cat | gmat | gre
TRANSCRIPT
A MATHS PRIMER FOR ALL EXAMS
CAT | GMAT | GRE
Chapter 1: Percentages
Percentage means ‘out of hundred’. A fraction having 100 in the denominator is called a
percentage.
To convert a fraction into a percentage, we need to multiply with 100. For example, the fraction
which is equivalent to doing
.
Some fractions and their conversions are shown below:
Fractio
n Percent Fraction Percent Fraction Percent Fraction Percent
Percentage change: Percentage change is calculated as the change between the initial and final
values calculated as a percentage of the initial value.
Given that the percentage change be n%, we have:
. Rearranging the terms,
we get:
Thus, in order to get the final value, the initial value is to be multiplied with
. Some
examples are shown below:
Percentage change Fraction
multiplied Percentage change Fraction multiplied
increase
decrease
increase
decrease
increase
decrease
increase
decrease
increase
decrease
Percentage change in product of 2 commodities: The net percentage change in the product of
two variables when one changes by and the other changes by is also given by the same
formula:
Percentage change in the product of 2 commodities: Let there be two commodities whose
values are and have percentage changes of and , the net percentage change in the
sum/difference of the two is given by:
Here and have to be taken with their sign, positive for increase and negative for decrease.
Chapter 2: Profit & Loss
Profit loss and discount involves the same concepts as in percentages. Some of the important
points are:
1. The price at which an article is bought is called Cost Price (C.P.)
2. The price at which the article is sold is called Selling Price (S.P.)
3. Profit = S.P. – C.P.
4. Profit % and Loss % are calculated on C.P.
5. Marked price (M.P.) is the price at which an article is sold without any discount.
6. Discount % is calculated on M.P.
7. When there is profit / loss:
;
;
When there is discount:
;
8. S.P. and C.P. in terms of number of articles:
If S.P. of n1 articles = C.P. of n2 articles, % loss or gain =
(If we buy x get y free, it means that x + y articles are sold at the cost price of x articles)
9. If two items are sold, each at the same price, one at a profit of k% and the other at a loss of
k%, there is an overall loss given by:
%.
Chapter 3: Ratio Proportion & Variation
Ratio: A ratio is defined as the relation a quantity of one kind bears with another quantity of the
same kind. The relation signifies what multiple or part the first quantity is of the second.
The ratio a : b can be written as
. In the ratio a : b, ‘a’ is the antecedent, and ‘b’ is the
consequent.
Properties:
1. Ratio remains same when the antecedent and consequent are multiplied/divided by the same
number:
.
2. If
, then k =
(Addendo)
3. If
, then
(Componendo & Dividendo)
4. If we have: , then
Chapter 4: Averages and Alligation
Alligation is the method by which we calculate the ratio in which two different ingredients of
known values are mixed to produce a mixture of a given value. The value of the ingredient may
refer to the price, composition, etc.
The alligation method:
When w1 quantity of cheaper ingredient A of cost C1 and w2 quantity of dearer ingredient B of
cost C2 are mixed to get an average cost of mixture Cm:
The rule can also be represented in the following way:
Price of cheaper quality per unit (C1) Price of dearer quality per unit (C2)
Price of mixture per unit (Cm)
(Price of dearer – Price of mixture) : (Price of mixture – Price of cheaper)
w1 : w2
To find the mean value of a mixture:
When x1 and x2 quantities of A and B of cost C1 and cost C2 respectively are mixed, cost of
mixture:
Chapter 5: Time & Distance
Basic Formulae:
(i) If the ratio of speeds of A and B is a : b, then the ratio of the times taken by them to
cover the same distance is b : a.
(ii) If the ratio of speeds of A and B is a : b, then the ratio of the distances travelled by
them in the same time is a : b.
(iii) If the ratio of time of travel of A and B is a : b, then the ratio of the distances travelled
by them with the same speed is a : b.
Average Speed:
If a man covers a some distances at different speeds, the average speed is given by:
(i) If a man covers a certain distance at x miles/h and an equal distance at y miles/h, the
average speed during the whole journey is
miles/h.
