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 solution to the inequality: 2x^2 - 8 < 0?

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

Solution

Step 1: Factor: 2(x^2 - 4) < 0. Step 2: Further factor: 2(x - 2)(x + 2) < 0. Step 3: The solution is (-2, 2).

Correct Answer: A — (-2, 2)

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Q. What is the solution to the inequality: 3(x + 2) ≤ 2(x + 5)?

A. x ≤ 1
B. x ≥ 1
C. x ≤ -1
D. x ≥ -1

Solution

Step 1: Distribute: 3x + 6 ≤ 2x + 10. Step 2: Subtract 2x: x + 6 ≤ 10. Step 3: Subtract 6: x ≤ 4.

Correct Answer: A — x ≤ 1

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Q. What is the solution to the inequality: 3(x - 2) > 2(x + 1)?

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

Solution

Step 1: Distribute: 3x - 6 > 2x + 2. Step 2: Subtract 2x: x - 6 > 2. Step 3: Add 6: x > 8.

Correct Answer: A — x > 1

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Q. What is the solution to the inequality: 3x + 2 > 5?

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

Solution

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

Correct Answer: A — x > 1

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Q. What is the solution to the inequality: 3x + 2 ≤ 5?

A. x ≤ 1
B. x ≥ 1
C. x < 1
D. x > 1

Solution

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

Correct Answer: A — x ≤ 1

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Q. What is the solution to the inequality: 3x + 4 ≤ 2x + 10?

A. x ≤ 6
B. x ≤ 4
C. x ≥ 6
D. x ≥ 4

Solution

Step 1: Subtract 2x from both sides: x + 4 ≤ 10. Step 2: Subtract 4 from both sides: x ≤ 6.

Correct Answer: A — x ≤ 6

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Q. What is the solution to the inequality: 4x + 1 ≥ 2x + 9?

A. x ≤ 4
B. x ≥ 4
C. x < 4
D. x > 4

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 9. Step 2: Subtract 1: 2x ≥ 8. Step 3: Divide by 2: x ≥ 4.

Correct Answer: B — x ≥ 4

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Q. What is the solution to the inequality: 4x - 1 < 3x + 2?

A. x < 3
B. x > 3
C. x < 1
D. x > 1

Solution

Step 1: Subtract 3x from both sides: x - 1 < 2. Step 2: Add 1 to both sides: x < 3.

Correct Answer: A — x < 3

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Q. What is the solution to the inequality: 4x - 7 ≤ 9?

A. x ≤ 4
B. x ≥ 4
C. x < 4
D. x > 4

Solution

Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.

Correct Answer: A — x ≤ 4

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Q. What is the solution to the inequality: 5 - 2x > 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 (reverse inequality): x < 1.

Correct Answer: B — x > 1

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Q. What is the solution to the inequality: 6 - x < 2?

A. x > 4
B. x < 4
C. x > 6
D. x < 6

Solution

Step 1: Subtract 6 from both sides: -x < -4. Step 2: Multiply by -1 (reverse the inequality): x > 4.

Correct Answer: A — x > 4

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Q. What is the solution to the inequality: 7 - 2x > 1?

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

Solution

Step 1: Subtract 7 from both sides: -2x > -6. Step 2: Divide by -2 (reverse the inequality): x < 3.

Correct Answer: A — x < 3

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Q. What is the solution to the inequality: x^2 + 2x - 8 > 0?

A. (-∞, -4) ∪ (2, ∞)
B. (-4, 2)
C. (-∞, 2) ∪ (4, ∞)
D. (-4, ∞)

Solution

Step 1: Factor the inequality: (x - 2)(x + 4) > 0. Step 2: Critical points are x = -4 and x = 2. Step 3: Test intervals: (-∞, -4), (-4, 2), (2, ∞). The solution is (-∞, -4) ∪ (2, ∞).

Correct Answer: A — (-∞, -4) ∪ (2, ∞)

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Q. What is the solution to the inequality: x^2 + 3x < 10?

A. x < 2 or x > -5
B. x > 2 and x < -5
C. x < -5 or x > 2
D. x > -5 and x < 2

Solution

Step 1: Rearrange: x^2 + 3x - 10 < 0. Step 2: Factor: (x + 5)(x - 2) < 0. Step 3: The solution is -5 < x < 2.

Correct Answer: D — x > -5 and x < 2

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Q. What is the solution to the inequality: x^2 + 3x - 4 ≤ 0?

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

Solution

Step 1: Factor the quadratic: (x + 4)(x - 1) ≤ 0. Step 2: Critical points are x = -4 and x = 1. Step 3: Test intervals: (-∞, -4), (-4, 1), (1, ∞). The solution set is [-4, 1].

Correct Answer: A — (-4, 1)

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Q. What is the solution to the inequality: x^2 + 4x < 5?

A. x < 1 or x > -5
B. x < -1 or x > 5
C. x < -5 or x > 1
D. x > -5 and x < 1

Solution

Step 1: Rearrange: x^2 + 4x - 5 < 0. Step 2: Factor: (x + 5)(x - 1) < 0. Step 3: Solution is -5 < x < 1.

Correct Answer: D — x > -5 and x < 1

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Q. What is the solution to the inequality: x^2 - 5x + 6 < 0?

A. 1 < x < 6
B. 2 < x < 3
C. x < 2 or x > 3
D. x > 2 and x < 3

Solution

Step 1: Factor the quadratic: (x - 2)(x - 3) < 0. Step 2: Test intervals: x < 2, 2 < x < 3, x > 3. Solution: 2 < x < 3.

Correct Answer: D — x > 2 and x < 3

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Q. What is the solution to the inequality: x^2 - 9 > 0?

A. x < -3 or x > 3
B. -3 < x < 3
C. x > -3 and x < 3
D. x < 3

Solution

Step 1: Factor: (x - 3)(x + 3) > 0. Step 2: Test intervals: solution is x < -3 or x > 3.

Correct Answer: A — x < -3 or x > 3

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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

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Q. What is the solution to the quadratic equation x^2 + 6x + 9 = 0?

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

Solution

The equation x^2 + 6x + 9 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. What is the solution to the quadratic equation x^2 - 5x + 6 = 0?

A. 1 and 6
B. 2 and 3
C. 3 and 4
D. 4 and 5

Solution

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

Correct Answer: B — 2 and 3

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Q. What is the solution to the system of equations: 2x + y = 10 and x - y = 2?

A. (4, 2)
B. (2, 6)
C. (6, 4)
D. (0, 10)

Solution

From x - y = 2, we get y = x - 2. Substituting into the first equation gives x = 4, y = 2.

Correct Answer: A — (4, 2)

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Q. What is the sum of the first 5 terms of the arithmetic progression 2, 5, 8, ...?

A. 15
B. 30
C. 20
D. 25

Solution

The first term a = 2, common difference d = 3. The sum of the first n terms S_n = n/2 * (2a + (n-1)d). For n = 5, S_5 = 5/2 * (2*2 + 4*3) = 5/2 * (4 + 12) = 5/2 * 16 = 40/2 = 20.

Correct Answer: C — 20

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Q. What is the sum of the first 5 terms of the arithmetic sequence 2, 5, 8, ...?

A. 30
B. 25
C. 20
D. 15

Solution

The first term a = 2, common difference d = 3. Sum = n/2 * (2a + (n-1)d) = 5/2 * (4 + 12) = 30.

Correct Answer: B — 25

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Q. What is the sum of the first 6 terms of the arithmetic progression 1, 4, 7, ...?

A. 60
B. 45
C. 30
D. 36

Solution

The first term a = 1, common difference d = 3. The sum of the first n terms S_n = n/2 * (2a + (n-1)d). For n = 6, S_6 = 6/2 * (2*1 + 5*3) = 3 * (2 + 15) = 3 * 17 = 51.

Correct Answer: B — 45

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Q. What is the sum of the first 6 terms of the geometric progression 1, 3, 9, ...?

A. 364
B. 364/2
C. 364/3
D. 182

Solution

The first term a = 1, and the common ratio r = 3. The sum of the first n terms of a GP is S_n = a(1 - r^n) / (1 - r). For n = 6, S_6 = 1(1 - 3^6) / (1 - 3) = (1 - 729) / -2 = -728 / -2 = 364.

Correct Answer: A — 364

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

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

Solution

The sum of the roots can be found using -b/a. Here, b = 6 and a = 1, so the sum is -6.

Correct Answer: A — -6

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Q. What is the sum of the roots of the equation x^2 - 8x + 15 = 0?

A. 8
B. 15
C. 7
D. 5

Solution

The sum of the roots can be found using -b/a. Here, b = -8 and a = 1. Thus, the sum is 8.

Correct Answer: A — 8

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Q. What is the sum of the roots of the polynomial x^2 + 6x + 8?

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

Solution

The sum of the roots of a quadratic ax^2 + bx + c is given by -b/a. Here, b = 6 and a = 1, so the sum is -6/1 = -6.

Correct Answer: A — -6

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Q. What is the sum of the roots of the polynomial x^2 + 6x + 9?

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

Solution

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

Correct Answer: A — -6

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Algebra (Secondary)

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