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 result of (3x^2 + 2x) + (4x^2 - 5x)?

A. 7x^2 - 3x
B. 7x^2 + 3x
C. x^2 - 3x
D. x^2 + 3x

Solution

Step 1: Combine like terms: (3x^2 + 4x^2) + (2x - 5x) = 7x^2 - 3x.

Correct Answer: A — 7x^2 - 3x

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Q. What is the result of (x + 2)(x - 2)?

A. x^2 - 4
B. x^2 + 4
C. 2x
D. x^2 + 2

Solution

Step 1: Recognize this as a difference of squares: (a + b)(a - b) = a^2 - b^2. Step 2: Here, a = x and b = 2. Result is x^2 - 4.

Correct Answer: A — x^2 - 4

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Q. What is the result of (x + 2)(x - 3)?

A. x^2 - x - 6
B. x^2 + x - 6
C. x^2 - 6
D. x^2 + 6

Solution

Use the distributive property: x^2 - 3x + 2x - 6 = x^2 - x - 6.

Correct Answer: A — x^2 - x - 6

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Q. What is the result of factoring the expression x^2 + 7x + 10?

A. (x + 5)(x + 2)
B. (x - 5)(x - 2)
C. (x + 10)(x - 1)
D. (x - 10)(x + 1)

Solution

Step 1: Find two numbers that multiply to 10 and add to 7: 5 and 2. Step 2: Factor: (x + 5)(x + 2).

Correct Answer: A — (x + 5)(x + 2)

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Q. What is the result of factoring the polynomial x^2 + 7x + 10?

A. (x + 5)(x + 2)
B. (x + 10)(x - 1)
C. (x - 5)(x - 2)
D. (x + 1)(x + 10)

Solution

Step 1: Find two numbers that multiply to 10 and add to 7: 5 and 2. Step 2: Factor: (x + 5)(x + 2).

Correct Answer: A — (x + 5)(x + 2)

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Q. What is the result of factoring the polynomial x^2 - 9?

A. (x - 3)(x + 3)
B. (x - 9)(x + 1)
C. (x + 3)(x + 3)
D. x(x - 9)

Solution

x^2 - 9 is a difference of squares, which factors to (x - 3)(x + 3).

Correct Answer: A — (x - 3)(x + 3)

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Q. What is the simplified form of the expression (x^2 - 1)/(x - 1)?

A. x + 1
B. x - 1
C. x^2 + 1
D. x^2 - 1

Solution

Step 1: Factor the numerator: (x - 1)(x + 1)/(x - 1). Step 2: Cancel (x - 1): x + 1.

Correct Answer: A — x + 1

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Q. What is the simplified form of the expression 2(x + 3) - 4?

A. 2x + 2
B. 2x + 6
C. 2x + 10
D. 2x - 4

Solution

Step 1: Distribute: 2x + 6 - 4. Step 2: Combine like terms: 2x + 2.

Correct Answer: B — 2x + 6

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Q. What is the simplified form of the expression 2√(8) + 3√(2)?

A. 4√(2)
B. 6√(2)
C. 8√(2)
D. 5√(2)

Solution

Step 1: Simplify 2√(8) = 2 * 2√(2) = 4√(2). Step 2: Combine: 4√(2) + 3√(2) = 7√(2).

Correct Answer: A — 4√(2)

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Q. What is the simplified form of the expression 2√18 + 3√8?

A. 6√2
B. 12√2
C. 8√2
D. 10√2

Solution

Step 1: Simplify √18 = 3√2 and √8 = 2√2. Step 2: Substitute: 2(3√2) + 3(2√2) = 6√2 + 6√2 = 12√2.

Correct Answer: A — 6√2

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Q. What is the simplified form of the expression 3(x + 2) - 2(x - 1)?

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

Solution

Step 1: Distribute: 3x + 6 - 2x + 2. Step 2: Combine like terms: (3x - 2x) + (6 + 2) = x + 8.

Correct Answer: B — x + 7

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Q. What is the simplified form of the expression 3x^2 - 2x + 4x^2 - 5?

A. 7x^2 - 5
B. x^2 - 5
C. x^2 + 5
D. 7x^2 + 5

Solution

Step 1: Combine like terms: (3x^2 + 4x^2) + (-2x) - 5 = 7x^2 - 5.

