Introduction
This section comprises of questions setup in the form of puzzles involving a certain number of items, be it persons or things. The candidate is required to analyse the given Information. And on the basis of it has to arrive at the conclusion.
Types of Puzzle Test
The questions on puzzle test may be of four types.
I Classification type questions
Comparison type questions
III. Family-based problems
Seating/placing arrangements
Classification Type Question
Classification type questions play an important role in a test of reasoning and aptitude. The example given below will help you to clear the concept regarding this.
Example
Read the following information carefully and answer the questions that follow:
(i) Five friends P, Q, R, S and T travelled to five different cities of Chennai, Kolkata, Delhi, Bangalore and Hyderabad by different modes of transport that is. Bus, Train, Aeroplane, Car and Boat from Mumbai.
(ii) The person who travelled to Delhi did not travel by boat.
(iii) R went to Bangalore by car and Q went to Kolkata by aeroplane.
(iv) S travelled by boat whereas T travelled by train.
(v) Mumbai is not connected by bus to Delhi and Chennai.
Which one of the following combinations of person and mode is not correct?
(a) P - Bus (b) Q – Aeroplane
(c) R - Car (d) T – Aeroplane
(e) None of these
Ans. (d)
Which one of the following combinations is true for S?
(a) Delhi-Bus
(b) Chennai – Bus
(c) Chennai – Boat
(d) Data inadequate
(e) None of these
Ans. (c)
Which one of the following combinations of place and mode is not correct?
(a) Delhi – Bus
(b) Kolkata – Aeroplane
(c) Bangalore – Car
(d) Chennai – Boat
(e) Hyderabad - Bus
Ans. (a)
The person travelling to Delhi went by which one of the following modes?
(a) Bus
(b) Train
(c) Aeroplane
(d) Car
(e) Boat
Ans. (b)
Who among the following travelled to Delhi?
(a) R (b) S
(c) T (d) Data inadequate
(e) None of these
Ans. (c)
Explanation:
The given information can be analysed as follow:
(a) Mode of Transport: R travels by Car, Q by Aeroplane, S by Boat and T by Train. Now, only P remains. So, P travels by Bus.
(b) Place of Travel: R goes to Bangalore, Q to Kolkata. Now, bus transport is not available for Delhi or Chennai. So, P who travels by bus goes to Hyderabad. S travels by boat and hence, did not go to Delhi but to Chennai. Now, only T remains. So, T goes to Delhi.
Ranking & Number Sequence Test
Learning Objectives
Introduction
Number Test
Raking Test
Time Sequence Test
Logical Sequence of Words
Number, Ranking and Time Sequence Test
In this chapter we deal with questions which are followed with a sequence consisting of numbers, ranking time and logical sequence of words. We have to find answers on the basis of given condition. The importance of such types of questions cannot be overemphasised as their presence in a test of reasoning is almost certain.
Though no explanations are required as how to attempt these questions in exams, in this chapter generally we deal with four types of questions. They are given section wise with solution as well.
1st: Number test
2nd: Ranking Test
3rd: Time sequence Test,
4th: Logical sequence of words
Number Test
Number test problems consists of number sequence/ problems with algebraic expressions, mathematical calculations and other compatible problems.
Example:
How many such 7s are there in the following number sequence which are followed by 4 but not immediately preceded by 8?
5 4 7 8 9 7 4 3 8 7 5 7 4 8 7 4 1 2 7 4 5 7 9 4
(a) Two (b) Three
(c) Four (d) Five
(e) None of these
Ans. (b)
Explanation: Number of 7s which are immediately followed by 4 but not immediately preceded by 8 are:
\[5\,\,4\,\,7\,\,8\,\,\underset{1}{\mathop{\underline{9\,\,7\,\,4}}}\,\,\,3\,\,8\,\,7\,\,\underset{2}{\mathop{\underline
5\,\,7\,\,4}}}\,\,\,8\,\,7\,\,4\,\,1\,\,\underset{3}{\mathop{\underline{2\,\,7\,\,4}}}\,\,\,5\,\,7\,\,9\,\,4\,\,\]
A number is 9 times twice the other number. The sum of two numbers is 133. The two numbers are :
(a) 9, 124 (b) 11, 122
(c) 17, 166 (d) 7, 126
(e) None of these
Ans. (d)
Explanation: If the smaller number is x, then
\[x+\left( 2x\times 9 \right)=133\]
\[x+18x=133\]
\[19x=133\]
\[x=133-19=7\]
If one number is 7, the other number is:
\[133-7=126\]
If all the numbers from 1 to 28 which are exactly divisible by 3 are arranged in descending order, which would come at the fifth place from the top?
(a) 12 (b) 21
(c) 15 (d) 18
(e) None of these
Ans. (c)
Explanation. The numbers divisible by 3 in descending order are: 27, 24 21 18 15, 12, 9, 6, 3 and the number at the fifth place is 15.
Commonly Asked Questions
How many 6’s are there in the following series of numbers which are preceded by 7 but not immediately followed by 9?
6 7 9 5 6 9 7 6 8 7 6 7 8 6 9 4 6 7 7 6 9 5 7 6 3
(a) one (b) Two
(c) Three (d) Four
(e) None of these
Ans. (c)
Explanation: \[6\text{ }7\text{ }9\text{ }5\text{ }6\text{ }9\text{ }\underset{1}{\mathop{\underline{7\text{ }6\text{ }8}}}\,\text{ }\underset{2}{\mathop{\underline{7\text{ }6\text{ }7}}}\,\text{ }8\text{ }6\text{ }9\text{ }4\text{ }6\text{ }7\text{ }7\text{ }6\text{ }9\text{ }5\text{ }\underset{3}{\mathop{\underline{7\text{ }6\text{ }3}}}\,\]
In a chess tournament each of six players will play every other player exactly once. How many matches more...
Introduction
In this section, question pattern is based on basic fundamentals of simple mathematical operations. It is divided into four types. Problems in this type of reasoning questions may be on the symbols used in basic mathematical operations, such as:
Additon: \[\mathbf{(+)}\]
Subtraction: \[\mathbf{(-)}\]
Multiplication: \[\mathbf{(\times )}\]
Division: \[\mathbf{(\div )}\]
Also (>, <, =) 'greater than' less than' and 'equal to etc.
Case - 1st
Basic BODMAS rule is applied to solve simple mathematical operations.
B = Brackets [first solve big bracket, followed middle and small]
O = Of
D = Division
M = Multiplication
A = Addition
S = Subtraction
Note: This chapter will also help the students to solve the problems of quantitative aptitude along with that of the reasoning.
If + means\[\mathbf{\div ,-}\]means \[\mathbf{\times ,\div }\] means + and \[\mathbf{\times }\] means -, then the value of \[\mathbf{36\times 12+4\div 6+2-3}\] when simplified is
(a) 12 (b) 38
(c) 42 (d) 56
(e) None of these
Ans. (c)
Explanation: Option (c) is correct. Using proper signs in the given expression,
We get \[36-12\div 4+6\div 2\times 3=36-3+3\times 3=36-3+9=42.\]
If P denotes\[\mathbf{\div }\], Q denotes\[\mathbf{\times }\], R denotes + and S denotes -, then 18Q12 P4 R5 S6 =?
(a) 46 (b) 53
(c) 64 (d) 75
(e) None of these
Ans. (b)
Explanation: Option (b) is correct. Using correct symbols, we get
\[18\times 12\div 4+5-6=18\times 3+5-6=54+5-6=53\]
Commonly Asked Questions
Direction: If \['+'\text{ }is\text{ }X\text{ }\!\!'\!\!\text{ }-'\text{ }is\text{ }'+'\,\,'\times '~is\,\,'\div '\,and\,\,'\div '\text{ }is\,\,'-'\] then answer the following questions.
