Complex numbers Questions and Answers

Use the polar grid function of Desmos Graphing Calculator to copy and paste the graph of the polar equation on the interval 2 2 Include any one point on the graph that is NOT on an axis and label the point

The graph shown is a transformation the graph of y sinx The pointed indicated has coordinates 7 0 5T 4T 3 2 E VT 16 ou 7 7 To w 19 The equation of the graph shown is y K 2 3 4 5 Enter a sine function

2 point Question From A to B a private plane flies 1 6 hours at 110 mph on a bearing of 34 It turns at point B and continues another 2 1 hours at the same speed but on a bearing of 124 to point C Round each answer to three decimal places At the end of this time how far is the plane from its starting point On what bearing from due north is the plane from its original location Miles

Alvin wants to know what his average paddling rate was for a canoe trip to a campsite On the way there it took him 6 hours against a river current of 2 km hr On the way back the same distance took him 4 hours with the same 2 km hr current Let r Alvin s average paddling rate To Campsite against the current From Campsite with the current Alvin s average paddling rate was d d d T t km hour If needed round to 2 decimal places

blocks east Which of the following statements are true More than one answer may be correct select all correct answers If Mary walked in a direct line to her friend s house she would walk a little less than 9 blocks For a right triangle the square of the hypotenuse is equal to the sum of the squares of the two legs O The distance Mary would walk in a direct line divided by 5 is 4 blocks no o

he graphs of y 2 sin z and y ave been plotted on the same axes below ind an approximation to the smallest positive solution to 2 sin 2 AM 2 A I 3 x 4t 5 T X 1 orrect to 10 significant figures nter your answer in the box below 25 e e 31 I 50 2

djust the value of a using the slider provided to complete the graph a 0 10 5 6 7n 6 2 3 4 3 2 TU 3 6 1 2 3 4 5 6 7 5 3 11 6

The graph shown is a vertical stretch of the graph of y sin z 5 1 7432 504 3 4 2 4 S Mk 2 1 2 56 7 4 2 374 5432714 2 The equation of the graph shown is y The graph could also be a transformation of the graph of y cos z In that case the equation of the graph would be y function Enter a sine function Enter a cosine

Find the arc lengths from P to Q if the radius is 30 5 feet and the arc spans an angle of 8 80 14 Rounds to 2 decimal places TIP You might try estimating first Then make sure your answer is in the ballpark of your estimate Number feet SZ G

1 x 3 1 4 Let f x and g x Find the composite function fog x and state its domain x 5

Find the product of the complex numbers Leave answer in polar form T Z 5 cos 3 I Z 6 cos 2 i sin i sin OA 11 cos 11 cos 5x 6 O C 11 cos 6 3 OD 30 cos 6 i sin OB 30 cos 30 co 5A 6 11 cos i sin 7 30 cos i sin 5x 6 i sin 5t 6

For the complex number w w5 w6 w7 W 8 10 11 3 i w w10 wo 20 27 Using the diagram click here for a larger view or by direct computation find the Cartesian form of the following w3 24 28 69 26 Im E w3 w 2010 W w12 wil we can compute directly that 2 1 31 2 3 i 2 Re wll w12 20 Note don t forget to use Maple notation eg sqrt 3 for the square root of 3 and and I capital i for the imaginary unit Hint Don t know how to use the diagram Think about how each of these roots are symmetrically placed around the unit circle

6 Prove that for any non zero and distinct real numbers 21 22 23 the Vandermond matrix 11 12 13 x x A

For what value of a does the SLLN hold for the sequence 1 o lija iii as Nime P X k

2 Consider R2 with the usual definition of lines but with distance function d x2 x Show that the triangle inequality is not true in this geometry by finding a triangle ABC such that AB BC AC

Find the difference quotient O a h 7 Ob h 7 Oc 2x h 7 Od 2x h 7h f x h f 2 for f x x 7x 6

Consider the simplex tableau given below X X X3 2 93 P 3 30 0 5 401 1 0 2 0 0 T 0 0 0 43 0 3 3 0 1 0 A Which variables are basic variables OX2 X3 S3 and P OX X2 X3 and 2 O S S2 S3 and x OX2 X3 S and P Which variables are nonbasic variables Ox and OX1 X2 and x3 Ox 5 and P OS S2 and s B Determine each of the values X P0001 11 27 30 15

