Additional Mathematics guide for O Levels

February 19, 2017 | Author: Umair Ahmed | Category: N/A
Share Embed Donate


Short Description

Download Additional Mathematics guide for O Levels...

Description

New Additional Mathematics Muhammad Hassan Nadeem

1. Sets A null or empty set is donated by { } or πœ™. P = Q if they have the same elements. P βŠ‡ Q, Q is subset of P. PβŠ†Q, P is subset of R. PβŠƒQ, Q is proper subset of P. PβŠ‚Q, P is proper subset of Q. PβŠ“Q, Intersection of P and Q. PβŠ”Q, union of P and Q. P’ compliment of P i.e. 𝓔-P

2. Simultaneous Equations βˆ’π‘ Β± 𝑏 2 βˆ’ 4π‘Žπ‘ π‘₯= 2π‘Ž

3. Logarithms and Indices Indices 1. π‘Ž0 = 1 2. π‘Ž βˆ’π‘ = 1 𝑝

3. π‘Ž = 𝑝 π‘ž

4. π‘Ž =

𝑝

1 π‘Žπ‘

π‘Ž 𝑝

π‘ž

π‘Ž 5. π‘Žπ‘š Γ— π‘Žπ‘› = π‘Žπ‘š +𝑛 6.

π‘Žπ‘š π‘Žπ‘›

= π‘Žπ‘š βˆ’π‘›

7. π‘Žπ‘š 𝑛 = π‘Žπ‘šπ‘›

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

8. π‘Žπ‘› Γ— 𝑏 𝑛 = π‘Žπ‘ 𝑛 9.

π‘Žπ‘› 𝑏𝑛

π‘Ž 𝑛

=

𝑏

Logarithms 1. π‘Ž π‘₯ = 𝑦 ≫ π‘₯ = π‘™π‘œπ‘”π‘Ž 𝑦 2. π‘™π‘œπ‘”π‘Ž 1 = 0 3. π‘™π‘œπ‘”π‘Ž π‘Ž = 1

4. π‘™π‘œπ‘”π‘Ž π‘₯𝑦 = π‘™π‘œπ‘”π‘Ž π‘₯ + π‘™π‘œπ‘”π‘Ž 𝑦 5. π‘™π‘œπ‘”π‘Ž

π‘₯

𝑦

= π‘™π‘œπ‘”π‘Ž π‘₯ βˆ’ π‘™π‘œπ‘”π‘Ž 𝑦

6. π‘™π‘œπ‘”π‘Ž 𝑏 = 7. π‘™π‘œπ‘”π‘Ž 𝑏 = 𝑦

π‘™π‘œπ‘” 𝑐 𝑏 π‘™π‘œπ‘” 𝑐 π‘Ž 1 π‘™π‘œπ‘” 𝑏 π‘Ž

8. π‘™π‘œπ‘”π‘Ž π‘₯ = π‘¦π‘™π‘œπ‘”π‘Ž π‘₯ 9. π‘™π‘œπ‘”π‘Ž 𝑏 π‘₯ = π‘™π‘œπ‘”π‘Ž π‘₯

1 𝑏

10. log 𝑏 π‘₯ = log 𝑏 𝑐log 𝑐 π‘₯ =

log 𝑐 π‘₯ log 𝑐 𝑏

4. Quadratic Expressions and Equations 1. Sketching Graph y-intercept Put x=0

x-intercept Put y=0

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

Turning point Method 1 x-coordinate: π‘₯ = y-coordinate: 𝑦 =

βˆ’π‘ 2π‘Ž 4π‘Žπ‘ βˆ’π‘ 2 4π‘Ž

Method 2 Express 𝑦 = π‘Žπ‘₯ 2 + 𝑏π‘₯ + 𝑐 as 𝑦 = π‘Ž π‘₯ βˆ’ 𝑕 2 + π‘˜ by completing the square. The turning point is 𝑕, π‘˜ .

