1. Number system:

(i) A number $x$ is called a rational number, if it can be written in the form $\frac{m}{n}$ , where $m & n$ are integers and $n \neq$ 0.

(ii) A number $x$ is called an irrational number, if it cannot be written in the form $\frac{m}{n}$ , where $m$ and $n$ are integers and $n \neq 0$ .

(iii) The decimal expansion of a rational number is either terminating or non-terminating recurring. Thus, we say that a number whose decimal expansion is either terminating or non-terminating recurring is a rational number.

(iv) The decimal expansion of an irrational number is non-terminating and non-recurring. Thus, we say that a number whose decimal expansion is non-terminating and nonrecurring is an irrational number.

(v) All the rational and irrational numbers taken together make up the collection of real numbers.

(vi) A real number is either rational or irrational.

(vii) There is a real number corresponding to every point on the number line. Also, corresponding to every real number there is a point on the number line.

(viii) If $r$ is rational and $s$ is irrational, then $r + s, r - s, r s$ and $\frac{r}{s}$ are irrational numbers.

(ix) If $n$ is a natural number other than a perfect square, then $\sqrt{n}$ is an irrational number.

If $' a'$ is a positive real number and $n$ is a positive integer, then the principal nth root of $' a'$ is the unique positive real number $x$ such that $x^{n} = a$ . The principal $n^{t h}$ root of a is denoted by $a^{1 / n} or \sqrt[n]{a}$ .

(x) Every composite number can be uniquely expressed as the product of powers of primes in ascending or descending order.

(xi) Let $a$ be a positive integer and $p$ be a prime number such that $p ∣ a^{2}$ then $p ∣ a$ .

(xii) There are infinitely many positive primes.

(xiii) Every positive integer different from $1$ can be expressed as a product of non-negative power of $2$ and an odd number.

(xiv) A positive integer $n$ is prime, if it is not divisible by any prime less than or equal to $\sqrt{n}$ .

(xv) If $p$ is a positive prime, then $\sqrt{p}$ is an irrational number. For example, $\sqrt{2}, \sqrt{3}$ $\sqrt{5}, \sqrt{7}, \sqrt{11}$ etc. are irrational numbers.

(xvi) Let $x = \frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^{m} \times 5^{n}$ where m, n are non-negative integers. Then, $x$ has a terminating decimal expansion which terminates after $k$ places of decimals, where $k$ is the larger of $m$ and $n$ .

(xvii) Let $x = \frac{p}{q}$ be a rational number, such that the prime factorisation of $q$ is not of the form $2^{m} \times 5^{n}$ , where $m, n$ are non-negative integers. Then, $x$ has non-terminating repeating decimal expansion.

2. Ratio and Proportion:

(i) Ratio: The ratio defines the relation between two quantities such as $a : b$ , where $a, b$ is not equal to $0$ . The sign for the ratio is $' :'$ . The first term is called the antecedent, the second term the consequent.

Example: The ratio of $5$ to $15$ is represented as $5 : 15 = 1 : 3$ . The comparison or simplest form of two quantities of the same kind is referred to as ratio. This relation gives us how many times one quantity is equal to the other quantity.

(ii) Some important properties of ratios:

(a) The ratio exists between the quantities of the same kind. While comparing two quantities, the units should be the same.

(b) The ratio of two fractions can be expressed as a ratio of two integers. Thus the ratio $\frac{a}{b} : \frac{c}{d}$ is measured by the fraction $\frac{a d}{b c}$ .

(c) If either, or both of the terms of a ratio be a surd quantity, then no two integers can be found which will exactly measure their ratio. Thus the ratio $\sqrt{2} : 1$ cannot be exactly expressed by any two integers.

(d) If the ratio of any two quantities can be expressed exactly by the ratio of two integers, the quantities are said to be commensurable, otherwise, they are said to be incommensurable.

(e) Ratios are compounded by multiplying together the fractions which denote them or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent.

(f) When the ratio $a : b$ is compounded with itself the resulting ratio is $a^{2} : b^{2}$ , and is called the duplicate ratio of $a : b$ . Similarly $a^{3} : b^{3}$ is called the triplicate ratio of $a : b$ . Also $a : b$ is called the sub duplicate ratio of $a^{2} : b^{2}$ .

(g) A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent.

(iii) Proportion: When two ratios are equal, the four quantities composing them are said to be proportional. Thus if $\frac{a}{b} = \frac{c}{d},$ then $a, b, c, d$ are proportionals. This is expressed by saying that $a$ is to $b$ as $c$ is to $d,$ and the proportion is written as $a : b : : c : d$ or $a : b = c : d .$

(a) The terms $a$ and $d$ are called the extremes, $b$ and $c$ the means.

(b) If four quantities are in proportion, the product of the extremes is equal to the product of the means.

