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# JEE Advanced 2023 Revision Notes for Algebra

Last updated date: 18th Mar 2023
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One of the crucial chapters of JEE Advanced syllabus is Vector Algebra. This chapter introduces the concepts of vector quantities in the form of algebraic expressions and equations. These fundamental principles will be very useful for aspirants in their higher studies. Simply put, this chapter involves the calculation of the magnitude and direction of vector quantities. To understand the concepts of this chapter better, download and refer to the Algebra JEE Advanced notes for free from Vedantu.

These revision notes are ideal for preparing this chapter without any difficulty. This extensive chapter is presented in a concise format for your quick preparation and revision. All the derivations and formulas are explained in a simpler manner to help you prepare this chapter for the JEE Advanced exam.

Competitive Exams after 12th Science

## Complex Number

A complex number is one that has the form $z = x + iy$, where $x,y \in R$ and $i = \sqrt { - 1}$and where $x$ is the real part and $y$ is the imaginary part, and it is expressed as $\operatorname{Re} (z) = x,\operatorname{lm} (z) = y$. If $x = 0$, the complex number is purely imaginary, and if $y = 0,$ it is purely real.

A complex number is also defined as an ordered pair of real numbers, and it is represented by the symbol $(a,b)$. If we write $z = (a,b)$ the real part is $a$ and the imaginary part is $b$ of the complex number $z.$

## Imaginary Numbers

Real numbers multiplied by the imaginary unit I yield imaginary numbers. In mathematics, the letter i (or 'j' in some books) represents the imaginary part of any complex number. It allows us to distinguish between any complex number's real and imaginary parts. Furthermore, it is not only not distinguishable but also has some value.

${\mathbf{i}}{\text{ }} = {\text{ }}\sqrt { - 1}$

## Algebra of complex numbers

### Complex Number Equality

Two Complex numbers ${z_1}$ and ${z_2}$ are equal if,

Condition 1) $\operatorname{Re} \left( {{z_1}} \right) = \operatorname{Re} \left( {{z_2}} \right)$

Condition 2) $\operatorname{Im} \left( {{z_1}} \right) = \operatorname{Im} \left( {{z_2}} \right)$

So, if, ${z_1} = x + 3i$ and ${z_2} = - 2 + {y_i}$ are equal, then according to the above conditions,

$\operatorname{Re} \left( {{z_1}} \right) = x$ and $\operatorname{Re} \left( {{z_2}} \right) = - 2$, so $x = - 2$

And Similarly

$\operatorname{Im} \left( {{z_1}} \right) = 3{\text{ and }}\operatorname{Im} \left( {{z_2}} \right) = y{\text{, so }}y = 3$

Algebraic Operations with Complex Numbers

• Addition: $(a + ib) + (c + id) = (a + c) + i(b + d)$

• Subtraction:$(a + ib) - (c + id) = (a - c) + i(b - d)$

• Multiplication : $(a + ib)(c + id) = ac + iad + ibc + {i^2}bd = (ac - bd) + i(ad + bc)$

