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Ch8 Polar

100

Convert the polar coordinates (6, π/4) to rectangular form.

(3√2,  3√2)  

200

Convert the rectangular point (−4, 4) to polar form. Give exact values.

(4√2, 135°)

300

Convert to rectangular r = 7 sin θ.

x² + y² = 7y

400

Find the modulus and argument of z = 5 + 12i and write z in polar form.

|z| = 13;   z = 13(cos 67.38° + i sin 67.38°)

500

Use DeMoivre's Theorem to compute (1 + i√3)⁶.

r = 2, 

θ = 60°.   2⁶ = 64;   6·60° = 360°.   

2⁶(cos 360° + i sin 360°)

Result: 64

Ch9 Vector

100

Let u have initial point P(3, 7) and terminal point Q(5, 2). Express u in terms of i and j.

u = 2i − 5j

200

Find |v| and direction θ for v = 3i + 3j.

|v| = 3√2;   θ = 45°

300

Given u = ⟨4, −3⟩ and v = ⟨−2, 5⟩, find the dot product u · v.

−23

400

Find the angle between

 u = ⟨2, −6⟩ and v = ⟨3, 1⟩.

90°  (the vectors are orthogonal)

500

Find the distance between P(1, 2, 3) and Q(4, 6, 3) in 3D space.

5

Ch10 Matrices

100

Solve the system: 

 y = 5 − 3x   and   y + 1 = x.

x = 1.5,   y = 0.5

200

Find the dimension of matrix 

A = [[1,2][3,4],[5,6],[7,8]].

4 x 2

300

Given A=[2  3], B=[[1,−2],[3,1]], C=[[1,0],[2,4]], D=[[3],[1]], find (AB)(CD).

23

400

Find the determinant of  [[3, 2],[1, 4]].

10

500

Given A=[[−2,1],[4,−3]] and B=[[3,0],[1,2]], solve X + A = 2B for X.

X = [[8, −1],[−2, 7]]

Ch11/12 Conics/Binomial

100

What conic section is (x+1)²/25 + (y−4)²/16 = 1? State the center and semi-axes.

Ellipse. Center (−1, 4),   a = 5,   b = 4.

200

Use Pascal's Triangle to expand (2x + 3y)⁴. Give the first and last terms only.

First: 16x⁴;   Last: 81y⁴

300

Find the 4th term of (2a + b)⁷ using the Binomial Theorem.

560 a⁴b³

400

Find the center and radius of the sphere 

 x² + y² + z² + 6x − 2z + 6 = 0.

Center (−3, 0, 1),   radius = 2.

500

A parabola has

 focus (0, 3) and directrix y = −3.

 Write its equation.

x² = 12y

Ch13/14 Limits/Probability

100

Evaluate lim(x→4) (2 − √x) / (4 − x).

1/4

200

Evaluate lim(x→∞) (

(2x^4 - 3x + 9)/(9x^4)

)

2/9

300

Evaluate

 lim(x→∞) (3x² + 1) / (x³ − x + 2).

0

400

A spinner has 8 positions. Each number is different.  You spin three times.  How many outcomes are possible?

512

500

Four independent trials with P(success) = 0.3. 

Find P(3 successes).

0.0756

Misc.

100

Simplify 10! / (8! · 2!) without a calculator.

45

200

Use the Binomial Theorem to write the first 3 terms of (x − √3)⁸.

x⁸ − 8√3 x⁷ + 84x⁶

300

What conic section is (x+1)²/25 - (y−4)²/16 = 1? State theeccentricity

Hyperbola   eccentricity = 

(sqrt 41)/5

400

Convert to a rectangular equation

2r = 

csc theta

y=1/2

500

z₁= 5(cos 30 + i sin 30)

z₂ = 6(cos 60 + i sin 60)

Calculate z₁z_{2 =}

z₁z₂  = 30(cos 90  + i sin90) 

z₁z₂  =  30i