# Higher Algebra by Hall and Knight

Hall and Knight Higher Algebra Solutions is one of the best study materials to learn and understand Algebra. Higher Algebra by Hall and Knight explicitly cover all the topics in an elaborate manner. After learning all the topics, the best understanding is developed when you practise a lot of questions. Hall and Knight solutions help you by providing the best tips and techniques to do so.

Higher Algebra Solutions cover all significant topics covered in the book such as ratio, proportion, variation, arithmetic progression, geometrical progression, harmonic progression and theorems connected with the progressions, scales of notation, and surds and imaginary quantities. Some more important chapters from this book include the theory of quadratic equations, miscellaneous equations, permutations and combinations, mathematical induction, binomial theorem, multinomial theorem, logarithms, exponential and logarithmic series, interest and annuities, inequalities, limiting values and vanishing fractions, and convergence and divergence of series.

Other vital topics covered here include undetermined coefficients, partial fractions, recurring series, continued fractions, indeterminate equations of the first degree, recurring continued fractions, indeterminate equations of the second degree, the summation of the series, the theory of numbers, the general theory of continued fractions, probability, determinants, miscellaneous theorems and examples, and theory of equations.

## Hall and Knight Higher Algebra Solutions: Chapter-wise

Hall and Knight Higher Algebra Book comprise 35 chapters. All of the questions asked in Hall and Knight Book have been answered comprehensively in our solutions. You will find them easy to understand and it will help you score good marks in class 12 exams and other competitive exams. The contents of the Higher Algebra by Hall and Knight are discussed below in detail.

## Higher Algebra by Hall and Knight Chapter 1 (Ratio) Solutions

This chapter teaches you commensurable and incommensurable quantities, the ratio of greater and less inequality, cross multiplication and eliminant of three linear equations. Ratio chapter has 28 questions. Our smart and systematic Hall and Knight Higher Algebra Solutions consist of answers to all questions in detail.

## Higher Algebra by Hall and Knight Chapter Chapter 2 (Proportions) Solutions

Chapter 2 of Hall and Knight Solutions teaches you about definitions and propositions of proportions, comparison between algebraical and geometrical definitions and the case of incommensurable quantities. There are 26 questions in this chapter. They cover all the concepts of the chapter sufficiently and help you become thorough with the chapter.

## Higher Algebra by Hall and Knight Chapter Chapter 3 (Variation) Solutions

In this chapter, you learn about the conditions if A α B, then A = mB. You also learn about inverse variation, joint variation and if A α B when C is constant, and if A α C when B is constant, then A = mBC and certain examples of joint variation. This chapter consists of 22 questions.

## Higher Algebra by Hall and Knight Chapter 4 (Arithmetical Progression) Solutions

Here you learn about the sum of n terms of an Arithmetical series, fundamental formulae, insertion of arithmetic means and discussion of roots of dn2 + (2a-d)n-2s = 0. This chapter of Hall and Knight Higher Algebra Solutions has 50 questions in 2 exercises.

## Higher Algebra by Hall and Knight Chapter 5 (Geometrical Progression) Solutions

Here you learn about the insertion of geometric means, the sum of n terms of a Geometrical series, the sum of n infinite Geometrical series, proof of rule for the reduction of a recurring decimal and sum of n terms of an arithmetic-geometric series. This chapter has 2 exercises and 51 questions.

## Higher Algebra by Hall and Knight Chapter 6 (Harmonical Progression. Theorems Connected with the Progressions) Solutions

This chapter teaches you about reciprocals of quantities in H.P are in A.P., harmonic mean, formulae connecting A.M, G.M., H.M. You also learn about hints for the solution of questions in progressions. Next, you learn about the sum of squares of the natural numbers and sum of cubes of the natural numbers. Next, you learn about ∑ notation, the number of shots in a pyramid on a square base, a pyramid on a triangular base and on a rectangular base and an incomplete pyramid. This chapter has 38 questions in two exercises.

