Series Formation

In mathematical terminology, the terms “sequence” and “series” have slightly different meaning. In simple words, a sequence is a comma-separated list of numbers or objects following a special order, whereas, a series is the sum of terms of a sequence. However, here the term “series formation” implies formation of both sequence and series.

In the questions on this section, usually a sequence of letters, numbers or symbols following some particular predefined rules will be given in the question. By understanding and applying these predefined rules, you are expected to find out the succeeding terms or a missing term in the given sequence.

Sequence can be categorized as:

  • Number Sequence
  • Letter Sequence
  • Symbol Sequence

1       Number Sequence

Some of the important types of number sequence are:

Arithmetic sequence

Here, successive numbers are obtained by adding (or subtracting) a fixed number to the previous number. The fixed number is called the common difference of the arithmetic sequence.

Examples:

  • 2, 4, 6, 8,….
  • 1, 6, 11, 16,….
  • 10, 8, 6, 4,…

Geometric Sequence

A geometric sequence is the one in which each successive number is obtained by multiplying (or dividing) a fixed number (can be fraction also) by the previous number. That fixed number is called the common ratio of the geometric sequence.

Examples:

  • 3, 6, 12, 24,…
  • 2048, 1024, 512, 256, 128, 64,…
  • 10, -40, 160, -640,…
  • 1024, 640, 400, 250,…

Trick: A simple trick to identify a geometric sequence is by dividing each pair of successive numbers in that sequence and checking whether we are getting a common ratio.

Prime Sequence

As we know, prime numbers are numbers which can be exactly divided only by one and that number. Prime sequence is the ordered set of prime numbers starting with any of the prime numbers.

Examples:

  • 7, 11, 13, 17, 19,…
  • 29, 31, 37, 41, 43,…

Power Sequence

In this sequence, every term will be the nth power of some consecutive numbers. It can be square, cube or any higher powers of a particular sequence of numbers.

Examples:

  • 1, 4, 9, 16, 25… – Square sequence
  • 1, 8, 27, 64, 125, … – Cube sequence
  • 1, 16, 81, 256… – 4th power sequence.

Reversal Sequence

In this sequence, reversing each term will give us a sequence, which can be any of the above-mentioned sequence.

Example:

  • 31, 51, 71, 91, 12…
  • 61, 52, 63, 94, 46, 18,..

Two-tier sequence

In a two-tier sequence, the difference of the successive terms can form any of the above-mentioned sequence like arithmetic sequence, prime sequence, power sequence etc.

Twin sequence

Twin sequence are the one in which two sequence are packed into one.

In a twin sequence any of the above-mentioned sequence like arithmetic sequence, prime sequence, etc. can occur.

Examples:

  • 1, 0, 3, 2, 5, 4, 7,…
  • 3, 1, 6, 3, 12, 5, 24,…

2      Letter Sequence

In this type of sequence, a number of letters are arranged in a sequence and we are supposed to find the next letter or letters in the sequence.

The easiest way to solve these questions is by converting the letters into their numerical equivalent.  That is, A = 1, B = 2, C = 3, … , X = 24, Y = 25, Z = 26.

After converting to numerical equivalent, the sequence becomes same as a number sequence. Finding the next number in a ‘Number Sequence’ is already discussed above.

Note that while finding the next number, if there occur a number, say x, greater than 26, subtract 26 from x to get the required number. In addition, if there occur a negative number, add 26 to it to get the required number.

Trick: For converting an alphabet to the corresponding number or vice-versa, it is convenient to remember one or two equivalents. It saves lot of time.

Say, remember that J ≡ 10 and T ≡ 20.

Now any letter can easily be compared to these two letters.

For example, to find the numerical equivalent of V:

V is the 2nd number from T.

Therefore, V is equivalent to 20 + 2 = 22.

3    Symbol Sequence

In a symbol sequence, a pattern is given for three or four figures; we have to decide which will be the next figure to complete the given sequence.

