new additional mathematics
TRANSCRIPT
New Additional Mathematics
Muhammad Hassan Nadeem
1. Sets
A null or empty set is donated by { } or π.
P = Q if they have the same elements.
P β Q, Q is subset of P.
PβQ, P is subset of R.
PβQ, Q is proper subset of P.
PβQ, P is proper subset of Q.
PβQ, Intersection of P and Q.
PβQ, union of P and Q.
Pβ compliment of P i.e. π-P
2. Simultaneous Equations
π₯ =βπ Β± π2 β 4ππ
2π
3. Logarithms and Indices
Indices
1. π0 = 1
2. πβπ =1
ππ
3. π1
π = ππ
4. ππ
π = ππ
π
5. ππ Γ ππ = ππ+π
6. ππ
ππ = ππβπ
7. ππ π = πππ
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Muhammad Hassan Nadeem
8. ππ Γ ππ = ππ π
9. ππ
ππ = π
π
π
Logarithms
1. ππ₯ = π¦ β« π₯ = πππππ¦
2. ππππ 1 = 0
3. πππππ = 1
4. πππππ₯π¦ = πππππ₯ + πππππ¦
5. ππππ π₯
π¦= πππππ₯ β πππππ¦
6. πππππ =πππ ππ
πππ ππ
7. πππππ =1
πππ ππ
8. πππππ₯π¦ = π¦πππππ₯
9. πππππ π₯ = πππππ₯1
π
10. logππ₯ = logππlogππ₯ =log ππ₯
log ππ
4. Quadratic Expressions and Equations
1. Sketching Graph
y-intercept
Put x=0
x-intercept
Put y=0
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New Additional Mathematics
Muhammad Hassan Nadeem
Turning point
Method 1
x-coordinate: π₯ =βπ
2π
y-coordinate: π¦ =4ππβπ2
4π
Method 2 2 2
square. The turning point is π, π .
2. Types of roots of πππ + ππ + π = π π2 β 4ππ β₯ 0 : real roots
π2 β 4ππ < 0 : no real roots
π2 β 4ππ > 0 : distinct real roots
π2 β 4ππ = 0 : equal, coincident or repeated real roots
5. Remainder Factor Theorems
Polynomials
1. ax2 + bx + c is a polynomial of degree 2.
2. ax3 + bx + c is a polynomial of degree 3.
Identities
π π₯ β‘ π π₯ βΊ π π₯ = π π₯ For all values of x
To find unknowns either substitute values of x, or equate coefficients of like powers
of x.
Express π¦ = ππ₯ + ππ₯ + π as π¦ = π π₯ β π + π by completing the
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New Additional Mathematics
Muhammad Hassan Nadeem
Remainder theorem
Factor Theorem
(x-a) is a factor of f(x) then f(a) = 0
Solution of cubic Equation
I. Obtain one factor (x-a) by trail and error method.
II. Divide the cubic equation with a, by synthetic division to find the quadratic
equation.
III. Solve the quadratic equation to find remaining two factors of cubic equation.
For example:
I. The equation π₯3 + 2π₯2 β 5π₯ β 6 = 0 has (x-2) as one factor, found by trail
and error method.
II. Synthetic division will be done as follows:
III. The quadratics equation obtained is π₯2 + 4π₯ + 3 = 0.
IV. Equation is solved by quadratic formula, X=-1 and X=-3.
V. Answer would be (x-2)(x+1)(x+3).
6. Matrices
If a polynomial f(x) is divided by (x-a), the remainder is f(a)
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New Additional Mathematics
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1. Order of a matrix
Order if matrix is stated as its number of rows x number of columns. For
example, the matrix 5 6 2 has order 1 x 3.
2. Equality
Two matrices are equal if they are of the same order and if their
corresponding elements are equal.
3. Addition
To add two matrices, we add their corresponding elements.
For example, 6 β23 5
+ β4 24 1
= 2 07 6
.
4. Subtraction
To subtract two matrices, we subtract their corresponding elements.
For example, 6 3 59 14 β5
β 2 7 5
β4 20 1 =
4 β4 012 β6 β6
.
5. Scalar multiplication
To multiply a matrix by k, we multiply each element by k.
For example, π 2 43 β1
= 2π 4π3π βπ
or 3 24 =
612
.
6. Matrix multiplication
To multiply two matrices, column of the first matrix must be equal to the row
of the second matrix. The product will have order row of first matrix X column
of second matrix.
For example: 2 41 32 β1
3 2 11 5 2
47 =
π π ππ π ππ π π
πππ
To get the first row of product do following:
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New Additional Mathematics
Muhammad Hassan Nadeem
a = (2 x 3) + (4 X 1) = 10 (1st row of first, 1st column of second)
b = (2 x 2) + (4 x 5) = 24 (1st row of first, 2st column of second)
c = (2 x 1) + (4 x 2) = 10 (1st row of first, 3st column of second)
d = (2 x 4) + (4 x 7) = 36 (1st row of first, 4st column of second)
e = (1 x 3) + (3 x 1) = 6 (2st row of first, 1st column of second)
f = (1 x 2) + (3 x 5) = 17 (2st row of first, 2st column of second)
g = (1 x 1) + (3 x 2) = 7 (2st row of first, 3st column of second)
h = (1 x 4) + (3 x 7) = 25 (2st row of first, 4st column of second)
i = (2 x 3) + (-1 x 1) = 5 (3st row of first, 1st column of second)
j = (2 x 2) + (-1 x 5) = -1 (3st row of first, 2st column of second)
k = (2 x 1) + (-1 x 2) = 0 (3st row of first, 3st column of second)
l = (2 x 4) + (-1 x 7) = 1 (3st row of first, 4st column of second)
7. 2 x2 Matrices
a. The matrix 1 00 1
is called identity matrix. When it is multiplied with any
matrix X the answer will be X.
b. Determinant of matrix π ππ π
will be = π ππ π
= ππ β ππ
c. Adjoint of matrix π ππ π
will be = π βπβπ π
d. Inverse of non-singular matrix (determinant is β 0) π ππ π
will be :
πππππππ‘
πππ‘ππππππππ‘=
1
ππ β ππ
π βπβπ π
8. Solving simultaneous linear equations by a matrix method
ππ₯ + ππ¦ = πππ₯ + ππ¦ = π
β«β« π ππ π
π₯π¦ =
ππ
π₯π¦ =
π ππ π
β1
Γ ππ
7. Coordinate Geometry
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New Additional Mathematics
Muhammad Hassan Nadeem
Formulas
π·ππ π‘ππππ π΄π΅ = π₯2 β π₯1 2 + π¦2 β π¦1 2
ππππππππ‘ ππ π΄π΅ = π₯1 + π₯2
2,π¦1 + π¦2
2
Parallelogram
If ABCD is a parallelogram then diagonals AC and BD have a common midpoint.
Equation of Straight line
To find the equation of a line of best fit, you need the gradient(m) of the line, and
the y-intercept(c) of the line. The gradient can be found by taking any two points
on the line and using the following formula:
ππππππππ‘ = π =π¦2 β π¦1
π₯2 β π₯1
The y-intercept is the y-coordinate of the point at which the line crosses the y-
axis (it may need to be extended). This will give the following equation:
π¦ = ππ₯ + π
Where y and x are the variables, m is the gradient and c is the y-intercept.
Equation of parallel lines
Parallel line have equal gradient.
If lines π¦ = π1π1 and π¦ = π2π2 are parallel then π1 = π2
Equations of perpendicular line
If lines π¦ = π1π1 and π¦ = π2π2 are perpendicular then π1 = β1
π2 and π2 = β
1
π1.
Perpendicular bisector
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New Additional Mathematics
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The line that passes through the midpoint of A and B, and perpendicular bisector of AB. For any point P on the line, PA = PB
Points of Intersection
The coordinates of point of intersection of a line and a non-parallel line or a curve
can be obtained by solving their equations simultaneously.
8. Linear Law
To apply the linear law for a non-linear equation in variables x and y, express the
equation in the form
π = ππ + π
Where X and Y are expressions in x and/or y.
9. Functions
Page 196
10. Trigonometric Functions
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New Additional Mathematics
Muhammad Hassan Nadeem
π is always acute.
Basics
sin π =ππππππππππ’πππ
ππ¦πππ‘πππ’π π
cos π =πππ π
ππ¦πππ‘πππ’π π
tan π =ππππππππππ’πππ
πππ π
tan π =sin π
cos π
cosec π =1
sin π
sec π =1
cos π
cot π =1
tan π
πππ β π£π
πππ + π£π
Sin
2
All
1
Tan
3
Cos
4
0,360 180
270
90
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New Additional Mathematics
Muhammad Hassan Nadeem
Rule 1
sin(90 βπ) = cos π
cos 90 β π = sin π
tan 90 β π =1
tan π= cot ΞΈ
Rule 2
sin(180 β π) = + sin π
cos 180 β π = βcos π
tan 180 β π = βtan π
Rule 3
sin(180 + π) = βsin π
cos 180 + π = βcos π
tan 180 + π = +tan π
Rule 4
sin(360 β π) = β sin π
cos 360 β π = +cos π
tan 360 β π = βtan π
Rule 5
sin(βπ) = βsin π
cos βπ = +cos π
tan βπ = βtan π
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New Additional Mathematics
Muhammad Hassan Nadeem
Trigonometric Ratios of Some Special Angles
cos 45 =1
2 cos 60 =
1
2 cos 30 =
3
2
sin 45 =1
2 sin 60 =
3
2 sin 30 =
1
2
tan 45 = 1 tan 60 = 3 tan 301
3
11. Simple Trigonometric Identities
Trigonometric Identities
sin2 π + cos2 π = 1
1 + tan2 π = sec2 π
1 + cot2 π = cosec2 π
12. Circular Measure
Relation between Radian and Degree
π
2 πππππππ = 90Β° π πππππππ = 180Β°
3π
2 πππππππ = 270Β° 2π πππππππ = 360Β°
π = ππ³ where s is arc length, r is radius and Ο΄ is angle of sector is radians
π΄ =1
2ππ =
1
2π2π³ where A is Area of sector
ππππ ππ π πππ‘ππ
ππππ ππ ππππππ=
πππππ ππ π πππ‘ππ
πππππ ππ ππππππ
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New Additional Mathematics
Muhammad Hassan Nadeem
13. Permutation and Combination
π! = π π β 1 π β 2 Γ β¦ Γ 3 Γ 2 Γ 1
0! = 1
π! = π π β 1 !
πππ=
π!
π β π !
ππΆπ=
π!
π β π ! π!
14. Binomial Theorem
π + π π = ππ + πΆ1πππβ1π + πΆ2
πππβ2π2 + πΆ3πππβ3π3 + β― + ππ
ππ+1 = ππΆπππβπππ
15. Differentiation
π
ππ₯ π₯π = ππ₯πβ1
π
ππ₯ ππ₯π + ππ₯π = πππ₯πβ1 + πππ₯πβ1
π
ππ₯ π’π = ππ’πβ1
ππ’
ππ₯
π
ππ₯ π’π£ = π’
ππ£
ππ+ π£
ππ’
ππ₯
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New Additional Mathematics
Muhammad Hassan Nadeem
π
ππ₯ π’
π£ =
π£ππ’ππ₯
β π’ππ£ππ₯
π£2
Where βvβ and βuβ are two functions
Gradient of a curve at any point P(x,y) is ππ¦
ππ₯ at x
16. Rate of Change
The rate of change of a variable x with respect to time is ππ₯
ππ‘
ππ¦
ππ‘=
ππ¦
ππ₯Γ
ππ₯
ππ‘
πΏπ¦
πΏπ₯β
ππ¦
ππ₯
πππππππ‘πππ ππππππ ππ π₯ =πΏπ₯
π₯Γ 100%
π π₯ + πΏπ₯ = π¦ + πΏπ¦ β π¦ +ππ¦
ππ₯πΏπ₯
17. Higher Derivative
ππ¦
ππ₯= 0 when x =a then point (a, f(a)) is a stationary point.
ππ¦
ππ₯= 0 and
π2π¦
ππ₯2 β 0 when x =a then point (a, f(a)) is a turning point.
For a turning point T
I. If π2π¦
ππ₯2 > 0, then T is a minimum point.
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New Additional Mathematics
Muhammad Hassan Nadeem
II. If π2π¦
ππ₯2 < 0, then T is a maximum point.