(ii) If a man covers a certain distance at x miles/h and another distance at y miles/h, such
that the time of travel is same, the average speed during the whole journey is
miles/h.
Relative Speed: Relative speed refers to the speed of a moving body with respect to another
moving body.
1. If 2 bodies move in the same direction with speeds V1 & V2 (V1 > V2), then their relative
speed is given as V1 – V2 => to cover a gap D between them, the time taken by the faster
body to catch up with the slower one is
– .
2. If 2 bodies are moving in opposite directions with speeds V1 & V2, then their relative speed
is given as V1 + V2 => To cover a gap D between them, the time taken is
.
Effective or Resultant Speed: Let the speed of an object without the effect of any medium be M
miles/h, and the speed of the medium be R miles/h:
(i) If an object travels in a medium against the flow of the medium: Upstream motion.
Speed Upstream = (M – R).
(ii) If an object travels in a medium with the flow of the medium: Downstream motion.
Speed of object Downstream = (M + R).
Time taken to meet: Let two objects P and Q start moving from two positions A and B in
opposite directions towards each other and meet after time Tm. After meeting they continue their
motion and reach their respective destinations T1 and T2 time after the meeting, then, we have:
Chapter 6: Time & Work
Basic Formulae:
1. If N1 people do a work W1 in D1 days and N2 equally efficient people do a work W2 in D2
days, then:
2. If 2 men take X and Y days to do the same work, their efficiencies are in the ratio
.
3. If A and B take P days more and Q days more than the time taken by A and B together to
complete a work, then time taken by P and Q together is give by the relation:
Chapter 7: Numbers
Introduction
Factors, Multiples, HCF and LCM
If the HCF of two numbers is H, the numbers can be assumed as H × a and H × b such
that a and b are co-prime.
While finding the largest/smallest number that leaves particular remainders when divided
by certain numbers, do not forget to first eliminate options based on the remainder and
divisor combinations, e.g. a number leaving a remainder of 5 on division by 8 has to be
odd; a number leaving a remainder of 3 on division by 15 has to have the unit digit as 3
or 8; etc.
LCM of multiples of 5 would end with just 0 or 5.
HCF of a set of numbers will be odd if even one of the numbers in the given set is odd.
ERROR PRONE AREA: While calculating HCF or LCM of fractions, the fractions
should be in the most reducible form.
The HCF of a set of numbers has to be a factor of the LCM of the set of numbers OR in
other words, the LCM of a set of numbers must be a multiple of the HCF of the set of
Numbers
Real Numbers
Rational Numbers
Recurring decimals
Terminating decimals
Irrational Numbers
Imaginary Numbers
numbers. Thus if HCF of a set of numbers is a multiple of 5, the LCM of the set of
numbers must also be a multiple of 5.
Do not forget that for two numbers, the product of numbers = HCF × LCM
If a series of numbers is of the type a × N + b, then consecutive numbers of this series
differ by N. Conversely any series of numbers differing by N (basically an AP with
common difference N) can be represented as a multiple of N ± x.
Test of Divisibility:
– A number is divisible by when the number formed by its last ‘k’ digits is divisible
by .
3/9 – A number is divisible by 3/9 when the sum of its digits is divisible by 3/9
respectively.
– A number is divisible by when the number formed by its last ‘k’ digits is divisible
by .
6 – A number is divisible by 6, if it is divisible by both 2 and 3.
11 – A number is divisible by 11, if the difference between the sum of the digits in the odd
places and in the even places of the number is divisible by 11.
7 – For 7, we have a series: {-2, -3, -1, 2, 3, 1}. We need to find the sum of the product of
consecutive digits of the number with the numbers in the series from the right to left. The
number is divisible by 7 if the above sum is divisible by 7.
13 – The rule is similar to that of 7, only that the series is {4, 3, -1, -4, -3, 1}.
Last Digit of a number: To find the last digits of a number raised to any exponent, we need to
consider only the last digit of the original number in the base. Thus, the last digit of is the
same as the last digit of .