2. Types of roots of π’‚π’™πŸ + 𝒃𝒙 + 𝒄 = 𝟎 𝑏 2 βˆ’ 4π‘Žπ‘ β‰₯ 0 : real roots 𝑏 2 βˆ’ 4π‘Žπ‘ < 0 : no real roots 𝑏 2 βˆ’ 4π‘Žπ‘ > 0 : distinct real roots 𝑏 2 βˆ’ 4π‘Žπ‘ = 0 : equal, coincident or repeated real roots

5. Remainder Factor Theorems Polynomials 1. ax 2 + bx + c is a polynomial of degree 2. 2. ax 3 + bx + c is a polynomial of degree 3.

Identities 𝑃 π‘₯ ≑𝑄 π‘₯ βŸΊπ‘ƒ π‘₯ =𝑄 π‘₯

For all values of x

To find unknowns either substitute values of x, or equate coefficients of like powers of x.

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

Remainder theorem If a polynomial f(x) is divided by (x-a), the remainder is f(a)

Factor Theorem (x-a) is a factor of f(x) then f(a) = 0

Solution of cubic Equation I. II. III.

Obtain one factor (x-a) by trail and error method. Divide the cubic equation with a, by synthetic division to find the quadratic equation. Solve the quadratic equation to find remaining two factors of cubic equation.

For example: I. II.

III. IV. V.

The equation π‘₯ 3 + 2π‘₯ 2 βˆ’ 5π‘₯ βˆ’ 6 = 0 has (x-2) as one factor, found by trail and error method. Synthetic division will be done as follows:

The quadratics equation obtained is π‘₯ 2 + 4π‘₯ + 3 = 0. Equation is solved by quadratic formula, X=-1 and X=-3. Answer would be (x-2)(x+1)(x+3).

6. Matrices www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

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 3

βˆ’2 βˆ’4 + 5 4

2 2 = 1 7

0 . 6

4. Subtraction To subtract two matrices, we subtract their corresponding elements. For example,

6 9

3 14

5 2 βˆ’ βˆ’5 βˆ’4

7 20

4 5 = 12 1

βˆ’4 βˆ’6

0 . βˆ’6

5. Scalar multiplication To multiply a matrix by k, we multiply each element by k. For example, π‘˜

2 3

4 2π‘˜ = βˆ’1 3π‘˜

2 6 4π‘˜ or 3 = . 4 12 βˆ’π‘˜

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. π‘Ž 𝑏 𝑐 𝑑 2 4 3 2 1 4 For example: 1 3 = 𝑒 𝑓 𝑔 𝑕 1 5 2 7 𝑖 𝑗 π‘˜ 𝑙 2 βˆ’1 To get the first row of product do following:

www.revision-notes.co.cc

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 1 0 is called identity matrix. When it is multiplied with any 0 1 matrix X the answer will be X. π‘Ž 𝑏 π‘Ž 𝑏 b. Determinant of matrix will be = = π‘Žπ‘‘ βˆ’ 𝑏𝑐 𝑐 𝑑 𝑐 𝑑 π‘Ž 𝑏 𝑑 βˆ’π‘ c. Adjoint of matrix will be = 𝑐 𝑑 βˆ’π‘ π‘Ž π‘Ž 𝑏 d. Inverse of non-singular matrix (determinant is β‰  0) will be : 𝑐 𝑑 π‘Žπ‘‘π‘—π‘œπ‘–π‘›π‘‘ 1 𝑑 βˆ’π‘ = π‘‘π‘’π‘‘π‘’π‘Ÿπ‘šπ‘–π‘›π‘Žπ‘›π‘‘ π‘Žπ‘‘ βˆ’ 𝑏𝑐 βˆ’π‘ π‘Ž a. The matrix

8. Solving simultaneous linear equations by a matrix method π‘Žπ‘₯ + 𝑏𝑦 = 𝑕 π‘Ž 𝑏 ≫≫ 𝑐π‘₯ + 𝑑𝑦 = π‘˜ 𝑐 𝑑 βˆ’1 π‘₯ π‘Ž 𝑏 𝑕 Γ— 𝑦 = 𝑐 𝑑 π‘˜

π‘₯ 𝑕 𝑦 = π‘˜

7. Coordinate Geometry www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

Formulas π·π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ 𝐴𝐡 =

π‘₯2 βˆ’ π‘₯1

π‘€π‘–π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘œπ‘“ 𝐴𝐡 =

2

+ 𝑦2 βˆ’ 𝑦1

2

π‘₯1 + π‘₯2 𝑦1 + 𝑦2 , 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: π‘”π‘Ÿπ‘Žπ‘‘π‘–π‘’π‘›π‘‘ = π‘š =