18. Derivative of Trigonometric Functions
π
ππ₯ sin π₯ = cos π₯
π
ππ₯ cos π₯ = β sin π₯
π
ππ₯ tan π₯ = sec2 π₯
π
ππ₯ sinn π₯ = π sinnβ1 π₯ cos π₯
π
ππ₯ cosn π₯ = βπ cosnβ1 π₯ sin π₯
π
ππ₯ tann π₯ = π tannβ1 π₯ sec2 π₯
19. Exponential and Logarithmic Functions
π
ππ₯ ππ’ = ππ’
ππ’
ππ₯
π
ππ₯ πππ₯ +π = ππππ₯ +π
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New Additional Mathematics
Muhammad Hassan Nadeem
A curve defined by y=ln(ax+b) has a domain ax+b>0 and the curve cuts the x-
axis at the point where ax+b=1
π
ππ₯ ππ π₯ =
1
π₯
π
ππ₯ ln π’ =
1
π’
ππ’
ππ₯
π
ππ₯ ππ ππ₯ + π =
π
ππ₯ + π
20. Integration
ππ¦
ππ₯= π₯ βΊ π¦ = π₯ ππ₯
π
ππ₯
1
2π₯2 + π = π₯ βΊ π₯ ππ₯ =
1
2π₯2 + π
ππ₯π ππ₯ =ππ₯π+1
π + 1+ π
ππ₯π + πππ ππ₯ =ππ₯π+1
π + 1+
ππ₯π+1
π + 1+ π
(ππ₯ + π)π ππ₯ = ππ₯ + π π+1
π(π + 1)+ π
π
ππ₯ πΉ π₯ = π(π₯) βΊ π π₯ ππ₯ = πΉ π β πΉ(π)
π
π
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New Additional Mathematics
Muhammad Hassan Nadeem
π π₯ ππ₯ + π π₯ ππ₯π
π
= π π₯ ππ₯π
π
π
π
π π₯ ππ₯ = β π π₯ ππ₯π
π
π
π
π π₯ ππ₯ = 0π
π
π
ππ₯ sin π₯ = cos π₯ βΊ cos π₯ ππ₯ = sin π₯ + π
π
ππ₯ βcos π₯ = sin π₯ βΊ sin π₯ ππ₯ = β cos π₯ + π
π
ππ₯ tan π₯ = sec2 π₯ βΊ π ππ2π₯ ππ₯ = π‘ππ π₯ + π
π
ππ₯ 1
πsin(ππ₯ + π) = cos(ππ₯ + π) βΊ cos(ππ₯ + π) ππ₯ =
1
πsin(ππ₯ + π) + π
π
ππ₯ β
1
πcos(ππ₯ + π) = sin(ππ₯ + π) βΊ sin(ππ₯ + π) ππ₯ = β
1
πcos(ππ₯ + π) + π
π
ππ₯ 1
πtan(ππ₯ + π) = sec2(ππ₯ + π) βΊ π ππ2(ππ₯ + π) ππ₯ =
1
ππ‘ππ (ππ₯ + π) + π
π
ππ₯ ππ₯ = ππ₯ βΊ ππ₯ ππ₯ = ππ₯ + π
π
ππ₯ βπβπ₯ = πβπ₯ βΊ πβπ₯ ππ₯ = βπβπ₯ + π
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New Additional Mathematics
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21. Applications of Integration
For a region R above the x-axis, enclosed by the curve y=f(x), the x-axis and the lines x=a and x=b, the area R is:
π΄ = π π₯ ππ₯π
π
For a region R below the x-axis, enclosed by the curve y=f(x), the x-axis and the lines x=a and x=b, the area R is:
π΄ = βπ π₯ ππ₯π
π
For a region R enclosed by the curves y=f(x) and y=g(x) and the lines x=a and x=b, the area R is:
π΄ = π π₯ β π(π₯) ππ₯π
π
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New Additional Mathematics
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22. Kinematics
π£ =ππ
ππ‘
π =ππ£
ππ‘
π = π£ ππ‘
π£ = π ππ‘
π΄π£ππππ π ππππ =π‘ππ‘ππ πππ π‘ππππ π‘πππ£πππππ
π‘ππ‘ππ π‘πππ π‘ππππ
π£ = π’ + ππ‘
π = π’π‘ +1
2ππ‘2
π =1
2 π’ + π£ π‘
π£2 = π’2 + 2ππ
23. Vectors
If ππ = π₯π¦ then ππ = π₯2 + π¦2
π = ππ and k > 0 a and b are in the same direction
π = ππ and k < 0 a and b are opposite in direction
Vectors expressed in terms of two parallel vectors a and b:
ππ + ππ = ππ + π π βΊ p = r and q = s
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New Additional Mathematics
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If A, B and C are collinear points βΊ AB=kBC
If P has coordinates (x, y) in a Cartesian plane, then the position vector of P is
ππ = π₯π + π¦π
where i and j are unit vectors in the positive direction along the x-axis and the y-
axis respectively.
Unit vector is the direction of ππ is 1
π₯2 + π¦2 π₯π + π¦π ππ
1
π₯2 + π¦2 π₯π¦
24. Relative velocity
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