\[\mathbf{9\div 5-4+3\times 2}\]
(a) 2 (b) -9
(c) -3 (d) 10
(e) None of these
Ans. (d)
Explanation: \[9-5+4\times 3\div 2=9-5+4\times \frac{3}{2}=9-5+6=15-5=10.\]
\[\mathbf{6+7\times 3-8\div 20=?}\]
(a) -3 (b) 7
(c) 2 (d) 1
(e) None of these
Ans. (c)
Explanation: \[6\times 7\div 3+8-20\]
\[6\times \frac{7}{3}+8-20=14+8-20=2\]
\[\mathbf{3\times 2+4-2\div 9=?}\]
(a) -1 (b) 1
(c) -2 (d) 3
(e) None of these
Ans. (a)
Explanation: \[3\div 2\times 4+2-9\]
\[=\frac{3}{2}\times 4+2-9=6+2-9=-1\]
\[\mathbf{6-9+8\times 3\div 20=?}\]
(a) -2 (b) 6
(c) 10 (d) 12
(e) None of these
Ans. (c)
Explanation: \[6+9\times 8\div 3-20=6+24-20=10\]
\[\mathbf{5\times 4-6\div 3+1=?}\]
(a) 5 (b) 4
(c) -I (d) 2
(e) None of these
Ans. (e)
If \[\mathbf{'+'}\] means \[\mathbf{'\times ','-'}\] means\[\mathbf{'\div ','\times '}\]means \[\mathbf{'-'}\] and\[\mathbf{'\div '}\], means\[\mathbf{'+'}\], then what will be the value of\[\mathbf{12\div 48-8\times 4+4=}\]?
(a) 8 (b) 4
(c) 20 (d) 2
(e) None of these
Ans. (d)
Explanation: \[12+48\div 8-4\times 4=12+6-4\times 4=12+6-16=18-6=2\]
Case - 2nd
Interchange of Signs and Numbers: In more...
Introduction
Numbers are the basic unit of Mathematics. After all, it with numbers that we perform the various functions which constitute Mathematics. For example: Addition, Subtraction, Multiplication & Division. The Number system is the backbone of any competitive exam. The correct understanding will help you to solve different and complex problems that appear in these examinations. First and for most/ let us have a look at the basic classification of numbers and its various kinds.
Classification of Numbers
Natural Numbers
Natural numbers are all of the whole numbers EXCEPT zero. 1, 1, 3. 4, 5, 6, 7, 8, 9, 10, 11.... They are also called counting numbers. The lowest natural number is 1.
Whole Numbers
Whole numbers are those numbers which start by 0 or we can say when 0 is included in the list of natural numbers then we call it whole numbers ; for example 0, 1, 2, 3, 4, 5.......
Integers
It is the series of both positive and negative numbers lying on the number line, it is the combination of both positive and negative whole and natural numbers.
Rational Numbers
Rational numbers are those numbers that can be written in the form of a ratio of x and y, where the denominator is not zero.
Real Numbers
The number which lies on the number line is a real number .The number can be positive or negative in nature, for example it may be like as \[3,4,5,6,-6,-5,-4,-3,-2.....\]
Irrational Numbers
Irrational numbers are those which are not rational, that is those numbers that cannot be written in the form of a ratio.
Counting Numbers
Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number in the number line is 1.
Complex Numbers
Includes real numbers and imaginary numbers are called complex numbers, eg. a + ib.
Prime numbers
The numbers which don't have any factor other than 1 or itself.
For example: 2, 3, 5, 7, 9, 29, 31, 43..................or we can say that the numbers which are not divisible by any number are called prime numbers. There are 24 prime numbers between 1 and 100.
2 is the only even prime number and the least prime number.
Percentage
Percentage
Percentage is a fraction whose denominator is 100. The numerator of the such fraction is called the rate percent. For example: 15 percent means\[\frac{15}{100}\]and denoted by 15 %.
% of A means \[\frac{A}{100}\] and simplifying it. Example: \[45\,%=\frac{45}{100}=\frac{9}{20}\]
For conversion of fraction \[\frac{p}{q}\] as percentage, we simply multiply it by 100 and out the sign of% or mathematically we can write \[=\frac{p}{q}=\left( \frac{p}{q}\times 100 \right)%.\]
Application Based Problem on Percentage
The following are the points to remember to solve the problem related to variation in the price of an article.
If the price of an article increases by x % then the reduction in consumption, so that expenditure remains unaffected, is\[\left( \frac{x}{100+x}\times 100 \right)%\]
If the price of an article decreases by x % then the increase in consumption, so that expenditure remains unaffected, is \[\left( \frac{x}{100-x}\times 100 \right)%\]
Problem Based on the Population of a Locality
Suppose the present population of a locality be W and let it increases by x % per annum then
Population after y years \[=A{{\left( 1+\frac{x}{100} \right)}^{y}}\]
Population before y years =\[\frac{A}{{{\left( 1+\frac{x}{100} \right)}^{y}}}\]
Ratio and Proportion
In this chapter we will study about the comparison of two or more quantities. When we compare only two quantities of same kind, it is called ratio and more than two quantities is called proportion.