(d)Then by definition $\frac{a}{b} = \frac{c}{d}$

(iv) Invertendo: If $a : b = c : d,$ then $b : a = d : c$

(v) Alternando: If $a : b = c : d,$ then $a : c = b : d$

(vi) Componendo: If $a : b = c : d,$ then $a + b : b = c + d : d$

$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a}{b} + 1 = \frac{c}{d} + 1 \Rightarrow \frac{a + b}{b} = \frac{c + d}{d}$

(vii) Dividendo: If $a : b = c : d,$ then $a - b : b = c - d : d$

$\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a}{b} - 1 = \frac{c}{d} - 1 \Rightarrow \frac{a - b}{b} = \frac{c - d}{d}$

(viii) $.$ Componendo and Dividendo: If $a : b = c : d,$ then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$

3. Surds and Indices:

(i) Any expression of the form $\frac{a}{\sqrt{b} + \sqrt{c}}$ can be rationalised by multiplying the numerator and the denominator by $\sqrt{b} - \sqrt{c}$ , the surd conjugate to denominator.

(ii) Similarly in the case of a fraction of the form $\frac{a}{\sqrt{b} + \sqrt{c} + \sqrt{d}}$ , first multiply both numerator and denominator by $\sqrt{b} + \sqrt{c} - \sqrt{d}$ , then denominator becomes ${(\sqrt{b} + \sqrt{c})}^{2} - {(\sqrt{d})}^{2}$ or $b + c - d + 2 \sqrt{b c}$ . Then multiply both numerator and denominator by $b + c - d - 2 \sqrt{b c}$

For e.g. Simplify $\frac{12}{3 + \sqrt{5} - 2 \sqrt{2}}$ .

$= \frac{12 (3 + \sqrt{5} + 2 \sqrt{2})}{{(3 + \sqrt{5})}^{2} - {(2 \sqrt{2})}^{2}}$

$= \frac{12 (3 + \sqrt{5} + 2 \sqrt{2})}{6 + 6 \sqrt{5}}$

$= \frac{2 (3 + \sqrt{5} + 2 \sqrt{2}) (\sqrt{5} - 1)}{(\sqrt{5} + 1) (\sqrt{5} - 1)}$

$= \frac{2 + 2 \sqrt{5} + 2 \sqrt{10} - 2 \sqrt{2}}{2}$

$= 1 + \sqrt{5} + \sqrt{10} - \sqrt{2}$ .

(iii) If $a, b$ are positive real numbers and $m, n$ are rational numbers, then

(a) $a^{m} \times a^{n} = a^{m + n}$

(b) $\frac{a^{m}}{a^{n}} = a^{m - n}$

(d) $a^{- n} = \frac{1}{a^{n}}$

(e) $(a b)^{n} = a^{n} b^{n}$

(f) ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

(g) $a^{\frac{m}{n}} = {(a^{m})}^{\frac{1}{n}} = {(a^{\frac{1}{n}})}^{m} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}, m, n \in N$

4. Algebraic inequalities

(i) A polynomial inequality is simply a type of equation which does not have an equal sign. Solving polynomial inequalities is same as solving polynomial equations.

(ii) Rational algebraic inequation: If $P (x)$ and $Q (x)$ are polynomial in $x$ , then the inequation $\frac{P (x)}{Q (x)} > 0, \frac{P (x)}{Q (x)} < 0, \frac{P (x)}{Q (x)} \geq 0$ and $\frac{P (x)}{Q (x)} \leq 0$ are known as rational algebraic inequations.

(iii) To solve these inequations we use the sign method as

(a) Obtain $P (x)$ and $Q (x)$ .
(b) Factorize $P (x)$ and $Q (x)$ into linear factors.
(c) Make the coefficient of x positive in all factors.
(d) Obtain critical points by equating all factors to zero.
(e) Plot the critical points on the number line. If these are n critical points, they divide the number line into $(n + 1)$ regions.
(f) In the right most region the expression $\frac{P (x)}{Q (x)}$ bears positive sign and in other region the expression bears positive and negative signs depending on the exponents of the factors.

(iv) Wavy Curve Method:

Let $f (x) = {(x - a_{1})}^{k_{1}} {(x - a_{2})}^{k_{2}} {(x - a_{3})}^{k_{3}} \dots {(x - a_{n - 1})}^{k_{n - 1}} {(x - a_{n})}^{k_{n}}$

where $k_{1}, k_{2}, k_{3}, \dots k_{n} \in N$ and $a_{1}, a_{2}, a_{3}, \dots ., a_{n}$ are fixed natural numbers satisfying the condition $a_{1} < a_{2} < a_{3} < \dots < a_{n - 1} < a_{n}$ .

First we mark the numbers $a_{1}, a_{2}, a, \dots, a_{n}$ on the real axis and the plus sign in the interval of the right of the largest of these numbers, i.e., on the right of $a_{n}$ .

If $k_{n}$ is even, we put plus sign on the left of $a_{n}$ and if $k_{n}$ is odd, then we put minus sign on the left of $a_{n}$ In the next interval we put a sign according to the following rule.

