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 value of x in the polynomial equation x^3 - 4x = 0?

A. x = 0, 2, -2
B. x = 1, -1, 0
C. x = 4, -4, 0
D. x = 3, -3, 0

Solution

Factor out x: x(x^2 - 4) = 0. Thus, x = 0 or x^2 - 4 = 0, giving x = 2 or x = -2.

Correct Answer: A — x = 0, 2, -2

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Q. What is the value of x in the polynomial equation x^3 - 4x^2 + 4x = 0?

A. x = 0, 2
B. x = 1, 3
C. x = 2, 4
D. x = -1, 5

Solution

Factor out x: x(x^2 - 4x + 4) = 0. This gives x = 0 or (x - 2)^2 = 0, so x = 2.

Correct Answer: A — x = 0, 2

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Q. What is the value of x in the polynomial equation x^3 - 4x^2 + x = 0?

A. x = 0, 1, 4
B. x = 0, 2, 3
C. x = 1, 2, 3
D. x = 0, -1, -4

Solution

Factor out x: x(x^2 - 4x + 1) = 0. Thus, x = 0 or solve x^2 - 4x + 1 = 0.

Correct Answer: A — x = 0, 1, 4

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Q. What is the value of x in the quadratic equation x^2 - 5x + 6 = 0?

A. x = 1 or x = 6
B. x = 2 or x = 3
C. x = 3 or x = 2
D. x = 0 or x = 5

Solution

Factor the equation: (x - 2)(x - 3) = 0. Thus, x = 2 or x = 3.

Correct Answer: B — x = 2 or x = 3

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Q. What is the vertex of the parabola given 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*1) = 2. Step 2: Substitute x back into the equation: y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1. Step 3: Vertex is (2, -1).

Correct Answer: A — (2, -1)

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Q. What is the vertex of the parabola represented by the equation y = x^2 - 4x + 1?

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

Solution

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

Correct Answer: A — (2, -3)

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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)

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

A. (2, 0)
B. (0, 4)
C. (4, 0)
D. (2, 4)

Solution

The vertex can be found using the formula x = -b/(2a). Here, a = 1, b = -4, so x = 2. Substitute x back to find y: y = 0.

Correct Answer: A — (2, 0)

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

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

Solution

The vertex can be found using the formula x = -b/(2a). Here, a = 1, b = -4. Thus, x = 2. Plugging x back into the equation gives y = 1. So, the vertex is (2, 1).

Correct Answer: A — (2, -1)

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Q. What is the vertex of the quadratic 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)

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Q. What is the vertex of the quadratic function y = 2x^2 - 8x + 5?

A. (2, -3)
B. (4, -3)
C. (2, 5)
D. (4, 5)

Solution

The vertex can be found using the formula x = -b/(2a). Here, a = 2 and b = -8, so x = 8/4 = 2. Plugging x = 2 into the equation gives y = 2(2)^2 - 8(2) + 5 = -3. Thus, the vertex is (2, -3).

Correct Answer: A — (2, -3)

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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)

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

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

Solution

To find the x-intercept, set y = 0.\n1. 4x - 2(0) = 8\n2. 4x = 8\n3. x = 2.\nThe x-intercept is 2.

Correct Answer: C — 4

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Q. What is the y-intercept of the line given by the equation 3x + 2y = 6?

A. 0
B. 3
C. 2
D. 6

Solution

Convert to slope-intercept form (y = mx + b).\n1. 2y = -3x + 6\n2. y = -1.5x + 3.\nThe y-intercept is 3.

Correct Answer: B — 3

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Q. What is the y-intercept of the line represented by the equation y = -3x + 6?

A. 6
B. 3
C. -3
D. -6

Solution

The y-intercept is the constant term when x = 0, which is 6.

Correct Answer: A — 6

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Q. What is the y-intercept of the line represented by the equation y = -3x + 7?

