Number Works:

Arithmetic Progression:

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

t_n = a + (n-1) d

S_n =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’.

t_n = a .r^n-1

S_n = a (rⁿ-1) / (r-1) if(r>1)

     = a (1-rⁿ) / (1-r) if (r <1)

…………a, ar, ar^{2………………}

…………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:

 1^c = (180 / π)^o

1^o =  (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: