Vikas Mathematics Practical Book 9th Class Answers

Mastering Mathematics: A Practical Guide to Vikas Mathematics Practical Book for 9th Class

The journey through mathematics in 9th class can be significantly enhanced by engaging with practical exercises. Vikas Mathematics Practical Book is a valuable resource that provides students with hands-on experience in applying mathematical concepts. This guide aims to provide comprehensive answers and explanations to the practical problems in the book, helping students develop a deeper understanding and appreciation for mathematics.

Introduction to Vikas Mathematics Practical Book

Vikas Mathematics Practical Book for 9th class is designed to supplement theoretical knowledge with practical applications. It includes a variety of activities, experiments, and projects that cover key topics in the 9th-grade mathematics curriculum. By working through these practical exercises, students can:

Reinforce their understanding of mathematical concepts.
Develop problem-solving skills.
Enhance their ability to apply mathematics in real-world scenarios.
Foster a greater interest in mathematics.

Chapter-wise Solutions and Explanations

Here is a detailed breakdown of the solutions and explanations for the practical exercises in each chapter of the Vikas Mathematics Practical Book for 9th class.

Chapter 1: Number Systems

Practical 1: Representation of Real Numbers on the Number Line

Objective: To represent real numbers (e.g., √2, √3, √5) on the number line.
Materials Required: Ruler, pencil, compass.
Procedure:
1. Draw a number line and mark integers at equal intervals.
2. To represent √2, construct a right-angled triangle with base and height equal to 1 unit. The hypotenuse will be √2 units long.
3. Use a compass to transfer the length of the hypotenuse onto the number line, starting from 0. The point where the arc intersects the number line represents √2.
4. Repeat the process for other real numbers like √3 and √5, using appropriate right-angled triangles.
Explanation: This practical helps students visualize real numbers and understand their position on the number line. It reinforces the concept of irrational numbers and their geometrical representation.

Practical 2: Verification of Laws of Exponents

Objective: To verify the laws of exponents using numerical examples.
Materials Required: Calculator.
Procedure:
1. Choose numerical values for variables a, b, m, and n.
2. Verify the following laws:
 - am * an = am+n
 - (am)n = amn
 - am / an = am-n (where a ≠ 0)
 - (ab)n = an * bn
 - (a/b)n = an / bn (where b ≠ 0)
3. Substitute the chosen values into each law and perform the calculations to verify that both sides of the equation are equal.
Explanation: This practical provides a hands-on approach to understanding and verifying the fundamental laws of exponents, which are essential for simplifying algebraic expressions.

Chapter 2: Polynomials

Practical 1: Factorization of Polynomials

Objective: To factorize polynomials using algebraic identities.
Materials Required: Paper, pencil.
Procedure:
1. Identify polynomials that can be factored using identities like:
 - a2 - b2 = (a + b)(a - b)
 - (a + b)2 = a2 + 2ab + b2
 - (a - b)2 = a2 - 2ab + b2
 - (a + b)3 = a3 + 3a2b + 3ab2 + b3
 - (a - b)3 = a3 - 3a2b + 3ab2 - b3
2. Apply the appropriate identity to factorize the polynomial.
3. Verify the factorization by expanding the factors to ensure they are equal to the original polynomial.
Explanation: This practical enhances students' ability to recognize and apply algebraic identities to factorize polynomials, a crucial skill for solving algebraic equations.

Practical 2: Verification of the Remainder Theorem

Objective: To verify the Remainder Theorem using numerical examples.
Materials Required: Paper, pencil.
Procedure:
1. Choose a polynomial P(x) and a linear polynomial (x - a).
2. Divide P(x) by (x - a) and find the remainder R.
3. Evaluate P(a) by substituting a into the polynomial P(x).
4. Verify that P(a) = R.
Explanation: This practical helps students understand the relationship between the remainder of a polynomial division and the value of the polynomial at a specific point, as stated by the Remainder Theorem.

Chapter 3: Coordinate Geometry

Practical 1: Plotting Points on the Cartesian Plane

Objective: To plot points with given coordinates on the Cartesian plane.
Materials Required: Graph paper, ruler, pencil.
Procedure:
1. Draw the x-axis and y-axis on the graph paper, intersecting at the origin (0, 0).
2. Mark the points with given coordinates (x, y) on the plane.
3. Label each point with its coordinates.
Explanation: This practical provides hands-on experience in plotting points on the Cartesian plane, which is fundamental for understanding coordinate geometry and graphing functions.

