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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(A**∪**B) = n(A) + n(B) — n(A**∩**B)**

**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(A**∪**B**∪**C) = n(A) + n(B) + n(C) — n(A**∩**B) — n(A**∩**C) -
n(B**∩**C) + n(A**∩**B**∩**C)
**

**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-1**C**r-1**

No group empty **n-1**C**r-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 = **n**C**r** 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.