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. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).

A. 5
B. 10
C. 7
D. 8

Solution

The determinant is calculated as \( 2*4 - 3*1 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

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

Solution

The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.

Correct Answer: A — 1

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Q. Calculate the determinant of the matrix: | 1 1 1 | | 2 2 2 | | 3 3 3 |

A. 0
B. 3
C. 6
D. 9

Solution

The rows are linearly dependent, hence the determinant is 0.

Correct Answer: A — 0

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Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).

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

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)

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

Solution

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: A — 1

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Q. Calculate the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)

A. ad - bc
B. ab + cd
C. ac - bd
D. bc - ad

Solution

The determinant is calculated as \( ad - bc \).

Correct Answer: A — ad - bc

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Q. Calculate the determinant | 1 0 0 | | 0 1 0 | | 0 0 1 |.

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

Solution

The determinant of the identity matrix is 1.

Correct Answer: B — 1

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Q. Calculate the determinant | 2 3 | | 4 5 | + | 1 1 | | 1 1 |.

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

Solution

The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.

Correct Answer: B — 1

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Q. Calculate the determinant: | 2 3 1 | | 1 0 2 | | 0 1 3 |.

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

Solution

The determinant evaluates to 0 as the rows are linearly dependent.

Correct Answer: A — -1

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Q. Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.

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

Solution

Using the determinant formula, we find that the determinant evaluates to 0.

Correct Answer: A — -1

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Q. Determine the coefficient of x^2 in the expansion of (3x - 4)^4.

A. 144
B. 216
C. 108
D. 96

Solution

The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.

Correct Answer: B — 216

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Q. Determine the coefficient of x^2 in the expansion of (3x - 4)^6.

A. 540
B. 720
C. 480
D. 360

Solution

The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.

Correct Answer: B — 720

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Q. Determine the coefficient of x^2 in the expansion of (x - 2)^6.

A. -60
B. -30
C. 15
D. 20

Solution

The coefficient of x^2 is C(6,2)(-2)^4 = 15 * 16 = 240.

Correct Answer: A — -60

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Q. Determine the solution for the inequality -2x + 6 > 0.

A. x < 3
B. x > 3
C. x < -3
D. x > -3

Solution

-2x + 6 > 0 => -2x > -6 => x < 3.

Correct Answer: B — x > 3

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Q. Determine the solution for the inequality -2x + 6 ≥ 0.

A. x ≤ 3
B. x ≥ 3
C. x ≤ -3
D. x ≥ -3

Solution

-2x + 6 ≥ 0 => -2x ≥ -6 => x ≤ 3.

Correct Answer: B — x ≥ 3

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Q. Determine the solution for the inequality -3x + 1 ≤ 4.

A. x ≥ -1
B. x ≤ -1
C. x ≥ 1
D. x ≤ 1

Solution

-3x + 1 ≤ 4 => -3x ≤ 3 => x ≥ -1.

Correct Answer: B — x ≤ -1

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Q. Determine the solution for the inequality -3x + 4 ≤ 1.

A. x ≥ 1
B. x ≤ 1
C. x ≥ -1
D. x ≤ -1

Solution

-3x + 4 ≤ 1 => -3x ≤ -3 => x ≥ 1.

Correct Answer: B — x ≤ 1

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Q. Determine the solution for the inequality 2x + 3 ≤ 7.

A. x ≤ 2
B. x ≥ 2
C. x ≤ 3
D. x ≥ 3

Solution

2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.

Correct Answer: A — x ≤ 2

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Q. Determine the solution for the inequality 6 - x > 2.

A. x < 4
B. x > 4
C. x < 6
D. x > 6

Solution

6 - x > 2 => -x > -4 => x < 4.

Correct Answer: A — x < 4

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Q. Determine the solution for the inequality 6 - x ≤ 3.

A. x ≥ 3
B. x ≤ 3
C. x ≥ 6
D. x ≤ 6

Solution

6 - x ≤ 3 => -x ≤ -3 => x ≥ 3.

Correct Answer: B — x ≤ 3

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Q. Determine the solution for the inequality 7 - 3x < 1.

A. x < 2
B. x > 2
C. x ≤ 2
D. x ≥ 2

Solution

7 - 3x < 1 => -3x < -6 => x > 2.

Correct Answer: A — x < 2

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Q. Determine the solution for the inequality 7x - 2 ≤ 5x + 6.

A. x ≤ 4
B. x ≥ 4
C. x ≤ 3
D. x ≥ 3

Solution

7x - 2 ≤ 5x + 6 => 2x ≤ 8 => x ≤ 4.

Correct Answer: A — x ≤ 4

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Q. Determine the solution set for the inequality 2(x - 1) ≥ 3.

A. x ≤ 2
B. x ≥ 2
C. x ≤ 3
D. x ≥ 3

Solution

2(x - 1) ≥ 3 => x - 1 ≥ 1.5 => x ≥ 2.

Correct Answer: B — x ≥ 2

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Q. Determine the solution set for the inequality 2x + 3 > 5.

A. x < 1
B. x > 1
C. x < 2
D. x > 2

Solution

2x + 3 > 5 => 2x > 2 => x > 1.

Correct Answer: B — x > 1

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Q. Determine the solution set for the inequality 2x + 3 > 7.

A. x < 2
B. x > 2
C. x < 3
D. x > 3

Solution

2x + 3 > 7 => 2x > 4 => x > 2.

Correct Answer: B — x > 2

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Q. Determine the solution set for the inequality 2x + 3 ≤ 7.

A. x ≤ 2
B. x ≥ 2
C. x < 2
D. x > 2

Solution

2x + 3 ≤ 7 => 2x ≤ 4 => x ≤ 2.

Correct Answer: A — x ≤ 2

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Q. Determine the solution set for the inequality 3x - 4 < 2x + 5.

A. x < 9
B. x > 9
C. x ≤ 9
D. x ≥ 9

Solution

3x - 4 < 2x + 5 => x < 9.

Correct Answer: A — x < 9

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Q. Determine the solution set for the inequality 4x - 1 > 3.

A. x < 1
B. x > 1
C. x ≤ 1
D. x ≥ 1

Solution

4x - 1 > 3 => 4x > 4 => x > 1.

Correct Answer: B — x > 1

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Q. Determine the solution set for the inequality 4x - 1 > 3x + 2.

A. x < 3
B. x > 3
C. x < 1
D. x > 1

Solution

4x - 1 > 3x + 2 => x > 3.

Correct Answer: B — x > 3

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Q. Determine the solution set for the inequality 4x - 1 < 3.

A. x < 1
B. x > 1
C. x ≤ 1
D. x ≥ 1

Solution

4x - 1 < 3 => 4x < 4 => x < 1.

Correct Answer: A — x < 1

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