## Higher Algebra by Hall and Knight Chapter 7 (Scales of notation) Solutions

Chapter 7 of Higher Algebra Book gives you an explanation of systems of notation. You also learn about the expression of an integral number in a proposed scale and expression of a radix fraction in a proposed scale. You also learn to determine the difference between a number and the sum of its digits is divisible by r-1, proof of rule for ‘casting out the nines’, and the test of divisibility by r+1. This chapter contains a total of 56 questions in 2 exercises.

## Higher Algebra by Hall and Knight Chapter 8 (Surds and Imaginary Quantities) Solutions

This chapter teaches you about rationalizing the denominators, rationalizing factors, and finding the square and cube roots of various mathematical quantities. Here you learn about handling some imaginary quantities and come to know that modulus of the product is equal to the product of moduli. You also learn about the square root of a=ib, powers of I, cube roots of unity – 1 + ω + ω2 and powers of ω. Our Hall and Knight Higher Algebra Solutions answer all 75 questions in 2 exercises of this chapter.

## Higher Algebra by Hall and Knight Chapter 9 (The Theory of Quadratic Equations) Solutions

In this chapter, you come to know that a quadratic equation cannot have more than two roots, conditions for real, equal and imaginary roots. You also come to know that sum of roots = -b/a and product of roots = c/a. You also learn here about the formation of equations when the roots are given, conditions that the roots of a quadratic should be (a) equal in magnitude and opposite in sign, (b) reciprocals. Next, you learn definitions of function, variable, rational integral function. Here you also learn about the conditions when ax2 + bx + c = 0 and aʹx2 + bʹx + cʹ = 0 may have a common root. Here you will find 54 questions in 3 exercises.

## Higher Algebra by Hall and Knight Chapter 10 (Miscellaneous Equations) Solutions

Chapter 10 contains questions based on equations involving one unknown quantity, reciprocal equations, equations involving two unknown quantities, homogeneous equations, equations involving several unknown quantities and indeterminate equations, and easy numerical examples. This chapter of Hall and Knight Higher Algebra Book contains 129 mixed questions in 4 exercises. Our Hall and Knight Higher Algebra Solutions answer all questions in details and cover all corners of the chapter.

## Higher Algebra by Hall and Knight Chapter 11 (Permutations and Combinations) Solutions

Higher Algebra chapter 11 teaches you about preliminary propositions, the number of permutations of n things r at a time, the number of combinations of n things r at a time. You also learn about the number of combinations of n things r at a time is equal to the number of combinations of n things n-r art a time. This chapter also teaches you about the number of ways in which m + n + p + …. things can be divided into classes containing m, n, p, …things severally.

Further, this chapter teaches you signification of the terms ʹlikeʹ and ʹunlikeʹ, the number of arrangements of n things taken all at a time, when p things are alike of one kind, q things are alike of a second kind. You also learn about the number of permutations of n things r at a time, when each may be repeated and the total number of combinations of n things. Next, you learn about the total number of combinations of n things. You will find 67 questions in 2 exercises in this chapter.

## Higher Algebra by Hall and Knight Chapter 12 (Mathematical Induction) Solutions

This chapter teaches you about illustrations of the methods of proof and product of n binomial factors of the form x + a. Our well-structured Hall and Knight Higher Algebra Solutions contain detailed answers to all 5 questions of this chapter.

## Higher Algebra by Hall and Knight Chapter 13 (Binomial Theorem: Positive Integral Index) Solutions

Chapter 13 teaches you about the expansion of (x + a)n, when n is a positive integer, general term of the expansion, the expansion may be made to depend upon the case in which the first term is unity, second proof of the binomial theorem. Further, you learn that the coefficients of terms equidistant from the beginning and end are equal. You also learn about, determination of the greatest term, the sum of the coefficients, the sum of coefficients of odd terms is equal to the sum of coefficients of even terms and expansion of multinomials. Here 58 questions are given in 2 exercises. Higher Algebra Solutions answer all questions comprehensively and an easy to understand language.

