Memory Aid | Mathematics — Secondary 5 (SN)

Simplify the following expression as much as possible: ||\dfrac{(27 a^3 b)^{\frac{1}{2}}}{27^{\frac{1}{3}}a^3}||

​Determine the simplified value of the following root : ||\sqrt{45}||

Simplify the following expression using the laws of logarithms: ||\log_4 6x^2 + \log_4 9xy - \log_4 2y||​

Factor the following algebraic expression: ||{\mid}-4x+8{\mid}|| of the form |a {\mid}x \pm h{\mid}.|

|f(x) = a (c) ^{bx} + k|

where

|b = | compounding frequency
|k = | asymptote
| c = 1 \pm| percentage change as a decimal number

When an investment is made with a banking institution, the return is generally evaluated according to an exponential function. However, a minimum amount of investment is required to benefit from more advantageous rates.

So, after how many years does an initial investment of $5 000, compounded every two years at an interest rate of 5% with a minimum investment of $3 000, return at least $8 000?
 

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point 

The solution set of an inequality is directly related to the equation associated with it.

|f(x) = a \log_c (b(x-h))|

where

|\dfrac{1}{b} + h =| the zero of the function
|h = | the asymptote of the function

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point

The solution set of an inequality is directly related to the equation associated with it.

|f(x) = a \sqrt{b(x-h)} + k|

where

|(h,k) = | coordinates of the vertex, 
|b = | generally |\pm 1| and
signs |a| and |b| depend on the orientation of the curve.

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point

The solution set of an inequality is directly related to the equation associated with it.

In standard form:  |f(x) = \displaystyle \frac{a}{b(x-h)} + k|

In general form: |f(x) = \displaystyle \frac{ax+b}{cx​+\ d}| 

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point

The solution set of an inequality is directly related to the equation associated with it.

|f(x) = a {\mid}x - h{\mid} + k|

where

|(h,k) =| vertex coordinate 
|\pm a =| slope of each straight line forming the function’s graph

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point

The solution set of an inequality is directly related to the equation associated with it.

We can choose from three trigonometric function models depending on the situation:

|f(x) = a \cos (b (x-h)) + k|

|g(x) = a \sin(b (x-h)) + k|

|h(x) = a \tan(b (x-h)) + k|

To keep your dog entertained, you decide to go outside and play his favourite game of “go fetch”. You are standing 10 metres from the house. You always throw the ball 30 metres from you. In addition, you noticed that at this distance, your dog takes 12 seconds to fetch the ball and bring it back to you. Of course, you throw the ball again as soon as he brings it back to you. You do this for five minutes.

However, since your dog is not perfectly trained, you are afraid that he will run away when he is more than 30 metres from the house. Taking this information into account, for how long during the game are you afraid that your dog will run away?

To solve an inequality related to this model, begin by following the same steps while adding one of these steps:

• Sketch a graph of the situation

• Check the inequality using a point

The solution set of an inequality is directly related to the equation associated with it.

Before solving this kind of equation, it is important to simplify them as much as possible using the different trigonometric identities.

What are the values of |x| which satisfy the following equation: ||3 \sin^2x + \sec x - 0{.}48 = \dfrac{1}{\cos x}||

Speculating on the stock market ​​is a real passion for some investors. To try to predict the values ​​of different stocks and the potential profits, they use different graphs and then associate them with mathematical models. To study a certain foreign company, we can use the following functions to model the different variables that influence the final return on each share: 

Number of shares on the market: |f(x) = 10x - 500|

Profit from a share: |g(x) = -x^2+160x - 6\ 400|

Number of shareholders: ​|h(x)= -2x^2 + 260x - 8\ 000|

where |x =| number of years since its creation

What function could be used to determine the average profit obtained by each shareholder?

To determine their budget for the following year, the Alloprof administration committee looked at the production costs of the virtual library files. They used two functions:

Function f : |t = \dfrac{5}{4} n|

Function g : |s = 124t + 2\ 000|

where |n = | number of files produced, |t=| the number of hours worked, and |s = | salary (in $) to be paid to employees.

Model this situation using a single function and then determine the total number of concept sheets that can be made with a budget of $13 625.
 

In order to maximize his business’ profits, the owner wants to know how many jackets and shirts he must sell each week. Due to production constraints, he knows that the maximum number of shirts corresponds to the difference between 21 and quadruple the number of jackets. For transportation reasons, the number of jackets must be greater than or equal to the difference between eight and three times the number of shirts. Finally, the difference between triple the number of jackets and double the number of shirts must be at least two.

Knowing that each jacket sold brings in a profit of $32 and the profit associated with selling a shirt is $17, what is the maximum weekly profit the business owner can expect to make? 

If an angle measures |\color{red}{227^\circ},| what is its measurement in radians?

Prove the following identity: ||\sec \theta - \cos \theta = \tan \theta \sin \theta||

In a Cartesian plane, draw |\color{red}{\overrightarrow u} = (-3, 8)| and then determine its norm and direction.

To navigate the various operations on vectors, it is important to clearly define the following concepts:

Addition and subtraction 
If |\color{blue}{\overrightarrow u = (a,b)}| and |\color{red}{\overrightarrow v = (c,d)}| , thus |\color{blue}{\overrightarrow u} + \color{blue}{\overrightarrow v} = (\color{blue}{a} + \color{red}{c}, \color{blue}{b}+ \color{red}{d}) |

Multiplication of a vector by a scalar 
If |\overrightarrow u = (\color{blue}{a}, \color{red}{b})| and |k| is a scalar, then |k \overrightarrow u = (k \color{blue}{a}, k \color{red}{b})|
 
The scalar product 
If |\color{blue}{\overrightarrow u = (a,b)}| and |\color{red}{\overrightarrow v = (c,d)}|, thus |\color{blue}{\overrightarrow u} \cdot \color{red}{\overrightarrow v} =  \color{blue}{a}\color{red}{c}+ \color{blue}{b}\color{red}{d}|

Linear combination of two vectors 
Given |\color{blue}{\overrightarrow u}| and |\color{red}{\overrightarrow v}| , it is possible to express |\color{green}{\overrightarrow w​}| as a linear combination knowing |\color{green}{\overrightarrow w} = k_1 \color{blue}{\overrightarrow u} + k_2 \color{red}{\overrightarrow v}| and |\{k_1,k_2\} \in \mathbb{R}.| 
 

Determine the values of the scalars |\{k_1,k_2\}| given that |\color{blue}{\overrightarrow w = (4,-12)}| is the result of a linear combination of |\color{red}{\overrightarrow u = (-1,4)}| and |\color{green}{\overrightarrow v = (2,5)}.| 

After a violent storm, a tree fell on the road leading to Julian's cottage. To clear the roadway, he ties a rope to the bottom of the tree to pull it out of the way.

How much work will Julien have to do to move the tree a distance of 12 m if he exerts a force of |150 \ \text{N}| and the rope he uses forms an angle of |21^\circ| relative to the horizontal (not accounting for friction)?

Demonstrate the following: ||(\overrightarrow{AC} + \overrightarrow{BD} - \overrightarrow{AC}) - (\overrightarrow {FG} +\overrightarrow{GE} + \overrightarrow {ED}) = \overrightarrow {BF}||

​Every morning, you have to go to the bus stop and wait for the bus to take you to school. In order for the stop to be centralized for the other students in the neighbourhood, you notice that it divides the street connecting your house to your school at a ratio of 1: 4.

​

Using the information available, determine the bus stop’s location coordinates. 

It is important to differentiate between the two types of notations used to illustrate the portion associated with a division point in order to use the appropriate notation for the formula: 

A ratio  | = a:b\ \Rightarrow\ \displaystyle \frac{a}{a+b} = |  A fraction

​

For her first fishing trip, Kelsey uses sonar to locate her potential catches. However, she wonders about the range of her sonar. Based on the information presented in the drawing below, determine the area, in |\text{km}^2,| covered by the radar.

Having adored her first fishing experience, Kelsey decides to get a magnificent canoe. However, she must determine the exact dimensions of the craft to ensure that she can transport it with her car. To check, she drew the canoe in a Cartesian plane to get the following information:

Using the information given, determine the maximum length and width of the canoe.

Regardless of the orientation of the hyperbola, the rate of change of the asymptotes (dotted lines) is equivalent to |\displaystyle \pm \frac{\color{fuchsia}{b}}{\color{red}{a}}.|

Kelsey decides to paddle in a faster current. To her great dismay, two boats begin to overtake her simultaneously. To avoid capsizing, she must pilot her boat away from the point where two wakes formed by the boats intersect. The situation can be illustrated as follows:

Using the data given, determine the equation associated with the mathematical model enabling Kelsey to better navigate.

Kelsey has noticed the curvature of her fishing rod when a fish bites the hook is a good indication of the size of the fish. Using her previously purchased sonar, she determines the following information:

Since it is a parabolic shape, Kelsey considers its equation that can model the situation.

Kelsey is a little tired of fishing and decides to pay for a trip to a region where it is possible to go boating with prehistoric-looking sharks that look like sea dinosaurs. The water is practically clear, so she can easily see them swimming, but she loses sight of them when they pass underneath the boat.

Assuming they are swimming in a straight line at a speed of 5 m/sec, determine how long the sharks are underneath the vessel.

What are the coordinates of the intersection points of the following two conics:

What are the coordinates of the point associated with an angle of |\displaystyle \frac{-17\pi}{4}| ?