Formulas In Add Maths

Formulas In Add Maths

1. Polynomials

• When dividing the polynomial expression by the linear divisor : The answer on the root is the quotient. (most cases its a quadratic that can be factorized)

• When doing remainder theorem: make the divisor = 0 then substitute the x value into the polynomial. r = P(x)

• In factor theorem if the poly expression P(x) is divided by a linear divisor which is a factor (divisible) of P(x) the remainder = 0. P(x) = 0

2.Quadratics 

Completing the square. 
 $ax^2+bx+c=0$  
  $a\left(x+h\right)^2+k$  

Formulas:

 $h=\frac{b}{2a}$    $k=\frac{4ac-b^2}{4a}$  

Quadratic Graph

• When a is positive the graph is minimum. When a is negative the graph is maximum.

• To find h & k: completing the square is used. (long way or formula way)

• To find the Line of Symmetry(L.O.S): it is -1 x h or -h. (N.B the h is from completing the square and if h is a negative number then the L.O.S is positive because a minus by a minus gives a positive)

• To find y - intercept: It is c from the quadratic given.

• To find the min/max value: It is k from completing the square.

• To find the min/max point: the coordinate is (-h , k).

• To find roots(x values): Make a (x + h) ² + k = 0 and solve for x.

Nature of Roots

• Steps: 1) Make f(x)=0 2) Identify a,b,c. 3)Work out b² & 4ac.

• b² > 4ac - Roots are Real & Distinct. (2 differ roots)

• b² = 4ac - Roots are Real & Equal. (2 roots are same)

• b² < 4ac - No Real Roots.

Inequalities

>,≥ , ≤, < , more than and equal to , less than and equal to

Quadratic inequalities

Steps:

1) Rearrange the inequality so that the right side = 0.

2) Factorize to find roots.

3)Sketch SMALL graph. (graph paper NOT needed)

4) Determine the region by shading.

> or ≥ 0 where the region is above the x-axis. (outside roots)

< or ≤ 0 where the region is below the x-axis.(within roots)

Rational Inequalities (fractions)

1) Step 1

2) Before steps 2-4, Multiply both sides of the equal sign by the Denominator squared (denominator of the fraction).

Quadratic : Roots & Coefficients -

 $\propto or\ \alpha\ \&\ \beta$  

•  $\propto or\ \alpha$  - Is called ALPHA &  $\beta$  is called BETA.  $\left(ax^2+bx+c\right)$    

• 1.  $\propto+\beta=\frac{-b}{a}$               6.  $\left(\propto+\beta\right)^2=squared\ of\ \propto+\beta$  

• 2.  $\propto\beta=\frac{c}{a}$                              7.  $\frac{\propto}{\beta}+\frac{\beta}{\propto}=\frac{\propto^2+\beta^2}{\propto\beta}$  

• 3.  $\propto^2\beta^2=\left(\propto\beta\right)^2$                                  $=\frac{\left(\propto+\beta\right)^2-2\propto\beta}{\propto\beta}$  

• 4.  $\propto^2\beta+\beta^2\propto\ =\ \propto\beta\left(\propto+\beta\right)\$  

• 5.  $\frac{6}{\propto}+\frac{6}{\beta}\ =\ \frac{6\beta+6\propto}{\propto\beta}=\frac{6\left(\propto+\beta\right)}{\propto\beta}$  (it can be any +ve number)

Finding the equation of New roots

Steps:
1) State coefficients a,b,c .
2) Find  $\propto+\beta\ \&\ \propto\beta$   from the given equation.

3) For new roots find: Sum Of Roots & Product Of Roots. 
4) Substitute  $\propto+\beta\ \&\ \propto\beta$  from step 2 into step 3 accordingly. 
 $5.\ \ x^2-\ \left(sum\ of\ new\ roots\right)x\ +\ \Pr oduct\ of\ new\ roots=0$ is the Equation.

