summary of quant

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Summary of Quantitative Ability

Wisdom Education

Summary of Quantitative ability

Average

I. Average =Sum of items

number of items

Sum of items = Average × Number of items

Number of items =Sum of items

Average

If A is the average of a1, a2, ..., an , then

(a1 – A ) + (a2 – A ) +... + (an – A ) = 0

II. The average of the numbers which are in arithmetic progression is the middle

number or the average of first and last numbers.

III. If the average of a1, a2, ..., an is A, Then

the average of a1 + k, a2 + k, .... an + k will be A + k.

the average of a1 − k, a2 − k, .... an – k will be A − k. the average of a1 k, a2 k, .... an k will be A k.

the average of a1 /k, a2 /k, .... an / k will be A /k.

If the average of N1 items is A1 , N2 items is A2,.... Nk items is Ak , then the average of all the items together isN1A1+ N2A2+ … + Nk Ak

N1+ N2+⋯.+ Nk

If the average of a group of N1 items is A1 and the average of an other group

of N2 items is A2 and the average of both the groups (i.e N1 + N2 items) is A, then

N1 : N2 =A − A2 ∶ A − A1 If ‘k’ items of average (A + x) are added to a group of ‘n’ items, whose average is A, then the new average will be A +

kx

n+k

o If ‘k’ items of average (A − x) are added to a group of ‘n’ items, whose average is A, then the new average will be

A −kx

n+k

From a group of ‘n’ items whose average is A, if ‘k’ items of average (A + x) are deleted, then the average of the remaining

items will be A −kx

n−k

o From a group of ‘n’ items whose average is A, if ‘k’ items of average (A − x) are deleted, then the average of the

remaining items will be A +kx

n−k

From a group of ‘n’ items whose average is A, If ‘k’ items of average (A + x) are re placed by p items of average (A + y),

Then the new average will beA +

py−kx

n+p−k




Wisdom Education

Numbers

Integers:

Integers: ..., – 3, – 2, – 1, 0, 1, 2, 3,....

Positive integers ( Natural numbers): 1, 2, 3, ....

Negative integers: . ..., – 3, – 2, – 1 Non negative integers(Whole numbers): 0, 1, 2, 3, ...

Even numbers:

The numbers which are divisible by 2 are called even numbers.

The general form of an even number is 2n, ( where n is an integer.)

Odd numbers:

The numbers which gives 1 remainder when divided by 2 are called odd numbers.

The general form of an odd number is 2n + 1, ( where n is an integer.)

Note : Even + Even = Even Even x Even = Even

Even + Odd = Odd Even x Odd = Even

Odd + Odd = Even Odd x Odd = Odd

The sum of ‘n’ odd numbers is odd, if ‘n’ is odd.

The sum of ‘n’ odd numbers is even, if ‘n’ is even.

Prime Numbers:

The numbers which are unbreakable (or) the numbers which have exactly two factors are called Prime numbers.




Wisdom Education

1 is neither prime nor composite.

2 is the only even number which is prime.

Every prime number greater than 3, when divided by 6 leaves either 1 or 5 remainder. i.e., every prime number greater than 3

can be written in the form of 6k ± 1, where k is an integer.

Composite Numbers:The numbers which are breakable, or the numbers which have more than 2 factors.

Perfect Numbers:

Let ‘N’ be a number such that the sum of all its factors is twice of ‘N’, then ‘N’ is a perfect number. Ex: 6, 28, 496, ..

Co- Primes or Relative primes:

The numbers which do not have any common factor (or) the pair of numbers whose HCF is 1 are called co -primes or relative

primes.

Ex: 10 & 21, 4 & 9




Wisdom Education

Rational Numbers:

The numbers which can be written in the form of a fraction (p

q ) ,where p & q are integers and q ≠ 0

Ex:3

4 ,6

5 , 0.24, 3, – 7 , ...

Irrational Numbers:

The numbers which cannot be represented in the form of a fraction

Ex: 3 , 113

, e, π, 2. 354691....