3 Use analytical or graphical methods to determine equilibria for each of the following differential equations a x 1 x 1 e 2x b y y 1 ye ay a 0 3R c R 1 1 d z Inz

9 Perform the following operations and write in standard form a 2 pts 15 4i 9 64 b 3 pts 8 3i 10 7i

Write the equation in general form for the graphs of the rational functions Verify the y intercepts TEST 2

1 point Evaluate the following expressions and write them in the form a bi for some a b E R 1 6i 3 8i 1 6i 3 8i 1 6i 3 8i 1 6i 1 6i 6il

llowing drawings are nets for polyhedra Find the surface area in square meters and volume in cubic meters for each polyhedron Hint Draw the polyhedron and label it with what you know Round your answers to one decimal p 10 m 12 m e area 477 8 7 10 m rectangular pyramid 14 m x m 10 m 10 m

The graph shown is of a trigonometric function of the form y aSinba or y aCosba where b 0 Determine the equation of the graph For

The graph to the right shows a comparison between two groups of people living with a certain disease in the United Staten during 1993 2000 Use this graph to work the following parts a Use the points 0 10 and 7 90 where x 0 at 1993 and x7 at 2000 to find an equation for a line approximating the data for the graph in blue Write it in standard form b Repeat part a using the points 0 25 and 7 74 for the graph in red c Solve the system formed by the two equations in parts a and b d Describe how the solution is related to the graph a The equation of the line approximating the data is Type your answer in standard form Simplify your answer con CEED Number Onthousand y 140 120 100 80 80 40 People Living with Disease 30 0 1993 1994 1995 1996 1997 1998 1999 20 Your

Assume is opposite side a l is opposite side b and yis opposite side c Determine whether there is no triangle one triangle or two triangles Then solve each triangle if t possible Round each answer to the nearest tenth If not possible enter IMPOSSIBLE Below enter your answers so that is smaller than a 9 b 4 26 8052 Ys73 730 C 8705 Draw a triangle labeling angles a and y and their opposite sides Using the Law of Sines which proportion relates and the sine of angle ir and its opposite side a to the sine of angle and its opposite side b7 Use the inverse sine function on the calculator to find possible angies of in degrees Are there two triangles based on these measures of angle and angle 17 IMPOSSIBLE Y IMPOSSIBLE IMPOSSIBLE x X X

From the generating function of Hermite polynomials is 2th 1 e obtain H Hn x t n o n the following 0

Find the amplitude of the sine curve shown below 2 O Select one O a 2 O b 8 O 4 3 O d 4 T 2 370

In AXYZ LY is a right angle and the value of sin X is given In AXYZ LY is a right angle and sin X 3 Find cos X cot X and sin Z in fraction and in 5 decimal form Round to the nearest hundredth if necessary a cos X 1 67 3 3 cot X 1 67 4 4 sin Z 0 8 3 b cos X 5 1 25 4 nimni 5 cot X 0 8 3 5 sin Z 1 33 4 Z Y 3 C cos X 0 8 5 4 cot X 1 33 3 sin Z 0 8 d cos X 4 1 33 3 3 cot X 0 8 4 nim sin Z 1 67

Def If a is a real number and if m and n are integers containing no common factor with n 2 then am n provided that Va exists nam

Consider the following subset of R W 5a 2b 4c b 2c b2c where a b c ER PART A State why W is a subspace of R and give a basis for W PART B Give a basis for W the orthogonal complement of W

i Write z 2 2 2 3i in polar form Clearly show all the working ii Find the value of Z 9 and write the answer in the form a bi

A B C a 115 00 b 86 93 c 76 67 Po d 83 33 80 50 100 Qo 100 200 200 P1 95 45 110 Q1 What is the value of a price weighted DJIA type index of the three stocks for the period t 0 100 200 200

7 3 Obtain a numerical estimate for the following definite integral sin x 1 sinx dx

The following graph shows the number of muffins sold at a store during a given week Muffins sold 16 129 14 10 6420 8 Monday Tuesday Thursday Wednesday Friday Sunday Saturday Muffins sold If you were to estimate the slope of the tangent line on Thursday what units would be attached to that slope Use full words and the word per in your answer For example if the units were m yr you would write meters per year Do NOT use the word sold in your answer The units on the tangent line are

An aerial photograph is taken of a building The photograph is made when the angle of elevation of the sun is 36 The shadow is determined to be 70 feet long How tall is the building The building is approximately feet tall Round to the nearest foot as needed GELEN

Use the graph to determine the following a the function s domain b the function s range c the x intercepts if any d the y intercept if any e the function values f 2 and f 2 Assume that the graph of the function continues its trend beyond the displayed coordinate grid a What is the function s domain Type your answer in interval notation Ay

2 Use the correct properties and methods to simplify Now let h 27 and j 4 and evaluate Answer x

Rewrite the equation in logarithmic form x 7 a log A B C with A B C Preview Preview Preview

Given the ellipse 9x 2 16y 2 144 0 A Determine the length of the arc of the 1st quadrant B Determine the volume generated if the area on the 1st and 2nd quadrants is revolved about the x axis with solution

05 02 MC The figure shows a line graph and two shaded triangles that are similar 5 4 B 3 2 1 O 0 15 12 9 6 30 3 6 9 12 15 A 4 2 3 4 5 4X Which statement about the slope of the line is true

a Graph f x OA 5x 6 x b Find f 1 and f 3 c State the domain of the function Ay a Choose the correct graph below 30 if x 2 if x 2 5 O B Ay O C Ay 30 G O D 30

2 2 by by 5 5 1 Determine the values of x for 0 x 2x where the maximum value s Find the period and amplitude of the cosine function graphed to the right on the viewing window 21 21 minimum value s and zeros occur BROS

Question 3 10 8 6 10 4 2 8 6 4 2 2 4 6 8 10 2 4 6 8 10 14 Write an equation for the graph where y depends on x

Use the graph below to determine the average rate of change of f x from x 1 to x 3 O a r o c 2 a r o c 1 Oa r o c 0 a r o c 1 www 3 i c

Move the slider to sketch the graph of the given polar equation When finished position the slider all the way to the right and submit the question to get it marked correct This graph is a circle Below is the curve r 5 sin 0 for 0 0 a Adjust the value of a using the slider provided to complete the graph T ST 6 71 6 a 0 10 27 3 47 3 T 2 XIV ST 2 37 2 2 T 3 34 5m 3 10 6 6 7 Al 6

1 Find the exact values of the cosine and sine of each angle Then find the decimal values to the nearest hundredth a b C 390 d 30 60 390

Interest compounded continuously Complete the following Initial Investment at t 0 Po 75 000 5 000 Interest Rate k 6 2 8 4 Doubling Time T in years 11 1 2 docin Amount after 5 yr 7 130 90 11 414 71 17 539 32 Note When calculating the interest rate round your decimal to 3 decimals then convert your result into a percent Note When calculating your time T round to 1 decimal Note When calculating your dollar amount round

In general as the unit price of a commodity increases the demand for that commodity decreases Also as a commodity s unit price increases the manufacturer normally increases the supply The point where supply is equal to demand is called the equilibrium point Find the number of DVDs and the price per DVD when supply equals demand The number of DVDs is The graph below shows the demand and supply equations for previously rented DVDs Q Q Cost of a Previously Rented DVD 26 24 22 20 18 16 14 12 10 0 Demand Supply Equilibrium Paint 1 2 3 Number of DVDs in thousands

raph by hand in the interval 0 to 2m Upload a photo Identify the asymptotes tane

Question 3 2 1 1 2 0 2 4 1 2 3 6 1 a Find the reduced row echelon form of A using Gauss Jordan elimination Let A 1 1 2 7 2 b If A is the augmented matrix of a linear system is that system consistent Explain 1 2 1 1 20 2 4 1 2 61 4 d Find a basis for the mill space of A 1 1 2 Recall that A e Find a basis for the column space of A f Determine the dimension of the row space of A Explain