3. Indices - Laws of Indices

• 1) PRODUCT :  $a^m\times a^n=a^{m+n}$  

• 2)  DIVISION  :   $a^m\div a^n=a^{m-n}$  

• 3) Power Raised To Another:   $\left(a^m\right)^n=a^{m\times n}$  

• 4) Negative Power :  $a^{-m}=\frac{1}{a^m}$  

• 5) Fractional :  $a^{\frac{m}{n}}=^{\ n}\sqrt{a^m}$  

• 6) Zero Power :   $a^0=1$  

• 7)  $\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m$  

4. Logarithms (Logs)- Laws Of Logs

• 1) PRODUCT :  $Log\ A\ +\ Log\ B\ =Log\ AB$  

• 2) QUOTIENT :  $Log\ A\ -\ Log\ B\ =\ Log\ \frac{A}{B}$  

• 3) Power :   $n\ \log\ x\ =\log\ x^n$  

• 4) Base = Number :   $Log\ _a\ a\ =1$  

• 5) Number = 1 :  $\log_a\ 1\ =0$  

Solving logarithmic equations

1) Take logs.

2) Use the laws of logs.

3) Make one log expression = another (1) log expression.

4) Drop logs.

5) Find x. (Factorize if needed)

Log Graphs / Linear Reduction

• Type 1 : y = kxⁿ becomes log y = n log x + log k

• Type 2 : y = Ab^x becomes log y = log b(x) + log A

• Type 1 both x and y values are logged

• Type 2 only y values are logged

• Type 1 k and n need to be found

• Type 2 A and b need to be found

5.Surds

• 1) Product :

   $\sqrt{xy}=\sqrt{x}\sqrt{y}$  

• 2) Quotient :  $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$  

• 3)   $\sqrt{x}\sqrt{x}=x$  

•  $Rationalization\ :\ \left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)\ =a-b$  

•  $\frac{a+\sqrt{b}}{c-\sqrt{d}}\times\frac{c+\sqrt{d}}{c+\sqrt{d}}$   or  $\frac{a-\sqrt{b}}{c+\sqrt{d}}\times\frac{c-\sqrt{d}}{c-\sqrt{d}}$  

6.Arithmetic Progression

• a - the first term , d - the common difference . d =  $T_2-T_1$  where T2 is the   2nd Term and T1 is the 1st Term.

• nth term =  $a+\left(n-1\right)d$  

• Sum of an A.P. :  $S_n=\frac{n}{2}\left(2a+\left(n-1\right)d\right)\ or\ S_n=\frac{n}{2}\left(a_1+a_n\right)$  where n = the number terms, a = the first term, d = the common difference,   $a_1$  = the first term and  $a_n$  = the nth term.

Geometric Progression

• a- first term, r - common ratio, r = $\frac{T_2}{T_1}$  where T2 means the 2nd term and T1 means the 1st term.

• nth term =  $ar^{n-1}$  

• Sum of a G.P. =  $S_n=\frac{a\left(1-r^n\right)}{1-r}\ when\ r<1\ \&\ \frac{a\left(r^{\ n}-1\right)}{r-1}\ when\ r>1$   

• Sum to Infinity =  $S_{\infty}=\frac{a}{1-r}\ when\ 0<r<1$  

7a.Box and Whiskers

• Median /  $Q_2$  =  $\left(\frac{n+1}{2}\right)^{th}$  

•  $Q_1\left(1st\ set\right)$  =  $\left(\frac{n+1}{2}\right)^{th}$  :  Lower Quartile 

•  $Q_3\left(2nd\ set\right)$  =  $\left(\frac{n+1}{2}\right)^{th}$  : Upper Quartile 

• Inter Quartile Range / IQR = Q3 - Q1

• Semi Inter Quartile Range / SIQR =  $\frac{Q3-Q1}{2}$  

•  Min and max value from the data is needed for the plot.

7b Stem and Leaf Plot

*The stem can have one or more digits and the leaf has the last digit.

* Uses all the values n the data and placed in ascending order

*Must have a key. " 1|2 represents 12..."

8.Standard Deviation (S.D)

1) Discrete Data -

 $Mean\ or\ \overline{x}=\ \frac{\in x}{n}\ \ \ \ S.D=\sqrt{\frac{\in\left(x-\overline{x}\right)^2}{n}}$ 

 2) S.D of a Frequency Distribution -
 $Mean\ or\ \overline{x}=\frac{\in xf}{f}\ \ \ \ \ S.D.=\sqrt{\frac{\in\left(x-\overline{x}\right)^2f}{\in f}}$  
3) S.D. FOR Grouped Data -
 $\overline{x\ }=\frac{\in fx\ }{\in f}\ \ \ \ S.D.=\sqrt{\frac{\in\left(x-\overline{x}\right)^2f}{\in f}}$  

  

 $9.\ Differentiation\ \cdot\ y=x^n\rightarrow\ \frac{dy}{dx}=nx^{n-1}$  
 $\cdot\ y=ax^n\ \rightarrow\ \frac{dy}{dx}=anx^{n-1}$  
 $\cdot\ y=+\ or\ -\ \left(whole\ number\right)x\ \rightarrow\ \frac{dy}{dx}=whole\ number$   $\cdot\ y=+\ or\ -\ whole\ number\ \ \rightarrow\ \frac{dy}{dx}=0$  
 $\cdot\ y=\sin x\ \rightarrow\ \frac{dy}{dx}=\cos\ x$  
 $y=\cos\ x\ \rightarrow\ \frac{dy}{dx}=-\ \sin x\$  

 $\cdot y=\sin\ \left(ax+b\right)\rightarrow\ \frac{dy}{dx}=a\cos\ \left(ax+b\right)$   $\cdot\ y=\cos\ \left(ax+b\right)=-\sin\ \left(ax+b\right)\times a$   $\cdot y=\cos\ \left(f\left(x\right)\right)\rightarrow\ \frac{dy}{dx}=-\sin\left(f\left(x\right)\right)\times f^1\left(x\right)$   $\cdot\ y=\sin\ \left(f\left(x\right)\right)\rightarrow\ \frac{dy}{dx}=\cos\ \left(f\left(x\right)\right)\ \times f^1\left(x\right)$  

continued..

• Product Rule :  $\frac{dy}{dx}=U\ \frac{dv}{dx}+V\ \frac{du}{dx}$   

• Quotient Rule :  $\frac{dy}{dx}=\frac{V\ \frac{du}{dx}-U\ \frac{dv}{dx}}{V^2}$  

• Chain Rule :  $y=f\ \left(g\left(x\right)\right)\ \rightarrow\ \frac{dy}{dx}=f^{-1}\left(g\left(x\right)\right)\ g^{-1}\left(x\right)$  where  $f^{-1}\&\ g^{-1}$  are the differentiated functions / the derivatives.

•  $\left(ax+\ b\right)^n=n\left(ax+b\right)^{n-1}\left(a\right)$  

•  $At\ a\ S.p\ \frac{dy}{dx}=0$  

10.Integration

•  $\int_{ }^{ }x^ndx\ =\ \frac{x^{n+1}}{n+1}+c$  

•  $\int_{ }^{ }ax^ndx=a\int_{ }^{ }x^ndx=a\left(\frac{x^{n+1}}{n+1}\right)+c$  : where a is a constant and placed outside the integral sign to be multiplied by the integrated function later.

•  $\int_{ }^{ }\left(ax+b\right)^n\ dx=\frac{\left(ax+b\right)^{n+1}}{n+1}\times\left(\frac{1}{a}\right)+c$  

•  $\int_{ }^{ }\cos\ x\ dx\ =\sin x\ +c$  

•  

continued..

•  $\int_{ }^{ }-\sin x\ dx=\cos\ x+c$  

•  $\int_{ }^{ }\sin x\ dx\ =-\cos x+c$  

•  $\int_{ }^{ }\sin\ \left(ax+b\right)dx=\frac{-\cos\left(ax+b\right)}{a}+c$  

•  $\int\ \cos\left(ax+b\right)dx=\frac{\sin\left(ax+b\right)}{a}+c$  

Definite Integrals

 $\int f\left(x\right)dx=\lceil g\left(x\right)\rceil_a^b\ =g\left(b\right)-g\left(a\right)$  
where f (x) dx  is the derivative to be integrated , g (x) is the integrated f (x) & a and b are numbers. (the answer is numerical)

Area Under A curve 
 $A=\int_a^by\ dx$  
Same process as definite integrals except the numerical answer is  $^2\ units\$  because the "area" was found. 

 

Volume of A Solid formed when a Plane Area is rotated about the x-axis.

 $V=\pi\int_a^b\ y^2\ dx\$  
where pi is constant , the function y given is squared before integrating and the numeric answer is in  $^3units$  because the  "volume " was found. The process is the same as definite integrals.

11.Trigonometry

• Radian Conversion :  $\pi^c=180\degree\ or\ 180\degree=\pi\ radians$  

• Length of Arc:  $s=r\theta$  where  $\theta$  is in radians.

• Area of Segment =Area of Sector - Area of Triangle 

• Area of Triangle =  $\frac{1}{2}ab\sin c$  

• Area of sector =  $\frac{1}{2}r^2\theta$ where  $\theta$  is in radians.

Signs of Sin, Tan , Cos (ASTC)

• In the First Quadrant all are Positive. 0-90 degrees.

• In the Second Quadrant Sin is Positive and Tan & Cos is Negative. 90 -180 degrees.

• In the Third Quadrant Tan is Positive and Sin & Cos is Negative. 180 - 270 degrees.

• In the Fourth Quadrant Cos is Positive and Sin & Tan is Negative. 270-360 degrees.

Expressing a large angle in terms of an acute angle .

• The alpha sign is used to represent the acute angle.

• In the 1st Quadrant : alpha = 180 - theta

• In the 2nd Quadrant : alpha = 180 - theta

• In the 3rd Quadrant : alpha = theta - 180

• In the 4th Quadrant : alpha = 360 - theta

Solving Trig Equations

• In the 1st Quadrant : theta = alpha

• In the 2nd Quadrant : theta = 180 - alpha

• In the 3rd Quadrant : theta = 180 + alpha

• In the 4th Quadrant : theta = 360 - alpha

Identities

•  $\operatorname{cosec}\ \theta=\frac{1}{\sin\theta}$  

•  $\sec\theta=\frac{1}{\cos\theta}$  

•  $1.\ \cot\theta=\frac{1}{Tan\ \theta}$  

•  $Tan\theta=\frac{\sin\theta}{\cos\theta}$  

•  $2.\ \cot\theta=\frac{\cos\ \theta}{\sin\theta}$  

•  $\sin^2\theta+\cos^2\theta=1$  

•  $1.\ \sin^2\theta=1-\cos^2\theta\ \ \&\ \cos^2\theta=1-\sin^2\theta$  

Compound Angle Formula

• Sin (A + B) = Sin A Cos B + Cos A Sin B

• Sin (A - B) = Sin A Cos B - Cos A Sin B

• Cos (A + B) = Cos A Cos B - Sin A Sin B

• Cos (A - B) = Cos A Cos B + Sin A Sin B

• Tan (A + B) = Tan A + Tan B / 1 - Tan A Tan B

• Tan (A - B) = Tan A - Tan B / 1 + Tan A Tan B

• / means divided by

Solving Equations using Double Angle Formula

•  $Sin\ 2\theta\ =\ 2\ Sin\theta\ Cos\theta\$  

•  $1.\ Cos\ 2\theta=\cos^2\theta-\sin^2\theta$  

•  $2.\ Cos\ 2\theta=1-2\sin^2\theta$  

•  $3.\ 2\cos^2\theta-1$  

•  $Tan=\frac{2\ Tan\theta}{1-\tan^2\theta}$  

Special Angles

•  $\sin30\degree=\sin\ \frac{\pi^c}{6}=\frac{1}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos30\degree=\cos\ \frac{\pi^c}{6}=\frac{\sqrt{3}}{2}$  

•  $\tan30^{\degree}=\tan\ \frac{\pi^c}{6}=\frac{1}{\sqrt{3}}or\ \frac{\sqrt{3}}{3}$  

•  $\sin60\degree=\sin\ \frac{\pi^c}{3}\ =\frac{\sqrt{3}}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos\ \frac{\pi^c}{3}=\ \cos60\degree=\frac{1}{2}$  

•  $\tan60\degree=\tan\ \frac{\pi^c}{3}=\sqrt{3}$  

•  $\sin45\degree=\sin\ \frac{\pi^c}{4}\ =\frac{\sqrt{2}}{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos45\degree=\cos\ \frac{\pi^c}{4}=\frac{\sqrt{2}}{2}$  

continued

•  $\tan45\degree=\tan\ \frac{\pi^c}{4}=1$  

•  $\cos90\degree=\cos\ \frac{\pi^c}{2}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sin90\degree=\sin\ \frac{\pi^c}{2}=1$  

•  $\sin180\degree=\sin\pi=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos\ 180\degree=\cos\pi=1$  

•  $\sin30\degree=\cos60\degree\ \ \ \ \ \ \ \ \ \ \sin60\degree=\cos30\degree\ \ \ \ \ \ \ \sin45\degree=\cos45\degree$  

•  $\sin\left(90-\theta\right)=\cos\theta\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos\left(90-\theta\right)=\sin\theta$  

12.Circles- Equation of a Circle

• Standard Form:  $\left(x-a\right)^2+\left(y-b\right)^2=r^2$     Center Coordinate: (a,b) and r = Radius of the circle

• General Form:  $\ x^2+y^2+2fx+2gy+C=0$  Center(-f,-g) Radius/r =  $\sqrt{f^2+g^2-C}$  

• Midpoint (center of circle)  $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$  

• Distance (diameter or Radius of circle depending on question)  $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$  

*grad of tangent = grad of curve at the point of contact

*grad of normal x grad of tangent = -1 (negative reciprocal)

*Equation of tangent / normal : y = mx + c

*Formula : y - y₁ = m (x - x₁)

13.Vectors

• Magnitude / Modulus of a Vector -  $\sqrt{x^2+y^2}$  

• Unit Vector= $\frac{vector}{magnitude\ }$  

• Angle between 2 vectors -  $a\ \times\ b\ =\left|a\right|\ \left|b\right|\ \cos\ \theta$  where  $\left|\right|$  means magnitude 

• Parallel Vectors :  $a\times b=\left|a\right|\ \left|b\right|$  

• Perpendicular Vectors :  $a\times b=0$  

14.Probability

•  $\cup:\ and\ \cap:or$  

• Types of Events : 

• 1. Mutually Exclusive Events -  $P\ \left(A\cup B\right)=P\left(A\right)+P\left(B\right)$  (Addition Rule)

• 2. Non-Mutually Exclusive Events -  $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$  

• 3. Independent Events -  $P\left(A\cap B\right)=P\left(A\right)\times P\left(B\right)$  (Multiplication Rule) 

• Conditional Probability - P (A / B) = $\frac{P\left(A\cap B\right)}{P\left(B\right)}$  where / means given .

14. Kinematics

 $average\ velocity\ =\ \frac{change\ in\ displacement\ }{tme\ taken}$  
 $average\ speed\ =\frac{total\ dis\tan ce\ travelled\ }{time\ taken\ }$  
 $acceleration\ =\frac{change\ in\ velocity\ }{time\ taken\ }$  

* $v=\frac{ds}{dt}\ or\ \ v=\ \int a\ dt$  
 $\cdot s=\int v\ dt$  
 $\cdot a=\frac{dv}{dt}=\frac{d^2s}{dt^2}$  
 v= velocity , s= displacement or distance , a = acceleration , t = time
At  maximum velocity, dv/dt = 0
 $Note:\$  the ones that apply integration has a constant c to be found.