Terminating decimal numbers:

The decimals which terminates ( 0.2, 0. 35, 1.234, ...)

Non terminating decimal numbers:

The decimals which do not terminate ( 0.3333..., 0. 345555..., 0. 132435... , ...)

Pure recurring decimals:

All the digits after the decimal terminates. ( 0.333.., 0. 454545..., .... )

Mixed recurring decimals:

Few digits after the decimal will not terminate but few digits terminates.( 0. 2343434..., 1. 2333..., ... )

Pure recurring to fraction conversion:

Ex: 0. ababaab... =ab

99 (

repeating digits

As many 9′s as number of repeating digits )

Mixed recurring to fraction conversion:

0.abcbcbc... =abc −a

990

( Total digits −non repeating digits

As many 9′s as number of repeating digits and As many 0′s as number of non repeating digits )

The product of ‘n’ consecutive numbers is always divisible by n!

(xn + yn) is divisible by x + y, when n is odd.

(xn – yn) is divisible by x – y, when n is odd.

(xn – yn) is divisible by (x – y) (x + y), when n is even.

Every number ‘N’ can be written in the form of a p × bq × cr × …, (where a, b, c are prime)

If p, q, r, … are even, then N is a perfect square. If p, q, r, … are multiples of 3, then N is a perfect cube. The number of factors of ‘N’ = (p + 1)(q + 1)(r + 1)…




Wisdom Education

The sum of all the factors of ‘N’ = (a0 + a1 + …a p)(b0 + b1 + …+bq)…

Number of co primes to ‘N’ which are less than ‘N’ = N 1 − 1

a 1 −

1

b…

Sum of all co primes to ‘N’ which are less than ‘N’ =N

2 × N 1 −

1

a 1 −

1

b…

Number of ways of writing ‘N’ as a product of two numbers which are co primes is 2n – 1 ,

(n is the number of different prime numbers in ‘N’.)

If ‘n’ is a prime number, then (n – 1)!, when divided by n leaves a remainder of n – 1.

1! + 2 × 2! + 3 × 3! + …. + n × n! = (n + 1)! – 1.

Divisibility rules:

2 or 5: check the last digit

4 or 25: check the number formed by the last two digits.

8 or 125: check the number formed by the last three digits.

3 or 9: check the sum of the digits

11: check the difference of the sum of the digits in even places and sum of

the digits in odd places

The last digit of any power of the numbers which end with 0, 1, 5, or 6 is 0, 1, 5, or 6 only.

The last digit of the numbers which end with 4 or 9 repeats after every 2 nd power

The last digit of the numbers which end with 2, 3,7 or 8 repeats after every 4 th power

The last two digits of any number is the remainder obtained by dividing the number with 100.

LCM & HCF

A factor of a number is an exact divisor of that number

o 1 is a factor of every number

o Every number is a factor of itself

o Every factor is less than or equal to the given number

o Every number is a multiple of itself.

o Every multiple of a number is greater than or equal to that number

LCM of Fractions =LCM of Numerators

HCF of Denominators

HCF of Fractions =HCF of Numerators

LCM of Denominators

The highest number which exactly divide a, b and c is HCF (a, b, c)

The highest number which when divides a, b and c leaves x, y, z remainders respectively is

HCF (a – x, b – y, c – z) The highest number which when divides a and b (a > b) gives same remainder is a – b.

The highest number which when divides a, b and c (a > b > c) gives same remainder is

HCF(a – b, b – c, c – a)

The smallest number which is exactly divisible by a, b and c is LCM (a, b, c)

The smallest number which when divided by a, b and c leaves ‘r’ remainder is LCM (a, b, c) + r

The smallest number which when divided by a, b and c leaves x, y, z remainders respectively

( Where a – x = b – y = c – z = k is LCM (a, b, c) – k




Wisdom Education

Percentage

To convert a percentage into a fraction or decimal, divide it with 100.

If y = k % of p, means y =k

100 × p

To convert a fraction or decimal into percentage, multiply it with 100. i.e.,

If y = ax, it means y is (a × 100)% of x.

percentage change = Actual changeinitial value

× 100

k% more than P = 1 +k

100 P

k% less than P = 1 − k

100 P

If y = bx, b >1, then y is (b – 1) 100% more than x.

If y = bx, b <1, then y is (1 – b) 100% less than x.

If Z = X.Y, and if X increases bya

b times, then Y should decrease by

a

b+a times to keep Z constant.

If X decreases by ab times, then Y should increase by a

b−a times to keep Z constant.

If Z = X.Y, and if X increases by a % and Y increases by b%, then the percentage increase in Z is a + b +ab

100

If Z = X + Y or X – Y, and if both X and Y are increased by a%, then Z also increases by

a%.

If X increases by a% and Y increases by b%, then we cannot find the percentage

increase in Z, unless we know the ratio of X and Y

Suppose if the ratio of X and Y is p : q, then the percentage increase in Z ispa%+ q(b%)

p+q

Profit & Loss

Profit = Selling price – Cost price.

Profit percentage =Profit

Cost price × 100

Margin = Selling price – Cost price.

Margin percentage =Margin

Selling price × 100

Discount = Marked price – Selling price

Discount percentage =discount

Marked price × 100

Markup =Marked price −Cost price

Cost price × 100.




Wisdom Education

Turnover or Revenue = Number of items sold × price of each item.

Ratio & Proportion

Suppose, if p : q = 2 : 3, it means thatp

q =

2

3 or p =

2

3 q

If it is given that the ratio of the ages of A and B is 3 : 4, it does not mean that A is 3 years

old and B is 4 years old. We can only say that Age of A

Age of B = 3

4, or we can assume that the

ages of A and B as 3k and 4k, where k is any real number.

If it is given that the ratio of boys and girls of a school is 2 : 3, it means that2

5 th of the

students of that school are boys and3

5 th of the students are girls.

If a : b =1

2 :

1

3 , it can be written as a : b =

1

2 × 6 ∶

1

3 × 6 = 3 : 2

If a : b : c =1

2 :

1

3 :

1

4 , it can be written as a : b : c =

1

2 × 12 ∶

1

3 × 12 ∶

1

4 × 12 = 6 : 4 : 3

If it is given that a : b :: c : d, it means a, b, c, d are in proportion or a : b = c : d.

If a, b, c are in continued proportion, then b is called the mean proportion of a and c.

o So the mean proportion of a and c is ac

Ifa

b =

c

d = r, then r =

a

b =

c

d =

a+c

b+d =

a−c

b−d =

k 1a+ k 2c

k 1b+ k 2d , where k 1, k 2 are any constants.

Ifa

b =

c

d =

e

f = r, then r =

a

b =

c

d =

e

f =

a+c+e

b+d+f =

k 1a+ k 2c+k 3e

k 1b+ k 2d+ k 3f , where k 1, k 2, k 3 are any constants.

The reciprocal ratio of a : b is b : a

The duplicate ratio of a : b is a2 : b2

The triplicate ratio of a : b is a 3 : b3

The sub duplicate ratio of a : b is a : b The sub triplicate ratio of a : b is a3 ∶ b3

If A and B run a business in partnership and earn some profit,

o profit of A

Profit of B=

Investment of A ×duration

Investment of B ×duration

Variation:

If A varies directly with B, then A = k × B orA1

B1=

A2

B2

If A varies inversely with B, then A =

k

B or A1B1 = A2B2

Time & Work

M1D1H1E1

W1 =

M2D2H2E2

W 2

(M =men, D = days, H = hours, E = efficiency, W = amount of work)

If a man can complete a task in ‘n’ days, then he can do 1/n th part of the work in a day.

If A can do a work in ‘a’ days and B can do the same work in ‘b’ days, then A and B together can do that work in1

1

a+

1

b

=ab

a+b

days.




Wisdom Education

Time & Distance

Distance = Speed × Time

Speed =Distance

Time Time =

Distance

Speed

Relative speed:

If two objects ‘A’ and ‘B’ move in the same direction with speeds Va, V b respectively, ( Va > V b), then the relative speed ofobject ‘A’ w.r.t object ‘B’ is Va – V b

If two objects ‘A’ and ‘B’ move in the opposite directions with speeds Va, V b respectively, ( Va > V b), then the relative speed

of object A’ w.r.t object ‘B’ is Va + V b

Resultant Speed:

If a boat is moving with a speed of B and the water is flowing with a speed of W, then

the resultant speed = B + W, ( same direction)

= B – W ( opposite direction)

Average Speed:

Average Speed = Total Distance travelledTotal Time taken

Circular motion:

If two persons A and B are running around a circular track of length ‘d’ meters with speeds Va, V b m/s respectively ( va >

V b), then

o They meet once in everyd

Va − Vb sec. (same direction)

o They meet once in everyd

Va + Vb sec. (opposite direction)

o They meet once at starting point after every LCM d

Va ,

d

Vb sec ( same direction / opposite direction)

If three persons A and B are running around a circular track of length ‘d’ meters with speeds Va, V b, Vc m/s respectively ( va

> V b > Vc), then

o They meet once in every LCM d

Va − Vb ,

d

Vb −Vc sec. (same direction)

o They meet once in every LCM d

Va − Vb ,

d

Vb +Vc sec. (A and B are in same direction, C in opposite direction)

o They meet once at starting point after every LCM d

Va ,

d

Vb ,

d

Vc sec (same direction / opposite direction)




Wisdom Education

Clocks

Angle between the hands when the time is HH:MM = 30H − 11

2M, where H is hours and M is minutes.

The speed of hour’s hand =1

2

o per minute

The speed of minute’s hand = 6o per minute

The relative speed of minutes hand w.r.t hours hand =11

2

or 5.5o per minute.

Calendars

1 ordinary year - 1 odd day

1 leap year - 2 odd days

100 years - 5 odd days (if there is no leap year century in between)

- 6 odd days (if there is leap year century in between)





400 years - 0 odd days

Equations

The general form of a linear equation is ax + b = 0 and if ax + b =0, x = − b

a

The set of equations a1x + b1y = c1 and a2x + b2y = c2 has

infinite solutions ifa1

a2 =

b1

b2 =

c1

c2

no solution ifa1

a2 =

b1

b2 ≠

c1

c2

unique solution ifa1

a2 ≠

b1

b2

The general form of a quadratic equation is ax2 + bx + c = 0, and its roots (solutions) are

x =−b± b2− 4ac

2a

If x1 and x2 are the roots of a quadratic equation ax2 + bx + c = 0, then

Sum of the roots x1 + x2 =−b

a

Product of the roots x1 . x2 =c

a

Let x1, x2 be the roots of the quadratic equation ax2 + bx + c = 0.

If b2 – 4ac = 0, The roots are real and equal. ( x1 = x2)

If b2 – 4ac > 0, The roots are real and distinct

If b2 – 4ac ≥ 0, The roots are real




Wisdom Education

If b2 – 4ac < 0, The roots are imaginary (complex).

Let x1, x2 (x1 > x2) be the roots of the quadratic equation x2 – ax + b = 0.

If a > 0, b > 0, both the roots are positive ( x 1 > 0, x2 > 0)

If a > 0, b < 0, The greater root is positive and smaller root is negative.

( x1 > 0, x2 < 0)

If a < 0, b < 0, The greater root is negative and smaller root is positive.

( x1 < 0, x2 > 0)

If a < 0, b > 0, both the roots are negative ( x1 < 0, x2 < 0)

Let f(x) = ax2 + bx + c.

If a > 0, then f(x) attains a minimum value of4ac−b2

4a at x =

−b

2a.

If a < 0, then f(x) attains a maximum value of4ac−b2

4a at x =

−b

2a.

Let x1, x2 and x3 be the roots of the cubic equation a1x3+ a2x

2 + a3x+ a4= 0.

x1 + x2 + x3 =−a2

a1

x1x2 + x2x3 + x3x1 =a3

a1

x1x2x3 =−a4

a1

Arithmetic Progression:

General form of A.P: a, a + d, a + 2d, …

( a = First term, d = Common difference)

nth term Tn = a + (n – 1)d

Sum of n terms =n

2 [ F. T + L. T]

=n

2 [ 2a + − 1]

number of terms =Last term – First term

Common difference + 1

Geometric Progression:

General form of G.P: a, ar, ar 2, …

( a = First term, r = Common ratio)

nth term Tn = a. rn−1

Sum of n terms Sn =arn − 1

r−1

Sum of infinite terms S∞= a

1−r

Harmonic Progression:

General form of H.P:1

a,

1

a+d ,

1

a+2d , …

nth term Tn =1

a+n−1d

Mean:

A.M. of a1, a2, … an =a1+a2+ ………+an

A.M of a, b =a+b

2




Wisdom Education

G.M. of a1, a2, …, an = a1a2 ………+an G.M. of a, b = ab

H.M. of a1, a2, … an =n

1

a1+

1

a2+⋯+

1

a n

H.M of a, b =2

1

a+

1

b

=2ab

a+b

n = 1 + 2 + 3 + … + n =n(n+1)

2

n2 = 12 + 22 + … + n2 =nn+1(2n+1)

6

n3 = 13 + 23 + … + n3 = n(n+1)

22

n(n + 1) = 1x2 + 2x3 + … + n( n + 1) =nn+1(n+2)

3

n(n + 1)(n + 2) = 1x2x3 + 2x3x4 + … + n(n+1)(n+2) =nn+1(n+2)(n+3)

4

( n) = 1 + 3 + 6 + … n terms =nn+1(n+2)

6

Functions

If f(x) = g(x), then f(a) = g(a)

f o g (x) = f { g(x) }

g o f (x) = g { f(x) }

If y = f (x), then x = f -1(y)

If f ( – x) = f (x), f(x) is an even function

If f ( – x) = – f (x), f(x) is an odd function

Logarithms

If ax = b ⟹ x =loga b (Or)

o If loga b = x ⟹ b = ax

o

Logarithm is defined only for positive numbers.

o loga 1 = 0 logbn am = mn logb a

o loga a = 1

o loga an = n

o logan a = 1n

o logan am = mn

o

logx a + logx b = logx ab

logx a – logx b = logxab

logy x =log x

log y=

1

log x y =

alog a x = x




Wisdom Education

Surd

Surd :

A number of the form a1/n or an, Which is not an integer is a surd.

Examples : 2, 3, 5, 53, 11

4,…….

( 4 is not a surd because 4 = 2. (an integer) similarly, 9, 83, 81

4,……. are not surds)

All surds are irrational numbers. But e, π,… are irrational numbers but not surds.

2, 3, 5,…. are called quadratic surds

23, 33

, 43,……. are called cubic surds

Surds like 2, 53, 74

,…….. are called pure surds or simple surds.

Surds like 2 3 , 3 2, 4 5, … are called mixed surds.

We can convert mixed surds in to pure surds.

For ex: 2 3 = 4 × 3 = 12

Similarly, 4 × 23

= 43

× 2

3

= 1283

Rationalizing factor :

Rationalizing factor is the surd with which if we multiply another surd, the result will be a rational number.

For example, We can take the rationalizing factor of 2 is 2 (because 2 × 2 = 2)

Our particular interest of surds are the surds of the form a + b or a – b.

The Rationalizing factor of a + b is a – b because (a+ b) (a – b) = a2 – b

Similarly, the Rationalizing factor of a – b is a+ b.

Whenever there is a surd in the denominator rationalize it.

a + b + 2 ab = a + b

Sets

A∪B = A + B – A∩B

A∪B∪C = A + B + C – A∩B – B∩C – C∩A + A∩B∩C

Interest

Simple Interest:

Simple Interest =P t r

100 ,

Amount (A) = P +P t r

100 = P1 +

t r

100

( P = principle, r = rate of interest, t = time)

Compound Interest:

Amount (A) = P 1 +r

100n

(A = amount, P = principle, r = rate of interest, n = number of compoundings)

C.I – S.I (after 2 years) = P r

1002

(C.I = Compound Interest, S.I = Simple Interest)




Wisdom Education

Mensuration

Square:

Area = a2 =1

2 d2

Perimeter = 4a

diagonal d = 2 a

Side a =d

2

Rectangle:

Area = l b

Perimeter = 2(l + b)

diagnol = l2+ b

2

(l = length, b = breadth)

Parallelogram:

Area = ah

= ab sinθ

Perimeter = 2(a + b)

( a, b are the sides)

Rhombus:

Area = ah

= a2 sinθ

=1

2d1d2

Perimeter = 4a

( a = side, d1, d2 are diagnols)

Circle:

Area = π r 2

Diameter d = 2r

Circumference = 2πr

Side = a

b

l

d = l2 + b2

a

b

θ

hb

a

a

a

θ

ha

a

d =

r




Wisdom Education

Sector:

Area =θ

360 × πr 2

θ

Length of arc ‘ l ’ =θ

360 × 2 π r

Triangle:

Area = ss − as − b(s − c)

s =a+ b+c

2

( a, b, c are sides)

Area =1

2 b × h ( b = base, h = height)

Equilateral Triangle:

Area = 34

a2 =h

2

3 h =

32

a

( a = side, h = height)

Regular hexagon:

Area = 6 × 34

a2

Volume

Cuboid:

Surface area = 2h(l + b)Total Surface area = 2(lb + bh + lh)

Volume = lbh

Diagonal = l2 + b + h2

r

A

B C

a a

a

h

a

h

bl




Wisdom Education

Cube:

Surface area = 4a2

Total Surface area = 6a2

Volume = a3

diagonal = a 3

Cylinder:

Surface area = 2πrh

Total Surface area = 2πr(r + h)

Volume = πr 2h

Cone:

Surface area = πrl

Total Surface area = πr(l + r)

Volume =

1

3 πr

2

h

Sphere:

Surface area = 4πr 2

Volume =4

3πr 3

Hemi-sphere:

Surface area = 2πr 2

Total Surface area = 3πr 2

Volume = 2

3πr 3

Geometry

Polygon:

Sum of all external angles of any polygon = 3600

Sum of all internal angles = (n – 2) 1800

Number of diagonals in a polygon = nC2 – n =n(n−3)

2

Triangle:

Sum of the three angles is 1800.

An exterior angle is equal to the sum of the interior opposite angles

The sum of any two sides is always greater than the third side

If two sides of a triangle are equal, then the angles oposte to those sides are also equal.

Scalene triangle: All three sides are different.

Isosceles triangle: Two sides are equal.

Equilateral triangle: All sides are equal. All angles are also equal. Each angle = 60o

Obtuse angled triangle: One angle is greater than 90o

a

aa

r

h




Wisdom Education

Acute angled triangle: All angles are less than 90o

Right Angled Triangle: One angle is exactly 90o

Obtuse angled Acute angled Right angled

In obtuse angled triangle, if AC2 > AB2 + BC2

In right angled, AC2 = AB2 + BC2

In acute angled triangle, AC2 < AB2 + BC2

In general, AC2 = AB2 + BC2 – 2(AB)(BC) cos B (Cosine Rule)

Trigonometric Ratios:

In a right triangle ABC, if θ be the angle between AC & BC

Sin θ =P

H, Cos θ =

B

H, Tan θ =

P

B ,Cosec θ =

H

P, Sec

H

B, Cot θ =

B

P

Similar Triangles:If two angles of ABC are equal to two angles of DEF, Then ABC and DEF are similar.

