A is an infinitely thin, infinitely long collection of points extending in two opposite directions. While drawing lines in geometry, an arrow at each end is put up to show that it can extend infinitely.
A line can be named either using two points on the line (for example\[\overleftrightarrow{AB}\] ) or simply by a letter, usually lowercase (for example, line m).
A line segment has two end points. It contains these endpoints and all the points of the line between them. One can measure the length of a segment, but not of a line.
A segment is named by its two endpoints, for example \[\overline{AB}\].
A ray is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.
A ray is named using its endpoint first, and then any other point on the ray (for example,\[\overline{BA}\])
An angle is formed when two lines (or rays or line-segments) meet.
The sum of two complementary angles is equal to\[90{}^\circ \].
The sum of two supplementary angles is equal to\[180{}^\circ \].
A right angle measures\[90{}^\circ \].
Two adjacent angles have a common vertex and a common arm but no common interior. Linear pair of angles are adjacent and supplementary to each other.
When two lines l and m meet, we say they intersect; the meeting point is called the point of intersection.
(i) When two lines intersect (looking like the letter X), two pairs of opposite angles are formed. They are called vertically opposite angles. They are equal in measure.
(ii) A transversal is a line that intersects two or more lines at distinct points, it gives rise to several types of angles as shown in the figure.
(iii) In the figure, transversal p intersects the lines I and m.
Mensuration
Mensuration is the branch of mathematics which deals with the study of geometrical shapes, their area, volume and related parameters. Here, we will discuss the areas and perimeter of plane figures.
Some important mensuration formula are listed in the table given below.
Ratio and Proportion
Ratio
We often have to compare quantities in our daily life. They may be heights, weights, salaries, marks etc. While comparing salaries of two persons i.e., salaries 6000 per month and 9000 per month, they may be written as the ratio 6000:9000 or 2 : 3. In general, the ratio of two quantities a and b in the same units is the fraction \[\frac{a}{b}\] and we write it as a:b. In the ratio a: b, we call a as the first term or antecedent and b, the second term or consequent. For example, the ratio 5:9 represents \[\frac{5}{9}\] with antecedent = 5, consequent = 9.
Note: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. For example, 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
Proportion
The equality of two ratios is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion. Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, \[a:b::c:d\Leftrightarrow (b\times c)\,\,=\,\,(a\times d)\].
Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional: a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional: Mean proportional between a and b is ab.
Comparison of Ratios
We say that (a : b) > (c : d) \[\Leftrightarrow \]\[\,\frac{\mathbf{a}}{\mathbf{b}}\,\,\mathbf{>}\,\,\frac{\mathbf{c}}{\mathbf{d}}\].
Compounded Ratio: The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
Duplicate Ratios:
Duplicate ratio of (a : b) is\[\left( {{a}^{2}}:\text{ }{{b}^{2}} \right)\].
Sub-duplicate ratio of (a : b) is \[\left( \sqrt{a},\text{ }\sqrt{b} \right)\] .
Triplicate ratio of (a : b) is \[\left( {{a}^{3}}:\text{ }{{b}^{3}} \right)\].
Sub-triplicate ratio of (a : b) is \[\left( {{a}^{1/3}}:\text{ }{{b}^{1/3}} \right)\].
If\[\,\,\frac{a}{b}\,\,=\,\,\frac{c}{d},\,\,then\,\,\frac{a+b}{c-b}\,\,=\,\,\frac{c+d}{c-d}\] [componendo and dividend].
Example
If 4x + 3y : 6x + 5y = \[\frac{\mathbf{11}}{\mathbf{17}}\], then find x : y.
(a) 0 : 1 (b) 2 : 1
(c) 1 : 0 (d) 5 : 0
(e) None of these
Ans. (b)
Explanation: \[\frac{4x+3y}{6x+5y}=\frac{11}{17}\,\,\,\,\Rightarrow \,\,\,\,\,17(4x+3y)\,\,\,\,=\,\,\,11(6x+5y)\]
\[~\Rightarrow \,\,\,\,\,68x + 51y = 66x + 55y\,\,\,\Rightarrow \,\,\,68x - 66x = 55y - 51y\]
\[\Rightarrow \] 2x = 4y \[\Rightarrow \]\[\frac{x}{y}\,\,=\,\,\frac{4}{2}\,\,\,\Rightarrow \] x : y = 2 : 1.
