Math Solved Questions

Z00001. Find the variation constant and an equation of variation where y varies inversely as x and y=0.9 when x=0.3.

Z00005. The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 8.197 kg and decreases continuously at a relative rate of 16% per day. Find the mass of the sample after four days. Write the exponential function for this situation. Do not round any intermediate computations, and round your answer to the nearest tenth.

Z00025. Find the equation for the line that passes through the point (-2,3), and that is perpendicular to the line with the equation x = -4.

Z00027. Write the equation of the hyperbola 4x² – 16y² – 32x – 32y – 16 = 0 in standard form.

Z00028. Consider a rabbit population P(t) satisfying the logistic equation dP/dt = kP(600-P). Assume the initial population is 100 rabbits and is then growing at the rate of 10 rabbits per day. Predict the rabbit’s population on day 25.

Z00029. Determine the interval(s) on which the function is (strictly) decreasing. Write your answer an an interval or list of intervals. When writing as a list of intervals, make sure to separate each interval with a comma and to use as few intervals as possible.

Z00033. A company is considering two insurance plans with coverage and premiums shown in the table. (For example, this means that $50 buys one unit of plan A, consisting of $10,000 fire and theft insurance and $180,000 of liability insurance.) The company needs at least $450,000 fire and theft insurance and at least $5,400,000 liability from these plans. How many units should be purchased from each plan to minimize the cost of the premiums? What is the minimum premium?

Z00034. Write a system of equations in two variables to solve the problem. In 2009, two tickets to a certain American band’s concert and two tickets to a certain British band’s concert cost, on average a total of $604. At those prices, four tickets to see this American band and two tickets to see this British band cost $866. What was the average cost of American band’s ticket and average cost of British band’s ticket?

Z00035. For the following exercises, draw an angle in standard position with the given measure. (1) – pi/10 (2) 22 pi/3

Z00036. A house sells for $139,000 and a 45% down payment is made. A 30-year mortgage at 8% was obtained. (i) Find the down payment. (ii) Find the amount of the mortgage. (iii) Find the monthly payment. (iv) Find the total interest paid.

Z00037. Use transformations to determine which graph below represents the equation y = – (x – 1)³ + 4 transformed from y = x³ . The original function is graphed in green and the transformed function is graphed in blue.

Z00038. You invested $18,000 in two accounts paying 5% and 6% annual interest, respectively. If the total interest earned for the year $920, how much was invested at each rate?

Z00039. For the given complex numbers a, b and c, find the following (a) Conjugate of complex number ‘a’ (b) Absolute value of complex number ‘a’ (c) Plot ‘a’ and ‘b’ on the Complex plane (d) Convert complex number ‘a’ to the exponential polar form.

Z00040. Factor the following expressions.

Z00041. In parts (a) and (b), use the given figure. (a) Solve the equation f(x) = g(x) (b) Solve the inequality g(x) ≤ f(x) < h(x).

Z00042. An automated car wash serves customers with the following serial process : pretreat, wash, rinse, wax, hand dry. Each of these steps is performed by a dedicated machine except for the hand-dry step, which is performed manually on each car by one of three workers. The steps of the process have the following processing times : (a) Which resource is the bottleneck of the process ? (b) If the car wash has a demand of 12 cars per hour, what is the flow rate of the process ? (c) If the car wash has a demand of 12 cars per hour, what is the utilization of the machine that performs the wax process ? (d) If the car wash has a demand of 8 cars per hour, what is the utilization of hand dry ?

Z00043. Out of 270 racers who started the marathon, 238 completed the race, 27 gave up, and 5 were disqualified. What percentage did not complete the marathon ?

Z00044. A company markets exercise DVDs that sell for $29.95, including shipping and handling. The monthly fixed costs (advertising, rent, etc.) are $38,130 and the variable costs (materials, shipping, etc.) are $9.45 per DVD. (A) Find the cost equation and the revenue equation. (B) How many DVDs must be sold each month for the company to break even ? (C) Graph the cost and revenue equations in the same coordinate system and show the break-even point. Interpret the regions between the lines to the left and to the right of the break-even point.

Z00045. A production department has 35 similar milling machines. The number of breakdowns on each machine averages 0.06 per week. Determine the probabilities of having (a) one, and (b) less than three machines breaking down in any week.

Z00046. Describe how the graph of the function g(x) = – 8(x)^0.5 can be obtained from the basic graph. Then graph the function.