## Higher Algebra by Hall and Knight Chapter 14 (Binomial Theorem: Any Index) Solutions

This chapter teaches you the Euler’s proof of the binomial theorem for any index, the general term of the expansion of (1 + x)n, expansion of (1 + x)n is only arithmetically intelligible when x < l. You also come to know that the expression (x /y)n can always be expanded by the binomial theorem, general term of the expansion of (1 – x)-n. You also learn about particular cases of the expansions of (1 – x)-n, approximations obtained by the binomial theorem, numerically greatest term in the expansion of (l + x)n, the number of homogeneous products of r dimensions formed out of n letters, the number of terms in the expansion of a multinomial, the number of combinations of n things r at a time. This chapter has 85 questions in 3 exercises. Higher Algebra Solutions by Hall and Knight answer all questions in a stepwise manner in an easy to understand manner.

## Higher Algebra by Hall and Knight Chapter 15 (Multinomial Theorem) Solutions

In this chapter you learn all about the general term in the expansion of (a + bx + cx2 + dx3 +…)p, when p is a positive integer and general term in the expansion of (a + bx + cx2 + dx3 +…)n when n is a rational quantity. This chapter of Hall and Knight Higher Algebra Book has 21 questions.

## Higher Algebra by Hall and Knight Chapter 16 (Logarithms) Solutions

In this chapter of our Hall and Knight Higher Algebra book, you learn the definition of the logarithm, the relation *N* = *a*loga*N*, and elementary propositions. You also learn about common logarithms, determination of the characteristic by inspection, advantages of logarithms to base 10, advantages of always keeping the mantissa positive and how to find the logarithms to base b, given the logarithms of all numbers to base a. Our Solutions explicitly answer all 56 questions of the 2 exercises of this book.

## Higher Algebra by Hall and Knight Chapter 17 (Exponential and Logarithmic Series) Solutions

In this chapter you learn about the expansion of ax, series for e when e is the limit of (1 + 1/n)n, when n is infinite. You also learn about the expansion of loge(1 + x), construction of tables of logarithms, rapidly converging series for loge(n + 1) – loge n and that the quantity e is incommensurable. This chapter has 24 questions.

## Hall and Knight Higher Algebra Chapter 18 (Interest and Annuities) Solutions

Interest and Annuities chapter teaches you about interest and amount of a given sum at simple interest, present value and discount of a given sum at simple interest, interest and amount of a given sum at compound interest, nominal and true annual rates of interest, case of compound interest payable every moment, present value and discount of a given sum at compound interest.

Further, you learn about annuities and related definitions, amount of unpaid annuity, simple interest, amount of unpaid annuity, compound interest, present value of an annuity, compound interest, number of years’ purchase, present value of a deferred annuity, compound interest and fine for the renewal of a lease. This chapter has 25 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 19 (Inequalities) Solutions

Inequalities chapter of Higher Algebra by Hall and Knight teaches you about elementary propositions, arithmetic mean of two positive quantities is greater than the geometric mean, the sum of two quantities being given, their product is greatest when they are equal: the product being given, the sum is least when they are equal.

Next, you get to know about the arithmetic mean of a number of positive quantities is greater than the geometric mean, how to find the greatest value of ambncp with a given sum of a, b, c, …. You also learn about easy cases of maxima and minima; the arithmetic mean of the mth powers of a number of positive quantities is greater than the mth power of their arithmetic mean except when m lies between 0 and 1. Here you will find 42 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 20 (Limiting Values and Vanishing Fractions) Solutions

Chapter 20 teaches you the definition of limit and limit of a0 + a1*x* + a2*x*2 + a3*x*3 + …. Is a0 when *x* is zero. You also learn that by taking *x* small enough, any term of the series a0 + a1*x* + a2*x*2 + a3*x*3 + …. may be made as large as we please compared with the sum of all that follow it; and by taking *x* large enough, any term may be made as large as we please compared with the sum of all that precedes it.

