1•Sum of first n natural numbers : 1 + 2 + 3 +……+ n = n ( n +1)/2
2• Sum of squares of first n natural numbers : 1² + 2² + 3² +……+ n² = n ( n +1) ( 2n +1) /6
3• Sum of cubes of first n natural numbers : 1³ + 2³ + 3³ +……+ n³ = [ n ( n +1)/2]²
4• Sum of first n odd numbers: n²
5• Sum of first n even numbers: n² + n
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In the figure, AB and CD are diameters of the circle with center O, and AB⊥CD and chord DF intersect AB at E and DE = 6, EF = 2. Find the area of the circle.
A. 25π
B. 24π
C. 36π
D. None of the above
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Find the smallest integer n such that √(n + 99) – √n < 1.
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ABCDEF is an equiangular hexagon ( each of its interior angle is 120°) . Find the EF + FA, if AB = 2, BC = 4, CD = 3 and DE = 5.
A. 4
B. 5
C. 6
D. 7
E. None of these
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Ali sells 120 shirts at cost price of 130 shirts after giving a discount same as cost price of 26 shirts. By how much percent Ali has marked his prices up than cost price?
1. 30%
2. 20%
3. 12%
4. 26%
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In the diagram which is not drawn to scale , the circles are tangent at A , the center of the larger circle is at B , CD = 12 , and EF = 8. What is the radius of the smaller circle ?
If a is a positive real number other than 1, and am = x, then we can write: m = logax, and we say that the value of log x to the base a is m.
Examples:
(i) 104 = 10000 log10 10000 = 4.
(ii) 35 = 243 log3 243 = 5.
(iii) (0.1)3 = 0.001 log (0.1) (0.001) = 3.
Properties of Logarithms
(a) In a plane, if there are n points of which no three are collinear, then
(b) In a plane, if there are n points out of which m points are collinear, then
(c) The number of diagonals of an n sided polygon is nC2 – n = n × (n – 3)/2
