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Xth Standard Formulae
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.
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.
∑n = n(n+1)/2
∑n2 = n(n+1)(2n+1)/6
∑n3 = [n(n+1)/2]2
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
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
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.
Volume = 4/3 πr3 cubic units
Surface area = 4πr2 sq. units.
Volume = 2/3 πr3 cubic units
Surface area = 2πr2 sq. units.
Total surface area = Curved surface area + area of circular base = 3πr2 sq. units.
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)
Total interest = PNR/100 where N = n(n + 1) /2 *12 years
Amount due = Amount deposited + interest
= Pn + PNR/100
Quarterly Interest = Pr/400
Half yearly Interest = Pr/200
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.
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
To find the equation of a straight line when slope ‘m’ and Y intercept ‘c’ are given
y = mx + c.
To find the equation of a line passing through a point (x1, y1) and with given slope ‘m’.
y – y1 = m(x – x1)
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)]
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
We have learnt the perpendicular bisectors of the sides of a triangle are concurrent.
The point of concurrence is called Circumcentre.
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.
Relation Between Degree and Radians:
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
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
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: