DEMYSTIFIEDFOUNDATIONS

OF

UNIVERSITY MATHEMATICS

© M.B.Abdullahi, 2015

All Rights Reserved

No part of this Publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system, without permission in writing from the Author.

ISBN:978 – 978 – 484 – 014-9

Published by Clear Resolutions,

Zaria, Kaduna State, Nigeria.

e-mail: kabzeesani@yahoo.com

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To my best friend since early years: Dr. Usman Said Bojude

Table of Contents

Title Page………………………………………………………………….............................i

Copy Right Page……………………………………………………… ….........................ii

Dedication……………………………………………………………........................…….iii

Table of Contents.......................................................................................................iv

Acknowledgements……………………………………………………........................ ix

Preface………………………………………………………………....................................x

Introduction……………………………………………………………….......................xii

Chapters:

1.0. LAWS OF INDICES..............................................................................................1

(a)Laws of Indices

2.0.SURDS....................................................................................................................12(a)Simplifications of Surds (b)Division of Surds(c)Rationalization of Numerator (d)Rationalization of Denominator by Conjugation

3.0.LOGARITHM OF NUMBERS..........................................................................20

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(a) Axioms of Logarithm (b)Logarithm of Numbers Using Table(c)Working with Bars

4.0.VARIATIONS.......................................................................................................33

(a)Types of Variations (1)Direct (2)Inverse(3)Join(4)Partial

5.0.ALGEBRA.............................................................................................................42

(a)Addition and Subtraction(b)Multiplication(c)Division(d)Difference of Two Squares(e)Pascal Triangle(f)Zero Fractions and Undefined Fractions

6.0.SUBJECT FORMULA.........................................................................................52

7.0.LINEAR INEQUALITY .....................................................................................59

(a)Line Graph(b)Cartesian Graph(c)Word Problems Leading to Linear Inequality

8.0.TRIGONOMETRY............................................................................................72

(a)SOHCAHTOA, (b)Quadrant(c)Special Angles(d)Radiant/ Degree Measure(e)Cotangent, Secant and Cosecant(f)Cosine as Counter Sine(g)Law of Sine(h)Law of Cosine(i)Sketching Sinusoidal Graphs(j)Trigonometric Identities(k)Sum and Difference Angles Formula(l)Double and Halve Angles Formula

9.0.SEQUENCE AND SERIES.................................................................................96

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9.1.Types of Sequences(a)A.P.(b)Sum of an A.P.(c)G.P.(d)Sum of G.P.(e)Sum to Infinity

10.0.QUADRATIC EQUATIONS........................................................................109

10.1Methods of Solving Q.E.(a)Factorization Method(b)Completing the Square Method(c)Formula Method(d)Graphical Method

11.0.SIMULTANEOUS EQUATIONS................................................................135

11.1Methods(a)Elimination(b)Substitution(c)Graphical (d)Inverse Matrix Method(e)Crammer’s Rule

12.0.PROBABILITY.............................................................................................148

(a)Coin Experiment(b)Dice(c)Cards(d)Box (e)Types of Probability(f)Addition and Multiplication of Probability(g)Probability with or without Replacement(h)Probability of Success and Failure(i)Mutually Exclusive Events(j)Conditional Probability(k)Axioms of Probability

13.0.SET THEORY..................................................................................................164

(a)Definitions of Terms(b)Operations with Venn Diagram (c)Tabular Form of Sets(d)Set Builder Form of Sets

14.0.LONGITUDES AND LATITUDES......................................................... 187

(a)Longitude Formula

15.0.BEARING AND DISTANCES ................................................................. 197

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(a)Alternate and Corresponding Angles(b)Cardinality of Bearings (c)Due Concept

16.0.CONSTRUCTION........................................................................................ 215

(a)Construction of Angles(b)Locus of a Point, 2 points and a Line (c)Inscribed and Outscribed Circle in a Triangle(d)Construction of Angles Multiple of 10