Tips and tricks for identifying the sequence quickly

  • If the sequence is increasing or decreasing with a small increase there must be an addition or subtraction involved, possibly an arithmetic sequence.

Example: 1, 3, 5, 7,…

  • If the sequence is increasing or decreasing with a large increase there must be a multiplication or division involved, possibly a geometric sequence.

Example: 4, 8, 16, 32,…

  • If the sequence is increasing or decreasing with a very large increase there must be square or cubes involved, possibly a power sequence.

Example: 36, 49, 64, 81,…

  • If the sequence is not linearly increasing and a pattern is not formed after solving for three terms then there is likely a combination of sequence, That is, a twin sequence.

Example: 1, 0, 3, 2, 5, 4, 7…

  • If a pattern is not formed after solving for 3 terms, then the pattern may involve an operation of first and second terms to give the third term. Even it may involve grouping of terms.

Example: 2, 3, 5, 6, 3, 4, 7, 12, 4, 5, 9…

Take groups of four. That is, (2,3,5,6), (3,4,7,12), (4,5,9,…)

  • If second term cannot be formed from first term or third term cannot be formed from first and second terms, then likely an operation is done on another sequence to get this sequence.

Example: 1, 4, 9, 16,… is a square of 1, 2, 3, 4,…

  • For symbol sequence, by observation, eliminate improbable options to save time.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Moving Locomotive Problem

A moving locomotive problem normally poses the challenge of finding out the time taken, the speed, or the distance covered while a particular object pass another. The moving locomotive can be a train, a car, a flight etc. and the objects can be a truck, a man, a platform, a pole, a bridge, a tunnel, a stationary train, a moving train etc.

 A moving locomotive is said to have crossed or passed an object (stationary or moving) only when the total length of the moving locomotive crosses the total length of the object.

For instance, for a train to cross an object,

Distance covered by the train = length of train + length of object.

Important Facts, Tips and Shortcuts for Calculation

  • km/hr. to m/s conversion:
    • x km/hr = (x × ) m/s
  • m/s to km/hr. conversion:
    • x m/s = (x ×  ) km/hr
  • Speed =
  • Suppose two objects are moving in the same direction at u m/s and v m/s, where
    u ˃ v, then their relative speed = (u – v) m/s.
  • Suppose two objects are moving in opposite directions at u m/s and v m/s, then their relative speed = (u + v) m/s.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Clock Problem

As you know, a clock is a device that measures and shows time. The clock consists of two arms called the minute hand and hour hand. (In clock problems, generally the existence of the second hand is neglected.) The dial of a clock is a circle (360O) whose circumference is divided into 12 parts, called hour spaces. Each hour space is divided into 5 parts, called minute spaces. Thus, the whole circumference of the clock is divided into 12 × 5 = 60 minute spaces.

Important facts and shortcuts for quick calculation

  • In 1 hour, hour hand will cover = 30o.
  • In 1 minute, minute hand will cover = 6o.
  • For the minute hand and hour hand to coincide, the angle between them should be 0o.
  • During every hour, minute hand and hour hand coincide once.
  • Both the hands coincide after every 65
  • For the minute hand and hour hand to lie opposite to each other, the angle between them should be 180o.
  • In 60 minutes, the minute hand gains 55 minutes on the hour hand.
  • In minutes, the minute hand gains 1 minute over the hour hand.
  • To convert minutes into hours, divide the corresponding minutes by 60.
  • To convert seconds into hours, divide the corresponding seconds by 3600.

Important formulas

  • Angle between H hours and M minutes = 1/2×|60H – 11M|

where, H is the time in hours, M is the time in minutes.

Note: |x| means absolute value or modulus value of x. It simply takes the positive value of the given number.

  • To find the mirror image of any given time (except any time between 11 and 1), subtract the given value from 11:60.
  • To find the mirror image of any time between 11 and 1, subtract the given value from 23:60.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Calendar Problem

  1. Odd Days:

In any year, there are 52 complete weeks.