Last digits of numbers follow a pattern. For example, last digits of 2n: 2
1 = 2, 2
2 = 4, 2
3 = 8, 2
4 =
16, 25 = 32, 2
6 = 64, 2
7 = 128, 2
8 = 256 ……. Note that they end in 2, 4, 8 and 6 and this pattern
gets repeated every fourth power. The patterns are shown below:
Number Pattern
1 1
2 2, 4, 8, 6
3 3, 9, 7, 1
4 4, 6
5 5
6 6
7 7, 9, 3, 1
8 8, 4, 2, 6
9 9, 1
0 0
Highest power of a number in the factorial of another number:
To find the highest power of x that divides N!, keep dividing N successively by x and the
addition of all the quotients (their integer parts) is your answer. (Successive division
means dividing the quotient of the earlier division). While this is the process of getting
the answer, do understand the concept behind find the highest power as it may be used in
other application. The interpretation of the highest power of x that divides a factorial is
that if from 1 × 2 × 3 × 4 × 5 × 6 × …… × N, if all the powers of x is segregated, how
many will they amount to. E.g. From 19!, if we segregate all powers of 3 we will have
something as follows:
1×2×3×4×5×(3×2)×7×8×(3×3)×10×11×(3×4)×13×14×(3×5)×16×17×(3×3×2)×19
= 38×N, N ≠ multiple of 3.
The same logic as above can be used for any product and not necessarily a factorial. Thus
if the questions is what is the highest power of 3 that can divide the product of squares of
all odd numbers from 1 to 20……one should identify that the powers of 3 would appear
only in the squares of 3, 9 and 15. Thus the largest power of 3 dividing the given product
is 2 + 4 + 2 = 8.
To find the highest power P of a number, say N (where N is a prime number) in the
factorial of another number F i.e. F!, we need to successively divide F by N and add the
integer parts of the quotients. Thus, the required highest power:
; here, [K] refers to the integer part of the number K.
When the number of zeroes at the end of a product of series of numbers is asked, think of
the highest power of 2 and 5 in the product.
If any number is expressed as 10n × m, where m is not a multiple of 10, then n is the
number of zeroes at the end of the given number. n is also the highest power of 10 that
divides the given number.
The above property is not just limited to 10. If a number N can be expressed as 7n × m,
where m is not a multiple of 7, then in this case also n is the largest power of 7 that can
divide the given number.
The above rules can be used effectively to factorize a factorial. Thus if one needs to find
the number of ways in which 15! can be written as a product of 2 numbers. Let us
factorize the given number means writing it in the form of 2a × 3
b × 5
c × 7
d × 11
e × ….
Thus we need to find the highest power of prime numbers that can divide the given
number. Thus in our case 15! can be written as 211
× 36 × 5
3 × 7
1 × 11
1 × 13
1.
Factors of a Natural Number: Any number can be expressed as a product of its prime factors
raised to suitable exponents.
Let N = , where p, q, r are distinct prime factors of N and a, b, c are the exponents of
those primes.
1. Let F be the number of factors of the number N. We have:
2. The number of ways in which N can be expressed as the product of two factors, including
N and 1 is given by:
3. The number of ways in which N can be expressed as a product of 2 co–prime factors is:
Here, k is the number of different prime factors of N.
4. Sum of all the factors of N is given by the following product of sum:
Remainders: There are multiple ways of calculating remainders. The rules are given below:
1. If the remainder when N is divided by D is R, then the remainder when is
divided by D is the same as the remainder when is divided by D, where k is some
positive integer.
For example: To find the remainder when is divided by 11:
We know that when divided by 11 leaves a remainder 10, which can be thought of as a
remainder of .
Thus:
is the final remainder.
2. Use of binomial theorem (for positive integer index) to find remainders: when
divided by ‘a’ leaves a remainder equal to the remainder when is divided by ‘a’.
For example: To find the remainder when is divided by 12:
is the final remainder (since 1728 = i.e. a multiple of
12).
