Algebra (Secondary) for Competitive Exam Success

Algebra (Secondary)

3D Geometry Arithmetic and Geometric Progressions Arithmetic and Geometric Progressions - Applications Arithmetic and Geometric Progressions - Case Studies Arithmetic and Geometric Progressions - Problem Set Binomial Theorem Binomial Theorem (Intro) Binomial Theorem (Intro) - Applications Binomial Theorem (Intro) - Case Studies Binomial Theorem (Intro) - Problem Set Complex Numbers and Quadratic Equations Coordinate Geometry - Circles Coordinate Geometry - Straight Lines Differential Equations Factorization Techniques Factorization Techniques - Applications Factorization Techniques - Case Studies Factorization Techniques - Problem Set Indices and Surds Indices and Surds - Applications Indices and Surds - Case Studies Indices and Surds - Problem Set Inequalities and Their Applications Inequalities and Their Applications - Applications Inequalities and Their Applications - Case Studies Inequalities and Their Applications - Problem Set Integral Calculus Limit, Continuity and Differentiability Linear Equations in One Variable Linear Equations in One Variable - Applications Linear Equations in One Variable - Case Studies Linear Equations in One Variable - Problem Set Linear Programming Basics Linear Programming Basics - Applications Linear Programming Basics - Case Studies Linear Programming Basics - Problem Set Mathematical Induction Matrices and Determinants Pair of Linear Equations - Word Problems Pair of Linear Equations - Word Problems - Applications Pair of Linear Equations - Word Problems - Case Studies Pair of Linear Equations - Word Problems - Problem Set Permutations and Combinations Polynomials - Introduction Polynomials - Introduction - Applications Polynomials - Introduction - Case Studies Polynomials - Introduction - Problem Set Polynomials - Roots and Factor Theorem Polynomials - Roots and Factor Theorem - Applications Polynomials - Roots and Factor Theorem - Case Studies Polynomials - Roots and Factor Theorem - Problem Set Quadratic Equations Quadratic Equations - Applications Quadratic Equations - Case Studies Quadratic Equations - Problem Set Quadratic Formula Applications Quadratic Formula Applications - Applications Quadratic Formula Applications - Case Studies Quadratic Formula Applications - Problem Set Sequences and Series Sets, Relations and Functions Simultaneous Linear Equations Simultaneous Linear Equations - Applications Simultaneous Linear Equations - Case Studies Simultaneous Linear Equations - Problem Set Trigonometry - Advanced Problems Vector Algebra

Q. What is the solution set for the inequality 3x - 4 < 5?

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

Solution

Add 4 to both sides: 3x < 9. Then divide by 3: x < 3.

Correct Answer: A — x < 3

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Q. What is the solution set for the inequality x + 2 > 3?

A. x > 1
B. x < 1
C. x > 5
D. x < 5

Solution

To solve x + 2 > 3, subtract 2 from both sides: x > 1.

Correct Answer: A — x > 1

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Q. What is the solution set for the inequality x + 2 > 5?

A. x < 3
B. x > 3
C. x < 7
D. x > 7

Solution

Subtract 2 from both sides: x > 3.

Correct Answer: B — x > 3

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Q. What is the solution set for the inequality x + 5 > 2?

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

Solution

To solve x + 5 > 2, subtract 5 from both sides: x > -3.

Correct Answer: A — x > -3

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Q. What is the solution set for the inequality x^2 - 4 > 0?

A. (-∞, -2) ∪ (2, ∞)
B. (-2, 2)
C. [-2, 2]
D. [2, ∞)

Solution

Factor the inequality: (x - 2)(x + 2) > 0. The solution set is x < -2 or x > 2.

Correct Answer: A — (-∞, -2) ∪ (2, ∞)

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Q. What is the solution set for the inequality x^2 - 5x + 6 > 0?

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

Solution

Factoring gives (x - 2)(x - 3) > 0. The roots are x = 2 and x = 3. The solution set is x < 2 or x > 3.

Correct Answer: A — x < 2 or x > 3

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Q. What is the solution set for the inequality: -4x + 1 < 3?

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

Solution

Step 1: Subtract 1 from both sides: -4x < 2. Step 2: Divide by -4 (reverse inequality): x > -1/2.

Correct Answer: B — x < -1/2

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Q. What is the solution set for the inequality: -4x + 1 ≤ 9?

A. x ≥ -2
B. x ≤ -2
C. x > -2
D. x < -2

Solution

Step 1: Subtract 1 from both sides: -4x ≤ 8. Step 2: Divide by -4 (reverse the inequality): x ≥ -2.

Correct Answer: A — x ≥ -2

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Q. What is the solution set for the inequality: 2x + 3 ≥ 11?

A. x ≥ 4
B. x ≤ 4
C. x > 4
D. x < 4

Solution

Step 1: Subtract 3 from both sides: 2x ≥ 8. Step 2: Divide both sides by 2: x ≥ 4.

Correct Answer: A — x ≥ 4

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Q. What is the solution set for the inequality: x^2 - 4 < 0?

A. (-2, 2)
B. (2, ∞)
C. (-∞, -2)
D. (-∞, 2)

Solution

Step 1: Factor: (x - 2)(x + 2) < 0. Step 2: Test intervals: solution is (-2, 2).

Correct Answer: A — (-2, 2)

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Q. What is the solution set for the inequality: x^2 - 5x + 6 < 0?

A. (2, 3)
B. (3, 2)
C. (1, 6)
D. (0, 5)

Solution

Step 1: Factor the quadratic: (x - 2)(x - 3) < 0. Step 2: Test intervals: solution is (2, 3).

