Paper or online is better for memory 🤔

🛡 Paper-Based: Better for Memory

Writing by hand improves memory – Studies show handwriting helps your brain process and remember information more deeply.

You can annotate – Underline, circle, or make notes on the question paper, which helps focus.

More time to think – No rush from clicking fast; you engage more slowly and deeply.


⚠️ Online: Sometimes Less Effective for Memory

Typing doesn't strengthen memory as much as writing by hand.

Distractions are easier – It’s tempting to rush or click randomly.

Less interaction with content – You can’t mark the screen easily like you can on paper.

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Matrices

☛ A matrix is a rectangular array of numbers arranged in rows and columns.

☛ A matrix is usually denoted by capital letters (e.g., A, B, C).

☛ Each number in a matrix is called an element or entry.


⊙Notation
☛ If a matrix A has m rows and n columns, we write it as Aₘₓₙ (m by n matrix).

✔️ Types of Matrices

☑Row matrix: Only one row (e.g., [2 3 5])

☑Column matrix: Only one column

☑Square matrix: Same number of rows and columns

☑Zero (Null) matrix: All elements are 0

☑Diagonal matrix: Only diagonal elements are non-zero

☑Scalar matrix: Diagonal matrix with same diagonal values

☑Identity matrix (Iₙ): Diagonal = 1, rest = 0

☑Symmetric matrix: A = Aᵗ (transpose)

☑Skew-symmetric matrix: A = -Aᵗ


✔ Matrix Operations

a) Addition and Subtraction

☛ Only possible if matrices have same order

☛ Add/subtract corresponding elements


b) Scalar Multiplication

☛ Multiply each element by a scalar (number)


c) Matrix Multiplication

Multiply Aₘₓₙ × Bₙₓₚ results in Cₘₓₚ

Rule: Multiply row by column

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For Grade 12

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Transformation on the plane

☛ A transformation of the plane refers to a rule or function that moves or changes the position, size, or orientation of points in the coordinate plane. These transformations are often studied in geometry and linear algebra.

📝There are three main types of transformations in the plane

1. Translation

✔Definition: Slides every point of a figure the same distance in the same direction.

✔Effect: No change in shape, size, or orientation.

Example

(x,y) → (x + a, y + b)

✅where a and b are constants.

2. Rotation

✔Definition: Turns a figure around a fixed point (called the center of rotation) by a certain angle.

✔Effect: Shape and size stay the same, but orientation changes.

🎱Common Rotations

90° counterclockwise: (x, y) → (-y, x)

180°: (x, y) → (-x,-y)

270° counterclockwise: (x, y) → (у, -x)

3. Reflection

✔️Definition: Flips a figure over a line (like a mirror).

🎱Common Reflections

✅Over the x-axis: (x, y) → (x, -y)

✅Over the y-axis: (x, y) → (-x, y)

✅Over the line y = x: (x,y) (y, x)


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At 3:30💥💥

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Questions 1(English)

Yabsera____ all her classmates at her party but only a few turned up.

A. is expected
B. expected
C. has expected
D. expects

Edit 🚫

Questions 3 (easy )

What is the longest river in the world 🌎

Winner 🏆
1 Habtamu
2 faysal the great
3 the wolf
4 roym Muhammad