Next, you learn about the method of determining the limits of vanishing fractions, discussion of some peculiarities in the solution of simultaneous equations and peculiarities in the solution of quadratic equations. This chapter has 18 questions.

## Hall and Knight Higher Algebra Chapter 21 (Convergency and Divergency of Series) Solutions

This chapter of Hall and Knight Solutions teaches you the case of terms alternately positive and negative and that a series is convergent if Lim *u*n / (*u*n -1) is less than 1. You also learn about auxiliary series and application of Binomial, Exponential and Logarithmic series and product of an infinite number of factors. It also teaches you about the condition for a series to be convergent and to be compared with other series and products of two infinite series. This chapter contains 34 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 22 (Undetermined Coefficients) Solutions

This chapter teaches you that if the equation f(*x*) = 0 has more than n roots, it is an identity. You also learn about proof of principle of undetermined coefficients for finite series and proof of principle of undetermined coefficients for infinite series. This chapter has 27 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 23 (Partial Fractions) Solutions

Partial Fractions chapter teaches you about decomposition into a partial fraction and use of partial fractions in expansions of equations. Chapter 23 of Higher Algebra Book consists of 26 questions.

## Hall and Knight Higher Algebra Chapter 24 (Recurring Series) Solutions

Here you learn about the scale of relation, the sum of a recurring series and about the generating function in recurring series. This chapter has 15 questions.

## Hall and Knight Higher Algebra Chapter 25 (Continued Fractions) Solutions

Higher Algebra Continued Fractions chapter teaches you the conversion of a fraction into a continued fraction, and that convergents are alternately less and greater than the continued fraction, law of formation of the successive convergents. You also learn that the convergents gradually approximate to the continued fraction, limits of the error in taking any convergent for the continued fraction and that each convergent is nearer to the continued fraction than a fraction with smaller denominator. You will find a total of 31 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 26 (Indeterminate Equations of the First Degree) Solutions

This chapter teaches you that the solution of *ax* – *by* = *c*, it also teaches you how to find the general solution when there is one solution given, the solution of *ax* + *by* = *c* and the general solution when there is one solution given. Next you learn the number of solutions of *ax* + *by* = *c *and the solution of *ax + by + cz = d*, *a’x + b’y + c’z = d’.* There are 24 questions in this chapter of Hall and Knight Higher Algebra Solutions.

## Hall and Knight Higher Algebra Chapter 27 (Recurring Continued Fractions) Solutions

In this chapter, you first find some numerical examples of continued fractions. Further, you learn that a periodic continued fraction is equal to a quadratic surd and about conversion of a quadratic surd into a continued fraction. Next, you learn that the quotients recur, the period ends with a partial quotient 2a1, the partial quotients equidistant from first and last are equal and the penultimate convergents of the periods. There are 49 total questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 28 (Indeterminate Equations of the Second Degree) Solutions

In this chapter you learn that the solution of *ax*2 + 2*hxy* + *by*2 + 2*gx* + 2*fy* + *c* = 0, the equation *x*2 – *Ny*2 = 1 can always be solved. Next you learn about the solution of* x*2 – *Ny*2 = -1, general solution of *x*2 – *Ny*2 = 1, solution of *x*2 – *n*2*y*2 = a and diophatine problems. This chapter of Hall and Knight Higher Algebra Book consists of 22 questions.

## Hall and Knight Higher Algebra Chapter 29 (Summation of Series) Solutions

This chapter firstly tells you the summary of previous methods. You learn to calculate *u*n the product of *n* factors in A.P., *u*n the reciprocal of the product of *n* factors in A.P., method of subtraction, expression of *u*n as sum of factorials, polygonal and figurate numbers, and Pascal’s triangles. Then you learn about the method of differences, and the method succeeds when *u*n is a rational integral function of *n* and further cases of recurring series. You also learn here about Bernoulli’s numbers. There are 85 questions in 3 exercises of this chapter.