𝑦2 βˆ’ 𝑦1 π‘₯2 βˆ’ π‘₯1

The y-intercept is the y-coordinate of the point at which the line crosses the yaxis (it may need to be extended). This will give the following equation: 𝑦 = π‘šπ‘₯ + 𝑐 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 𝑦 = π‘š1 𝑐1 and 𝑦 = π‘š2 𝑐2 are parallel then π‘š1 = π‘š2 Equations of perpendicular line If lines 𝑦 = π‘š1 𝑐1 and 𝑦 = π‘š2 𝑐2 are perpendicular then π‘š1 = βˆ’

1 π‘š2

and π‘š2 = βˆ’

Perpendicular bisector

www.revision-notes.co.cc

1 π‘š1

.

New Additional Mathematics Muhammad Hassan Nadeem

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‘Œ = π‘šπ‘‹ + 𝑐 Where X and Y are expressions in x and/or y.

9. Functions Page 196

10. Trigonometric Functions

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

πœƒπ‘–π‘  + 𝑣𝑒 90

Sin 2

All 1

180

0,360

Tan 3

Cos 4 270 πœƒπ‘–π‘  βˆ’ 𝑣𝑒

πœƒ is always acute.

Basics sin πœƒ = cos πœƒ = tan πœƒ = tan πœƒ =

π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ π‘•π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘π‘Žπ‘ π‘’ π‘•π‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’ π‘π‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘–π‘π‘’π‘™π‘Žπ‘Ÿ sin πœƒ

π‘π‘Žπ‘ π‘’

cos πœƒ 1

cosec πœƒ = sec πœƒ = cot πœƒ =

sin πœƒ 1

cos πœƒ 1 tan πœƒ

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

Rule 1 sin(90 βˆ’ πœƒ) = cos πœƒ cos 90 βˆ’ πœƒ = sin πœƒ tan 90 βˆ’ πœƒ =

1 tan πœƒ

= cot ΞΈ

Rule 2 sin(180 βˆ’ πœƒ) = + sin πœƒ cos 180 βˆ’ πœƒ = βˆ’cos πœƒ tan 180 βˆ’ πœƒ = βˆ’tan πœƒ

Rule 3 sin(180 + πœƒ) = βˆ’sin πœƒ cos 180 + πœƒ = βˆ’cos πœƒ tan 180 + πœƒ = +tan πœƒ

Rule 4 sin(360 βˆ’ πœƒ) = βˆ’ sin πœƒ cos 360 βˆ’ πœƒ = +cos πœƒ tan 360 βˆ’ πœƒ = βˆ’tan πœƒ

Rule 5 sin(βˆ’ πœƒ) = βˆ’sin πœƒ cos βˆ’πœƒ = +cos πœƒ tan βˆ’πœƒ = βˆ’tan πœƒ

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

Trigonometric Ratios of Some Special Angles cos 45 = sin 45 =

1

cos 60 =

2 1

1 2

3 2 1 sin 30 = 2 1 tan 30 3

cos 30 =

3 2 tan 60 = 3 sin 60 =

2 tan 45 = 1

11. Simple Trigonometric Identities Trigonometric Identities sin2 πœƒ + cos 2 πœƒ = 1 1 + tan2 πœƒ = sec 2 πœƒ 1 + cot 2 πœƒ = cosec 2 πœƒ

12. Circular Measure Relation between Radian and Degree πœ‹ 2 3πœ‹ 2

π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  = 90Β°

πœ‹ π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  = 180Β°

π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  = 270Β°

2πœ‹ π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘  = 360Β°

𝑠 = π‘Ÿπ›³ where s is arc length, r is radius and Ο΄ is angle of sector is radians 1

1

2

2

𝐴 = π‘Ÿπ‘  = π‘Ÿ 2 𝛳

where A is Area of sector π‘Žπ‘Ÿπ‘’π‘Ž π‘œπ‘“ π‘ π‘’π‘π‘‘π‘œπ‘Ÿ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘“ π‘ π‘’π‘π‘‘π‘œπ‘Ÿ = π‘Žπ‘Ÿπ‘’π‘Ž π‘œπ‘“ π‘π‘–π‘Ÿπ‘π‘™π‘’ π‘Žπ‘›π‘”π‘™π‘’ π‘œπ‘“ π‘π‘–π‘Ÿπ‘π‘™π‘’

