Algebra Review Constructed Response Qs
25 a) CSET! Know how to solve a system of two equations with two unknowns with inverse matrices.
...
14. A farmer has 70 acres of land available for planting either soybeans or wheat. The costs, workdays, and profit per acre are shown below: soybeans | wheat preparation cost/acre | $60 | $30 | workdays required/acre | 3 | 4 | profit/acre | $180 | $100 | The farmer cannot spend more that $1800 in preparation costs nor more than a total of 120 workdays. How many acres of each crop should be planted in order to maximize profit? What is the profit?
24 acres of soybeans and 12 acres of wheat for a maximum profit of $5520 no video solution
45. CSET! I've heard the CSET asks you to "identify three types of division involving zero." I was a little annoyed that of all the questions they could ask you, they chose this, and they chose to phrase it so vaguely.
My interpretation -- and correct me if you have a better one -- is for any non-zero number a Case 1: 0/a = 0, Case 2: a/0 is undefined Case 3: 0/0 is indeterminate (a calculus concept!). Here's a good excerpt I got from www.singaporemath.com about why case 2 and case 3 are undefined. https://drive.google.com/file/d/0B7UrQE4OBGiLcHdubzBFeFBGaFE/view?usp=sharing
30. Three forces act simultaneously on an object with the following magnitudes and directions: 70 lbs at -30°, 40 lbs at 45°, and 60 lbs at 135°. find the magnitude and direction of the resultant force, R.
No video solution at this time. Please refer to written solutions
15. CSET! Use mathematical induction to prove 1 + 3 + 5 + ....+ (2n - 1) = n^2
No video solution at this time. Please refer to written solutions.
27. CSET! Graph f(x) = (2/3)^-x . State the equation of the inverse and plot it on the same axis. Give the domain, range, and asymptotes for both functions.
No video solution at this time. Please refer to written solutions.
31. Use vectors to find angle A of the triangle with the given vertices A(1, 2), B(3, 4), C(2, 5).
No video solution at this time. Please refer to written solutions.
34. Given vectors a = <1,- 2, 1> and b = <2, 3, 5> find: a) the cosine of the angle between the vectors. b) (you can skip this part -- not on test) all nonzero vectors perpendicular to the vector a + b c) all nonzero vectors perpendicular to both vectors a and b
No video solution at this time. Please refer to written solutions.
37. Prove the Conjugate Roots Theorem. Not a bad one to know, though from what I'm hearing, it's not on the test.
No video solution at this time. Please refer to written solutions.
7. Use the Euclidean algorithm to find the greatest common divisor, d, of the numbers 726 and 275. Then show that d can be expressed as a linear combination in the form d = 726m + 275n, with m and n integers.
No video solution. Refer to written solutions.
63. CSET! Prove n^3 - n is divisible by 3 for any integer n.
No written solution at this time. https://www.youtube.com/watch?v=O-1rQ4J4yrM
49. Be familiar with writing complex numbers in polar form See MC #61 for more targeted problem
Practice Problems: https://drive.google.com/file/d/0B7UrQE4OBGiLQjJ6RS1HZkEwckk/view?usp=sharing Solutions: https://youtu.be/LPw0nbLSUhA?list=PLhcYKxFPtkB4tnIqVo4G3btGpAbNVUVRW
53. CSET! Functions vs. Relations
Relations relate input values with output values, usually x-values with y-values. A relation is a function if every input value maps to exactly one output value. Example: y = x^2 is a function: every x value input gives exactly one predictable y value output. When x =2, the only possible y value is y = 4. When x = 3, I get y = 9 only, etc. When graphed, functions pass the vertical line test (any vertical line will cross a function only once). However, y^2 = x is not a function. There are x's that when inputted, give two potential y-value outputs. For example when x = 4, y could equal 2 or -2. What's with that?! You'll see that when graphed, relations that are not functions do not pass the vertical test. This particular example is a sideways parabola. If we wanted to write the relation y^2 = x as two related functions, we could split it up y = √x and y = -√x. Test the graphs for yourself with Desmos.
54. CSET! If a and b are non-zero complex numbers and, state the conditions for 3 cases: |a+b| = |a| + |b| |a+b| < |a| + |b| |a+b| > |a| + |b|
Rough Solution: Lol, I finally figured this one out! Here's my rough solution. I'll make my own video sometime soon. This is The Triangle Theorem of Inequality for Complex Numbers that extends from our usual theorem from the real numbers. A graphical interpretation of complex numbers helps to visualize. The absolute value of the sum is always less than or equal to the sum of the absolute values. The equal case is only true if the angle between a and b is 0. The greater than case is never true. I'll make my own video sometime soon. In the meantime, these two videos I found on YouTube by teacher Eddie Woo helped me visualize: https://youtu.be/8n3jLEh5uq0 https://youtu.be/NFRUvs6K2FQ
22. Solve the following the system using augmented matrices and Gauss-Jordan elimination. If there are no solutions, state so. If there are multiple solutions, write a general solution and give two specific solutions. x - y + 2z = -3 2x + y - z = 0 - x + 2y - 3z = 7
http://youtu.be/1RqQyDlPHWA
25. Solve the following the system using augmented matrices and Gauss-Jordan elimination. If there are no solutions, state so. If there are multiple solutions, write a general solution and give two specific solutions. x + 2y - z = 5 x - y + 2z = 2 5x -8y + 13z = 7
http://youtu.be/5XA5EriGpFk
24. Solve the following the system using augmented matrices and Gauss-Jordan elimination. If there are no solutions, state so. If there are multiple solutions, write a general solution and give two specific solutions. x - 2y + 2z = -9 2x + y - z = 7 3x - y + z = 5
http://youtu.be/5iSnumX20Xs
28. CSET! Given f(x) = x^3 + x^2 - 6x, sketch each of the following functions. Show the coordinates of the intercepts and the equation of any asymptotes. a) f(x) b) √f(x) c) 1/f(x)
http://youtu.be/7h20mWp1JkM
17. CSET! Use mathematical induction to prove 1*2 + 2*3 + 3*4 ....+ n(n+1) = n(n + 1)(n + 2)/3
http://youtu.be/9EUEMU0O4o0
18. CSET! Prove a number is divisible by 3 if the sum of its digits is divisible by 3.
http://youtu.be/B7nlMUcKOTA
33. CSET! Given vectors a = <5, 1, 4> and b = <-1, 0, 2> find: a) a vector orthogonal to both vectors and b) the cosine of the angle between the vectors.
http://youtu.be/BoD5VSNcjYs
39. CSET! Given the following matrices A, B, and C, find matrix K such that AKB = C. A = [1 2][0 -3] B = [-1 3] C = [3 -9][-6 18]
http://youtu.be/CNHZ6kyqZdk
41. CSET! Given f(x) = 2x^3 - 5x^2 - 8x + 6, list the possible rational roots. Then show that f(x) has an irrational root between x = 2 and x = 4. (This one came straight from a constructed response on the previous exam!)
http://youtu.be/JZGwAAvdeVU
23. Solve the following the system using augmented matrices and Gauss-Jordan elimination. If there are no solutions, state so. If there are multiple solutions, write a general solution and give two specific solutions. x - 2y + 3z = 9 -x + 3y = -4 2x - 5y + 5z = 17
http://youtu.be/LKndMNdd1G8
19. CSET! Prove a number is divisible by 4 if the number formed by the last two digits is also divisible by 4.
http://youtu.be/PRAC_WFw1Yo
42. CSET! Given two perpendicular lines y = m1x + b1 and y = m2x + b2, with m1, m2 ≠ 0, represent the system in matrix form and use the determinant function to evaluate the solvability of the system.
http://youtu.be/S3yfxOmkG0E
29. A helicopter pilot wants to travel in a path due north, but the wind is blowing from the southeast at a speed of 25√2 mph. If the helicopter has an airspeed of 50mph in still air, in what direction must he "aim" the helicopter and what will be the helicopter's actual speed?
