Maths Magic OBJECTIVE MATHEMATICS For all competitive Exams
J. P. Dixit
Maths Magic : Objective Mathematics by J. P. Dixit Š Amar Ujala Publications Ltd. Published by Amar Ujala Publications Ltd. and printed at Microtone Printec Pvt. Ltd., B-45, Okhla Industrial Area, Fase-I, New Delhi-110020 Second Edition : 2013 Price : ` 350/ISBN : 978-93-82948-10-0 z
Due care and diligence has been taken while publishing this book. However, the publisher does not hold any responsibility for any mistake that may have inadvertently crept in. The publisher does not accept responsibility for any loss arising out of the use of this book.
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All rights reserved. Neither this publication nor any part of it may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher.
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All disputes are subject to the exclusive jurisdiction of competent courts and forums in Noida only.
I N D E X
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.
Number Series Numbers or Number System Decimal Fraction Simplification Problem Based on Numbers Surds or Radicals H.C.F. and L.C.M. Square Roots and Cube Roots Average Profit and Loss True Discount/Discount Percentage Simple Interest Compound Interest Alligation or Mixture Ratio and Proportion Partnership Time and Work Time and Distance Problem on Trains Boat and Stream Pipes and Cistern Area and Perimeter Volume and Surface Areas (3-Dimensial Figures) Problems Based on Ages Stock and Shares Races and Games Clocks Calendar Tabulation Bar - Graphs Pie - Diagram Line Graphs Statistics Permutations and Combinations Probability Trigonometry Height and Distance
15 33 46 62 82 98 111 120 132 146 164 175 193 205 222 229 247 257 272 285 299 305 315 337 364 378 387 392 397 401 415 425 437 452 461 467 474 488
Preface
This book is intended for the aspirants of various preliminary as well as final competitive examinations conducted by different service selection commissions and boards such as UPSC, PSC, PCS, SSC, IBPS, RRB, SBI and managements institutes. The book contains a huge collection of questions asked in respective examinations and these questions are solved by shortcut as well as fundamental methods. Nowadays although short cut methods are required for solving question papers within given time but fundamental methods are absolutely essential for learning mathematics. I strongly believe that students learn mathematics well only when they construct their own mathematical thinking. Information can be transmitted from books to the needful but mathematical understanding and knowledge come from within the learner as that individual explores, discovers, and makes connections. This understanding is developed only by clearing the basics of mathematics. Keeping this in mind due weightage is given to the fundamental and explanatory methods. Consequently this book does not simply provide content; rather, it facilitates readers’ construction of their own knowledge of mathematics. Here are the salient features of the book: z
The syllabus and question pattern of almost all the popular exams is covered.
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Original examination questions are solved with short-cut as well as fundamental methods according to the needs of examinees and fitting in demand of the particular question.
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Short cut methods fully explained and adequately illustrated.
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To avoid unnecessary elaboration and tediousness only appropriate and notable questions are included.
Second revised, updated and enlarged edition: z
10 to 15 new questions from the latest examinations are added in the beginning of the each chapter.
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More tricky formulae are added wherever required.
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Three new chapters on calendar, Trigonometry and Height and Distance are added.
Break-UP of Mathematics for Various Examinations
SSC-FCI
Total No. of Question
Question of math
Separate paper
50
Area of Priority
Trigonometry's, Algebra and Geometry basic concepts Number theory: Divisibility, remainders LCM and HCF, Fractions-comparisons. Basic Maths: Simplification (BODMAS) Surds, indices, Roots, squares, Cubes Algebra: Linear equation, Quadratic equations, Polynomials, Avg and Ratios Mean, Median, Mode Wine-Water mix ture (Allegations, Weighted avg.) Ratio-Proportion-variations, Partnership STD, Time speed distance, Trains, plat forms, Boats-streams, Time and Work, Pipes and Cisterns Geometry: Angles, sides, bisectors, circles etc. Mensuration (area, volume, perimeter) Trigonometry, Basic % (increase, decrease in consumption, population) Data-interpretation cases. Profit, loss, discount, marked price. Simple and compound interest rate PCP, Permutation, Combination, Probability, Coordinate Geometry Progression: Arithmetic+ Geometry Logarithms.
