Introduction
Analogy' means 'Equivalence'. In questions based on analogy, a particular relationship is given and another similar relationship has to be identified from the alternatives provided. Analogy tests are therefore meant to test a candidate's overall vocabulary, power of reasoning and abiliy to think concisely and accurately. Below are given some common relationships which will help you to know analogies better. 
Introduction
Sometimes an 'unseen passage' may be set and students may be asked to answer certain questions based on the passage. They can write good answers only
(i) When they fully grasp the meaning of the passage,
(ii) When they are able to express the meaning of the passage in their own simple, clear and direct English, and
(iii) When they clearly understand the questions asked.
Comprehension means 'Understanding' whatever you read and answering Question related to it. Answering question to a given passage depends actually on the following abilities of a student.
How good you are in understanding the meaning of the entire passage; in finding the answer in the passage; command you have in English Language.
Points to Keep in Mind While Doing Comprehension
1. Read the paragraph carefully and underline important points.
2. Read the questions one by one and try to find the answers in the paragraph.
3. Write answers in your own words.
4. Read the passage carefully two or three times, till your understand clearly (1) its subject and (2) what is said about the subject. Ask yourself, 'What is the main theme of the passage I am reading? What does the author say about that subject?
5. Read carefully the questions carefully, one by one, and try to understand them.
6. Now identify the part of the passage the question refers.
7. Do not add unnecessary details.
8. Finally write answers to the questions in your own words. Do not copy the language of the passage. The answer should be brief and to the point.
9. Revise your answers and examine them carefully
10. Be precise and to the point.
Passage
Read the following passage and answer the questions: The men who were governing Athens summoned Socrates to appear before them and to stand trial. His friends and pupils begged him to escape or to hide until the storm had blown over. But Socrates was no coward. He knew that he had done nothing wrong and that he had only taught what he believed to be just, true and honorable, and so he went to the court, an undersized, ugly old man, dust and travel-stained, but with a noble heart beating under the shabby garments which everyone knew so well. He made a powerful, dignified speech, answering every question, explaining that, although the Athenians knew it not, he was really their friend. He told them that they would gain nothing by taking away the last few years of his life, but that he was willing to die many deaths for what he believed to be right. The judges listened to him and condemned him to death. The old man made no complaint. He leaned on his staff, looking round the crowded court, "No evil can happen to a good man," he said, "either in life or more... 
Line of symmetry
Line of symmetry is a line that divides a figure into two identical parts, each of which is the mirror image of the other. In the above figures the imaginary line is known as the line of symmetry, so the line of symmetry is the imaginary line along which we can fold the figures so that both the halves coin cide each other.
Note:
Which of the following figures does not have any line of symmetry?
(a) 
(b) 
(c) 
(d) 
How many line(s) of symmetry does a regular octagon can have?
Solution:
A regular octagon have 8 lines of symmetry, which is shown below
Introduction
In our day to life we come across so many things around us which seems to be divided into two equal halves and both the halves look like in exact correspondence or position along an imaginary line. Take a look at the following figures.
If we fold these figures along an imaginary line, we will find that both the halves are mirror images of each other. The very common term used in our day to day life for these type of figures is "symmetry". So, symmetry occurs when one shape becomes exactly the same or both the figures overlap each other. Such types of figures are called symmetrical figures. 
Triangles
A triangle has three sides and three vertices. The sum of the three angles of a triangle is equal to 180° and one angle of a triangle \[={{180}^{o}}-\] (Sum of two angles).
In ABC, \[\angle A+\angle B+\text{ }\angle C={{180}^{o}}\]
In APQR, \[\angle P+\angle Q+\angle R={{180}^{o}}\]
Find the measurement of A if B and C are given when A, B and C are the angles of the triangle.
(a) \[~A={{180}^{o}}-(B+C)\]
(b) \[A={{180}^{o}}+(B-C)\]
(c) \[A={{180}^{o}}+C-B\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
The sum of angles of the triangle \[={{180}^{o}}.\]
Hence, the angle \[A={{180}^{o}}-(B+C).\]
Exterior
Angle of a Triangle The angle between the produced side and its adjacent side of triangle is called exterior angle. The exterior angle is equal to the sum of two opposite interior angles of that triangle. In the picture given below the angles \[\angle ABD\] and \[\angle ACE\] are the exterior angles of the triangle.