## Hall and Knight Higher Algebra Chapter 30 (Theory of Numbers) Solutions

Theory of Numbers chapter of Higher Algebra by Hall and Knight, you learn the statement of principles. You also learn that number of primes is infinite, and no rational algebraical formula can represent primes only, a number can be resolved into prime factors in only one way, number of divisors of a given integer, number of ways an integer can be resolved into two factors, and the sum of the divisors of a given integer.

You also learn Fermat’s theorem, the definition of congruent and Wilson’s theorem. There are 2 exercises and 59 questions in this chapter.

## Hall and Knight Higher Algebra Chapter 31 (The General Theory of Continued Fractions) Solutions

Chapter 31 tells you about the law of formation of successive convergents. It also tells you about convergent may have positive proper fractions in ascending order of magnitude, general values of convergent when *a*n and *b*n are constant. You also learn about cases where the general value of convergent can be found, and series expressed as continued fractions and conversion of one continued fraction into another. This chapter has a total of 33 questions in 2 exercises.

## Hall and Knight Higher Algebra Chapter 32 (Probability) Solutions

Higher Algebra Probability chapter gives you definitions and illustrations of simple and compound events. Next, you learn about the probability that two independent events will both happen is *pp*‘, and that the given formula also holds for dependent events. Further, you get to know the chance of an event which can happen in mutually exclusive ways, the chance of an event happening exactly r times in n trials, expectation and probable value – “Problem of points”.

You also learn about inverse probability, statement of Bernoulli’s theorems, concurrent and traditionary testimony, and local probability by geometrical methods. There are 109 questions in 5 exercises of this chapter.

## Hall and Knight Higher Algebra Chapter 33 (Determinants) Solutions

In Hall and Knight Determinants chapter, you come to know about elimination of two and three homogeneous linear equations. You also learn that the determinant is not altered by interchanging rows and columns, the development of determinant of the third order, and that the sign of a determinant is altered by interchanging two adjacent rows or columns. Further, you learn that if two rows or columns are identical, the determinant vanishes, a factor common to any row or column may be placed outside and about cases where constituents are made up of a number of terms.

You also learn about the reduction of determinants by simplification of rows or columns, the product of two determinants, application to the solution of simultaneous equation and determinant of fourth/any order and notations. There are 46 questions in 2 exercises of this chapter.

## Hall and Knight Higher Algebra Chapter 34 (Miscellaneous Theorems and Examples) Solutions

Chapter 34 reviews the fundamental laws of Algebra and the method of detached coefficients. It also discusses Horner’s method of synthetic division, symmetrical and alternating function, and worked-out examples of identities. Further, you learn here about identities proved by properties of cube roots of unity, elimination, elimination by symmetrical functions, Euler’s method of elimination, Sylvester’s Dialytic Method, Bezout’s method and miscellaneous examples of elimination. This chapter of Hall and Knight Higher Algebra consists of 84 questions given in 3 exercises.

## Hall and Knight Higher Algebra Chapter 35 (Theory of Equations) Solutions

You learn in Theory of Equations chapter that every equation of the nth degree has n roots and no more, relations between the roots and the coefficients are not sufficient for the solution, cases of a solution under given conditions, easy cases of symmetrical functions of the roots, imaginary and surd roots occur in pairs, formation and solution of equations with surd roots, Descartes’ rule of signs, derived functions, and calculation of f(x+h) by Horner’s process.

Further, you learn that an equation of an odd degree has one real root, an equation of an even degree with its last term negative has two real roots, determination of equal roots, the sum of an assigned power of the roots, transformation of equations. Next, you learn some discussion of reciprocal equations, removal of an assigned term, cubic equations, Cardan’s solution, discussion of the solution, solution by trigonometry in the irreducible case. You also learn about Biquadratic Equations, Ferrari’s Solution, and Descartes’ solution and undetermined multipliers. There are 5 exercises and 130 questions in this chapter.

In the end, the Hall and Knight Higher Algebra book contains a total of 300 Miscellaneous Examples covering all the concepts explained in the book. Our Hall and Knight Higher Algebra Solutions cover them all with great conceptual length.