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. Which inequality represents the solution set for x + 4 < 10?

A. x < 6
B. x > 6
C. x ≤ 6
D. x ≥ 6

Solution

To solve the inequality, subtract 4 from both sides: x < 6.

Correct Answer: A — x < 6

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Q. Which inequality represents the solution set for x^2 - 4 < 0?

A. x < -2 or x > 2
B. -2 < x < 2
C. x > -2 and x < 2
D. x < 2

Solution

The inequality x^2 - 4 < 0 can be factored as (x - 2)(x + 2) < 0. The solution set is -2 < x < 2.

Correct Answer: B — -2 < x < 2

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Q. Which inequality represents the solution to the equation 4x - 8 = 0?

A. x < 2
B. x > 2
C. x = 2
D. x ≤ 2

Solution

To solve the equation, add 8 to both sides: 4x = 8. Then divide by 4: x = 2. The solution is x = 2.

Correct Answer: C — x = 2

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Q. Which of the following inequalities represents the solution to 2x + 3 > 7?

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

Solution

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

Correct Answer: A — x > 2

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Q. Which of the following inequalities represents the solution to 3x + 2 > 11?

A. x > 3
B. x < 3
C. x > 4
D. x < 4

Solution

Subtracting 2 from both sides gives 3x > 9. Dividing by 3 results in x > 3.

Correct Answer: C — x > 4

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Q. Which of the following inequalities represents the solution to x + 4 > 2?

A. x > -2
B. x < -2
C. x > -6
D. x < -6

Solution

Subtract 4 from both sides: x > -2.

Correct Answer: A — x > -2

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Q. Which of the following is a factor of the polynomial x^2 + 4x + 4?

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

Solution

The polynomial can be factored as (x + 2)(x + 2) or (x + 2)^2. Therefore, x + 2 is a factor.

Correct Answer: A — x + 2

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Q. Which of the following is a factor of the polynomial x^2 - 4?

A. x - 2
B. x + 2
C. x^2 + 4
D. x - 1

Solution

The polynomial x^2 - 4 can be factored as (x - 2)(x + 2). Therefore, x - 2 is a factor.

Correct Answer: A — x - 2

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Q. Which of the following is a factor of the polynomial x^2 - 5x + 6?

A. (x - 2)
B. (x - 3)
C. (x + 2)
D. (x + 3)

Solution

Step 1: Factor the polynomial: (x - 2)(x - 3). Step 2: The factors are (x - 2) and (x - 3).

Correct Answer: A — (x - 2)

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Q. Which of the following is a factor of the polynomial x^3 - 3x^2 - 4x + 12?

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

Solution

Using synthetic division or the factor theorem, we can test x = 2. Substituting gives us 2^3 - 3(2^2) - 4(2) + 12 = 0, confirming that x - 2 is a factor.

Correct Answer: A — x - 2

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Q. Which of the following is a factor of the polynomial x^3 - 4x^2 + 4x?

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

Solution

We can factor out x from the polynomial: x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x - 2)^2. Thus, x is a factor.

Correct Answer: C — x

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Q. Which of the following is a factor of x^2 - 9?

A. (x - 3)
B. (x + 3)
C. (x - 3)(x + 3)
D. (x^2 + 9)

Solution

x^2 - 9 is a difference of squares and can be factored as (x - 3)(x + 3).

Correct Answer: C — (x - 3)(x + 3)

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Q. Which of the following is a polynomial?

A. 3x^2 + 2x - 1
B. 1/x + 2
C. sqrt(x) + 3
D. ln(x)

Solution

A polynomial is a sum of terms with non-negative integer exponents. 3x^2 + 2x - 1 fits this definition.

Correct Answer: A — 3x^2 + 2x - 1

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Q. Which of the following is a root of the polynomial x^2 + 3x - 10?

A. -5
B. 2
C. 1
D. -2

Solution

Factoring gives (x + 5)(x - 2) = 0. The roots are x = -5 and x = 2, so 2 is a root.

