MoyMath – Worksheets & Question Banks | IB MYP-DP, IGCSE, A-Level, CBSE, ICSE, AP

Mission Statement: Creating new ideas and implementing them is our major concern. Upgrading us, to support our next generation by training them to earn values, ethics, and power of enthusiasm is our key motto. This is a question bank website for middle school, high school, college, and university students. We have collected questions from various sources, websites, and subject specialists to help students practice more.

IB MYP 2 Math Task Sheet: Order of Operations || Criterion D: Applying Mathematics in Real-Life Contexts

IB MYP 2 Math – Criterion D Task: Order of Operations

IB MYP 2 Mathematics – Criterion D Task

Topic: Order of Operations

Criterion D: Applying Mathematics in Real-Life Contexts

Real-Life Context: Supermarket Shopping

You have a budget of $100 for your shopping. You plan to buy the following items:

3 packs of juice boxes, each costing $4.50
2 family pizzas, on sale as Buy 1 Get 1 50% off (original price: $11.00 each)
4 bags of apples, each weighing 1.5 kg and costing $2.60 per kg
A 10% discount voucher applies only after calculating total before tax
Sales tax is 8%

Task Instructions

Write a single mathematical expression using numbers, operations, and brackets to calculate your total bill before tax and discount.
Apply the order of operations correctly to calculate the total cost after applying the 10% discount and then the 8% sales tax.
Show all your steps clearly, explaining why you perform each operation in that order.
State whether your total cost is within your $100 budget.
Write a brief reflection (2–3 sentences) on how using the correct order of operations helps in budgeting or real-life purchases.

Sample Expression (for reference)

For example, the expression before discount and tax could look like this:
$3 \times 4.50 + (11 + 11 \times 0.5) + (4 \times 1.5 \times 2.60)$
Then apply discount and tax in correct order.

Extension (Optional Challenge)

Suppose you want to buy 2 movie tickets after shopping. Each ticket costs $12.75.

Recalculate your total cost including movie tickets.
Decide if your budget is sufficient.
If over budget, suggest how you could adjust your shopping to stay within $100.

Assessment Rubric – Criterion D

Level	Description
0	The student does not reach a standard described by any of the descriptors below.
1–2	The student applies mathematical strategies to some extent, with limited effectiveness in a real-life situation.
3–4	The student appropriately selects and applies mathematical strategies, providing a partially accurate solution to a real-life situation.
5–6	The student effectively applies relevant mathematical strategies and gives a mostly accurate solution with some justification.
7–8	The student consistently applies relevant, sophisticated mathematical strategies to reach a valid, accurate solution and provides a clear, well-justified explanation connected to the real-life context.

IB MYP 2 Math Task Sheet: Order of Operations || Criterion C – Communicating

MYP 2 Mathematics – Order of Operations Task (Criterion C)

MYP 2 Mathematics – Task Sheet

Topic: Order of Operations

Level: MYP Year 2 (Grade 7)

Assessment Focus: Criterion C – Communicating

Global Context

Scientific and Technical Innovation: How can we ensure accurate results in scientific and real-life calculations?

Learning Objectives

Use and explain the correct order of operations (BIDMAS/PEMDAS).
Communicate mathematical steps clearly and logically using appropriate notation.

Part A: Solve and Show Your Work

Solve each expression using the correct order of operations. Show ALL steps clearly.

$ 8 + 6 \div 2 \times (3 + 1) $
$ (5 + 3)^2 - 4 \times 2 $
$ 18 \div 3 + 2 \times (4 + 5) $
$ [2 + 3 \times (6 - 4)]^2 \div 5 $
$ 7 + \{8 - [3 + 2 \times (1 + 1)]\} \times 2 $

Part B: Explain Your Reasoning

Choose one problem from Part A and explain your reasoning in full sentences. Example:

“First, I solved the brackets because they come first in the order of operations. Then I did the multiplication before the addition…”

Criterion C – Rubric (Communicating)

Level	Descriptor
0	The student does not reach a standard described by any of the descriptors below.
1–2	The student attempts to organize work using a limited logical structure. Attempts to use some appropriate mathematical language or representation.
3–4	The student organizes work using a logical structure. Uses appropriate mathematical language and representation.
5–6	The student organizes work using a coherent structure and appropriate mathematical language. Uses a variety of representations effectively.
7–8	The student organizes work using a complete, coherent structure. Uses accurate and appropriate mathematical language. Uses varied, appropriate representations and includes clear explanations.

Challenge (Optional)

Write your own multi-step expression that uses brackets, exponents, multiplication/division, and addition/subtraction. Solve it and explain why the order matters.

IBDP AAHL Mathematics Worksheet || Functions, Sequences, Algebra, Trigonometry, Differentiation, Vectors, Complex Numbers

IB Mathematics AA HL – Full Practice Set

DP Mathematics AA HL – Practice Set

Note: This worksheet excludes Integration, Statistics, and Probability.

🔢 Section A: Algebra & Functions

Solve for $ x \in \mathbb{R} $:

$ \log_2(x^2 - 5x + 6) = 1 $
Given $ f(x) = \dfrac{2x + 3}{x - 1} $:
(a) Find the inverse function $ f^{-1}(x) $
(b) State the domain and range of $ f(x) $
Let $ f(x) = ax^3 + bx^2 + cx + d $, with a local maximum at $ x = 1 $ and an inflection point at $ x = 2 $.
Given $ f(1) = 4 $, find $ a, b, c, d $.

🔁 Section B: Sequences & Series

Given $ u_n = 2^n + 3n $, find the value of $ \sum_{n=1}^{6} u_n $.
The first three terms of a geometric sequence are $ x, 2x - 3, 4x - 6 $.
Find the value of $ x $ and the common ratio $ r $.

📐 Section C: Trigonometry

Solve for $ \theta \in [0^\circ, 360^\circ] $:

$ 2\sin\theta - \sqrt{3} = 0 $
Prove the identity:

$ \dfrac{1 + \tan^2x}{1 - \tan^2x} = \sec(2x) $
The angle of elevation from the ground to a tower is $ 30^\circ $.
After walking 50 m closer, the angle becomes $ 45^\circ $.
Find the height of the tower.

✏️ Section D: Differentiation

Differentiate:

$ f(x) = \dfrac{x^2 \sin x}{e^x} $
Given $ f'(x) = 6x^2 - 18x $, find the stationary points and determine their nature.
Find the derivative of:

$ f(x) = \ln\left(\sqrt{x^2 + 4}\right) $

🧭 Section E: Vectors

Let $ \vec{a} = (3, -1, 2), \vec{b} = (1, 4, -2) $:

(a) Find $ \vec{a} \cdot \vec{b} $
(b) Find the angle between vectors $ \vec{a} $ and $ \vec{b} $
A line passes through point $ A(1, 2, -1) $ with direction vector $ \vec{d} = (2, -1, 3) $.
Find the Cartesian equation of the line.

🔮 Section F: Complex Numbers

Solve for $ z \in \mathbb{C} $:

$ z^2 + (3 - 2i)z + (4 + i) = 0 $
Express $ \dfrac{3 + 4i}{1 - 2i} $ in the form $ a + bi $, where $ a, b \in \mathbb{R} $.

📌 Instructions

Show all working clearly and neatly.
Use GDC for verification only—full reasoning must be shown.
Use correct mathematical notation at all times.

IB Mathematics AA HL – Derivatives: Product, Quotient & Chain Rule Practice Set

IB Mathematics AA HL – Derivatives Practice Set

Topic: Product Rule, Quotient Rule, Chain Rule

Level: Higher Level (AA HL)

Total Questions: 14

Section A: Short Questions (Fundamental Applications of Rules)

Apply the appropriate differentiation rule: product, quotient, or chain rule. Give your answers in simplified form.

Differentiate the function:

$ f(x) = (3x^2 + 5)(\sin x) $
Find $ \dfrac{dy}{dx} $ if

$ y = x^2 e^x $
Differentiate the function:

$ f(x) = \ln x \cdot \tan x $
Differentiate:

$ f(x) = \dfrac{x^2 + 1}{\sqrt{x}} $
Find the derivative of:

$ y = \dfrac{\sin x}{x^2 + 1} $
If $ f(x) = \dfrac{\ln x}{x^3} $, find $ f'(x) $
Differentiate the following function:

$ f(x) = \cos(3x^2 + 2) $
Find $ \dfrac{dy}{dx} $ if

$ y = \sqrt{1 + \tan x} $
Differentiate:

$ y = e^{(x^2 + 4x)} $
Let $ f(x) = \ln(\sqrt{1 + e^x}) $. Find $ f'(x) $

Section B: Extended Response Questions (IB-Style)

These problems require extended reasoning and clear, structured solutions. Use correct mathematical notation throughout.

