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XII Important Formulas

Chapter # 1

FUNCTIONS AND LIMITS

1. Open Interval (a,b): We consider all the values between a and b excluding a and b.

2. Close Interval [a, b]: We consider all the values between a and b including a and b.

3. Even Function: A function ( ) is said to be even function if ( ) ( ).

For example, | |

4. Odd Function: A function ( ) is said to be odd function if ( ) ( ).

For example, | |

| |

5. Neither Even nor Odd: The sum or difference of even and odd function is called neither

even nor odd function.

For example, ( )

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17. (

)

‘𝑥’ measure in radian

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Chapter # 2 and 3

STRAIGHT LINE AND THE GENRAL EQUATION OF

STRAIGHT LINE

1. The distance between two points ( ) and ( ) is given by

√( ) ( )

2. The distance from ( ) to origin (0,0) is

√

3. The distance from ( ) to -axis is

| |

4. The distance from ( ) to -axis is

| |

5. The co-ordinates of a point which divide the join of ( ) and ( ) in the given

ratio is

For internally:

(

)

For externally:

(

)

6. The co-ordinates of midpoint of the join of ( ) and ( ) is

(

)

7. Median of a Triangle: The line joining the vertex and

mid-point of opposite side is called median of a triangle.

8. Centroid of Triangle: The point of intersection (point of

concurrency) of the medians is called centroid of a triangle. The co-ordinates of centroid

are

(

)

C

E D

A F B

𝐶(𝑥 𝑦 )

𝐴(𝑥 𝑦 ) 𝐵(𝑥 𝑦 )

𝐴(𝑥 𝑦 )

𝐵(𝑥 𝑦 )

𝐴(𝑥 𝑦 )

𝐵(𝑥 𝑦 )

𝑚 𝑚

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9. In center of a triangle: The point of intersection of the internal

bisectors of the angles is called in center.

The co-ordinates of in center are

(

)

10. The diagonal of parallelogram bisect each other.

11. Inclination of line: The line making an angle

with positive -axis.

12. Slope of a line (gradient): The tangent of inclination is

called slope of line and denoted by m.

For Slope i. When line parallel to -axis slope=0

ii. When line parallel to -axis slope=

iii. When line perpendicular to -axis slope=

iv. When line perpendicular to -axis slope=0

v. When line bisect the first and third quadrant

vi. When line bisect the second and fourth quadrant

vii. When line passes through two points ( ) and ( ) then

viii. When equation of line is given

ix. When two lines are parallel then their slopes are same.

i.e.

x. When two lines are perpendicular then the product of their slope is -1.

i.e.

𝑆𝑙𝑜𝑝𝑒 𝑚 𝑇𝑎𝑛𝜃

𝑆𝑙𝑜𝑝𝑒 𝑇𝑎𝑛𝜃 𝑇𝑎𝑛45°

45°

𝜃

𝑆𝑙𝑜𝑝𝑒 𝑇𝑎𝑛𝜃 𝑇𝑎𝑛 5° 5°

𝐶(𝑥 𝑦 )

𝐴(𝑥 𝑦 ) 𝐵(𝑥 𝑦 )

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xi. When the line and are parallel then

xii. When the line and are perpendicular then

xiii. The angle between two lines is

13. Equation of line: Every linear equation represents a straight line.

i. Slope-Intercept form: Equation of line is , where

ii. Two-Intercept form: Equation of line is

where

iii. When line passes through one point ( ), equation is

( )

iv. When line passes through two points ( ) and ( ) equation is

v. Equation of is

vi. Equation of is

vii. Equation of line parallel to is

viii. Equation of line parallel to is

ix. Equation of line perpendicular to is

x. Equation of line perpendicular to is

𝑎 𝑎 𝑏 𝑏

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14. The line is perpendicular to the line

15. Ortho-center: The point of intersection of altitude is called ortho-center

16. Concurrent lines: When two or more than two

lines pass through a point,

lines are concurrent.

17. If

and

are concurrent line then

|

|

18. The distance form a point ( ) to a line is

| |

√

19. If , point ( ) is above the line.

20. If , point ( ) is below the line.

21. If , point ( ) is on the line.

22. The distance between two parallel lines and is

| |

√

23. Area of triangle:

|

|

24. If area of triangle is zero then points are collinear.

25. Homogenous Equation: An equation which is in the form of is

called homogeneous equation of degree 2. Homogenous equation of degree 2 represent

a pair of line passes through origin having slopes

√

and

√

26. The angle between pair of line is

√

27. If then lines are perpendicular where

𝑙 𝑙 𝑙

𝐶(𝑥 𝑦 )

𝐴(𝑥 𝑦 ) 𝐵(𝑥 𝑦 )

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Chapter # 4

DIFFERENTIABILITY

1. First Principal:

( )

( ) ( )

2.

( )

3.

4.

( )

5.

6.

( )

7.

8.

( )

9.

10.

( )

11.

12.

( )

13.

14.

( )

15.

16.

17.

√

18.

√ ( )

19.

√

20.

√ ( )

21.

22.

( )

23.

24.

( )

25.

| |√

26.

| |√

27.

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Chapter # 5

APPLICATION OF DIFFERENTIAL CALCULUS

1. Slope of tangent at ( )

( )

2. Slope of curve at ( )

( )

3.

( )

4. Magnitude of acceleration =

( )

=

( )

5. For extreme points

( )

6. If

( ) > 0, then point is minimum.