Z00047. Give a geometric description of the following system of equations. (1) -5x – 5y = 1 ; 4x + y = 7; -21x – 9y = -27 (2) -5x – 5y = 1 ; 4x + y = 7; -21x – 9y = -29 (3) 12x – 12y = -4 ; 6x – 6y = -2; -18x + 18y = 6

Z00048. Determine the coordinates of the x and y-intercepts for each of the following linear functions. Use the intercepts, and another point if needed to sketch the graph of each. (a) f(t) = 4/5t – 3 (b) 3x + 4y = -2 (c) -x/3 + y = 0

Z00049. Solve the following systems of linear equations using Gaussian Elimination and back substitution. Check your answers by substituting them into the original system. (1.1.1) 6x₁ – 3x₂ – 6x₃ = – 63 ; 2x₁ – 3x₂ + 5x₃ = 17 ; – x₁ – 2x₂ – x₃ = 15 (1.1.2) – 5x₁ – 4x₂ – 4x₃ = 33 ; 3x₁ + x₂ = – 25 ; – 4x₂ + x₃ = – 7

Z00050. Do the following for the function g(x) = 10x² + 5x (a) Express the slope of the secant line in terms of x and h. (b) Find m_sec for h = 0.5, 0.1 and 0.01 at x = 1. What value does m_sec approach as h approaches 0? (c) Find the equation for the secant line at x = 1 with h = 0.01. (d) Graph g and the secant line found in part (c) on the same viewing window.

Z00051. Solve for x: |(1 – 2x)| = 3

Z00052. Determine the value of h such that the matrix is the augmented matrix of a linear system with infinitely many solutions.

Z00053. My wife has 3 email accounts (Yahoo, gm–ail, and Work email). Her yahoo account is her junk email account. Currently, she has 15,000 unread emails in her yahoo account, and she receives an additional 30 emails each day. Her gm–ail account has 5 unread emails, and receives 40 new emails a day. Her work email account has 10 unread emails, and it receives 60 new emails a day. If no emails are read, write equations that describe the number of unread emails in each account at day x.

Z00054. Michael and Jacqueline work at a computer store. Michel sold five less than twice the number of computers that Jacqueline sold. If j represents the number of computers Jacqueline sold, write an expression for the number of computers Michael sold. Note : An expression does not have an equal sign.

Z00055. Find the principal amount, the broker-assisted commission, and the automated commission for 60 shares at $1 per share. Use the tables for the typical discount commission structure.

Z00056. Find one non-trivial solution of Ax = 0 by inspection. [Hint : Think of the equation Ax = 0 written as a vector equation.]

Z00057. The pressure of a gas varies jointly as the amount of the gas (measured in moles) and temperature and inversely as the volume of the gas. If the pressure is 1,215 kiloPascals (kPa) when the number of moles is 8, the temperature is 270^o Kelvin and the volume is 960 cc, find the pressure when the number of moles is 5, the temperature is 290^o Kelvin and the volume is 600 cc.

Z00058. Verify the following for x = 45^o : sin² (x) – cos² (x) – tan² (x) =

Z00059. The number of  hours of daylight, H, on day t of any given year (on January 1, t = 1) in a particular city can be modeled by the below given function. Find the number of hours of daylight in the city on a particular given day.

Z00060. Use the given statements of similar triangles to solve for unknown variable x.

Z00061. (1) Give an example of a system of equations with two variables that has infinitely many solutions. (2) Change one number in your system of equations from part 1 to make it a system that has no solutions. (3) Solve the two systems to verify they have the correct outcomes.

Z00062. Evaluate the discriminant and determine the number and type of solutions to the equation. 6x² – 4x + 6 = 0

Z00063. Calculate the female-to-male earnings ratio for each of the given years, and enter the ratios in the table. Round the ratios to three decimal places.

Z00064. Suppose an isosceles triangle has base angles that each measure twice the number of degrees as the third angle. Determine the measure of each angle.

Z00065. (1) Explain how the graph of g is obtained from the graph of f. (a) f(x)=x^0.5, g(x) = -x^0.5+5 (b) f(x)=x^0.5, g(x) = (-x)^0.5+5 (2) Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. (a) y = -f(x) + 9 (b) 8 f(x) – 9

Z00066. What is incorrect about solving x² = 3x by dividing by x on both sides? Explain.

Z00067. The ratio of Jake’s age and Alan’s age is 10:9. The ratio of Erin’s age and Alan’s age is 29:27. Alan is over 40 years old but under 70 years old. What is Alan’s age?