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

13. Permutation and Combination 𝑛! = 𝑛 𝑛 βˆ’ 1 𝑛 βˆ’ 2 Γ— … Γ— 3 Γ— 2 Γ— 1 0! = 1 𝑛! = 𝑛 𝑛 βˆ’ 1 !

π‘›π‘ƒπ‘Ÿ = π‘›πΆπ‘Ÿ =

𝑛! π‘›βˆ’π‘Ÿ ! 𝑛! 𝑛 βˆ’ π‘Ÿ ! π‘Ÿ!

14. Binomial Theorem π‘Ž+𝑏

𝑛

= π‘Žπ‘› + 𝐢1𝑛 π‘Žπ‘›βˆ’1 𝑏 + 𝐢2𝑛 π‘Žπ‘› βˆ’2 𝑏 2 + 𝐢3𝑛 π‘Žπ‘› βˆ’3 𝑏 3 + β‹― + 𝑏 𝑛 π‘‡π‘Ÿ+1 = π‘›πΆπ‘Ÿ π‘Žπ‘›βˆ’π‘Ÿ 𝑏 π‘Ÿ

15. Differentiation 𝑑 𝑛 π‘₯ = 𝑛π‘₯ π‘›βˆ’1 𝑑π‘₯ 𝑑 π‘Žπ‘₯ π‘š + 𝑏π‘₯ 𝑛 = π‘Žπ‘šπ‘₯ π‘š βˆ’1 + 𝑏𝑛π‘₯ π‘›βˆ’1 𝑑π‘₯ 𝑑 𝑛 𝑑𝑒 𝑒 = π‘›π‘’π‘›βˆ’1 𝑑π‘₯ 𝑑π‘₯ 𝑑 𝑑𝑣 𝑑𝑒 𝑒𝑣 = 𝑒 +𝑣 𝑑π‘₯ 𝑑𝑐 𝑑π‘₯

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

𝑑𝑒 𝑑𝑣 𝑣 βˆ’π‘’ 𝑑 𝑒 = 𝑑π‘₯ 2 𝑑π‘₯ 𝑑π‘₯ 𝑣 𝑣 Where β€˜v’ and β€˜u’ are two functions Gradient of a curve at any point P(x,y) is

𝑑𝑦 𝑑π‘₯

at x

16. Rate of Change The rate of change of a variable x with respect to time is

𝑑π‘₯ 𝑑𝑑

𝑑𝑦 𝑑𝑦 𝑑π‘₯ = Γ— 𝑑𝑑 𝑑π‘₯ 𝑑𝑑 𝛿𝑦 𝑑𝑦 β‰ˆ 𝛿π‘₯ 𝑑π‘₯ π‘ƒπ‘’π‘Ÿπ‘π‘’π‘›π‘‘π‘Žπ‘”π‘’ π‘π‘•π‘Žπ‘›π‘”π‘’ 𝑖𝑛 π‘₯ =

𝛿π‘₯ Γ— 100% π‘₯

𝑓 π‘₯ + 𝛿π‘₯ = 𝑦 + 𝛿𝑦 β‰ˆ 𝑦 +

𝑑𝑦 𝛿π‘₯ 𝑑π‘₯

17. Higher Derivative 𝑑𝑦 𝑑π‘₯ 𝑑𝑦 𝑑π‘₯

= 0 when x =a then point (a, f(a)) is a stationary point. = 0 and

𝑑2𝑦 𝑑π‘₯ 2

β‰  0 when x =a then point (a, f(a)) is a turning point.

For a turning point T

I.

If

𝑑2𝑦 𝑑π‘₯ 2

> 0, then T is a minimum point.

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

II.

If

𝑑2𝑦 𝑑π‘₯ 2

< 0, then T is a maximum point.

