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 (r^{n}-1) / (r-1) if(r>1)
= a (1-r^{n}) / (1-r) if (r <1)
…………a, ar, ar^{2………………}
…………a/r, a ar………… are G.P numbers.
Special series:
∑n = n(n+1)/2
∑n^{2 }= n(n+1)(2n+1)/6
∑n^{3 }= [n(n+1)/2]^{2}
Mensuration:
Right circular cylinder:
Base area of the cylinder = πr^{2} sq. units
Volume = Base area * height = πr^{2 }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 (R^{2 }- r^{2}) 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 = ⅓πr^{2 }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=√(r^{2}+h^{2})^{ }
^{ }
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(R^{2} + Rr + r^{2})^{ } cubic units.
Sphere:
Volume = 4/3 πr^{3 } cubic units
Surface area = 4πr^{2} sq. units.
Hemisphere:
Volume = 2/3 πr^{3 } cubic units
Surface area = 2πr^{2} sq. units.
Total surface area = Curved surface area + area of circular base = 3πr^{2} 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. = (Pr^{2}/100^{2})(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(x_{1}, y_{1}) and B(x_{2}, y_{2}) :
d = √[(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}]
Mid-point formula: If M is is the mid-point of the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) then
M = [(x_{1} + x_{2})/2 , (y_{1} + y_{2})/2]
Area of a Triangle: ½ [ x_{1} ( y_{2} – y_{3}) + x_{2} ( y_{3} – y_{1}) + x_{3} ( y_{1} – y_{2})] or
Area of a Triangle = ½ [ (x_{1 }y_{2} + x_{2 }y_{3} + x_{3} y_{1}) - (x_{2 }y_{1} + x_{3 }y_{2} + x_{1} y_{3})]
Area of the quadrilateral = ½ [ (x_{1 }y_{2} + x_{2 }y_{3} + x_{3} y_{4} + x_{4} y_{1}) - (x_{2 }y_{1} + x_{3 }y_{2} + x_{4} y_{3} + x_{1} y_{4})]
Condition for collinearity of three points:
∆ = ½ [ (x_{1 }y_{2} + x_{2 }y_{3} + x_{3} y_{1}) - (x_{2 }y_{1} + x_{3 }y_{2} + x_{1} y_{3})] = 0
Slope or Gradient of a Straight Line:
Slope = tan θ = m
Slope of a line joining two points A(x_{1}, y_{1}) and B(x_{2}, y_{2})
m = (y_{2} – y_{1}) / (x_{2} – x_{1})^{ } or
m = (y_{1} – y_{2}) / (x_{1} – x_{2})
Condition for two lines to be parallel:
m_{1} = tan θ_{1} and m_{2} = tan θ_{2}
tan θ_{1} = tan θ_{2}
m_{1 }= m_{2 }
Hence,_{ }if two lines are parallel, they have the same slope
Condition for two lines to be perpendicular:
m_{1} m_{2 }= -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 (x_{1}, y_{1}) and with given slope ‘m’.
y – y_{1} = m(x – x_{1})
Two-Points form
To find the equation of a straight line passing through two points (x_{1}, y_{1}) and (x_{2}, y_{2}):
[(y – y_{1}) / (y_{2} – y_{1})]^{ } = [(x – x_{1}) / (x_{2} – x_{1})]
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 m_{1 }and m_{2 }are the slopes of two parallel lines then m_{1 }= m_{2 }
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 m_{1 }and m_{2 }are the slopes of two perpendicular lines, then m_{1} m_{2 }= -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 a_{1}x + b_{1}y + c_{1} =0, a_{2}x + b_{2}y + c_{2} = 0, and a_{3}x + b_{3}y + c_{3} =0 may be concurrent.
a_{3} (b_{1}c_{2} – b_{2}c_{1}) + b_{3}(c_{1}a_{2} – c_{2}a_{1}) + c_{3}(a_{2}b_{2} – a_{2}b_{1}) = 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 = [(x_{1} + x_{2 }+ x_{3})/3 , (y_{1} + y_{2 }+ y_{3})/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 |
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:
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 |