When investing money, people want to know their earnings over time and compare investment performances. Comparing investments is easier when the same amount is invested in each, but this is not usually the case. Return on investment (ROI) shows the earnings of an investment based on equal initial investments. For example, an investment making $50 in a month with a $100 investment, and another making $75 with a $120 investment, ROI helps determine which has a better return.

To calculate ROI, follow these steps:

1. Determine the investment's beginning and ending balances for the month. For instance, a stock worth $14 on January 1 and $18 on January 31.

2. Subtract the beginning price from the ending price. In this example, $18 - $14 = $4.

3. Divide the result from step 2 by the beginning price to find the rate of return. Here, $4 ÷ $14 = 0.286 or 28.6 percent, which is the rate of return for the month. This calculation allows for the comparison of different investments based on their ROI, making it easier to decide which investment performed better.


# Choose one of the factors of x^{6} + 1000.

The given expression can be factorized using the (a + b)^{3} identity. By rewriting the identity as a^{3} + b^{3} = (a + b)^{3} - 3ab^{2} - 3a^{2}b, we can express x^{6} + 1000 as (x^{2})^{3} + (10)^{3} = (x^{2} + 10)^{3} - 3(x^{2})(10)^{2} - 3(x^{2})^{2}(10).

Rewriting this equation, we have (x^{2} + 10)^{3} - 300x^{2} - 30x^{4}, which simplifies to (x^{2} + 10)^{3} - 30x^{2}(x^{2} + 10). Taking (x^{2} + 10) as the common factor, we get (x^{2} + 10){(x^{2} + 10)^{2} - 30x^{2}}.

Expanding this expression, we have (x^{2} + 10)(x^{4} + 100 + 20x^{2} - 30x^{2}), which simplifies to (x^{2} + 10)(x^{4} - 10x^{2} + 100). Therefore, the factors of x^{6} + 1000 are (x^{2} + 10) and (x^{4} - 10x^{2} + 100).

## Choose one of the factors of x^{6} + 1000.

x^{2}, x^{2} - 10, x^{4} - 10x^{2} + 100, x^{4} + 10x^{2} + 100

The correct answer is x^{4} - 10x^{2} + 100.


Grade 2 maths skills are organized into categories. The key categories and skills include:

* Counting and number patterns: A.1, A.5, A.6
* Comparing and ordering, names of numbers, addition, and subtraction with one digit: D.4, D.9, E.4
* Multi-digit operations: addition and subtraction with two and three digits
* Properties: J.3
* Patterns: K.2
* Place values and estimation and rounding: M.7
* Mixed operations and logical reasoning: O.1
* Money: P.1, P.6, P.17
* Time: Q.6, Q.7, Q.8, Q.9, Q.10, Q.11, Q.12, Q.13, Q.14, Q.16
* Spatial sense and data and graphs: S.10
* Units of measurement: T.1, T.2, T.3
* Two-dimensional and three-dimensional shapes: U.7
* Geometric measurement: W.1, W.3

To practice these skills, click on any link to start. The questions will increase in difficulty as you improve, and your score will be tracked.


Proof has a significant place in Primary mathematics classrooms, but it's essential to clarify what is meant by proof at this level. To explore this concept, consider the following problems and activities:

1. **Investigating Squared Numbers**: Take a number, square it, and subtract the original number. What kind of number do you get? Can you prove it?

2. **Multiplication Structure**: Choose an example to demonstrate the general structure of multiplication. How does one instance reveal properties of multiplication?

3. **Adding Odd Numbers**: Examine the property of adding two odd numbers together. What pattern emerges?

4. **Counting Neighbours**: Select three consecutive counting numbers, add them together, and observe any patterns.

5. **Odd and Even Statements**: Evaluate statements about odd and even numbers. Are they always true, sometimes true, or never true?

6. **Adding Three Numbers**: When adding three numbers, will the result be odd or even? How can you be certain?

7. **Grid Clues**: Given a set of clues, some of which are true but unhelpful, can you identify the useful clues to find a chosen number on a grid?

8. **Describing Marbles**: Use provided information to describe marbles that sparkle on the outside but are dark inside. What colors must they have?

9. **Generic Proof**: This concept introduces the idea of using one example to prove a general result by understanding its underlying structure.

10. **Calculation and Shape Properties**: Assess statements about calculation and properties of shapes. Are they always true, sometimes true, or never true?

11. **Logical Reasoning**: Apply logical reasoning to determine the number of cows and sheep in each field based on given conditions.

12. **Problem Solving**: Problem solving is central to mathematical learning. It involves using and developing mathematical concepts and skills to overcome challenges.

13. **Ladybird's Walk**: A ladybird walks around a triangle. How much has it turned when it returns to the starting point?

14. **Letter Game**: Play a game where two players take turns selecting words to find all occurrences of the same letter.

15. **Paradoxes**: Explore statements that seem both true and untrue at the same time. Investigate these paradoxes to understand their nature.

16. **Logic and Mathematical Logic**: Discuss what logic means and how it differs from mathematical logic.

17. **Strategic Game - Sprouts**: Learn about "Sprouts," a game simple enough for younger children yet challenging for experienced mathematicians.

18. **Teaching Proof**: Understand how to guide learners from experimental arguments to mathematical arguments and deductive reasoning in the context of proof.

19. **3D Shapes**: Visualize whether given nets can fold into 3D shapes. Use videos to verify your predictions.

20. **Dice Challenge**: Place three dice in a row and find ways to turn them so that the numbers on top and front of the dice total the same. Can you find all possible solutions?


To create a formula in Smartsheet that counts cells containing "OOS" and displays the total in another cell, use the CountIf function.

In the cell where you want to display the answer, enter the following formula:
=CountIf([Column One]:[Column One],"OOS")

This formula assumes that "OOS" is associated with cells in only one column. If "OOS" appears in multiple columns, you will need to adjust the formula accordingly.

For practice working with formulas directly in Smartsheet, consider using the Formula Handbook template. This resource can help you become more familiar with creating and using formulas in Smartsheet.


Minesweeper is a classic game that can be solved using logical deduction and linear algebra. The game can be represented as a matrix $M\in \mathbb{R} ^{m\times n}$, where $m$ and $n$ are the number of rows and columns, respectively. Each cell in the matrix can be represented by a variable $v_{ij}$, where $i$ and $j$ are the row and column indices, respectively.

The game can be solved by setting up a system of linear equations, where each equation represents a constraint on the values of the variables. For example, if a cell has a value of $k$, then the sum of the values of the cells in its neighborhood must be equal to $k$. This can be represented by the equation:

$E_{ij} ~:~ \sum _{p=i\pm 1, q = j\pm 1}^{1\leq p \leq m, 1\leq q \leq n} \delta _{pq}v_{pq} = c_{ij}$

where $c_{ij}$ is the number of mines surrounded by the cell at position $(i,j)$, and $\delta _{ij} = 1$ if the cell at position $(i,j)$ is concealed, 0 otherwise.

The system of linear equations can be solved using linear algebra techniques, such as Gaussian elimination or LU decomposition. The solution to the system of equations will give the values of the variables $v_{ij}$, which represent the locations of the mines in the game.

One algorithm for solving Minesweeper using linear algebra is as follows:

Algorithm 1:

Input: A partially solved Minesweeper matrix $M\in \mathbb{R}^{m\times n}$

Output: Any definite deduction

1. Convert the puzzle into linear equations
2. Perform row reduction to solve the system of equations
3. Interpret the solution
4. Return any conclusion

For example, consider a classic 1-2 combination:

$\begin{pmatrix}

\square & \square & \square & \square & \square & \square & \square & \square \\

\square & 2 & 1 & 1 & 2 & 2 & 1 & \square

\end{pmatrix}$

This can be represented by the following system of linear equations:

$\begin{pmatrix}

1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2\\

0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1\\

0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 2\\

0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 2\\

0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1

\end{pmatrix}

\xrightarrow[~]{rref}

\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 & 1\\

0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 1\\

0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0\\

0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\

0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & -1 & 1\\

0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \end{pmatrix}$

The solution to this system of equations can be interpreted to give the locations of the mines in the game.

However, this approach has some limitations. The deduction of the solution relies heavily on the specific structure of the puzzle, and it may not be possible to automate the process of interpreting the solution. Additionally, the system of linear equations may be too complex to solve efficiently, especially for larger puzzles.

To improve the efficiency of the algorithm, it may be possible to use a different approach, such as working over a finite field instead of the real numbers. This could potentially simplify the system of linear equations and make it easier to solve. However, this approach would require a deeper understanding of the underlying mathematics and may not be straightforward to implement.

In conclusion, solving Minesweeper using linear algebra is a complex task that requires a deep understanding of the underlying mathematics. While it is possible to develop algorithms for solving the game, these algorithms may have limitations and may not be efficient for larger puzzles. Further research is needed to develop more efficient and effective algorithms for solving Minesweeper.


In this chapter, we introduce fundamental mathematical concepts related to coordinates and coordinate systems. We will assume some simplifications to support the techniques and topics addressed in subsequent chapters.

We locate objects in space by specifying their coordinates, which are pairs of numbers that uniquely identify a point. A coordinate system employs one or more numbers to specify the position of a point. There are different coordinate systems, such as Cartesian, polar, and others, and we can transform coordinates from one system to another. We will use the Cartesian coordinate system.

In the Cartesian coordinate system for two dimensions, a coordinate is defined by two numbers that measure the signed distance to two perpendicular axes, x and y. The origin is set to a specific point, and the axis orientation is defined. For example, in geographic coordinates, the origin is set to the point where the equator and the zero meridian cross. Depending on the origin and axis orientation, coordinates for a specific point can be different.

A coordinate system may also define the orientation of the axis. In the Cartesian coordinate system, the x coordinate increases as we move to the right, and the y coordinate increases as we move upwards. However, we could define an alternative Cartesian coordinate system with different axis orientation, resulting in different coordinates.

To work with a set of coordinates, we must use the same coordinate space, which includes the origin and axis orientation. Fortunately, we can transform coordinates from one space to another by performing translations and rotations.

In 3D coordinates, we need an additional axis, the z axis, and 3D coordinates are formed by a set of three numbers (x, y, z). As in 2D Cartesian coordinate spaces, we can change the orientation of the axes in 3D coordinate spaces as long as the axes are perpendicular.

3D coordinates can be classified into two types: left-handed and right-handed. To determine the type, we can use the following method: form an "L" shape with the thumb and index finger, and the middle finger should point in a direction perpendicular to the other two. If this can be done with the left hand, it's left-handed; if it requires the right hand, it's right-handed.

2D coordinate spaces are all equivalent, as we can transform from one to another by applying rotation. However, 3D coordinate spaces are not all equal, and we can only transform from one to another by applying rotation if they both have the same handedness.

When dealing with 3D graphics, we use different 3D coordinate spaces, each with its own context and purpose. A set of coordinates is meaningless unless it refers to something specific. For example, the coordinates (40.438031, -3.676626) are geometric coordinates (latitude and longitude) that represent a place in Madrid.

When loading 3D objects, we get a set of 3D coordinates expressed in a 3D coordinate space called object coordinate space. Graphics designers create 3D models using a coordinate space relevant only to the model, without knowing the 3D scene it will be displayed in.

When drawing a 3D scene, all 3D objects are relative to the world space coordinate space. We need to transform from 3D object space to world space coordinates, which may involve rotation, stretching, enlargement, or translation. We also need to restrict the range of the 3D space shown, similar to moving a camera through the 3D space, and transform world space coordinates to camera or view space coordinates. Finally, these coordinates need to be transformed to screen coordinates, which are 2D, by projecting 3D view coordinates to a 2D screen coordinate space.

In OpenGL, coordinates are between -1 and +1, and the z axis is perpendicular to the screen. These concepts will be revisited in subsequent chapters with practical examples.


# Unit: Magnetic effects of electric current

## About this unit
Magnets have a close relationship with electric currents, and this connection is used to build essential devices like motors and generators. This chapter explores this relationship and its applications.

## Key Topics
To master this unit, focus on the following skills:
- Magnetic field lines: direction, special properties, and field strength
- Right-hand thumb rule
- Magnetic field around a straight current-carrying wire
- Direction of magnetic field due to a current-carrying circular loop
- Magnetic field due to a current-carrying solenoid
- Force on a current in a magnetic field
- Fleming's left-hand rule
- Electric motor: parts, functions, and working
- Electromagnetic induction

## Practice and Mastery
Level up on each skill by answering questions correctly. The requirements and rewards for each set of skills are as follows:
- Magnetic field lines: Get 3 of 4 questions correct to level up and collect up to 240 Mastery points.
- Right-hand thumb rule and magnetic field around a straight current-carrying wire: Get 3 of 4 questions correct to level up and collect up to 160 Mastery points.
- Direction of magnetic field due to a current-carrying circular loop and magnetic field due to a current-carrying solenoid: Get 3 of 4 questions correct to level up and collect up to 160 Mastery points.
- Force on a current in a magnetic field and Fleming's left-hand rule: Get 3 of 4 questions correct to level up and collect up to 160 Mastery points.
- Electric motor: parts and functions, and electric motor: working: Get 3 of 4 questions correct to level up and collect up to 160 Mastery points.
- Electromagnetic induction: Get 5 of 7 questions correct to level up.

## Unit Completion
Level up on all skills in this unit and collect up to 1200 Mastery points to complete it.


Lego Maths with Y6

This autumn, wet playtimes have led to an increased interest in Lego bricks among children. As they build and play, they instinctively apply mathematical concepts, such as using 8-stud bricks and substituting them with combinations of smaller bricks, like two 4-stud bricks or a 6-stud and a 2-stud. This demonstrates an understanding of arrays and other mathematical principles, even if they are not aware of it.

To capitalize on this enthusiasm, an investigation into the numerical properties of Lego bricks was conducted with a Year 6 class. The students were encouraged to identify and record numerical patterns, without building, and discovered evidence of factors, number bonds, arrays, divisibility, sequencing, and square numbers. They also gained experience with data handling and discussing numerical properties.

Key findings included:
- A student discovering that 100 is not divisible by 8
- Another student identifying patterns of square numbers using 4-stud bricks, exploring the relationship between the number of studs and bricks
- A group investigating the relationship between the number of studs on top of a brick with 2 rows of studs and the holes underneath, leading to the realization that the number of holes is one less than one 'line' of studs

After organizing their results and checking their ideas, the students were able to explain their findings using mathematical vocabulary and, with some guidance, created a formula. This sparked further investigation, as students sought to find the 'rule' for a flatter base with rows of 4 studs instead of 2. Although time ran out, the discussion continued into the next lesson, with one student determined to explain the mathematical rule and formula they had discovered overnight.

The Lego maths lesson was a great success, demonstrating the potential for Lego bricks to facilitate engaging and interactive mathematical learning experiences.


**Linear Programming**

Linear programming is a technique for selecting the best alternative from available alternatives, where the objective function and constraint function can be expressed in quantitative terms.

**Minimization Case**
The minimization case involves finding the minimum value of an objective function, subject to certain constraints.

**Maximization Case**
The maximization case involves finding the maximum value of an objective function, subject to certain constraints. For example, a firm producing two products, A and B, wants to maximize profit. The objective function and constraint function can be expressed as linear mathematical functions.

**Prerequisites for Linear Programming**
There are certain prerequisites for applying linear programming:
1. There should be a well-defined objective function.
2. The objective function and constraint function should be linear.
3. There should be limited resources.

**Motivation Theories**

**Carrot and Stick Approach**
The Carrot and Stick Approach is a traditional motivation theory that uses rewards and punishments to motivate people. Rewards can be financial or non-financial benefits, while punishments are used to push individuals towards desired behavior.

**Reinforcement Theory**
The Reinforcement Theory, proposed by B.F. Skinner, states that behavior is a function of its consequences. Individuals develop behaviors based on the outcomes of their actions.

**Adam's Equity Theory**
Adam's Equity Theory, proposed by John Stacey Adams, states that people maintain a fair relationship between performance and rewards compared to others. Employees get demotivated if their inputs exceed their outputs.

**Vroom's Expectancy Theory**
Vroom's Expectancy Theory, proposed by Victor H. Vroom, states that people are motivated to perform activities to achieve goals to the extent they expect certain actions to help them achieve those goals.

**Alderfer's ERG Theory**
Alderfer's ERG Theory is an extension of Maslow's Needs Hierarchy, categorizing needs into Existence, Relatedness, and Growth needs.

**McClelland's Needs Theory**
McClelland's Needs Theory, proposed by David McClelland, states that specific needs are acquired over time and molded by life experiences. The theory identifies three basic motivating needs: achievement, affiliation, and power.

**Herzberg's Motivation-Hygiene Theory**
Herzberg's Motivation-Hygiene Theory, proposed by Fredrick Herzberg, states that certain factors are desirable to achieve goals, while others are undesirable and should be avoided. The theory categorizes factors into motivators and hygiene factors.


The number 108 has been considered sacred in various cultures and traditions. In Hindu and Buddhist beliefs, there are 108 virtues to cultivate and 108 defilements to avoid, emphasizing the importance of balance. Hindu and Buddhist Malas, or rosary beads, also contain 108 beads. In Islam, the number 108 is sometimes used as a substitute for the name of God. Additionally, there are 108 movements in Tai chi, highlighting the frequency of this number across different disciplines.

The significance of the number 9 is also noteworthy, as it is the last and highest single-digit number. In Hindu tradition, the number 9 is used to express various concepts and practices. For example, Rishi Vyasa created 9 Puranas, and there are 9 universal substances or elements in the Vaisheshika branch of Hindu philosophy. In Indian aesthetics, there are 9 kinds of Rasa, and the number 9 is considered the principle number in the time dimension.

According to Indian mythology, there are 4 Yugas, or ages, each with a specific duration: SatYuga (172,800 years), TretaYuga (1,296,000 years), DwaparYuga (864,000 years), and KaliYuga (432,000 years). Interestingly, the sum of the digits of each of these numbers equals 9. The letter 9 is also associated with Lord Brahma, the creator of the universe, and is considered the only positive perfect power.

The connection between 9 and 108 is evident in Vedic mathematics, where the energy of a number is evaluated by summing its individual digits. For instance, the Mahabharat has 18 chapters (1+8=9), the Geeta has 18 chapters (1+8=9), and there are 108,000 Shloks (verses) in the Mahabharat (1+0+8+0+0+0=9). The Sanskrit alphabet has 54 letters (5+4=9), each with masculine and feminine, Shiva and Shakti (Yin and Yang) aspects. The product of 54 and 2 is 108 (1+0+8=9).

The number 108 is also significant in various religious and cultural contexts. In Shaiva religions, particularly among Lingayats, malas with 108 beads are used for prayer and meditation. In Gaudiya Vaishnavism, Lord Krishna had 108 followers known as gopis, and the recital of Lord Krishna's 108 names is often accompanied by the counting of a 108-beaded mala. Additionally, there are 108 forms of dance in Indian traditions, and Adiguru Shiva or Nataraja is considered the creator of the cosmic dance.

The union between the microcosm and macrocosm is reflected in the number 108. In Vedic astrology, the Sun is treated as God and has 12 signs (Zodiac signs). The diameter of the Sun is approximately 108 times that of the Earth, and the average distance between the Earth and the Sun is 108 times the Sun's diameter. The average distance from the Earth to the Moon is about 108 times the Moon's diameter, which is why the Moon appears the same size as the Sun during eclipses.

The ratio of 108 may be the key to finding planets hospitable to life outside of our solar system. This number is also found in the Fibonacci sequence, which describes the growing rate of a nautilus shell. The shell's chambers grow at a rate of 1.08 times larger than the previous one, creating a logarithmic spiral. This pattern is observed in nature and is a testament to the magic of the natural world.

In yoga, the number 108 refers to spiritual completion. The human body has nine doors, and the heart chakra is formed by the convergence of 108 energy lines or nadis. There are also 108 marmas (pressure points) or sacred parts of the body. The internal temperature at which the human body's vital organs begin to fail from overheating is 108 degrees Fahrenheit.

The practice of sun salutations in yoga is often performed in nine rounds of 12 postures, totaling 108. This practice is meant to honor and activate the sun energy within the individual. By practicing chanting, breath work, or asana in rounds of 108, the ancient yogis believed that one could align themselves with the rhythm of creation and ultimately bring an end to the cycle of reincarnation.

The Vedic system describes the universe as being created by the five elements: space, air, fire, water, and earth. These elements give rise to the three attributes: Raj (birth), Sat (protection), and Tam (destruction or death). The mathematical or geographical evidence proves that one circle has 360 degrees in space. By dividing the circle using the four elements and three attributes, the logic of time can be observed. The circle is divisible by the product of four elements multiplied by three attributes, which involves the belief that the three attributes exist in the circle.

The division of the circle gives birth to the 12 months and the 12 hours (half of the day). Adding the time principle number 9 to the 12 divisions results in 108, which is the code in our life circle. All measuring systems are merely reference frames, and the number 108 is symbolic of a bigger picture. As Galileo Galilei said, the universe is written "in mathematical language," and the mysteries of creation can be unraveled through numbers and equations. The code is 108.


I have made a conclusion while investigating a vector-related problem and I am trying to prove it. My conclusion is as follows:

"If you have a number of vectors (n) with a resultant of Zero, and you cross each one of them with a non-zero vector, then the resultant of the cross results will also be zero."

I would appreciate it if you could confirm whether my conclusion is correct or not. If needed, I can provide further clarification.


The composition of accelerations involves measuring the change of direction and the acceleration of the actual velocity along a curve. Acceleration can be resolved into components along and perpendicular to the radius-vector.

If the velocity and its direction are given at any time or position, the determination of the path and other circumstances of the motion is a question of pure mathematics. The same applies if the total acceleration and its direction are given, provided the velocity, direction of motion, and position of the point are known at a particular instant.

The equations of motion can be integrated to determine the path described by the point. In the case of given velocity, we have three simultaneous equations that can be integrated to determine the coordinates of the point in terms of time. By eliminating time, we obtain two equations that represent surfaces on which the path lies, and their intersection determines the path.

In the case of given acceleration, we have similar equations, but each has to be twice integrated. The arbitrary constants introduced by integration can be determined if the coordinates and components of the velocity of the point are known at a given epoch.

Several interesting results can be deduced from the principles of velocity and acceleration. For example, if the velocity of a moving point is uniform and its direction revolves uniformly in a plane, the path described is a circle. If a point moves in a plane and its component velocity parallel to each of two rectangular axes is proportional to its distance from that axis, the path is an ellipse or hyperbola.

When the velocity is uniform but revolves uniformly in a right circular cone, the motion of the point is in a circular helix whose axis is parallel to that of the cone. With uniform acceleration in the direction of motion, a point describes spaces proportional to the squares of the times elapsed since the commencement of the motion.

In the case of uniform acceleration in a constant direction, the path described is a parabola whose axis is parallel to that direction. This is the case of a projectile moving in a vacuum. When there is uniform acceleration in a tortuous curve, the point describes a helix on a cylinder, and the acceleration is directed perpendicularly towards the axis of the cylinder.

The acceleration is also related to the force required to produce it. For example, a finite force perpendicular to the line of motion, in a direction revolving with an infinitely great angular velocity, maintains constant infinitely small deflection from the line of undisturbed motion, finite curvature, and infinite tortuosity.

When the acceleration is perpendicular to a given plane and proportional to the distance from it, the path is a plane curve, which is the harmonic curve if the acceleration is towards the plane, and a more or less fore-shortened catenary if from the plane.

If the acceleration is directed to a fixed point, the path is in a plane passing through that point, and in this plane, the areas traced out by the radius-vector are proportional to the times employed. This includes the case of a satellite or planet revolving about its primary.

Multiple choice questions:
a. What is the path described by a point with uniform velocity and direction revolving uniformly in a plane?
A) Circle
B) Ellipse
C) Hyperbola
D) Parabola

b. What is the path described by a point with uniform acceleration in a constant direction?
A) Circle
B) Ellipse
C) Hyperbola
D) Parabola

c. What is the path described by a point with acceleration directed to a fixed point?
A) Circle
B) Ellipse
C) Hyperbola
D) Plane curve

Answers:
a. A) Circle
b. D) Parabola
c. D) Plane curve


**Divide and Conquer Algorithm Design**

The Divide and Conquer strategy has been used by generals for centuries to defeat stronger opponents by dividing them into smaller, weaker sections. This approach is still used today, not only in the military but also in programming, to solve complex problems efficiently.

**What is Divide and Conquer Algorithm?**

The Divide and Conquer algorithm works by dividing a problem into smaller sections and solving each one individually. This approach is used to reduce the complexity of a problem and solve it more efficiently.

**Why is it Used?**

The Divide and Conquer strategy is used because it causes less casualties, takes less time, and is an efficient way to defeat a larger opponent. In programming, efficiency is crucial, and this approach helps achieve it by reducing the speed, space-time complexity, and power consumed.

**The Main Concepts:**

1. **Divide**: Split the problem into smaller components.
2. **Conquer**: Solve the smaller components.

**Use of Divide and Conquer in Programming**

A common use of Divide and Conquer in programming is to find a specific number in a sorted array of size N. The most obvious solution is to use a linear search, which has a time complexity of O(N).

**Linear Search**

The linear search algorithm works by iterating through the array and comparing each element with the target number. The time complexity of linear search is:

* **Best Case**: O(1) - the element is found at the first index.
* **Worst Case**: O(N) - the element is found at the last index.

**Binary Search**

Binary search is an algorithm that uses the Divide and Conquer technique to find an element in a sorted array more efficiently. The algorithm works by repeatedly dividing the search interval into half until the element is found. The time complexity of binary search is:

* **Best Case**: O(1) - the element is found at the middle index.
* **Worst Case**: O(log N) - the element is found after dividing the array into smaller sections.

**Binary Search Code**

The binary search algorithm can be implemented using a while loop or recursive calls. The code for both methods is provided below:

**While Loop Method**

```c
int Binary_Search(int Arr[], int F, int L, int S) {
    while (L >= F) {
        int M = F + (L - F) / 2;
        if (Arr[M] == S)
            return M;
        else if (Arr[M] > S)
            L = M - 1;
        else
            F = M + 1;
    }
    return -1;
}
```

**Recursive Method**

```c
int Binary_Search(int Arr[], int F, int L, int S) {
    if (L >= F) {
        int M = F + (L - F) / 2;
        if (Arr[M] == S)
            return M;
        else if (Arr[M] > S)
            return Binary_Search(Arr, F, M - 1, S);
        else
            return Binary_Search(Arr, M + 1, L, S);
    }
    return -1;
}
```

**Time Complexity**

The recurrence relation for the binary search algorithm is T(n) = T(n/2) + O(1). The time complexity is O(log N) in the worst case.

**Other Applications of Divide and Conquer**

The Divide and Conquer technique is used in many other algorithms, including:

* **Merge Sort**
* **Quicksort**
* **Strassen's Algorithm**
* **Closest Pair Points**

These algorithms use the Divide and Conquer technique to solve complex problems more efficiently.


R Statistics refers to the number of people who have had a physical injury and the average length of time they've been injured. The two most commonly used statistics are the total number of injuries sustained by a person and the average length of time they've been injured.

These statistics are used to understand a person's physical experience and average injury length. The average injury length is calculated by dividing the daily number of hits by the average injury length.

R Statistics is a simplified form of numbers used by most people. It is designed to be used by computers in various ways. The key words in R Statistics are average, average injury, distance, and average.

For example, if a person is injured in a car collision, the first two statistics are the average injury length and the average distance between the injury and the person. The third most commonly cited statistic is the average injury speed per person, which measures the average speed of the person when injured.

R Statistics can be used to compare data extracted from source documents and other documents. It is a useful tool for understanding physical experiences and average injury lengths.

In statistics, the R Statistical is a document typically produced by a computer. It is used to calculate the average of injuries sustained in a car crash, among other things.

To understand R Statistics, one needs to know how the statistics are calculated. A common definition of a statistic is that it is composed of variables called data or events. A person is an event if they are involved in an accident, and a person is a data point if they are injured.

The average of a person's previous injuries and their current injuries is used to calculate the first statistic. For example, a person who was injured in a motorcycle accident is the average length of time they have been injured.

A person who is injured in an automobile accident is the most common injury at the point of injury. The average of the injury lengths, average injuries, and average distances between the injury and the person are used to calculate the second and third most commonly referenced statistics.

R Statistics is used to make maps and understand the size of an area. The average size of a small town is used as a reference point. The Town Board can provide statistics about a town, and the City Council can submit a report to fill in the information needed.

Real numbers are a form of statistics that are used to create facts. They have their origin in the physical world but can be subject to mistakes. Real numbers are equal to zero when they are equal to real numbers.

The difference between two real numbers is the real difference. Numerical analysis can be used to say that a number is equal to a real number if it is equal in real number to a real. The difference between real and real numbers is 1 if the value of the difference is 1.

Results show that the difference between both real numbers is the difference in the difference between Real Number and Difference. The difference of two numbers is the real change of difference in difference between two numbers.

Examples of real differences include the difference of 1, which takes one more number and the difference of 1 is 2. The difference of two numbers and Difference of Real Numbers have the same Difference.

In conclusion, R Statistics is a useful tool for understanding physical experiences and average injury lengths. It is used to calculate the average of injuries sustained in a car crash and to compare data extracted from source documents. Real numbers are a form of statistics that are used to create facts, and the difference between two real numbers is the real difference.


**Section 1: Cryptography**

Choose one question from Q.1(a, b) and Q.2(a, b).

1(a) Compute the inverse of 17 in mod 23 arithmetic, showing steps clearly. (6 marks)

1(b) State Euler's theorem. (4 marks)

2(a) Prove that 13 is a primitive root of 19, providing proper working. (6 marks)

2(b) Apply the Diffie-Hellman key exchange between two parties A and B, where A's secret is 9 and B's secret is 6, using 13 as the primitive root of 19. Show the shared secret and math working steps clearly. (4 marks)

**Section 2: Cryptography and Security**

Choose one question from Q.3(a, b) and Q.4(a, b).

3(a) Define cryptanalysis and mention the applications of public key cryptography. (6 marks)

3(b) List the problems associated with the one-time pad. (4 marks)

4(a) Describe the purpose of the S-box in DES. (6 marks)

4(b) Give examples of types of attacks. (4 marks)

**Section 3: Web Security**

Choose one question from Q.5 and Q.6(a, b).

5. Describe how SSL counters the following web security threats:
    a) Brute force attacks
    b) Known plaintext attacks
    c) Replay attacks
    d) Man-in-the-middle attacks
    e) Password sniffing
    f) IP spoofing
    g) IP hijacking
    h) SYN flooding (16 marks)

6(a) Explain the difference between tunnel and transport mode in IPSEC and how it defends against replay attacks. (8 marks)

6(b) What protocols comprise SSL? What is the difference between an SSL connection and an SSL session? (8 marks)

**Section 4: Security Threats and Protocols**

Choose one question from Q.7(a, b) and Q.8(a, b).

7(a) Explain statistical anomaly detection and rule-based instruction detection, highlighting their differences. (8 marks)

7(b) Consider a machine M1 hosting a https-based public website. Is it possible for an intruder with no access to M1 to set up a fake website with the same URL on their machine M2 and go unnoticed? (8 marks)

8(a) Briefly explain trap doors, trojan horses, worms, and zombies. (8 marks)

8(b) With a neat diagram, explain the process of digital signing and digital verification. (8 marks)

**Section 5: Cybersecurity and Cybercrime**

Choose one question from Q.9 and Q.10.

9. Write notes on:
    a) Cyberstalking
    b) Cybercrime and cloud computing
    c) Phishing (18 marks)

10. Write notes on:
    a) Social engineering attacks
    b) ITA 2000
    c) DoS and DDoS attacks (18 marks)


## Lesson: Identify Plane Shapes and Solid Shapes

Developing the Concept

This lesson builds on the "Introducing the Concept" lesson, where children practiced sorting and classifying plane shapes. Now, they will compare solid shapes and relate plane shapes to the faces of solid shapes.

**Materials:**
- Geometric solids or manipulatives: cubes, pyramids, rectangular prisms, cones, cylinders, and spheres (1 manipulative for every 4-5 children)
- Objects: small rubber ball, unopened tissue box (rectangular prism), cube-shaped box, can, cone-shaped party hat, pyramid
- Worksheets: Solids (PDF file) and Face Match (PDF file) for each child and an overhead copy

**Preparation:**
1. Make copies of the worksheets for each child and an overhead copy.
2. Set out the objects for visibility.
3. Divide children into groups of 4-5.

**Prerequisite Skills and Concepts:**
Children should know the names and attributes of plane shapes.

**Introduction:**
Ask children to name four basic plane shapes and describe their corners and sides.

**Activity:**
Give each child a Solids worksheet and display the transparency copy on the overhead.

**Wrap-Up and Assessment Hints:**
Assess children's knowledge with "I'm thinking of…":
1. Give each child a set of shape cards.
2. Describe a solid or plane shape's attributes, e.g., "I'm thinking of a solid shape with one round face."
3. Check if children hold up the correct card (e.g., cone for the description).
4. Repeat for each shape.


In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back.

Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space are Cartesian, cylindrical, and spherical polar coordinates. A Cartesian coordinate surface in this space is a plane, while the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, which is curved.

Curvilinear coordinates are often used to define the location or distribution of physical quantities, such as scalars, vectors, or tensors. Mathematical expressions involving these quantities can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors.

Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system.

Spherical coordinates are one of the most used curvilinear coordinate systems in fields such as Earth sciences, cartography, and physics, and engineering. They are particularly useful for describing the motion of a particle in a sphere or for problems with spherical symmetry. Other curvilinear coordinate systems, such as cylindrical coordinates, are used for problems with cylindrical symmetry.

The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. It allows for the transformation of mathematical expressions between different coordinate systems, making it a powerful tool for solving problems in physics, engineering, and other fields.


In this article, we aim to provide a straightforward solution to a generic first-order differential equation. This equation often appears in various contexts, such as simple RC circuits, and can be frustrating to encounter repeatedly. To make the solution accessible to those with limited mathematical knowledge, we will break it down into manageable steps.

A generic first-order linear differential equation can be represented as:

To solve this equation, we start by multiplying all terms by a function called u(x), resulting in equation (1).

The product rule of the derivative states that:

Applying this rule to u(x)*y, we get:

Notice that the second part of formula (2) is identical to the first part of formula (1). To make these two parts equal, we need to dimension P(x) correctly using the formula:

Since P(x) is known, but u(x) is not, we must solve equation (3) by integrating both parts, which gives us:

Recall that:

Integrating both parts, we obtain:

Using equations (4) and (5), we can derive:

From the definition of the logarithm, we know that:

Applying this definition to equation (6) and setting C=0, we get:

Setting:

and using equations (1) and (2), we can integrate both parts to obtain:

The solution to the differential equation is:

In the next section, we will explore particular cases of this generic solution.


To find common multiples, we need at least two numbers. Common multiples are the multiples that both numbers share. The first three common multiples of 2 and 7 and 8 are not directly provided, but we can analyze the information given to understand how to approach this.

The first three multiples of 6 are 6, 12, and 18. The first three multiples of 456 are 456, 912, and 1368. For 4, 5, and 6, the first three common multiples are 60, 120, and 180.

Multiples of a number include that number and all its products when multiplied by integers (excluding zero). For example, multiples of 5 include 5, 10, 15, 20, and so on. To be common, these multiples need to be compared to another list of multiples.

The first three multiples of 3 and 5 are 15, 30, and 45. The first three multiples of 25 are 25, 50, and 75.

The concept of common multiples involves finding numbers that are multiples of two or more given numbers. The least common multiple (LCM) is the smallest number that is a multiple of both numbers. For example, the LCM of 6 and 10 is 30, and their common multiples are 30, 60, 90, 120, 180, etc.

Given the options for the first three common multiples of 2, 7, and 8:
- 30, 60, and 90
- 88, 176, 264
- 42, 84, 126

We need to identify the correct set. The LCM of 2, 7, and 8 would give us the first common multiple. The LCM of 2 and 7 is 14, and the LCM of 14 and 8 is 56. Thus, the first three common multiples of 2, 7, and 8 would be multiples of 56, which are 56, 112, and 168. None of the provided options directly match this calculation, indicating a need to directly calculate or verify the LCM and its multiples for 2, 7, and 8.

The correct approach to finding common multiples involves identifying the LCM of the given numbers and then listing its multiples. For 2, 7, and 8, the correct first three common multiples, based on the LCM, would indeed be 56, 112, and 168, which does not match any of the provided choices directly.

Common multiples of numbers like 9 and 10 occur exactly once in their list, with their LCM being 90. All numbers have multiples, and common multiples are those shared between two or more sets of multiples. The least common multiple is the smallest number in the list of common multiples.

In summary, to find common multiples, especially for the question regarding 2, 7, and 8, one must calculate the LCM of these numbers and then list its first three multiples. The provided options do not directly include the correct answer based on the LCM calculation, which would be 56, 112, and 168 for 2, 7, and 8.


## NCERT Solutions for Class 10 Maths Chapter 14 Statistics (Ex 14.2) Exercise 14.2

**Introduction to Statistics**

Statistics is a subject concerned with the collection, organization, analysis, interpretation, and presentation of data. It is used in various fields for data analysis purposes and requires technical and cognitive skills. The government uses statistics to calculate indicators such as unemployment and stock market trends.

**Importance of NCERT Solutions**

NCERT Solutions provide quality solutions to every question in the exercise, helping students gain experience and score better marks in the final exams. These solutions are productive in student preparation and can be downloaded from the Vedantu website or app.

**Completing Exercise 14.2**

To complete Exercise 14.2, students should allocate at least one hour to understand the concepts. Continuous practice is essential to attain desired results. NCERT Solutions can be referred to for easier studying.

**Ungrouped Data**

Ungrouped data is represented in its original or raw form, where observations are not classified into groups. For example: 5, 3, 4, 3, 5, 4, 4, 3, 4, 4, 3, 3, 3, 4, 3, 27. Analyzing ungrouped data is easier with small sets, but becomes challenging with larger quantities.

**Direct Method of Finding the Mean**

To find the mean using the direct method:

1. Classify data into intervals and find the corresponding frequency.
2. Find the class mark by considering the midpoint of the upper-class limits and the lower class limits.
3. Group and tabulate the product of the class mark and its corresponding frequency.
4. Divide the acquired sum by the sum of the frequencies to attain the mean.

**Assumed Mean Method**

To find the mean using the assumed mean method:

1. Classify data into intervals and find the corresponding frequency.
2. Find the class mark by considering the midpoints of the upper-class limits and the lower class limits.
3. Assume one of the class marks as the mean.
4. Find the deviation of each data point from the assumed mean (di = xi - a).
5. Calculate the mean of the deviations.

**Availability of NCERT Solutions**

NCERT Solutions for Class 10 Maths Chapter 14 are available and can be downloaded from the Vedantu website or app. These solutions contain detailed and explained answers, making the learning process smoother and simpler for students. Regular practice of these exercises will enable students to solve any question from this chapter.


To determine if a number is divisible by a certain number, the following rules can be applied:

1. Divisibility by 1: All numbers are divisible by 1.
2. Divisibility by 2: If the number is even, it is divisible by 2.
3. Divisibility by 3: If the sum of its digits is divisible by 3, then the number is also divisible by 3.
4. Divisibility by 4: If the number ends in 0 or a number that is divisible by 4, then it is divisible by 4.
5. Divisibility by 5: If the number ends in 0 or 5, it is divisible by 5.
6. Divisibility by 6: If the number is divisible by both 3 and 2, it is divisible by 6.
7. Divisibility by 7: Take the last digit, double it, and subtract it from the rest of the number. If the result is divisible by 7 (including 0), then the number is also divisible by 7. For example, 861: 86 - 1x2 = 84, and 84/7 = 12.
8. Divisibility by 8: If the last three digits of the number are divisible by 8, then the whole number is divisible by 8.
9. Divisibility by 9: If the sum of its digits is divisible by 9, then the number is also divisible by 9.
10. Divisibility by 10: If the number ends in 0, it is divisible by 10.
11. Divisibility by 11: Add the digits in the odd positions from left to right, and add the digits in the even positions. Find the difference between these two sums. If the difference is 0 or a multiple of 11, then the original number is divisible by 11. For example, 1320: (1+3) - (2+0) = 2, but a correct example is needed.
12. Divisibility by 12: If the number is divisible by both 3 and 4, it is divisible by 12.

These rules provide a quick way to determine if a number is divisible by a certain number without performing the actual division.


The angle sum of a polygon is related to the number of its sides. For a triangle, the sum of the interior angles is 180 degrees. This fact is crucial in geometry and is used in various theorems, including the exterior angle theorem and the polygon angle sum formula.

The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180 degrees. This theorem is used in many algebra and geometry problems. To prove this, consider a triangle with angles 65 degrees, x degrees, and 30 degrees. Since these angles form a straight line, their sum must be 180 degrees. Therefore, 65 + x + 30 = 180, and solving for x gives x = 85.

The angle sum identities are used to find the function value for the sum of two angles. For example, using the identity for the sine of a sum, we can find the sine of 75 degrees by determining two angles whose sum is 75 and for which we know the values of both sine and cosine.