Chapter 8: Linear Equations
Linear Equations: Linear equations are equations in which the highest index of the variables is
1. Consider the equation 3x + 5y = 9. Here, x and y are the variables. Index of both x and y is 1.
Simultaneous Linear Equations (in two variables): The two equations represent two straight
lines and the solution represents their point of intersection.
Ex. a1x + b1y + c1 = 0 ……………… Equation (1)
a2x + b2y + c2 = 0 ……………… Equation (2)
Here, a1, b1, c1, a2, b2, c2 are the constants and x, y are the variables.
Different cases for simultaneous equations:
(i)
: This implies that both the equations are identical equations and hence,
there are infinite solutions.
(ii)
: This implies that both the equations are equations of parallel lines and
hence, there exists no possible solution since parallel lines never intersect.
(iii)
: This refers to the case where a unique solution can be obtained from the two
equations.
a.
: Since the ratio of coefficients of y is the same as the ratio of
the constant terms, the value of y in the equation is zero.
b.
: Since the ratio of coefficients of x is the same as the ratio of
the constant terms, the value of x in the equation is zero.
Chapter 9: Quadratic Equations
Quadratic Equations: An equation is said to be quadratic when the variables contain the highest
exponent of 2, thus ax2 + bx + c is a quadratic expression in one variable i.e. of x.
Roots of a Quadratic Equation: After suitable reduction, every quadratic equation can be
written in the form: ax2 + bx + c = 0 and the solution of the equation is:
and
where p and q are the roots of the equation.
Discriminant: The value given by – (the quantity under the square root) is called the
discriminant and depending upon its value we can determine the nature of the roots of the
quadratic equations.
(i) If : The roots are real
(ii) If The roots are real and equal
(iii) If : The roots are real and unequal
(iv) If : The roots are imaginary
(v) If D is a perfect square: The roots are real and rational
(vi) If D is not a perfect square: The roots are real and irrational (one root is conjugate of
the other)
Sum and Product of the Roots: If the roots of the quadratic equation ax2
+ bx + c = 0 are p and
q, then we have:
Suppose we have to form the equation whose roots are and . So (x – ) = 0 and (x – ) = 0
=> (x – )(x – ) = 0 => x2 – ( + )x + = 0 i.e. x
2 – (sum of roots)x + product of roots =
0.
Maxima and Minima: Any quadratic expression can be expressed in the form
where k is the value of x where the expression takes a minimum or maximum
value and m is the corresponding minimum or maximum value of the expression.
c
p q
Minimum
value
m
k
(a > 0)
x
y
Real roots; value of m is
negative, k is the mid-point of
the roots p and q
Real roots; value of m is
positive, k is the mid-point of
the roots p and q
c
p q
Maximum
value
m
k
(a < 0)
x
y
Chapter 10: Inequalities
Inequalities: It deals with cases where variables or numbers are less than or more than other
variables or numbers. The following symbols are used:
>: More than <: Less than
: Less than or equal to : More than or equal to
1. Polynomial inequalities: For example: .
First we plot the roots and decide the sign of the expression between any two of the three
roots. Since the sign of the expression will alternate between consecutive roots, we can
decide the region. The above expression at x = 0 gives a value 8 > 0. Thus, we have the
following diagram:
The shaded region in the diagram depicts the given inequality. Thus: – 2 < x < 1 OR x > 4.
2. Modulus: Modulus of a number is a function that returns the magnitude of the number
ignoring the sign. Thus, |2| = |– 2| = 2. Hence, we have: .
Another way of interpreting modulus is distance from a point on the number line. For
example, |x – 2| = 6 implies that the distance of the point x from the point 2 on the number
line is 6 units. Hence, we attain the points 8 and – 4 on the number line (shown below).
Thus, from the above concept, we have:
a. |x – a| > b => x > (a + b) OR x < (a – b)
b. |x – a| < b => (a – b) < x < (a + b)
– 2 1 4 x
– 4 2 8
6 6
Important points:
0 is negative i.e. < 0a a or a a a a
0a a or a a not possible
0 could be any real numbera a or a a a
Basically a a means a a (implying a is negative) OR a a (Implying a is positive
or zero). Thus could be any real number.
