Algebra and Functions: Master Advanced Mathematical Concepts

Question 1

Expand the binomial expression (x + y)^3.

Accepted Answer

x^3 + 3x^2y + 3xy^2 + y^3

Answer

x^3 + y^3

Answer

x^2y + 3xy^2 + y^3

Answer

x^3 + 2x^2y + xy^2 + y^3

Question 2

Determine the complex conjugate of the number (3 - 4i).

Accepted Answer

(3 + 4i)

Answer

(3 - 4i)

Answer

(-3 + 4i)

Answer

(-3 - 4i)

Question 3

Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...

Accepted Answer

210

Answer

165

Answer

255

Answer

300

Question 4

Calculate the area of the triangle with vertices at (0, 0), (3, 0), and (0, 4).

Accepted Answer

12

Answer

6

Answer

18

Answer

24

Question 5

Find the length (modulus) of the vector <2, -3>.

Accepted Answer

√13

Answer

1

Answer

√5

Answer

5

Question 6

What is the general form of a 4th-degree polynomial?

Accepted Answer

ax^4 + bx^3 + cx^2 + dx + e

Answer

ax^3 + bx^2 + cx + d

Answer

ax^4 + bx^2 + cx + e

Answer

ax^2 + bx + c

Question 7

Calculate the derivative of the function f(x) = e^(3x - 2).

Accepted Answer

3e^(3x - 2)

Answer

e^(3x - 2)

Answer

e^(3x - 2) * 3x

Answer

0

Question 8

Determine the general term of the arithmetic sequence with first term 5 and common difference 3.

Accepted Answer

a_n = 5 + 3n

Answer

a_n = 3 + 5n

Answer

a_n = 5 - 3n

Answer

a_n = 3 - 5n

Question 9

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

f'(x) = 3x^2 + 4x - 5

Answer

f'(x) = x^2 + 4x - 5

Answer

f'(x) = 3x^2 + 2x - 5

Answer

f'(x) = x^2 + 2x - 5

Question 10

Expand the binomial (x + y)^n to find the term with an index of r. Which of the following represents this term?

Accepted Answer

nx^(n-r)y^r

Answer

(n-1)x^(n-r-1)y^(r+1)

Answer

x^ny^r

Answer

y^nx^r

Question 11

Which of the following expressions represents a polynomial of degree 4?

Accepted Answer

x^4 - 2x^3 + x^2 - 3x + 2

Answer

x^2 + 2x + 1

Answer

x^3 - 3x^2 + 2x - 1

Answer

x^5 + x^3 - x + 1

Question 12

Determine the sum to infinity of the geometric series 1 + 0.5 + 0.25 + .... Given that the common ratio is 0.5.

Accepted Answer

2

Answer

1

Answer

1.5

Answer

2.5

Question 13

Find the sine of an angle measuring 30 degrees.

Accepted Answer

0.5

Answer

0

Answer

1

Answer

2

Question 14

Find the derivative of the function f(x) = x^3 - 2x^2 + 1.

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

x^3 - 4x^2 + 1

Answer

x^3 - 4x + 1

Question 15

Determine the integral of the function f(x) = e^x.

Accepted Answer

e^x + C

Answer

xe^x + C

Answer

x^2e^x + C

Answer

e^x^2 + C

Question 16

Find the value of (a + b)^3 when a = 1 and b = 2.

Accepted Answer

27

Answer

9

Answer

81

Answer

216

Question 17

Which of the following is NOT a complex number?

Accepted Answer

1

Answer

3 + 4i

Answer

2 - 5i

Answer

√-9

Question 18

Which of the following vectors is orthogonal to the vector [1 2 3]?

Accepted Answer

[2 -1 0]

Answer

[-1 2 -3]

Answer

[1 -2 3]

Answer

[3 2 -1]

Question 19

Find the degree of the polynomial p(x) = x^5 - 2x^3 + 4x - 1.

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 20

Find the value of sin(π/3).

Accepted Answer

√3/2

Answer

0

Answer

1/2

Answer

1

Question 21

Find the inverse function of f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3)/2

Answer

f^-1(x) = (x + 3)/2

Answer

f^-1(x) = 2x - 3

Answer

f^-1(x) = 3x + 2

Question 22

Find the range of the function f(x) = |x - 1|.

Accepted Answer

[0, ∞)

Answer

(0, ∞)

Answer

[1, ∞)

Answer

(-∞, ∞)

Question 23

Calculate the determinant of the matrix A = | 2 3 |

Accepted Answer

5

Answer

-5

Answer

1

Answer

-1

Question 24

Which vector is parallel to v = <2, -1, 3>?

Accepted Answer

<4, -2, 6>

Answer

<1, 0, 1.5>

Answer

<2, 1, -3>

Answer

<-2, 1, -3>

Question 25

The correct formula for the binomial theorem is:

Accepted Answer

(a + b)^n = nCa^n + nC₁a^(n-1)b + nC₂a^(n-2)b² + ... + nCₙb^n

Answer

(a - b)^n = nCa^n - nC₁a^(n-1)b + nC₂a^(n-2)b² - ... - nCₙb^n

Answer

(a + b)^n = nCa^n - nC₁a^(n-1)b + nC₂a^(n-2)b² - ... - nCₙb^n

Answer

(a - b)^n = nCa^n + nC₁a^(n-1)b + nC₂a^(n-2)b² + ... + nCₙb^n

Question 26

The identity matrix is:

Accepted Answer

[[1, 0], [0, 1]]

Answer

[[0, 1], [1, 0]]

Answer

[[1, 1], [1, 1]]

Answer

[[0, 0], [0, 0]]

Question 27

The degree of the polynomial 2x^3 - 5x^2 + 3x - 7 is:

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 28

The sum of the first 10 terms of the geometric sequence 3, 6, 12, ... is:

Accepted Answer

1893

Answer

1890

Answer

1903

Answer

1900

Question 29

The value of sin(π/3) is:

Accepted Answer

√3/2

Answer

1/2

Answer

1

Answer

√2/2

Question 30

The equation of the line passing through the points (2, 5) and (4, 9) is:

Accepted Answer

y = 2x + 1

Answer

y = 2x - 1

Answer

y = x + 3

Answer

y = -x + 7

Question 31

Find the domain and range of the function f(x) = √(x - 1).

Accepted Answer

Domain: [1, ∞), Range: [0, ∞)

Answer

Domain: (-∞, 1], Range: [0, ∞)

Answer

Domain: [-∞, 1), Range: [0, ∞)

Answer

Domain: (1, ∞), Range: (0, ∞)

Question 32

Find the inverse function of f(x) = 2x + 1.

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

f^-1(x) = (x + 1)/2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) = 1 - 2x

Question 33

Find the modulus of the complex number (3 + 4i).

Accepted Answer

7

Answer

1

Answer

5

Answer

9

Question 34

Expand the binomial expression (2x - 3y)^5.

Accepted Answer

32x^5 - 240x^4y + 600x^3y^2 - 600x^2y^3 + 270xy^4 - 27y^5

Answer

32x^5 + 240x^4y + 600x^3y^2 + 600x^2y^3 + 270xy^4 - 27y^5

Answer

32x^5 - 240x^3y^2 + 600x^2y^3 - 600xy^4 + 270y^5

Answer

32x^5 - 240x^4y + 600x^2y^3 - 600xy^4 + 270y^5

Question 35

Prove the trigonometric identity sin 2θ = 2 sin θ cos θ.

Accepted Answer

Use the double angle formula for sine.

Answer

Use the Pythagorean identity.

Answer

Use the half angle formula for cosine.

Answer

Use the product-to-sum formula.

Question 36

Which of the following accurately describes an application of the Binomial Theorem?

Accepted Answer

Expanding (a + b)^n into a sum of terms

Answer

Identifying the roots of a polynomial expression

Answer

Solving logarithmic equations

Answer

Simplifying complex trigonometric expressions

Question 37

Which of the following is NOT a fundamental property of complex numbers?

Accepted Answer

Conjugates have different moduli

Answer

Commutative under addition

Answer

Associative under multiplication

Answer

Distributive property holds

Question 38

Calculate the determinant of the matrix [[2, 1], [-3, 5]]

Accepted Answer

11

Answer

-1

Answer

-11

Answer

1

Question 39

Which of the following sequences converges?

Accepted Answer

a_n = 1/n

Answer

a_n = n/2

Answer

a_n = (-1)^n

Answer

a_n = n^2

Question 40

Expand the binomial expression (x + 2)^4.

Accepted Answer

x^4 + 8x^3 + 24x^2 + 32x + 16

Answer

x^4 + 4x^3 + 6x^2 + 4x + 1

Answer

x^4 + 4x^3 + 12x^2 + 16x + 16

Answer

x^4 + 4x^3 + 8x^2 + 8x + 4

Question 41

Determine the degree of the polynomial x^3 - 2x^2 + 5x - 1.

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 42

Determine the range of the function f(x) = sqrt(x - 1).

Accepted Answer

[1, infinity)

Answer

[0, infinity)

Answer

[-1, infinity)

Answer

[0, 1]

Question 43

Expand (2x + 3y)^3:

Accepted Answer

8x^3 + 36x^2y + 54xy^2 + 27y^3

Answer

8x^3 - 36x^2y - 54xy^2 - 27y^3

Answer

4x^3 + 12x^2y + 18xy^2 + 3y^3

Answer

4x^3 - 12x^2y - 18xy^2 - 3y^3

Question 44

Find the complex conjugate of z = 3 + 4i:

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 45

Calculate the product of matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]:

Accepted Answer

[[11, 14], [23, 30]]

Answer

[[5, 6], [15, 18]]

Answer

[[2, 4], [6, 8]]

Answer

[[1, 2], [3, 4]]

Question 46

Write the equation of the plane passing through points (1, -2, 3) and (2, 1, 4), perpendicular to the vector [1, 1, 1]:

Accepted Answer

x - y + z - 2 = 0

Answer

x + y - z - 2 = 0

Answer

x + y + z - 2 = 0

Answer

x - y - z - 2 = 0

Question 47

Determine the degree of the polynomial f(x) = 2x^4 - 3x^3 + 2x^2 - 5x + 1:

Accepted Answer

4

Answer

2

Answer

3

Answer

5

Question 48

Calculate sin(π/3):

Accepted Answer

√3/2

Answer

1/2

Answer

1

Answer

0

Question 49

Specify the domain of the function f(x) = 1/(x^2 - 1):

Accepted Answer

x ∈ R, x ≠ ±1

Answer

x ∈ R, x ≠ 0

Answer

x ∈ R

Answer

x ∈ R, x ≠ 1

Question 50

Find the derivative of f(x) = x^3 + 2x^2 - 5x + 1:

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^2 + 2x - 5

Answer

x^3 + 2x^2 - 5