• Division : $\dfrac{{a + ib}}{{c + id}}{\text{ }} = \dfrac{{(a + ib)(c - id)}}{{(c + id)(c - id)}}$$= \dfrac{{(ac + bd)}}{{{c^2} + {d^2}}} + i\dfrac{{(bc - ad)}}{{{c^2} + {d^2}}} ## Addition of Complex Numbers If {z_1} = a + ib and {z_2} = c + id, then the sum of these two complex numbers, {z_1} + {z_2}, is calculated as: {z_1} + {z_2} = (a + ib) + (c + id) = (a + c) + i(b + d) Therefore, {z_1} + {z_2} = \operatorname{Re} \left( {{z_1} + \operatorname{Re}{z_2}} \right) + \operatorname{Im} \left( {{z_1} + {z_2}} \right) Addition of complex numbers can be another complex number. Addition of complex numbers satisfies the following properties • Closure Law: That is, the sum of two complex numbers is another complex number. If {z_1} + {z_2} where {z_1} and {z_2} are complex numbers, then z will be a complex number as well. • Commutative Law: According to commutative law, for any two complex numbers {z_1} and {z_2},{z_1} + {z_2} = {z_2} + {z_1}. • Associative Law: For any three complex numbers say \left( {{z_1} + {z_2}} \right) + {z_3} = {z_1} + \left( {{z_2} + {z_3}} \right). • Existence of Additive Identity: Additive identity, also known as zero complex number, is denoted by 0 (or 0 + i0), which means that for any complex number z,z + 0 = z. • Existence of Additive Inverse: The additive inverse or negative of any complex number z is a complex number with the opposite sign in both its real and imaginary parts. It is symbolised by - z and z + ( - z) = 0. ## Difference of Two Complex Numbers If {z_1} = a + ib and {z_2} = c + id the difference of these two complex numbers is {z_1} - {z_2} is calculated as follows: {z_1} - {z_2} = (a + ib) - (c + id){\text{ }} = (a - c) + i(b - d) Therefore, {z_1} - {z_2} =\operatorname{Re}\left( {{Z}_{1}} \right)+\operatorname{Re}\left( {{Z}_{2}} \right) + \operatorname{Im} \left( {{z_1} - {z_2}} \right) A complex number's difference can be another complex number. ## Multiplication of Two Complex Numbers If z_{1}=a+i b \text { and } z_{2}=c+i d then the product of these two complex numbers is. {z_1} \times {z_2} is calculated as follows: {z_1} \times {z_2} = (a + ib) \times (c + id){\text{ }} {{z}_{1}}{{z}_{2}}~=\text{ }ac\text{ }+\text{ }i\left( ad \right)\text{ }+\text{ }i\left( bc \right)\text{ }+\text{ }{{i}^{2}}\left( bd \right) {{z}_{1}}{{z}_{2}}~=\text{ }ac\text{ }+\text{ }i\left( ad \right)\text{ }+\text{ }i\left( bc \right)\text{ }+\text{ }\left( -1 \right)\left( bd \right) {z_1} \times {z_2} = (ac - bd) + i(ad + bc) Therefore, {z_1} \times {z_2} = \left[ {\operatorname{Re} \left( {{z_1}} \right)\operatorname{Re} \left( {{z_2}} \right) - \operatorname{Im} \left( {{z_1}} \right)\operatorname{Im} \left( {{z_2}} \right)} \right] + i\left[ {\operatorname{Re} \left( {{z_1}} \right)\operatorname{Im} \left( {{z_2}} \right) + \operatorname{Im} \left( {{z_2}} \right)\operatorname{Re} \left( {{z_2}} \right)} \right] ## Division of Two Complex Numbers If {z_1} = a + ib and {z_2} = c + idthen the division of these two complex numbers, \dfrac{z_1}{z_2}, is calculated as follows: \dfrac{{{z_1}}}{{{z_2}}} = \dfrac{{(a + ib)}}{{(c + id)}} On Rationalization: \dfrac{{{z_1}}}{{{z_2}}} = \dfrac{{(a + ib)}}{{(c + id)}} \times \dfrac{{(c - id)}}{{(c - id)}}{\text{ }} \dfrac{{{z_1}}}{{{z_2}}} = \dfrac{{(ac + bd)}}{{\left( {{c^2} + {d^2}} \right)}} + \dfrac{{i(bc - ad)}}{{\left( {{c^2} + {d^2}} \right)}} ## Conjugate of Complex Number A complex number's conjugate z = a + ib is obtained by changing the sign of the imaginary part of z, which is denoted as \bar z \bar z = a - ib z + \bar z = 2a z - \bar z = 2bi ## Modulus of a Complex Number The modulus of a complex number is the distance from the origin of the point on the argand plane representing the complex number z. Let P be the point corresponding to the complex number z = x + iy. Then OP = \left| z \right| = \sqrt {({x^{2\;}} + {\text{ }}{y^2}\;)} . Modulus Properties (i) |z| = 0 \Leftrightarrow z = 0 (ii) |z|=\left| {\bar{z}} \right|=|-z|=\left| -\bar{z} \right| (iii) - |z| \leqslant \operatorname{Re} (z) \leqslant |z| (iv) - |z| \leqslant \operatorname{Im} ({\text{z}}) \leqslant |{\text{z}}| (v) z{\bar{z}} = |z{|^2} If {z_1},{z_2} are two complex numbers, then (vi) \left| {{z_1}{z_2}} \right| = \left| {{z_1}} \right| \cdot \left| {{z_2}} \right| (vii) \left| \dfrac{z_1}{z_2} \right| = \left| \dfrac{z_1}{z_2}\right|, if {z_2} \ne 0 (viii) {{\left| {{z}_{1}}+{{z}_{2}} \right|}^{2}}={{\left| {{z}_{1}} \right|}^{2}}+{{\left| {{z}_{2}} \right|}^{2}}+{{\bar{z}}_{1}}{{\bar{z}}_{2}}+{{z}_{1}}{{\bar{z}}_{2}}={{\left| {{z}_{1}} \right|}^{2}}+{{\left| {{z}_{2}} \right|}^{2}}+2\operatorname{Re}\left( {{z}_{1}}{{{\bar{z}}}_{2}} \right) (ix) {{\left| {{z}_{1}}+{{z}_{2}} \right|}^{2}}+{{\left| {{z}_{1}} \right|}^{2}}-{{\left| {{z}_{2}} \right|}^{2}}-{{\bar{z}}_{1}}{{\bar{z}}_{2}}-{{z}_{1}}{{\bar{z}}_{2}}={{\left| {{z}_{1}} \right|}^{2}}+{{\left| {{z}_{2}} \right|}^{2}}-2\operatorname{Re}\left( {{z}_{1}}{{{\bar{z}}}_{2}} \right) (x) {\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} - {z_2}} \right|^2} = 2\left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right) ## Modulus and Argument of a Complex Number Let z = x + iy = (x,y) for all x,y \in {\text{R}} and {\text{i}} = \sqrt { - 1} (Image will be uploaded soon) Modulus of a Complex Number The length OP is known as the modulus of the complex number z, which is denoted by \left| z \right|. i.e. {\text{OP}} = {\text{r}} = |{\text{z}}| = \sqrt {\left( {{x^2} + {y^2}} \right)} If (x,y) \ne (0,0) \theta is known as the argument or amplitude of z, i.e. \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) [angle made by OP with positive X-axis] or \arg (z) = {\tan ^{ - 1}}\dfrac{y}{x} Also, the argument of a complex number is not unique since if \theta is a value of the argument, so also in 2n\pi \text{ }+\text{ }\theta Furthermore, the argument of a complex number is not unique, because if \theta is a value of the argument, then it is also in 2n\pi + \theta , where n \in I But usually, we take only that value for which 0 < \theta < 2\pi . Any two arguments of a complex number differ by 2{\text{n}}\pi . Argument of z will be \theta ,\pi - \theta ,\pi + \theta and 2\pi - \theta according as the point z lies in I, II, III and IV quadrants respectively, where \theta = {\tan ^{ - 1}}\left| {\dfrac{y}{x}} \right| ## The Argument of a Complex Number If z is a non-zero complex number represented by the complex plane's point P, then the argument of z is the angle formed by OP with the positive direction of the real axis. Argument of a Complex Number Principal Value of the Argument for Different Positions of z in the Complex Plane 1. When z lies in the first quadrant Argument of a Complex Number in 1st Quadrant \arg (z) = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) 2. When z lies in the second quadrant Argument of a Complex Number in 2nd Quadrant \arg (z) = \pi - {\tan ^{ - 1}}\left( {\dfrac{y}{{|x|}}} \right) 3. When z lies in the third quadrant Argument of a Complex Number in 3rd Quadrant \arg (z) = - \pi + {\tan ^{ - 1}}\left( {\dfrac{{|y|}}{{|x|}}} \right) 4. When z lies in the fourth quadrant ( Image will be uploaded soon) Argument of a Complex Number in 4th Quadrant \arg (z) = - {\tan ^{ - 1}}\left( {\dfrac{{|y|}}{x}} \right) ## Polar Form of a Complex Number Let z be a non-zero complex number, then we can write z = r(\cos \theta + i\sin \theta ) where r = |z| and \theta = \arg (z). ## Euler's Formula The complex number \cos \theta + i\sin \theta is denoted by {e^{i\theta }} or \operatorname{cis} \theta . That is{e^{i\theta }} = \operatorname{cis} \theta = \cos \theta + i\sin \theta ## De Moivre's Theorem and its Applications ### (a) The Integral Index De Moivre's Theorem If n is an integer, then {(\cos \theta + i\sin \theta )^n} = \cos (n\theta ) + i\sin (n\theta ) ### (b) De Moivre's Theorem for Rational Index. If n is a rational number, then one of the values of {(\cos \theta + i\sin \theta )^n} is \cos (n\theta ) + i\sin (n\theta ). In fact, if n = \dfrac{p}{q}, p,q \in {\mathbf{I}},q > 0 and p,q share no factors, then {(\cos \theta + i\sin \theta )^n} has q distinct values, one of which is \cos (n\theta ) + i\sin (n\theta ). {\mathbf{nth}} Roots of Unity By a root of unity we mean any complex number which satisfies the equation We mean any complex number z that satisfies the equation by a nth root of unity. {z^n} = 1 Because an equation of degree n has n roots, there are n,z values that satisfy the equation (1). To get these n,z values, we write 1 = \cos (2k\pi ) + i\sin (2k\pi ) where k \in {\mathbf{I}} and \Rightarrow z = \cos \left( {\dfrac{{2k\pi }}{n}} \right) + i\sin \left( {\dfrac{{2k\pi }}{n}} \right) {\text{where }}\quad k = 0,1,2, \ldots ,n - 1. ## Cube Roots of Unity Cube roots of unity are given by 1,\omega ,{\omega ^2}, where \omega = \cos \left( {\dfrac{{2\pi }}{3}} \right) + i\sin \left( {\dfrac{{2\pi }}{3}} \right) = \dfrac{{ - 1 + \sqrt 3 i}}{2}{\text{ and }}{\omega ^2} = \dfrac{{ - 1 - \sqrt 3 i}}{2} Some Results Involving Complex Cube Root of Unity (\omega ) (i) {\omega ^3} = 1,\bar \omega = {\omega ^2} and \dfrac{1}{\omega } = {\omega ^2} (ii) 1 + \omega + {\omega ^2} = 0 (iii) {x^3} - 1 = (x - 1)(x - \omega )\left( {x - {\omega ^2}} \right) (iv) \omega and {\omega ^2} are roots of {x^2} + x + 1 = 0 (v) {a^3} - {b^3} = (a - b)(a - b\omega )\left( {a - b{\omega ^2}} \right) = (a - b)\left( {a\omega - b{\omega ^2}} \right)\left( {a{\omega ^2} - b\omega } \right) (vi){a^2} + {b^2} + {c^2} - bc - ca - ab = \left( {a + b\omega + c{\omega ^2}} \right)\left( {a + b{\omega ^2} + c\omega } \right) (vii) {a^3} + {b^3} + {c^3} - 3abc = (a + b + c)\left( {a + b\omega + c{\omega ^2}} \right)\left( {a + b{\omega ^2} + c\omega } \right) (viii) {x^3} + 1 = (x + 1)(x + \omega )\left( {x + {\omega ^2}} \right) (ix) {a^3} + {b^3} = (a + b)(a + b\omega )\left( {a + b{\omega ^2}} \right) (x) Cube roots of real number a are {{a}^{\dfrac{1}{3}}},{{a}^{\dfrac{1}{3}}}\omega ,{{a}^{\dfrac{1}{3}}}{{\omega }^{2}}. Geometric Interpretation of Multiplying a Complex Number By {{\text{e}}^{i\alpha }} Let z be a complex number that is not zero. In the polar form, z can be written as follows: z = r(\cos \theta + i\sin \theta ) = r{e^{i\theta }} where r = |z| and \arg (z) = \theta We have z{e^{i\alpha }} = r{e^{i\theta }}{e^{i\alpha }} = r{e^{i(\theta + \alpha )}} Thus, z{e^{i\alpha }} denotes a complex number with modulus r and argument \theta + \alpha . Geometrically, z{e^{i\alpha }}can be obtained by rotating the segment joining O and P(z) through an anticlockwise angle \alpha . ## Solved Example: 1. Let p and q be two complex numbers such that q \ne 0. If the roots of the equation {x^2} + px + {q^2} = 0 have the same absolute value, then p/q is 1. an integer 2. a rational number 3. a real number 4. a complex number. Ans. (c) Solution: Let {x_1},{x_2} be the roots of {x^2} + px + {q^2} = 0 and \left| {{x_1}} \right| = \left| {{x_2}} \right| = r. Note that {x_1}{x_2} = {q^2} \ne 0. Now, \dfrac{{{p^2}}}{{{q^2}}} = \dfrac{{{{\left( {{x_1} + {x_2}} \right)}^2}}}{{{x_1}{x_2}}} = \dfrac{{{x_1}}}{{{x_2}}} + \dfrac{{{x_2}}}{{{x_1}}} + 2 = \dfrac{{{x_1}{{\bar x}_2}}}{{{{\left| {{x_2}} \right|}^2}}} + \dfrac{{{{\bar x}_1}{x_2}}}{{{{\left| {{x_1}} \right|}^2}}} + 2 = \dfrac{1}{{{r^2}}}\left( {{x_1}{{\bar x}_2} + {{\bar x}_1}{x_2}} \right) + 2 = \dfrac{2}{{{r^2}}}\operatorname{Re} \left( {{x_1}{{\bar x}_2}} \right) + 2 \Rightarrow \dfrac{{{p^2}}}{{{q^2}}}{\text{ is a real number}}{\text{.}} {\text{ Next, as }} - \operatorname{Re} \left( {{x_1}{{\bar x}_2}} \right) \leqslant \left| {{x_1}{{\bar x}_2}} \right| = \left| {{x_1}{x_2}} \right| = \left| {{x_1}} \right|\left| {{x_2}} \right| = {r^2} \Rightarrow \dfrac{{{p^2}}}{{{q^2}}} \geqslant - \dfrac{{2{r^2}}}{{{r^2}}} + 2 = - 2 + 2 = 0 Therefore, \dfrac{p}{q} is a real number. 2. If \omega is an imaginary cube root of unity, then value of the expression 2(1 + \omega )\left( {1 + {\omega ^2}} \right){\text{ }} + 3(2 + \omega )\left( {2 + {\omega ^2}} \right) + \ldots . + (n + 1)(n + \omega )\left( {n + {\omega ^2}} \right) is 1. \dfrac{1}{4}{n^2}{(n + 1)^2} + n 2. \dfrac{1}{4}{n^2}{(n + 1)^2} - n 3. \dfrac{1}{4}n{(n + 1)^2} - n 4. \dfrac{1}{4}{n^2}{(n + 1)^2} - 2n Ans. (a) Solution: r th term of the given expression is (r + 1)(r + \omega )\left( {r + {\omega ^2}} \right) = {r^3} + 1 \therefore value of the given expression is \sum\limits_{r = 1}^n {\left( {{r^3} + 1} \right)} = \dfrac{1}{4}{n^2}{(n + 1)^2} + n ## Quadratic Equations A formula of the quadratic form is a{x^2} + bx + c = 0........(1) A quadratic equation is defined as a \ne 0,a,b,c \in {\mathbf{C}}, the set of real numbers. The discriminant of the equation is the quantity D = {b}^2 - 4ac A complex number \alpha such that a{\alpha ^2} + b\alpha + c = 0 is a root of the quadratic equation (1). The quadratic formula gives the roots of equation (1) x = \dfrac{{ - b \pm \sqrt D }}{{2a}} ## Nature of Roots 1. If a,b,c \in {\mathbf{R}} and a \ne 0. Then the following are true: • If and only if D > 0 the equation (1) has real and distinct roots. • If and only if D = 0, the equation (1) has real and equal roots. • If and only if D < 0, the equation (1) has complex roots with non-zero imaginary parts. • If and only if p + iq(p,q \in {\mathbf{R}},q \ne 0) is a root then p - iq is a root of equation (1). 2. If a,b,c \in {\mathbf{Q}} are rational numbers and D is a perfect square of a rational number, then equation (1) has rational roots. 3. If a,b,c \in {\mathbf{Q}} and p + \sqrt q (p,q \in {\mathbf{Q}}) are irrational roots of equation (1), then p - \sqrt q is also an irrational root of equation (1). 4. If the set of integers, a = 1,b,c \in {\mathbf{I}}, and the roots of equation (1) are rational numbers, then these roots must be integers. 5. If more than two distinct complex numbers satisfy equation (1), equation (1) becomes an identity, that is, a = b = c = 0. ## Relation Between Roots and Coefficients If \alpha and \beta are the quadratic equation's roots (1), then \alpha + \beta = - \dfrac{b}{a} and \alpha \beta = \dfrac{c}{a} It is worth noting that a quadratic equation with roots \alpha and \beta is given by (x - \alpha )(x - \beta ) = 0 ## Quadratic Expression and Its Graph Let f(x) = a{x^2} + bx + c, where a,b,c \in {\mathbf{R}} and a \ne 0 We have f(x) = a\left[ {{x^2} + \dfrac{b}{a}x + \dfrac{c}{a}} \right] = a\left[ {{x^2} + \dfrac{b}{a}x + \dfrac{{{b^2}}}{{4{a^2}}} + \dfrac{c}{a} - \dfrac{{{b^2}}}{{4{a^2}}}} \right] = a\left[ {{{\left( {x + \dfrac{b}{{2a}}} \right)}^2} + \dfrac{{4ac - {b^2}}}{{4{a^2}}}} \right]\,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ............(2) When is a Quadratic Expression always positive and when is it always negative? It follows from (2) that f(x) > 0( < 0)\forall x \in {\mathbf{R}} if and only if a > 0( < 0) and D = {b^2} - 4ac < 0. Also, it follows from (2) that f(x) \geqslant 0( \leqslant 0)\forall x \in {\mathbf{R}} if and only if a > 0( < 0) and D = {b^2} - 4ac = 0. In this case f\left( x \right) > 0\left( { < 0{\text{ }}} \right)for each x \in {\mathbf{R}},x \ne - b/2a. Quadratic Expression ## Position of Roots of a Quadratic Equation Let f(x) = a{x^2} + bx + c be a quadratic expression in a,b,c \in {\mathbf{R}} and let k be a real number. Conditions for both roots to be greater than a Real Number k. If then the parabola opens upwards and intersects the -axis in where If a > 0 and {b^2} - 4ac > 0 the parabola y = a{x^2} + bx + c rises and intersects the x-axis in \alpha and \beta \alpha ,\beta = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} In this case both the roots \alpha and \beta will be more than k if k lies to the left of both \alpha and \beta. Case Conditions for a number k to lie between the roots of a quadratic equation A real number k lies between the roots of the quadratic equation f(x) = a{x^2} + bx + c = 0 if and only if a and f(k) are of opposite signs, that is, if and only if (i) a > 0 (ii) D > 0 (iii) f(k) < 0 (i) a < 0 (ii) D > 0 (iii) f(k) > 0 Combining, we may say k lies between the roots of f(x) = a{x^2} + bx + c = 0 if and only if (i) D > 0 (ii) af(k) < 0 ## Descartes Rule of Signs for the Roots of a Polynomial • Rule 1: A polynomial equation's maximum number of positive real roots f(x) = {a_0}{x^n} + {a_1}{x^{n - 1}} + {a_2}{x^{n - 2}} + \ldots + {a_{n - 1}}x + {a_n} = 0 is the number of times the sign of a coefficient changes from positive to negative and from negative to positive. For Example, in the equation {x^3} + 3{x^2} + 7x - 11 = 0 the coefficient signs are ++- Because there is only one sign change, the number of positive roots of {x^3} + 3{x^2} + 7x - 11 = 0 can only be one. • Rule 2: The polynomial equation's maximum number of negative roots is f\left( x \right) = 0. denotes the number of changes in the signs of the coefficients of the equation f\left( { - x} \right) = 0from positive to negative and negative to positive. ## Polynomial and Polynomial Equation An expression of the form {a_0}{x^n} + {a_1}{x^{n - 2}} + ...... + {a_{n - 1}}x + {a_n} where {a_0},{a_1},{a_2}......