Ratio
A ratio is a relation between two quantities of same kind. Comparison is made between the two quantities by considering what part of one quantity is that of the other quantity. The two quantities are called the terms of ratio.
If x and y are two quantities of same kind then the ratio of x to y is x/y or \[x:y.\] It is represented by\[x:y.\]
Important Points Related to Ratio
The first term of ratio is called antecedent and the second term is called the consequent.
If \[\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=.......\]then each ratio is equal to \[\frac{a+c+e.......}{b+d+f......}\]
Multiplication and division by the same number (except zero) with antecedent and consequent of the ratios are equal in values, the resultant ratio remains unchanged.
Proportion
It is the equality of two ratios i.e. if a : b = c : d, then ad = cd that implies product of extremes = product of means. Four quantities p, q, r, s are in proportion if ps = qr.
Important Points Related to Proportion
If\[\frac{a}{b}=\frac{c}{d}\]then
\[\frac{a+b}{b}=\frac{c+d}{d}\](componendo)
\[\frac{a-b}{b}=\frac{c-d}{d}\](dividendo)
\[\frac{a+b}{a-b}=\frac{c+d}{c-d}\](componendo and dividendo)
If three numbers a, b, c are in continued proportion and written as a : b :: b : c then\[\frac{a}{b}=\frac{b}{c}\Rightarrow {{b}^{2}}=ac\Rightarrow b=\sqrt{ac}\] hence, b is called mean.
Profit and Loss
Cost Price
It is the price of an article at which the shopkeeper purchases the goods from manufacturer or wholesaler. In short it more...
Geometry
Geometry is the visual study of shapes, sizes, patterns, and positions. It occurred in all cultures, through at least one of these five strands of human activities: The following formulas and relationships are important in solving geometry problems.
Angle Relationships
The base angles of an isosceles triangle are equal
The sum of the measures of the interior angles of any n-sided polygon is 180(n - 2) degrees.
The sum of the measures of the exterior angles of any n-sided polygon is 360°.
If two parallel lines are cut by a transversal, the alternate interior angles are equal, and the corresponding angles are equal.
Angle Measurement Theorems
A central angle of a circle is measured by its intercepted arc.
An inscribed angle in a circle is measured by one-half of its intercepted arc,
An angle formed by two chords intersecting within a circle is measured by one-half the sum of the opposite intercepted arcs.
An angle formed by a tangent and a chord is measured by one-half its intercepted arc.
An angle formed by two secants, or by two tangents, or by a tangent and a secant, is measured by one-half the difference of the intercepted arcs.
Proportion Relationships
A line parallel to one side of triangle divides the other two sides proportionally.
In two similar triangles, corresponding sides, medians, altitudes, and angle bisectors are proportional.
If two chords intersect within a circle, the product of the segments of one is equal to the product of the segments of the other.
If a tangent and a secant are drawn to a circle from an outside point, the tangent is the mean proportional between the secant and the external segment.
In similar polygons the perimeters have the same ratio as any pair of corresponding sides.
Triangle Relationships
If an altitude is drawn to the hypotenuse of a right triangle, it is the mean proportional between the segments of the hypotenuse, and either leg is the mean proportional between the hypotenuse and the segment adjacent to that leg.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. (Remember the Pythagorean triples: 3, 4, 5; 5, 12, 13.)
In a 30°- 60° right triangle, the leg opposite the 30° angle is one-half the hypotenuse, and the leg opposite the 60° angle is one-half the hypotenuse times \[\sqrt{3}\,.\]
In a right isosceles triangle the hypotenuse is equal to either leg times \[\sqrt{2}\,.\]
In an equilateral triangle of sides, the altitude equals \[\frac{s}{2}\sqrt{3}\,.\]
Commonly Asked Questions
If the sides of a triangle are produced then the sum of the exterior angles i.e.
\[\angle DAB+\angle EBC+\angle FCA\]is equal to:
(a) 180° (b) 270°
(c) 360° (d) 240°
(e) None of these
Ans. (c)
Explanation: Sum of exterior angles = 360°
2. In a more...
Factorial
The factorial/ symbolized by an exclamation mark (!), is a quantity defined for all integers greater than or equal to 0. Mathematically/ the formula for the factorial is as follows.
