Q. If z = re^(iθ), what is the value of |z|?
Show solution
Solution
The modulus |z| = r, as |re^(iθ)| = r.
Correct Answer: A — r
Learn More →
Q. If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?
A.
x^2 + y^2 = 100
B.
x^2 + y^2 = 10
C.
x^2 + y^2 = 50
D.
x^2 + y^2 = 25
Show solution
Solution
The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.
Correct Answer: A — x^2 + y^2 = 100
Learn More →
Q. If z = x + yi, find the real part of z^3.
A.
x^3 - 3xy^2
B.
3x^2y - y^3
C.
x^3 + 3xy^2
D.
3x^2 - y^3
Show solution
Solution
Using the binomial expansion, z^3 = (x + yi)^3 = x^3 - 3xy^2 + (3x^2y - y^3)i.
Correct Answer: A — x^3 - 3xy^2
Learn More →
Q. If z1 = 1 + i and z2 = 1 - i, find z1 * z2.
Show solution
Solution
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
Learn More →
Q. If z1 = 1 + i and z2 = 2 - 3i, find z1 * z2.
A.
7 - i
B.
7 + i
C.
5 - i
D.
5 + i
Show solution
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i + 3 = 5 - i.
Correct Answer: A — 7 - i
Learn More →
Q. If z1 = 1 + i and z2 = 2 - 3i, what is z1 * z2?
A.
5 - i
B.
8 - i
C.
7 + i
D.
1 + 5i
Show solution
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i.
Correct Answer: A — 5 - i
Learn More →
Q. If z1 = 1 + i and z2 = 2 - i, find z1 * z2.
A.
3 + i
B.
3 - i
C.
2 + 3i
D.
2 - 3i
Show solution
Solution
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - 1 = 3 + i.
Correct Answer: A — 3 + i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - 5i, then z1 + z2 is?
A.
6 - 2i
B.
6 + 2i
C.
2 - 2i
D.
2 + 2i
Show solution
Solution
z1 + z2 = (2 + 3i) + (4 - 5i) = 6 - 2i.
Correct Answer: A — 6 - 2i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - i, find z1 + z2.
A.
6 + 2i
B.
6 + 4i
C.
2 + 6i
D.
8 + 2i
Show solution
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
Learn More →
Q. If z1 = 2 + 3i and z2 = 4 - i, what is z1 + z2?
A.
6 + 2i
B.
6 + 4i
C.
2 + 4i
D.
2 + 2i
Show solution
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
Learn More →
Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), find \( |A| \).
Show solution
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
Learn More →
Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |2A| \)?
Show solution
Solution
The determinant of \( 2A \) is \( 2^2 * |A| = 4 * (-2) = -8 \).
Correct Answer: B — 8
Learn More →
Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |A| \)?
Show solution
Solution
The determinant is calculated as (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
Learn More →
Q. If \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), what is the determinant of A?
A.
ad - bc
B.
bc - ad
C.
a + d
D.
b + c
Show solution
Solution
The determinant of matrix A is given by the formula \( ad - bc \).
Correct Answer: A — ad - bc
Learn More →
Q. If \( B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \), what is \( |B| \)?
Show solution
Solution
The determinant is calculated as 1(1*1 - 0*3) - 2(0*1 - 1*2) + 1(0*3 - 1*2) = 1 - 4 - 2 = -5.
Correct Answer: B — 2
Learn More →
Q. If \( B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \), what is \( \det(B) \)?
Show solution
Solution
Using the determinant formula, we find \( \det(B) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 \).
Correct Answer: A — -24
Learn More →
Q. If \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is \( |B| \)?
Show solution
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
Learn More →
Q. If \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |B| \)?
Show solution
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
Learn More →
Q. If \( B = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \), what is \( |B| \)?
Show solution
Solution
The determinant is \( 1*5 - 2*3 = 5 - 6 = -1 \).
Correct Answer: B — 1
Learn More →
Q. If \( B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \), find \( |B| \).
Show solution
Solution
The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer: A — 5
Learn More →
Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( |C| \).
Show solution
Solution
The determinant is 0 because the first column is a linear combination of the others.
Correct Answer: A — 0
Learn More →
Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( \det(C) \).
Show solution
Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer: A — 0
Learn More →
Q. If \( C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), what is the determinant of C?
A.
ad - bc
B.
bc - ad
C.
a + d
D.
b + c
Show solution
Solution
The determinant of C is given by the formula \( ad - bc \).
Correct Answer: A — ad - bc
Learn More →
Q. If \( D = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{vmatrix} \), find \( D \).
Show solution
Solution
Calculating gives \( 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = -4 + 24 + 1 = 21 \).
Correct Answer: A — -10
Learn More →
Q. If \( F = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), what is the value of the determinant?
Show solution
Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer: A — 0
Learn More →
Q. If \( J = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} \), what is the value of the determinant?
Show solution
Solution
The determinant is calculated as \( 1(1*3 - 0*1) - 2(0*3 - 1*2) + 1(0*1 - 1*2) = 3 + 4 - 2 = 5 \).
Correct Answer: A — 0
Learn More →
Q. If \( y = \cot^{-1}(x) \), what is \( \frac{dy}{dx} \)?
A.
\( -\frac{1}{1+x^2} \)
B.
\( \frac{1}{1+x^2} \)
C.
0
D.
undefined
Show solution
Solution
The derivative of \( y = \cot^{-1}(x) \) is \( -\frac{1}{1+x^2} \).
Correct Answer: A — \( -\frac{1}{1+x^2} \)
Learn More →
Q. If \( y = \sec^{-1}(x) \), what is \( \frac{dy}{dx} \)?
A.
\( \frac{1}{
B.
x
C.
\sqrt{x^2-1}} \)
D.
\( \frac{1}{x\sqrt{x^2-1}} \)
.
0
.
undefined
Show solution
Solution
The derivative of \( y = \sec^{-1}(x) \) is \( \frac{1}{|x|\sqrt{x^2-1}} \).
Correct Answer: B — x
Learn More →
Q. If \( y = \sin^{-1}(x) + \cos^{-1}(x) \), what is the value of \( y \)?
A.
0
B.
1
C.
\( \frac{\pi}{2} \)
D.
undefined
Show solution
Solution
Since \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \) for all \( x \) in the domain of \( \sin^{-1} \) and \( \cos^{-1} \), the answer is \( \frac{\pi}{2} \).
Correct Answer: C — \( \frac{\pi}{2} \)
Learn More →
Q. If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?
A.
0
B.
1
C.
\( \frac{\pi}{4} \)
D.
undefined
Show solution
Solution
When \( x = 1 \), \( y = \tan^{-1}(1) + \tan^{-1}(y) \) leads to \( y = \frac{\pi}{4} \).
Correct Answer: C — \( \frac{\pi}{4} \)
Learn More →
Showing 1711 to 1740 of 2847 (95 Pages)
