PDHMA 9th Class Mathematics Model Paper 2019
Solved Mathematics Model Question Paper with Step-by-Step Solutions, MCQs, Chapter-Wise Analysis, Topic Analysis and Difficulty Level
PDHMA 9th Class Mathematics 2019 Model Paper
Welcome to the PDHMA 9th Class Mathematics Model Paper 2019. This practice paper has been designed in a realistic school-examination style for Class 9 Mathematics revision.
The paper contains mathematical questions followed immediately by their solutions. After each solution, a detailed analysis explains the topic, chapter area, concept, skill and difficulty level.
The final part contains additional MCQs, topic-wise analysis, difficulty analysis and examination preparation guidance.
This is a newly created 2019-style practice/model paper. It is not presented as a verbatim reproduction of an original archived PDHMA examination paper.
Mathematics Model Paper Overview
| Category | Details |
|---|---|
| Class | 9th Class |
| Subject | Mathematics |
| Model Year | 2019 Style |
| Paper Type | Full-Length Practice Model Paper |
| Question Style | Numerical, Algebraic, Geometry and Application-Based |
| Solutions | Step-by-Step Solutions |
| Analysis | Topic, Chapter, Concept, Skill and Difficulty |
How to Attempt the Mathematics Model Paper
- Read the complete question before starting the calculation.
- Identify the mathematical concept required to solve the problem.
- Write the known information clearly.
- Select the correct formula, theorem or method.
- Show the important calculation steps instead of writing only the final answer.
- Check signs, units, arithmetic and simplification before moving forward.
- After completing the paper, compare your work with the solutions and study the question analysis.
SECTION A – MATHEMATICS QUESTIONS WITH SOLUTIONS
Every question is followed by its solution and detailed analysis.
3x + 7 – 2x + 5
Group the like terms:
3x – 2x = x
7 + 5 = 12
Therefore,
3x + 7 – 2x + 5 = x + 12
Skill Tested: Simplification of algebraic expressions. The main requirement is recognising like terms and combining their coefficients correctly.
2x + 5 = 17
Given:
2x + 5 = 17
Subtract 5 from both sides:
2x = 17 – 5
2x = 12
Divide both sides by 2:
x = 6
Therefore, the value of x is 6.
Skill Tested: Solving a one-variable linear equation through basic inverse operations.
(3² + 4²)
3² = 9
4² = 16
Therefore:
9 + 16 = 25
Skill Tested: Correct evaluation of powers and basic numerical calculation.
x² + 5x + 6
We need two numbers whose product is 6 and whose sum is 5.
The numbers are 2 and 3.
Therefore:
x² + 5x + 6
= x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
Skill Tested: Recognising suitable factor pairs and splitting the middle term correctly.
a² + 2ab + b²
Substitute a = 3 and b = 5.
a² + 2ab + b²
= 3² + 2(3)(5) + 5²
= 9 + 30 + 25
= 64
Alternatively, using the identity:
a² + 2ab + b² = (a + b)²
= (3 + 5)² = 8² = 64.
Skill Tested: Substitution and application of a standard algebraic identity.
Formula:
Perimeter = 2(l + b)
Here:
l = 12 cm
b = 7 cm
Therefore:
Perimeter = 2(12 + 7)
= 2 × 19
= 38 cm
Skill Tested: Selecting and applying the correct perimeter formula with given measurements.
Formula:
Area of triangle = ½ × base × height
= ½ × 10 × 8
= 5 × 8
= 40 cm²
Skill Tested: Formula recall and numerical substitution.
The sum of the angles of a triangle is 180°.
Therefore:
50° + 60° + x° = 180°
110° + x° = 180°
x° = 180° – 110°
x = 70°
Skill Tested: Application of the angle-sum property of a triangle.
When a transversal intersects two parallel lines, corresponding angles are equal.
Therefore, if one corresponding angle is 65°, the other corresponding angle is also:
65°
Skill Tested: Recognition and application of angle relationships formed by parallel lines.
Formula:
Mean = Sum of observations ÷ Number of observations
Sum:
6 + 8 + 10 + 12 + 14 = 50
There are 5 observations.
Mean = 50 ÷ 5
Mean = 10
Skill Tested: Calculating the arithmetic mean from a small data set.
Let the original number be x.
An increase of 20% means the new value is:
x + 20% of x
= x + 0.20x
= 1.20x
Therefore:
1.20x = 240
x = 240 ÷ 1.20
x = 200
Therefore, the original number was 200.
Skill Tested: Reverse calculation using percentage increase.
2/3 + 5/6
The LCM of 3 and 6 is 6.
Convert 2/3 into a denominator of 6:
2/3 = 4/6
Therefore:
4/6 + 5/6 = 9/6
= 3/2
Answer = 3/2
Skill Tested: Addition of rational numbers with unlike denominators.
Formula:
Circumference = 2πr
Here r = 7 cm.
C = 2 × 22/7 × 7
C = 2 × 22
C = 44 cm
Skill Tested: Applying the circumference formula and simplifying the numerical expression correctly.
Cost Price = ₹500
Selling Price = ₹575
Profit = SP – CP
= 575 – 500
= ₹75
Profit percentage:
= (Profit ÷ CP) × 100
= (75 ÷ 500) × 100
= 15%
Profit = 15%
Skill Tested: Identifying cost price, selling price and profit before calculating the percentage.
Given:
x + y = 10 …….. (1)
x – y = 4 …….. (2)
Add equations (1) and (2):
2x = 14
x = 7
Substitute x = 7 into equation (1):
7 + y = 10
y = 3
Therefore:
x = 7 and y = 3
Skill Tested: Solving simultaneous linear equations using elimination and substitution.
Consider rectangle ABCD.
In triangles ABC and BAD:
AB is common.
BC = AD because opposite sides of a rectangle are equal.
∠ABC = ∠BAD = 90°.
Therefore, by SAS congruence:
△ABC ≅ △BAD
Hence, corresponding sides are equal:
AC = BD
Therefore, the diagonals of a rectangle are equal.
Skill Tested: Constructing a logical geometric proof using properties of rectangles and triangle congruence.
SECTION B – MATHEMATICS MCQs
Objective questions for quick revision and concept testing.
What is the value of 5²?
Topic: Powers | Chapter: Number System | Difficulty: Easy
Which of the following is a factor of 24?
Topic: Factors | Chapter: Number System | Difficulty: Easy
If 3x = 21, what is x?
Topic: Linear Equation | Chapter: Algebra | Difficulty: Easy
What is the coefficient of x in 7x + 4?
Topic: Algebraic Expressions | Chapter: Algebra | Difficulty: Easy
What is the sum of the angles of a triangle?
Topic: Triangle | Chapter: Geometry | Difficulty: Easy
What is the area of a rectangle with length 8 cm and breadth 5 cm?
Topic: Area | Chapter: Mensuration | Difficulty: Easy
What is the mean of 4, 6 and 8?
Topic: Statistics | Chapter: Data Handling | Difficulty: Easy
If the cost price is ₹200 and selling price is ₹220, the profit is:
Topic: Profit & Loss | Chapter: Commercial Arithmetic | Difficulty: Easy
Which expression represents the perimeter of a square with side a?
Topic: Perimeter | Chapter: Mensuration | Difficulty: Easy
If two angles are complementary and one is 35°, the other angle is:
Topic: Angles | Chapter: Geometry | Difficulty: Moderate
Which of the following is a rational number?
Topic: Rational Numbers | Chapter: Number System | Difficulty: Easy
What is the factorisation of x² – 9?
Topic: Factorisation | Chapter: Algebra | Difficulty: Moderate
Complete Paper Analysis
The model paper covers several major areas of Class 9 Mathematics. The questions are deliberately varied so that students practise direct numerical calculation, algebraic manipulation, geometry, mensuration, statistics and commercial arithmetic.
Detailed Questions
MCQs
Major Mathematics Areas
Question-Wise Difficulty Analysis
| Question | Main Topic | Difficulty | Reason |
|---|---|---|---|
| 1 | Algebraic Expressions | Easy | Simple collection of like terms. |
| 2 | Linear Equation | Easy | Requires basic inverse operations. |
| 3 | Powers | Easy | Direct square calculation. |
| 4 | Factorisation | Moderate | Requires suitable factor pairs and middle-term splitting. |
| 5 | Algebraic Identity | Moderate | Requires substitution and identity recognition. |
| 6 | Perimeter | Easy | Direct application of rectangle formula. |
| 7 | Triangle Area | Easy | Direct formula-based question. |
| 8 | Triangle Angles | Easy | Uses the 180° angle-sum property. |
| 9 | Parallel Lines | Moderate | Requires recognition of corresponding angles. |
| 10 | Statistics | Easy | Simple mean calculation. |
| 11 | Percentage | Moderate | Requires reverse percentage calculation. |
| 12 | Rational Numbers | Easy | Requires LCM and fraction addition. |
| 13 | Circle | Easy | Direct circumference formula. |
| 14 | Profit & Loss | Moderate | Requires profit and percentage calculation. |
| 15 | Linear Equations | Moderate–Difficult | Requires simultaneous equation method. |
| 16 | Geometry Proof | Difficult | Requires logical proof and congruence reasoning. |
Chapter-Wise Analysis
🔴 Algebra
Algebra receives significant attention through simplification, linear equations, identities and factorisation. Students should practise manipulating expressions carefully and checking signs after every step.
🟡 Number System
The paper includes powers and rational-number operations. Students should be comfortable with fractions, signs, powers and basic numerical properties.
🔴 Geometry
Triangle properties, parallel lines and quadrilateral reasoning require students to remember theorems and understand how geometric relationships work.
🟡 Mensuration
Perimeter, area and circumference questions test formula recall and correct substitution of measurements.
🔴 Commercial Arithmetic
Percentage and profit-and-loss problems require students to translate a word problem into mathematical operations.
🟡 Statistics
Mean-based questions test the ability to add observations and divide by the number of observations.
Difficulty Distribution
| Level | Approximate Questions | What It Tests |
|---|---|---|
| Easy | 8 | Formula recall, direct calculation and basic concepts. |
| Moderate | 6 | Application, algebraic manipulation and multi-step calculation. |
| Difficult | 2 | Logical reasoning and geometry proof. |
The model paper is designed at an overall Easy to Moderate level, with a small number of questions requiring deeper reasoning.
Which Questions Need More Practice?
Students who are comfortable with direct formulas should still pay attention to questions that require multiple steps. In this model paper, factorisation, percentage increase, simultaneous equations and geometry proof demand more careful reasoning.
These questions should not be memorised. Instead, students should practise similar examples using different numbers.
For example, after solving the simultaneous equations question, create another pair of equations and solve them using the same elimination method.
Similarly, after learning profit percentage, practise examples containing different cost prices and selling prices.
How to Improve Mathematics Score
- Revise formulas regularly rather than trying to memorise all formulas on the last day.
- Solve examples without looking at the solution.
- Write every important step in multi-step numerical problems.
- Check negative signs carefully in algebraic calculations.
- Practise geometry diagrams and theorem-based reasoning.
- Keep a separate list of mistakes made during practice tests.
- Repeat difficult question types using new numerical values.
- Take timed mock tests after completing chapter revision.
Common Mistakes in Class 9 Mathematics
Wrong Formula Selection
Students sometimes know several formulas but select the wrong one for the given problem. Read the question carefully before writing the formula.
Sign Errors
Negative signs are particularly important in algebra. Recheck every step after expanding or transposing terms.
Skipping Steps
Writing only a final answer makes it difficult to identify where an error occurred. Show important calculations.
Incorrect Units
Area, perimeter and circumference have different units. Always check whether the answer should be in cm, cm² or another appropriate unit.
Calculation Errors
Even when the method is correct, arithmetic mistakes can change the final answer. Review calculations before moving forward.
Weak Geometry Reasoning
Geometry questions require understanding relationships between angles, sides and figures. Memorising theorem names without understanding them can cause difficulty.
Seven-Day Mathematics Revision Plan
Day 1 – Number System
Revise rational numbers, powers and basic numerical operations.
Day 2 – Algebra
Practise expressions, identities, factorisation and linear equations.
Day 3 – Geometry
Revise angle properties, triangles, parallel lines and quadrilateral concepts.
Day 4 – Mensuration
Practise perimeter, area, circle and related formula-based problems.
Day 5 – Statistics and Arithmetic
Revise mean, percentages, profit and loss and application-based problems.
Day 6 – MCQ Practice
Attempt objective questions without looking at the answers.
Day 7 – Full Model Paper
Attempt a complete paper under examination-style conditions and analyse every mistake afterward.
How to Analyse Your Mathematics Test
Do not judge your preparation only by the final score. A detailed analysis can tell you exactly where marks were lost.
Type 1 – Concept Error
You did not understand the mathematical rule or theorem. Return to the chapter and study the concept again.
Type 2 – Formula Error
You remembered the wrong formula or substituted values incorrectly. Revise the formula and practise several examples.
Type 3 – Calculation Error
The method was correct but arithmetic went wrong. Practise slower, clearer calculations.
Type 4 – Careless Error
You knew the concept but misread the question or missed a sign or unit. Develop a final checking habit.
Type 5 – Time Management Error
You understood the questions but could not complete them within the available time. Use timed practice papers.
Frequently Asked Questions
What is this PDHMA 9th Class Mathematics Model Paper?
It is a newly prepared 2019-style Class 9 Mathematics practice paper created for revision and examination preparation.
Is this the exact original PDHMA 2019 paper?
No. It is a model/practice paper and is not presented as a verbatim copy of an archived original paper.
Does every question have a solution?
Yes. The main numerical and mathematical questions are followed immediately by step-by-step solutions.
What does the question analysis include?
The analysis includes topic, chapter area, concept, skill tested and difficulty level.
Which questions are considered difficult?
Multi-step algebra and geometry proof questions generally require more reasoning than direct formula-based questions.
How should I use the solutions?
First solve the question independently. Then compare your method with the solution and identify the exact step where your approach differed.
Are the MCQs useful for revision?
Yes. They are useful for quickly checking formulas, concepts and basic mathematical properties.
How can I improve my Mathematics marks?
Practise regularly, understand formulas, solve multi-step problems, analyse mistakes and take timed model tests.
Why is showing calculation important?
Writing the main steps makes the mathematical method clear and helps you identify mistakes during checking.
Should I memorise solved questions?
No. Understand the method and practise new questions using the same concept.
Final Analysis of the Model Paper
This Class 9 Mathematics model paper provides practice across several important mathematical skills. The first group of questions focuses on basic algebra and numerical concepts, while later questions gradually introduce more application and reasoning.
The geometry section tests properties of angles, parallel lines and quadrilaterals. Mensuration questions focus on formula application, while percentage and profit-and-loss problems connect mathematics with practical situations.
The final geometry proof is intentionally more demanding because students must organise a logical argument rather than simply substitute numbers into a formula.
The MCQ section provides a faster way to revise basic concepts. Students can use it after completing the main paper to check whether important formulas and properties have been remembered correctly.
Learn Concept → Solve Example → Attempt Without Help → Check Solution → Analyse Error → Practise Similar Question → Take Mock Test