If more...
Data Collection
Data collection is important as it helps us to collect, study and record information, make decisions about important issues as well as to pass Information on to others. It enables us to provide information regarding a specific topic or area of study.
The data that are collected need to be organised in a proper table, so that it becomes easy to understand and interpret.
Lets understand some important terms:
Frequency: The frequency of a particular data value is the number of times the data value occurs. For example, if four students have a score of 80 in mathematics, and then the score of 80 is said to have a frequency of 4. The frequency of a data value is often represented by f.
Variable: The measurable characteristics like age, height. Income etc. are called variables.
Mean: The mean or arithmetic mean is the average of the numbers. Average is a number that represents or shows the central tendency of a group of observations or data. It can be calculated by using the formula given below:
Mean =\[\frac{\sum{fixi}}{\sum{fi}}\]. (where f is frequency and x is variable).
Mode: It is another form of central tendency or representative value. The mode in a list of numbers refers to the list of numbers that occur most frequently.
For example, to find the mode of 9, 11, 3, 44, 17, 11, 17, 15, 15, 15, 27, 40, 8
Put the numbers in increasing order as shown below:
3, 8, 9, 15, 15, 15, 17, 17, 17, 17, 27, 40, 44
The number which occur most frequently is the mode. In the given data, the mode is 17 which occurs the most at 4 times.
The median: The Median is the ‘middle value’ in your list. It is also a form of representative value. When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order If a set of n observations is given, where n is odd, then
Median = \[{{\left( \frac{n+1}{2} \right)}^{th}}\]observation
When observations are arranged in an increasing order.
When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two.
If a set of n observations is given, where n is even, then
Median = \[\frac{{{\left( \frac{n}{2} \right)}^{th}}observation+{{\left( \frac{n}{2}+1 \right)}^{th}}observation}{2}\]
When observations are arranged in increasing order.
Range: The difference between the smallest and largest observation is called the range.
Example
The mean, median, mode and range for the following list of values are:
8, 9, 10, 10, 10, 11, 11, 11, more...
Analogy
Introduction
Analogy is a similarity between like features of two things on which comparison may be based in terms of reasoning, the meaning of analogy is logical similarity in two or more things. This similarity may be on the basis of properties, kinds, traits, shapes etc.
Analogy plays a significant role in problem solving such as decision making, perception, memory, creativity, emotion, explanation and communication, it lies behind basic tasks such as the identification of places, objects, etc.
Example
Money : Bank : : Book: Library
Explanation: 1st pair- Money: Bank (Thing and a place where it is kept).
2nd pair- Book: Library (Thing and a place where it is kept).
As ‘Money’ is deposited in ‘Bank’, similarly a ‘Book’ is deposited in a ‘Library’.
Clearly, both pairs show similar relationship in a logical way. Hence, both pairs are analogous.
Example
Nose : Smell : : Skin : Feel
Explanation: 1st pair - Nose: Smell (Body part and its utility relationship).
2nd pair – Skin : Feel (Body part and its utility relationship).
We smell (a type of sensation) with nose (a body part); similarly we feel (a type of Sensation something when it touches our skin (a body part),
Clearly, both pairs show similar relationship (body part and its utility relationship). Hence, both pairs exhibit analogy.
Types of Problems
Problems based on analogy may be of different types.
Problems Based on Synonymous Relationship
In such problems, the words given in one pair having same meaning and relationship has to be found in another pair.
Example:
Hard : Tough : : Wrong : Untrue
Explanation; 1st pair-Hard: Tough (Synonymous relationship).
2nd pair-Wrong: Untrue (Synonymous relationship).
Example:
Sad : Sorrow : : Happy : Joy
Explanation: 1st pair – Sad : Sorrow (Synonymous relationship).
2nd pair – Happy : Joy (Synonymous relationship).
Commonly Asked Questions
Select the pair which is related in the same way as the pair of words given in the question:
Baby : Infant : : _________ : ________
(a) Cub : Bear (b) Country : Continent
(c) Girl : Woman (d) Film : Movie
(e) None of these
Ans. (d)
Explanation: Option (d) is correct because “Baby” and ‘Infant’ are synonymous words. In the same manner ‘Film’ and ‘Movie’ are also the synonymous words. Rest of the options is incorrect because words in options (a), (b) and (c) have small- big relationship. Option (e) is useless because of the correctness of option (d).