http://youtu.be/m5iJaMTy6-M
43. CSET! (Linear Programming: easier practice in CR 10 - 14) A political candidate wants to advertise his campaign. He wants at least 80 minutes and at most 120 minutes of total ad time. Research shows that 50,000 listeners are reached per minute when advertising on television and 10,000 listeners are reached per minute through radio advertising. He wants to reach at least 1,500,000 listeners. The cost of advertising per minute on television is $800 and the cost for radio advertising is $200 per minute. Let t be the number of minutes spent on television advertising and r be the number of minutes spent on radio advertising. Write and graph a system of inequalities representing the constraints of the given situation. What would be the minimum cost of advertising satisfying all the given constraints.
http://youtu.be/mG1lcC7Qd-E
44. CSET! A diet consists of two food sources, whose nutrition facts are given below. If the diet must meet the required daily intake of carbohydrates, fat, and protein, draw a graph representing the possible amounts of each food source a person on this diet could consume. See packet for graphic chart.
http://youtu.be/nf0AHYFC-g8
40. CSET! A line passes through the points (-5, 10) and (4, 28). A parabola passes through the points (-4/3, 0), (2, 0), (0, -16). Find the equation of the line and the parabola. Then sketch them on the same coordinate axis and find the points of intersection.
http://youtu.be/qJJuiVX0lLI
20. CSET! Prove √2 is irrational.
http://youtu.be/rUo3zFOSPSo
16. CSET! Use mathematical induction to prove 1 + 4 + 7 + ....+ (3n - 2) = n(3n - 1)/2
http://youtu.be/remtqYWJ7D4
26. CSET! Graph f(x) = (2/3)^x . State the equation of the inverse and plot it on the same axis. Give the domain, range, and asymptotes for both functions.
http://youtu.be/t42_KX_6CMA
64 a) CSET! More Rings and Fields! Given the set S = {1,2,3,4} represents a ring under addition and multiplication as described in the tables below, what additional property/properties of a field does it have? Is it a field? If not, what property/properties does it fail to demonstrate? See packet for table.
https://1drv.ms/w/s!Apq1xqsTphFDkC1GiFiq-NKRD7Al?e=hIeAN9
64 b) CSET! More Rings and Fields! Given the set S = {1,2,3,4} represents a ring under addition and multiplication as described in the tables below, what additional property/properties of a field does it have? Is it a field? If not, what property/properties does it fail to demonstrate? See packet for table.
https://1drv.ms/w/s!Apq1xqsTphFDkC1GiFiq-NKRD7Al?e=hIeAN9
57. CSET! Prove gcd(p,q) = gcd(p + q, lcm(p,q))
https://drive.google.com/file/d/0B7UrQE4OBGiLTzZMbnpoSGEyZFE/view?usp=sharing https://youtu.be/fPOIufqvyKc
60. CSET! A man drives 90 miles at a certain rate, then drives the next 50 miles at ¾ his original rate. If he had driven the entire distance at the slower rate, it the trip would have been 22.5 minutes longer. What was his original speed?
https://drive.google.com/file/d/0B7UrQE4OBGiLWkxseF9JeHFoYjA/view?usp=sharing
55. CSET! Explain why 4(-3) = -12 and use this result to explain why -4(-3) = 12.
https://drive.google.com/file/d/0B7UrQE4OBGiLX3V4WkFrOGZ3XzA/view?usp=sharing https://youtu.be/_V_JojgAUO0
58. CSET! More practice with transformations of functions A) If f(x) = x-3, find f-1(x). Give the domain and range of f(x) and f-1(x), and graph both on the same set of axis. B) If f(x) = 5-x, find f-1(x). Give the domain and range of f(x) and f-1(x), and graph both on the same set of axis.
https://drive.google.com/file/d/0B7UrQE4OBGiLY1ozbzZiWVo4ZnM/view?usp=sharing https://youtu.be/fYtXW_Oi3Cg
56. CSET! Show that the division of fractions results in the product of the division of the numerators and the division of the denominators. Write as a complex fraction and prove.
