Xth Standard formula - Chennai Tuition Centre


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Xth Standard Formulae

Number Works:

Arithmetic Progression:

The first term of an A.P is generally denoted by ‘a’ and the common difference is denoted by‘d’.

 

tn = a + (n-1) d

Sn =n/2 [2a + (n-1) d] = n/2[2a+l]

n = (l-a)/d +1

……….a, a+d, a+2d………..

………a-d, a, a+d…………. are A.P. numbers.

 

Geometric Progression:

The first term of a G.P is generally denoted by ‘a’ and the common ratio is denoted by ‘r’.

 

tn = a .rn-1

Sn = a (rn-1) / (r-1) if(r>1)

     = a (1-rn) / (1-r) if (r <1)

…………a, ar, ar2………………

…………a/r, a ar………… are G.P numbers.

 

 

Special series:


∑n = n(n+1)/2

∑n2  = n(n+1)(2n+1)/6

∑n3  = [n(n+1)/2]2

 

Mensuration:

 

Right circular cylinder:

Base area of the cylinder = πr2 sq. units

Volume = Base area * height = πr2 h cubic units

Curved surface area = Circumference of the base * height = 2πrh sq. units

Total surface area = Curved surface area + 2 base areas = 2πr (h+r) sq. units

 

Hollow Cylinder:

Area of each end = π (R2 - r2) sq. units

Volume = Exterior volume – Interior volume = πh(R+r)(R-r) cubic units

Curved surface area = External surface area + Internal surface area = 2πh(R+r) sq. units

Total surface area = Curved surface area + 2 area of base rings = 2π(R+r) (R-r+h) sq. units

 

Right circular cone:

Volume = 1/3 * Base area * height = ⅓πr2 h cubic units

Curved surface area = ½ * perimeter of the base * slant height = πrl sq. units

Total surface area = Curved surface area +  base areas = πr (l+r) sq. units

 Slant_height=l=√(r2+h2)                                                                                                                  

 

 

Hollow Cone:

Radius of the sector = Slant height

R = l

Length of the arc = θ/360 * 2πR

Circumference of the base of cone =  Length of the arc.

2πr =  θ/360 * 2πR

Radius of a cone = r =  θ/360 *  Radius of a sector

r =  θ/360 *  l

Volume of Frustum of a cone  =  ⅓πh(R2 + Rr + r2)  cubic units.

 

 

Sphere:

Volume = 4/3 πr3  cubic units

Surface area = 4πr2 sq. units.

Hemisphere:

Volume = 2/3 πr3  cubic units

Surface area = 2πr2 sq. units.

Total surface area = Curved surface area + area of circular base =  3πr2 sq. units.

 

 

 

 

Consumer Arithmetic:

 

The formula for calculating Amount:

A = P (1 + r/100)n

The formula for calculating Compound interest:

C.I. = A – P = P (1 + r/100)n  - P

The formula for calculating simple interest:

S.I. = PNR/100

 

Formula to calculate difference between C.I. and S.I.

When n = 2 years

C.I. – S.I. = P(r/100)2 

When n = 3 years

C.I. – S.I. = (Pr2/1002)(3 + r/100)

 

Recurring Deposit:

Total interest = PNR/100 where N = n(n + 1) /2 *12  years

Amount due = Amount deposited + interest

                      = Pn + PNR/100 

 

Fixed Deposit:

Quarterly Interest = Pr/400

Half yearly Interest = Pr/200

 

 

Co-ordinate Geometry:

Distance between two points A(x1, y1) and  B(x2, y2) :

     d = √[(x2 – x1)2 + (y2 – y1)2]

Mid-point formula: If M is is the mid-point of the line segment joining A(x1, y1) and  B(x2, y2) then

                   M =  [(x1 + x2)/2 , (y1 + y2)/2]

Area of a Triangle: ½ [ x1 ( y2 – y3) + x2 ( y3 – y1) + x3 ( y1 – y2)]  or

 

Area of a Triangle = ½ [ (x1 y2 + x2 y3 + x3 y1) - (x2 y1 + x3 y2 + x1 y3)]

 

 

Area of the quadrilateral =  ½ [ (x1 y2 + x2 y3 + x3 y4 + x4 y1) - (x2 y1 + x3 y2 + x4 y3 + x1 y4)]

 

 

Condition for collinearity of three points:

 

∆ = ½ [ (x1 y2 + x2 y3 + x3 y1) - (x2 y1 + x3 y2 + x1 y3)]  = 0

 

Slope or Gradient of a Straight Line:

Slope = tan θ  = m

Slope of a line joining two points A(x1, y1) and  B(x2, y2)

m = (y2 – y1) / (x2 – x1)     or

m =  (y1 – y2) / (x1 – x2)

 

Condition for two lines to be parallel:

m1 = tan θ1  and m2 = tan θ2 

tan θ1 = tan θ2 

m1 =  m2 

Hence, if two lines are parallel, they have the same slope

 

Condition for two lines to be perpendicular:

 m1 m2  =  -1

product of the slopes of two lines is -1 n then the lines are perpendicular.