Thus,
\[\angle ACE=\angle BAC+\angle ABC\]
\[\angle ABD=\angle BAC+\angle ACB\]
\[\angle CAF=\angle ABC\text{ }+\angle ACB\]
Find the measurement of the angles x and y from the picture given below.
(a) \[x={{80}^{o}}\]and\[y={{10}^{o}}\]
(b) \[x={{110}^{o}}\]and\[~y={{120}^{o}}\]
(c) \[x={{70}^{o}}\]and\[y={{90}^{o}}\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
\[x\]and\[y\] are the exterior angle therefore,
\[x=\angle PRQ\text{ }+\angle PQR={{60}^{o}}+{{50}^{o}}={{110}^{o}}\]
\[y=\angle QPR+\angle PQR={{70}^{o}}+{{50}^{o}}={{120}^{o}}\]
Area of the Triangle
The area of a triangle is the area bounded by the sides of the triangle.
Heron's Formula To find the area of a triangle by Heron's Formula, First find the semi perimeter of a triangle by the following formula: \[s=\frac{a+b+c}{2}\] (Here "s" stands for half of perimeter or semi perimeter and a, b and c refers to 3 sides of the triangle) \[Area=\sqrt{s(s-a)(s-b)(s-c)}\]by heron's formulae.
If the sides of a triangle are 18 cm, 24 cm and 30 cm then find the area of the triangle.
(a) \[216c{{m}^{2}}\]
(b) \[210c{{m}^{2}}\]
(c) \[21c{{m}^{2}}\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
Let sides A = 18 cm, B = 24 cm, C = 30 cm \[s=\frac{a+b+c}{2}=\frac{18+24+30}{2}=\frac{72}{2}=36\] \[Area=\sqrt{s(s-a)(s-b)(s-c)}\]by heron's formulae \[=\sqrt{36\left( 36-18 \right)\left( 36-24 \right)\left( 36-30 \right)}\] \[=\sqrt{36\times 18\times 12\times 6}\] \[=\sqrt{{{6}^{2}}\times 9\times 2\times 6\times 2\times 6}\] \[=\sqrt{{{6}^{2}}\times {{3}^{2}}\times {{2}^{2}}\times {{6}^{2}}}\] \[=\sqrt{{{(6\times 3\times 2\times 6)}^{2}}}=6\times 3\times 2\times 6=216\,c{{m}^{2}}\]
Area of the Right Angled
Triangle The area of a right angle triangle is the half of the Product of height and base. more... 
Operations on Algebraic Expressions
When constant and variables are linked with any of the following fundamental arithmetic operations, addition, subtraction, multiplication and division. The solution of the expression is obtained by simplification of the expression.
Simplify,\[\text{2}{{\text{X}}^{\text{2}}}\text{-5}{{\text{X}}^{\text{2}}}\text{+6}\]?
(a) \[\text{2}{{\text{X}}^{\text{2}}}-\text{5}{{\text{X}}^{\text{2}}}\]
(b) \[\text{5}{{\text{X}}^{\text{2}}}\text{+6}\]
(c) \[-3{{x}^{2}}+6\]
(d) All of these
(e) None of these
Answer: (c)
Explanation
\[2{{X}^{2}}-5{{X}^{2}}+6=-3{{X}^{2}}+6.\]
Addition of Algebraic Expression
Addition is possible even if terms are like. The addition of two unlike terms is possible and their addition is in the same form. Addition of \[2x+3x=5x\] but the addition of \[2x+3y=2x+3y.\]
Add the following polynomials,\[~{{\text{X}}^{\text{3}}}-\text{3}{{\text{X}}^{\text{2}}}-\text{6X+10}\]and \[\text{4}{{\text{X}}^{\text{3}}}\text{+10}{{\text{X}}^{\text{2}}}\text{+15X}-20\]?
