Algebra Quick Refresh, Formulas and forms for Mathematics. Visvesvaraya Technological University
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Algebra Quick Refresh, Formulas and forms for Mathematics. Visvesvaraya Technological University

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Math Formulas & Fundas

Shortcuts, Formulas & Tips

Vol. 2: Algebra & Modern Math

present

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Quadratic and Other Equations

For a quadratic equation, ax2 + bx + c = 0, its roots

 �� �������� �� = −���� ± √���� 2−4��������

2����

 Sum of roots = �� + �� = −���� ����

 Product of roots = ���� = ���� ����

Discriminant ∆ = ����2 − 4������

Condition Nature of Roots ∆ < 0 Complex Conjugate

∆ = 0 Real and equal

> 0 and a perfect square Rational and unequal

> 0 and not a perfect square Irrational and unequal

Cubic equation ax3+bx2+cx+d = 0

 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

Biquadratic equation ax4+bx3+cx2+dx+e = 0

 Sum of the roots = - b/a  Sum of the product of the roots taken two at a

time = c/a  Sum of the product of the roots taken three at a

time = -d/a  Product of the roots = e/a

Tip: If c = a, then roots are reciprocal of each other Tip: If b =0, then roots are equal in magnitude but opposite in sign. Tip: Provided a, b and c are rational  If one root is p + iq, other root will be p – iq  If one root is p + �����, other root will be p – �����

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Inequalities

If a > b and c > 0,

 a + c > b + c  a - c > b - c  ac > bc  a/c > b/c

If a, b ≥ 0, then an > bn and 1/an< 1/bn, where n is positive.

a < b and x > 0, then ����+�� ����+��

> ���� ����

a > b and x > 0, then ����+�� ����+��

< ���� ����

Modular Inequalities

|x- y| = |y - x| |x. y| = |x| . |y| |x+ y| < |x| + |y| |x+ y| > |x| - |y|

Quadratic Inequalities

(x – a) (x – b) > 0 {a < b}

 (x < a) U (x > b)

(x – a) (x – b) < 0 {a > b}

 a < x < b

For any set of positive numbers: AM≥GM≥HM

 (a1+a2+ ….+an)/n ≥(a1.a2. …..an)1/n

If a and b are positive quantities, then

 ����+���� 2 ≥ √��������

If a,b,c,d are positive quantities, then

 ���� ����

+���� ���� +���� ����

+���� ����

≥ 4  ����4 + ����4 + ��4 + ��4 ≥ 4������������

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If a,b,c …. k are n positive quantities and m is a natural number, then

 ��������+��������+��������….+������

���� > �����+����+����…+��

���� � ����

�����+����+����+⋯+�� ����

� ����

> ����. ����. ��.��… .��

Tip: ��������+��������

2 >�����+����

2 � ����

[���� ≤ 0 �������� ���� ≥ 1]

��������+��������

2 < �����+����

2 � ����

[0 < ���� < 1]

Tip: For any positive integer n, 2≤ �1 + 1 ���� � ����

≤ 3

Tip: ambncp……..will be greatest when ���� ����

= ���� ����

=���� ����

Tip: If a > b and both are natural numbers, then

 �������� < �������� {Except 32 > 23& 42 = 24}

Tip: (n!)2 ≥ nn

Tip: If the sum of two or more positive quantities is constant, their product is greatest when they are equal and if their product is constant then their sum is the least when the numbers are equal.

 If x + y = k, then xy is greatest when x = y  If xy = k, then x + y is least when x = y

Continued >>

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Logarithm

��������(��������) = ��������(����) + ��������(����)

�������� ����� ���� � = ��������(����)− ��������(����)

��������(��������) = ���� ��������(����)

������������(����) = ��������(����) ��������(����)

������������ ���� = 1

������������ 1 = 0

������������ ������ = ����

Ln x means log�� ����

���� = ����log�� ��

Functions

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Domain: Set of real and finite values that the independent variable can take.

Range: Set of real and finite values that the dependent variable can have corresponding to the values of the independent variable

Co-Domain: Set of real and finite values that the dependent variable can have.

One to One: Every element in the Domain has one and

only one image in the Co-Domain. Every element in Co- Domain has one and only one pre-image in the Domain.

Many to One: If at least two elements in Domain have the same image in the co-domain.

Onto Function: If for every element in the Co-Domain there is at least one pre-image in the Domain. In this case, Range = Co-Domain

Into Function: If there is at least one element in the Co- Domain which does not have a pre-image in the Domain. In this case, Range is a proper subset of Co-Domain.

Even Function: f(x) is even if and only if f(-x) = f(x) for all values of x. The graph of such a function is symmetric about the Y-Axis

Odd Function: f(x) is odd if and only if f(-x) = - f(x) for all values of x. The graph is symmetric about the origin

Graphs

Tip: Range is a subset of Co-Domain. Co-domain may or may not have values which do not have a pre- image in the domain.

Tip: It is not a function if for some value in the domain, the relationship gives more than one value. Eg: f(x) = √���� (At x = 4, f(x) could be both +2 and -2)

Tip: Domain cannot have any extra value ie the values at which the function does not exist.

Tip: If f(x) is an odd function and f(0) exists  f(0) = 0

Continued >>

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f(x) = |x|

If we consider –f(x), it gets mirrored in the X-Axis.

If we consider f(x+2), it shifts left by 2 units

If we consider f(x-2), it shifts right by 2 units.

If we consider f(x) + 2, it shifts up by 2 units.

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If we consider f(x) – 2, it shifts down by 2 units.

If we consider f(2x) or 2f(x) ,the slope doubles and the rise and fall become much sharper than earlier

If we consider f(x/2) or ½ f(x), the slope halves and the rise and fall become much flatter than earlier.

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Set Fundamentals

The number of elements in a set is called its cardinal number and is written as n(A). A set with cardinal number 0 is called a null set while that with cardinal number ∞ is called an infinite set.

Set A is said to be a subset of Set B if each and every element of Set A is also contained in Set B. Set A is said to be a proper subset of Set B if Set B has at least one element that is not contained in Set A. A set with ‘n’ elements will have 2n subsets (2n – 1 proper subsets)

The Universal set is defined as the set of all possible objects under consideration.

Union of two sets is represented as A ∪ B and consists of elements that are present in either Set A or Set B or both. Intersection of two sets is represented as A ∩ B and consists of elements that are present in both Set A and Set B. n(AB) = n(A) + n(B) — n(AB)

Venn Diagram: A venn diagram is used to visually represent the relationship between various sets. What do each of the areas in the figure represent?

I – only A; II – A and B but not C; III – Only B; IV – A and C but not B; V – A and B and C; VI – B and C but not A; VII – Only C

n(ABC) = n(A) + n(B) + n(C) — n(AB) — n(AC) - n(BC) + n(ABC)

Tip: Any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X. The empty set is always a proper subset, except of itself. Every other set is then a subset of the universal set.

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Binomial Theorem

For some basic values:

(���� + ����)0 = 1 (���� + ����)1 = ���� + ���� (���� + ����)2 = ����2 + 2�������� + ����2 (���� + ����)3 = ����3 + 3����2���� + 3��������2 + ����3 (���� + ����)4 = ����4 + 4����3���� + 6����2����2 + 4��������3 + ����4 (���� + ����)5 = ����5 + 5����4���� + 10����3����2 + 10����2����3 + 5��������4 + ����5 Theorem (��+ ��)���� = �������� ���������� + �������� ������−������ + �������� ������−������… + ���������� ����������

(���� + 1)���� = �������� + ������������−1 + ��2���� ��������−2… + �������� + 1

��0���� + ��1���� + ��2���� … + ���������� = 2����

��0���� + ��2���� + ��4���� … = ��1���� + ��3���� + ��5���� … = 2����

2 = 2 ����−1

Some basic properties

Tip: There is one more term than the power of the exponent, n. That is, there are terms in the expansion of (a + b)n.

Tip: In each term, the sum of the exponents is n, the power to which the binomial is raised.

Tip: The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.

Tip: The coefficients start at 1 and increase through certain values about “half”-way and then decrease through these same values back to 1.

Tip: To find the remainder when (x + y)n is divided by x, find the remainder when yn is divided by x.

Tip: (1+x)n ≅ 1 + nx, when x<<1

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Permutation & Combination

When two tasks are performed in succession, i.e., they are connected by an 'AND', to find the total number of ways of performing the two tasks, you have to MULTIPLY the individual number of ways. When only one of the two tasks is performed, i.e. the tasks are connected by an 'OR', to find the total number of ways of performing the two tasks you have to ADD the individual number of ways.

Eg: In a shop there are‘d’ doors and ‘w’ windows. Case1: If a thief wants to enter via a door or window, he can do it in – (d+w) ways. Case2: If a thief enters via a door and leaves via a window, he can do it in – (d x w) ways. Linear arrangement of ‘r’ out of 'n' distinct items (nPr):

The first item in the line can be selected in 'n' ways AND the second in (n — 1) ways AND the third in (n — 2) ways AND so on. So, the total number of ways of arranging 'r' items out of 'n' is

(n)(n - 1)(n — 2)...(n - r + 1) = ����!

(����−����)!

Circular arrangement of 'n' distinct items: Fix the first item and then arrange all the other items linearly with respect to the first item. This can be done in (n — 1)! ways.

Tip: In a necklace, it can be done in (����−��)! ��

ways.

Selection of r items out of 'n' distinct items (nCr): Arrange of r items out of n = Select r items out of n and then arrange those r items on r linear positions.

nPr=nCr x r! → nCr = Prn

��! = ����! ��!(����−��)!

Continued >>

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Dearrangement If 'n' things are arranged in a row, the number of ways in which they can, be deranged so that none of them occupies its original place is

����!� 1 0!−

1 1! +

1 2!− … +

(−1)����

����! �

Partitioning

‘n’ similar items in ‘r’ distinct groups

No restrictions n+r-1Cr-1

No group empty n-1Cr-1

‘n’ distinct items in ‘r’ distinct groups

No restrictions rn

Arrangement in a group important

(����+ ���� − ��)! (���� − ��)!

‘n’ similar items in ‘r’ similar groups

List the cases and then find out in how many ways is each case possible

‘n’ similar items in ‘r’ similar groups

List the cases and then find out in how many ways is each case possible

Tip: Number of ways of arranging 'n' items out of which `p' are alike, 'q' are alike, 'r' are alike in a line

is given by = ���� ! ����!����!����!

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Probability

P(A) = ���������������� ���� ������������������������ �������������������� ������������ ������������������ ���� ��������������������

For Complimentary Events: P(A) + P(A’) = 1

For Exhaustive Events: P(A) + P(B) +P(C)… = 1

Addition Rule:

P (A U B) = P(A) + P(B) – P(A B)

For Mutually Exclusive Events P(A ∩ B) = 0

P (A U B) = P(A) + P(B)

Multiplication Rule:

P(A B) = P(A) P(B/A) = P(B) P(A/B)

For Independent Events P(A/B) = P(B) and P(B/A) = P(B)

P(A B) = P(A).P(B) P (A U B) = P(A) + P(B) – P(A).P(B)

Odds

Odds in favor = ���������������� ���� ������������������������ �������������������� ���������������� ���� ������������������������������ ��������������������

Odds against = ���������������� ���� ������������������������������ �������������������� ���������������� ���� ������������������������ ��������������������

Tip: If the probability of an event occurring is P, then the probability of that event occurring ‘r’ times in ‘n’ trials is = nCr x P

r x (1-P)n-r

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Sequence, Series & Progression

Arithmetic Progression

�������� = ����1 + (���� − 1)��

������ = ���� 2

(����1 + ��������) = ���� 2

[2����1 + (���� − 1)��]

Geometric Progression

�������� = ������������−1

������ = ����(1−������) 1−��

Sum till infinite terms = ���� 1−��

(Valid only when r<1)

Sum of first n natural numbers

 1 + 2 + 3 … + ���� = ����(����+��) ��

Sum of squares of first n natural numbers

 12 + 22 + 32… + ����2 = ����(����+��)(������+��) ��

Sum of cubes of first n natural numbers

 13 + 23 + 33… + ����3 = �����(����+��) �� � ��

Tip: Number of terms = ��������−����1

���� + 1

Tip: Sum of first n odd numbers  1 + 3 + 5 … + (2���� − 1) = ����2

Tip: Sum of first n even numbers  2 + 4 + 6 … 2���� = ����(���� + 1)

Tip: If you have to consider 3 terms in an AP, consider {a-d,a,a+d}. If you have to consider 4 terms, consider {a-3d,a-d,a+d,a+3d}

Tip: If all terms of an AP are multiplied with k or divided with k, the resultant series will also be an AP with the common difference dk or d/k respectively.

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