When passing through the point $a_{n - 1}$ the polynomial $f (x)$ ) changes sign. if $k_{n - 1}$ is an odd number and the polynomial $f (x)$ has same sign if $k_{n - 1}$ is an even number. Then, we consider the next interval and put a sign in it using the same rule.

Thus, we consider all the intervals. The solution of $f (x) > 0$ is the union of all interval in which we have put the plus sign and the solution of $f (x) < 0$ is the union of all intervals in which we have put the minus Sign.

5. Partial fractions:

Partial Fractions of $\frac{f (x)}{g (x)}$ , when $g (x)$ Contains Non–repeated Linear factors

$\frac{p x + q}{(x - a) (x - b)} = \frac{A}{(x - a)} + \frac{B}{(x - b)}$ , $A$ and $B$ are real numbers and their values should be determined suitably.

Partial Fractions of $\frac{f (x)}{g (x)}$ , where Both $f (x)$ and $g (x)$ are Polynomials

$\frac{p x^{2} + q x + r}{{(x - a)}^{2} (x - b)}$ = $\frac{A}{(x - a)} + \frac{B}{{(x - a)}^{2}} + \frac{C}{(x - b)}$

$\frac{p x^{2} + q x + r}{(x - a) (x^{2} + b x + c)}$ = $\frac{A}{(x - a)} + \frac{B x + C}{(x^{2} + b x + c)}$ , Where $x^{2} + b x + c$ can not be factorised further and $A, B$ and $C$ are real numbers and their values should be determined suitably.

Partial Fractions of $\frac{f (x)}{f (x)}$ , when $g (x)$ Contains Repeated and/or Non-repeated Linear Factors

$\frac{p x + q}{{(x - a)}^{2}}$ = $\frac{A}{(x - a)} + \frac{B}{{(x - a)}^{2}}$

$\frac{p x^{2} + q x + r}{{(x - a)}^{2} (x - b)}$ = $\frac{A}{(x - a)} + \frac{B}{{(x - a)}^{2}} + \frac{C}{(x - b)}$ , Where $A, B,$ and $C$ are real numbers and their values should be determined suitably.

Partial Fractions of $\frac{f (x)}{g (x)}$ , when $g (x)$ Contains Irreducible Factors

$\frac{p x^{2} + q x + r}{(x - a) (x^{2} + b x + c)}$ = $\frac{A}{(x - a)} + \frac{B x + C}{(x^{2} + b x + c)}$ , Where $x^{2} + b x + c$ cannot be factorised further and $A, B$ and $C$ are real numbers and their values should be determined suitably.

6. Variation:

One quantity A is said to vary directly as another $B$ , when the two quantities depend upon each other in such a manner that if $B$ is changed, $A$ is changed in the same ratio.

Note. The word directly is often omitted, and $A$ is said to vary as $B$ .

For instance: if a train moving at a uniform rate travels $40$ $miles$ in $60$ $minutes$ , it will travel $20$ $miles$ in $30$ $minutes$ , $80$ $miles$ in $120$ $minutes$ , and so on; the distance in each case being increased or diminished in the same ratio as the time. This is expressed by saying that when the velocity is uniform the distance is proportional to the time, or the distance varies as the time.

(i) The symbol $\propto$ is used to denote variation; so that $A \propto B$ is read “ $A$ varies as $B$ ”.

(ii) If $A$ varies as $B,$ then $A$ is equal to $B$ multiplied by some constant quantity.

(iii) For suppose that $a, a_{1}, a_{2}, a_{3} \dots, b, b_{1}, b_{2}, b_{3} \dots$ are corresponding values of $A$ and $B .$

Then, by definition, $\frac{a}{a_{1}} = \frac{b}{b_{1}}; \frac{a}{a_{2}} = \frac{b}{b_{2}}; \frac{a}{a_{3}} = \frac{b}{b_{3}};$ and so on,

$∴ \frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}} = \dots,$ each being equal to $\frac{a}{b}$

Hence $\frac{a n y v a l u e o f A}{t h e c o r r e s p o n d i n g v a l u e o f B}$ is always the same; that is $\frac{A}{B} = m,$ where $m$ is constant.

$∴ A = m B .$

7. Exponential Series:

(i) $e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots .$

(ii) $e^{- x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \frac{x^{4}}{4!} \dots .$

(iii) $a^{x} = e^{x \log_{e} a} = 1 + x \log_{e} a + \frac{x^{2} {(\log_{e} a)}^{2}}{2!} + \frac{x^{3} {(\log_{e} a)}^{3}}{3!} + \dots$

8. Logarithmic Series:

(i) $\log_{e} (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \dots$

(ii) $\log_{e} (1 - x) = - x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4} - \dots$

The solution set of the rational inequality x+9x-6≤0 is

Important Questions on Fundamentals of Mathematics

Learn and Prepare for any exam you want !

The solution set of the rational inequality $\frac{x + 9}{x - 6} \leq 0$ is