(a) \[\text{5}{{\text{X}}^{\text{3}}}\text{+7}{{\text{X}}^{\text{2}}}\text{+9X}-10\]
(b) \[\text{5}{{\text{X}}^{\text{2}}}\text{+6+45}\]
(c) \[\text{5}{{\text{X}}^{\text{2}}}\text{+3}{{\text{X}}^{2}}\text{+6}\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
\[\begin{align} & {{\text{X}}^{\text{3}}}-\text{3}{{\text{X}}^{\text{2}}}-\text{6X+10} \\ & \frac{\text{4}{{\text{X}}^{\text{3}}}\text{+10}{{\text{X}}^{\text{2}}}\text{+15X}-20}{\text{5}{{\text{X}}^{\text{3}}}\text{+7}{{\text{X}}^{\text{2}}}\text{+9X}-10} \\ \end{align}\] Alternative Method \[({{X}^{3}}-3{{X}^{2}}-6X+10+\left( 4{{X}^{3}}+10{{X}^{2}}+15X-20 \right)\]\[=X3-3{{X}^{2}}-6X+10+4{{X}^{3}}+10{{X}^{2}}+15X-20\] \[={{X}^{3}}+4{{X}^{3}}-3{{X}^{2}}+10{{X}^{2}}-6X+15X+10-20\]\[=5{{X}^{3}}+7{{X}^{2}}+9X-10\]
Subtraction of Algebraic Expression
Subtraction of two like terms is same as the subtraction of 2 mangoes from 4mangoes. Number of mangoes are constant and the name, mangoes are like terms for both the numbers 2 and 4. The subtraction of 2 bananas from 4 mangoes is not possible.
Subtract: \[4{{X}^{2}}Y-3XY+5X\]from \[10{{X}^{2}}6XY+15X-25\] ?
(a) \[8{{x}^{3}}+2{{x}^{2}}+9x\]
(b) \[6{{x}^{2}}y-3xy+10x-25\]
(c) \[~5x-3{{X}^{2}}+6\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
\[=~(10{{X}^{2}}Y-6XY+15X-25)-(4{{X}^{2}}Y-3XY+5X)\]
\[=~10{{X}^{2}}Y-6XY+15X-25-4{{X}^{2}}Y+3XY-5X\] \[=10{{X}^{2}}4{{X}^{2}}Y-6XY+3XY+15X-5X-25\]
\[6{{X}^{2}}Y-3XY+10X-25\]
Multiplication of Algebraic Expression
The following steps are used to perform the multiplication of algebraic expression.
Ist : Write the sign of the resulting product according to the following rules,
\[\left( \frac{+x+=+,+x-=-}{-x-=+,-x+=-} \right)\]
IInd : Write the product of constant.
IIIrd : Write the product of variable according to the following rule, \[({{a}^{m}}\times {{a}^{n}}={{a}^{m+n}})\]
Multiply, \[({{a}^{2}}+ab+{{b}^{2}})({{a}^{2}}-ab-{{b}^{2}})\]
(a) \[{{a}^{4}}{{a}^{2}}{{b}^{2}}-2a{{b}^{3}}-{{b}^{4}}\]
(b) \[6{{a}^{2}}b-3ab+10a-25\]
(c) \[5a-3{{a}^{2}}+6~\]
(d) All of these
(e) None of these
Answer: (a)
Explanation
\[({{a}^{2}}+ab+{{b}^{2}})({{a}^{2}}-ab-{{b}^{2}})\] \[={{a}^{2}}({{a}^{2}}-ab-{{b}^{2}})+ab({{a}^{2}}-ab-{{b}^{2}})+{{b}^{2}}({{a}^{2}}-ab-{{b}^{2}})\] \[={{a}^{4}}-{{a}^{3}}b-{{a}^{2}}{{b}^{2}}+{{a}^{3}}b-{{a}^{2}}{{b}^{2}}-a{{b}^{3}}+{{a}^{2}}{{b}^{2}}-a{{b}^{3}}-{{b}^{4}}\] \[={{a}^{4}}-{{a}^{2}}{{b}^{2}}-a{{b}^{3}}-a{{b}^{3}}-{{b}^{4}}\] \[={{a}^{4}}-{{a}^{4}}{{b}^{2}}-2a{{b}^{3}}-{{b}^{4}}\]
Division of Algebraic Expression
The following steps are used to perform the division of the algebraic expression
Ist : First keep the polynomials which is to be divided in division form.
IInd: Divide first term of dividend by 1st term of divisor and write quotient.
IIIrd: Write the product of quotient x divisor, below the dividend and subtract it from dividend.
IV th: Repeat this process until the degree of remainder becomes less than divisor
Divide:\[2{{x}^{2}}+3x+1\]by \[(x+1)\]?