18. Derivative of Trigonometric Functions 𝑑 sin π‘₯ = cos π‘₯ 𝑑π‘₯ 𝑑 cos π‘₯ = βˆ’ sin π‘₯ 𝑑π‘₯ 𝑑 tan π‘₯ = sec 2 π‘₯ 𝑑π‘₯

𝑑 sinn π‘₯ = 𝑛 sinnβˆ’1 π‘₯ cos π‘₯ 𝑑π‘₯ 𝑑 cos n π‘₯ = βˆ’π‘› cos nβˆ’1 π‘₯ sin π‘₯ 𝑑π‘₯ 𝑑 tann π‘₯ = 𝑛 tannβˆ’1 π‘₯ sec 2 π‘₯ 𝑑π‘₯

19. Exponential and Logarithmic Functions 𝑑 𝑒 𝑑𝑒 𝑒 = 𝑒𝑒 𝑑π‘₯ 𝑑π‘₯ 𝑑 π‘Žπ‘₯ +𝑏 𝑒 = π‘Žπ‘’ π‘Žπ‘₯ +𝑏 𝑑π‘₯

www.revision-notes.co.cc

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 xaxis at the point where ax+b=1 𝑑 1 𝑙𝑛 π‘₯ = 𝑑π‘₯ π‘₯ 𝑑 1 𝑑𝑒 ln 𝑒 = 𝑑π‘₯ 𝑒 𝑑π‘₯ 𝑑 𝑙𝑛 π‘Žπ‘₯ + 𝑏 𝑑π‘₯