17.0.NUMBER SYSTEM...................................................................................... 233

(a)Conversion of Bases(b)Conversion of Other Bases to Base 10(c)Conversion of Another Base to Another(d)Conversion by Grouping Method

18.0.STATISTICS..................................................................................................243

(a)Importance of Statistics(b)Applications of Statistics(c)Data(d)Types of Averages

19.0.THEORY OF LOGIC ...............................................................................247

(a)Sentences(b)Simple Statements(c)Examples of Compound Statement(d)Some Logical Connectives(e)Boolean Polynomial

20.0OPERATIONS RESEARCH........................................................................257

(a)Steps in O.R.(b)Models of O.R.(c)Linear Programming Problems(LPP)(d)Simplex Method(e)Transportation Model(f)Assignment Model

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21.0DIFFERENTIATIONS .................................................................................273

(a)Chain Rule(b)Product Rule(c)Quotient Rule(d)Implicit Function(e)Function of a Function (f)Trigonometric Function(g)Differentiation of e and log(h)Differentiation of Some Typical Functions(i)Differentiation of Infinite Terms

22.0. INTEGRATIONS...........................................................................................286

(a)Integration Formula (b)Integration by Part(c)Order of Integration(d)Integration by Substitution

23.0.MATRICES....................................................................................................290

(a)Types of Matrices (b)Equality of Matrices(c)Operations on Matrices(d)Scalar Multiplication(e)Transpose of Matrices(f)Multiplication of Matrices(g)Adjoint of Matrices(h)Determinant of Matrices(i)Inverse of Matrices(j)Solutions of System of Linear Equations(k)Crammer’s Rule

24.0THEORIES ON NUMBERS........................................................................307

24.1 Classification of Numbers (a)Natural Numbers (b)Integer Numbers (c) Rational Numbers (d) Real Number (e) Extended Real Numbers 24.2 Word Alternatives for Numbers 24.3 Some Patterns of Numbers 24.4 Loubere Magic Squares

ANSWERS TO EXERCISES .................................................................................341

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Acknowledgements

My warm extension ofdeep and profound gratitude to: The present Governor of Gombe State: His Excellency, Alh. Ibrahim Hassan DanKwambo, (OFN); the former Vice-Chancellor of the Gombe State University: Prof. Abdullahi Mahdi(CON); the Vice-Chancellor of the Federal University, Kashere: Prof. Mohammed Kabir Farook; the former D.G.of the NMC, Abuja: Prof. Sam. O. Ale and the present D.G.: Prof. A.R.T. Solarinof the National Mathematical Centre, Abuja; the Rector: Jigawa State Polytechnic: Prof. G.U.Garba, the Commissioner for Education, Gombe State: Mrs Aisha Ahmed and the Honourable Commissioner for Higher Education: Dr. Isah Muhammadu Wade.

I have to record thanks to the following people: The VC of the Gombe State University: Prof. Ibrahim Umar(OON), the Registrar of the Federal University, Kashere: Dr. Abubakar Aliyu Ba Feto, the Chiarman, Tinka Point, Gombe State: (Dr) Bala Bello Tinka, the Honourable Speaker of Gombe State: Alh. Inuwa Garba, the former Dean Faculty of Science, FUK,Gombe: Dr Mohammed Sani Gumel and the Dean Faculty of Science, FUK,Gombe: Prof. M.B. Abdullahi and the SSG, Gombe State.

I equally owed thanks to: The Gombe State University Alumni Association; the Gombe International School; the Chairman of Akko Local Government Area; Alh. Suleiman Yahuza of Ashaka Cement Company; Alh. Y.M.Baba; a Permanent Secretary; Imam Abubakar Lamido of Bolari Central Mosque; Dr Jaudo of FOS, FUK,Gombe; Dr Mishra of GSU, Gombe; and Americanah, my Geometry Teacher, of the NMC, Abuja.

