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 sum of the roots of the polynomial x^2 - 4x + 4?

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

Solution

The sum of the roots of a quadratic equation ax^2 + bx + c is given by -b/a. Here, -(-4)/1 = 4.

Correct Answer: B — 4

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Q. What is the sum of the roots of the polynomial x^2 - 5x + 6?

A. 5
B. 6
C. 3
D. 0

Solution

The sum of the roots of a quadratic equation ax^2 + bx + c is given by -b/a. Here, -(-5)/1 = 5.

Correct Answer: A — 5

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Q. What is the sum of the roots of the quadratic equation 2x^2 + 4x - 6 = 0?

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

Solution

Step 1: Use the formula -b/a. Here, b = 4 and a = 2. Step 2: Sum of roots = -4/2 = -2.

Correct Answer: A — -2

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Q. What is the sum of the roots of the quadratic equation x^2 + 3x - 10 = 0?

A. -3
B. 3
C. 10
D. -10

Solution

The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a. Here, b = 3 and a = 1, so the sum is -3/1 = -3.

Correct Answer: A — -3

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Q. What is the sum of the roots of the quadratic equation x^2 + 4x + 4 = 0?

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

Solution

The sum of the roots is given by -b/a = -4/1 = -4.

Correct Answer: A — -4

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Q. What is the sum of the roots of the quadratic equation x^2 + 6x + 9 = 0?

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

Solution

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

Correct Answer: A — -6

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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

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Q. What is the value of k if the equation 2x^2 + kx + 3 = 0 has one solution?

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

Solution

For one solution, the discriminant must be zero: k^2 - 4(2)(3) = 0. Thus, k^2 = 24, so k = ±√24 = ±2√6. The integer value is -4.

Correct Answer: B — -4

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Q. What is the value of k if the equation x^2 + kx + 16 = 0 has one real solution?

A. k = 8
B. k = -8
C. k = 0
D. k = 4

Solution

For one real solution, the discriminant must be zero: k^2 - 4(1)(16) = 0. Thus, k^2 = 64, so k = ±8.

Correct Answer: B — k = -8

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Q. What is the value of k if the polynomial x^2 + kx + 16 has roots 4 and -4?

A. -8
B. 0
C. 8
D. 16

Solution

Using Vieta's formulas, the sum of the roots (4 + (-4)) = 0, so k = 0.

Correct Answer: A — -8

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Q. What is the value of k if the polynomial x^2 + kx + 16 has roots that are both 4?

A. -8
B. -16
C. 8
D. 16

Solution

Using the fact that the sum of the roots is -k, we have 4 + 4 = -k, so k = -8.

Correct Answer: A — -8

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Q. What is the value of k if the polynomial x^2 + kx + 9 has roots 3 and -3?

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

Solution

Using Vieta's formulas, the sum of the roots (3 + (-3)) = -k, so k = 0. The product of the roots (3 * -3) = 9, which is consistent. Thus, k = 0.

Correct Answer: B — 6

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Q. What is the value of k if x^2 + kx + 16 has roots 4 and -4?

A. -8
B. 0
C. 8
D. 16

Solution

Using Vieta's formulas, the sum of the roots is -k = 4 + (-4) = 0, so k = 0.

Correct Answer: A — -8

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Q. What is the value of k if x^2 - kx + 12 has roots 3 and 4?

A. 7
B. 12
C. 9
D. 8

Solution

Using the sum of roots, 3 + 4 = k, we find k = 7.

Correct Answer: A — 7

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Q. What is the value of x in the equation 2(x + 5) = 20?

A. x = 5
B. x = 10
C. x = 15
D. x = 20

Solution

Step 1: Divide both sides by 2: x + 5 = 10. Step 2: Subtract 5 from both sides: x = 5.

Correct Answer: A — x = 5

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Q. What is the value of x in the equation 2(x - 3) = 8?

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

Solution

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

Correct Answer: B — x = 6

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Q. What is the value of x in the equation 2x^2 - 4x - 6 = 0?

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. x = (4 ± √64) / 4 = (4 ± 8) / 4. Thus, x = 3 or x = -1.

Correct Answer: A — x = 3

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Q. What is the value of x in the equation 2x^2 - 8 = 0?

A. 2
B. 4
C. 0
D. 8

Solution

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

Correct Answer: B — 4

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Q. What is the value of x in 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 from both sides: x = 4.

Correct Answer: A — x = 1

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Q. What is the value of x in the equation 3(x + 2) = 21?

A. x = 5
B. x = 7
C. x = 3
D. x = 1

Solution

Divide both sides by 3: x + 2 = 7. Then subtract 2: x = 5.

Correct Answer: A — x = 5

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

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

Solution

Step 1: Distribute: 3x - 3 = 2x + 4. Step 2: Subtract 2x from both sides: x - 3 = 4. Step 3: Add 3 to both sides: x = 7.

Correct Answer: B — 0

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

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

Solution

Step 1: Distribute: 3x - 3 = 2x + 4. Step 2: Subtract 2x from both sides: x - 3 = 4. Step 3: Add 3 to both sides: x = 7.

Correct Answer: B — x = 2

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Q. What is the value of x in the equation 3(x - 2) = 12?

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

Solution

Divide both sides by 3: x - 2 = 4. Then add 2: x = 6.

Correct Answer: C — 6

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Q. What is the value of x in the equation 3x + 2 = 11?

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

Solution

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

Correct Answer: A — 3

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

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

Solution

Divide the equation by 3: x^2 + 4x + 4 = 0. Factor: (x + 2)(x + 2) = 0, giving x = -2.

Correct Answer: B — x = -4

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

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

Solution

Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Take the square root: x = 2 or x = -2.

Correct Answer: A — 2

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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

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Q. What is the value of x in the equation 4(x - 1) = 2x + 6?

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

Solution

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

Correct Answer: B — x = 3

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Q. What is the value of x in the equation 4x + 5 = 2x + 17?

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

Solution

Step 1: Subtract 2x from both sides: 2x + 5 = 17. Step 2: Subtract 5 from both sides: 2x = 12. Step 3: Divide by 2: x = 6.

Correct Answer: A — 3

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Algebra (Secondary)

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