Correct Answer: B — 2

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Q. Which of the following is a root of the polynomial x^2 - 4?

A. -2
B. 0
C. 2
D. Both -2 and 2

Solution

Step 1: Factor the polynomial: (x - 2)(x + 2). Step 2: Set each factor to zero: x - 2 = 0 or x + 2 = 0. Roots are x = 2 and x = -2.

Correct Answer: D — Both -2 and 2

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Q. Which of the following is a root of the polynomial x^2 - 7x + 10?

A. 1
B. 2
C. 5
D. 10

Solution

The polynomial can be factored as (x - 2)(x - 5). Setting each factor to zero gives us x = 2 and x = 5, so 2 is a root.

Correct Answer: B — 2

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Q. Which of the following is a root of the polynomial x^3 - 6x^2 + 11x - 6?

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

Solution

Using the Rational Root Theorem, we can test x = 1, 2, and 3. Testing x = 3 gives 3^3 - 6(3^2) + 11(3) - 6 = 0, so x = 3 is a root.

Correct Answer: C — 3

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Q. Which of the following is a solution to the equation 2x^2 - 8 = 0?

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

Solution

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

Correct Answer: B — 2

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Q. Which of the following is a solution to the equation 3x + 7 = 16?

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

Solution

Step 1: Subtract 7 from both sides: 3x = 9. Step 2: Divide both sides by 3: x = 3.

Correct Answer: B — x = 2

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Q. Which of the following is a solution to the equation 4x + 1 = 13?

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

Solution

Subtract 1 from both sides: 4x = 12. Then divide by 4: x = 3.

Correct Answer: A — x = 2

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Q. Which of the following is a solution to the equation 4x + 1 = 17?

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

Solution

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

Correct Answer: C — x = 2

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Q. Which of the following is a solution to the equation 4x - 7 = 5?

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

Solution

Add 7 to both sides: 4x = 12. Then divide by 4: x = 3.

Correct Answer: A — x = 1

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Q. Which of the following is a solution to the equation x^2 + 2x - 8 = 0?

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

Solution

Factor the equation: (x + 4)(x - 2) = 0. The solutions are x = -4 and x = 2.

Correct Answer: A — -4

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Q. Which of the following is a solution to the equation x^2 + 6x + 9 = 0?

A. x = -3
B. x = 3
C. x = 0
D. x = -6

Solution

The equation can be factored as (x + 3)(x + 3) = 0. Thus, the solution is x = -3.

Correct Answer: A — x = -3

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Q. Which of the following is a solution to the equation x^2 - 7x + 10 = 0?

A. 1
B. 2
C. 5
D. 10

Solution

Factoring gives (x - 5)(x - 2) = 0, so the solutions are x = 5 and x = 2.

Correct Answer: C — 5

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Q. Which of the following is a solution to the inequality 2x + 3 > 7?

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

Solution

Step 1: Subtract 3 from both sides: 2x > 4. Step 2: Divide by 2: x > 2. The solution includes all values greater than 2.

Correct Answer: B — 2

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Q. Which of the following is a solution to the inequality 3x - 4 < 5?

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

Solution

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

Correct Answer: A — x < 3

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Q. Which of the following is a solution to the inequality 3x - 5 < 4?

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

Solution

Add 5 to both sides: 3x < 9. Then divide by 3: x < 3.

Correct Answer: A — x < 3

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Q. Which of the following is a solution to the inequality 5x - 7 < 8?

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

Solution

Add 7 to both sides: 5x < 15. Divide by 5: x < 3.

Correct Answer: A — x < 3

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Q. Which of the following is a solution to the inequality: -2x + 4 > 0?

A. x < 2
B. x > 2
C. x ≤ 2
D. x ≥ 2

Solution

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

Correct Answer: A — x < 2

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Algebra (Secondary)

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