Tag: APSC Maths Reasoning notes

Geometry and Mensuration Formulas & Properties – APSC PrelimCSAT Paper Notes, formulas

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Geometry and Mensuration Important Formulas

& Properties – 

Numerical Aptitude Notes for APSC Prelims CSAT Paper, SSC and Competitive Exams

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Circle
  • Area of a circle = R2, where R is the radius.
  • Circumference of a circle = 2πR.
  • Length of an arc =  2 π r ×  (θ/360°), where θ is the central angle.
  • Area of a sector = θ/2 × r2.
  • Perimeter of a semi-circle =πR + 2R.

 

Quadrilaterals

Parallelogram

  • Area of parallelogram = (Base x Height).
  • The diagonals of a parallelogram bisect each other.
  • Each diagonal of a parallelogram divides it into triangles of the same area.
  • A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
  • Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

Rectangle

  • Area of a rectangle = (Length x Breadth).
  • Perimeter of a rectangle = 2(Length + Breadth).
  • The diagonals of a rectangle are equal and bisect each other.

Square

 

  • Area of a square = (side)2 = (diagonal)2.
  • Area of 4 walls of a room = 2 (Length + Breadth) x Height.
  • The diagonals of a square are equal and bisect each other at right angles.

Rhombus

  • Area of a rhombus = x (Product of diagonals).
  • The diagonals of a rhombus are unequal and bisect each other at right angles.

Trapezium

  • Area of a trapezium =  (sum of parallel sides)/2 x distance between them.

 

Triangle

 

  • Sum of the angles of a triangle is 180°.
  • The sum of any two sides of a triangle is greater than the third side.
  • Area of a triangle = x Base x Height.
  • Area of a triangle = s(s-a)(s-b)(s-c), where a, b, c are the sides of the triangle and s = (a + b + c).
  • Area of an equilateral triangle = sqrt(3)/4 x (side)2.

Pythagoras Theorem

In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2.

  • Median – The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.
  • Centroid – The point where the three medians of a triangle meet. The centroid divided each of the medians in the ratio 2 : 1.
  • In an isosceles triangle, the altitude from the vertex bisects the base.
  • The median of a triangle divides it into two triangles of the same area.
  • The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.

Probability – Important Formulas – APSC Prelim CSAT Paper Notes

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Probability – Important Formulas & Notes

Numerical Aptitude Notes for APSC Prelims CSAT Paper, SSC and Competitive Exams

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Probability

A quantitative measure of the chance of occurrence of a particular event. The likelihood of the occurrence of a particular event. The probability of event A is often written as P(A).

The probability of an event can only be between 0 and 1 and can also be written as a percentage. 

Probability of Occurrence of an Event

Let S be the sample and let E be an event.

P(E) = n(E) /n(S)


Results on Probability

P(S) = 1

0 <= P (E) <= 1

P() = 0

For any events A and B, P(A B) = P(A) + P(B) – P(A  B)

 

Experiment – An operation which can produce some well-defined outcomes is called an experiment.

Random Experiment – An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment. Like drawing a card from a pack of well-shuffled cards.

Sample Space – When we perform an experiment, then the set S of all possible outcomes is called the sample space.

In tossing a coin, S = {H, T}

If two coins are tossed, S = {HH, HT, TH, TT}.

In rolling a dice, S = {1, 2, 3, 4, 5, 6}.

Event – Any subset of a sample space is called an event.

Equally Likely Events  – Events are said to be equally likely if there is no preference for a particular event over the other.

Mutually Exclusive Events  – Two or more than two events are mutually exclusive if the occurrence of one of the events excludes the occurrence of the other.

Independent Events – If the occurrence or non-occurrence of one event does not influence the occurrence or non-occurrence of the other.

Simple Events – Events where one experiment happens at a time and it has a single outcome.

Compound Event – An event in which there is more than one possible outcome.

Exhaustive Events – The mutually exclusive events that form the sample space collectively are called the exhaustive events. Example, when a coin is tossed, either Head or Tail appears and they collectively form the sample space.

 

 

L.C.M and H.C.F based Questions – Important Formulas for APSC Prelim CSAT Paper

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L.C.M  and H.C.F based Questions – Important Formulas

Numerical Aptitude Notes for APSC Prelim CSAT Paper, SSC and Competitive Exams

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Factors and Multiples:

If number a divided another number b exactly, we say that a is a factor of b.

b is also a multiple of a.

 

Highest Common Factor (H.C.F.) or Greatest Common Divisor (G.C.D.)

The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.

Two methods of finding the H.C.F. of a given set of numbers:

Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

Division Method: To find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the H.C.F.

 

Least Common Multiple (L.C.M.)

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of finding the L.C.M. of a given set of numbers:

Factorization Method Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

Division Method: Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

 

H.C.F. and L.C.M. of Fractions

H.C.F. = (H.C.F. of Numerators)/(L.C.M. of Denominators)

L.C.M. = (L.C.M. of Numerators)/(H.C.F. of Denominators)

 

Product of two numbers = Product of their H.C.F. and L.C.M.

Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.

 

 

Comparison of Fraction numbers

  1. Find the L.C.M. of the denominators of the given fractions.
  2. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number.
  3. The resultant fraction with the greatest numerator is the greatest.

 

Downstream/Upstream Boat Speed – Important Formulas – APSC PrelimCSAT Paper Notes

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Downstream/Upstream Boat Speed – Important Formulas

Numerical Aptitude Notes for APSC Prelim CSAT Paper, SSC and Competitive Exams

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Still Water: If the speed of the water is zero, i.e. water is stationary, then it is called still water.

Stream: The moving water in the river is known as a stream.

  • Upstream: If a boat or a swimmer moves in the opposite direction of the stream, then it is called upstream, i.e. direction against the stream.
  • Downstream: If a boar or a swimmer moves in the same direction of the stream, then it is called downstream, i.e. direction along the stream

 

If the speed of a boat in still water is u km/hr and Speed of the stream is v km/hr,

  • Speed downstream = (u + v) km/hr.
  • Speed upstream = (u – v) km/hr.


If the speed downstream is a km/hr and the speed upstream is b km/hr, then

  • Boat Speed in still water = ½ (Downstream Speed + Upstream Speed) = 1/2 x (a + b) km/hr.
  • Rate of stream = ½ (Downstream Speed – Upstream Speed) = 1/2 x (a – b) km/hr.
  • Average Speed of Boat = {(Upstream Speed × Downstream Speed) / Boat’s Speed in Still Water}

 

 

If it takes “T” hours for a boat to reach a point in still water and comes back to the same point then,
the distance between the two points = {(u2-v2) × T} / 2u,
where “u” is the speed of the boat in still water and “v” is the speed of the stream.


If it takes “T” hours more to go to a point in upstream than downstream for the same distance, then
the distance = {(u2-v2) × t} / 2v,
where “u” is the speed of the boat in still water and “v” is the speed of the stream.

If a boat travels a distance downstream in “t1” hours and returns the same distance upstream in “t2” hours, then,
the speed of the man in still water = [v × {(t2+t1) / (t2-t1)}] km/hr,
where “v” is the speed of the stream

Calendar, Odd days, Leap years – Important Formulas – APSC Prelim CSAT Paper Notes, Qs, Formulas

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Calendar, Odd days, Leap years – Important Formulas & Notes

Numerical Aptitude Notes for APSC Prelims CSAT Paper, SSC and Competitive Exams

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Odd Days

To find the day of the week on a given date, use the concept of ‘odd days’.

Number of days more than the complete weeks are called odd days in a given period.

 

Finding No. of odd Days

1 ordinary year = 365 days = (52 weeks + 1 day.) i.e. 1 ordinary year has 1 odd day.

1 leap year = 366 days = (52 weeks + 2 days) i.e. 1 leap year has 2 odd days.

1 Century = 100 years = 76 ordinary years + 24 leap years = (76 x 1 + 24 x 2) odd days = 124 odd days. = (17 weeks + days) 5 odd days = 5 odd days in 100 years

Odd days in 200 years = (5 x 2) %7 =3 odd days.

Number of odd days in 300 years = (5 x 3) %7 = 1 odd day.

Number of odd days in 400 years = (5 x 4 + 1) %7 = 0 odd day.

Similarly, 800 years, 1200 years, 1600 years, 2000 years  has 0 odd days.

Last day of a century cannot be Tuesday or Thursday or Saturday as odd days of any century can be one 0, 1, 3 and 5 days. 

For the calendars of two different years to be the same, the conditions to be satisfied are

  • Both years must be of the same type. i.e., both years must be ordinary years or both years must be leap years.
  • 1st January of both the years must be the same day of the week.

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