Understanding the modulus function is an important part of the Class 11 mathematics curriculum. Also known as the absolute value function, the modulus function deals with non-negative outputs regardless of whether the input is positive or negative. Students often find this topic intriguing due to the nature of the function, its graph, and its application in solving equations and inequalities. The function is symbolized asf(x) = |x|, and plays a significant role in both algebra and calculus. In this topic, we explore various types of questions on the modulus function for Class 11 students, including conceptual questions, multiple-choice questions, graphical interpretations, and application-based problems.
Definition and Properties of the Modulus Function
What is the modulus function?
The modulus or absolute value function is defined as follows
|x| = x, ifx ⥠0|x| = -x, ifx < 0
It ensures that the output is always a non-negative value. For example,|5| = 5and|-5| = 5.
Important properties
|x| ⥠0for all real numbers|x| = 0if and only ifx = 0|a à b| = |a| à |b||a + b| ⤠|a| + |b|(Triangle inequality)
Basic Conceptual Questions
1. Evaluate the following
|3| = 3|-7| = 7|0| = 0
2. Solve the equation|x| = 8.
The solution is
x = 8orx = -8
3. Solve|x - 3| = 5
Break the modulus into two cases
x - 3 = 5 â x = 8x - 3 = -5 â x = -2
Graphical Questions on Modulus Function
What does the graph off(x) = |x|look like?
The graph is V-shaped with the vertex at the origin (0, 0). It is symmetric about the y-axis. Forx ⥠0, the graph is identical toy = x, and forx < 0, it isy = -x.
Transformations on modulus functions
Understand how the graph changes when the modulus function is transformed
f(x) = |x - 2|shifts right by 2 unitsf(x) = |x + 3|shifts left by 3 unitsf(x) = |x| + 4shifts up by 4 unitsf(x) = -|x|reflects across the x-axis
Questions Based on Modulus Inequalities
1. Solve|x| < 5
This inequality is true when
-5 < x < 5
2. Solve|x + 2| ⥠7
Two cases
x + 2 ⥠7 â x ⥠5x + 2 ⤠-7 â x ⤠-9
Therefore,x â (-â, -9] ⪠[5, â)
3. Solve|2x - 3| ⤠1
Break into compound inequality
-1 ⤠2x - 3 ⤠1
Add 3 throughout
2 ⤠2x ⤠4â Divide by 2 â1 ⤠x ⤠2
Application-Based Questions
1. A point lies at a distance of less than 4 units from origin. Represent this using modulus.
|x| < 4
2. The distance between two points x and 5 is less than 2. Write the inequality.
|x - 5| < 2
3. A ptopic moves along a line such that its distance from the origin is more than 3 units. Represent using modulus.
|x| > 3
True or False Questions
|x| is always non-negative. â True|x| = xfor all values of x. â False (not valid when x is negative)|a à b| = |a| + |b|â False (should be|a| à |b|)|x - 3| = 3 - xfor x ⤠3 â True
Multiple Choice Questions (MCQs)
1. What is the value of|-8| + |3 - 7|?
- (a) 4
- (b) 8
- (c) 12
- (d) 9
Answer (d) 9
2. The solution of|x - 1| < 2is
- (a)
x < -1 - (b)
-1 < x < 3 - (c)
x > 3 - (d)
-3 < x < 1
Answer (b)
Short Answer Questions
1. Define modulus function and draw its graph.
The modulus function is defined asf(x) = |x|. Its graph is a V-shaped figure with vertex at the origin.
2. Evaluate|x^2 - 9|when x = 2.
x^2 - 9 = 4 - 9 = -5â|x^2 - 9| = 5
3. Find the value of x if|x + 1| = 0
Since modulus is zero only when the expression inside is zero,x + 1 = 0 â x = -1
The modulus function is a foundational topic for students in Class 11, forming a bridge between algebra and graphical interpretation. Understanding the behavior of absolute values, how they influence equations and inequalities, and how to represent them graphically are key skills. Through a variety of practice questions and clear explanations, students can develop confidence in solving problems related to the modulus function. These questions help build conceptual clarity and prepare learners for more advanced topics in calculus and real analysis. Practicing questions on modulus function regularly will greatly enhance mathematical problem-solving skills and analytical thinking.