(a) \[~3x+2\]
(b) \[~2x+1\]
(c) \[5x-3\]
(d) All of these
(e) None of these
Answer: (b)
Explanation
\[x+1\overset{2x+1}{\overline{\left){\begin{align} & 2{{x}^{2}}+3x+1 \\ & \frac{\pm 2{{x}^{2}}\pm more... 
Identification of Terms of the Algebraic expression
Literals or Variables
Alphabetical symbols are used in mathematics called variables or literals, \[a,\text{ }b,\text{ }c,~~~d,\text{ }m,\text{ }n,\text{ }x,\text{ }y,\text{ }z\text{ }..........,\]etc. are some common letters used for variables.
Constant terms
The symbols which itself indicate a permanent value is called constant. All numbers are called constant. \[6,10,\frac{10}{11},15,-6,\sqrt{3}.......\]etc. are constant because, the value of the number does not change or remains unchanged. Therefore it is called constant.
Variable Terms
A term which contains various numerical values is called variable term. Product of 4 and\[\text{ }\!\!~\!\!\text{ X=4 }\!\!\times\!\!\text{ X=4X}\] Product of \[\text{2,X,}{{\text{Y}}^{\text{2}}}\]and \[Z=2\times X\times {{Y}^{2}}\times Z=2X{{Y}^{2}}\] Product of ?3, m and \[n=-3\times m\times n=-3mn\] Thus, \[4X,2X{{Y}^{2}}Z-3mn,\]are variable terms We also know that \[1\times X=X,1\times Y\times \text{ }z=YZ,-1\times {{a}^{2}}\times b\times c=-{{a}^{2}}\text{ }bc\]Thus\[\text{X,YZ,-}{{\text{a}}^{\text{2}}}\text{ bc}\]are variable terms
Types of Terms
There are two types of terms, like and unlike. Terms are classified by similarity of their variables.
Like Term The terms having same variables are called like terms. \[\text{6X, X,-2X, }\frac{\text{4}}{\text{9}}\text{X,}\], are like terms, \[\text{ab,-ab,4ab,9ab,}\]\[\text{ab},-\text{ab,4ab,9ab,}\] are like terms. \[\text{2}{{\text{X}}^{\text{2}}}\text{,3}{{\text{X}}^{\text{2}}}\text{Y,}{{\text{X}}^{\text{2}}}\text{Y,}\frac{\text{10}}{\text{7}}{{\text{X}}^{\text{2}}}\text{Y}\]are like terms.
Unlike Term The terms having different variables are called unlike terms. \[\text{6X, 2}{{\text{Y}}^{\text{2}}}\text{,}-\text{9}{{\text{X}}^{\text{2}}}\text{YZ, 4XY,}\]are unlike terms. \[\text{9a,}-\text{b,3}{{\text{a}}^{\text{2}}}\text{,4ab,}\]are unlike terms. \[\text{6}{{\text{X}}^{\text{2}}}\text{,7ab,4}{{\text{a}}^{\text{2}}}\text{b,}\]are unlike terms.
Coefficient
The coefficient of every term is multiplied with the term. In term, \[-6{{m}^{2}}\text{ }np,\]coefficient of\[-6=m{{n}^{2}}\]p because \[m{{n}^{2}}\text{ }p\]is multiplied with ? 6 to form \[\text{-- 6m}{{\text{n}}^{\text{2}}}\text{p}\] similarly. Coefficient of \[{{m}^{2}}=-6np,\]coefficient of \[n=-6{{m}^{2}}p\] Coefficient of \[{{\text{m}}^{\text{2}}}\text{n}\,\,\text{p=}-6\]and Coefficient of \[-6\text{=}{{\text{m}}^{\text{2}}}\text{np}\text{.}\]
Variable or Literal Coefficient
The variable part of the term is called its variable or literal coefficient. In term \[-\frac{\text{5}}{\text{4}}\text{abc,}\]variable coefficient is abc.