In a given period, the number of days more than the complete weeks are called odd days.

  1. Leap Year:

For a year to be a leap year,

  1. It should be divisible by 4.
  2. If it is divisible by 100, it should also be divisible by 400.

Note: A leap year has 366 days.

Examples:

  • The years 884, 1564, 1600, 2000, 2012 are leap years.
  • The years 431, 777, 1722, 2001, 2007, 2200 are not leap years.
  1. Ordinary Year:

The year, which is not a leap year, is called an ordinary year or a non-leap year.

An ordinary year has 365 days.

Examples:

  • The years 573, 998, 2009 are ordinary years.
  • The years 1700, 1800, 1900, 2100 are ordinary years.
  1. Counting of Odd Days:

First thing to remember is that the, 1st January of the year AD 1 was a Monday and therefore, we must set Sunday as the 0th day, Monday as the 1st day,…, Saturday as the 6th day.

Day of the week related to Odd days:

No. of odd days: 0 1 2 3 4 5 6
Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

Month of a year related to Odd days:

Month Jan Feb Mar April May June July Aug Sep Oct Nov Dec
Days 31 28/29 31 30 31 30 31 31 30 31 30 31
Odd days 3 0/1 3 2 3 2 3 3 2 3 2 3

That is, 3 odd days for every month with 31 days, 2 odd days for every month with 30 days and 0/1 odd days for February.

Therefore, 1 ordinary year has 1 odd day.

Therefore, 1 leap year has 2 odd days.

(Why not 25 leap years? Because the 100th year is a non-leap year.)

 

Tricks and shortcuts for quick calculation

  • For making calculations quicker, the number of days/odd days in a year is given by
Months Ordinary year Leap year
Days Odd days Days Odd days
January 1 to March 31

April 1 to June 30

July 1 to September 30

October 1 to December 31

90

91

92

92

6

0

1

1

91

91

92

92

0

0

1

1

For example

To get the number of odd days of November 17th of a particular year:

Odd days from January 1 to September  odd days.

Odd days from October 1 to October 31 = 3 odd days.

Odd days from November 1 to November  odd days.

Thus, total odd days of November 17th  odd days.

  • Number of odd days of 1900 years = 1 odd day. Memorizing this is useful, as in most of the questions the year will be some year after 1900.
  • The last day of a century must be Sunday, Monday, Wednesday or Friday. (The reason is given in Illustration 4.)
  • An ordinary year always begins and ends on the same day of the week.
  • For an ordinary year, February has exactly 4 weeks, so February and March always consecutively start on the same day of the week.
  • Any date in March is the same day of the week as the corresponding date in November of that year. Another pair showing similar corresponding feature is April-July.

Conditions for calendars of two different years to be same

  1. Both the years must be of the same type. That is, either both must be non-leap years or both must be leap years.
  2. The net number of odd days for the period between these two years should be zero. That is, 1st January of both the years must fall on the same day of the week.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Logical Puzzles

Logical puzzles are one of the most interesting set of questions in the CSIR-NET exams. Just like the detective Sherlock Holmes, who solves puzzling questions pertaining to a crime by deducing answers from observations, these questions too demand deducing the correct option from the observations made.

Step 1: Write down all the observations from the given question.

Step 2: Analyse the observations one by one.

Step 3: Eliminate the impossible options for each observation.

Step 4: Continue this process until you weed out three of the options so that you are left with the obvious correct option.

Note that while analysing, there may be some irrelevant observations with respect to the given question. Avoid them to save time.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Observational Ability

Analytical and observational type questions are those that can only be worked out with your observational and analytical skills. For the same reason, there is no particular theory for this section. All we can do is, get a command over these tricky questions through practice. It should be noted that most of the questions in this section can be done in less than a minute and practicing more questions will help you acquire that speed.

In some questions, drawing Venn-diagrams can help you visualize the given condition easily. First, let’s discuss how to draw a Venn-diagram.

To draw a Venn diagram:

Step 1: Identify the elements in the question.