0 is positive or zero i.e. 0a a or a a a a
Basically a a means a a (which is not possible) OR a a (implying a is positive
or zero). Thus a a is nothing but just a a .
The graph of |x + y| = k and |x – y| = k will be
And the graph of |x| ± |y| = k be also be exactly similar to the above!
And would it now be a surprise if we understand that the graph of |x – k| ± |y – h| = k will
also be similar except that the co-ordinates will change. The center of the diamond
(square) shape rather than being (0, 0) will just be (k, h) and accordingly the other co-
ordinates will also change but the size of the diamond will remain the same i.e. a square
of side √2 × k.
3. An inequality (with respect to zero) involving a ratio of two quantities is equivalent to the
same inequality involving the product of the same quantities:
a.
b.
(k, 0)
(0, k)
(-k, 0)
(0, -k)
Chapter 11: Functions & Graphs
Functions: When y is expressed in terms of x such that for any x, a unique value of y is
obtained, is called a function of x. Functions of x are usually denoted by symbols of the form
f(x), g(x), h(x), etc.
Domain and Range: Let f: A B, Then the set A is known as the domain of ‘f’ and the set B
is known as the co-domain of ‘f’. The set of all f- images of element of A is known as the range
of ‘f’ or image set of A under ‘f’ and is denoted by f(A). Thus, f(a) = {f(x) : x A} = range of
‘f’.
For example, Let A = {-1, 0, 1}. Consider f(x) = x2. Then f(-1) = (-1)
2 = 2, f(1) = (1)
2 = 1, f(0)
= 0.
Here domain = {-2, -1, 0, 1, 2}. Range = {0, 1}.
Inverse of a function: To generate the inverse of a function, we need to follow the following
steps:
1. Express x in terms of y
2. Interchange x and y in the final expression
f(x) = , find the inverse function .
.
Thus, the inverse function
.
Three important points:
Even function: If f(- x) = f (x), it is even. Even functions are symmetric about Y-axis.
Odd function: If f(- x) = - f(x), it is odd. Odd functions are symmetric about Origin.
A function and its inverse is symmetric about the line x = y.
Composite functions:
Let f(x) = 2x + 1 and g(x) = .
Then, f(g(x)), f(f(x)), g(g(x)) and g(f(x)) are composite functions.
For example, f(g(x)) = 2g(x) + 1 = ; g(f(x)) = .
Periodic function: A function f(x) is periodic if there exists a real number N so that f(x + N) =
f(x) for all real x. N is the period of the function.
For example, f(x) = sin(x) is a periodic function with period since sin( ) = sin(x).
Piece-wise functions: functions which have different expressions over different values of x are
piece-wise functions. An example is shown below:
=
Max – Min functions:
f(x) = max (a, b) implies that f(x) = a if a > b OR f(x) = b if b > a.
f(x) = min (a, b) implies that f(x) = a if a < b OR f(x) = b if b < a.
Chapter 12: Sequences
Sequence: A series of numbers a1, a2 … an formed by some definite rule is a sequence or
progression.
Arithmetic Progression: A succession of numbers is said to be in arithmetic progression (A.P.)
if the difference between any two consecutive terms is always constant (called the common
difference).
a, a + d, a + 2d, … (where first term is a & common difference is d)
Geometric Progression: A progression, in which every term bears a constant ratio with its
preceding term, is called a geometrical progression (G.P.). The constant ratio is called the
common ratio.
a, ar, ar2, … (where first term is a & common ratio is r)
Some important results:
1. Sum of first ‘n’ natural numbers:
2. Sum of first ‘n’ odd numbers: 3 5 2n 1 n2
3. Sum of the first ‘n’ even numbers: n n n 1
4. Sum of the squares of first ‘n’ natural numbers: 12 2
2 32 n2
n n 1 2n 1
6
5. Sum of the cubes of first ‘n’ natural numbers: 13 2
3 33 n3
n n 1
2 2
6. Arithmetic-co-geometric series: An example of how to proceed is shown below:
Chapter 13: Geometry
Triangles: A triangle is a figure bounded by three lines (AB, BC and CA) in a plane. A, B, C are
called the vertices of the triangle.