Constant Coefficient
The constant part of the term is called constant coefficient. In term \[-\frac{\text{5}}{\text{4}}\text{ }\!\!~\!\!\text{ abc,}\] constant coefficient is \[-\frac{\text{5}}{\text{4}}\text{ }\!\!~\!\!\text{ }\text{.}\]
Polynomials
An expression having two or more terms is known as polynomials. The expression \[3+5x\]is a polynomial and degree of the polynomial is the highest power of variable which presents in the term. In the expression, \[3+5x,x\]is the variable and its power is 1 therefore, the degree of the polynomial is 1.
\[5{{x}^{2}}+3{{y}^{3}}\](It is a polynomial in \[x\] and \[y\]and its degree is 3)
\[5{{x}^{2}}+3{{y}^{-3}}\] (It is not a polynomial as exponent if y is negative integer)
Monomials
An expression which has one term is called monomials, ie. \[4y,3{{b}^{2}}\]
Binomials
An expression which has two terms is called binomials, ie. \[3{{b}^{2}}-4ac.\]
Trinomials
An expression which has three terms is called trinomials, ie. \[{{x}^{2}}-ac+3z\]
Quadrinomials
An expression which has four terms is called Quadrinomials. ie.\[~{{a}^{2}}-bc+x-5\] 
Introduction
An equation with variable is known as an algebraic expression. The value of unknown variable is obtained by simplification of the expression. All even numbers cannot be represented because there are infinite number of even numbers, therefore, it can be represented by an equation called algebraic expression. Even numbers \[=2\times n\]whereas, n is a variable and the value of n may have 1, 2, 3, 4,........,etc. 
Introduction
LCM of two or more numbers is their least common multiple. LCM of 4 and 6 is 12, it means, 12 is the common multiple of 4 and 6, therefore, 12 is exactly divisible byeach 4 and 6. The least common multiple or LCM of 4 and 6 is not other than 12. HCF or highest Commonfactoroftwoormore numbers is obtained by factorization of the numbers. Therefore, the factors of 12 and 14 are 1, 2, 3, 4, 6, 12 and 1, 2, 7, 14 respectively, but the HCF or highest common factor is only 2. 
HCF (highest Common Factor)
Highest Common Factor is also called as Greatest Common Measure (GCM) or Greatest Common Divisor (GCD). H.C.F of two or more numbers is the greatest number which exactly divides each of the number. Therefore, HCF of two or more numbers is its highest common factors. Let us consider the HCF of 4, 8, 16 and 32. Factors of\[4=2\times 2,\text{ }8=2\times 2\times 2,16=2\times 2\times 2\times 2,32=2\times 2\times 2\times 2\times 2\] Therefore, \[2\times 2=4\]is the highest common factor which can exactly divide the numbers, 4, 8, 16, and 32.so, 4 is the HCF of 4, 8, 16 and 32. H.C.F is calculated by prime factorization and continued division methods.
HCF by Prime Factorization Method
The HCF of two or more numbers is obtained by the following steps:
Step 1: Find the prime factors of each one of the given number.
Step 2 : Find the common prime factors from prime factors of all the given numbers.
Step 4 : The product of the common prime factors is the HCF of given numbers. Let us consider the Greatest Common Measure of 36, 90, 72.
Step 1: The prime factors of \[36=2\times 2\times 3\times 3,90=2\times 5\times 3\times 3\text{ }\]and\[72=2\times 2\times 2\times 3\times 3\]
Step 2 : The common prime factors from all the prime factors \[=2\times 3\times 3\]
Step 3 : Therefore, HCF of 36, 90 and \[72=2\times 3\times 3=18\]
Find the HCF of 101, 573, and 1079 by prime factorization method?
(a) 4
(b) 1
(c) 2
(d) All of these
(e) None of these
Answer: (b)
Explanation
The prime factors of 101 = 1 and 101,573 = 1,3,191 but the common factor is 1. Therefore, the HCF of 101, 573 and 1079 = 1.
HCF by Continued Division Method
The HCF of two or more numbers is obtained by continued division. The greatest number is considered as dividend and smallest number is divisor. Follow the following steps to perform the HCF of the given numbers:
Step 1: Divide the greatest number by smallest.
Step 2: If remainder is zero, then divisor is the HCF of the given number.
Step 3: If remainder is not zero then, divide again by considering divisor as ne dividend and remainder as new divisor till remainder becomes zero.
Step 4 : The HCF of the numbers is last divisor which gives zero remainder.
The HCF of 45, 76 = ?
(a) 1
(b) 2
(c) 3
(d) 0
(e) None of these
Answer: (a)