Step 2: Identify the categories you have to put these items into.

Step 3: List out the items that fall into both categories, each category, and neither.

Step 4: Draw universal set, represent it as a rectangle. Mark number of elements in the universal set.

Step 5: Draw circles corresponding to each category you identified inside the universal set. Label each circle according to the category you want it to represent. Enter number of items common to each category in the overlapping portion of circles with the respective labels. Insert other details in other parts of the circle. Also mark number of items not belonging to any (= number of objects in universal set – sum of items in categories) in the space outside of these circles and inside universal set (if required).

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Data Interpretation

Data interpretation is the process through which inferences are drawn about the data available for analysis. In other words, the process of drawing inferences and conclusions through the interpretation of data is what data interpretation is all about.

In the case of questions related to data interpretation, data will be given in the form of any of the below mentioned tools followed by questions pertaining to the same. These questions are intended to test the ability of the student to interpret the information presented and to select the appropriate data for answering a question.

Tips to solve data interpretation questions

  • Read the data very carefully. Even the minutest word must not be overlooked since many a times even a single word or phrase could become critical.
  • The easiest way to solve a data analysis question is to weed out the incorrect options from the given options. While doing so, stop the weeding out process at once you get the correct option. Do not try to check the next option if you get one correct answer. Always remember that time is very important.
  • If there are more than one graphs or charts or tables, understand the relationship between them clearly before you proceed to solve the questions asked.
  • Answer only the questions asked. Do not answer or calculate things which have not been asked for.
  • The given data may be insufficient for interpreting some answer options. Avoid those options at once.

For the data interpretation questions, data will be represented in any of the data presentation tools like tables, pie-charts, bar graphs etc. Therefore, we can have an overview of the different types of questions using these data presentation tools.

1      Table

A table is a display of data arranged into rows and columns. Almost any quantitative information can be organized into a table. A table consists of horizontal rows and vertical columns. The heading for each row and column helps the reader understand the data and the units used for the same.

2      Bar chart

A bar chart or bar graph is a chart with rectangular bars with lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally. Bar charts are used for plotting discrete (or discontinuous) data; that is data which has discrete value and is not continuous. Bar graphs consist of an axis and a series of labelled horizontal or vertical bars that show different values for each bar. The numbers along a side of the bar graph are called the scale. The important point to note about bar graphs is their bar length or height—the greater their length or height, the greater their value.

3      Line graph

A line graph is a useful data presentation tool for showing a long series of data. Line graphs are also useful for comparing several different series of data in the same graph. Line graphs display data in two dimensions. We call the dimensions the x-axis and the y-axis.

By convention the dependent variable or y variable is on the vertical axis and the independent or x variable is on the horizontal axis. When reading a line graph, you will notice that rise and falls in the line show how one variable is affected by the other.

4      Histograms

A histogram is a graphical display of data using bars of different heights. It shows a result of continuous data, such as: weight, height etc. It is used to summarize discrete or continuous data that are measured on an interval scale. A histogram divides up the range of possible values in a data set into classes or groups.

5      Pie charts

A pie chart is a circular chart divided into sectors. In a pie chart, the arc length of each sector is proportional to the quantity it represents. When a pie chart is formed from a data, it breaks up a whole into its parts. The share of each part in a pie chart is proportionate to its share of the whole data.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Geometry

Geometry is the branch of mathematics that deals with the properties, measurement and relations of points, lines, angles, surfaces and solids.

Geometry is such a vast topic and therefore it is almost an impossible endeavour to gather or compile all the concepts pertaining to this in a single book. Therefore, we will be discussing only the relevant concepts with respect to the type of questions that are posed in the CSIR-NET examination. With a proper understanding of these concepts, you will be able to attempt any geometrical questions in the CSIR-NET examination.

Geometry can be divided into plane geometry and solid geometry.

Plane Geometry

A plane is any two-dimensional surface such as circles and squares. Plane geometry is all about figures in a plane.