The sum of the interior angles of a polygon with n sides is given by the formula: Sum of Interior Angles = (n-2) * 180. This formula is used to find the sum of the interior angles of any polygon.

In the case of a quadrilateral, the sum of the interior angles is 360 degrees, since (4-2) * 180 = 360. For a triangle, the sum of the interior angles is 180 degrees, since (3-2) * 180 = 180.

The angle sum theorem has various applications in geometry and trigonometry. It is used to find missing angles in triangles and polygons, and to prove other theorems, such as the exterior angle theorem.

Some key concepts related to angle sum include:

* Interior angles: angles inside a shape
* Exterior angles: angles outside a shape
* Straight line: a line with an angle of 180 degrees
* Sine, cosine, and tangent: trigonometric functions used to find the values of angles

Some important formulas and theorems related to angle sum include:

* Sum of Interior Angles = (n-2) * 180
* Triangle Angle Sum Theorem: the sum of the measures of the interior angles of a triangle is 180 degrees
* Exterior Angle Theorem: the sum of the exterior angles of a polygon is 360 degrees

Multiple choice questions:

1. What is the sum of the interior angles of a triangle?
a) 90 degrees
b) 180 degrees
c) 270 degrees
d) 360 degrees

Answer: b) 180 degrees

2. What is the formula for the sum of the interior angles of a polygon with n sides?
a) (n-2) * 180
b) (n+2) * 180
c) (n-1) * 180
d) (n+1) * 180

Answer: a) (n-2) * 180

3. What is the sum of the interior angles of a quadrilateral?
a) 180 degrees
b) 270 degrees
c) 360 degrees
d) 450 degrees

Answer: c) 360 degrees


### Statement Snap
To win a game of Statement Snap, knowledge of number properties is necessary.

### Factors and Multiples Game
In this game, players take turns choosing a number, with the goal of blocking their opponent.

### Multiples Sudoku
Each clue in this Sudoku is the product of two numbers in adjacent cells.

### How much can we spend?
A country uses 3z and 5z coins. The possible totals that can be made and whether there is a largest total that cannot be made are to be determined.

### Diminishing Returns
The proportion of the square that is colored blue and how the pattern will continue are to be determined.

### Sieve of Eratosthenes
This recipe for sieving numbers reveals interesting patterns.

### Gabriel's Problem
Gabriel multiplied numbers together and erased them. The original numbers are to be determined.

### American Billions
The divisibility game involves creating numbers where the first two digits are divisible by 2, the first three digits are divisible by 3, and so on.

### Factors and Multiples Puzzle
Using knowledge of number properties, all squares on the board are to be filled.

### Power mad!
The behavior of powers of numbers is to be explained.

### Alison's quilt
A rectangle is formed by nine squares. The dimensions of the rectangle are to be found.

### Counting Factors
An efficient method to determine the number of factors of a large number is to be found.

### Funny Factorisation
The number 4396 can be written as the product of two numbers using the digits 1 to 9. The factors are to be determined.

### Take Three From Five
Caroline and James pick sets of five numbers, and Charlie tries to find three that add up to a multiple of three. A strategy to stop Charlie is to be determined.

### Stars
The step size to visit all dots on a circle is to be determined.

### Product Sudoku
The clues for this Sudoku are the product of numbers in adjacent squares.

### Route to infinity
The route to infinity is to be described, and the next step is to be determined.

### Differences
A guarantee that the product of the differences of any three numbers is always even is to be determined.

### Shopping Basket
The prices of items in a shopping basket that add and multiply to the same amount are to be determined.

### Beelines
The relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses is to be determined.

### LCM Sudoku
This Sudoku uses information about lowest common multiples to help solve it.

### Expenses
The largest number that, when divided into 1905, 2587, 3951, 7020, and 8725, leaves the same remainder each time is to be determined.


**Theory**

The principal distance, denoted as **f**, is the distance from the projection centre to the picture plane, equivalent to the focal length in photography. This distance is represented by **f** = (**OO°**), where **OO°** is the principal distance.

A distance circle with a radius equal to the principal distance represents the projection centre, allowing it to be placed in space relative to the picture plane. The centre of projection is located on the perpendicular to the picture plane, conducted at a distance equal to the radius of the distance circle, and passing through the principal point.

**Practice**

To apply this concept, draw a circle with a diameter close to the width of the paper. The projection centre **O** will be on the perpendicular to the paper, passing through the principal point **O°**, which is the centre of the distance circle.

**Notes**

For a restricted sight, the eye must be placed at point **O**, in the space in front of the paper. In this position, the eye can rotate freely, allowing for various perspectives.


## How to use the FALSE function

The FALSE function returns the logical value FALSE. It has no arguments and is used for compatibility with other software.

### FALSE Function Syntax

=FALSE()

### FALSE Function Example

The boolean value FALSE is interpreted the same as =FALSE(). A formula in cell C3 can be used to demonstrate this:

1=2 returns FALSE.

The IF function can also be used with the FALSE function:

IF(*logical_test*, [*value_if_true*], [*value_if_false*])

IF(1=2, TRUE(), FALSE()) returns FALSE.

### Uses of the FALSE function

The FALSE function is often used in combination with other functions, such as the IF function, to test logical conditions. It can also be used to create unique distinct lists, search text in cells, and filter records based on date ranges.

### Examples of the FALSE function in use

* Creating a list of unique words from a cell range
* Applying Conditional Formatting formulas to a cell range
* Highlighting lookups across relational tables
* Using the VLOOKUP function with multiple conditions
* Using the INDEX and MATCH functions to lookup and return multiple results
* Merging or combining two data sets using a condition
* Creating a calendar that automatically highlights dates based on date ranges

### Functions in the 'Logical' category

The FALSE function is one of many functions in the 'Logical' category, which includes functions such as the IF function, the AND function, and the NOT function. These functions can be used to test logical conditions and make decisions based on those conditions.

### Best practices for using the FALSE function

* Use the FALSE function in combination with other functions to test logical conditions
* Use the FALSE function to create unique distinct lists and filter records
* Use the FALSE function to highlight lookups across relational tables and merge data sets
* Use the FALSE function in conjunction with other 'Logical' category functions to make decisions based on logical conditions.


# Contribution Margin

### Rearranging the Profit Equation:

The profit equation is: Profit = Selling Price (x) – Variable Costs (x) – Total Fixed Costs. Rearranging it yields: Profit = (Selling Price – Variable Costs)x – Total Fixed Costs. The part in the bracket, Selling Price – Variable Costs, is known as the contribution margin.

### What Is The Contribution Margin?

The contribution margin is the amount each sale contributes towards recovering fixed costs. Every sale generates Selling Price – Variable Cost, which contributes to recovering fixed costs. Once fixed costs are recovered, the breakeven point is reached, and the contribution margin contributes to increasing profit.

### Contribution Margin Ratio:

The contribution margin can be any number, but it's more useful in context. Consider two products: one with a $50 selling price and $15 contribution margin, and another with a $100 selling price and $25 contribution margin. The contribution margin ratio is calculated as: Contribution Margin Ratio = Contribution Margin/Selling Price = (Selling Price – Variable Costs)/Selling Price.

### Calculating the Contribution Margin Ratio:

Using the formula, the contribution margin ratio for the first product is $15/$50 = 30%, and for the second product is $25/$100 = 25%.

### Interpreting the Contribution Margin Ratio:

A higher contribution margin ratio means a faster breakeven point. Companies should prioritize selling high-contribution products, but not exclusively, as products with lower contribution margins may sell faster and in higher volumes. Neglecting these sales can have adverse effects on profitability.

The contribution margin ratio is a key metric in understanding a company's profitability and making informed decisions. By calculating and interpreting this ratio, companies can optimize their sales strategies and improve their financial performance.


Mathematics is a subject that stays with us throughout our lives, and it is essential in various daily activities such as counting money, telling time, and dialing phone numbers. One of the branches of mathematics is solving radical equations.

**What is a Radical Equation?**

A radical equation is an equation in which at least one variable expression is inside a radical, usually a square root. The symbol **n****x** represents the nth root of x, and an equation containing a radical expression is called a radical equation. Examples of radical expressions include 5, 4z, 17a+1, 664, and 5±29.

**How to Solve Radical Equations?**

To solve radical equations, follow these steps:

1. Isolate the radical expression concerning the variable.
2. Raise both sides of the equation to the power of the index of the radical.
3. Eliminate the radical.
4. If there is still a radical equation, repeat steps 1, 2, and 3. Otherwise, solve the resulting equation and check its answer in the original equation.

**Examples of Solving Radical Equations**

Example 1: Solve 5x-4 – 9 = 0.

Solution:

Step 1: Isolate the radical by adding 9 to both sides.
5x-4 = 9

Step 2: Raise both sides of the index to the power of the index, i.e., 2.
[5x-4]² = (9)²

Step 3: Solve the equation for the variable.
5x = 81 + 4
5x = 85
x = 85/5
x = 17

Example 2: Solve 7 + a-3 = 1.

Solution:

Step 1: Isolate the radical by subtracting 7 on both sides.
a-3 = -6

Step 2: Raise both sides of the index to the power of the index, i.e., 2.
[a-3]² = (-6)²

Step 3: Solve the equation for the variable.
a = 36 + 3
a = 39

**Solving Radical Equations in Case of Odd Index**

In the case of radicals with odd indexes, it is possible to have negative answers. To solve such problems, follow these steps:

1. Isolate the radical expression.
2. Raise both sides of the equation to the index of the radical.
3. Solve to get the value of the variable.

Example: Solve 32x+3 + 5 = 2.

Solution:

Step 1: Isolate the radical expression by subtracting 5 on both sides.
32x+3 = -3

Step 2: Raise both sides of the index to the power of the index, i.e., 3.
[32x+3]³ = (-3)³

Step 3: Solve the equation for the variable.
2x + 3 = -27
2x = -27 - 3
2x = -30
x = -15

**Solving Radical Equations in Case of More than One Radical Expression**

To solve radical equations with more than one radical expression, repeat the steps for each radical expression.

Example: Solve 2x-5 – x-1 = 1.

Solution:

Step 1: Isolate the radical expressions.
2x-5 = 1 + x-1

Step 2: Raise both sides of the index to the power of the index, i.e., 2.
[2x-5]² = [1 + x-1]²

Step 3: Solve for the value of the variable.
2x - 5 = x + 2*x-1

**Frequently Asked Questions**

1. How do you solve radicals step by step?
Answer: Solve radicals by finding the square root of both sides, moving negative signs, multiplying by the conjugate, and simplifying.