{a_n} are constants ({a_0} \ne 0) and n is a positive integer is called a polynomial in x of degree n. If f (x) is a real or complex polynomial, then f(x) = 0 is known as a polynomial equation. ## Sequences A sequence is a function with the set of natural numbers as its domain. A sequence is called a real sequence if its range is a subset of real numbers (complex numbers) (complex sequence). If f:{\mathbf{N}} \to {\mathbf{C}} is a sequence, it is usually denoted by \langle f(1),f(2),f(3), \cdots \rangle = \langle f(n)\rangle . If {t_n} = f(n), the sequence is written as \left\langle {{t_1},{t_2},{t_3}, \cdots } \right\rangle . ## Arithmetic Progression When the difference {t_n} - {t_{n - 1}} is a constant for all n \in {\mathbf{N}},n > 1a sequence of numbers \left\langle {{t_n}} \right\rangle is said to be in arithmetic progression (A.P.). This constant is known as the A.P. common difference, usually denoted by the letter d. If a is the first term and d the common difference of the A.P., then its nth term {t_n} is given by {t_n} = a + (n - 1)d {t_n} = {t_m} + (n - m)d The sum {S_n} of the first n terms of such an A.P. is given by {S_n} = \dfrac{n}{2}[2a + (n - 1)d] = \dfrac{n}{2}(a + l) where l is the last term, that is, the nth term of the A.P. ## Properties of Arithmetic Progression • If a constant is added or subtracted from each term of an AP, then the resulting sequence is also an AP with the same common difference. • If each term of a given AP is multiplied or divided by a non-zero constant k, then the resulting sequence is also an AP with common difference kdor \dfrac{d}{k}, where d is the common difference of the given AP. • A sequence is an AP if its nth term is of the form An + B, i.e. a linear expression in n. • In a finite AP the sum of the terms equidistant from the beginning and end is always the same. i.e. {a_1} + {a_n} = {a_2} + {a_{n - 1}} = {a_3} + {a_{n - 2}} = \ldots ## Geometric Progression (G.P.) A Geometric Progression is a sequence in which the initial term is non-zero and each subsequent term is r times the preceding term, where r is some fixed non-zero number known as the G.P. 's common ratio. {n^{th}}term of G.P. \left( {{t_n}} \right) {t_n} = a{(r)^{n - 1}},n \in N Sum of first n terms of a GP \left( {{S_n}} \right): {S_n} = \dfrac{{a\left( {{r^n} - 1} \right)}}{{r - 1}},r \ne 1 = na{\text{ }},r = 1 = 0{\text{ }},r = - 1{\text{ and }}n{\text{ is even}} = a{\text{ ,}}r = - 1{\text{ and }}n{\text{ is odd}} Sum of infinite terms in G.P. If - 1 < r < 1, then the sum of the infinite G.P. a + ar + a{r^2} + \ldots {S_\infty } = \dfrac{a}{{1 - r}} ## Harmonic Progression The sequence {a_1},{a_2}, \cdots ,{a_n}, \cdots where {a_i} \ne 0 for each i is said to be in harmonic progression (H.P.) if the sequence \dfrac{1}{{{a}_{1}}},\dfrac{1}{{{a}_{2}}},........,\dfrac{1}{{{a}_{n}}}, the nth term of the H.P., is given by {a_n} = \dfrac{1}{{a + (n - 1)d}} where a = \dfrac{1}{{{a_1}}} and d = \dfrac{1}{{{a_2}}} - \dfrac{1}{{{a_1}}} ## Summation of Some Series Of Natural Numbers • \sum\limits_{k = 1}^n k = 1 + 2 + 3 + \cdots + n = \dfrac{1}{2}n(n + 1) • \sum\limits_{k = 1}^n {{k^2}} = {1^2} + {2^2} + \cdots + {n^2} = \dfrac{1}{6}n(n + 1)(2n + 1) • \sum\limits_{k = 1}^n {{k^3}} = {1^3} + {2^3} + \cdots + {n^3} = {\left( {\sum\limits_{k = 1}^n k } \right)^2} = {(1 + 2 + 3 + \cdots + n)^2} • = {\left[ {\dfrac{1}{2}n(n + 1)} \right]^2} = \dfrac{1}{4}{n^2}{(n + 1)^2} ## Sum of the Products of Two Terms of a Sequence To obtain the sum \sum\limits_{i < j} {{a_i}} {a_j}, we use the identity 2\sum\limits_{i < j} {{a_i}} {a_j} = {\left( {{a_1} + {a_2} + \ldots + {a_n}} \right)^2} - \left( {a_1^2 + a_2^2 + \ldots + a_n^2} \right) ## Method of Difference for Summation of Series If possible, express r th term as the difference of two terms as follows: {t_r} = f(r) - f(r \pm 1). ## Logarithms ### Exponential Function For x \in {\mathbf{R}}, we define {e^x} = 1 + x + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + \ldots + \dfrac{{{x^n}}}{{n!}} + \ldots = \sum\limits_{n = 0}^\infty {\dfrac{{{x^n}}}{{n!}}} It can be proved for every x \in {\mathbf{R}}, the number {e^x} is a finite number. We call {e^x} the exponential function. Properties • {e^x} is defined for each x \in {\mathbf{R}}. • {e^x} > 0 for all x \in {\mathbf{R}}. • {e^0} = 1. • If a,b \in {\mathbf{R}},a < b then {e^a} < {e^b}. • {e^x} \times {e^y} = {e^{x + y}} for all x,y \in {\mathbf{R}}. • \dfrac{{{e^x}}}{{{e^y}}} = {e^{x - y}} for all x,y \in {\mathbf{R}} • {\left( {{e^x}} \right)^y} = {e^{xy}} for all x,y \in {\mathbf{R}}. • For each positive real number x there exists one and only one real number y such that {e^y} = x. • {e^x} is one to one function. ## Logarithm Given a positive real number x, there is only one real number y such that {e^y} = x; we refer to this number y as \log x or {\log _e}x. The symbol \ln x also represents the logarithm. In general, the logarithm to the base of the number x(x > 0) is the number y such that {a^y} = x. Thus, y = {\log _a}x denotes {a^y} = x. {\log _{10}}x is known as the common logarithm for a = 10, and {\log _e}x is known as the natural logarithm for a = e,. ## Properties of Logarithms • {\log _a}1 = 0,a > 0,a \ne 1. • {\log _a}a = 1,a > 0,a \ne 1. • {\log _e}{e^x} = x\forall x \in {\mathbf{R}} and {e^{{{\log }_e}x}} = x\forall x > 0 i.e. {\log _e}x is the inverse function of exponential function {e^x}. • {\log _a}(xy) = {\log _a}x + {\log _a}y for all x,y > 0 and a > 0,a \ne 1. • {{\log }_{a}}(\dfrac{1}{x})=-{{\log }_{a}}x\forall x>0,a>0,a\ne 1. • {{\log }_{a}}(\dfrac{x}{y})={{\log }_{a}}x-{{\log }_{a}}y,x,y>0,a>0, a \ne 1. • If a \in {\mathbf{R}} and x > 0, then we define {x^a} = {e^{a{{\log }_e}x}} = \sum\limits_{n = 0}^\infty {{{\left( {a{{\log }_e}x} \right)}^n}} • \log {x^y} = y\log x\forall x > 0,y \in {\mathbf{R}}. • (Change of base formula) If a > 0,a \ne 1 then {\log _a}x = \dfrac{{{{\log }_e}x}}{{{{\log }_e}a}}. In fact if a,b,c > 0 and a,c \ne 1 then {\log _a}b = \dfrac{{{{\log }_c}x}}{{{{\log }_c}a}}, • If a > 1 and 0 < x < y then {\log _{ax}}x < {\log _a}y (i.e. {\log _a}x is an increasing function). In particular if x > 1 then {\log _a}x > 0 and if 0 < x < 1 then {\log _a}x < 0. • Logarithmic Inequalities • For \varphi (x) > 1,{\log _{\varphi (x)}}f(x) \geqslant {\log _{\varphi (x)}}g(x) \Leftrightarrow f(x) \geqslant g(x) > 0 • For 0 < \varphi (x) < 1,{\log _{\varphi (x)}}f(x) \geqslant {\log _{\varphi (x)}}g(x) \Leftrightarrow 0 < f(x) \leqslant g(x) • For \varphi (x) > 1,{\log _{\varphi (x)}}f(x) \geqslant a \Leftrightarrow f(x) \geqslant {a^{\varphi (x)}} • For 0 < \varphi (x) < 1,{\log _{\varphi (x)}}f(x) \geqslant a \Leftrightarrow 0 < f(x) \leqslant {a^{\varphi (x)}} ## Permutation and combination ### Fundamental Principles of counting ### The Sum Rule Assume that A and B are two disjoint (mutually exclusive) events that never occur together. Assume that A occurs in m ways and B occurs in n ways. Then A or B can happen in m + n different ways. This rule can also be applied to events that are not mutually exclusive. ### The Product Rule Assume an event X can be divided into two stages, A and B. Assume that stage A occurs in m different ways and that these stages are unrelated in the sense that stage B occurs in n different ways regardless of the outcome of stage A. Then event X occurs in mn different ways. This rule applies even if event X can be divided into more than two stages. ## Permutation Each arrangement which can be made by taking some or all of several things is called a permutation. Thus, in permutation, the order of arrangement is important. When the order is changed, a different permutation is obtained. e.g The permutation of three letters A, B and C taking all at a time are ABC,ACB,{\text{ }}BCA,BAC,CBA and CAB. ## Conditions of Permutation: 1) The number of all permutations of n different objects taking r at a time. (i) when a particular object is always included is r {}^{n - 1}{P_{r - 1}}. (ii) when a particular object is never included is {}^{n - 1}{P_r}. 2) Number of permutations of n different things, taken r at a time, when s particular things are to be always included in each arrangement, is {}^{n-s}{{C}_{r-s}} 3) Number of permutation of n different things, taken all at a time , when m specified things always come together , is m! \times (n - m + 1)! 4) Number of permutation of n different things, taken all at a time , when m specified things never come together n! - m! \times (n - m + 1)! 5) The number of permutation of n things , of which {p_1} are alike of one kind ;{p_2}are alike of second kind; {p_3} are alike of third kind;….;{p_r} are alike of r 3th kind such that {p_1} + {p_2} + \ldots + {p_r} = n{\kern 1pt} {\kern 1pt} {\kern 1pt} \,is{\kern 1pt} \,{\kern 1pt} \,\dfrac{{n!