Algebra Study Resources for Competitive Exam Success

Algebra

Binomial theorem Complex numbers Determinants Inverse Trigonometric Functions Linear Inequalities Logarithms Matrices Permutations & combinations Quadratic equations Sequence & series Sets & Relations

Q. If z = re^(iθ), what is the value of r if z = 4 + 3i?

A. 5
B. 7
C. 4
D. 3

Solution

r = |z| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.

Correct Answer: A — 5

Learn More →

Q. If z = re^(iθ), what is the value of r if z = 4 + 4i?

A. 4√2
B. 8
C. 4
D. 2√2

Solution

r = |z| = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2.

Correct Answer: A — 4√2

Learn More →

Q. If z = re^(iθ), what is the value of z^2?

A. r^2e^(i2θ)
B. re^(iθ)
C. 2re^(iθ)
D. r^2e^(iθ)

Solution

Using the property of exponents, z^2 = (re^(iθ))^2 = r^2e^(i2θ).

Correct Answer: A — r^2e^(i2θ)

Learn More →

Q. If z = re^(iθ), what is the value of z^3?

A. r^3 e^(i3θ)
B. r^3 e^(iθ)
C. 3re^(iθ)
D. r^3 e^(iθ^3)

Solution

Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).

Correct Answer: A — r^3 e^(i3θ)

Learn More →

Q. If z = re^(iθ), what is the value of |z|?

A. r
B. θ
C. re
D. 1

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

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

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.

A. 2
B. 0
C. 1
D. -1

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

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

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

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

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

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

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 \( |A| \).

A. -2
B. 2
C. 0
D. 1

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 \( |2A| \)?

A. -8
B. 8
C. 4
D. 16

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 \( |A| \)?

A. -2
B. 2
C. 0
D. 4

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

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 \( |B| \)?

A. 1
B. 2
C. 3
D. 4

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 \( \det(B) \)?

A. -24
B. 24
C. 0
D. 12

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 \( |B| \)?

A. 0
B. 1
C. 2
D. 3

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 \( |B| \)?

A. -2
B. 2
C. 0
D. 1

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 \( |B| \)?

A. -1
B. 1
C. 2
D. 3

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 \( |B| \).

A. 5
B. 6
C. 7
D. 8

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 \( |C| \).

A. 0
B. 1
C. 2
D. 3

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 \( \det(C) \).

A. 0
B. 1
C. 2
D. 3

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

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 \( D \).

A. -10
B. 10
C. 0
D. 5

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?

A. 0
B. 1
C. 2
D. 3

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?

A. 0
B. 1
C. 2
D. 3

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 →

Showing 481 to 510 of 862 (29 Pages)

Algebra

On a scale of 0–10, how likely are you to recommend The Soulshift Academy?