properties of the graph of the logarithm? e) When did the average life expectancy at birth reach 79.3 years? 66. The following scatterplot shows the price of a company share at the beginning of each month of a given year. p 60 Share Price ($) Jan Apr Jul Oct Jan Month a) What is the degree of the polynomial suggested to fit the data plotted? Justify your choice. b) Use a calculator to find the equation defining a cubic polynomial function that models the graph. c) Verify the validity of the model developed by comparing the calculated share price to the one deduced from the graph for different years within the given range. d) Use the graph of the model to estimate the maximum share price over the given data range. e) Use the graph of the model to estimate the minimum share price over the given data range. f) Assuming the validity of the model in the following year, what is the expected share price at the beginning of February of the following year?

12 EOY Revision Packet Mathematics, IM3 Page 1 of The yearly depreciation, D, of a washer depends on its assumed lifetime n. The following table lists the values of D for different values of n. n (years) D ($) a) Produce a scatterplot for the data presented in the table. b) What model is suggested to represent the given data? c) Use a calculator to define a function that models the relation between n and D. d) What does the constant in the equation defining the function represent? 68. The rate of ventilation of a closed space is the time required to exchange the air in a space. The rate of ventilation of a hall depends on the number of people in this hall. The following table gives this rate for different numbers of people. Number of persons in the room n Rate of ventilation P (hours) a) Draw a scatterplot of the data in the table. b) Comment on the rate of change of n with respect to P as n increases. c) Suggest a function to model the data in the table. d) The function given by n = a + b ln P is suggested to model the relationship between n and P. Estimate the value of a and b. e) What is the practical domain of the function? f) What is the practical interpretation of the domain within the context?

13 EOY Revision Packet Mathematics, IM3 Page 13 of Due to the implemented economic reforms, the financial deficit of a country is declining by the year. The following table shows the deficit, in billions of dollars, during the period from Year Deficit (billions of dollars) a) Draw a scatterplot of the data. b) Derive an exponential model to represent the decrease of the deficit along the years. c) Verify the model by comparing its graph to the scatterplot of the data. d) What is the average yearly rate of the decrease of the deficit? e) When will the deficit reach half its value in 01? 70. A resonant circuit consists of an ideal conductor connected in series with an ideal capacitor. The resonance frequency of such a circuit is the frequency at which their impedance tends to zero. In a lab test, a capacitor of capacitance C is connected in series with an inductor of inductance L. The resonance frequency is a function of the product LC. The following results were obtained to estimate the resonance frequency f at different products LC. The estimates also depended on the leakage resistances in the circuit. LC f (hertz) a) Draw a scatterplot of the data. b) What model do you suggest to represent the data? c) Use a calculator to generate a power function that models the data. d) By using a proper scale, compare the graph of the model and the scatterplot of the data. e) What is the resonant frequency when the value of the product LC is ? 71. Express the given radian measure of the angle in degrees. 3 7 a) 4 b) 6

14 EOY Revision Packet Mathematics, IM3 Page 14 of To three decimal places, sin 5 = 0.588, cos 5 = 0.809, and cot 5 = Use this information to evaluate the trigonometric ratios given in 7 1. Verify your answers using a calculator. 11π a) sin 10 b) cos 10 c) tan Find the exact values of x between 0 and π, in radians, that satisfy the given equation. a) sin x = 3 b) cos x = 1 c) tan x = 74. Use the graph of y = sin x to deduce the graph of the function defined and determine the amplitude, the period, and the phase shift. Also state the range of the function. a) y = sin x b) y = 1 + sin x c) y = sin π x Use the graph of y = cos x to deduce the graph of the function defined and determine the amplitude, the period, and the phase shift. Also state the range of the function. a) y = 1 cos π 3x 3 b) y = 3 cos x π A consulting firm concluded that the temporary employment in a certain county exhibits a cyclic behavior throughout the years. The firm suggested a model to simulate the demand on temporary hiring given by y = 1.8sin(0.8t + 3.5) + 3., where y is the number of temporary employees hired in thousands and t is the time in years since the beginning of 00. a) Find the period, the amplitude, the phase shift, and the vertical offset. b) If the model remains valid until 00, during which years of the period extending between 00 and 00 is the demand for temporary hiring maximum and during which years is it minimum? Calculate the extreme values for the demand. c) What is the expected demand in 015? d) Sketch the graph of the sinusoidal function over the period between 00 and 00 and describe its features.