If ABC and DEF are similar.AB

DE=

BC

EF=

AC

DF

Area of ABC

Area of DEF = AB

DE2

= BC

EF2

= AC

DF2

The line joining the vertex to the mid point of the opposite side is called median.

The point of intersection of the medians is called centroid.

The centroid divides each median from the vertex in the ratio 2 : 1

A

HP

CB

θ

B

A

CB

A

CB

A

B C




Wisdom Education

Apollonius theorem:

AB2 + AC2 = 2(AD2 + BD2),

( AB, BC, AC are sides of a triangle, AD is median. D is mid point of BC)

The point of intersection of the internal bisectors of the angles of a triangle is called Incentre.

In ABC, if AD is the internal angular bisector, thenBD

DC=

AB

AC

In ABC, if AD is the internal angular bisector and ‘I’ is the incentre, then AIID

= AB+ACBC

In ABC, if D divides AB in the ratio m:n and E divides the side AC in the ratio m : n, then DE is parallel to BC and DE =m

m+n

BC

A line joining the mid points of any two sides of a triangle must be parallel to the third side and half of third side.

If ‘r’ is the in radius of the triangle ABC, then the area of the triangle A = r × s, where s =a+ b+c

2

If ‘R’ is the circum radius of the triangle ABC, then the area of the triangle A =abc

4R

Circle:.

A perpendicular drawn from the centre on to a chord bisects the chord

Equal chords of a circle are equidistant from the centre. If the two circles touch externally, the distance between their centres is equal to sum of their radii.

If the two circles touch internally, the distance between their centres is equal to difference of their radii.

Angle at the centre made by an arc is equal to twice the angle made by the arc at any point on the remaining part of the

circumference.

The angle in a semicircle is 900.

If two chords AB and CD intersect externally or internally at P, then PA × PB = PC × PD

If PAB is a secant and PT is a tangent, then PT2 = PA × PB

Co-ordinate geometry

The distance between two points A (x1 ,y1) and B (x2 , y2) is x2 − x1

2

+ y2 − y12

The point which divides the line joining two points A(x1, y1) and B(x2, y2) in the ratio m1 : m2 internally is m1x2+m2x1

m1+m2,

m1y2+m2y1

m1+m2,

The point which divides the line joining two points A(x1, y1) and B(x2, y2) in the ratio m1 : m2 externally is m1x2−m2x1

m1−m2,

m1y2−m2y1

m1−m2

The mid-point of the line joining A (x1 y1) and B (x2, y2) is x1+x2

2,

y1+y2

2

If A (x1, y1), B (x2, y2) and C (x3, y3) are three vertices of a triangle, then

o centroid = x1+x2+x3

3,

y1+y2+y3

3

o Incentre = ax1+ bx2+cx3

a+ b+c,

ay1+ by2+cy3

a+ b+c

Where a = BC, b = CA and c = AB. Equation of a line through the point (x1, y1) and having slope m is y – y1= m (x – x1)

Equation of a line through the points (x1, y1) and (x2, y2) is y – y1 =y2−y1

x2−x1 (x – x1)




Wisdom Education

Equation of a line with intercepts a and b on x-axis and y-axis respectively isx

a+

y

b= 1.

Slope of a line parallel of x-axis is zero

Slope of a line parallel to y-axis is not defined.

Slope of a line equally inclined to equal the axis is – 1 or 1

Slope of the line through the points A (x1, y1) and B (x2, y2) isy2−y1

x2−

x1

Slope of the line ax + by + c = 0, is – a

b

Slopes of two parallel are equal.

