General Fundas For Algebra
Whenever there appears any term of the type a3 + b3 + c3, do check for a + b + c being equal to zero. If a + b + c is indeed zero, then a3 + b3 + c3 = 3abc.
•The series 1, 3, 6, 10, 15 should immediately be recognized as series of sum of first n natural numbers.
•To form all natural numbers from 1 to N by adding any natural numbers, one would just need 1, 2, 4, 8, 16, 32, 64…..2n, where 2n is the largest power of 2 smaller than or equal to N. E.g. What is the minimum number of weights needed to be able to measure all natural numbers weights till 80, if weights can be kept only on one pan of the balance. One would need weights 1, 2, 4, 8, …64
•To form all natural numbers from 1 to N by adding or subtracting any natural numbers, one would just need 1, 3, 9, 27, 81, …..3n. Be careful of the largest number needed in this case. E.g. What is the minimum number of weights needed to be able to measure all natural numbers weights till 80, if weights can be kept on both pans of the balance. One would need weights 1, 3, 9, 27, 81.
•In questions of the type where certain flowers/sweets/points etc gets diminished and then increases and again diminishes and again increases……rather than forming an equation, see if one can work backwards if final quantity is given or else work with options. Also in such cases, if 1/3rd the objects are given away, work on the objects that are remaining i.e. 2/3rd to save time.
If working with options, select options intelligently e.g. I pick 1/3rd of the chocolates in a bowl and then return 3, next I pick 1/5th of the chocolates and then return five, next ……What is the number of chocolates in the bowl initially? The initial number of chocolates has to be a multiple of 3. Also 2/3rd of the initial number of chocolates plus 3 should be divisible by 5. This should be enough to reduce the possible options to just about 2.
•Remember that a2, b2, c2 or |a|, |b|, |c| are either zero or positive quantity. Thus solution to a2 + b2 + c2 = 0 or |a| + |b| + |c| = 0 is a = b = c =0
Algebra equations and inequalities
While solving simultaneous equations of the form x2 + y2 = 65 and x + y = 3, there is no real need to solve it theoretically, which turns out to be a laborious process. Consider perfect squares and by hit and trial see if two perfect squares add up to 65. x2 cannot be 4, 9, 25, 36 as then y2 has to be 61, 56, 40, 29 which are not perfect squares. Thus the possible values for x and y are (±1, ±8) and (±4, ±7). The values that satisfy x + y = 3 are 7 and -4.
The only assumption in the above is that x and y are rational. And this can be ascertained by looking at option choices or certain other data in the question or else one can simply take chances. If one does not get perfect squares adding up to 65, only then work theoretically.
Consider this question:
If x2 + y2 = 0.1 and |x – y| = 0.2, what is the value of |x| + |y|?
1. 0.3 2. 0.4 3. 0.6 4. 0.2
Consider the square of 0.1, 0.2, 0.3, 0.4 i.e. 0.01, 0.04, 0.09, 0.16 and it is obvious that 0.01 + 0.09 = 0.1. Thus x and y are 0.1 and 0.3. And these values also satisfy the second equation. Also the answer choices make it evident that one should not be thinking of multiple values or solutions. Thus answer has to be 0.4
• In simultaneous equations please remember that we need as many independent equations as the number of variables used ONLY when we have to find the unique value of all the variables. If we do not need the values of all the variables but only the value of an equation using the variables, we can possibly find the answer even when the number of equations is less than the number of variables. So do not be in a hurry to mark “Cannot be determined” as the answer in such case.
Even after having as many equations as the number of variables, it is not necessary that we will be able to fid the values of all the variables. To be able t do so, all the equations should be independent. Check to see if by adding or subtracting two of the equations (including use of multiples if needed) we can derive one of the given equation. If we can do this, all the given equations are not independent.
• Whenever the question deals with roots of any polynomial, axn + bxn-1 + cxn-2 + …… + px + q = 0, remember to check options such that sum of roots = -b/a and product of roots = ±q/a. Thus if a = 1 (as is usually the case or else you could make a = 1 by dividing the entire equation by a) and q is an integer and it is also known that the roots are also integers, then the roots have to be factors of q.
• For a quadratic expression used in an inequality and that cannot be factorized, find the determinant and see if the quadratic expression takes only positive or negative values or takes all values:
For a quadratic ax2 + bx + c with imaginary roots i.e. b2 – 4ac < 0,
(ax2 + bx + c) is always positive if a is positive OR
(ax2 + bx + c) is always negative if a is negative
