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mathematics
linear algebra
Questions and Answers of
Linear Algebra
Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. a. y ≤ 4 - x2, x - y ≤ 2 b. x2 + y2 ≤ 16, x + y < 4
The vertex of the parabola y2 - 4y - 12x - 8 = 0 is which of the following? A. (1, -2) C. (2, -1) B. (-1, 2) D. (-2, 1)
The graph of x2 + 4y2 = 4 is which of the following?
Find two numbers whose product is 4 and the sum of whose reciprocals is 65/56.
Find an equation of the circle that passes through the points (10, 7), (-6, 7), and (-8, 1).
Find an equation of ellipse containing the point (-1/2, 3√3/2) and with vertices (0, -3) and (0, 3).
Two radio transmitters positioned 400 mi apart along the shore send simultaneous signals to a ship that is 250 mi offshore, sailing parallel to the shoreline. The signal from transmitter A reaches
Is a circle a special type of ellipse? Why or why not?
In Exercises 6-13, match the equation with one of the graphs (a)-(h), which follow.1. y2 = 5x 2. y2 = 9 - x2 3. 3x2 + 4y2 = 12 4. 9y2 - 4x2 = 36 5. x2 + y2 + 2x - 3y = 8
In Exercises, match the equation with one of the graphs (a)-(d), which follow.1. 4x2 - y2 = 4 2. x2 - 2x - 3y = 5 3. x2 + 4x + y2 - 2y - 4 = 0 4. 9x2 + 4y2 = 36
Find the center, the vertices, the foci, and the asymptotes of the hyperbola. Then draw the graph. a. 4x2 - y2 = 4 b. (y - 2)2/4 - (x + 1)2/9 = 1
Find the asymptotes of the hyperbola given by 2y2 - x2 = 18.
A satellite dish has a parabolic cross section that is 18 in. wide at the opening and 6 in. deep at the vertex. How far from the vertex is the focus?
Leisurescape is planting a rectangular flower garden with a perimeter of 18 ft and a diagonal of √41 ft. Find the dimensions of the garden.
It will take 210 ft of fencing to enclose a rectangular playground with an area of 2700 ft2. Find the dimensions of the playground.
Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. y ≥ x2 - 4, y < 2x - 1
Find an equation of the circle for which the endpoints of a diameter are (1, 1) and (5, -3).
Find the vertex, the focus, and the directrix of the parabola. Then draw the graph. a. x2 = 12y b. y2 + 2y - 8x - 7 = 0
Find the center and the radius of the circle given by x2+y2+2x - 6y - 15 = 0. Then draw the graph.
Find the center, the vertices, and the foci of the ellipse. Then draw the graph. a. 9x2 + 16y2 = 144 b. (x + 1)2/4 + (y - 2)2/9 = 1
In each of the following, the nth term of a sequence is given. Find the first 4 terms, a10, and a15.(a) an = 4n - 1(b) an = (n - 1) (n - 2) (n - 3)
Find the indicated term of the given sequence. (a) an = 5n - 6; a8 (b) an = (3n - 4) (2n + 5); a7 (c) an = (2n + 3)2; a6
Use a graphing calculator to construct a table of values and a graph for the first 10 terms of the sequence.
Predict the general term, or nth term, an, of the sequence. Answers may vary. a. 2, 4, 6, 8, 10, c b. 3, 9, 27, 81, 243, c c. -2, 6, -18, 54, c
Find the indicated partial sums for the sequence. a. 1, 2, 3, 4, 5, 6, 7, c; S3 and S7 b. 1, -3, 5, - 7, 9, - 11, c; S2 and S5 c. 2, 4, 6, 8, c; S4 and S5
Find and evaluate the sum.a.b. c.
Write sigma notation. Answers may vary. a. 5 + 10 + 15 + 20 + 25 + g b. 7 + 14 + 21 + 28 + 35 + g c.2 - 4 + 8 - 16 + 32 - 64
Find the first 4 terms of the recursively defined sequence. a. a1 = 4, an + 1 = 1 + 1 / an b. a1 = 256, an + 1 = √an c. a1 = 6561, an+1 = (-1)n √an
Suppose that $1000 is invested at 6.2%, compounded annually. The value of the investment after n years is given by the sequence model an = $100011.0622n, n = 1, 2, 3, c. a) Find the first 10 terms
The value of a post-hole digger is $5200. Its salvage value each year is 75% of its value the year before. Give a sequence that lists the salvage value of the post-hole digger for each year of a
Adahy is paid $9.80 per hour for working at Red Freight Limited. Each year he receives a $1.10 hourly raise. Give a sequence that lists Adahy's hourly wage over a 10-year period.
Suppose that a single cell of bacteria divides into two every 15 min. Suppose that the same rate of division is maintained for 4 hr. Give a sequence that lists the number of cells after successive
One of the most famous recursively defined sequences is the Fibonacci sequence. In 1202, the Italian mathematician Leonardo da Pisa, also called Fibonacci, proposed the following model for rabbit
The table below lists the average weekly earnings of U.S. production workers in recent years.Sources: U.S. Bureau of Labor Statistics, U.S. Department of Labor a) Use a graphing calculator to fit a
The table below lists world motor vehicle production in recent years.Sources: Automotive New Data Center; R. L. Polk a) Use a graphing calculator to fit a quadratic sequence regression function an =
In 2010, the United States, Japan, and China had a total of 89,348 patent applications. The number of applications in China was 3741 less than one-half the number of applications in Japan. The number
Find the center and the radius of the circle with the given equation. (a) x2 + y2 - 6x + 4y = 3 (b) x2 + y2 + 5x - 8y = 2
Find the first 5 terms of the sequence, and then find S5. a. an = 1 / 2n log 1000n b. an = in, i = √ -1 c. an = ln (1∙2 ∙ 3∙ ∙ ∙ ∙ ∙n)
For the sequence whose nth term is an = (1-)n (2n + 1), find a21.
Find the sum of the first 20 terms of the series 17 + 13 + 9 + g .
Find the 11th term of the geometric sequence 10, -5, 5 / 2 , -5 / 4 ,.......... .
For a geometric sequence, r = 0.2 and S4 = 1248. Find a1.r = 0.2, S4 = 1248
The value of an office machine is $10,000. Its salvage value each year is 80% of its value the year before. Give a sequence that lists the salvage value of the machine for each year of a 6-year
Tamika accepts a job, starting with an hourly wage of $8.50, and is promised a raise of 25¢ per hour every three months for 4 years. What will Tamika's hourly wage be at the end of the 4-year period?
To create a college fund, a parent makes a sequence of 18 equal yearly deposits of $2500 in a savings account on which interest is compounded annually at 5.6%. Find the amount of the annuity.
Find the first 5 terms of the sequence with general term an = n + 1 / n + 2 .
Use mathematical induction to prove that, for every natural number n,
How many 4-digit numbers can be formed using the digits 1, 3, 5, 6, 7, and 9 without repetition?
How many code symbols can be formed using 4 of the 6 letters A, B, C, X, Y, Z if the letters: a) Can be repeated? b) Are not repeated and must begin with Z?
The Bay Woods Scuba Club has 28 members. How many sets of 4 officers can be selected from this group?
On a test with 20 questions, a student must answer 8 of the first 12 questions and 4 of the last 8. In how many ways can this be done?
Find the 5th term of the binomial expansion (x - y)7.
Suppose we select, without looking, one marble from a bag containing 6 red marbles and 8 blue marbles. What is the probability of selecting a blue marble?
Ethan has 6 pennies, 5 dimes, and 4 quarters in his pocket. Six coins are drawn at random. What is the probability of getting 1 penny, 2 dimes, and 3 quarters?
The graph of the sequence whose general term is an = 2n - 2 is which of the following?
Use a graphing calculator to construct a table of values and a graph for the first 10 terms of the sequence with general term an = n + 1 / n + 2 .
Find the first 4 terms of the recursively defined sequence a1 = 3, an + 1 = 2 + 1 van .
Find the 15th term of the arithmetic sequence 2, 5, 8, c.
The 1st term of an arithmetic sequence is 8 and the 21st term is 108. Find the 7th term.
Find the first term and the common difference. a. 3, 8, 13, 18, c b. $1.08, $1.16, $1.24, $1.32, c c. 9, 5, 1, -3, c
Find the sum.
How many poles will be in a stack of telephone poles if there are 50 in the first layer, 49 in the second, and so on, with 6 in the top layer?
Max, an investment counselor, sets up an investment situation for a client that will return $5000 the first year, $6125 the second year, $7250 the third year, and so on, for 25 years. How much is
Theaters are often built with more seats per row as the rows move toward the back. Suppose that the first balcony of a theater has 28 seats in the first row, 32 in the second, 36 in the third, and so
If 10¢ is saved on October 1, 20¢ is saved on October 2, 30¢ on October 3, and so on, how much is saved during the 31 days of October?
When a parachutist jumps from an airplane, the distances, in feet, that the parachutist falls in each successive second before pulling the ripcord to release the parachute are as follows: 16, 48,
A formation of a marching band has 10 marchers in the first row, 12 in the second row, 14 in the third row, and so on, for 8 rows. How many marchers are in the last row? How many marchers are there
A gardener is making a planting in the shape of a trapezoid. It will have 35 plants in the first row, 31 in the second row, 27 in the third row, and so on. If the pattern is consistent, how many
Find the vertices and the foci of the ellipse with equation 9x2 + 16y2 = 144.