Z00068. Find an angle θ with 0^o < θ < 360^o that has the same (a) Sine function value as 220^o (b) Cosine function value as 220^o .

Z00069. In R³ with the usual dot product. (a) Find the angle between the vectors (1, 0, -2) and (1, 1, 1) (b) Find the distance from (1, 0, -2) to (1, 1, 1).

Z00070. A population numbers 17,000 organisms initially and grows by 6.9% each year. Suppose P represents population, and t the number of years of growth. Write an exponential model for the population in the form P = a ⋅ b^t.

Z00071. Evaluate the following. (a) sin (5π/4) (b) cos (2π/3) (c) tan (-3π/4) (d) sec (-5π/4).

Z00072. Find the values of x, y and z as shown in the given geometrical figure.

Z00073. Convert the angle 240^o to radians. Give the exact value and use pi for π.

Z00074. For the given function, find the number b so that the average rate of change of f is 3 on the interval [b, 2]. f(x) = x² + 3x.

Z00075. Given the functions : f(x) = x³ – 5x ; g(x) = (5x)^0.5 ; h(x) = 2x+5. Evaluate the function (g o f) (x) for x = -2. Write your answer in exact simplified form. Select “Undefined” if applicable.

Z00076. If log₂P = x, log₂Q = y and log₂R = z, write in terms of x, y and z: (a) log₂(PR) (b) log₂(RQ²) (c) log₂(PR/Q) (d) log₂(P²Q^0.5)

Z00077. BoxedNGone truck rentals calculates that its price function is p(x) = 180 – 3x, where p is the price (in dollars) at which exactly x trucks will be rented per day. Find the number of trucks that BoxedNGone should rent and the price it should charge to maximize revenue. Also find the maximum revenue.

Z00078. For each pair of functions f and g below, find f(g(x)) and g(f(x)). Then, determine whether f and g are inverses of each other. (a) f(x) = 1/(2x), x ≠ 0 ; g(x) = 1/(2x), x ≠ 0 (b) (a) f(x) = x – 6 ; g(x) = x – 6.

Z00079. Verify each of the following trigonometric identity. (1) csc x / cot x = tan x / sin x (2) 1 / tan x cot² x = cot x tan² x (3) cos² x / tan x = cot x / sec² x

Z00080. Find the area of the triangle ABC. a = 12.2 m b=5.6 m C = 15.7^o.

Z00081. A formula for the length of a diagonal from the upper corner of a box to the opposite lower corner is D = (L² + B² + H²)^0.5 where, L, W and H are the length, width and height respectively. Find the length of the diagonal of the box if the length is 23 inches, width is 15 inches and height is 7 inches. Leave your answer in simplified radical form.

Z00082. Find the area of the shaded region enclosed in a semicircle of diameter 7 centimeters. The length of chord AB is 4 centimeters.

Z00083. A die is rolled 50 times with the following results. Compute the empirical probability that the die comes up a 5.

Z00084. Describe how the graph of g(x)=1/(x+5) – 2 can be obtained from the graph of f(x)=1/x ?

Z00085. Find the value of sin θ for given value of tan θ if cos θ > 0.

Z00086. Find sin(x/2), cos(x/2) and tan(x/2) from the given information. tan(x)=√2, 0^o<x<90^o.

Z00087. Solve triangle ABC if a = 8, b = 10 and C = 5. Round angle measures to the nearest degree.

Z00088. After simplification, how many terms will be there in 4x³ + 9y² – 3x + 2 -1?

Z00089. f(x) = 1.9 (2.29)^x . (i) The value of the function when x = 0 is ……………… (ii) Find the ratio of output values that correspond to increases of 1 in the input value in order to determine the 1-unit growth factor. (iii) Determine the 1-unit percent change.

Z00090. A student borrows $56,500 at 7.2% compounded monthly. Find the monthly payment and total interest paid over a 25 year payment plan.

Z00091. Three partners are dividing a plot of land among themselves using the lone-divider method. After the divider D divides the land into three shares S₁, S₂ and S₃, the choosers C₁ and C₂ submit their bids for these shares. (a) Suppose that the choosers’ bids are C₁ : {S₂, S₃}; C₂ : {S₁}. Describe two different fair divisions of the land. (b) Suppose that the choosers’ bids are C₁ : {S₁, S₂}; C₂ : {S₂, S₃}. Describe three different fair divisions of the land.

For Concepts of Various Topics, Please Refer Math Notes.