I feel gratified to trust and respect the following sheikhs/scholars: Adamu Dakoro, Tahir Inuwa Ibrahim, Yahaya Bello, Musa Adam Mai Hula, Isah Ali

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Pantami, Ibrahim Zamfara, Muhammad Lawan, Aminu Daurawa, Ash Ali,and Dogara Aliyu Kaya.

To my publisher: To hide knowledge to the Blacks, put it into writing. It is a conventional wisdom that students read for only examination. Trying and failing is better than failing to try.

Preface

Many students involving themselves into the mathematics programmes

outside Nigeria used to be inspired that there is need for improvement in

the Nigerian Syllabus. This is my first contribution. The next volume will

be on Miscellaneous Problems in Geometry, and the third one will be on

Further Mathematics for Secondary Schools. My job, along with other

things however also importantly, is originality; and non verbatim

virtuosoing.

This is the first book on production in line with the present review on the

subject matter.Hence, terminologies chosen on the present subject are

being simplified. The book covered the new additional syllabus of the

Nigerian Research and Development Council, Sheda-Kwali, Abuja:

Matrices, Operations Research, and Calculus.

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It will interest the user of this material to realize that 100% of the

examples presented in the book cannot be found exactly in any other O-

Level Textbook on the globe, 100% of the exercises cannot be found in

any other O-Level Textbook in Nigeria, less than 20% of the book was not

written off heart. The Operations Research constituted more than 80% of

the 20%. It is included in this book to cover the new additions.

It will also interest the user of this material to realize that this is the first

book on the Earth to: Convert00 to 00 , discuss nature of roots of the

equations of motion s=ut+12a t 2 and its like, observe that n distributes in

s= n2(2a+(n−1)d ) quadraticly and other equations of its like, present

Fulani Counting Number System, discover fastest and accurate new

method of tackling logarithm of numbers less than one, demonstrate

constructions of angles multiples of approximely 100(mathematicians

concluded that it is not possible(from the application of extension field

equation, but the author constructed their approximates) ),and define

equality”=” as a verb.

This is also the first book in Nigeria to: Demonstrate practical

applications of variations to physics equations, demonstrate the xi

differences of distances in sketch(to avoid common mistakes in Bearings

and Distances), include both Grouping Method of Conversions(this

appeared in only very few Nigerian O-Level textbooks), and Conversions

of Fractional Bases(this also appeared in only very few O-Level Textbooks

), define harmonic, quadratic and generalized f – mean, use Inverse Matrix

Method, go with practical applications, use a different proof of Mighty

Formulae and graph quadratic equations on y-axis, include both Dice and

Card Probabilities. These are needed for successful University

Mathematics Honourary Programmes.

M.B.Abdullahi

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Introduction

Mathematics is not a popular subject, even though its importance may be

generally conceded. The reason for this is to be found in the common

superstition that Mathematics is but a continuation, a further

development, of Fine Art of Arithmetics, of juggling with numbers (David

Hilbert and S. Cohn-Vossen).

You need to muster O-Level Mathematics including miscellaneouses to be

sure of having a background for University Mathematics.You should get

inside of miscellaneous topics very reasonably for many mystic

miscellaneous problems do manifest in undergraduate university

Mathematics Programmes.

At the first glance, the problems in this book may appear really mystic

and unreasonable; but after delving into the solutions of the problems,

you may realize that it is just a common trick. These tricks are

widespread in A-Level Mathematics Texts.

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Can a Mathematician See Red?

Consider the sphere

A hollow rounded surface

With no thickness.

Each point that we see

From the outside

Is also a point we see can see

From the inside.

If I paint red

All over the outside,

Is the inside red?

The mathematician says NO,

For the layer of paint

Form a new sphere

That is new sphere

That is outside the outside

And not inside

A mathematician takes safe pleasure

In surface mysteries

A poet will see red inside.

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Joanne Growney,

Department of Mathematics and Computer

Science,

Blooms Borg University, Bloomsburg.

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