Find an equation of the ellipse with vertices 10, -52 and 10, 52 and minor axis of length 4.
Straight-Line Depreciation. A company buys an office machine for $5200 on January 1 of a given year. The machine is expected to last for 8 years, at the end of which time its trade-in value, or
Find three numbers in an arithmetic sequence such that the sum of the first and third is 10 and the product of the first and second is 15.
Find the first term and the common difference for the arithmetic sequence for which a2 = 40 - 3q and a4 = 10p + q. If p, m, and q form an arithmetic sequence, it can be shown that m = 1 p + q2>2. The
Insert three arithmetic means between -3 and 5.
Insert four arithmetic means between 4 and 13.
Find the common ratio. a. 2, 4, 8, 16,......... b. 18, -6, 2, - 2/3,....... c. -1, 1, -1, 1,........
Find the indicated term of the geometric sequence. a. 2, 4, 8, 16, c; the 7th term b. 2, -10, 50, -250, c; the 9th term c. 2, 223, 6, c; the 9th term
Find the nth, or general, term. a. 1, 3, 9,...... b. 25, 5, 1,....... c. 1, -1, 1, -1, .....
Determine whether the statement is true or false. a. The sequence 2, -2 √2, 4, -4√2, 8,....... is geometric. b. The sequence with general term 3n is geometric. c. The sequence with general term
Find the sum, if it exists. a. 4 + 2 + 1 + ......... b. 7 + 3 + 9 / 7 + ........ c. 25 + 20 + 16 + ........
Find fraction notation. a. 0.131313......, or 0. b. 0.2222........., or 0. c. 53. 8.999
Suppose someone offers you a job for the month of February (28 days) under the following conditions. You will be paid $0.01 the 1st day, $0.02 the 2nd, $0.04 the 3rd, and so on, doubling your
A ping-pong ball is dropped from a height of 16 ft and always rebounds 14 of the distance fallen. a) How high does it rebound the 6th time? b) Find the total sum of the rebound heights of the ball.
A bungee jumper always rebounds 60% of the distance fallen. A bungee jump is made using a cord that stretches to 200 ft.a) After jumping and then rebounding 9 times, how far has a bungee jumper
Had leytown has a present population of 100,000, and the population is increasing by 3% each year. a) What will the population be in 15 years? b) How long will it take for the population to double?
To save for the down payment on a house, the Clines make a sequence of 10 yearly deposits of $3200 each in a savings account on which interest is compounded annuallyat 4.6%. Find the amount of the
To create a college fund, a parent makes a sequence of 18 yearly deposits of $1000 each in a savings account on which interest is compounded annually at 3.2%. Find the amount of the annuity.
A piece of paper is 0.01 in. thick. It is cut and stacked repeatedly in such a way that its thickness is doubled each time for 20 times. How thick is the result?
A sequence of yearly payments of P dollars is invested at the end of each of N years at interest rate i, compounded annually. The total amount in the account, or the amount of the annuity, is V.a)
A sequence of payments of $300 is invested over 12 years at the end of each quarter at 5.1%, compounded quarterly. Find the amount of the annuity. Use the formula in Exercise 64(b).
A sequence of yearly payments of $750 is invested at the end of each of 10 years at 4.75%, compounded annually. Find the amount of the annuity. Use the formula in Exercise 64(a).
Great Grains Cereal Company is about to market a new low-carbohydrate cereal in a city of 5,000,000 people. They plan an advertising campaign that they think will induce 30% of the people to buy the
Suppose the government is making a $13,000,000,000 expenditure for educational improvement. If 85% of this is spent again, and so on, what is the total effect on the economy?
For each pair of functions, find (f ͦ g)x2 and 1g ͦ f 21x2. a. f (x) = x2, g(x) = 4x + 5 b. f (x) = x - 1, g(x) = x2 + x + 3
Assume that a1, a2, a3,... is a geometric sequence. Prove that ln a1, ln a2, ln a3, ..... is an arithmetic sequence.
Find the sum of the first n terms of 1 + x + x2 +.....
Find the sum of the first n terms of x2 - x3 + x4 - x5 + ......
The sides of a square are 16 cm long. A second square is inscribed by joining the midpoints of the sides, successively. In the second square, we repeat the process, inscribing a third square. If this
List the first five statements in the sequence that can be obtained from each of the following. Determine whether each of the statements is true or false. a. n2 < n3 b. n2 - n + 41 is prime. Find a
The following formulas can be used to find sums of powers of natural numbers. Use mathematical induction to prove each formula.
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