Let us discuss some important aspects in plane geometry.

  1. Line

Types of lines

  • Intersecting lines

Two or more lines that meet at a point are called intersecting lines. The common point is called the point of intersection.

  • Concurrent Lines

Three or more lines in a plane are concurrent if they intersect at a single point. 

  • Perpendicular Lines

A line is said to be perpendicular to another line if the two lines intersect at a right angle (90O angle).

Parallel Lines
intersect or touch at any point are called parallel lines.

Parallel lines and the transversal

A transversal is a line that intersects two or more other lines. When it intersects parallel lines, the following angles are congruent (equal).

Corresponding angles – Angles that are in the same position on each line.

Alternate interior angles – Angles on the opposite sides of the transversal and on the interior of the parallel lines.

Alternate exterior angles – Angles on opposite sides of the transversal and on the exterior of the parallel lines.

Vertically opposite angles – The pairs of opposite angles formed between each line and the transversal.

Note: Angles on one side of a straight line will always add to 180 degrees [Linear Pair].

  1. Triangle

A triangle is a closed figure with three sides and three vertices.

Types of triangles

  • Right triangles

A triangle with a right angle (That is, 90o) is called a right triangle.

Pythagoras Theorem

In a right triangle,

(Hypotenuse)2 = (Base)2 + (Height)2

The numbers that satisfy this relation are called Pythagorean triplets.

Example: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61),

(12, 35, 37), (15, 20, 25) etc.

Note: If you see a triangle with sides exactly as any of these Pythagorean triplets, you can be sure that the triangle is a right triangle.

  • Isosceles triangles

Triangles with two equal sides are called isosceles triangles.

The angles opposite to the equal sides are also equal.

  • Equilateral Triangles

Triangles with three equal sides are called equilateral triangles.

For equilateral triangles, each of the three angles are also equal to 60o.

Important facts about triangles

  • Sum of the interior angles of a triangle is 180o (Angle Sum Property)
  • The sum of any two sides of a triangle is greater than the third side.
  • The diagonal of a square divides the square into two isosceles right triangles.
  • Median – A median of a triangle is a line segment from one vertex to the mid-point on the opposite side of the triangle.

Centroid – A centroid of a triangle is the point where the three medians of the triangle meet.

    • Orthocenter – In a triangle, the orthocenter is the point of intersection of the altitudes.
      • Incircle – The incircle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle’s incenter.
      • Circumcircleof a triangle is a circle, which passes through all the vertices of the triangle. The center of this circle is called the circumcenter and its radius is called the circumradius.
  • Circle

Circle is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle.

  • Chord

Chord is the line that joins two points on a circle. Diameter is the largest chord.

Arc

An arc of a circle is two points on the circle and the continuous part of the circle between the two points.

If the arc measures xo, the arc length is (2πr).

  • Tangent

A tangent to a circle is a line that touches the circle at exactly one point, never entering the circle’s interior.

Important circle theorems

  1. A tangent and radius meet at right angles.
  2. Two tangents from a common exterior point to a circle are equal in length.
  3. The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference of the circle.
  4. Angles subtended by an arc in the same segment of a circle are equal.
  5. The angle subtended in a semicircle is a right angle (90o).
  • Parallelogram

A parallelogram is a quadrilateral (closed figure with four sides) with two pairs of parallel sides.

Important properties of Parallelogram

  • Opposite sides are parallel and equal
  • Opposite angles are equal.
  • The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
  • Diagonals of a parallelogram bisect each other. That is, they are divided in their intersection point into two.
  • Parallelogram law – The sum of squares of diagonals is equal to the sum of squares of four sides.
  1. Rhombus

A parallelogram with all sides equal is called a rhombus.

Important properties of Rhombus

  • Opposite sides are parallel and all sides are equal.
  • Opposite angles are equal.
  • The diagonals of a rhombus are perpendicular to each other.
  • Diagonals of a rhombus bisect each other.
  • As Rhombus is a special parallelogram, it also follows the parallelogram law.

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Monetary Problems

Monetary problems are those related to money. This section includes Simple Interest, Compound Interest, profit and loss etc.

1      Profit and Loss:

Cost Price (C.P.): The price at which an article is purchased is called Cost Price (C.P.)

Sale Price (S.P.): The price at which an article is sold is called Sale Price (S.P.) or selling price.

Marked Price (M.P.): The price put on the article by the seller or manufacturer is called Marked Price (M.P). Normally marked price is the price we see as the M.R.P or maximum retail price.

Profit or Gain: If Sale Price is greater than Cost Price then the seller is said to have a profit or gain.

Loss: If Cost Price is greater than Sale Price then the seller is said to have incurred a loss.

Important Formulas:

Profit or Gain = S.P. – C.P.

Loss = C.P. – S.P.

Profit % or Gain% = (Profit/CP)×100

Loss% = (Loss/CP)×100

Alternative Logical Formulas:

In the case of profit, SP = (100% + Profit%) of CP

In the case of loss,    SP = (100% – Loss%) of CP

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

Average

1      Arithmetic Mean:

In simple language, an average is the sum of a list of numbers divided by the number of elements in the list.

Example: The average of four observations {50, 55, 70, 85} = (50 + 55 + 70 + 85)/4 = 65.

Although the arithmetic mean is not the only “mean”, it is by far the most commonly used. Therefore, if the term “mean” is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.

Important facts about Arithmetic mean:

  • If each of the given element is increased or decreased by x, then their average is also respectively increased or decreased by x.

Example: The average of 45, 55, 80 is 60. If 20 is added to each of the elements the new set is 65, 75, 100 whose average is 80. Clearly 80 is 20 more than the initial average 60.

  • If each of the given elements is multiplied by x, then their average gets multiplied by x.
  • If each of the given elements is divided by a nonzero x, then their average gets divided by x.
  • If x1and x2 are the arithmetic means of two samples of sizes n1and n2 respectively, then the arithmetic mean z of the distribution combining the two samples can be calculated as

z = (x1n+ x2n2)/2

  • The value of the arithmetic mean of two elements is always in between the value of the two elements.

2      Geometric Mean:

The geometric mean is defined as the nth root of the product of n numbers.

That is, to find the geometric mean of a set of ‘n’ numbers, multiply the ‘n’ numbers and then take the nth root of the product.

Example 1: Geometric mean of two numbers, say 3 and 27 is the square root of their product; i.e.,9.

Example 2:  Geometric mean of three numbers, say 4, 6 and 9 is the cube root of their product; i.e.,6.

Note: Geometric mean is always less than or equal to the Arithmetic mean.

3         Median

Median is the middle value that separates the higher half from the lower half of the set of observations.

To find the median, we arrange the observations in the increasing (or descending) order. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

Example 1: To find the median of the set of observations {10, 15, 12, 3, 55}, arrange the set of observations in the increasing order as {3, 10, 12, 15, 55}.

Since there are 5 observations (odd), the median is the 3rd term, that is 12.

Example 2: To find the median of the set of observations {5, 20, 10, 35}, arrange the set of observations in the increasing order as {5, 10, 20, 35}.

Since there are 4 observations (even), the median is the average of 2nd and 3rd terms.

That is, Median = (10+20)/2  = 15.

4      Mode:

The mode is the most frequently appearing value in the set of observations.

Example: The mode of the set of observations {1, 5, 4, 5, 7, 2, 2, 5}

Here 5 is the most frequently appearing value.

Hence, 5 is the mode.

5      Standard Deviation:

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Variance for N observations is defined as:

Variance = (xi – Mean)2

Standard Deviation = root(variance) 

Example: Consider the set of observations {1, 3, 5, 7}

Here the Mean = 4.

Variance = ((1– 4)2 + (3 – 4)2 + (5 – 4)2 + (7 – 4)2)/4  = 5.

Standard Deviation = root(5)

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”