SSC-CGL (I) (II) 200 Separate paper
50 (I) 100 (II)
RBI Grade -B
200
30
Linear equations, Average, Ratios, ProfitLoss, %, Permutation, Venn Diagrams, Area and Volume Combination, Probability
Civil Services (Pre) II
80
7-10
CAT
Separate paper
30
Linear Algebra, Theorems, Geometry, Ratio & Proportion, Quadratic equations, Prime number Permutation Combinations, BODMAS, HCF and LCM,Data Sufficiency, Data] Interpretation, Probability Geometry Ratios and Proportion, Ratios, Quadratic Probability, Permutations Combinations and linear equations, Algebra, Profit-Loss, Averages, Percentages, Partnership Number system: HCF, LCM, Geometric Progression, Arithmetic Progression, Arithmetic mean, Geometric mean, Harmonic mean, Median, Mode, Number Base System and BODMAS etc.
NDA (I) and (II)
Separate paper
120
ALGEBRA: Concept of set, Operations of sets, Venn Diagrams, De Morgan laws.
Complex number-basic propeties, MATRICES and DETERMINANTS : Types of matrices, Operations on Matrices, Determinant of a matrix, Basic properties of determinants TRIGONOMETRY: Angles and their measures in degrees and in radians, Trigonometrical ratios. Trigonometric identities. Sum and difference formulae. Multiple and Sub multiple angles. Etc. ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSION:Rectangular Cartesian coordinate System. Equation of a line in various forms. Angle between two lines. Distance of point from a line. Equation of a circle in stan dard and in general form. Eccentricity and axis of a conic. Etc.DIFFERETIAL CALCULUS: Concept of real value function. Composite range and graph of a function. Composite functions, One to One and inverse INTEGRAL CALCULUS AND DIFFER ENTIAL EQUATION: Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, Evaluation of definite integrals- determination of areas of plane regions bounced by curves applications.VECTOR ALGEBRA: Vectors in two and three dimensions, Magnitude and direction of a vectors, Scalar Multiplication of a vector, Scalar product or dot prodct of two vectors. Vector product or cross product of two vectors. Etc. STATICTICS AND PROBABILITY: Classification of data, Frequency distribution, Cumulative frequency, Histogram, Pie Chart, Frequency Polygon, Measures of Central tendecymean, median, mode. Random experiment, Outcomes and associated sample space, events, mutually exclusive and exhaustive events. Impossible and certain Events. Union and Intersection and composite events. Complementary, Elementary and com posite events. Elementary theorems on probability, Bayes theorem- Simple Problems. Random variable as function on a sample space. Etc. CDS- (I) (II)
Separate Paper
100
ARITHMETIC: Number system-Natural numbers, Integers, Rational and real Numbers, Fundamental Operations addition, Subtraction, Multiplication, division, Square
roots, Decimal, fractions. Unitary method, time and distance, time and work, percentages, applications to simple and compound interest, profit and loss, ratio and proportion, variation etc. ALGEBRA: Concept of set, Operations of sets, Venn Diagrams, De Morgan laws. Complex number-basic proper t i e s , H.C.F., L.C.M., theory of polynomials, solution of quadratic equations, relation between its roots and coefficients, Simultaneous linear equations in two unknown-analytical and Graphical solution. Etc. TRIGONOMETRY: sine x, cosine x, Tangent x when 0° <x< 90° Values of sin x, cos x and tan x, for x=0°, 30°,45°, 60°, and 90° . Simple trigonometry identities. Use of trigonometric tables. Simple cases of Heights and distances. GEOMETRY: Line and angles, plane and plane figures. Theorems on properties of angles at a point, parallel lines, sides and angles of a triangle, Congruency of triangles, Similar triangles, Concurrence of medians and altitudes, properties of diagonals of a par allelogram, rectangle and squares, diagonals of a parallelogram, rectangle and square, Circle and its properties including tangents and normal's etc. MENSURATION: Area of squares, rectangles, parallelograms, triangle and circle. Areas of figures which can be split up into these figure, Surface area and Volume of cuboids, lateral surface and volume of right circular cones and cylinders, Surface area and volume of spheres. STATISTICS: Collection and tabulation of statistical data, Graphical representation, frequency polygons, histograms, bar charts, pie charts etc. Measures of central tendency.