Correct Answer: A — (2, 3)

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Q. What is the solution to the equation 2(x - 3) = 4?

A. x = 1
B. x = 2
C. x = 4
D. x = 7

Solution

2(x - 3) = 4. Divide by 2: x - 3 = 2. Add 3: x = 5.

Correct Answer: D — x = 7

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Q. What is the solution to the equation 2x + 3 = 11?

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

Step 1: Subtract 3 from both sides: 2x = 8. Step 2: Divide both sides by 2: x = 4.

Correct Answer: C — x = 4

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Q. What is the solution to the equation 2x - 5 = 3x + 1?

A. x = -6
B. x = -4
C. x = 4
D. x = 6

Solution

Step 1: Rearrange: 2x - 3x = 1 + 5. Step 2: Combine like terms: -x = 6. Step 3: Multiply by -1: x = -6.

Correct Answer: A — x = -6

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Q. What is the solution to the equation 2x^2 - 4x - 6 = 0 using the quadratic formula?

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

Solution

Using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a. Here, a = 2, b = -4, c = -6. Discriminant = (-4)² - 4(2)(-6) = 16 + 48 = 64. Thus, x = (4 ± √64) / 4 = (4 ± 8) / 4. Solutions are x = 3 and x = -1.

Correct Answer: A — x = 3

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Q. What is the solution to the equation 2x^2 - 8 = 0?

A. x = 2, -2
B. x = 4, -4
C. x = 0, 4
D. x = 4, 0

Solution

First, add 8 to both sides: 2x^2 = 8. Then divide by 2: x^2 = 4. Taking the square root gives x = ±4.

Correct Answer: B — x = 4, -4

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Q. What is the solution to the equation 3(x + 2) = 2(x + 5)?

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

Solution

Step 1: Distribute: 3x + 6 = 2x + 10. Step 2: Subtract 2x from both sides: x + 6 = 10. Step 3: Subtract 6: x = 4.

Correct Answer: A — x = 1

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Q. What is the solution to the equation 3(x - 1) = 12?

A. x = 3
B. x = 4
C. x = 5
D. x = 6

Solution

Step 1: Divide both sides by 3: x - 1 = 4. Step 2: Add 1 to both sides: x = 5.

Correct Answer: B — x = 4

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Q. What is the solution to the equation 3(x - 1) = 2(x + 2)?

A. x = 5
B. x = 1
C. x = 3
D. x = 2

Solution

3(x - 1) = 2(x + 2). Expanding gives 3x - 3 = 2x + 4. Subtract 2x: x - 3 = 4. Add 3: x = 7.

Correct Answer: C — x = 3

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Q. What is the solution to the equation 3(x - 2) = 12?

A. x = 6
B. x = 4
C. x = 2
D. x = 0

Solution

Step 1: Divide both sides by 3: x - 2 = 4. Step 2: Add 2 to both sides: x = 6.

Correct Answer: A — x = 6

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Q. What is the solution to the equation 3(x - 2) = 9?

A. x = 1
B. x = 2
C. x = 5
D. x = 6

Solution

Step 1: Divide both sides by 3: x - 2 = 3. Step 2: Add 2 to both sides: x = 5.

Correct Answer: C — x = 5

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Q. What is the solution to the equation 3x + 2 = 11?

A. x = 3
B. x = 2
C. x = 4
D. x = 5

Solution

Subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3.

Correct Answer: A — x = 3

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Q. What is the solution to the equation 3x + 4 = 10?

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

To solve for x, subtract 4 from both sides: 3x = 6. Then divide by 3: x = 2.

Correct Answer: A — x = 2

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Q. What is the solution to the equation 4(x - 1) = 12?

A. x = 4
B. x = 3
C. x = 5
D. x = 2

Solution

Divide both sides by 4: x - 1 = 3. Add 1 to both sides: x = 4.

Correct Answer: B — x = 3

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Q. What is the solution to the equation 4(x - 1) = 2(x + 3)?

A. x = 5
B. x = 1
C. x = 2
D. x = 3

Solution

Distribute: 4x - 4 = 2x + 6. Subtract 2x: 2x - 4 = 6. Add 4: 2x = 10. Divide by 2: x = 5.

Correct Answer: C — x = 2

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Q. What is the solution to the equation 4x + 1 = 17?

A. x = 3
B. x = 4
C. x = 5
D. x = 6

Solution

Step 1: Subtract 1 from both sides: 4x = 16. Step 2: Divide both sides by 4: x = 4.

Correct Answer: A — x = 3

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Q. What is the solution to the equation 4x + 1 = 2x + 9?

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

Solution

Subtract 2x from both sides: 2x + 1 = 9. Then subtract 1: 2x = 8. Finally, divide by 2: x = 4.

Correct Answer: A — x = 4

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Q. What is the solution to the equation 4x - 1 = 3x + 2?

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

Solution

Step 1: Subtract 3x from both sides: x - 1 = 2. Step 2: Add 1 to both sides: x = 3.

Correct Answer: A — x = 1

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Q. What is the solution to the equation 4x - 12 = 0?

A. x = 3
B. x = 4
C. x = 2
D. x = 0

Solution

Step 1: Add 12 to both sides: 4x = 12. Step 2: Divide by 4: x = 3.

Correct Answer: A — x = 3

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Q. What is the solution to the equation 4x - 2 = 10?

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

Add 2 to both sides: 4x = 12. Then divide by 4: x = 3.

Correct Answer: B — x = 3

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Algebra (Secondary)

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