Properties of triangles:
1. Sum of any 2 sides in a triangle is greater than the third side.
2. The greatest side in a triangle has the greatest angle opposite to it.
3. Sum of the interior angles is and sum of its exterior angles is
.
4. Any exterior angle in a triangle is equal to sum of the interior opposite angles.
5. If two sides of a triangle are equal it is called isosceles. The angles opposite to the equal
sides are also equal.
6. If all the three sides are equal it is called equilateral. Each angle is equal to .
7. A triangle with no two sides equal is called scalene.
8. If one angle of a triangle is it is called right-angled.
If ABC is right-angled at A, then: AB2 + AC
2 = BC
2 (Pythagoras’ theorem).
9. If one angle is more than it is called obtuse-angled.
10. If all the angles are less than it is called acute-angled.
11. In a triangle ABC, if a, b & c are the 3 sides, where c is the greatest side, then,
(i) If c2 < a
2 + b
2, is acute.
(ii) If c2 = a
2 + b
2, is right angle.
(iii)If c2 > a
2 + b
2, is obtuse.
12. The perimeter of a triangle is the sum of its three sides.
A
B C
13. The medians of a triangle refer to the side bisectors drawn from the opposite vertex. All
three medians always intersect at a single point called the centroid. The centroid divides
each median in the ratio 2:1 (2 units from the vertex and 1 unit from the side).
14. The altitudes of a triangle refer to the perpendicular drawn from a vertex to the opposite
side. All three altitudes always intersect at a single point called the orthocentre.
15. The in-centre refers to the point of intersection of the three lines which bisect the angles at
the vertex of the triangle. A circle can be drawn with its centre as the in-centre which
touches the three sides of the triangle. The radius of the circle is called the in-radius.
16. The circum-centre refers to the point of intersection of the perpendicular side bisectors of
the three sides of the triangle. A circle can be drawn with its centre as the circum-centre
which passes through the three vertices of the triangle. The radius of the circle is called the
circum-radius.
Similar Triangles: Similar triangles are similar in shape, do not have same size but are
proportional i.e. the ratio of their sides is constant.
Two or more triangles can be called similar if:
i) There are two angles of the same measure in all the triangles (since sum of angles is ,
if two angles are equal, the third angle must also be equal)
ii) The ratio of two corresponding sides of both triangles are equal and the angle included
between the above two sides of the two triangles are equal.
If two triangles are similar:
a) Ratio of their corresponding sides = ratio of any corresponding length measure of the
triangles; example: ratio of their heights or medians etc.
b) Ratio of their areas = square of the ratio of their corresponding sides.
c) Ratio of their perimeters = ratio of their corresponding sides.
Properties of Regular Polygons: If the number of sides of a polygon in n:
a) Sum of all interior angles = n 2 x 1800
b) Sum of all exterior angles =
c) Number of diagonals is given by n n 3
2
d) Each interior angle in a regular polygon = n 2 x 1800
n
e) Each exterior angle in a regular polygon = 360
0
n
f) Area of a regular polygon =
x (perimeter) x (perpendicular from centre to any
side)
Circles: Some properties of circles are mentioned below:
1. O = Centre of the circle, OA = Radius, AB = Tangent, XY = Chord
2. Tangent is perpendicular to the radius i.e. AB OA.
3. Perpendicular from the centre bisects the chord i.e. OZ bisects XY.
4. Similarly, if Z is the mid-point of XY, then OZ is perpendicular to XY.
Some special points:
1. Among all quadrilaterals with the same area, the square has the least perimeter.
2. Among all quadrilaterals with the same perimeter, the square is has the greatest area.
3. Among all figures with the same perimeter, the circle has the maximum area.
4. Among all figures with the same area, the circle has the least perimeter.
O
Radius
A
B
X Y Z
Chapter 14: Mensuration and Solid Geometry
We consider different solid shapes in this section. The properties of different solids are
mentioned below.