NUMBERS OR NUMBER SYSTEM BASIC FORMULAE b)2
a2
b2
l
(a + = + 2ab + (a – b)2 = a2 – 2ab + b2 a2 – b2 = (a + b) (a – b) (a + b)2 – (a – b)2 = 4ab (a + b)2 + (a – b)2 = 2 (a2 + b2) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca a3 + b3 = (a + b) (a2 – ab + b2)
l
a3 – b3 = (a – b) (a2 + ab + b2)
l l l l l l
(a3 + b3 + c3 – 3abc) = (a + b + c) (a2 + b2 + c2 – ab – bc – ca) l If a + b + c = 0 then a3 + b3 + c3 = 3abc. l
l
Dividend = (Divisor × Quotient) + Remainder. a + b + c = 0 then a3 + b3 + c3 = 3abc.
SIMPLIFICATION l
SURDS OR REDICALS Law of Radicals The law of indices which are applicable to the surds also are :
( x) n
(ii) (iii)
a if a > o |a| = a if a < o
n
x .n y
x = y
n
x
n
y
( x)
m
n
(iv)
m n
(v)
=x
xy =
n
=
x =
(
n
xm
mn
)
x =n
m
a− b
)
x
Law of Surds (vi) The conjugate surds of
(
BODMAS RULE : This rule depicts the
correct sequence in which the operations are to be executed, so as to find the value of a given expression n B stands for = Bracket n O stands for = Of n D stands for = Division n M stands for = Multiplication n A stands for =Addition n S stands for = Subtraction l Brackets are of four types and they are operate in the following order, l Bar or line, (), {}, [] l Thus the complete order of operation will be —, (), {}, [] of ÷, ×, +, – l Order of + and – can be interchanged or they can be operated simultaneously too. Modulus of a Real Number : Modulus of a real number a is defined as
n
n
(i)
)
a + b is+
(
a+ b = c+ d (vii) If Then, a – c and b= d a a
(viii) (ix)
a a
(x) If Then,
a......∞ = a
1 a..... n times = a1 − n 2 a+ a+
a......∞ = x x(x – 1) = a
H.C.F AND L.C.M. Method of Prime factors Write each of the given number as Product of Prime factors l Find the product of the highest powers of all the factors that occur in the given numbers l This product, then, is the required L.C.M. of the given numbers l
(2)
Important Formulae
L.C.M. and H.C.F. of fractions : H.C.F. of Numerators H.C.F. = L.C.M. of Denominators L.C.M. of Numerators L.C.M. = H.C.F. of Denominators l L.C.M. of (a, b) × H.C.F. of (a, b) = (a × b) l If a number N, when divided by x, y or z leaves respective remainder P in each case then N is of the form K × L.C.M. of (x, y, z) + P
AVERAGE Average of Position includes Median and Mode. l Arithmetic Average is used of all averages. for example: average income, average profit, average age, average marks etc. l It is defined as the sum of all values of items divided by the total number of items. l In Individual series. Average =
Sum of observations Number of observations
x1 + x2 + x3 ........... + xn n l In Discreate series
x =
Or
x1 f1 + x2 f 2 + ........... + xn f n f1 + f 2 + ........... + f n l Geometric Mean : Geometric Mean of x1, x2 .....xn is denoted by