=

π‘Ž π‘Žπ‘₯ + 𝑏

𝑑𝑦 =π‘₯ ⟺ 𝑦= 𝑑π‘₯

π‘₯ 𝑑π‘₯

20. Integration

𝑑 1 2 π‘₯ +𝑐 =π‘₯ ⟺ 𝑑π‘₯ 2

1 π‘₯ 𝑑π‘₯ = π‘₯ 2 + 𝑐 2

π‘Žπ‘₯ 𝑛+1 π‘Žπ‘₯ 𝑑π‘₯ = +𝑐 𝑛+1 𝑛

𝑛

π‘Žπ‘₯ + π‘Žπ‘

π‘š

π‘Žπ‘₯ 𝑛+1 𝑏π‘₯ π‘š +1 𝑑π‘₯ = + +𝑐 𝑛+1 π‘š+1

π‘Žπ‘₯ + 𝑏 𝑛+1 (π‘Žπ‘₯ + 𝑏) 𝑑π‘₯ = +𝑐 π‘Ž(𝑛 + 1) 𝑛

𝑑 𝐹 π‘₯ 𝑑π‘₯

𝑏

= 𝑓(π‘₯) ⟺

𝑓 π‘₯ 𝑑π‘₯ = 𝐹 𝑏 βˆ’ 𝐹(π‘Ž) π‘Ž

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem 𝑏

𝑐

𝑓 π‘₯ 𝑑π‘₯ + π‘Ž

𝑐

𝑓 π‘₯ 𝑑π‘₯ = 𝑏

𝑓 π‘₯ 𝑑π‘₯ π‘Ž

𝑏

π‘Ž

𝑓 π‘₯ 𝑑π‘₯ = βˆ’ π‘Ž

𝑓 π‘₯ 𝑑π‘₯ 𝑏

π‘Ž

𝑓 π‘₯ 𝑑π‘₯ = 0 π‘Ž

𝑑 sin π‘₯ = cos π‘₯ ⟺ 𝑑π‘₯

cos π‘₯ 𝑑π‘₯ = sin π‘₯ + 𝑐

𝑑 βˆ’cos π‘₯ = sin π‘₯ ⟺ 𝑑π‘₯

sin π‘₯ 𝑑π‘₯ = βˆ’ cos π‘₯ + 𝑐

𝑑 tan π‘₯ = sec 2 π‘₯ ⟺ 𝑑π‘₯

𝑠𝑒𝑐 2 π‘₯ 𝑑π‘₯ = π‘‘π‘Žπ‘› π‘₯ + 𝑐

𝑑 1 sin(π‘Žπ‘₯ + 𝑏) = cos(π‘Žπ‘₯ + 𝑏) ⟺ 𝑑π‘₯ π‘Ž

cos(π‘Žπ‘₯ + 𝑏) 𝑑π‘₯ =

1 sin(π‘Žπ‘₯ + 𝑏) + 𝑐 π‘Ž

𝑑 1 βˆ’ cos(π‘Žπ‘₯ + 𝑏) = sin(π‘Žπ‘₯ + 𝑏) ⟺ 𝑑π‘₯ π‘Ž

1 sin(π‘Žπ‘₯ + 𝑏) 𝑑π‘₯ = βˆ’ cos(π‘Žπ‘₯ + 𝑏) + 𝑐 π‘Ž

𝑑 1 tan(π‘Žπ‘₯ + 𝑏) = sec 2 (π‘Žπ‘₯ + 𝑏) ⟺ 𝑑π‘₯ π‘Ž

𝑠𝑒𝑐 2 (π‘Žπ‘₯ + 𝑏) 𝑑π‘₯ =

𝑑 π‘₯ 𝑒 = 𝑒π‘₯ ⟺ 𝑑π‘₯ 𝑑 βˆ’π‘’ βˆ’π‘₯ = 𝑒 βˆ’π‘₯ ⟺ 𝑑π‘₯

1 π‘‘π‘Žπ‘› (π‘Žπ‘₯ + 𝑏) + 𝑐 π‘Ž

𝑒 π‘₯ 𝑑π‘₯ = 𝑒 π‘₯ + 𝑐 𝑒 βˆ’π‘₯ 𝑑π‘₯ = βˆ’π‘’ βˆ’π‘₯ + 𝑐

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

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: 𝑏

𝐴=

𝑓 π‘₯ 𝑑π‘₯ π‘Ž

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: 𝑏

𝐴=

βˆ’π‘“ π‘₯ 𝑑π‘₯ π‘Ž

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: 𝑏

𝐴=

𝑓 π‘₯ βˆ’ 𝑔(π‘₯) 𝑑π‘₯ π‘Ž

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

22. Kinematics 𝑣=

𝑑𝑠 𝑑𝑑

π‘Ž=

𝑑𝑣 𝑑𝑑

𝑠=

𝑣 𝑑𝑑

𝑣=

π‘Ž 𝑑𝑑

π΄π‘£π‘’π‘Ÿπ‘”π‘’ 𝑠𝑝𝑒𝑒𝑑 =

π‘‘π‘œπ‘‘π‘Žπ‘™ π‘‘π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ π‘‘π‘Ÿπ‘Žπ‘£π‘’π‘™π‘™π‘’π‘‘ π‘‘π‘œπ‘‘π‘Žπ‘™ π‘‘π‘–π‘šπ‘’ π‘‘π‘Žπ‘˜π‘’π‘›

𝑣 = 𝑒 + π‘Žπ‘‘ 1 𝑠 = 𝑒𝑑 + π‘Žπ‘‘ 2 2 𝑠=

1 𝑒+𝑣 𝑑 2

𝑣 2 = 𝑒2 + 2π‘Žπ‘ 

23. Vectors π‘₯ If 𝑂𝑃 = 𝑦 then 𝑂𝑃 =

π‘₯2 + 𝑦2

𝒃 = π‘˜π’‚ and k > 0 a and b are in the same direction 𝒃 = π‘˜π’‚ and k < 0 a and b are opposite in direction Vectors expressed in terms of two parallel vectors a and b: 𝑝𝒂 + π‘žπ’ƒ = π‘Ÿπ’‚ + 𝑠𝒃 ⟺ p = r and q = s

www.revision-notes.co.cc

New Additional Mathematics Muhammad Hassan Nadeem

If A, B and C are collinear points ⟺ AB=kBC If P has coordinates (x, y) in a Cartesian plane, then the position vector of P is 𝑂𝑃 = π‘₯π’Š + 𝑦𝒋 where i and j are unit vectors in the positive direction along the x-axis and the yaxis respectively. Unit vector is the direction of 𝑂𝑃 is 1 π‘₯π’Š + 𝑦𝒋 π‘œπ‘Ÿ π‘₯2 + 𝑦2

1 π‘₯2 + 𝑦2

π‘₯ 𝑦

24. Relative velocity

www.revision-notes.co.cc

View more...

Comments

Copyright ©2017 KUPDF Inc.
SUPPORT KUPDF