A. 7
B. -3
C. 3
D. 0

Solution

The y-intercept is the constant term when x = 0, which is 7.

Correct Answer: A — 7

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Q. Which expression represents the difference of squares for a^2 - b^2?

A. (a + b)(a - b)
B. (a - b)(a + b)
C. (a + b)^2
D. (a - b)^2

Solution

The difference of squares is factored as (a + b)(a - b).

Correct Answer: A — (a + b)(a - b)

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Q. Which expression represents the difference of squares for x^2 - 16?

A. (x - 4)(x + 4)
B. (x - 8)(x + 8)
C. (x - 2)(x + 2)
D. (x - 16)(x + 16)

Solution

The expression can be factored as (x - 4)(x + 4) since 16 is 4^2.

Correct Answer: A — (x - 4)(x + 4)

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Q. Which expression represents the expansion of (x + 2)(x - 3)?

A. x^2 - x - 6
B. x^2 - x + 6
C. x^2 + x - 6
D. x^2 + x + 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. Which expression represents the expansion of (x + 2)^2?

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

Solution

Use the formula (a + b)^2 = a^2 + 2ab + b^2: (x + 2)^2 = x^2 + 4x + 4.

Correct Answer: B — x^2 + 4x + 4

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Q. Which expression represents the factored form of x^2 + 5x + 6?

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

Solution

The expression factors to (x + 2)(x + 3) since 2 and 3 add to 5 and multiply to 6.

Correct Answer: A — (x + 2)(x + 3)

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Q. Which expression represents the factored form of 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

The expression factors to (x + 5)(x + 2) since 5 and 2 add to 7 and multiply to 10.

Correct Answer: A — (x + 5)(x + 2)

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Q. Which expression represents the factored form of x^2 - 16?

A. (x - 4)(x + 4)
B. (x - 8)(x + 2)
C. (x - 2)(x + 2)
D. (x + 4)(x + 4)

Solution

This is a difference of squares: x^2 - 4^2 = (x - 4)(x + 4).

Correct Answer: A — (x - 4)(x + 4)

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Q. Which expression represents the factored form of x^2 - 9?

A. (x - 3)(x + 3)
B. (x - 1)(x + 1)
C. (x - 2)(x + 2)
D. (x - 4)(x + 4)

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)

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Q. Which expression represents the polynomial 3x^2 + 5x - 2 factored?

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

Solution

To factor, find two numbers that multiply to -6 and add to 5: (3x - 1)(x + 2).

Correct Answer: A — (3x - 1)(x + 2)

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Q. Which expression represents the sum of the first n terms of an arithmetic series with first term a and common difference d?

A. n/2 * (2a + (n-1)d)
B. n * (a + d)
C. n * a + d
D. n/2 * (a + d)

Solution

The formula for the sum of the first n terms is S_n = n/2 * (2a + (n-1)d).

Correct Answer: A — n/2 * (2a + (n-1)d)

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Q. Which expression represents the sum of the polynomials (2x^2 + 3x) and (4x^2 - x)?

A. 6x^2 + 2x
B. 2x^2 + 4x
C. 6x^2 + 4x
D. 2x^2 + 2x

Solution

Step 1: Combine like terms: (2x^2 + 4x^2) + (3x - x) = 6x^2 + 2x.

Correct Answer: A — 6x^2 + 2x

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Q. Which expression represents the sum of the polynomials (3x^2 + 2x) and (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. Which expression represents the sum of the roots of the equation x^2 - 6x + 8 = 0?

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

Solution

Step 1: The sum of the roots is given by -b/a: 6/1 = 6.

Correct Answer: A — 6

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Q. Which expression represents the sum of the roots of the quadratic equation x^2 + 6x + 8 = 0?

A. -6
B. 6
C. -8
D. 8

Solution

The sum of the roots is given by -b/a. Here, b = 6, so the sum is -6.

Correct Answer: A — -6

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Algebra (Secondary)

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