Loss % =
(v)
S.P. =
(100 + Gain%) × C.P. 100
(vi)
S.P. =
(100 − Loss%) × C.P. 100
(vii)
C.P. =
100 × S.P. (100 + Gain%)
100 × S.P. (100 − Loss%) (ix) If an article is sold at a gain of 35% then S.P. = 135% of C.P. (x) If an article is sold at a loss of 20% then S.P. = 80% of C.P. (xi) When a person sells two similar items, one at a gain of say, x% and the other at a loss of x %, then the seller always incurs a loss given by : (viii)
C.P. =
2
Error Gain % = True value − Error × 100 % l
G.M. = n x1 × x2 × ....... × xn l Harmonic Mean : Harmonic Mean of x1, x2 .....xn is denoted by
If two articles are bought for the same price (i.e. the cost price are equal) and one sold at a profit of x1% and the second is sold at a profit of x2%, then the overall percentage of profit
x1 + x2 = 2 × 100 %
H.M =
PROFIT AND LOSS Formulae : (i) (ii) (iii)
Gain = S.P. – C.P. Loss = C.P. – S.P.
Gain × 100 Gain % = C.P.
2
Common loss and Gain% x Loss% = = 10 10 (xii) If a trader professes to sell his goods at C.P., but uses false weights, then
x =
1 1 1 1 1 + ......... + + n x1 x2 xn l Mean : Relation among AM, GM and HM (GM)2 = (AM) (HM)
Loss × 100 C.P.
(iv)
TRUE DISCOUNT/ DISCOUNT l
Discount = (Market price – Selling price)
MP − SP × 100 MP l T.D. = Interest on the P. W. at a given rate (r) and time (t)
l
Discount % =
T.D. =
P.W. × r × t 100
Important Formulae
(3)
⇒
=
100 × T.D. + T. D. r ×t
P.W. × r × t = P. W. + 100
100 + 1 = T.D. r ×t
r ×t = P. W. 1 + 100
100 + r × t = T.D. r ×t
100 × r × t = P. W. 100
⇒ l
100 × S.D. P.W. = 100 + r × t
⇒
Also S. D. = P. W. + T. D.
l
100 × T.D. P.W. = r ×t S. D. = P. W. + T. D.
A× r ×t 100 + r × t When the sum is put at compound interest, T.D. =
then P.W =
S.I.×T.D Sum = S.I. –T.D.
Amount t
r 1 + 100
[sum due = Amount]
S. I. – T. D. = S. I. on T. D.
PERCENTAGE Fractional Equivalents of Important Percentages 1% =
1 = 0.01 100
2% =
1 = 0.02 50
4% =
1 = 0.04 25
8% =
2 = 0.08 25
16% =
4 25
64% =
16 = 0.64 25
96% =
24 = 0.96 25
5% =
1 = 0.05 20
10% =
1 = 0.1 10
20% =
1 5
40% =
2 = 0.4 5
60% =
3 = 0.6 5
80% =
4 5
120% =
6 = 1.2 5
25% =
1 = 0.25 4
3 1 37 % = 8 2
50% =
1 = 0.5 2
2 1 16 % = 3 6
1 1 33 % = 3 3
1 1 12 % = = 0.125 8 2
100% = 1 1 5 83 % = = 0.833 3 6
1 1 8 % = 3 12
1 1 6 % = 16 4 7 1 87 % = 8 2 2 2 66 % = 3 3
1 4 133 % = 3 3
Actual increase (i) Percentage increase = Original quantity × 100 Actual decrease (ii) Percentage decrease = Original quantity × 100 (iii) If percentage increase is x% then the new value
(
)
x = 1 + 100 × initial value (iv) If x1, x2, x3, x4........are successive percentage of change, then
Final value = [ (1 ± x1 %) (1 ± x2 %) (1 ± x3 %).....] × (initial vlaue)