If n is an integer greater than or equal to I, then
\[n\,\,!=n\left( n-1 \right)\left( n-2 \right)\left( n-3 \right)...(3)(2)(1).\]
Example
\[1!=1,\,\,2!=2,\,\,3!=6,\,\,4!=4\cdot 3\cdot 2\cdot 1=24,\,\,5!=5\times 4\times 3\times 2\times 1=120\]
\[6!=6\times 5\times 4\times 3\times 2\times 1=720,\,\,7!=5040\,\,and\,\,8!=40320\,etc.\]
The special case 0! is defined to have value 0! = 1.
Permutation
The different arrangements which can be made by taking some or all of the given things or objects at a time is called Permutation.
Example:
All permutations (arrangements) made with the letters a, b, c by taking two at a time will be (ab, be, ca, ba, ac, cb).
Number of Permutations: Number of all permutations of n things, taking r at a time is:
\[^{n}{{P}_{r}}=\frac{n!}{n-r!}=n(n-1)(n-2)(n-3)...\,\,...(n-r+1).\]
Note: This is valid only when repetition is not allowed.
Permutation of n different things taken r at a time. When repetition is Allowed: \[n\times n\times n...\,\,...\,\,...\,\,.\,\,.r\]times = \[{{n}^{r}}\]
Permutation of n things taking all n things at a time= n!
Out of n objects \[{{n}_{1}}\] are alike one type,\[{{n}_{2}}\] are alike another type, \[{{n}_{3}}\] are alike third type, nr are alike another type such that \[({{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.......nr)=n\]
Number of permutations of these n things are =\[\frac{n!}{{{n}_{1}}!{{n}_{2}}!\,\,...\,\,...\,\,.{{n}_{r}}!}\]
Combination
Each of the different selections or groups which are made by taking some or all of a number of things or objects at a time is called combination.
The number of combinations of n dissimilar things taken r at a time is denoted by \[^{n}{{C}_{r}}\,\,or\,\,C\,\,(n,\,\,r).\]
\[^{n}{{C}_{r}}=\frac{n!}{r!(n-r)!}=\frac{n(n-1)(n-2).....(n-r+1)}{1.2.3......r}\]
Also \[^{n}{{C}_{0}}=1;\,\,\,\,\,\,\,\,\,\,\,{{\,}^{n}}{{C}_{n}}=1;\]
Note: (i) \[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{(n+1)}}{{C}_{r}}\]
(ii) \[^{n}{{C}_{r}}{{=}^{n}}{{C}_{n-r}}\]
Important Formula
In a group of n-members if each member offers a shake hand to the remaining members then the total number of handshakes = \[^{n}{{C}_{2}}=\frac{n(n-1)}{1.2}=\frac{n(n-1)}{2}\]
The number of diagonals in a regular polygon of ‘n’ sides is \[\frac{n(n-3)}{2}.\]
From a group of m-men and n-women, if a committee of remembers \[(r\le m+n)\]is to be formed, then the numbers of ways it can be done is equal to \[^{(m+n)}{{C}_{r}}.\]
The number of ways a group of r-boys (men) and s-girls (women) can be made out of m boys (men) and n-girls (women) is equal to \[{{(}^{m}}{{C}_{r}}{{\times }^{n}}{{C}_{s}}).\]
From a group of m-boys and n-girls the number of different ways that a committee of remembers can be formed so that the committee will have at least one girl is \[^{(m+n)}{{C}_{r}}{{-}^{m}}{{C}_{r}}.\]
Commonly Asked Questions
If a die is cast and then a coin is tossed, find the number of all possible outcomes.
(a) 11 (b) 12
(c) 10 (d) 15
(e) None of these
Ans. (b)
Explanation: A die can fall in 6 different ways showing six different points 1, 2, 3, 4, 5, 6, ... and a coin can more...
Introduction
Mensuration is a science of measurement of the lengths of lines, area of surfaces and volumes of solids.
Some Important Definitions and Formulae:
If any closed figure has three sides then it is called a triangle.
In a triangle the sum of three angles is 180°.
In a triangle the sum of the lengths of any two sides should be more than the third side.
Similarly the difference between any two sides of a triangle is less than the third side.
The side on which a triangle rests is called the base. The length of the perpendicular drawn on the base from opposite vertex is called the height of the triangle.
