Last Minute Revision Notes

Algebra (Factorization And Expansion)

• Remember the expansion of a² – b²
  • It is always (a+b)(a-b)
  • Most people forget it when expanding or factorizing.
  • For example
    • If you are given  1 – 9 and you are asked to factorize it, remember both 1 and 9 are perfect squares. So basically you are being asked
      • 1² – 3²
        • It will expand as (1+3)(1-3)
      • Now you can use this expression to further work on your algebraic manipulation as required.

Simultaneous Equations

• Elimination Method
  • Remember, in elimination method, if you have selected the values for the variable x to eliminate
    • And Both the values are either BOTH Positive or BOTH negative
    • You will change signs of the second equation.
  • Remember, in elimination method, if you have selected the values for the variable x to eliminate
    • And both the values have different signs i.e. one is positive and the other is negative
    • You DO NOT change the signs.
  • Submission Method
    • If you make the THIRD equation from equation ONE, you will insert the THIRD equation in EQUATION TWO.
    • Once you have the value of one of the variables i.e. either x or y, you can use any equation i.e. Equation ONE, TWO or THREE to find the other value by inserting the value you have already found in the equation.
    • How to make equation 3?
      • Simply make either X or Y the subject of the formula for either equation 1 or equation 2.
      • Then insert this new equation 3 in the place of the OTHER equation where you see x or y (depending on which letter you made the subject of the formula.

Ordering

• Remember, NEVER try to order the values that are in different formal
  • Always bring them in the same format as decimals and then order them
  • COMMON MISTAKE ALERT
    • You need to write the actual values as the final answer.
    • Do not write the converted decimal values you have made for the working as your final answer. Major blunder.

Vectors

• Remember, both the perpendicular, perpendicular bisector and parallel rules that you studied in coordinate geometry also work in vectors
  • For example, if the examiner tells you two vectors are parallel, you will automatically know the gradient or the rise and run of both vectors based on the information provided for any one vector
  • COMMON MISTAKE: parallel does not mean equal. Parallel vectors can have different lengths. Remember this aspect.

Inequalities

• One common rule
  • A dotted line is always either < or >. It is never ≤ or ≥.
  • A firm line (a normal line without dotted gaps) is always ≤ or ≥. It is never < or >.
  • Exception to the rule
    • IF the examiner gives you one inequality already solved and given, and he uses ≤ or ≥ for a dotted line, then for that particular question, you will use ≤ or ≥ for the dotted line.
    • Similarly, IF the examiner gives you one inequality already solved and given, and he uses < or > for a firm line, then for that particular question, you will use < or > for the firm line.
  • Where is lesser than/ lesser than equal to and where is greater than/ greater than equal to.
    • If you have a line that is going upward from left to right.
      • Its left side is greater than/ greater than equal to
      • Its right side is lesser than/ lesser than equal to
    • If you have a line that is going downward from left to right
      • Its left side is lesser than/ lesser than equal to
      • Its right side is greater than/ greater than equal to
    • What to do if you are given just the line on the graph paper and asked to determine the inequality.
      • Use coordinate geometry rules of y = mx + c to find its equation.
        • Then, make “c” the subject of the equation by keeping it one side and everything else on the other side.
      • Then use the left and right rule above to determine what sign to use in the place of “=” between the equation.

Transformation

• Remember, if multiplying a Matrix A with Matrix B produces the image of the transformation C, with A being the object matrix and B being the transformation matrix.
  • Then Multiplying C with B Inverse () matrix will map the image C back onto object A.
• Common mistake issue
  • Many students tend to miss out the required items when they are given an object and image, and are asked to write describe the transformation that marked the object onto image. Here are the requirements for each transformation
    • Reflection: Line of Reflection.
    • Rotation: Center of Rotation, Angle of Rotation, Clockwise or Anticlockwise Direction of Rotation (Most students miss this one and lose marks)
    • Enlargement: Center of Enlargement and Factor of Enlargement (Scale Factor)
    • Translation: Translation Vector
  • Remember
    • Carefully check which item given is object and which one is image. Many students tend to lose marks on this aspect.

Coordinate Geometry

• First issue is to remember what each of these terms mean when doing coordinate geometry questions.
  • Parallel: The gradient of the two lines is the same.
  • Equal: The lines are equal in length.
  • Perpendicular: The lines insect and make an angle of 90 degrees. It also means that gradient of line 1 and gradient of line 2 have a specific relation:
    • Gradient of Line 1 * Gradient of Line 2 = -1
  • Bisector: One line that cuts another line exactly in half. Not necessarily perpendicular.
  • Perpendicular Bisector: One line that cuts the other line exactly in half and also makes a 90 degree angle in the process.
• If you are given a graph, remember that the equation y = mx +c has the following meaning
  • m is the gradient
  • c is the point where the line cuts the y axis. It is the y-intercept of the line.
  • So if the image is given on graph, you can simply look and find some components of the equation of the line.
• If you are asked to find the area.
  • First check if the coordinates satisfy the requirements of any particular geometric shape like right angled triangle or square or rectangle. If yes, use the direct formula for these to find the solution. NEVER DETERMINE that a geometric shape is formed just because it looks like it from visuals. Always check the coordinates.
  • If no shape is directly formed.
    • Check if you can form small coordinate shapes around the region for which you have to find the area. For example, maybe the region is enclosed by a large rectangle and a few right angled triangles. If so, find the area of the large shape and subtract the areas of the not required geometric small regions to find the area of the region you need.