2. What are examples of radical equations?
Answer: Examples of radical equations include √x + 3 = 5, 2√3x - 4 = 10, and (6√3) + 2 = 10.

3. How do you solve a square radical?
Answer: Solve a radical square problem by finding the perfect square root, getting rid of non-perfect squares, and taking the sum of the remaining terms.

4. What are the three steps to solving a radical equation?
Answer: The steps to solving a radical equation are factoring, using the rational root test, and using the remainder property of algebraic fractions.

5. How do you simplify radicals easily?
Answer: Simplify radicals by finding the root, multiplying by itself, and pulling out the root. If the answer is irrational, multiply it by itself until it becomes rational.


A student is invited to "act out" motion corresponding to a plot of effective potential vs. distance, playing the role of the "Earth" while the instructor plays the "Sun". This activity typically begins with a lecture about the classical equations of orbital motion, specifically a derivation that reformulates 2-D equations of orbital motion to 1-D by using the effective potential.

Before initiating the activity, a reference plot is made on the board, which can be contrasted with the harmonic oscillator potential energy plot for comparison. The activity is approximately 8 minutes long and can be referenced through a video clip.

The key takeaway from this activity is that the plot of the effective potential contains information about the angular momentum, while the corresponding energy graph only describes radial motion. To reinforce this concept, it is valuable to discuss the conservation of energy equation:

\[E = {{1}\over{2}} \mu \dot{r}^2 + {{1}\over{2}} {l^2 \over \mu r^2} - {k \over r}\]

Explicitly identifying the effective potential \({{1}\over{2}} {l^2 \over \mu r^2} - {k \over r}\), the radial kinetic energy \({{1}\over{2}} \mu \dot{r}^2\), and the \(\phi\) component of the kinetic energy \({{1}\over{2}} {l^2 \over \mu r^2}\) helps to solidify understanding of the concept.


Colors are used to indicate the opposing sides of conjugate pairs, allowing for potential candidate removal. A chain of conjugate pairs exists when multiple pairs are linked together, with cells colored in two alternating colors. This means cells from a single conjugate pair are never colored the same.

When a conjugate pair tree is formed using a single value, all cells of one color must contain the considered value, while cells of the other color cannot. The simple coloring process does not define which set of cells the value will end up in, but it can be used for candidate removal on other cells.

In Simple Coloring Type 1, a colored tree of conjugate pairs is created based on a value. For example, using the value 4, the tree is formed by coloring one cell of the first conjugate pair blue and the other green. The process continues until no more pairs can be found. Then, cells that can contain the value 4 are checked for a cell from the tree of each color in the same unit. If a cell is found to have the same color as a cell in the same unit, the value 4 can be removed from that cell.

The conjugate pair chain properties mean that all cells of one color must contain the value, or all cells of the other color must. This strategy does not determine which color will contain the value, but it can be used to remove candidates. For instance, if the value 4 is in the blue cells, it cannot be in a cell that shares a row or column with a blue cell. If the value 4 is in the green cells, it cannot be in a cell that shares a row or column with a green cell.

Simple Coloring Type 2 involves creating a colored chain of conjugate pairs based on a value, such as 8. The tree is formed by starting with any conjugate pair and building up the colored tree. Once complete, cells of the same color in the same unit are looked for. If two cells of the same color are found in the same unit, the value can be removed from all cells of that color, as it would result in duplicate values in the same unit.

The properties of conjugate pair chains mean that all cells of one color must contain the considered value, or all cells of the other color must. If all cells of one color contain the value, it cannot be in cells of the other color. This strategy can be used to remove candidates and solve Sudoku puzzles.


SoLE System Quiz

1. The equations y = 0.5x + 2 and y = -2x - 3 have ____ solution(s).
A. no
B. one
C. two
D. three

2. A system of two linear equations may have infinitely ____ solutions.
A. one
B. two
C. three
D. many

3. A system of two linear equations may have no _______.
A. variables
B. solutions
C. a and b
D. none of the above

4. A system of linear equations is a set of ___ or ____ linear equations.
A. one or two
B. two or three
C. two or more
D. none of the above

5. If lines intersect at zero points, they are _____.
A. independent
B. intersecting
C. parallel
D. a and b

6. Solve the system: 5+2y=x-3; 3x+2y=11
A. (13/4,19/3)
B. (-19/4,13/8)
C. (-13/8,19/4)
D. (19/4,-13/8)

7. To eliminate x from: 6x + 10y = 15 and 10x - 2y = 11, we could
A. multiply the first equation by -5 and the second by 3
B. multiply the first equation by 11 and the second by -15
C. multiply the first equation by 10 and the second by 6
D. leave the first equation unchanged and multiply the second by 5

8. Solve the system: y = -x + 7 and 9 + x + y = 0
A. (2,1)
B. (-3,4)
C. (-1,0)
D. this system has no solutions

9. Solve the system: 2x + 3y = 5 and y + 2x = 7
A. (4,-2)
B. (-2,4)
C. (4,-1)
D. (-2,-5)

10. For the system: x + y = 4 and -3x + 5y = 14, use
A. Substitution
B. Addition
C. a or b
D. none of the above

11. A solution to a system of equations
A. must satisfy one or more equations
B. must not satisfy any equations
C. is a collection of points for each equation
D. must satisfy every equation

12. If two lines are parallel, they have
A. one point of intersection
B. either no points or infinitely many points of intersection
C. no points of intersection
D. infinitely many points of intersection

13. For the system: 3x - 2y = 14 and 11x + 12y = 33, use
A. Addition
B. Substitution
C. a and b
D. none of the above


# Cantor's Theorem/Proof 1

## Theorem

There is no surjection from a set $S$ to its power set for any set $S$, meaning $S$ is strictly smaller than its power set.

## Proof

To prove this by contradiction, assume $S$ has a surjection $f: S \to \powerset S$. Then, for all $x \in S$, $\map f x \in \powerset S$ and $\map f x \subseteq S$ by the definition of a power set.

For every $x \in S$, there are two possibilities due to the Law of Excluded Middle:
- $x \in \map f x$
- $x \notin \map f x$

Let $T = \set {x \in S: x \notin \map f x}$. Since $f$ is a surjection, there exists $a \in S$ such that $T = \map f a$. This leads to two contradictory statements:
- $a \in \map f a \implies a \notin \map f a$
- $a \notin \map f a \implies a \in \map f a$

This contradiction implies the initial assumption of a surjection from $S$ to its power set is false.

#### Law of the Excluded Middle

This proof relies on the Law of the Excluded Middle, a fundamental axiom in classical logic first identified by Aristotle. However, the intuitionist school of logic rejects this law, which in turn invalidates this proof from their perspective.

## Sources

- 1968: A.N. Kolmogorov and S.V. Fomin, *Introductory Real Analysis*, Theorem 6.
- 1971: Allan Clark, *Elements of Abstract Algebra*, Chapter 1, Section 14.
- 1996: H. Jerome Keisler and Joel Robbin, *Mathematical Logic and Computability*, Appendix A.6, Theorem A.6.1.
- 2010: Raymond M. Smullyan and Melvin Fitting, *Set Theory and the Continuum Problem*, Chapter 1, Section 3.


Lemma 17.17.5 states that for a ringed space $(X, \mathcal{O}_ X)$ and an open subset $U \subset X$, the sheaf $j_{U!}\mathcal{O}_ U$ is a flat sheaf of $\mathcal{O}_ X$-modules.

The proof involves analyzing the stalks of $j_{U!}\mathcal{O}_ U$, which are either zero or equal to $\mathcal{O}_{X, x}$. This observation allows us to apply Lemma 17.17.2 to conclude the proof.

No multiple choice questions are provided in the text.


The International Mathematics Assessment for Schools (IMAS) consists of two rounds. The first round is for all participants, while the second round is for students who perform well in the first round.

**First Round of IMAS**
The first round comprises 25 problems, to be completed within 75 minutes. The problems are categorized as follows:
- Problems 1-10: EASY (3 points each)
- Problems 11-20: AVERAGE (4 points each)
- Problems 21-25: CHALLENGE (6 points each)
The first 20 problems are multiple-choice, while the last 5 require integer answers between 0 and 999.

To recognize students' achievements, IMAS provides awards based on performance relative to other students in their country and year level. The award scheme is as follows:
- **High Distinction**: Above 95 percentile
- **Distinction**: 85-95 percentile
- **Credit**: 50-85 percentile
- **Participation**: Below 50 percentile
Each participant receives a certificate with a personal report, detailing their achievement and percentile rank.

**Second Round of IMAS**
The second round is conducted for the top 10% of participants in each country. This test consists of 15 problems:
- Problems 1-5: Multiple-choice (4 points each)
- Problems 6-13: Short answers (5 points each)
- Problems 14-15: Detailed solutions (20 points each, with partial marks for incomplete answers)
Participants have 120 minutes to complete the test.

Awards for the second round are as follows:
- **Gold Medal**: Top 5% percentile
- **Silver Medal**: 6-15% percentile
- **Bronze Medal**: 16-30% percentile
If a country invites more than the top 10% of participants to the second round, the IMAS Executive Council will only award medals based on the top 10% of all participants.


# Implementation of Egyptian Fraction using Greedy Algorithm in C++

A positive fraction can be represented as the sum of unique unit fractions, where a unit fraction has a numerator of 1 and a denominator of a positive number. For example, 1/8 is a unit fraction.

Examples of Egyptian Fractions include:
- 5/6 = 1/2 + 1/3 (not 2/3 + 1/2 as previously mentioned, this seems to be an error)
- 8/15 = 1/3 + 1/5 (correct example)

## Egyptian Fraction using Greedy Algorithm in C++

To implement the Egyptian Fraction using a Greedy Algorithm in C++, follow these steps:

1. Get the numerator (n) and denominator (d) of the fraction.
2. Check for corner cases where d is 0 or n is 0.
3. If d is divisible by n, print 1/(d/n).
4. If n is divisible by d, print n/d.
5. Compare n and d:
   - If n > d, print n/d and recursively call the egyptianFraction() function with (n%d, d).
   - If d > n, find the smallest unit fraction that does not exceed the given fraction and recursively call the egyptianFraction() function.