Practical 2: Distance Formula Application

Objective: To calculate the distance between two points using the distance formula.
Materials Required: Graph paper, ruler, pencil.
Procedure:
1. Plot two points (x1, y1) and (x2, y2) on the Cartesian plane.
2. Apply the distance formula: d = √((x2 - x1)2 + (y2 - y1)2).
3. Calculate the distance d between the two points.
Explanation: This practical reinforces the application of the distance formula, enabling students to calculate the distance between any two points in a coordinate plane.

Chapter 4: Linear Equations in Two Variables

Practical 1: Graphing Linear Equations

Objective: To graph linear equations in two variables.
Materials Required: Graph paper, ruler, pencil.
Procedure:
1. Rewrite the linear equation in the form y = mx + c, where m is the slope and c is the y-intercept.
2. Find at least two points that satisfy the equation by substituting different values for x and solving for y.
3. Plot these points on the graph paper and draw a straight line through them.
Explanation: This practical helps students visualize linear equations and understand the relationship between the equation and its graph.

Practical 2: Solving Linear Equations Graphically

Objective: To solve a system of linear equations graphically.
Materials Required: Graph paper, ruler, pencil.
Procedure:
1. Graph each linear equation on the same Cartesian plane.
2. Identify the point of intersection of the two lines.
3. The coordinates of the point of intersection represent the solution to the system of equations.
Explanation: This practical provides a visual method for solving systems of linear equations, reinforcing the concept that the solution is the point where the lines intersect.

Chapter 5: Introduction to Euclid’s Geometry

Practical 1: Verification of Euclid’s Postulates

Objective: To verify Euclid’s postulates through practical demonstrations.
Materials Required: Ruler, pencil, compass.
Procedure:
1. Postulate 1: A straight line may be drawn from any one point to any other point.
  - Take two points on a paper and draw a straight line connecting them using a ruler.
2. Postulate 2: A terminated line can be produced indefinitely.
  - Draw a line segment and extend it on both sides using a ruler.
3. Postulate 3: A circle can be drawn with any center and any radius.
  - Use a compass to draw a circle with any chosen center and radius.
4. Postulate 4: All right angles are equal to one another.
  - Draw two right angles using a protractor and compare them.
5. Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
  - Draw two lines and a transversal such that the sum of the interior angles on one side is less than 180 degrees. Extend the lines to observe that they intersect on that side.
Explanation: This practical allows students to visualize and understand Euclid's postulates, which form the foundation of Euclidean geometry.

Practical 2: Angle Sum Property of a Triangle

Objective: To verify that the sum of the angles of a triangle is 180 degrees.
Materials Required: Paper, ruler, pencil, protractor.
Procedure:
1. Draw a triangle on a piece of paper.
2. Measure the three angles of the triangle using a protractor.
3. Add the measures of the three angles.
4. Verify that the sum is approximately equal to 180 degrees.
Explanation: This practical provides empirical evidence for the angle sum property of a triangle, reinforcing a fundamental concept in geometry.

Chapter 6: Lines and Angles

Practical 1: Verification of Properties of Parallel Lines

Objective: To verify the properties of parallel lines and transversals.
Materials Required: Paper, ruler, pencil, protractor.
Procedure:
1. Draw two parallel lines and a transversal.
2. Measure the corresponding angles, alternate interior angles, and co-interior angles.
3. Verify that:
  - Corresponding angles are equal.
  - Alternate interior angles are equal.
  - Co-interior angles are supplementary (add up to 180 degrees).
Explanation: This practical helps students understand the relationships between angles formed by parallel lines and a transversal.

Practical 2: Angle Bisector Theorem

Objective: To construct and verify the angle bisector theorem.
Materials Required: Paper, ruler, pencil, compass.
Procedure:
1. Draw an angle and construct its bisector using a compass.
2. Choose a point on the angle bisector and draw perpendiculars to the arms of the angle.
3. Measure the lengths of the perpendiculars.
4. Verify that the lengths of the perpendiculars are equal.
Explanation: This practical demonstrates the angle bisector theorem, which states that any point on the angle bisector is equidistant from the arms of the angle.

Chapter 7: Triangles

Practical 1: Congruency of Triangles

Objective: To verify the conditions for congruency of triangles (SAS, ASA, SSS, RHS).
Materials Required: Paper, ruler, pencil, protractor, compass.
Procedure:
1. Draw two triangles that satisfy one of the congruency conditions:
  - SAS (Side-Angle-Side): Two sides and the included angle are equal.
  - ASA (Angle-Side-Angle): Two angles and the included side are equal.
  - SSS (Side-Side-Side): All three sides are equal.
  - RHS (Right-Hypotenuse-Side): The hypotenuse and one side of two right-angled triangles are equal.
2. Measure the corresponding sides and angles of the triangles.
3. Verify that the triangles are congruent by confirming that the congruency condition is satisfied.
Explanation: This practical reinforces the understanding of the conditions required for triangles to be congruent.

Practical 2: Properties of Isosceles Triangles

Objective: To verify that the angles opposite to equal sides of an isosceles triangle are equal.
Materials Required: Paper, ruler, pencil, protractor.
Procedure:
1. Draw an isosceles triangle with two equal sides.
2. Measure the angles opposite to the equal sides using a protractor.
3. Verify that the measures of these angles are equal.
Explanation: This practical demonstrates a key property of isosceles triangles, reinforcing the relationship between sides and angles.

Chapter 8: Quadrilaterals

Practical 1: Angle Sum Property of a Quadrilateral

Objective: To verify that the sum of the angles of a quadrilateral is 360 degrees.
Materials Required: Paper, ruler, pencil, protractor.
Procedure:
1. Draw a quadrilateral on a piece of paper.
2. Measure the four angles of the quadrilateral using a protractor.
3. Add the measures of the four angles.
4. Verify that the sum is approximately equal to 360 degrees.
Explanation: This practical provides empirical evidence for the angle sum property of a quadrilateral.

Practical 2: Properties of Parallelograms

Objective: To verify the properties of parallelograms.
Materials Required: Paper, ruler, pencil.
Procedure:
1. Draw a parallelogram.
2. Measure the opposite sides and opposite angles.
3. Verify that:
  - Opposite sides are equal.
  - Opposite angles are equal.
  - Diagonals bisect each other.
Explanation: This practical helps students understand and verify the key properties of parallelograms.

Chapter 9: Areas of Parallelograms and Triangles

Practical 1: Area of a Parallelogram

Objective: To find the area of a parallelogram.
Materials Required: Paper, ruler, pencil.
Procedure:
1. Draw a parallelogram.
2. Measure the base and the height of the parallelogram.
3. Calculate the area using the formula: Area = base * height.
Explanation: This practical allows students to calculate the area of a parallelogram using the formula.

Practical 2: Area of a Triangle

Objective: To find the area of a triangle using Heron’s formula.
Materials Required: Paper, ruler, pencil.
Procedure:
1. Draw a triangle.
2. Measure the lengths of the three sides (a, b, c).
3. Calculate the semi-perimeter (s) using the formula: s = (a + b + c) / 2.
4. Calculate the area using Heron’s formula: Area = √(s(s - a)(s - b)(s - c)).
Explanation: This practical enables students to apply Heron’s formula to find the area of a triangle when the lengths of all three sides are known.

Chapter 10: Circles

Practical 1: Relationship Between Angle Subtended by an Arc at the Center and at the Circumference

Objective: To verify that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
Materials Required: Paper, ruler, pencil, protractor, compass.
Procedure:
1. Draw a circle with center O.
2. Choose an arc AB on the circle.
3. Choose a point C on the remaining part of the circle.
4. Measure angle AOB (angle at the center) and angle ACB (angle at the circumference).
5. Verify that angle AOB = 2 * angle ACB.
Explanation: This practical demonstrates an important theorem related to circles, reinforcing the relationship between angles subtended at the center and circumference.

Practical 2: Cyclic Quadrilaterals

Objective: To verify that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
Materials Required: Paper, ruler, pencil, protractor, compass.
Procedure:
1. Draw a circle.
2. Draw a quadrilateral inside the circle such that all its vertices lie on the circle (cyclic quadrilateral).
3. Measure the opposite angles of the quadrilateral.
4. Verify that the sum of each pair of opposite angles is 180 degrees.
Explanation: This practical helps students understand and verify the properties of cyclic quadrilaterals.

Chapter 11: Constructions

Practical 1: Construction of Angle Bisectors

Objective: To construct the bisector of a given angle.
Materials Required: Paper, ruler, pencil, compass.
Procedure:
1. Draw an angle.
2. With the vertex as the center, draw an arc intersecting the arms of the angle at two points.
3. With these points as centers and a radius greater than half the distance between them, draw arcs intersecting each other.
4. Draw a line from the vertex to the point of intersection of the arcs. This line is the angle bisector.
Explanation: This practical provides hands-on experience in constructing angle bisectors.

Practical 2: Construction of Perpendicular Bisectors

Objective: To construct the perpendicular bisector of a given line segment.
Materials Required: Paper, ruler, pencil, compass.
Procedure:
1. Draw a line segment.
2. With each endpoint as the center and a radius greater than half the length of the line segment, draw arcs intersecting each other on both sides of the line segment.
3. Draw a line through the points of intersection of the arcs. This line is the perpendicular bisector.
Explanation: This practical allows students to construct perpendicular bisectors of line segments.