7. If

( ) < 0, then point is maximum

Chapter # 6

ANTIDERIVATES (INTERGRALS)

1. If

∫[ ( )] ( ) [ ( )]

2. If

∫[ ( )] ( ) ( )

3. ∫ ( )

( ) ( )

4. ∫

5. ∫

6. ∫

7. ∫

8. ∫

9. ∫

10. ∫

11. ∫

12. ∫

13. ∫

14. ∫ ( )

15. ∫ ( )

16. ∫

17. ∫

√

18. ∫

√

19. ∫

20. ∫√

√

21. ∫ ( ) ( )

22. ∫ [ ( ) ( )] ( )

23. ∫

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24. Differential Equation: An equation involving derivates is called differential equation.

Chapter # 7

Circle

1. Circle: The set of all points whose distances from fixed point are same, called circle.

Fixed point is called center.

2. Diameter: The chord passes through the center of a circle is called diameter.

3. Equations of circle:

i. When center and radius are given, we use standard equation of circle, which is:

( ) ( )

where ( )

ii. When circle passes through two or more than two points, we use general equation of

circle, which is

where ( )

√

4. When circle passes through origin then there will be no constant term

Equation is

5. When two circles are concentric (same center), then only the value of “c” changes.

6. When circle touches (i.e. the line )

of centre

| |

√ | |

Squaring on both sides

𝑔 𝑐

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7. When circle touches (i.e. the line )

of center

| |

√ | |

Squaring on both sides

8. When center at origin, then equation of circle is

9. The length of tangent from ( ) to a line is

√

Chapter # 8 Parabola, Ellipse and Hyperbola

1. Parabola: The set of all points whose distances from fix point and fix line are same is called parabola. Fix point is called FOCUS and fixed line is called DIRECTRIX.

2. Vertex of Parabola: The mid-point of focus and directrix is called vertex of parabola.

3. Axis of Parabola: The line passes through the focus and perpendicular to directrix is called the axis of parabola.

4. Latus Rectum (L.R): The chord passes through the focus and perpendicular to axis of parabola is called Latus Rectum.

|4 |

5. If then figure is Parabola 6. If then figure is Ellipse 7. If then figure is Hyperbola 8. ( ) figure is Ellipse 9. ( ) figure is Hyperbola

𝑓 𝑐

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ELLIPSE

1. Centre of Ellipse: The mid-point of foci is called center of ellipse.

2. Vertices: The point of intersection of ellipse and the line passes through the foci is called vertices.

3. Major Axis: The line joining the vertices is called major axis, length of major axis is

4. Minor Axis: The line passes through the center and perpendicular to major axis is called minor axis, length of minor axis is

5. Eccentricity: The ratio

is called eccentricity, denoted by ‘e’.

6. Directrix: The line which is at a distance

from the center is called directrix.

7. Latus Rectum: The chord passes through the focus and perpendicular to major axis is

called Latus Rectum, length of

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PARABOLA

y

( ) ( )

Same figure as equation 1

( ) ( )

Same figure as equation 2

Vertex ( ) Vertex ( ) Vertex ( ) Vertex ( ) Focus ( ) Focus ( ) Focus ( ) Focus ( ) Equation of directrix: Equation of directrix: Equation of directrix: Equation of directrix: Equation of axis: Equation of axis: Equation of axis: Equation of axis:

Length of L.R |4 | Length of L.R |4 | Length of L.R |4 | Length of L.R |4 |

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Note: 1. The distance between foci

2. The distance between directrix

( )

( )

( )

( )

Same figure as equation 1 Same figure as equation 2

Centre ( ) Centre ( ) Centre ( ) Centre ( ) Foci ( ) Foci ( ) Foci ( ) Foci ( ) Vertex ( ) Vertex ( ) Vertex ( ) Vertex ( ) Minor ( ) Minor ( ) Minor ( ) Minor ( )

Directrix

Directrix

Directrix

Directrix

Length of major axis Length of major axis Length of minor axis Length of minor axis

Length of L.R=

Length of L.R=

( ) ( )

ELLIPSE

𝐹 ( 𝑐 )

𝐵( 𝑏)

𝐴 ( 𝑎 ) 𝐴(𝑎 )

𝐵( 𝑏)

𝐹 ( 𝑐 )

𝐵( 𝑏)

𝐴 ( 𝑎 ) 𝐴(𝑎 )

𝐵( 𝑏)

𝐴( 𝑎)

𝐴 ( 𝑎)

𝐹( 𝑐)

𝐹 ( 𝑐)

𝐵(𝑏 ) 𝐵 ( 𝑏 )

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HYPERBOLA

1. 2. 3. ( )

4.

5. The distance between foci

6. The distance between directrix

7. Length of Latus Rectum

8. If , hyperbola is rectangular. Note: For the chart of hyperbola, change the sign of the general equation of Ellipse, rest of the chart is same.

Chapter # 9

VECTORS 1. Let

then | | √

2. Let ( )

( )

then ( )

3. Unit Vector

4. | || |

5.

| || |

6. Let

then

7.

8. If dot product is zero then vectors are perpendicular.

9. | || |

where is a vector which is perpendicular to the plane to the vector.

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10.

| || |

11.

12.

13. Cross product is a vector which is perpendicular to the both given vectors.

14. Let

then

|

|

15. Work done

Compiled By: Ahsan Shahid

New Fazaia Intermediate College, Faisal, Karachi,

Pakistan.

Practical Centre.

E-mail: ahsanshahid55@yahoo.com

Contact: +923451392526

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