The provided C++ code implements this algorithm:
```cpp
#include <bits/stdc++.h>
using namespace std;

void egyptianFraction(int n, int d){
    if (d == 0 || n == 0) return;
    if(d%n == 0){
        cout<<"1/"<<d/n;
        return;
    }
    if(n%d == 0){
        cout<<n/d;
        return;
    }
    if (n>d){
        cout<<n/d<<" + ";
        egyptianFraction(n%d, d);
        return;
    }
    int x = d/n+1;
    cout<<"1/"<<x<<" + ";
    egyptianFraction(n*x-d, d*x);
}

int main(){
    int n, d;
    cout<<"Enter the numerator of fraction: ";
    cin>>n;
    cout<<"Enter the denominator of fraction: ";
    cin>>d;
    cout<<"Egyptian Fraction representation of "<<n<<"/"<<d<<" is"<<endl;
    egyptianFraction(n, d);
    return 0;
}
```

Output examples:
- Input: n = 2, d = 3; Output: 1/2 + 1/6
- Input: n = 1, d = 6; Output: 1/6
- Input: n = 18, d = 19; Output: 1/2 + 1/3 + 1/9 + 1/342


### Polynomials in Standard Form

A **polynomial** is made up of different terms that contain **positive integer** powers of the variables. For example:

\begin{align*}4x^3+2x^2-3x+1\end{align*}

Each part of the polynomial that is added or subtracted is called a **term** of the polynomial. The numbers appearing in each term in front of the variable are called the **coefficients**. The number appearing all by itself without a variable is called a **constant**.

### Degrees of Polynomials and Standard Form

Each term in the polynomial has a different **degree**. The degree of the term is the power of the variable in that term. The degree of a polynomial is the same as the degree of the term with the highest degree.

For example:

\begin{align*}& 4x^3 && \text{has degree} \ 3 \ \text{and is called a cubic term or} \ 3^{rd} \ \text{order term}.\\ & 2x^2 && \text{has degree} \ 2 \ \text{and is called a quadratic term or} \ 2^{nd} \ \text{order term}.\\ & -3x && \text{has degree} \ 1 \ \text{and is called a linear term or} \ 1^{st} \ \text{order term}.\\ & 1 && \text{has degree} \ 0 \ \text{and is called the constant}.\end{align*}

Polynomials can have more than one variable. The degree of a term is the sum of the powers on each variable in the term.

### Identifying Constants and the Degree of a Polynomial

To identify the constants and degree of a polynomial, we need to look at each term and determine its degree.

For example:

\begin{align*}x^5-3x^3+4x^2-5x+7\end{align*}

The coefficients of each term in order are 1, -3, 4, and -5 and the constant is 7.

The degrees of each term are 5, 3, 2, 1, and 0. Therefore the degree of the polynomial is 5.

### Identifying Polynomials

To identify if an expression is a polynomial, we need to check if it has any negative exponents, radicals, or variables in the denominator.

For example:

a) \begin{align*}5x^5-2x\end{align*} is a polynomial.

b) \begin{align*}3x^2-2x^{-2}\end{align*} is not a polynomial because it has a negative exponent.

c) \begin{align*}x\sqrt{x}-1\end{align*} is not a polynomial because it has a radical.

d) \begin{align*}\frac{5}{x^3+1}\end{align*} is not a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator of a fraction.

e) \begin{align*}4x^\frac{1}{3}\end{align*} is not a polynomial because it has a fractional exponent.

f) \begin{align*}4xy^2-2x^2y-3+y^3-3x^3\end{align*} is a polynomial.

### Standard Form

A polynomial that is arranged in order of decreasing power is called **standard form**.

For example:

\begin{align*}4x^4-3x^3+2x^2-x+1\end{align*}

The first term of a polynomial in standard form is called the **leading term**, and the coefficient of the leading term is called the **leading coefficient**.

### Writing Polynomials in Standard Form

To write a polynomial in standard form, we need to rearrange the terms in order of decreasing power.

For example:

a) \begin{align*}7-3x^3+4x\end{align*} becomes \begin{align*}-3x^3+4x+7\end{align*}.

b) \begin{align*}ab-a^3+2b\end{align*} becomes \begin{align*}-a^3+ab+2b\end{align*}.

c) \begin{align*}-4b+4+b^2\end{align*} becomes \begin{align*}b^2-4b+4\end{align*}.

### Simplifying Polynomials

A polynomial is simplified if it has no terms that are alike. **Like terms** are terms in the polynomial that have the same variable(s) with the same exponents.

For example:

\begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms.

To simplify a polynomial, we need to combine like terms.

For example:

\begin{align*}x^2+\underline{6xy} - \underline{4xy} + y^2\end{align*}

We can simplify this polynomial by combining the like terms \begin{align*}6xy\end{align*} and \begin{align*}-4xy\end{align*} into \begin{align*}(6-4)xy\end{align*}, or \begin{align*}2xy\end{align*}. The new polynomial is \begin{align*}x^2+2xy+y^2\end{align*}.

### Example

Simplify and rewrite the following polynomial in standard form. State the degree of the polynomial.

\begin{align*}16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}

Start by simplifying by combining like terms:

\begin{align*} 16x^2y^3-3xy^5-2x^3y^2+2xy-7x^2y^3+2x^3y^2\end{align*}

is equal to

\begin{align*}(16x^2y^3-7x^2y^3)-3xy^5+(-2x^3y^2+2x^3y^2)+2xy\end{align*}

which simplifies to

\begin{align*}9x^2y^3-3xy^5+2xy.\end{align*}

In order to rewrite in standard form, we need to determine the degree of each term. The first term has a degree of \begin{align*}2+3=5\end{align*}, the second term has a degree of \begin{align*}1+5=6\end{align*}, and the last term has a degree of \begin{align*}1+1=2\end{align*}. We will rewrite the terms in order from largest degree to smallest degree:

\begin{align*}-3xy^5+9x^2y^3+2xy\end{align*}

The degree of a polynomial is the largest degree of all the terms. In this case that is 6.

### Review

Indicate whether each expression is a polynomial.

1. \begin{align*}x^2+3x^{\frac{1}{2}}\end{align*} - Not a polynomial
2. \begin{align*}\frac{1}{3}x^2y-9y^2\end{align*} - Polynomial
3. \begin{align*}3x^{-3}\end{align*} - Not a polynomial
4. \begin{align*}\frac{2}{3}t^2-\frac{1}{t^2}\end{align*} - Not a polynomial
5. \begin{align*}\sqrt{x}-2x\end{align*} - Not a polynomial
6. \begin{align*}\left ( x^\frac{3}{2} \right )^2\end{align*} - Polynomial

Express each polynomial in standard form. Give the degree of each polynomial.

1. \begin{align*}3-2x\end{align*} - \begin{align*}-2x+3\end{align*}, degree 1
2. \begin{align*}8-4x+3x^3\end{align*} - \begin{align*}3x^3-4x+8\end{align*}, degree 3
3. \begin{align*}-5+2x-5x^2+8x^3\end{align*} - \begin{align*}8x^3-5x^2+2x-5\end{align*}, degree 3
4. \begin{align*}x^2-9x^4+12\end{align*} - \begin{align*}-9x^4+x^2+12\end{align*}, degree 4
5. \begin{align*}5x+2x^2-3x\end{align*} - \begin{align*}2x^2+2x\end{align*}, degree 2


**Part A: Instruction**

Only one of the choices A, B, C, D is correct. For each correct response, you get 1 mark. For each incorrect response, you lose 1/2 mark.

**NMTC 2017 Paper For Sub Junior Level Ques No 1**

The fraction 4/37 is written in the decimal form 0.a1a2a3….. The value of a2017 is

**Options:**

A. 8
B. 0
C. 1
D. 5

**Solution:**
4/37 = 0.108108108…
2017 is divisible by 3. Thus, the remainder is 1. Therefore, a2017 digit is 1.

**NMTC 2017 Paper For Sub Junior Level Ques No 2**

The number of integers x satisfying the equation (x2 – 3x + 1)x+1 = 1 is

**Options:**

A. 2
B. 3
C. 4
D. 5

**NMTC 2017 Paper For Sub Junior Level Ques No 3**

The number of two-digit numbers ab such that the number ab – ba is a prime number is

**Options:**

A. 0
B. 1
C. 2
D. 3

**Solution:**
ab – ba = 10a + b – 10b – a = 9a – 9b = 9(a – b). So, 9 is always the factor of ab – ba. So, there is no such number present.

**NMTC 2017 Paper For Sub Junior Level Ques No 4**

If A = 5425/1444 + 2987/3045 – 493/4284

**Options:**

A. 1 < A < 2
B. 2 < A < 3
C. 3 < A < 4
D. A < 1

**NMTC 2017 Paper For Sub Junior Level Ques No 5**

What is the 2017th letter in ABRACADABRAABRACADABRA…, where the word ABRACADABRA is repeatedly written?

**Options:**

A. A
B. B
C. C
D. R

**NMTC 2017 Paper For Sub Junior Level Ques No 6**

How many of the following statements are true?

1. A 10% increase followed by another 5% increase is equivalent to a 15% increase.
2. If the radius of a circle is doubled then the ratio of the area of the circle to the circumference is doubled.
3. If a positive fraction is subtracted from 1 and the resulting fraction is again subtracted from 1 we get the original fraction.

**Options:**

A. 0
B. 1
C. 2
D. 3

**NMTC 2017 Paper For Sub Junior Level Ques No 7**

In the adjoining figure, the breadth of the rectangle is 10 units. Two semicircles are drawn on the breadth as diameter. The area of the shaded region is 100 sq units. The shortest distance between the semicircles is

**Options:**

A. 5Pi/2
B. 5Pi
C. 5Pi/3
D. 3Pi/4

**NMTC 2017 Paper For Sub Junior Level Ques No 8**

When you arrange the following in descending order: (A) 15% of 30, (B) 8% of 15, (C) 20% of 20, (D) 26% of 10, (E) 9% of 25. The middle one is

**Options:**

A. 15% of 30
B. 8% of 15
C. 20% of 20
D. 26% of 10

**NMTC 2017 Paper For Sub Junior Level Ques No 9**

A boy aims at a target shown in the figure. When he hits the center circle he gets 7 points, first annular region 3 points. He shoots six times. Which one of the following is a possible score?