 

Note:

If the line is parallel to X axis (perpendicular to y axis ), then the slope is 0.

If the line is perpendicular to X axis (parallel to y axis ), then the slope is not defined.

If the slope of a line is ‘m’ , then the slope of the line perpendicular to it is – 1/m.

 

Equation of a Line:

Equations of coordinate axes:

The x coordinate of every point Y axis is 0 therefore equation of Y axis is  x = 0

The y coordinate of every point X axis is 0 therefore equation of X axis is  y = 0

Equation of a straight line parallel to X-axis  is y = b

Equation of a straight line parallel to Y-axis  is x = a

 

Slope-Intercept form

To find the equation of a straight line when slope ‘m’ and Y intercept ‘c’ are given

                                          y = mx + c.

 

 Slope-Point form

To find the equation of a  line passing through a point (x1, y1) and with given slope ‘m’.

                                   y – y1 = m(x – x1)

 

Two-Points form

To find the equation of a straight line passing through two points (x1, y1) and (x2, y2):

[(y – y1) / (y2 – y1)]  = [(x – x1) / (x2 – x1)]

 

Intercept Form:

Equation of a straight line which makes intercepts a and b on the coordinate axes is

                                      x/a + y/b = 1

 

General form of a straight line:

The linear equation ax + by + c = 0 always represents a straight line. This is the general form of a straight line.

 

Slope of the line = - a/b

In general, slope = - Coefficient of x / Coefficient of y

 

Condition for parallelism of two straight lines:

We know that two straight lines are parallel, if their slopes are equal. If m1 and m2 are the slopes of two parallel lines then m1 =  m2 

The equation of all lines parallel to the line ax + by + c can be put in the form ax + by + k = 0 for different values of k.

 

 

Condition for perpendicularity of two straight lines:

We know that two straight lines are perpendicular, if the product of their slopes is -1. If m1 and m2 are the slopes of two perpendicular lines, then m1 m2  =  -1

The equation of all lines perpendicular to the line ax + by + c can be put in the form bx - ay + k = 0 for different values of k.

 

Concurrency of three lines:

Condition that the lines a1x + b1y + c1 =0, a2x + b2y + c2 = 0, and  a3x + b3y + c3 =0 may be concurrent.

a3 (b1c2 – b2c1) + b3(c1a2 – c2a1) + c3(a2b2 – a2b1) = 0

  

Circumcentre:

We have learnt the perpendicular bisectors of the sides of a triangle are concurrent.

The point of concurrence is called Circumcentre.

 

Centroid:

We have learnt that the medians of a triangle meet at a point. The point is known as Centroid.

 

Centroid is = [(x1 + x2 + x3)/3 , (y1 + y2 + y3)/3]

 

Orthocenter of a Triangle:

We have learnt in theoretical geometry, the altitudes of a triangle meet at a point. This point is called Orthocenter.

  

Trigonometry:


Relation Between Degree and Radians:

Angles in Degrees

    0

30

45

60

90

Angles in Radians

0

π / 6

π / 4

π / 3

π / 2

 

 1c = (180 / π)o

 

1o =  (π / 180)c

 

Trigonometric Ratios of an acute angle of a right triangle:

 

Sin θ = Length of opposite side / Length of hypotenuse side

Cos θ = Length of adjacent side / Length of hypotenuse side

Tan θ = Length of opposite side / Length of adjacent side

Sec θ = Length of hypotenuse side / Length of adjacent side

Cosec θ = Length of hypotenuse side / Length of opposite side

Cot θ = Length of adjacent side  / Length of opposite side 

 

Reciprocal Relations:

 

Sin θ = 1 / Cosec θ                       Sec θ = 1 / Cos θ

 

Cos θ = 1 / Sec θ                          Cosec θ = 1 / Sin θ

 

Tan θ = 1 / Cot θ                           Cot θ = 1 / Tan θ

  

Quotient Relations:

 

Tan θ = Sin θ / Cos θ        

 

Cot θ = Cos θ / Sin θ

 

 

Trigonometric ratios of Complementary angles:

 

Sin (90 – θ) = Cos θ                          Sec (90 – θ) = Cosec θ

 

Cos (90 – θ) = Sin θ                          Cosec (90 – θ) = Sec θ

 

Tan (90 – θ) = Cot θ                           Cot (90 – θ) = Tan θ

 

Trigonometric ratios for angle of measure:

 

   θ

0

30

45

60

90

Sin θ

0

½

1/√2

√3/2

1

Cos θ

1

√3/2

1/√2

½

0

Tan θ

0

1/√3

1

√3

Cot θ

√3

1

1/√3

0

Sec θ

1

2/√3

√2

2

Cosec θ

2

√2

2/√3

1