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Algebra & Functions

100

Find the inverse of f(x)=3x−5

 f^-1(x)=x+5/3 

200

State the domain of f(x)=1/x²−9

x≠±3

300

Solve log⁡₃(x)+log⁡₃(2x)=4

x=3

400

The range of f(x) = √5 - 2x

y≥0

500

Find the exact solution to e^2x−5e^x+6=0

x=ln⁡2, x=ln⁡3

Calculus

100

Differentiate y=7x³

21x²

200

Find d/dx(ln⁡(3x))

1/x

300

Evaluate ∫(2x+4) dx

x²+4x+C

400

Find the derivative of y=x^x 

y′=x^x(lnx+1)

500

Evaluate ₀¹∫xe^x2 dx

e−1/2

Trigonometry

100

 Simplify sin⁡0^∘

0

200

Solve sin ⁡x=1/2  for 0≤x<2π

x=π/6,5π/6

300

Prove the identity:
tan⁡x=sin⁡x/cos⁡x

Use definition tan⁡ x=opp /adj=sin⁡ x/cos⁡ x

400

Find the exact value: sin⁡(75^∘)

square root of 6 + square root of 2 divided by 4

500

Solve: 2cos⁡²x−3cos⁡x+1=0 

cosx=1 or cos⁡x=1/2

Complex Numbers

100

Write 3−4i in modulus-argument form

r=5, θ=−tan^-1(4/3)

200

Multiply: (2+i)(3−2i)

8−i

300

Find all solutions of z²=−16

z=4i, −4i

400

Express 1/3−i in the form a + bi

3+i/10

500

Solve z⁴=81 in polar form

z=3e^i(πk/2), k=0,1,2,3

Vectors

100

Find the magnitude of ⟨3,4⟩

5

200

Compute the dot product: ⟨1,2,3⟩⋅⟨4,5,6⟩

32

300

Determine the angle between vectors if u⋅v=0

90^o

400

Find the vector equation of a line through (1,2,3) in direction (2,-1,4)

r=⟨1,2,3⟩+t⟨2,−1,4⟩

500

Find the distance from point (3,1,2) to the plane 2x−y+2z=10.

∣2(3)−1(1)+2(2)−10∣/ square root of 2² + (-1)² +2² = 1