Ch8 Polar
Convert the polar coordinates (6, π/4) to rectangular form.
(3√2, 3√2)
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
Solve the system:
y = 5 − 3x and y + 1 = x.
x = 1.5, y = 0.5
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.
Evaluate lim(x→4) (2 − √x) / (4 − x).
1/4
Simplify 10! / (8! · 2!) without a calculator.
45
Convert the rectangular point (−4, 4) to polar form. Give exact values.
(4√2, 135°)
Find |v| and direction θ for v = 3i + 3j.
|v| = 3√2; θ = 45°
Find the dimension of matrix
A = [[1,2][3,4],[5,6],[7,8]].
4 x 2
Use Pascal's Triangle to expand (2x + 3y)⁴. Give the first and last terms only.
First: 16x⁴; Last: 81y⁴
Evaluate lim(x→∞) (
(2x^4 - 3x + 9)/(9x^4)
)
2/9
Use the Binomial Theorem to write the first 3 terms of (x − √3)⁸.
x⁸ − 8√3 x⁷ + 84x⁶
Convert to rectangular r = 7 sin θ.
x2 + y2 = 7y
Given u = ⟨4, −3⟩ and v = ⟨−2, 5⟩, find the dot product u · v.
−23
Given A=[2 3], B=[[1,−2],[3,1]], C=[[1,0],[2,4]], D=[[3],[1]], find (AB)(CD).
23
Find the 4th term of (2a + b)⁷ using the Binomial Theorem.
560 a⁴b³
Evaluate
lim(x→∞) (3x² + 1) / (x³ − x + 2).
0
What conic section is (x+1)²/25 - (y−4)²/16 = 1? State theeccentricity
Hyperbola eccentricity =
(sqrt 41)/5
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°)
Find the angle between
u = ⟨2, −6⟩ and v = ⟨3, 1⟩.
90° (the vectors are orthogonal)
Find the determinant of [[3, 2],[1, 4]].
10
Find the center and radius of the sphere
x² + y² + z² + 6x − 2z + 6 = 0.
Center (−3, 0, 1), radius = 2.
A spinner has 8 positions. Each number is different. You spin three times. How many outcomes are possible?
512
Convert to a rectangular equation
2r =
csc theta
y=1/2
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
Find the distance between P(1, 2, 3) and Q(4, 6, 3) in 3D space.
5
Given A=[[−2,1],[4,−3]] and B=[[3,0],[1,2]], solve X + A = 2B for X.
X = [[8, −1],[−2, 7]]
A parabola has
focus (0, 3) and directrix y = −3.
Write its equation.
x² = 12y
Four independent trials with P(success) = 0.3.
Find P(3 successes).
0.0756
z1= 5(cos 30 + i sin 30)
z2 = 6(cos 60 + i sin 60)
Calculate z1z2 =
z1z2 = 30(cos 90 + i sin90)
z1z2 = 30i