| **Re: Math Formlae and Shortcuts - **
January 23rd, 2013
**THEORY OF EQUATIONS:**
Shortcuts to getting information about the roots
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(1) If an equation contains all positive co-efficients of any powers of x, then it has no positive roots.[/i]
e.g. x^4+3x^2+2x+6=0 has no positive roots .
(2) If all the even powers of x have same sign coefficients and all the odd powers of x have the opposite sign coefficients, then the equation has no negative roots.
e.g. x^2 - x +2 = 0
(3) Summarizing DESCARTES rules of signs:
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).
(4)Consider the two equations
ax + by = c
dx + ey = f
Then, If {a}/{d} = {b}/{e} = {c}/{f}, then we have infinite solutions for these equations.
If \frac{a}{d} = \frac{b}{e} \neq \frac{c}{f} , then we have no solution for these equations.
If \frac{a}{d} \neq \frac{b}{e} , then we have a unique solutions for these equations.
(5) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to be 2-3i and if there are three possible roots of the equation, we can conclude that the last root is real. This real roots could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum.
(6) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign coefficients then the equation has no real roots in each case, except for x=0 in the second case.
(7) Besides Complex roots, even irrational roots occur in pairs. Hence if 2+root(3) is a root, then even 2-root(3) is a root . (All these are very useful in finding number of positive, negative, real, complex etc roots of an equation )
(8) |x| + |y| >= |x+y| (|| stands for absolute value or modulus ) (Useful in solving some inequations)
(9) For 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
(10) 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
Geometry
General Notions and useful shortcuts:
Polygons:
(1) For any regular polygon, the sum of the interior angles is equal to 360 degrees
(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.
(2.1)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]
(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sies equal).
(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .
Triangles
(1) In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
(2) In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
(3) The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .
(4.1)In any triangle
a=b*CosC + c*CosB
b=c*CosA + a*CosC
c=a*CosB + b*CosA
(4.2)In any triangle
a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
cosC = (a^2 + b^2 - c^2)/2ab
sin2A = 2 sinA * cosA
cos2A = cos^2(A) - sin^2 (A)
(5.1)APPOLLONIUS THEOREM:
In a triangle , if AD be the median to the side BC , then
AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .
(5.2) Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.
(6) The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)
is((a+c+e)/3 , (b+d+f)/3) .
(7) Let a be the side of an equilateral triangle . then if three circles be drawn
inside this triangle touching each other then each's radius = a/(2*(root(3)+1))
(8) Let W be any point inside a rectangle ABCD .
Then WD^2 + WB^2 = WC^2 + WA^2
(9) Some pythagorean triplets:
3,4,5 (3^2=4+5)
5,12,13 (5^2=12+13)
7,24,25 (7^2=24+25)
8,15,17 (8^2 / 2 = 15+17 )
9,40,41 (9^2=40+41)
11,60,61 (11^2=60+61)
12,35,37 (12^2 / 2 = 35+37)
16,63,65 (16^2 /2 = 63+65)
20,21,29(EXCEPTION)
Quadrilateral
(1) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the
internal opposite angle.
(2) If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal
to the sum of the other pair .
(3) the quadrilateral formed by joining the angular bisectors of another quadrilateral is
always a rectangle.
Areas:
(1)Area of a triangle
1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c))
where s=a+b+c/2
=a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the
triangle
(2.1) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2
(2.2) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is 0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.
(3.1) Area of a regular hexagon : root(3)*3/2*(side)*(side)
(3.2) Area of a hexagon = root(3) * 3 * (side)^2
(4) Area of a parallelogram = base * height
(5) Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.
Stereometry
(1) for similar cones , ratio of radii = ratio of their bases.
(2) Volume of a pyramid = 1/3 * base area * height
Number properties
(1) Product of any two numbers = Product of their HCF and LCM .
Hence product of two numbers = LCM of the numbers if they are prime to each other .
(2) The HCF and LCM of two nos. are equal when they are equal .
(3) For any 2 numbers a>b
a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa
respectively)
(4) (GM)^2 = AM * HM
(5) For three positive numbers a, b ,c
(a+b+c) * (1/a+1/b+1/c)>=9
(6) For any positive integer n
2<= (1+1/n)^n <=3
(7) a^2+b^2+c^2 >= ab+bc+ca
If a=b=c , then the equality holds in the above.
(8) a^4+b^4+c^4+d^4 >=4abcd
(9) If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximum
if a/p = b/q = c/r = d/s
(10) (m+n)! is divisible by m! * n! .
(11.1)If n is even , n(n+1)(n+2) is divisible by 24
(11.2)If n is any integer , n^2 + 4 is not divisible by 4
(12) 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)
(13) when a three digit number is reversed and the difference of these two
numbers is taken , the middle number is always 9 and the sum of the other two
numbers is always 9 .
(14) Let 'x' be certain base in which the representation of a number is 'abcd' , then
the decimal value of this number is a*x^3 + b*x^2 + c*x + d
(15) 2<= (1+1/n)^n <=3
(16) (1+x)^n ~ (1+nx) if x<<<1
(17) |a|+|b| = |a+b| if a*b>=0 else |a|+|b| >= |a+b|
(18) In a GP (Geometric Progression?) the product of any two terms equidistant from a term is always constant .
(19)The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .
(20)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+..........) |