IOQM Number Theory Modulus Function Question for Practice

Number theory is one of the most important topics in the Indian Olympiad Qualifier in Mathematics (IOQM). Many questions in the IOQM exam come from number theory, especially those involving the modulus function. Solving an IOQM Number Theory Modulus Function Question requires a strong understanding of absolute value properties, logical reasoning, and systematic case analysis.

Students preparing for IOQM modulus function questions often encounter modulus-based equations where they must break the problem into different cases depending on the sign of the expression. These problems test conceptual clarity and analytical thinking, which are essential skills for Olympiad mathematics.

Here, we will explain the concept of the modulus function, understand how it appears in IOQM number theory problems, and learn the correct approach to solving an IOQM Number Theory Modulus Function Question.

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What is the Modulus Function in Number Theory?

The modulus function, also known as the absolute value function, represents the distance of a number from zero on the number line. Since distance cannot be negative, the modulus of a number is always non-negative.

The modulus function is defined as follows:

|x| = x, if x ≥ 0
|x| = –x, if x < 0

This means that the modulus removes the negative sign from a number.

Examples:

|5| = 5
|-5| = 5

In many IOQM Number Theory Modulus Function Questions, the modulus expression appears inside equations or inequalities. To solve such questions, students must divide the number line into intervals and solve each case separately.

IOQM Number Theory Modulus Function Questions

Practice these IOQM Number Theory modulus function questions to strengthen your concepts on absolute values, inequalities, and equations. These problems will help improve your problem-solving speed and accuracy. Check the questions below and test your understanding effectively.

1. The root(s) of equation √(x+5) + √(x+21) = √(6x+40) is/are given by
   (A) x = 4
   (B) x = -14/3
   (C) Both A & B
   (D) None of these

2. Solution set of √(5x² - 6x + 8) - √(5x² - 6x - 7) = 1 is
   (A) 2, 3
   (B) 4, 3/5
   (C) 4, -14/5
   (D) 9, -16/5

3. Solution set of equation √(x² + 4x - 21) + √(x² - x - 6) = √(6x² - 5x - 39) is
   (A) 2
   (B) 3
   (C) -5/3
   (D) All of these

4. Sum of all the real solutions of the inequality ((x² + 4)(√(x² - 16))) / ((x² + 2)(x² - 9)) ≤ 0 is
   (A) 5
   (B) 4
   (C) 8
   (D) 0

5. Find the domain of the function f(x) = √((x - 2)/(x + 2)) + √((1 - x)/(1 + x))
   (A) (-∞, -2] ∪ [2, ∞)
   (B) [-1, 1]
   (C) ∅
   (D) None of these

6. Solve √(x² - 5x + 6) > √(x - 4)
   (A) x ∈ (-∞, 2] ∪ [3, ∞)
   (B) x ∈ [4, ∞)
   (C) x ∈ [3, 10/3)
   (D) x ∈ [2, 3]

7. Find the number of integral values of x satisfying √(-x² + 10x - 16) < x - 2
   (A) 7
   (B) 5
   (C) 2
   (D) 3

8. Solution set of √(x² - 3x) ≥ 2 is
   (A) x ∈ (-∞, -1] ∪ [4, ∞)
   (B) x ∈ (-∞, 0] ∪ [3, ∞)
   (C) x ∈ (-∞, -1) ∪ [4, ∞)
   (D) None of these

9. Solution set of x - 4 < √(x² + 4x - 12) is
   (A) x ∈ [4, ∞)
   (B) x ∈ (-∞, -6] ∪ [2, ∞)
   (C) x ∈ R
   (D) x ∈ (-8, -3] ∪ [6, ∞) ∪ {4}

10. Solution set of √(5 - x²) ≤ x + 1 is
    (A) x ∈ [1, √5)
    (B) x ∈ [-√5, √5]
    (C) x ∈ [1, √5]
    (D) x ∈ [-1, 1] 

IOQM Number Theory Modulus Function Question PDF

IOQM Number Theory Modulus Function Question is an important topic for students preparing for olympiad exams. These problems test logical thinking and number theory concepts. Practicing such questions helps improve problem-solving skills. Below we have provided the PDF for additional practice and better understanding.

Why Modulus Function Questions Are Important in IOQM?

The IOQM exam focuses on testing logical thinking rather than memorization. Modulus-based questions are useful for testing a student's understanding of mathematical concepts.

An IOQM Number Theory Modulus Function Question often requires students to:

Identify the points where the expression inside the modulus becomes zero
Break the equation into different cases
Solve each case carefully
Check whether the solution satisfies the interval condition

Because of this structured approach, modulus problems are commonly included in olympiad-level mathematics exams.

Example of an IOQM Number Theory Modulus Function Question

Consider the following problem.

Find all integers x satisfying the equation

|x − 3| + |x + 2| = 7

This is a typical IOQM-style modulus function problem.

Step-by-Step Solution Approach

To solve this IOQM Number Theory Modulus Function Question, we first identify where the expressions inside the modulus become zero.

x − 3 = 0 → x = 3
x + 2 = 0 → x = −2

These two values divide the number line into three regions.

x < −2
−2 ≤ x < 3
x ≥ 3

We now solve the equation in each interval.

Case 1: x < −2

In this region, both expressions inside the modulus are negative.

|x − 3| = −(x − 3)
|x + 2| = −(x + 2)

Substituting into the equation:

−(x − 3) − (x + 2) = 7

−2x + 1 = 7

x = −3

Since −3 < −2, this solution is valid.

Case 2: −2 ≤ x < 3

In this region:

|x − 3| = −(x − 3)
|x + 2| = x + 2

Substitute in the equation:

−(x − 3) + (x + 2) = 7

5 = 7

This is not possible. Therefore, there is no solution in this interval.

Case 3: x ≥ 3

In this region:

|x − 3| = x − 3
|x + 2| = x + 2

Substituting into the equation:

(x − 3) + (x + 2) = 7

2x − 1 = 7

x = 4

Since 4 ≥ 3, this solution is valid.

Final Answer

The integer solutions are

x = −3 and x = 4

Important Properties of the Modulus Function

Students solving an IOQM Number Theory Modulus Function Question should remember some key properties of the modulus function.

Non-negative property

|x| ≥ 0

The modulus of any number is always non-negative.

Multiplication property

|ab| = |a| × |b|

This property is useful when simplifying expressions.

Triangle inequality

|a + b| ≤ |a| + |b|

This property is often used in olympiad-level number theory problems.

Common Mistakes Students Make

Students preparing for IOQM often make certain mistakes while solving modulus questions.

Not splitting the equation into cases

Many students try to solve modulus equations directly without considering different cases.

Ignoring interval conditions

Even if a solution is obtained, it must satisfy the interval condition for that case.

Calculation errors

Since modulus problems involve multiple steps, small arithmetic mistakes can lead to incorrect answers.

Avoiding these mistakes can greatly improve accuracy in olympiad exams.

Tips to Solve IOQM Number Theory Modulus Function Questions

Students should follow a structured method while solving modulus problems. Below are the tips that students need to keep in mind - 

• First identify the values where the expression inside the modulus becomes zero.

• Divide the number line into intervals using these values.

• Solve the equation separately for each interval.

• Check whether the obtained solution satisfies the interval condition.

• Practicing similar problems regularly can help students develop confidence in solving IOQM Number Theory Modulus Function Questions.