Select the pair which is related in the same way as the pair of words given in the question.
Catch : Capture : : ________ : more...
Blood Relation
Blood Relation
Blood relation is a biological relation. Although husband and wife are not biologically related to each other but they are biological parents of their own children. Similarly, brother, sister, paternal grandfather, paternal grandmother maternal grandfather, maternal grandmother, grandson, grandmother, niece, cousin etc. are our blood relatives.
Blood relations are mainly classified into two categories;
Relations of Paternal Side
SeriesIntroduction
A series is a sequence of many elements following a particular pattern or rule. Such sequence is formed by putting the elements one after another from left to right. There are mainly three types of series - Letter series, Number series and Mixed series.
Letter Series
A letter series is a sequence of many elements made of English alphabets. Such sequence is formed by putting the letters one after another from left to right.
Type 1
This type of letter series is a sequence of Setters related to each other by some specific pattern.
Example
Which one of the following letters will come in place of the question mark (?)?
A E I M Q?
(a) T (b) U
(c) V (d) W
(e) None of these
Ans. (b)
Explanation: The letters in the series are moved four steps forward alternatively.
Replace the question mark (?) in the given letter series with the correct letter.
C E H J M O?
(a) Q (b) S
(c) R (d) T
(e) None of these
Ans. (c)
Explanation: The letters in this series are moved two and three steps forward alternately.
Commonly Asked Questions
Replace the question mark (?) in the given letter series with the correct letter.
U R P M K?
(a) G (b) E
(c) H (d) F
(e) None of these
Ans. (c)
Explanation: The letters in this series are moved three and two steps backward alternately.
Replace the question mark (?) in the given letter series with the correct letter.
A B F G K L?
(a) P (b) M
(c) R (d) O
(e) None of these
Ans. (a)
Explanation: The letters in this series are moved one and four steps forward alternately.
Type 2
Example
In this type, a group of English alphabets forms a series according to a certain rule.
WV, PO, IH, BA?
(a) ST (b) RS
(c) UT (d) UV
(e) None of these
Ans. (c) more...
Classification
Introduction
Classification is a process of grouping various objects on the basis of their common properties. In the Questions that are generally asked on classification we have to assort the items of a given group on the basis of certain common quality they possess and then spot the stranger out. For sample, if we have to separate out the odd one from the group of four animals- lion, tiger, bear and goat, then definitely that animal will be goat because goat is the only animal in the group which is a domestic animal. Rests of the animals (lion, tiger and bear) are wild animals.
Letter Based Classification
In these type, a letter or a group of letters is to be sort out from the others on the others on the basis of a specific pattern.
Example
Which of the following pair of letters is different from the other three?
(a) JM-KL (b) QP-RS
(c) EH-FG (d) MP-NO
(e) None of these
Ans. (b)
Explanation: In all other groups, the two letters on the right fit between the two letters on the left.
Which one of the following pairs of letters is different from the other three?
(a) AEC (b) PTR
(c) FJH (d) KPM
(e) None of these
Ans. (d)
Explanation: In all other groups, the first, third and second letters are alternate.
Commonly Asked Questions
Which one of the following pairs of letters is different from the other three?
(a) AYT-BZU (b) FNG-EMF
(c) RWO-QVN (d) HJD-GIC
(e) None of these
Ans. (a)
Explanation: In all other groups, the letters in the first part are one step forward than the corresponding letters in the second part.
Which one of the following pairs of letters is different from the other tree?
(a) DEVW (b) GHMN
(c) JKNO (d) CBED
(e) None of these
Ans. (d)
Explanation: The letters are arranged in increasing order, while in (d) they are not
Word Based Classification
In these type of questions/ a word is to be sorted out from the others on the basis of some properties or a particular pattern
Example
Choose the word which is not like the other words in the group.
(a) Paper (b) Pencil
(c) Rubber (d) Stationery
(e) None of these
Explanation: All others are the various stationery materials.
Choose the word which is not like the other words in the group.
(a) Brinjal (b) Potato
(c) Orange (d) Lady Finger
(e) None of these
Ans. (c)
Explanation: Orange is fruit while others are vegetables.