https://drive.google.com/file/d/0B7UrQE4OBGiLYXZUN0RLSGs5bnM/view?usp=sharing https://youtu.be/k9wJiLlQ0BE
59. CSET! The value of a car decreases by 19% annually. If the car's price when new was $32,000, write an equation to model the value, v(x), of the car after x years. With the knowledge that log3 ≈ 0.45 estimate after how many years the car's value be one-third its original price.
https://drive.google.com/file/d/0B7UrQE4OBGiLZ05fQmhCbjBjVFE/view?usp=sharing https://youtu.be/b2qTGSQcipY
46. CSET! Be familiar with sigma notation Prove (i=1) to n, Σ a(x sub i + y sub i) = (i=1) to n, Σ (a*x(sub i)) +(i=1) to n, Σ (a*y(sub i)) Prove (i=1) to n, Σ a(x(sub i)+y(sub i)) = a*(i=1) to n, Σ(x(sub i)) + a*(i=1) to n, Σ(y(sub i))
https://drive.google.com/file/d/0B7UrQE4OBGiLa0dvcm5KZEpjSEU/view?usp=sharing https://youtu.be/DQD7DGm_0rQ https://youtu.be/plEADLRuyUc
48. CSET! More Number Theory CSET! How many pairs of numbers have a GCF of 6 and an LCM of 90? (#2 on written solution below) CSET! How many pairs of numbers have a GCF of 15 and an LCM of 315? (#3 on written solution below) CSET! How many pairs of numbers have a GCF of 20 and an LCM of 600? #4 on written solution below) CSET! How many pairs of numbers have a GCF of 18 and a GCF of 756? (#5 on written solution below)
https://drive.google.com/file/d/0B7UrQE4OBGiLa0k5LVVmRWlfRjA/view?usp=sharing https://youtu.be/s8hY8Ryhujs
47. CSET! Use properties of exponents to prove why a0 = 1, a ≠ 0.
https://drive.google.com/file/d/0B7UrQE4OBGiLc0lWYXlDVWRDZDg/view?usp=sharing
13. An owner of a fruit orchard hires a crew of workers to prune at least 25 of his 50 fruit trees. Each newer tree requires one hour to prune, while each older tree needs 1 ½ hours. The crew contracts to work for at least 30 hours, and charges $15 for each newer tree and $20 for each older tree. How many of each kind of tree will the owner have pruned to minimize his cost?
https://drive.google.com/file/d/0B7UrQE4OBGiLeDRXUURCdng2NzQ/view?usp=sharing
50. CSET! Vistaprint sells cards at 5 cents per card for the first 500 cards, then 2 cents per cardard thereafter. The cost of manufacturing is 1 cent per card. Write an equation for the profit if over 500 cards are sold.
https://drive.google.com/file/d/0B7UrQE4OBGiLekRiUDktRV9zLWc/view?usp=sharing https://youtu.be/0xBVlbHytwk
51. CSET! Hoverboards sell for $500 each. The cost of materials is $100 for the first 1000 and $75 thereafter. The cost of labor is $200 for the first 2000 and $150 thereafter. Find a profit equation if over 2000 hoverboards are sold.
https://drive.google.com/file/d/0B7UrQE4OBGiLekRiUDktRV9zLWc/view?usp=sharing https://youtu.be/KlyYVrTN6T8
52. CSET! On Day 1, a gas station sells a total of 1000 gallons of regular, plus, and premium gas in a 2:5:3 ratio. On Day 2, the same station sells 20% more gas, at a 5:3:2 ratio. Which of the following is true? a) Three times as much regular gas was sold on day 2. b) Twice as much plus gas was sold on day 1. c) The total of plus and premium on day 1 is the same as that on day 2.
https://drive.google.com/file/d/0B7UrQE4OBGiLekRiUDktRV9zLWc/view?usp=sharing https://youtu.be/rWyUKUPwx0c
61. CSET! Express z= -5+5√3 and its conjugate in polar form and graph both on the complex plane. (MC 49 has extra practice with polar form, though I believe the problem you'll encounter on the test will look like this one)
https://drive.google.com/file/d/1bWPQA1sPS8YbvJO5EKOBusiUs58cSzEg/view?usp=sharing
6. Use the Euclidean algorithm to find the greatest common divisor, d, of the numbers 936 and 666. Then show that d can be expressed as a linear combination in the form d = 936m + 666n, with m and n integers.