If the three sides of a triangle have three different lengths then it is called a scalene triangle.
If exactly two side of a triangle are equal and the third side has different length then it is called an isosceles triangle.
If all the three sides of a triangle are equal then it is called an equilateral triangle.
Area of Plane Geometrical Figures
Triangle
(i) Right Triangle
(ii) Scalene Triangle (Heron's formula)
(iii) Isosceles Triangle
(iv) Equilateral Triangle
Right Triangle: Area of right triangle = \[\frac{1}{2}~\left( perpendicular \right)\times Base=\frac{1}{2}\times AB\times BC\]
Scalene Triangle (Heron's formula): Let, a, b, c be the length of sides of a triangle
then area =\[\sqrt{s(s-a)(s-b)(s-c)}\] sq. unit, where s = \[\frac{1}{2}(a+b+c)\] Isosceles Triangle:
Area of isosceles triangle \[=\frac{1}{2}\times BC\times AD=\frac{1}{4}b\sqrt{4{{a}^{2}}-{{b}^{2}}}\]
Equilateral triangle:
Area = \[\frac{\sqrt{3}}{4}{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\]
Circle
A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.
A circle with centre O and radius r is generally denoted by C (0, r).
Circle Formulas
1. The circumference C of a circle of radius r is given by the formula \[C=2{}^\circ \pi r.\]
2. The area A of a circle of radius r is given by the formula \[A=\pi {{r}^{2}}.\]
3. The areas of two circles are to each other as the squares of their radii.
4. The length L of an arc of n° in a circle of radius r is given by the formula\[L=\frac{n}{360}\times 2\pi r.\]
5. The area A of a sector of a circle of radius r with central angle of n° is given by\[L=\frac{n}{360}\times \pi {{r}^{2}}.\]
Quadrilateral We know that a geometrical figure bounded by four lines segment is called quadrilateral. In this section we will study about area and perimetre of different quadrilaterals.
Rectangle
more...
Probability
A mathematically measure of uncertainty is known as probability. If there are ‘a’ elementary events associated with a random experiment and 'b' of them are favourable to event 'E';
Then the probability of occurrence of event E is denoted by P(e).
Experiment
An operation which can produce some well- defined outcomes is called an experiment.
Random Experiment: An experiment in which all possible outcomes are known and exact outcome cannot be predicted is called a random experiment.
Example: Rolling an unbiased dice has all six outcomes (1, 2, 3, 4, 5, 6) known but exact outcome can be predicted.
Outcome: The result of a random experiment is called an outcome.
Sample Space: The set of all possible outcomes of a random experiment is known as sample space.
Example: The sample space in throwing of a dice is the set (1, 2, 3, 4, 5, 6).
Trial: The performance of a random experiment is called a trial.
Example: The tossing of a coin is called trial.
Event
An event is a set of experimental outcomes, or in other words it is a subset of sample space.
Example: On tossing of a dice, let A denotes the event of even number appears on top A: {2, 4, 6}.
Mutually Exclusive Events: Two or more events are said to be mutually exclusive if the occurrence of any one excludes the happening of other in the same experiment. E.g. On tossing of a coin is head occur, then it prevents happing of tail, in the same single experiment.
Exhaustive Events: All possible outcomes of an event are known as exhaustive events. Example: In a throw of single dice the exhaustive events are six {1, 2, 3, 4, 5, 6}.
Equally Likely Event: Two or more events are said to be equally likely if the chances of their happening are equal.
Example: On throwing an unbiassed coin, probability of getting Head and Tail are equal.
Playing Cards
Total number of card are 52.
There are 13 cards of each suit named Diamond, Hearts, Clubs and Spades.
Out of which Hearts and diamonds are red cards.
Spades and Clubs are black cards.
There are four face cards each in number four Ace, king, Queen and jack.
Black Suit (26)
Red Suit (26)
Spades (13) & Club (13)
Diamond (13) & Heart (13)
Each Spade, Club, Diamond, Heart has 9 digit cards 2, 3, 4, 5, 6, 7, 8, 9, and 10.
There are 4 Honour cards each Spade, Club, Diamond, Heart Contains 4 numbers of Honours cards Ace, king, Queen and jack.
India-Location
You have already seen the map of India In the previous classes. Now you closely examine the map of India (Figure 1.1).