Locus

• One common mistake
  • Always read the question carefully to check if you are asked to color the region INSIDE the given locus or OUTSIDE it.
• Second issue
  • When you are finding the locus from 2 distinct points, make sure your compass is opened exactly the same length when you put it on point B, as it was when you put it on point A to mark the line. Varying lengths would give improper outputs.
• Do locus step by step
  • You are given 3-4 locus to perform. Perform the first one first. They are given in a way to ensure that you can easily do them without messing things up.

Sequences

• In sequences, one important trick to remember is.
  • They will give you one sequence completely formulated in part a or b of the question.
  • Then, they will ask you in one of the next parts to find another sequence.
  • The trick is to avoid trying forming the new sequence entirely by yourself. It could take a lot of time.
  • Instead, you the sequence they have already given in the same question’s previous parts and experiment with it by changing a few values, adding something, subtracting something, multiplying or dividing something. You will find that the new sequence you need to form is connected to and made from the one already given. It will save time.
• Always test ANY sequence you make with at least 3 values to see if it gives the correct answer.
  • Remember, sequences may produce 2 values correct in row, but they can’t produce 3 if they are not the correct sequence (at this level of education).

Trigonometry

• There will always be a part in lengthy trigonometry questions that will ask you that ‘there is a plane, helicopter, bird etc flying over point A, and a person is standing at point B observing them.
  • The trick is to take your pencil or pen, and place it right on top of point A. Like put the pen on it as if you were trying to mark a dot on the paper. Now the top of your pencil or pen represents the flying bird, the line made by your pencil’s top (the bird) to the point A is perpendicular to the line from Point A to point B.
    • Thus, you can use both the sin,cos, tan and the angle of elevation/depression rules for this part now.
  • Requirements for cosine rule
    • Two sides and included angle
  • Requirements for sine rule
    • One side and its corresponding angle, along with either the side or the angle of the other pair.
  • Bearing is ALWAYS in three digits. So prefer writing 058 degrees for a bearing of 58 degrees rather than just 58 degrees.
  • Remember, bearing is ALWAYS the clockwise movement.
    • Students make a massive blunder that sometimes if they think one point is closer to the other point using anticlockwise movement, they write they write the anticlockwise angle. THAT IS WRONG. Bearing always has a clockwise angle. Always calculate the clockwise measurement.
  • Remember
    • From the same points
      • The outer angle of depression from the upper point
      • is the same as the inner angle of elevation from the lower point

Mensuration

• Remember, as I taught before as well.
  • For any shape that seems uncertain, try to break the shape down into smaller, simpler, proper shapes that you know the formulae of and then use them to calculate the area/ volume.
• Huge Blunder.
  • Always remember, perimeter is ONLY the outside boundary of the shape. Anything inside it that can not be touched from outside is NOT included in the perimeter.
  • Similarly, Surface area is only the OUTSIDE, TOUCHABLE surface. For surface area, the trick is to find the area of every surface that can be touched from outside of the given shape and add it.
  • Why tricky.
    • A cylinder that is closed from both sides has 3 surfaces that can be touched from outside: the curved surface, the circle surface on one closed end and the circle surface on the other closed end. You need to find the area of each and add them together to find the surface area of the complete cylinder.
    • A cylinder that is closed from one side (like a glass used to drink water) has 2 surfaces that can be touched from outside: the curved surface and the circle surface on one closed end. The circle surface on the other closed end DOES NOT EXIST as it is an open side.  You need to find the area of each and add them together to find the surface area of the complete cylinder.
    • A cylinder that is open from both sides has 1 surface that can be touched from outside: the curved surface. The circle surface on one closed end and the circle surface on the other closed end BOTH DON’T EXIST HERE. You need to find the area of each of the curved surface only and that will be the surface area of this cylinder.

Bounds (Upper and Lower)

• If you are asked to find the maximum difference between two values in any situation of a – b
  • You will take the upper bound of a and the lower bound of b
• If you are asked to find the minimum difference between two values in any situation of a – b
  • You will take the lower bound of a and the upper bound of b.
• This applies specifically to the situations where you are given two shapes, one inside another, and are asked what is the largest area between both shapes and the smallest possible area between both shapes etc.

Time

• When dealing with questions of different time zones
  • Solve questions in two steps.
  • First find the time difference between two locations based on the time zone of one location (location of departure is preferable because it makes the calculation easy).
  • Then subtract or add the time difference to your answer from the previous step. This reduces the chances of mistake instead of trying to both find the natural time difference and the change due to the time zone at the same time.

Money

• Questions involving money have 2 decimal places. NEVER 3 until specifically asked in the question.
• Remember, if you are doing a question where you have to convert one currency in another, but you don’t know the direct rate between them, use the third currency given as intermediary.
  • For example 1 USD = 5 PKR and 1 USD = 10 GBP, then you can convert GBP to PKR by using USD as an intermediary so 10 GBP = 5 PKR.
  • However, make sure the intermediary that you use has a value of “1” in both equations. If not, divide the values over each other separately in each equation to first make both equations have the value of the intermediary as “1” in their equation.

Probability

• Just remember the AND and OR rule I shared 2 weeks ago in the group. That will make sure you can solve every question perfectly.
• Always look for REPLACEMENT or WITHOUT REPLACEMENT.
• Always look for independent/ mutually exclusive events etc.

Significant Figures

• Remember
  • If the value starts with nothing before decimal i.e. 0.
    • Then every 0 after the decimal is insignificant until there is a value after them. So.
    • 000000000 has 0 significant values as there is no value after the zeros placed after decimal.
    • However, 0.0000000001 has 1 significant figures
    • Now the twist is if there comes a zero between some significant values, it will be significant.
      • So 0.000000000101 has 3 significant figures because the 0 between both ones is significant.
    • If the value starts with something before the decimal (a non zero value).
      • Then every 0 after the decimal is significant even if there is no value after them. So.
      • 000000000 has  11 significant figures

Standard Form

• The main trick is if you are given very large calculations in standard form, and you have to answer in standard form as well
  • Do not convert them in normal form and then calculate.
  • Instead, use the rule of indices of adding powers when multiplying two values with same base and subtracting powers when dividing two values with same base to find the answer within significant form quickly.

Symmetry

• Let’s cram
  • Wait
    • If you cant cram, then for rotational symmetry just turn the paper around in a complete circle and check how many times you see the same shape
    • For line of symmetry, remember, if we fold the shape on the line of symmetry, it must land perfectly on the other side.
  • Rectangle line of symmetry is 2 and rotational symmetry is also of order 2
  • Square line of symmetry 4 and rotational symmetry of order 4
  • Rhombus has 2 lines of symmetry and rotational symmetry of order 2
  • Kite has a rotational symmetry of order 1 and and 1 line of symmetry
  • A parallelogram has 0 lines of symmetry and rotational symmetry of order 2
  • Trapezium has rotational symmetry of order 1 and zero line of symmetry.
• Remember, every shape has at least rotational symmetry of order 1 because if you turn the shape 360 degrees, it will once come back to the starting shape that you saw.

Diagrams

• For any diagram, like bar chart, do not forget to label the axis.
  • Remember, labeling axis requires both the name and unit. For example, if you are counting time, then mention if it is in seconds, minutes or hours etc.
• In histogram.
  • Make sure you remember that histograms are usually made on frequency densities unless otherwise asked, not the frequency itself.
• In pie chart.
  • Remember, the total angles are 360 degree. So you have to use the same for the conversions between pie chart and real values.
  • Also, when drawing pie chart, go either completely clockwise or completely anti-clockwise. Otherwise, you will mess up the markings.
• In a cumulative frequency curve.
  • Major blunder is when taking quartiles.
    • Remember, if the total cumulative frequency is 200 (it will be marked on the y axis), and you are asked to find the interquartile range
      • First, find the upper quartile.
        • How? 200 *0.75 = 150. Check 150 on the y axis, connect it to the curve and find the value at the point for x axis.
      • Then, find the lower quartile
        • How? 200*0.25 = 50. Check 50 on the y axis, connect it to the curve and find the value at the point for x axis.
      • Then subtract the lower quartile value of x axis from the upper quartile value of x axis.
    • REMEMBER, the values are of x axis not the y axis.

Matrices

• How to check if A*B is possible where A and B are both matrices?
  • The number of columns in A must be the same as the number of rows in B
  • The Final answer will have the number of rows of A and the number of columns of B
• REMEMBER, changing the position of the matrix in the multiplication can change the answer and possibility of multiplication.
  • So for B*A to be possible.
    • The number of columns in B must be the same as the number of rows in A
    • The Final answer will have the number of rows of B and the number of columns of A
  • What is Null Matrix?
    • A matrix where all values are 0.
  • Unity matrix is
  • Remember, some transformation questions require matrices as explained before as well. So in any transformation question that you are struck, think about matrices.
    • How to make transformation matrix?
      • Simple, apply the transformations step by step as given on the unity matrix. The final matrix that ends up is the matrix for that transformation.
      • In other words, apply the transformation to (1,0) coordinate and (0,1) coordinate. Then put the final points in matrix form.
      • REMEMBER each coordinate is represented by the vertical column.
        • So it is like
          • Where x and y are the x and y coordinates of each point.
        • COMMON MISTAKE
          • Thinking that one horizontal row represents a coordinate is the biggest blunder. The blunder is shown here:  THIS IS WRONG.

Lesson Tags

Revision | Detailed Notes For Preparation & Revision | O Level Mathematics D 4024

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