*TRICKS 4 EXAM- Sum of n natural numbers -> The sum of first n natural numbers = n (n+1)/2 -> The sum of squares of first n natural numbers is n (n+1)(2n+1)/6 -> The sum of first n even numbers= n (n+1) -> The sum of first n odd numbers= n^2 Finding Squares of numbers To find the squares of numbers near numbers of which squares are knownTo find 41^2 , Add 40+41 to 1600 =1681 To find 59^2 , Subtract 60^2-(60+59) =3481 Finding number of Positive Roots If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then. Eg: x^4+3x^2+2x+6=0 has no positive roots . Finding number of Imaginary Roots For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) . Hence the remaining are the minimum number of imaginary roots of the equation(Since wealso know that the index of the maximum power of x is the number of roots of an equation.) Reciprocal Roots The equation whose roots arethe reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a Roots Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1 Finding Sum of the rootsFor a cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the product of the roots taken two at a time = c/a product of the roots = -d/a For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = - b/a sum of the product of the roots taken three at a time = c/a sum of the product of the roots taken two at a time = -d/a product of the roots = e/a Maximum/Minimum -> If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4 -> If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) . Inequalities -> x + y >= x+y ( stands for absolute value or modulus ) (Useful in solving some inequations) -> a+b=a+b if a*b>=0 else a+b >= a+b -> 2GM>HM>b (where AM, GM,HM stand for arithmetic, geometric , harmonic menasa respectively) (GM)^2 = AM * HM Sum of Exterior Angles For any regular polygon , the sum of the exterior angles is equal to 360 degrees hence measure of any external angle is equal to 360/n. ( where n is the number of sides) For any regular polygon , the sum of interior angles=(n-2)180 degrees So measure of one angle in Square-----=90 Pentagon--=108 Hexagon---=120 Heptagon--=128.5 Octagon---=135 Nonagon--=140 Decagon--=144 Problems on clocks Problems on clocks can be tackled as assuming two runners going round a circle ,one 12 times as fast as the other . That is , the minute hand describes 6 degrees /minute the hour hand describes 1/2 degrees /minute . Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute . The hour and the minute handmeet each other after every 65(5/11) minutes after being together at midnight. (This canbe derived from the above) . Co-ordinates Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2] Ratio If a1/b1 = a2/b2 = a3/b3 =.............. , then each ratio is equal to (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to (a1+a2+a3+............./b1+b2+b3+..........) Finding multiples x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) )......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3) Exponents e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity 2 GP -> In a GP the product of any two terms equidistant from a term is always constant . -> The sum of an infinite GP =a/(1-r) , where a and r are resp. the first term and common ratio of the GP . Mixtures If Q be the volume of a vesselq qty of a mixture of water and wine be removed each time from a mixture n be the number of times this operationbe done and A be the final qtyof wine in the mixture then , A/Q = (1-q/Q)^n
Posted on: Sat, 31 Aug 2013 03:18:05 +0000
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