# Mathematics 9 (Science Group)

Mathematics 9 is written by Dr. Karamat H. Dar and Prof. Irfan-ul-Haq and this book is published by Carvan Book House, Lahore, Pakistan. This book consist of 302 pages and there are 17 units. Notes of Unit 1 and 3 are provided by **Engr. Moin Latif**. We are very thankful to him for providing these notes. The soft form of this book can be downloaded from PITB webiste from HERE

# Definitions

- Definitions by Amir Shehzad | Download PDF

- Definitions by Bahadar Ali Khan | Download PDF

## Unit 01: Matrix

## Unit 02: Real and Complex Numbers

The following MCQs was send by Amir Shehzad. We are very thankful to him for sending these notes.

- Solutions | Download PDF

- MCQs | Download PDF

## Unit 03: Logarithms

These notes are prepared by Engr. Moin Latif. MCQs are made by Mr. Amir Shehzad

- Solutions | Download PDF

- MCQs | Download PDF

## Unit 04: Algebraic Expressions and Algebraic Formulas

In this unit, following topics has been covered:

- Algebraic Expressions
- Algebraic Formulas
- Surds and their Application
- Rationalization

The following notes of this chapter are provided by Mr. Adil Aslam

## Unit 05: Factorization

In this unit, following topics has been covered: After studying this unit , the students will be able to:

- Recall factorization of expressions of the following types.
- $ka + kb + kc$
- $ac + ad + bc + bd$
- $a^2 + 2ab + b^2$
- $a^2 – b^2$
- $a^2 + 2ab + b^2 – c^2$

- Factorize the expressions of the following types.
- Type I: $a^4 + a^2b^2 + b^4$ or $a^4 + 4b^4$
- Type II: $x^2 + px + q$
- Type III: $ax^2 + bx + c$
- Type IV: $(ax^2 + bx + c) (ax2 + bx + d) + k$

$(x + a) (x + b) (x + c) (x + d) + k$

$(x + a) (x + b) (x + c) (x + d) + kx^2$ - Type V: $a^3 + 3a^2b + 3ab^2 + b^3$

$a^3 − 3a^2b + 3ab^2 − b^3$ - Type VI: $a^3 + b^3$

- State and prove remainder theorem and explain through examples.
- Find Remainder (without dividing) when a polynomial is divided by a linear polynomial.
- Define zeros of a polynomial.
- State and prove Factor theorem.
- Use Factor theorem to factorize a cubic polynomial.

The following notes of this chapter are provided by Mr. Adil Aslam

## Unit 06: Algebraic Manipulation

In this unit, following topics has been covered:

- Highest common factor and least common multiples
- Basic operations on algebraic fractions
- Square root of algebraic expression

There are total four exercises in unit 6. Notes of this chapter will be added soon.

## Unit 07: Linear Equations and Inequalities

These notes are send by Mr. **Malik Faisal Rafiq**. We are very thankful to him for sending these notes.

**Linear Equations:** A linear equation is one unknown variable $x$ is an equation of the form $ax+b=0$, where $a,b\in \mathbb{R}$ and $a\neq 0$.

**Radical Equations:** When the variable in an equation occurs under a radical, the equation is called radical equation.

**Equation Involving Absolute Value:** The absolute value of real number $a$ denoted by $|a|$ is defined as
$$
|a|=\left\{\begin{matrix}
a & \text{if } a\geq 0, \\
-a & \text{if } a<0.
\end{matrix}\right.
$$
e.g. $|6|=6$, $|0|=0$, $|-3|=3$.

**Defining inequalities:** Let $a,b$ be real numbers. Then $a$ is greater than $b$ if the difference $a-b>0$ and we denote this order relation by the inequality $a>b$. An equivalent statement is that $b$ is less than $a$, symbolized by $b<a$.

- Exercise 7.1 | View Online | Download PDF

- Exercise 7.2 | View Online | Download PDF

- Exercise 7.3 | View Online | Download PDF

## Unit 08: Linear Graph and their Application

Notes of this unit are available at the following page: https://www.mathcity.org/matric/9th_science/unit08

## Unit 09: Introduction to Coordinate Geometry

After studying this unit, the students will be able to:

- define coordinate geometry.
- derive distance formula to calculate distance between two points given in Cartesian plane.
- use distance formula to find distance between two given points.
- use distance formula to show that given three (or more) points are colinear.
- Use distance formula to show that the given three non-collinear points for
- and equilateral triangle.
- an isosceles triangle.
- a right angled triangle.
- a scalene triangle.

- Use distance formula to show that given four non-collinear points form
- a square,
- a rectangle,
- a parallelogram.

- Recognize the formula to find the midpoint of the line joining two given points
- Apply distance and mid point formulae to solve/verify different standard results related to geometry.

No notes available yet

## Unit 10: Congruent Triangles

After studying this unit, the students will be able to:

- Prove that in any correspondence of two triangles, if one side and any two angles of one triangle are congruent to the corresponding side and angles of the other, then the triangles are congruent.
- Prove that if two angles of a triangle are congruent, then the sides opposite to them are also congruent.
- Prove that in a correspondence of two triangles, if three sides of one triangle are congruent to the corresponding three sides of the other, the two triangles are congruent.
- Prove that if in the correspondence of two right-angled triangles, the hypotenuse and one side of one are congruent to the hypotenuses and the corresponding side of the other, then the triangles are congruent.

- Exercise 10.1 | Download PDF

- Exercise 10.2 | Download PDF

- Exercise 10.3 | Download PDF

- Exercise 10.4 | Download PDF

- Review Exercise 10 | Download PDF

## Unit 12: Line Bisectors and Angle Bisectors

After studying this unit, the students will be able to:

- prove that any point on the right bisector of a line segment is equidistant from its end points.
- prove that any point equidistant from the end points of a line segment is on the right bisector of it.
- prove that the right bisectors of the sides of a triangle are concurrent.

The following theorems was send by Bahadar Ali Khan. We are very thankful to him for sending these notes.

- Important Theorems | Download PDF

# Unit 15: Pythagoras' Theorem

After studying this unit, the students will be able to:

- Pythagoras theorem

The following Solutions & MCQs was send by Amir Shehzad. We are very thankful to him for sending these notes.

- Solutions | Download PDF

- MCQs | Download PDF

# Unit 16: Theorem Related With Area

After studying this unit, the students will be able to:

- prove that parallelograms on the same base and lying between the same parallel lines (or of the same altitude) are equal in area.
- prove that parallelograms on equal bases and having the same altitude are equal in area.
- prove that triangles on the same base and of the same altitude are equal in area.
- prove that triangles on equal bases and of the same altitude are equal in area.

The following MCQs was send by Amir Shehzad. We are very thankful to him for sending these notes.

- MCQs | Download PDF