- Quantitative Ability – POINTS TO REMEMBER
1. If an equation (i.e. f(x) = 0) contains all positive co-efficients of any powers of x, it has no positive roots.
Eg: x3+3x2+2x+6=0 has no positive roots
2. For an equation, if all the even powers of x have same sign coefficients and all the odd powers of x have the opposite sign coefficients, then it has no negative roots.
3. 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. 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 root could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum.
a.For a cubic equation ax3+bx2+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
b.For a bi-quadratic equation ax4+bx3+cx2+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
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. Consider the two equations
a. If a1/a2 = b1/b2 = c1/c2, then we have infinite solutions for these equations.
b. If a1/a2 = b1/b2 <> c1/c2, then we have no solution.
c. If a1/a2 <> b1/b2, then we have a unique solution.
8. Roots of x2 + x + 1=0 are 1, w, w2 where 1 + w + w2=0 and w3=1
9. a + b = a + b if a*b>=0
else, a + b >= a + b
10. The equation ax2+bx+c=0 will have max. value when a<0>0. The max. or min. value is given by (4ac-b2)/4a and will occur at x = -b/2a
11. a. If for two numbers x + y=k (a constant), then their PRODUCT is MAXIMUM if x=y (=k/2). The maximum product is then (k2)/4.
b. If for two numbers x*y=k (a constant), then their SUM is MINIMUM if x=y (=root(k)). The minimum sum is then 2*root (k).
12. 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.
13. For any 2 numbers a, b where a>b
a. a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic means respectively)
b. (GM)^2 = AM * HM
14. For three positive numbers a, b, c
(a + b + c) * (1/a + 1/b + 1/c)>=9
15. For any positive integer n
2<= (1 + 1/n)^n <=3
16. a2 + b2 + c2 >= ab + bc + ca
If a=b=c, then the case of equality holds good.
17. a4 + b4 + c4 + d4 >= 4abcd (Equality arises when a=b=c=d=1)
18. (n!)2 > nn
19. 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
20. If n is even, n(n+1)(n+2) is divisible by 24