Correct Answer: A — 7x^2 - 5

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Q. What is the slope of the line represented by the equation 2y - 4x = 8?

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

Solution

First, rewrite the equation in slope-intercept form (y = mx + b).\n1. 2y = 4x + 8\n2. y = 2x + 4.\nThe slope (m) is 2.

Correct Answer: A — 2

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Q. What is the slope of the line represented by the equation 2y = 4x + 6?

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

Solution

Step 1: Rewrite in slope-intercept form: y = 2x + 3. Step 2: The slope (m) is 2.

Correct Answer: A — 2

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Q. What is the slope of the line represented by the equation 4x - 2y = 8?

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

Solution

Rearranging to slope-intercept form: y = 2x - 4. The slope is 2.

Correct Answer: C — -2

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Q. What is the solution set for the equation 2x + 3 = 7?

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

Solution

Subtract 3 from both sides: 2x = 4.\nThen divide by 2: x = 2.

Correct Answer: A — x = 2

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

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

Solution

To solve for x, subtract 2 from both sides: 3x = 9. Then divide by 3: x = 3.

Correct Answer: B — x = 4

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

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

Solution

To solve 3x + 4 = 10, 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 set for the equation 4x + 1 = 2x + 5?

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

Solution

Subtract 2x from both sides: 2x + 1 = 5. Then subtract 1: 2x = 4. Finally, divide by 2: x = 2.

Correct Answer: A — x = 2

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Q. What is the solution set for 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 set for the equation 4x + 8 = 0?

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

Solution

Subtract 8 from both sides: 4x = -8. Then divide by 4: x = -2.

Correct Answer: A — x = -2

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Q. What is the solution set for the equation x^2 - 9 = 0?

A. x = 3, -3
B. x = 9
C. x = 0
D. x = 1, -1

Solution

Factoring gives (x - 3)(x + 3) = 0. Thus, the solutions are x = 3 and x = -3.

Correct Answer: A — x = 3, -3

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Q. What is the solution set for the inequality 2x + 1 ≥ 3?

A. x ≥ 1
B. x ≤ 1
C. x > 1
D. x < 1

Solution

Subtract 1 from both sides: 2x ≥ 2. Then divide by 2: x ≥ 1.

Correct Answer: A — x ≥ 1

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Q. What is the solution set for the inequality 2x + 1 ≥ 5?

A. x ≥ 2
B. x ≤ 2
C. x > 2
D. x < 2

Solution

Subtract 1 from both sides: 2x ≥ 4. Then divide by 2: x ≥ 2.

Correct Answer: A — x ≥ 2

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Q. What is the solution set for the inequality 2x + 3 ≥ 7?

A. x ≤ 2
B. x ≥ 2
C. x < 2
D. x > 2

Solution

Subtract 3 from both sides: 2x ≥ 4. Then divide by 2: x ≥ 2.

Correct Answer: B — x ≥ 2

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Q. What is the solution set for the inequality 2x + 5 > 3?

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

Solution

Step 1: Subtract 5 from both sides: 2x > -2. Step 2: Divide by 2: x > -1.

Correct Answer: A — x > -1

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Q. What is the solution set for the inequality 2x - 3 < 5?

A. x < 4
B. x > 4
C. x < 1
D. x > 1

Solution

First, add 3 to both sides: 2x < 8. Then divide by 2: x < 4.

Correct Answer: A — x < 4

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

A. x < 2
B. x > 2
C. x = 2
D. x ≤ 2

Solution

Solving the inequality: 2x < 4, thus x < 2.

Correct Answer: A — x < 2

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Q. What is the solution set for the inequality 2x - 4 < 6?

A. x < 5
B. x > 5
C. x < 2
D. x > 2

Solution

Add 4 to both sides: 2x < 10. Then divide by 2: x < 5.

Correct Answer: A — x < 5

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Q. What is the solution set for the inequality 2x - 5 < 3?

A. x < 4
B. x > 4
C. x < 2
D. x > 2

Solution

Add 5 to both sides: 2x < 8. Then divide by 2: x < 4.

Correct Answer: A — x < 4

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Algebra (Secondary)

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