Q. If x = tan^(-1)(√3), what is the value of sin(2x)?
-
A.
√3/2
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B.
1
-
C.
√2/2
-
D.
0
Solution
Since tan^(-1)(√3) = π/3, then 2x = 2π/3 and sin(2x) = sin(2π/3) = √3/2.
Correct Answer: B — 1
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Q. If x^2 + 2x + 1 = 0, what is the value of x?
Solution
This is a perfect square: (x + 1)^2 = 0 => x = -1.
Correct Answer: A — -1
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Q. If x^2 - 6x + 9 = 0, what is the value of x?
Solution
This is a perfect square: (x - 3)^2 = 0, so x = 3.
Correct Answer: A — 3
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Q. If y = cos^(-1)(1/2), what is the value of y?
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A.
π/3
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B.
π/4
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C.
π/6
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D.
π/2
Solution
cos^(-1)(1/2) = π/3, since cos(π/3) = 1/2.
Correct Answer: A — π/3
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Q. If y = cos^(-1)(x), then dy/dx is:
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A.
-1/√(1-x^2)
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B.
1/√(1-x^2)
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C.
0
-
D.
1
Solution
The derivative of cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = cos^(-1)(x), then what is dy/dx?
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A.
-1/√(1-x^2)
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B.
1/√(1-x^2)
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C.
1/x
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D.
-1/x
Solution
The derivative of y = cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = cos^(-1)(x), what is dy/dx?
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A.
-1/√(1-x^2)
-
B.
1/√(1-x^2)
-
C.
0
-
D.
1
Solution
The derivative of cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = sin^(-1)(x), then what is the derivative dy/dx?
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A.
1/√(1-x^2)
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B.
1/(1-x^2)
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C.
√(1-x^2)
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D.
1/x
Solution
The derivative of y = sin^(-1)(x) is dy/dx = 1/√(1-x^2).
Correct Answer: A — 1/√(1-x^2)
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Q. If y = sin^(-1)(x), then x = sin(y) implies:
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A.
y = x
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B.
y = -x
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C.
y = 1-x
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D.
y = 1+x
Solution
By definition, if y = sin^(-1)(x), then x = sin(y).
Correct Answer: A — y = x
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Q. If y = sin^(-1)(x), what is dy/dx?
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A.
1/sqrt(1-x^2)
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B.
1/(1-x^2)
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C.
sqrt(1-x^2)
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D.
1/x
Solution
The derivative of y = sin^(-1)(x) is dy/dx = 1/sqrt(1-x^2).
Correct Answer: A — 1/sqrt(1-x^2)
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Q. If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?
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A.
0
-
B.
1/√(1-x^2)^3
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C.
-1/√(1-x^2)^3
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D.
undefined
Solution
The second derivative d^2y/dx^2 = -1/√(1-x^2)^3.
Correct Answer: C — -1/√(1-x^2)^3
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Q. If y = tan^(-1)(x), then the range of y is:
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A.
(-π/2, π/2)
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B.
(0, π)
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C.
(-π, π)
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D.
[0, 1]
Solution
The range of y = tan^(-1)(x) is (-π/2, π/2).
Correct Answer: A — (-π/2, π/2)
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Q. If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?
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A.
0
-
B.
-2/(1+x^2)^2
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C.
2/(1+x^2)^2
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D.
1/(1+x^2)
Solution
The first derivative dy/dx = 1/(1+x^2). The second derivative d^2y/dx^2 = -2/(1+x^2)^2.
Correct Answer: B — -2/(1+x^2)^2
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Q. If z = 1 + i, find the conjugate of z.
-
A.
1 - i
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B.
1 + i
-
C.
-1 + i
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D.
-1 - i
Solution
The conjugate of z = 1 + i is z̅ = 1 - i.
Correct Answer: A — 1 - i
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Q. If z = 1 + i, find the value of z^3.
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A.
-2 + 2i
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B.
2 + 2i
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C.
0
-
D.
1 + 3i
Solution
z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.
Correct Answer: A — -2 + 2i
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Q. If z = 1 + i, find the value of z^4.
Solution
z^4 = (1 + i)^4 = (2e^(iπ/4))^4 = 16e^(iπ) = -16.
Correct Answer: A — -4
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Q. If z = 1 + i, find z^2.
-
A.
2i
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B.
2
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C.
1 + 2i
-
D.
0
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: C — 1 + 2i
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Q. If z = 1 + i, find z^3.
-
A.
-2 + 2i
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B.
2 + 2i
-
C.
0
-
D.
1 + i
Solution
z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.
Correct Answer: A — -2 + 2i
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Q. If z = 1 + i, find z^4.
Solution
z^4 = (1 + i)^4 = 4i.
Correct Answer: A — -4
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Q. If z = 1 + i, what is z^2?
-
A.
2i
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B.
2
-
C.
1 + 2i
-
D.
0
Solution
z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: C — 1 + 2i
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Q. If z = 1 + i√3, find the argument of z.
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A.
π/3
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B.
2π/3
-
C.
π/6
-
D.
5π/6
Solution
The argument θ = tan^(-1)(√3/1) = π/3.
Correct Answer: A — π/3
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Q. If z = 1 + i√3, find the modulus of z.
Solution
|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.
Correct Answer: A — 2
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Q. If z = 1 + i√3, find the value of |z|^2.
Solution
|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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Q. If z = 1 + i√3, find |z|.
Solution
|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.
Correct Answer: A — 2
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Q. If z = 1 + i√3, find |z|^2.
Solution
|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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Q. If z = 1 + i√3, then the argument of z is?
-
A.
π/3
-
B.
π/6
-
C.
2π/3
-
D.
5π/6
Solution
The argument θ = tan^(-1)(√3/1) = π/3.
Correct Answer: A — π/3
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Q. If z = 1 + i√3, what is |z|^2?
Solution
|z|^2 = 1^2 + (√3)^2 = 1 + 3 = 4.
Correct Answer: A — 4
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Q. If z = 2 + 2i, find the argument of z.
-
A.
π/4
-
B.
π/2
-
C.
3π/4
-
D.
0
Solution
The argument is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.
Correct Answer: A — π/4
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Q. If z = 2 + 2i, find the conjugate of z.
-
A.
2 - 2i
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B.
2 + 2i
-
C.
-2 + 2i
-
D.
-2 - 2i
Solution
The conjugate of z = 2 + 2i is z̅ = 2 - 2i.
Correct Answer: A — 2 - 2i
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Q. If z = 2 + 2i, find the modulus of z.
Solution
|z| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Correct Answer: A — 2√2
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