Commonly Asked Questions
Choose the word which is not like the other words in more...
Understanding the Concept of Directions
There are four main directions – East, West, North and South as shown below:
Note: On paper North is always on the top while South is always as the bottom.
There are four cardinal directions – North-East (N - E), North-West (N - W), South-East (S - E) and South-West (S - W) as shown below.
Important Points Regarding Directions
If our face is towards North, then after left turn our face will be towards West while after right turn it will be towards East.
If our face is towards South, then after left turn our face will be towards East and after right turn it will be towards West.
If our face is towards East, then after left turn our face will be towards North and after right turn it will be towards South.
If our face is towards West, then after left turn our face will be towards South and after right turn it will be towards North.
If our face is towards North-West, then after left turn our face will be towards South-West and after right turn it will be towards North-East.
If our face is towards South-West, then after left turn our face will be towards South-East and after right turn it will be towards North-West.
If our face is towards South-East, then after left turn our face will be towards North- East and after right turn it will be towards South-West.
If our face is towards North-East, then after left turn our face will be towards North- West and after right turn it will be towards South-East.
Important Points Regarding Shadow of a Person
At the time of sunrise if a man stands facing the East, his shadow will be towards West.
At the time of sunset the shadow of an object is always in the East.
If a man stands facing the North, at the time of sunrise his shadow will be towards his left and at the time of sunset it will be towards his right.
At 12:00 noon, the rays of the sun are vertically downward hence there will be no shadow.
Example
A cat runs 20 metre towards East and turns right, runs 10 metre and turns to right, runs 9 metre and again turns to left, runs 5 metre and then turns to left, runs 12 metre and finally turns to left and runs more...
Time, Clock and Calendar
The face or dial of a watch is a circle whose circumference is divided into 60 equal parts minute spaces.
A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand.
In 60 minutes, the minute hand gains 55 minutes on the hour hand.
In every hour, both the hands coincide once.
The hands are in the same straight line when they are coincident or opposite to each other.
When the two hands are at right angles, they are 15 minute spaces apart.
When the hand’s are in opposite directions, they are 30 minute spaces apart
Angle traced by hour hand in 12 hours \[=360{}^\circ \].
Angle traced by hour hand in one hour \[={{30}^{{}^\circ }}\].
Angle traced by hour hand in one minute \[=0.5{}^\circ \].
Angle traced by minute hand in 60 minutes \[=360{}^\circ \].
Too Fast and Too Slow: If a watch or a clock indicates 8.15, when the correct time 8 is said to be 15 minutes too fast.
On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.
Example
A clock shows the time as 3:30 pm if the minute hand gains 2 minutes every hour, how many minutes will the clock gain by 6 am?
(a) 20 minutes (b) 29 minutes
(c) 30 minutes (d) 35 minutes
(e) None of these
Ans. (b)
Explanation: Hours between 3:30 pm and 6 am are = \[14\frac{1}{2}\]
So, numbers of minutes gained will be = \[14\frac{1}{2}\times 2=29\]minutes.
Two watches, one of which gained at the rate of 1 minute and other lost at the rate of 1 minute daily, were set correctly at noon on the 1st January 1978. When did the watches indicate the same time?
(a) Dec 30, 1978 noon (b) Dec 25, 1978 noon
(c) Dec 27, 1978 noon (d) Dec 26, 1978 noon
(e) None of these
Ans. (c)
Explanation: The first watch gains on the second watch 1 + 1 = 2 minutes in a day.
The watch will indicate the same time when the one has gained 12 hours on the other
As 2 minutes is gained in one day.
So 12 hours is gained \[= 1 / 2\times 12~60 = 360 days\]Counting 360 days from 1st Jan more...
A line can be named either using two points on the line (for example\[\overleftrightarrow{AB}\] ) or simply by a letter, usually lowercase (for example, line m).
A segment is named by its two endpoints, for example \[\overline{AB}\].
A ray is named using its endpoint first, and then any other point on the ray (for example,\[\overline{BA}\])
(iii) In the figure, transversal p intersects the lines I and m.
Type 2
Example
In this type, a group of English alphabets forms a series according to a certain rule.
WV, PO, IH, BA?
(a) ST (b) RS
(c) UT (d) UV
(e) None of these
Ans. (c)