**Options:**

A. 16
B. 26
C. 19
D. 41

**NMTC 2017 Paper For Sub Junior Level Ques No 10**

After simplifying the fraction, We get a term independent of

**Options:**

A. a, y
B. b, x
C. a, b
D. x, y

**NMTC 2017 Paper For Sub Junior Level Ques No 11**

If 7 Rasagullas are distributed to each boy of group, 10 Rasagullas would be left. If 8 are given to each boy then 5 Rasagullas would be left. So the person who distributes the Rasagullas brought 15 more Rasagullas and distributed the same number (x) Rasagullas to each. There is no Rasagullas left. Then x is

**Options:**

A. 10
B. 11
C. 12
D. 14

**NMTC 2017 Paper For Sub Junior Level Ques No 12**

In the adjoining diagram, all squares are of the same size. The total area of the figure is 288 square cms. The perimeter of the figure (in cm) is

**Options:**

A. 86
B. 96
C. 106
D. 92

**NMTC 2017 Paper For Sub Junior Level Ques No 13**

When Newton was a primary school student he had to multiply a number by 5. But by mistake he divided the number by 5. The percentage error he committed is

**Options:**

A. 95%
B. 96%
C. 50%
D. 75%

**NMTC 2017 Paper For Sub Junior Level Ques No 14**

ABC is an isosceles triangle with sides AB = AC = 3x – 4 = 3x/4 + 32. The area of the equilateral triangle with side length x is

**Options:**

A. 32
B. 36
C. 54
D. 64

**NMTC 2017 Paper For Sub Junior Level Ques No 15**

Two distinct numbers a and b are selected from 1, 2, 3, . . . 60. The maximum value of ab/(a – b) is

**Options:**

A. 6750
B. 5270
C. 4850
D. 3540

**NMTC 2017 Paper For Sub Junior Level Ques No 16**

Two cogged wheels of which one has 16 cogs and the other 27 cogs, mesh into each other. If the latter turns 80 times in three quarters of a minute, the number of turns made by the other in 8 seconds is

**NMTC 2017 Paper For Sub Junior Level Ques No 17**

If n is a positive integer such that a2n = 2, then 2a6n – 16 is

**NMTC 2017 Paper For Sub Junior Level Ques No 18**

The least number of children in a family such that every child has at least one sister and one brother is

**Solution:**
Family should have at least four children (2 male and 2 female).

**NMTC 2017 Paper For Sub Junior Level Ques No 19**

A water tank is 4/5 full. When 40 liters of water is removed, it becomes 3/4 full. The capacity of the tank in liters is

**NMTC 2017 Paper For Sub Junior Level Ques No 20**

ABC is an equilateral triangle. Squares are described on the sides AB and AC as shown. The value of x is

**NMTC 2017 Paper For Sub Junior Level Ques No 21**

ABCD is a trapezium with AB = 6 cm, AD = 8 cm and CD = 18 cm. The sides AB and CD are parallel and AD is perpendicular to AB. P is the point of intersection of AC and BD. The difference between the areas of the triangle PCD and PAB is

**NMTC 2017 Paper For Sub Junior Level Ques No 22**

The price of cooking oil has increased by 25%. The percentage of reduction that a family should effect in the use of oil so as not to increase the expenditure is

**NMTC 2017 Paper For Sub Junior Level Ques No 23**

The number of natural numbers between 99 and 999 which contains exactly one zero is

**NMTC 2017 Paper For Sub Junior Level Ques No 24**

In the adjoining figure, we have semicircle and AB = BC = CD. The ratio of the unshaded area to the shaded area is

**NMTC 2017 Paper For Sub Junior Level Ques No 25**

Gold is 19 times as heavy as water and copper is 9 times as heavy as water. The ratio in which these two metals be mixed so that the mixture is 15 times as heavy as water is

**NMTC 2017 Paper For Sub Junior Level Ques No 26**

Five angles of a heptagon (seven sides polygon) are 160 degree 135 degree, 185 degree, 145 degree and 125 degree. If the other two angles are both equal to x degree, then x is

**NMTC 2017 Paper For Sub Junior Level Ques No 27**

ABCD is a trapezium with AB parallel to CD and AD perpendicular to AB. If AB = 23 cm. CD = 35 cm and AD = 5 cm. The perimeter of the given trapezium in cm is

**NMTC 2017 Paper For Sub Junior Level Ques No 28**

The number of three-digit numbers which are multiple of 11 is

**NMTC 2017 Paper For Sub Junior Level Ques No 29**

If a, b are the digits, ab denotes the number 10a + b. Similarly, when a, b, c are digits, abc denotes the number 100a + 10b + c. If X, Y, Z are digits such that XX + YY + ZZ = XYZ, then (XX)x(YY)x(ZZ) is

**NMTC 2017 Paper For Sub Junior Level Ques No 30**

The positive integer n has 2, 5 and 6 as its factor and the positive integer m has 4, 8, 12 as its factors. The smallest value of m + n is


**Properties of Union of Sets**

1. **Idempotent Law**: A ∪ A = A
2. **Identity Law**: A ∪ φ = A
3. **Commutative Property**: A ∪ B = B ∪ A
4. **Associative Property**: (A ∪ B) ∪ C = A ∪ (B ∪ C)
5. **Law of U**: U ∪ A = U

**Examples of Union of Sets**

1. If A = {1, 2, 3, 4} and B = {2, 3, 5, 6, 7}, then A ∪ B = {1, 2, 3, 4, 5, 6, 7}.
2. If P = {1, 2, 3, 4} and Q = {x | x ∉ N, 1 < x < 8}, then Q = {2, 3, 4, 5, 6, 7} and P ∪ Q = {1, 2, 3, 4, 5, 6, 7}.
3. If A = {x | x is a multiple of 2} and B = {x | x is an odd natural number}, then A = {2, 4, 6, 8, 10, …} and B = {1, 3, 5, 7, 9, 11, …}, so A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …}.

**Key Concepts in Set Theory**

* Sets
* Representation of Set
* Cardinal Number
* Types of Sets
* Pairs of Sets
* Subset
* Complement of Set
* Union of Sets
* Intersection of Sets
* Operations on Sets
* De Morgan's Law
* Venn Diagrams
* Problems on Intersection of Sets


Innumeracy refers to the inability of people to understand numbers, statistics, and probability. The term "numeracy" was first used in a 1959 report by Geoffrey Baron Crowther, and "innumeracy" was coined by Douglas Hofstadter and popularized by John Allen Paulos in his book "Innumeracy". Numeracy is defined as an understanding of basic mathematical concepts, including addition, subtraction, multiplication, division, fractions, percents, and decimals.

Numeracy is essential for making informed decisions in everyday life, such as calculating the cost per item or understanding discounts. However, many people struggle with basic numeracy, and this can have significant consequences personally and for society as a whole. The potential problems caused by innumeracy are numerous, and it is crucial to address this issue.

One criticism of emphasizing numeracy is that calculators and technology can perform mathematical tasks, making basic math skills less necessary. However, working with numbers trains the mind for logical and critical thinking, allowing individuals to make better decisions. Basic numeracy enables people to make everyday comparisons, understand statistics, and take informed risks.

Numeracy also plays a critical role in major life decisions, such as buying a house or managing finances. Without a basic understanding of mathematics, individuals may struggle to make informed decisions, even with the aid of calculators. A simple test of numeracy can be administered using questions such as:

* A baseball and a bat cost $1.10 together, with the bat costing one dollar more than the ball. How much does the ball cost?
* A patch of lily pads doubles in size every day and covers the entire lake in 48 days. How long would it take for the patch to cover half of the lake?

The answers to these questions are $0.05 for the ball and 47 days for the patch to cover half of the lake, respectively. These questions highlight the importance of basic numeracy in everyday life.

Contrary to popular claims, studies have shown that numeracy rates among younger generations are improving. Research by W. L. Nelson et al. and Rashid et al. found that numeracy rates have been consistently improving each new generation, with recent generations showing more substantial gains in numeracy. Therefore, it is essential to continue promoting numeracy and addressing innumeracy to ensure that individuals have the necessary skills to make informed decisions in their personal and professional lives.

Key concepts and facts:

* Innumeracy refers to the inability to understand numbers, statistics, and probability.
* Numeracy is defined as an understanding of basic mathematical concepts.
* Basic numeracy is essential for making informed decisions in everyday life.
* Numeracy rates among younger generations are improving, contrary to popular claims.
* Simple tests of numeracy can be administered to assess basic math skills.

Important numbers:

* 1959: The year the term "numeracy" was first used.
* 48: The number of days it takes for the patch of lily pads to cover the entire lake.
* 47: The number of days it takes for the patch to cover half of the lake.
* $1.10: The total cost of the baseball and bat.
* $0.05: The cost of the ball.

Core concepts:

* Numeracy
* Innumeracy
* Basic mathematical concepts
* Critical thinking
* Informed decision-making

Multiple choice questions:

* What is the cost of the ball in the baseball and bat problem?
A) $0.10
B) $0.05
C) $1.00
D) $1.05

Answer: B) $0.05

* How long would it take for the patch of lily pads to cover half of the lake?
A) 24 days
B) 47 days
C) 48 days
D) 49 days

Answer: B) 47 days


The term "percent" originates from the words "per cent", meaning "out of a hundred". For instance, 35 percent equals 35 out of 100, or 35 hundredths. To convert a percentage to a decimal, divide by 100, which is equivalent to moving the decimal point two places to the left. Conversely, to convert a decimal to a percentage, reverse this process.

Exercise: Convert decimals to percentages and percentages to decimals.


University of North Carolina at Chapel Hill
offers a BS in Physics and Mathematics. Key topics in the mathematics curriculum include:
- Trigonometry: Introduction to Trigonometry
- Functions:
  - Linear Functions
  - Polynomials
  - Rational Functions


# Key Concepts

A measure describes the "tailedness" of the probability distribution of a real-valued random variable, related to geometric mean size and particles logarithmic mean size.

A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. In physics, Larmor precession is the precession of the magnetic moment of any object with a magnetic moment about an external magnetic field.

Far from a magnet, its magnetic field is described by a dipole field characterized by its total magnetic moment. Rotating unbalance refers to the uneven distribution of mass around an axis of rotation, where a rotating mass is out of balance when its center of mass does not coincide with the axis of rotation.

The arithmetic mean size of particles in a soil's size distribution can be calculated using the arithmetic method of moments. This shows how much variation or dispersion from the average exists, with the logarithmic mean size needing to be precalculated.

A cantilever is a beam anchored at one end, carrying a load to the support where it is forced against by a moment and shear stress. Variation or dispersion from the average in a soil's particle size distribution can be measured, with arithmetic mean size being a key factor.

Formulas can be created to calculate these measures, including geometric mean size, particles logarithmic mean size, and arithmetic mean size, to understand and analyze various physical and statistical phenomena.