Let $ f(x) = \sin(x^2 \cdot e^x) $

(a) Find $ f'(x) $.
(b) Comment on the effect of the chain rule in this derivative.
The curve is defined implicitly by the equation:

$ x^2 y + y^3 = \tan x $

Find $ \dfrac{dy}{dx} $ in terms of $ x $ and $ y $, using implicit differentiation.
Let $ f(x) = \dfrac{\ln(\sin x)}{x^2 + 1} $, where $ 0 < x < \pi $.

(a) Find $ f'(x) $.
(b) Hence, find the equation of the tangent to the curve at $ x = \dfrac{\pi}{4} $
A balloon rises vertically such that its height in meters at time $ t $ seconds is given by:

$ h(t) = 50 \ln(2t + 1) $

A camera is placed 100 meters horizontally from the point where the balloon was released. Let $ D(t) $ be the distance between the camera and the balloon.

(a) Express $ D(t) $ in terms of $ t $.
(b) Find $ \dfrac{dD}{dt} $ when $ t = 3 $, giving appropriate units.
(c) Interpret your result in the context of the motion of the balloon.

IBDP Mathematics - Analysis and Approaches (HL) || Unit 1 || Topic 2 : Problem in Exponent and Logarithm

International Baccalaureate Diploma Program (IBDP) - Analysis and Approaches (AAHL) || Unit Circle and Trigonometric Ratios Beyond Acute Angles

Unit Circle and Trigonometric Ratios Beyond Acute Angles - Worksheet

Name: ___________________

Date: ___________________

Instructions:

Answer the following questions.
Show all your work, including any necessary diagrams.
Ensure you clearly explain your steps where applicable.

Questions

Evaluate the following expressions using the unit circle:
- a) $ \sin\left(\frac{5\pi}{4}\right) $
- b) $ \cos\left(\frac{7\pi}{3}\right) $
Given $ \theta = \frac{5\pi}{6} $, find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $ using the unit circle.
Find the exact value of $ \csc\left(\frac{3\pi}{2}\right) $.
The point $ P $ lies on the unit circle at an angle of $ \theta = \frac{4\pi}{3} $. What are the coordinates of $ P $?
Prove the identity $ 1 + \tan^2(\theta) = \sec^2(\theta) $ using the unit circle.
For $ \theta = \frac{5\pi}{3} $, determine the value of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $. Then, verify that $ \sin(\theta) $ and $ \cos(\theta) $ satisfy the Pythagorean identity $ \sin^2(\theta) + \cos^2(\theta) = 1 $.
Given $ \tan(\theta) = -\frac{1}{\sqrt{3}} $, and $ \theta $ is in the fourth quadrant, find the exact values of $ \sin(\theta) $ and $ \cos(\theta) $.
A point $ P $ on the unit circle forms an angle of $ \theta = \frac{7\pi}{6} $. Determine the reference angle for $ \theta $, and find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $.
A pendulum swings back and forth following the sine function $ y(t) = 5 \sin\left( \frac{\pi}{2} t \right) $, where $ t $ is in seconds. Find the period of the pendulum’s motion and the amplitude of the sine wave.
Using the unit circle, determine the exact value of $ \tan\left(\frac{11\pi}{6}\right) $ and explain your reasoning.

Bonus Question (Optional)

Show that $ \cos(\theta + \pi) = -\cos(\theta) $ using the unit circle, and use this identity to find $ \cos\left( \frac{5\pi}{3} \right) $.

End of Worksheet

IBDP AAHL/AASL - Understanding Unit circle in Trigonometry

Unit Circle in Trigonometry

Understanding the Unit Circle in Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the fundamental concepts in trigonometry is the unit circle, a powerful tool that provides a visual way to understand trigonometric functions. Whether you're new to trigonometry or looking to deepen your understanding, the unit circle is essential for working with sine, cosine, and tangent functions.

In this blog, we’ll explore what the unit circle is, how it’s constructed, and how it connects to trigonometric functions.

What is the Unit Circle?

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of a coordinate plane. In the Cartesian coordinate system, the equation for a circle is:

x² + y² = r²

For a unit circle, r = 1, so the equation becomes:

x² + y² = 1

This circle plays a crucial role in defining the six main trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) and helps us understand their values for any given angle.

The Basics of the Unit Circle

Imagine a point on the unit circle that makes an angle θ with the positive x-axis. The position of this point is determined by two coordinates: x and y.

The x-coordinate of the point corresponds to the cosine of the angle θ.
The y-coordinate corresponds to the sine of the angle θ.

In other words, for any angle θ, the coordinates of the point on the unit circle are:

(cos(θ), sin(θ))

This relationship is key to understanding the values of sine and cosine for any angle. The unit circle provides us with a way to easily visualize these values without relying on memorizing individual values for each angle.

Understanding Angles on the Unit Circle

Angles on the unit circle are typically measured in radians, though degrees can also be used. Here’s how radians and degrees relate to each other:

360° = 2π radians
180° = π radians
90° = π/2 radians

On the unit circle, we typically work with positive angles in the counterclockwise direction and negative angles in the clockwise direction. The most common angles you’ll encounter on the unit circle include 0°, 90°, 180°, and 270°, as well as their corresponding radian measures.

Key Points on the Unit Circle

Let’s now examine some key angles and their coordinates on the unit circle, as well as their sine and cosine values:

At 0° (or 0 radians):
- Coordinates: (1, 0)
- cos(0) = 1, sin(0) = 0
At 90° (or π/2 radians):
- Coordinates: (0, 1)
- cos(π/2) = 0, sin(π/2) = 1
At 180° (or π radians):
- Coordinates: (-1, 0)
- cos(π) = -1, sin(π) = 0
At 270° (or 3π/2 radians):
- Coordinates: (0, -1)
- cos(3π/2) = 0, sin(3π/2) = -1

These are just a few of the important points on the unit circle, but the circle contains infinitely many points corresponding to every possible angle.

Trigonometric Functions and the Unit Circle

Once we understand how the coordinates of the unit circle relate to sine and cosine, we can move on to the other trigonometric functions, which are also based on the unit circle.

Tangent: The tangent of an angle θ is the ratio of the sine to the cosine:
tan(θ) = sin(θ) / cos(θ)
Cosecant: The cosecant function is the reciprocal of the sine:
csc(θ) = 1 / sin(θ)
Secant: The secant function is the reciprocal of the cosine:
sec(θ) = 1 / cos(θ)
Cotangent: The cotangent function is the reciprocal of the tangent:
cot(θ) = 1 / tan(θ)

These additional functions are essential when solving trigonometric equations or understanding complex wave patterns in fields such as physics and engineering.

Why is the Unit Circle So Important?

The unit circle provides a way to visualize and understand trigonometric functions for any angle, not just the angles commonly found in right triangles. Here’s why it’s crucial:

Simplifies Trigonometric Calculations: Instead of memorizing trigonometric values for specific angles, the unit circle allows you to compute these values easily by finding the corresponding point on the circle.
Helps with Graphing: The unit circle helps us understand the behavior of trigonometric functions, especially when graphing them. The periodic nature of these functions becomes clear when you see the repeated patterns on the circle.
Solving Complex Problems: In calculus, physics, and engineering, the unit circle is used to solve complex problems, such as those involving periodic motion, waves, or rotations.

Conclusion

The unit circle is an essential concept in trigonometry, providing a straightforward and visual way to understand the relationships between angles and trigonometric functions. By mastering the unit circle, you’ll have a deeper understanding of sine, cosine, and other trigonometric functions, enabling you to solve a wide range of problems in mathematics and its applications.

So next time you encounter a trigonometric function, remember the unit circle—it’s the key to unlocking the mysteries of angles and their relationships!

Click and visit

IB MYP 2 Mathematics – Criterion D Task

Topic: Order of Operations

Real-Life Context: Supermarket Shopping

Task Instructions

Sample Expression (for reference)

Extension (Optional Challenge)

Assessment Rubric – Criterion D

MYP 2 Mathematics – Task Sheet

Topic: Order of Operations

Global Context

Learning Objectives

Part A: Solve and Show Your Work

Part B: Explain Your Reasoning

Criterion C – Rubric (Communicating)

Challenge (Optional)

DP Mathematics AA HL – Practice Set

🔢 Section A: Algebra & Functions

🔁 Section B: Sequences & Series

📐 Section C: Trigonometry

✏️ Section D: Differentiation

🧭 Section E: Vectors

🔮 Section F: Complex Numbers

📌 Instructions

IB Mathematics AA HL – Derivatives Practice Set

Topic: Product Rule, Quotient Rule, Chain Rule

Section A: Short Questions (Fundamental Applications of Rules)

Section B: Extended Response Questions (IB-Style)

Instructions:

Questions

Bonus Question (Optional)

What is the Unit Circle?

The Basics of the Unit Circle

Understanding Angles on the Unit Circle

Key Points on the Unit Circle

Trigonometric Functions and the Unit Circle

Why is the Unit Circle So Important?

Conclusion