Right Prism: A prism is a shape that has the same uniform cross-section at any point of its
height. A right prism is a prism where the lateral edges are perpendicular to the base.
1. Volume of a prism = Area of base x Height
2. Lateral surface area = Perimeter of base x Height
3. Total surface area = Lateral surface area + Base area(s)
Right Pyramid: A pyramid is a shape whose area of cross-section decreases at a uniform rate till
it converges to a point, called the vertex. A pyramid whose base is a regular polygon, the centre
of which coincides with the foot of the perpendicular dropped from the vertex on base is called
right pyramid.
1. Volume of a pyramid =
x Area of base x Height
2. Lateral surface area =
x Perimeter of base x Slant height
3. Total surface area = Lateral surface area + Base area
Solids inside other solids:
1. If a largest possible sphere is circumscribed by a cube of edge ‘a’, the radius of the sphere
= a
2.
2. If a largest possible cube is inscribed in a sphere of radius ‘a’, then the edge of the cube
=2a
3.
3. If a largest possible sphere is inscribed in a cylinder of radius ‘a’ and height ‘h’, then for h
> a, the radius of the sphere = a, and for a > h, radius = h
2.
Chapter 15: Co-ordinate Geometry
Rectangular axes: The figure shows the XY Cartesian plane. The line XOX is called the X axis
and YOY the Y axis. A point P(x, y) in the plane denotes the x value or abscissa of P and the y
value or the ordinate of P.
The plane is divided into four equal parts by the two axes, called the Quadrants (I, II, III, and
IV).
Points to remember:
i) The x-axis divides the line joining the points (x1, y1) and (x2, y2) in the ratio y1 : y2.
ii) The y-axis divides the line joining the points (x1, y1) and (x2, y2) in the ratio x1 : x2.
iii) Two lines are parallel if their inclinations are equal
iv) Two lines are perpendicular if the product of their slopes is .
v) The equation of a straight line parallel to the x-axis is y = a, where a is a constant.
vi) The equation of a straight line parallel to the y-axis is x = b, where b is a constant.
vii) The slope of a line parallel to x axis is zero and that of a line parallel to y axis is
undefined.
Circle: If P(x, y) be a point on the circle having:
Centre O(0, 0) and radius r.
The equation of the circle is:
(+, -) (-, -)
(-, +)
Y
Y’
X’ X
P (x, y) I II
III IV
O
(+, +)
r
Y
X
P(x, y)
y
x
Chapter 16: Set Theory
1. Union: The Union of two sets A and B, (i.e. A B), is a set that contains all the elements
contained in A or B.
A = {2, 3}, B = {1, 3, 5}
Then A B = {1, 2, 3, 5}
2. Intersection: Intersection of two sets A and B, (i.e. A B), that contains the elements
common to both A and B.
A = {2, 3}, B = {1, 3, 5}
Then A B = {3}
3. Difference of Two Sets: Difference of two sets A and B, (i.e. A – B), is a set of elements
present in A but not in B.
A = {2, 4}, B = {2, 6, 5}
Then: A – B = {4} (as shown in the diagram beside).
Similarly: B – A = {6, 5}
4. Complement: Complement of a set A, (i.e. or ), is a set that contains the elements
outside A.
If A = {1, 2} and ξ = {1, 2, 7, 8}
Then A’ = {7, 8}
A
B
A
A
B
A B
Chapter 17: Permutation & Combination
Basic Principle of Counting:
If an event P can be performed in ‘x’ ways and another event Q can be performed in ‘y’ ways
and both operations are mutually exclusive, i.e. there is nothing common to P and Q, then P OR
Q can be performed in x + y ways.
If an event P can be performed in ‘x’ ways and for each of these ‘x’ ways, there are ‘y’ different
ways of performing another independent event Q, then P AND Q can be performed in m x n
ways.