}}{{{p_1}!{p_2}!{p_3}! \ldots {p_r}!}} ## Combinations: Each of the different selections made by taking some or all of a number of objects,(irrespective of there arrangements) is called a combinations , e.g. The different combinations formed of two out of three letters A,B,Care AB,AC,BC. The number of combinations of n different things taken r at a time is denoted by {}^n{C_r}\,{\kern 1pt} \,{\text{or}}{\kern 1pt} \,\,\,C(n,r){\kern 1pt} \,{\kern 1pt} Then, {}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}} = \dfrac{{{}^n{P_r}}}{{r!}}[0 \leqslant r \leqslant n] = \dfrac{{n(n - 1)(n - 2).....(n - r + 1)}}{{r(r - 1)(r - 2)...2,1}},n \in N\,{\kern 1pt} {\text{and }}r \in W If , r>n ,then {}^n{C_r} = 0 Properties: (i) {}^n{C_r} is a natural number. (ii) {}^n{C_r} = {}^n{C_{n - r}}, for 0 \leqslant r \leqslant n (iii) {}^n{C_0} = {}^n{C_n} = 1,{}^n{C_1} = n (iv) Let n and r be non-negative integers such that r \leqslant n. Then, {}^n{C_r} + {}^n{C_{r - 1}} = {}^{n + 1}{C_r}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} [{\text{Pascal's law}}] {}^n{C_r} = \dfrac{n}{r}{}^{n - 1}{C_{r - 1}}[1 \leqslant r \leqslant n] (v)If 1 \leqslant r \leqslant n ,then n.{}^{n - 1}{C_{r - 1}} = (n - r + 1){}^n{C_{r - 1}}. (vi) {}^n{C_x} = {}^n{C_y} \Rightarrow x = yorx + y = n (vii) \dfrac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\left( \dfrac{n-r+1}{r} \right) (viii) {}^n{C_0} + {}^n{C_1} + {}^n{C_2} + ..... + {}^n{C_n} = {2^n} (ix) {}^n{C_0} + {}^n{C_2} + {}^n{C_4} + ..... = {}^n{C_1} + {}^n{C_3} + {}^n{C_5}..... = {2^{n - 1}} (x) {}^{2n + 1}{C_0} + {}^{2n + 1}{C_1} + {}^{2n + 1}{C_2} + .... + {}^{2n + 1}{C_n} = {2^{2n}} ## Circular Permutation There are (n-1)! ways to arrange n distinct objects around a circle. (i) Here, anti-clockwise and clockwise order of arrangements are considered as distinct permutations. (ii) If anti-clockwise and clockwise order of arrangements are not distinct, e.g. arrangement of beads in a necklace, arrangement of flowers in a garland etc. then number of circular permutations of n distinct items is \dfrac{(n-1)!}{2} . (iii) The number of circular permutations of n different things taken r at a time, when clockwise and anti-clockwise orders are taken as different is \dfrac{\left( ^{\text{n}}{{\text{P}}_{\text{r}}} \right)}{\text{r}}, for 1\le \text{r}\le \text{n} (iv) The number of circular permutations of n different things taken r at a time, when clockwise and anti-clockwise orders are taken as same is \dfrac{\left( ^{n}{{P}_{\text{r}}} \right)}{2\text{r}}. ## Division of Identical Objects The number of ways to distribute n identical objects among r people while giving zero or more to each is equal to the number of ways to arrange n identical objects of one kind and (r-1) identical objects of the second kind (separators) in a row =\dfrac{(n+r-1) !}{n !(r-1) !}={ }^{\mathrm{n}+\mathrm{r}-1} C_{\mathrm{r}-1} ## Derangement The number of ways to arrange n objects numbered 1,2,3, \ldots, nat nplaces numbered 1,2,3, \ldots, n so that no object goes to the place corresponding to its number is n !\left(\dfrac{1}{2 !}-\dfrac{1}{3 !}+\dfrac{1}{4 !}-\cdots+(-1)^{n-1} \dfrac{1}{n !}\right) ## Binomial Theorem Binomial theorem for a positive integral index Suppose n is a positive integer and x,yare two complex numbers. {{(x+y)}^{n}}{{=}^{n}}{{C}_{0}}{{x}^{n}}{{+}^{n}}{{C}_{1}}{{x}^{n-1}}y{{+}^{n}}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}+\ldots {{+}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{y}}+\ldots {{+}^{n}}{{C}_{n-1}}x{{y}^{n-1}}{{+}^{n}}{{C}_{n}}{{y}^{n}}................(1) The coefficients { }^{n} C_{0},{ }^{n} C_{1}, \cdots,{ }^{n} C_{n} are called binomial coefficients. ## Properties of Binomial Expansion 1. The expansion contains (n+1)terms (1). 2. The sum of the exponents of xand yin any term of (1) is always n. 3. Binomial coefficients of terms equidistant from the beginning and end are equal, because { }^{n} C_{r}={ }^{n} C_{n-r} (0 \leq r \leq n) 4. The term { }^{n} C_{r} x^{n-r} y^{n} is the (r+1) th term from the beginning of the expansion. It is usually denoted by T_{r+1} and is called the general term of the expansion. ## Middle Terms 1. \boldsymbol{n} is even If n is even, then the expansion (1) has just one middle term, viz. \left(\dfrac{n}{2}+1\right) th term. It is given by ^{n}{{C}_{n/2}}{{x}^{\dfrac{n}{2}}}{{y}^{\dfrac{n}{2}}}. 2. \boldsymbol{n} is odd If n is odd, then the expansion (1) has two middle terms, viz. \left(\dfrac{n+1}{2}\right) th term and \left(\dfrac{n+1}{2}+1\right) th term. The Greatest Coefficient If n is an even number, the greatest coefficient in the expansion of (x+y)^{n} is ^{n}{{C}_{\dfrac{n}{2}}}. If n is an odd number, the expansion of (x+y)^{n} has two greatest coefficients. These are ^{n}{{C}_{\dfrac{n-1}{2}}} and ^{n}{{C}_{\dfrac{n+2}{2}}}, respectively. ## The Greatest Term To find the greatest term in the expansion of (x+y)^{n}, x, y>0, n \in N put \alpha=\dfrac{(n+1) y}{x+y} If \alpha is not an integer, then the expansion of (x+y)^{n} has just one greatest term and it is given by k=[\alpha] where [\alpha] denotes the greatest integer \leq \alpha. If \alpha is an integer, then expansion of (x+y)^{n} has two greatest terms viz. \alpha th and (\alpha+1) th terms. ## Some Other Useful Expansions In the following expansions C_{r} stands for { }^{n} C_{r} 1. (x+y)^{n}+(x-y)^{n} =2\left[C_{0} x^{n}+C_{2} x^{n-2} y^{2}+C_{4} x^{n-4} y^{4}+\ldots\right] 2. (x+y)^{n}-(x-y)^{n} =2\left[C_{1} x^{n-1} y+C_{3} x^{n-3} y^{3}+C_{5} x^{n-5} y^{5}+\ldots\right] 3. n(n-1)(n-2)\ldots (n-k+1){{(x+y)}^{n-k}}=\sum\limits_{r=k}^{n}{r}(r-1)(r-2)\ldots (r-k+1){{C}_{r}}{{x}^{r-k}}{{y}^{n-r}} 4. (1+x)^{n}+(1-x)^{n}=2\left[C_{0}+C_{2} x^{2}+C_{4} x^{4}+\ldots\right] 5. (1+x)^{n}-(1-x)^{n}=2\left[C_{1} x+C_{3} x^{3}+\ldots .\right] 6. C_{1}+2 C_{2} x+3 C_{3} x^{2}+\ldots .+n C_{n} x^{n-1} =n(1+x)^{n-1} 7. {{C}_{0}}x+{{C}_{1}}\dfrac{{{x}^{2}}}{2}+{{C}_{2}}\dfrac{{{x}^{3}}}{3}+\ldots .+{{C}_{n}}\dfrac{{{x}^{n+1}}}{n+1}=\int_{0}^{x}{{{(1+t)}^{n}}}dt ## Some Properties of the Binomial Coefficients 1. C_{0}+C_{1}+C_{2}+\ldots .+C_{n}=2^{n} 2. C_{0}+C_{2}+C_{4}+\ldots=C_{1}+C_{3}+C_{5}+\ldots=2^{n-1} 3. C_{0}-C_{1}+C_{2}-\ldots+(-1)^{n} C_{n}=0 4. For n>1,C_{1}-2 C_{2}+3 C_{3}-\ldots .+(-1)^{n-1} n C_{n}=0 5. If P_{n}=C_{0} C_{1} C_{2} \ldots C_{n}, then \dfrac{P_{n+1}}{P_{n}}=\dfrac{(n+1)^{n}}{n !} ## Multinomial Theorem If we want to expand (x+y+z)^{n} (where n is a positive integer and x,y,zare complex numbers), we can take y+z=a,expand the binomial (x+a)^{n}, and then expand (y+z)^{r} in each term of (x+a)^{n}. In general, we get the following outcome: \left(a_{1}+a_{2}+\ldots+a_{m}\right)^{n}=\sum \dfrac{n !}{n_{1} ! n_{2} ! \cdots n_{m} !} a_{1}^{n_{1}} a_{2}^{n_{2}} \cdots a_{m}^{n_{m}}$$\text{where the summation is taken over all non-negative integers}$${{n}_{1}},{{n}_{2}},\ldots ,{{n}_{m}}such that n_{1}+n_{2}+\cdots+n_{m}=n. The number of distinct terms in the expansion (1) is { }_{n+m-1} C_{m-1} ## Some Particular Expansions For |x|<1 1. (1+x)^{-1}=1-x+x^{2}-x^{3}+x^{4}-\ldots . 2. (1-x)^{-1}=1+x+x^{2}+x^{3}+x^{4}+\ldots . 3. (1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+5 x^{4}-\ldots . 4. (1-x)^{-2}=1+2 x+3 x^{2}+4 x^{3}+5 x^{4}+\ldots 5. (1+x)^{-3}=1-3 x+6 x^{2}-10 x^{3}+15 x^{4}-\ldots . 6. (1-x)^{-3}=1+3 x+6 x^{2}+10 x^{3}+15 x^{4}+\ldots 7. (1-x)^{-n}=1+{ }^{n} C_{1} x+{ }^{n+1} C_{2} x^{2}+{ }^{n+2} C_{3} x^{3}+\ldots where if n is a natural number. ## Matrices The algebra of matrices A matrix is a numerical array that is rectangular in shape. The numbers could be simple or complex. It could be written as A = \left[ {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}} \\{{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}} \\\vdots &{}&{}&{} \\{{a_{m1}}}&{{a_{m2}}}& \cdots &{{a_{mn}}}\end{array}} \right] or as A = {\left( {{a_{ij}}} \right)_{m \times n}}. A matrix with m rows and n columns is known as a m \times nmatrix, and its size (or dimension) is also known as m \times n. A matrix A = {\left( {{a_{ij}}} \right)_{m \times n}} is said to be a (i) square matrix \quad if m = n (ii) row matrix \quad if m = 1 (iii) column matrix \quad if n = 1 (iv) Null or zero matrix \quad if {a_{ij}} = 0\forall i and j (v) diagonal matrix \quad if m = n and {a_{ij}} = 0 (vi) scalar matrix \quad if m = n and {a_{ij}} = 0 \forall i \ne j and {a_{ii}} = \lambda \forall i (vii) unit or identity matrix if m = n and {a_{ij}} = 0 \forall i \ne j and{a_{ii}} = 1\forall i (viii) upper triangular matrix if m=n and a_{i j}=0 \forall i>j (ix) lower triangular matrix if m=n and a_{i j}=0$$\forall i<j$