https://www.youtube.com/watch?v=9rtCjmKYdmA&feature=youtu.be
5. Let y = f(x) = x2 + bx + c, with b and c real numbers. If one of the zeros is 3 + 5i, find the vertex of the graph of f(x).
https://www.youtube.com/watch?v=A5Y7TMaGD1Y&feature=youtu.be
2. CSET! Prove the quadratic formula.
https://www.youtube.com/watch?v=HqiiU9y88XI&feature=youtu.be note: practice completing the square
3. CSET! Rectangle ABCD is inscribed in a semi-circle. The radius of the semi-circle is √6. Find the area of rectangle ABCD written as a function of x. If the area is 2√5, find the possible dimensions of the rectangle.
https://www.youtube.com/watch?v=QcA9q-wPnBE&feature=youtu.be
1. CSET! Prove that the rational numbers are a field under addition and multiplication, but that the integers are not.
https://www.youtube.com/watch?v=eHY3XebK6hU&feature=youtu.be
8. Use the Euclidean algorithm to find the greatest common divisor, d, of the numbers 81 and 237. Then show that d can be expressed as a linear combination in the form d = 81m + 237n, with m and n integers.
https://www.youtube.com/watch?v=exOaZyrvWlo&feature=youtu.be
9. CSET! Use the Euclidean algorithm to find a solution to 7x + 11y = 13, such that x and y are integers.
https://www.youtube.com/watch?v=qAWHENTVyzQ&feature=youtu.be
4. CSET! A window is in the shape of a semicircle surmounting a rectangle. If the perimeter of the window is P, find the dimensions of the rectangle and semicircle for which maximum light would be allowed.
https://www.youtube.com/watch?v=rNKhS4XomgU&feature=youtu.be
36. CSET! Prove the Factor Theorem. Yes! You should know this one for sure!
https://youtu.be/-cG0Bh-8UB0
62. CSET! Graphing Integer Functions (also known as step functions or floor functions) Notation varies also: int(x), [x], ⌊x⌋, floor(x) Graph the integer functions f(x) = [x] f(x) = [x] +3 f(x) = [x+3] f(x) = 3[x] f(x) = 1/3[x] f(x) = [3x] f(x) = [1/3x]
https://youtu.be/12X4YsaFy1Q
32. Use vectors to find angle A of the triangle with the given vertices A(-3, 0), B(2, 2), C(0, 6).
https://youtu.be/5qCtN6bWG3A
10. A farmer can plant up to 8 acres of land with wheat and barley. His use of a necessary pesticide is limited by federal regulations to 10 gallons. Wheat requires 2 gallons of pesticide for every acre planted and barley requires 1 gallon per acre. If he can earn $5000 for every acre he plants with wheat and $3000 for every acre he plants with barley, how many acres of each should he plant to maximize his profits?
https://youtu.be/ClgA1-xnmPA
21. Prove that there are an infinite number of primes.
https://youtu.be/ctC33JAV4FI
12. Suppose $25,000 is available to invest. The financial planner advises at least $15,000 in T-bills at 6% interest, and no more than $5000 in bonds at 9% interest. How much should be invested in each account to obtain a maximum yield?
https://youtu.be/vHMRhL4TqFo
11. A toy factory has 20 days to manufacture its latest products. It can produce Talking Teddies at a rate of 60 per day and Screaming Sarahs at a rate of 70 per day. Due to a limited number of eyeballs available, a maximum of 1300 dolls can be made. If it makes a profit of $30 from each Talking Teddy and $26 from each Screaming Sarah, how many of each should the factory make to maximize its profits?
solution one: https://youtu.be/RkF2Mr_WrUo solution two: https://youtu.be/nePSMIdPt14