If m1 and m2 are the slopes of two perpendicular lines, then m1m2 = – 1

Length of perpendicular from the point (x1, y1) to the line ax + by + c = 0 isax1+ by1+c

a2+ b2

Distance between parallel lines; ax + by + c = 0 and ax + by + d = 0c−d

a2+ b2

Area of triangle:

If A (x1, y1), A (x2, y2) and C (x3, y3) are three vertices of a triangle, then its area is

1

2 x1 − x2 x1 − x3

y1− y

2y

1− y

3 = 1

2 x1− x2)(y

1 − y3 − (y

1 − y2)(x1 − x3)

The equation of circle is (x – h)2 + (y – k)2 = r 2, centre = (h , k) and radius = r.

Permutations & Combinations

n! = 1 ×2 × 3 ×.... × n

ncr =

n!

n−r !r !

n pr =

n!

n−r !

n pr = ncr

× r!

ncr = ncn−r

nc0 = ncn

= 1,

nc1 = n.

n p0 = 1,

n p1 = n,

n pn−1 = n pn

= n!

nc0+ nc1

+ nc2 + nc3

+ .... + ncn = 2n

If an event ‘A’ can be done in ‘m’ ways and an other event ‘B’ can be done in ‘n’ ways, then both the events ‘A’ and ‘B’

together can be done in a given order in m × n ways.




Wisdom Education

‘n’ objects can be arranged in ‘n’ places in n! ways.

‘n’ objects out of which ‘r’ are identical, can be arranged in ‘n’ places inn!

r ! ways.

‘n’ objects can be distributed to ‘n’ persons such that each gets one in n! ways.

‘n’ objects can be distributed to ‘n’ persons (such that more than one can be given to a person) in nn ways.

‘r’ objects can be arranged in ‘n’ places (r < n) in n pr ways.

‘r’ objects can be distributed to ‘n’ persons (r < n) such that no person gets more than one in n pr

ways.

‘r’ objects can be distributed to ‘n’ persons (such that more than one can be given to a person) in nr ways.

‘n’ persons can stand in a row in n! ways.

‘n’ persons can sit around a circular table in (n – 1)! ways.

‘r’ objects can be selected out of ‘n’ objects (r < n) in ncr ways.

‘n’ identical objects can be distributed to ‘r’ persons in (n + r − 1)c(r −1) ways.

The number of ways of selecting one or more objects out of ‘n’ objects is

2n – 1 .

(Since nc0+ nc1

+ nc2+ .... + ncn

= 2n)

The maximum number of lines that can be drawn using ‘n’ non collinear points is nc2.

The maximum number of lines that can be drawn using ‘n’ points out of which ‘r’ points are collinear is

nc2 − r c2+ 1 The maximum number of triangles that can be formed using ‘n’ non collinear points is nc3

.

The maximum number of triangles that can be formed using ‘n’ points out of which ‘r’ points are collinear

is nc3− r c3

The maximum number of intersection points formed by drawing ‘n’ lines is nc2.

The maximum number of intersection points formed by drawing ‘n’ circles is 2 × nc2.

‘n’ lines, no two of which are parallel and no three of them pass through the same point are drawn on a

plane, Then the number of regions that the plane would be divided into is ( n) + 1

The number of diagonals that can be drawn in a polygon of ‘n’ sides is nc2− n =

n(n−3)

2 .

Probability

The probability (or) chance of an event ‘A’ occurs is represented by P(A)

=The number of outcomes satisying the event A

The total number of possible outcomes

The probability that an event ‘A’ does not occur is represented by P(Ā) = 1 – P(A).

P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 1 – p(A ∩ B)

o if A and B are mutually exclusive events, P(A ∩ B) = 0,

Then P(A∪B) = P(A) + P(B).

P(A

∪ B

∪ C) = P (A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A) + P (A ∩ B ∩ C)

= 1 – P(A ∩ B ∩ C).

Two events A and B are said to be independent, if the occurrence of one event does not affect the occurrence of other event.

If A and B are two independent events, the probability of occurring both A and B, represented by P(A∩B) = P(A) × P(B).

The odds in favor of an event A is P(A) : P(Ā). The odds against to an event A is P(Ā) : P(A).

summary of quant

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