Chapter 12: Heron’s Formula

Practical 1: Application of Heron’s Formula

Objective: To calculate the area of different triangles using Heron’s formula.
Materials Required: Paper, ruler, pencil.
Procedure:
1. Draw various triangles with different side lengths.
2. Measure the lengths of the three sides (a, b, c) of each triangle.
3. Calculate the semi-perimeter (s) using the formula: s = (a + b + c) / 2.
4. Calculate the area using Heron’s formula: Area = √(s(s - a)(s - b)(s - c)).
Explanation: This practical reinforces the application of Heron’s formula in finding the areas of different triangles.

Chapter 13: Surface Areas and Volumes

Practical 1: Surface Area of a Cube and Cuboid

Objective: To find the surface area of a cube and a cuboid.
Materials Required: Paper, ruler, pencil.
Procedure:
1. Cube:
 - Measure the length of one side (a).
 - Calculate the surface area using the formula: Surface Area = 6a2.
2. Cuboid:
 - Measure the length (l), width (w), and height (h).
 - Calculate the surface area using the formula: Surface Area = 2(lw + wh + lh).
Explanation: This practical provides hands-on experience in calculating the surface areas of cubes and cuboids.

Practical 2: Volume of a Cylinder

Objective: To find the volume of a cylinder.
Materials Required: Paper, ruler, pencil, cylindrical object.
Procedure:
1. Measure the radius (r) of the base of the cylinder and the height (h).
2. Calculate the volume using the formula: Volume = πr2h.
Explanation: This practical allows students to calculate the volume of a cylinder using the formula.

Chapter 14: Statistics

Practical 1: Collection and Representation of Data

Objective: To collect data and represent it graphically.
Materials Required: Paper, pencil, ruler, graph paper.
Procedure:
1. Collect data on a topic of interest (e.g., heights of students in a class).
2. Organize the data in a frequency table.
3. Represent the data using:
  - Bar graph
  - Histogram
  - Frequency polygon
Explanation: This practical provides experience in collecting, organizing, and representing data graphically.

Practical 2: Calculation of Mean, Median, and Mode

Objective: To calculate the mean, median, and mode for a given set of data.
Materials Required: Paper, pencil, calculator.
Procedure:
1. Collect a set of data.
2. Mean: Calculate the sum of the data values and divide by the number of values.
3. Median: Arrange the data in ascending order and find the middle value. If there are an even number of values, the median is the average of the two middle values.
4. Mode: Identify the value that occurs most frequently in the data set.
Explanation: This practical reinforces the understanding of measures of central tendency (mean, median, and mode).

Chapter 15: Probability

Practical 1: Experimental Probability

Objective: To determine the experimental probability of an event.
Materials Required: Coin, die, paper, pencil.
Procedure:
1. Coin Tossing:
  - Toss a coin a large number of times (e.g., 100 times).
  - Record the number of heads and tails.
  - Calculate the experimental probability of getting a head (number of heads / total number of tosses).
2. Die Rolling:
  - Roll a die a large number of times (e.g., 100 times).
  - Record the number of times each face (1 to 6) appears.
  - Calculate the experimental probability of getting each face (number of times the face appears / total number of rolls).
Explanation: This practical helps students understand the concept of experimental probability through hands-on experimentation.

Tips for Success in Mathematics Practical Exams

Understand the Concepts: Before attempting the practical exercises, ensure you have a solid understanding of the underlying mathematical concepts.
Follow the Procedure: Carefully follow the procedure outlined in the practical book or provided by your teacher.
Neat and Accurate Diagrams: Draw neat and accurate diagrams using the appropriate tools (ruler, pencil, compass, protractor).
Show Calculations: Clearly show all calculations and steps involved in solving the problems.
Label Everything: Label all diagrams, graphs, and calculations appropriately.
Practice Regularly: Practice the practical exercises regularly to improve your skills and confidence.
Seek Help: Don't hesitate to ask for help from your teacher or classmates if you encounter any difficulties.
Review and Revise: Review and revise all the practical exercises before the exam to ensure you are well-prepared.

Conclusion

Vikas Mathematics Practical Book for 9th class provides a valuable opportunity for students to reinforce their understanding of mathematical concepts through hands-on activities. By working through the practical exercises, students can develop problem-solving skills, enhance their ability to apply mathematics in real-world scenarios, and foster a greater interest in the subject. This guide provides comprehensive answers and explanations to help students master the practical problems in the book and achieve success in their mathematics studies.