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. Solve the inequality 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 both sides by 2: x ≥ 2.

Correct Answer: A — x ≥ 2

Q. Solve the inequality 3x - 5 < 4.

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

Solution

Step 1: Add 5 to both sides: 3x < 9. Step 2: Divide by 3: x < 3.

Correct Answer: B — x < 2

Q. Solve the inequality 5x - 2 < 3.

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

Solution

Step 1: Add 2 to both sides: 5x < 5. Step 2: Divide both sides by 5: x < 1.

Correct Answer: A — x < 1

Q. What are the roots of the quadratic equation x^2 - 5x + 6 = 0?

A. x = 1, 6
B. x = 2, 3
C. x = -2, -3
D. x = 0, 6

Solution

Step 1: Factor the equation: (x - 2)(x - 3) = 0. Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0. Step 3: Solutions are x = 2 and x = 3.

Correct Answer: B — x = 2, 3

Q. What are the solutions to the quadratic equation x^2 + 4x + 4 = 0?

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

Solution

Step 1: Factor the equation: (x + 2)(x + 2) = 0. Step 2: Set the factor to zero: x + 2 = 0. Step 3: Solution is x = -2 (double root).

Correct Answer: A — x = -2, -2

Q. What is the factored form of the polynomial x^2 + 5x + 6?

A. (x + 2)(x + 3)
B. (x - 2)(x - 3)
C. (x + 1)(x + 6)
D. (x + 3)(x + 2)

Solution

Step 1: Find two numbers that multiply to 6 and add to 5: 2 and 3. Step 2: Factor: (x + 2)(x + 3).

Correct Answer: A — (x + 2)(x + 3)

Q. What is the factored form of the polynomial x^2 - 9?

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

Solution

Step 1: Recognize it as a difference of squares: x^2 - 3^2. Step 2: Factor: (x - 3)(x + 3).

Correct Answer: A — (x - 3)(x + 3)

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)

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

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

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

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

Q. What is the solution to the equation 4x - 7 = 5?

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

Solution

Step 1: Add 7 to both sides: 4x = 12. Step 2: Divide both sides by 4: x = 3.

Correct Answer: B — x = 2

Q. What is the solution to the equation 5x + 2 = 17?

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

Solution

Step 1: Subtract 2 from both sides: 5x = 15. Step 2: Divide by 5: x = 3.

Correct Answer: B — x = 4

Q. What is the solution to the equation 5x - 2 = 3x + 6?

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

Solution

Step 1: Subtract 3x from both sides: 2x - 2 = 6. Step 2: Add 2 to both sides: 2x = 8. Step 3: Divide by 2: x = 4.

Correct Answer: B — x = 3

Q. What is the solution to the equation 7x + 2 = 3x + 18?

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

Solution

Step 1: Subtract 3x from both sides: 4x + 2 = 18. Step 2: Subtract 2: 4x = 16. Step 3: Divide by 4: x = 4.

Correct Answer: A — x = 4

Q. What is the solution to the quadratic equation x^2 + 4x + 4 = 0?

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

Solution

Step 1: Factor the equation: (x + 2)(x + 2) = 0. Step 2: Set the factor to zero: x + 2 = 0. Solution is x = -2.

Correct Answer: A — x = -2

Q. What is the sum of the roots of the quadratic equation x^2 - 6x + 8 = 0?

A. 6
B. 8
C. 4
D. 2

Solution

Step 1: Use the sum of roots formula -b/a. Here, b = -6, a = 1. Step 2: Sum = 6.

Correct Answer: A — 6

Q. What is the value of x in the equation 4(x - 1) = 2(x + 3)?

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

Solution

Step 1: Distribute: 4x - 4 = 2x + 6. Step 2: Subtract 2x from both sides: 2x - 4 = 6. Step 3: Add 4: 2x = 10. Step 4: Divide by 2: x = 5.

Correct Answer: C — x = 3

Q. What is the value of x in the inequality 3x - 7 < 2?

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

Solution

Step 1: Add 7 to both sides: 3x < 9. Step 2: Divide by 3: x < 3.

Correct Answer: A — x < 3

Q. What is the value of x in the polynomial equation 3x^2 + 12x = 0?

A. x = 0, -4
B. x = 0, 4
C. x = -3, -4
D. x = 3, 4

Solution

Step 1: Factor out 3x: 3x(x + 4) = 0. Step 2: Set each factor to zero: 3x = 0 or x + 4 = 0. Step 3: Solutions are x = 0 and x = -4.

Correct Answer: A — x = 0, -4

Q. What is the vertex of the parabola represented by the equation y = x^2 - 4x + 3?

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

Solution

Step 1: Use the vertex formula x = -b/2a: x = 4/2 = 2. Step 2: Substitute x back into the equation: y = 2^2 - 4(2) + 3 = -1. Vertex is (2, -1).

Correct Answer: A — (2, -1)

Q. What is the vertex of the quadratic function y = x^2 - 4x + 3?

A. (2, -1)
B. (2, -4)
C. (1, 2)
D. (3, 0)

Solution

Step 1: Use the vertex formula x = -b/2a. Here, a = 1, b = -4. Step 2: x = 4/2 = 2. Step 3: Substitute x back: y = 2^2 - 4*2 + 3 = -1. Vertex is (2, -1).

Correct Answer: A — (2, -1)

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)

Q. Which of the following is the correct factorization of x^2 - 9?

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

Solution

Step 1: Recognize it as a difference of squares: a^2 - b^2 = (a - b)(a + b). Step 2: Here, a = x, b = 3. So, (x - 3)(x + 3).

Correct Answer: A — (x - 3)(x + 3)

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