A matrix is said to be triangular if it is either lower or upper triangular matrix.

Matrices Addition: If and only if two matrices $A$ and $B$ are the same size, they can be added. When two matrices are the same size, addition is performed term by term. As an example,

$\left[ {\begin{array}{*{20}{l}}a&b&c \\d&e&f\end{array}} \right] + \left[ {\begin{array}{*{20}{l}}p&q&r \\s&t&u\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{a + p}&{b + q}&{c + r} \\{d + s}&{e + t}&{f + u}\end{array}} \right]$

Scalar Multiplication:  If $A$ is a matrix and $\alpha$ is a scalar, $\alpha A$ is the matrix formed by multiplying each element of$A$ by $\alpha$. As an example,

$3\left[ {\begin{array}{*{20}{c}}1&{ - 2}&5 \\7&8&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}3&{ - 6}&{15} \\{21}&{24}&3\end{array}} \right]$

Properties of Scalar Multiplication:  If $A$ and $B$ are two matrices of the same size, and $\alpha ,\beta$ are two scalars, then

\begin{align}& (\alpha +\beta )A\text{ }=\alpha A+\beta A \\& (\alpha \beta )A\text{ }=\alpha (\beta A) \\& \alpha (A+B)\text{ }=\alpha A+\alpha B\text{ } \\\end{align}

Matrix Multiplication: Let $A = {\left( {{a_{ij}}} \right)_{m \times n}}$ and $B = {\left( {{b_{ij}}} \right)_{r \times s}}$ be two matrices. We say that $A$ and $B$ are comparable for the product $AB$if $n = r$, that is, if the number of columns of $A$ is equal to the number of rows of $B$.

## Properties of Matrix Multiplication

If $A = {\left( {{a_{ij}}} \right)_{m \times n}},B = {\left( {{b_{ij}}} \right)_{n \times p}}$ and $C = {\left( {{c_{ij}}} \right)_{p \times q}}$ then

1. $(AB)C = A(BC)$  [Associative law]

2. $A{I_n} = {I_m}A = A$

3. $AB$may not be equal to $BA$

4. $k(AB) = (kA)B = A(kB)$ where $k$ is a scalar

5. If $A$ is a square matrix, then

${A^m}{A^n} = {A^{m + n}}\forall m,n \in {\mathbf{N}}$

${\left( {{A^m}} \right)^n} = {A^{mn}}\quad \forall m,n \in {\mathbf{N}}.$

6. If $A$ is an invertible matrix then ${\left( {{A^{ - 1}}BA} \right)^m} = {A^{ - 1}}{B^m}A{\text{and }}{A^{ - m}} = {\left( {{A^{ - 1}}} \right)^m}\forall m \in {\mathbf{N}}.$

## Transpose of a matrix

Assume $A = {\left( {{a_{ij}}} \right)_{m \times n}}$ is a matrix. The $A$ transpose, denoted by ${A^\prime }$or ${A^t}$, is the matrix ${A^\prime } = {\left( {{b_{ij}}} \right)_{n \times m}}$, where ${b_{ij}} = {a_{ji}}\forall i$ and $j$.

By $\bar A$ we mean a matrix $B = {\left( {{b_{ij}}} \right)_{m \times n}}$ where ${b_{ij}} = {\bar a_{ij}}$ where $\bar a$ denotes the conjugate of $a$.

By ${A^*}$ we mean

${A^*} = {(\bar A)^\prime } = \left( {{{\bar A}^\prime }} \right)$

## Properties of Transpose of Matrix

1. ${(A + B)^\prime } = {A^\prime } + {B^\prime }$

2. ${(kA)^\prime }\quad = k{A^\prime }$ where $k$ is a scalar

3. ${(AB)^\prime }\quad = {B^\prime }{A^\prime }\quad$ [Reversal law]

4. If $A$ is an invertible matrix, then ${\left( {{A^{ - 1}}} \right)^\prime } = {\left( {{A^\prime }} \right)^{ - 1}}$

## Minors and cofactors

### Minor

Allow $A = {\left( {{a_{ij}}} \right)_{n \times n}}$ to be a square matrix of order n. The determinant of the square sub-matrix of order $(n - 1)$ obtained by deleting the ith row and $j$th column of matrix $A$ is then the minor ${M_{ij}}$ of the element ${a_{ij}}$ of matrix $A$.

### Cofactors

Let $A = {\left( {{a_{ij}}} \right)_{n \times n}}$ be a square matrix of order $n$. Then the cofactor of the element ${a_{ij}}$ of the matrix $A$ is denoted by ${C_{ij}}$ and is equal to ${( - 1)^{i + j}}{M_{ij}}$ where ${M_{ij}}$ is the minor of the element ${a_{ij}}$ of the matrix $A$.

Note that

$\left| {\begin{array}{*{20}{l}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\{{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right| = {a_{11}}{M_{11}} - {a_{12}}{M_{12}} + {a_{13}}{M_{13}} = {a_{11}}{C_{11}} + {a_{12}}{C_{12}} + {a_{13}}{C_{13}}$

## Adjoint and inverse of the matrix

Let $A = {\left( {{a_{ij}}} \right)_{n \times n}}$ be the square matrix. The adjoint of $A$ is defined as the matrix $adjA = {\left( {{b_{ij}}} \right)_{n \times n}}$ where ${b_{ij}} = {A_{ji}}$ and ${A_{ji}}$ is the cofactor of the $(j,i)$th element of $A$.

1. $A(\operatorname{adj} A) = (\operatorname{adj} A)A = |A|{I_n}$

2. $\operatorname{adj} (kA) = {k^{n - 1}}(\operatorname{adj} A)$

3. $\operatorname{adj} (AB) = (\operatorname{adj} B)(\operatorname{adj} A)$

Definition: A square matrix $A$ is said to be singular if $|A| = 0$ and non-singular if $|A| \ne 0$.

Definition: Inverse of a square matrix $A = {\left( {{a_{ij}}} \right)_{n \times n}}$ is the matrix $B = {\left( {{b_{ij}}} \right)_{n \times n}}$ such that

$AB = BA = {I_n}.$

In fact ${A^{ - 1}} = \dfrac{1}{{|A|}}(\operatorname{adj} A)$ if $|A| \ne 0\$

## Properties of Inverse

Let $A$ and $B$ be two invertible matrices of the same size.

1. Inverse of a matrix if it exists is unique.

2. $A{A^{ - 1}} = {A^{ - 1}}A = {I_n}$

3. ${\left( {{A^{ - 1}}} \right)^{ - 1}} = A$

4. ${(kA)^{ - 1}} = {k^{ - 1}}{A^{ - 1}}$ if $k \ne 0$

5. ${(AB)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}$

## Special Type of Matrix:

Symmetric matrix:  It is defined as the square matrix which is equal to its transpose

A square matrix $A = \left[ {{a_{ij}}} \right]$ is called symmetric matrix, if ${a_{ij}} = {a_{ji}}$ for all i, j or ${A^T} = A$

e.g.,

$\left[ {\begin{array}{*{20}{l}}a&h&g \\h&b&f \\g&f&c\end{array}} \right]$

Hence, it is a symmetric matrix.

Skew- symmetric matrix:  It is defined as a square matrix which is equal to the negative of its transpose.

A square matrix $A = \left[ {{a_{ij}}} \right]$ is called skew-symmetric matrix, if ${a_{ij}} = - {a_{ji}}$ for all $i,j$ or ${A^T} = - A$.

$\left[ {\begin{array}{*{20}{c}}0&h&g \\{ - h}&0&f \\{ - g}&{ - f}&0\end{array}} \right]$

All principal diagonal elements of a skew-symmetric matrix are always zero because of any diagonal element.

${{a}_{ij}}=-{{a}_{ij}}$

For  i = j(i.e for diagonal elements)

$\Rightarrow {{a}_{ii}}=0$

Thus, it is a skew-symmetric matrix.

## Properties of Symmetric and Skew-symmetric Matrices

(i) If $A$ is a square matrix, then $A + {A^T},A{A^T},{A^T}A$ are symmetric matrices, while $A - {A^T}$ is skew-symmetric matrix.

(ii) If ${\text{A}},{\text{B}}$ are two symmetric matrices, then

(a) $A \pm B, AB + BA$ are also symmetric matrices,

(b) $AB - BA$ is a skew-symmetric matrix,

(c) $AB$ is a symmetric matrix, when $AB = BA$.

(iii) If $A,B$ are two skew-symmetric matrices, then

(a) $A \pm B,AB - BA$ are skew-symmetric matrices,

(b) $AB + BA$ is a symmetric matrix.

(iv) Every square matrix $A$ can uniquely be expressed as sum of a symmetric and skew-symmetric matrix

i.e., $A = \left[ {\dfrac{1}{2}\left( {A + {A^T}} \right)} \right] + \left[ {\dfrac{1}{2}\left( {A - {A^T}} \right)} \right]$.

## Determinants

### Evaluation of Determinants

A determinant of order two is written as

$\left| {\begin{array}{*{20}{l}}{{a_{11}}}&{{a_{12}}} \\{{a_{21}}}&{{a_{22}}}\end{array}} \right|\quad \left( {{a_{ij}} \in {\mathbf{C}}\forall i,j} \right)$

and is equal to ${a_{11}}{a_{22}} - {a_{12}}{a_{21}}$.

A determinant of order three is written as

$\left| {\begin{array}{*{20}{l}}{{a_{11}}}&{{a_{12}}} \\{{a_{21}}}&{{a_{22}}}\end{array}} \right|\quad \left( {{a_{ij}} \in {\mathbf{C}}\forall i,j} \right)$

and is equal to

$a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \\ \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \\ \end{vmatrix}$

$= {a_{11}}{a_{22}}{a_{33}} + {a_{12}}{a_{23}}{a_{31}} + {a_{13}}{a_{32}}{a_{21}} - {a_{13}}{a_{31}}{a_{22}} - {a_{32}}{a_{23}}{a_{11}} - {a_{12}}{a_{21}}{a_{33}}$

## Properties of Determinants

1. Reflection Property

If the determinant's rows are changed into columns and the columns into rows, the determinant remains unchanged.

2. All-zero Property The determinant is zero if all of the elements in a row (column) are zero.

3. Proportionality [Repetition] Property

If the elements of a row (column) are proportional, or if the elements of another row (column) are proportional, then the determinant is zero.

4. Switching Property: The interchange of any two determinant rows (columns) changes its sign.

5. Scalar Multiple Property: If all the elements of a determinant's row (column) are multiplied by a non-zero constant, the determinant is multiplied by the same constant.

6. Property of Invariance

$\left| {\begin{array}{*{20}{l}}{{a_1}}&{{b_1}}&{{c_1}} \\{{a_2}}&{{b_2}}&{{c_2}} \\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right| = \left| {\begin{array}{*{20}{l}}{{a_1} + \alpha {b_1} + \beta {c_1}}&{{b_1}}&{{c_1}} \\{{a_2} + \alpha {b_2} + \beta {c_2}}&{{b_2}}&{{c_2}} \\{{a_3} + \alpha {b_3} + \beta {c_3}}&{{b_3}}&{{c_3}}\end{array}} \right|$

So, a determinant remains unaltered under an operation of the form ${C_i} \to {C_i} + \alpha {C_j} + \beta {C_k}$, where $j,k \ne i$, or an operation of the form ${R_i} \to {R_i} + \alpha {R_j} + \beta {R_k}$, where $j$, $k \ne i$.

7. Sum Property

$\left| {\begin{array}{*{20}{l}}{{a_1} + {b_1}}&{{c_1}}&{{d_1}} \\{{a_2} + {b_2}}&{{c_2}}&{{d_2}} \\{{a_3} + {b_3}}&{{c_3}}&{{d_3}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{{a_1}}&{{c_1}}&{{d_1}} \\{{a_2}}&{{c_2}}&{{d_2}} \\{{a_3}}&{{c_3}}&{{d_3}}\end{array}} \right| + \left| {\begin{array}{*{20}{c}}{{b_1}}&{{c_1}}&{{d_1}} \\{{b_2}}&{{c_2}}&{{d_2}} \\{{b_3}}&{{c_3}}&{{d_3}}\end{array}} \right|$

8. Factor Property: If a determinant $\Delta$ becomes zero when $x = \alpha$ is entered, then $(x - \alpha )$ is a factor of $\Delta$.

9. Triangle Property: If all of a determinant's elements above or below the main diagonal are zeros, the determinant is equal to the product of diagonal elements. That is to say,

$\left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}} \\0&{{b_2}}&{{b_3}} \\0&0&{{c_3}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{{a_1}}&0&0 \\{{a_2}}&{{b_2}}&0 \\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right| = {a_1}{b_2}{c_3}$

## Linear Equations

The system of linear homogeneous equations

${a_1}x + {b_1}y + {c_1}z = 0$

${a_2}x + {b_2}y + {c_2}z = 0$

${a_3}x + {b_3}y + {c_3}z = 0$

has a non-trivial solution (i.e., at least one of the $x,y,z$ is different from zero) if and only if $\Delta = 0,$ where

$\Delta = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{b_1}}&{{c_1}} \\{{a_2}}&{{b_2}}&{{c_2}} \\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right|$

If $\Delta \ne 0$, then the only solution of the above system of equations is $x = 0,y = 0$ and $z = 0$.

## Cramer's Rule

If $\Delta = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}} \\{{a_3}}&{{b_3}}&{{c_3}}\end{array}} \right| \ne 0$ then the solution of the system of linear equations

${a_1}x + {b_1}y + {c_1}z = {d_1}$

${a_2}x + {b_2}y + {c_2}z = {d_2}$

${a_3}x + {b_3}y + {c_3}z = {d_3}$  is given by $x = \dfrac{{{\Delta _1}}}{\Delta },y = \dfrac{{{\Delta _2}}}{\Delta },z = \dfrac{{{\Delta _3}}}{\Delta }$

where

${\Delta _1} = \left| {\begin{array}{*{20}{l}}{{d_1}}&{{b_1}}&{{c_1}} \\{{d_2}}&{{b_2}}&{{c_2}} \\{{d_3}}&{{b_3}}&{{c_3}}\end{array}} \right|,{\Delta _2} = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{d_1}}&{{c_1}} \\{{a_2}}&{{d_2}}&{{c_2}} \\{{a_3}}&{{d_3}}&{{c_3}}\end{array}} \right|$ and ${\Delta _3} = \left| {\begin{array}{*{20}{l}}{{a_1}}&{{b_1}}&{{d_1}} \\{{a_2}}&{{b_2}}&{{d_2}} \\{{a_3}}&{{b_3}}&{{d_3}}\end{array}} \right|$

## Probability

### Definitions

1. A random experiment is one in which all of the experiment's outcomes are known ahead of time, and (ii) the exact outcome of any specific performance of the experiment is unpredictable, i.e. not known ahead of time.

2.  A sample space, denoted by $S,$associated with a random experiment is a set of points such that each element of $S,$ denotes an experiment outcome, and (ii) any experiment performance results in an outcome that corresponds to exactly one element of $S$. In other words, the sample space contains all of the possible outcomes of the random experiment.  The elements of the sample space associated with a random experiment are called the elementary or simple or indecomposable events of that experiment.

3.  A random experiment is associated with an event, which is a subset of the sample space $S$ (associated with that experiment). We say that an event $E$ ( $\subseteq S$ ) has occurred if the experiment's outcome $\omega$ is an element of $E$. If the experiment's outcome $\omega$ is such that $\omega \notin E$, we say that the event $E$did not occur.

The empty set $\emptyset$, which is a subset of every set, also represents an event and is said to be an impossible event. The set $S$ is a subset of $S$ and represents the sure event.

## Classical definition of Probability

If a random experiment produces $N$ mutually exclusive, equally likely, and exhaustive outcomes, of which $n$ are favourable to the occurrence of an event $A$, then the probability of occurrence of $A$ is given by $P(A)$.

$P(A) = \dfrac{n}{N}$

## A few theorems on probability

1. If $A$ and $B$ are two mutually exclusive events, then $P(A \cup B) = P(A) + P(B)$.

2. If $A$ is any event, then $P\left( {{A^\prime }} \right) = 1 - P(A)$

Here, P is the probability. Also, $P\left( {{A^\prime }} \right)$ is termed as a complement of P(A).

3. If $A$ and $B$ are two events, then

$P\left( {A \cap {B^\prime }} \right) = P(A) - P(A \cap B)$

4. If $A$ and $B$ are two events, then $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

5. If $A$ and $B$ are two events, then $P$ (exactly one of $A,{\text{ }}B$ $occurs)$

Event A and B

From figure,

$= P\left[ {\left( {A \cap {B^\prime }} \right) \cup \left( {{A^\prime } \cap B} \right)} \right]$

$= P(A) - P(A \cap B) + P(B) - P(A \cap B)$

$= P(A) + P(B) - 2P(A \cap B)$

$= P(A \cup B) - P(A \cap B){\text{ }}$

${\text{Also, }}P({\text{ exactly one of }}A,B{\text{ occurs }})$

$= P\left( {A \cap {B^\prime }} \right) + P\left( {{A^\prime } \cap B} \right)$

$= P\left( {{B^\prime }} \right) - P\left( {{A^\prime } \cap {B^\prime }} \right) + P\left( {{A^\prime }} \right) - P\left( {{A^\prime } \cap {B^\prime }} \right)$

$= P\left( {{A^\prime }} \right) + P\left( {{B^\prime }} \right) - 2P\left( {{A^\prime } \cap {B^\prime }} \right)$

$= P\left( {{A^\prime } \cup {B^\prime }} \right) - P\left( {{A^\prime } \cap {B^\prime }} \right)$

6. If $A$ and $B$ are two events

$P\left( {{A^\prime } \cup {B^\prime }} \right) = 1 - P(A \cap B\,\,{\kern 1pt} {\text{and }}P\left( {{A^\prime } \cap {B^\prime }} \right) = 1 - P(A \cup B)$

## Independent Events

We say that two events $A$ and $B$ are independent if the occurrence or non-occurrence of  does not affect the probability of occurrence or non-occurrence of , that is, if  provided $P(A) \ne 0$

We say that two things happened. $A$ and $B$ are independent if the occurrence or non-occurrence of $A(B)$ has no effect on the probability of occurrence or non-occurrence of $B(A)$, that is, if $P(B\mid A) = P(B)$if  provided $P(A) \ne 0$

$\Rightarrow \dfrac{{P(B \cap A)}}{{P(A)}} = P(B) \Rightarrow P(A \cap B) = P(A)P(B)$

Thus, two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A)P(B)$

If two events are not independent, they are said to be dependent.

Mean and Variance of a Random Variable

The expectation (mean) of the random variable $X$ is defined as

$E(X) = \sum\limits_{i = 1}^n {{p_i}} {x_i}$ and the variance of $X$ is defined as $\operatorname{var} (X) = \sum\limits_{i = 1}^n {{p_i}} {\left( {{x_i} - E(X)} \right)^2} = \sum\limits_{i = 1}^n {{p_i}} x_i^2 - {(E(X))^2}$

## Bayes Rule

Let $S$ be a sample space and $H_{1}, \cdots, H_{n}$ be $n$ mutually exclusive events such that $\bigcup_{j=1}^{n} H_{j}=S$ and $P\left(H_{j}\right)>0$ for $j=1,2, \cdots, n$. We can think of the $H_{j}$ 's as the 'causes' (or the hypotheses) that lead to the outcome of an experiment. The probabilities $P\left(H_{j}\right), j=1,2, \cdots, n$ are called prior probabilities. Suppose the experiment results in an outcome of event $A$, where $P(A)>0$.

## Bayes Theorem

Let $\left\{H_{j}\right\}$ be mutually exclusive events such that $P\left(H_{j}\right)>0$ for $j=1,2, \cdots, n$ and $S=\bigcup_{j=1}^{n} H_{j}$. Let $A$ be an event with $\mid P(A)>0 .$ Then for $j=1,2, \cdots, n$

$P\left(H_{j} \mid A\right)=\dfrac{P\left(H_{j}\right) P\left(A \mid H_{j}\right)}{\sum_{k=1}^{n} P\left(H_{k}\right) P\left(A \mid H_{k}\right)}$

## Probability Distribution of a Random Variable

Description giving the values of the random variable along with the corresponding probab is called the probability distribution of the random variable $X$.

In general, the probability distribution of a random variable $X$ is defined as follows:

The probability distribution of a random variable $X$ is the system of numbers

$\mathrm{X}: x_{1}, x_{2}, \cdots x_{n}$

$\mathrm{P}(\mathrm{X}) : p_{1}, p_{2}, \cdots, p_{n}$

where,

$p_{i}>0, \quad \sum_{i=1}^{n} p_{i}=1, i=1,2, \ldots, n$

The real numbers $x_{1}, x_{2}, \ldots, x_{n}$ are the possible values of the random variable $\mathrm{X}$ and $p_{\mathrm{i}}(i=1,2, \ldots, n)$ is the probability of the random variable $\mathrm{X}$ taking the value $x_{i}$ i.e., $\mathrm{P}\left(\mathrm{X}=x_{i}\right)=p_{i}$

## Bernoulli Trials

• The outcome of any trial is independent of the outcome of any other trial. In such trials, the probability of success or failure remains constant. Such independent trials, which have only two outcomes, are usually referred to as 'success' or 'failure' are called Bernoulli trials.

• Trials of a random experiment are called-Bernoulli trials if they satisfy the following conditions:

There should be a finite number of trials.

• The trials should be independent.

• Each trial has exactly two outcomes: success or failure.

• The probability of success remains the same in each trial.

Example: 30 Six balls are drawn successively from an urn containing 7 red and 9 black balls. Tell whether or not the trials of drawing balls are Bernoulli trials when after each draw, the ball drawn is (i) replaced (ii) not replaced in the urn.

Solution

(i) The number of trials is finite. When the driving is done with replacement, the probability of success (say, red ball) is $\mathrm{P}=\dfrac{7}{16}$, which is same for all six trials (draws). Hence, the drawing of balls with replacements are Bernoulli trials.

(ii) When the drawing is done without replacement, the probability of success (i.e., red ball) in first trial is $\dfrac{7}{16}$, in 2nd trial is $\dfrac{6}{15}$ if the first ball drawn is red or $\dfrac{7}{15}$ if the first ball drawn is black and so on. Clearly, the probability of success is not the same for all trials. hence the trials are not Bernoulli trials

## Importance of Maths Vector Algebra

When a physical quantity has magnitude and direction, it is called a vector quantity. Previously, we have learned how to calculate only the magnitude of different scalar quantities using simple algebraic formulas. In this chapter, the introduction of direction and magnitude will alter the formulas and their relevant operations in a certain way.

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## FAQs on JEE Advanced 2023 Revision Notes for Algebra

1. What is a scalar quantity?

A physical quantity that has only magnitude but does not have direction is called a scalar quantity. Mass is a scalar quantity as it has magnitude and does not have direction.

2. What is the difference between vector and scalar quantity?

The only difference between a vector and a scalar quantity is that the former has direction along with magnitude. For instance, velocity is vector quantity but speed is not.

3. What is the difference between scalar and vector addition?

In scalar addition, similar quantities are added but in vector addition, the similar physical quantities in a particular direction can only be added.

4. What is a 3-dimensional vector?

A physical quantity in a 3D space having components on three axes, x, y, and z, is called a 3D vector.