Combination: Combination refers to selection. If there are n distinct objects from which r
objects need to be selected, it is referred to as combination of r objects from n (where n r).
It is written as
1.
2.
3.
4. is maximum in value if
(n even) OR
or
–
(n odd)
Selection of some or all of n different things: Of the n things, one can select either 1 or 2 or 3
or … n things. This is calculated as:
.
Selection of some or all of n identical things: If there are n identical things, then selection of
any number of them can be done in only 1 way as there cannot be a choice between identical
objects. Thus, the number of ways in which we can select either none or all of n identical things
is (n + 1). The number of ways in which we can select some or all of (p + q) things, of which p
are alike of one kind, q are alike of another kind is therefore given by (p + 1)(q + 1). However,
this includes the case in which none of the things are chosen and hence needs to be removed.
Thus, the final answer is – .
Distribution of distinct items equally in identical groups: Let there be ‘n’ distinct objects that
need to be distributed in ‘r’ identical groups.
This is possible only if n is divisible by r.
Let
, i.e. each group has k items.
Thus, the number of ways of achieving the distribution
Distribution of distinct items equally in distinct groups: Let there be ‘n’ distinct objects that
need to be distributed in ‘r’ distinct groups:
Distribution of distinct items in identical groups: The number of ways of distributing ‘n’
distinct objects in ‘r’ groups (not necessarily equal distribution) is given by:
Distribution of distinct items in distinct groups: The number of ways of distributing ‘n’
distinct objects in ‘r’ groups (not necessarily equal distribution) is given by:
Permutation: Permutation is the number of arrangements of r things which can be made by
taking some or all of n different things is called permutation of n things taken r at a time (where
n r). It can be written in the form of formula as
Permutations of n items where some of them are identical:
If there are n things of which p are identical things of one kind, q are identical things of another
kind, r are identical things of another kind and so on , and all of them need to
arranged:
The number of ways is:
.
Permutations of n different items in a circle:
Since a circle has no starting point, we need to first assign one of the items in any of the
available positions to denote the starting point. The rest (n – 1) items can be arranged in (n – 1)!
ways.
Thus, the number of arrangements in a circle (people sitting around a circular table) is (n – 1)!
However, for cases where clockwise and anti-clockwise arrangements become identical (for
example, arranging different flowers in a garland), the answer becomes
ways.
Chapter 18: Probability
Random Experiment: A random experiment is an experiment where all the possible outcomes
of the experiment are known in advance and the exact outcome is unpredictable.
For example: while tossing a coin, the possible outcome is either a head or a tail. However, we
cannot predict what the outcome will be.
Sample Space: This constitutes the set of all outcomes than can possibly occur in an experiment.
However, it must be maintained that no two of the above outcomes can occur simultaneously.
For example, the sample space for throwing a dice is {1, 2, 3, 4, 5, 6}.
Event: This constitutes a set of outcomes which are a part of the sample space.
In the above example of throwing a dice, a possible event could be the occurrence of prime
numbers. This event is denoted by {2, 3, 5}. Thus, we can see that the event is a subset of the
sample space.
Mutually Exclusive Events: If two or more events have nothing in common, then they are said
to be mutually exclusive events.
For example, if we choose a card from a deck of 52 cards, getting a diamond and a spade are
mutually exclusive events since both cannot happen simultaneously. On the contrary, getting a
diamond and a queen are not mutually exclusive since a card can be a queen of diamonds.
Exhaustive Events: A set of events is exhaustive if there is no scope of any other situation
coming up. For example, on asking a question, a man’s reply can either be a true statement or a
false statement. There is no other possible scenario.
Independent Events: Two events are said to be independent if one event does not affect the
other event.
For example, students’ taking a test, the performance of one student does not affect the
performance of other students. Hence, these are independent events.
Classical Definition of Probability: Probability is the measure of chances of occurrence of an
event.
If a random experiment, the probability of the occurrence of A, usually denoted by P(A) is given
by:
It should be remembered that:
’
Expected value: Expected value of an event is the sum of the product of probabilities of every
outcome with the value associated with the outcome.