Mark the southernmost and northernmost latitudes and the easternmost and westernmost longitudes.
The mainland of India, extends from Kashmir in the north to Kanniyakumari in the south and Arunachal Pradesh in the east to Gujarat in the west. India's territorial limit further extends towards the sea upto 12 nautical miles (about 21.9 km) from the coast.
(See the box for conversion).
Statute mile
63,360 inches
Nautical mile
72,960 inches
1 Statute mile
about 1.6 km (1.584 km)
1 Nautical mile
about 1.8 km (1.852 km)
Our southern boundary extends upto \[6{}^\circ 45'\text{ }N\]latitude in the Bay of Bengal. Let us try to analyses the implications of having such a vast longitudinal and latitudinal extent.
If you work out the latitudinal and longitudinal extent of India, they are roughly about 30 degrees, whereas the actual distance measured from north to south extremity is 3,214 km, and that from east to west is only 2,933 km. What is the reason for this difference? Consult Chapter 3 on the topic Latitude, Longitude and Time in the book Practical Work in Geography - Part I (NCERT, 2006) to find out.
This difference is based on the fact that the distance between two longitudes decreases towards the poles whereas the distance between two latitudes remains the same everywhere. Find out the distance between two latitudes?
From the values of latitude, it is understood that the southern part of the country lies within the tropics and the northern part lies in the sub-tropical zone or the warm temperate zone. This location is responsible for large variations in land forms, climate, soil types and natural vegetation in the country.
Now, let us observe the longitudinal extent and its implications on the Indian people. From the values of more...
\[\angle DAB+\angle EBC+\angle FCA\]is equal to:
(a) 180° (b) 270°
(c) 360° (d) 240°
(e) None of these
Ans. (c)
Explanation: Sum of exterior angles = 360°
2. In a more... 
Scalene Triangle (Heron's formula): Let, a, b, c be the length of sides of a triangle
then area =\[\sqrt{s(s-a)(s-b)(s-c)}\] sq. unit, where s = \[\frac{1}{2}(a+b+c)\] Isosceles Triangle:
Area of isosceles triangle \[=\frac{1}{2}\times BC\times AD=\frac{1}{4}b\sqrt{4{{a}^{2}}-{{b}^{2}}}\]
Equilateral triangle:
Area = \[\frac{\sqrt{3}}{4}{{(side)}^{2}}=\frac{\sqrt{3}}{4}{{a}^{2}}\]
Circle
A circle is a geometrical figure consisting of all those points in a plane which are at a given distance from a fixed point in the same plane. The fixed point is called the centre and the constant distance is known as the radius.
A circle with centre O and radius r is generally denoted by C (0, r).
Circle Formulas
1. The circumference C of a circle of radius r is given by the formula \[C=2{}^\circ \pi r.\]
2. The area A of a circle of radius r is given by the formula \[A=\pi {{r}^{2}}.\]
3. The areas of two circles are to each other as the squares of their radii.
4. The length L of an arc of n° in a circle of radius r is given by the formula\[L=\frac{n}{360}\times 2\pi r.\]
5. The area A of a sector of a circle of radius r with central angle of n° is given by\[L=\frac{n}{360}\times \pi {{r}^{2}}.\]
Quadrilateral We know that a geometrical figure bounded by four lines segment is called quadrilateral. In this section we will study about area and perimetre of different quadrilaterals.
Rectangle
more... 
Rational Numbers
Rational numbers are those numbers that can be written in the form of a ratio of x and y, where the denominator is not zero.
Real Numbers
The number which lies on the number line is a real number .The number can be positive or negative in nature, for example it may be like as \[3,4,5,6,-6,-5,-4,-3,-2.....\]
Irrational Numbers
Irrational numbers are those which are not rational, that is those numbers that cannot be written in the form of a ratio.
Counting Numbers
Counting numbers are those numbers which are well managed on the number line with the difference of 1. The smallest counting number in the number line is 1.
Complex Numbers
Includes real numbers and imaginary numbers are called complex numbers, eg. a + ib.
Prime numbers
The numbers which don't have any factor other than 1 or itself.
For example: 2, 3, 5, 7, 9, 29, 31, 43..................or we can say that the numbers which are not divisible by any number are called prime numbers. There are 24 prime numbers between 1 and 100.
