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Xth Standard FormulaeNumber 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 + (n1) d S_{n} =n/2 [2a + (n1) d] = n/2[2a+l] n = (la)/d +1 ……….a, a+d, a+2d……….. ………ad, 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^{n1} S_{n} = a (r^{n}1) / (r1) if(r>1) = a (1r^{n}) / (1r) 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)(Rr) 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) (Rr+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
Coordinate 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}] Midpoint formula: If M is is the midpoint 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 Xaxis is y = b Equation of a straight line parallel to Yaxis is x = a
SlopeIntercept form To find the equation of a straight line when slope ‘m’ and Y intercept ‘c’ are given y = mx + c.
SlopePoint 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}) TwoPoints 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:
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:
