“A person who can, within a year, solve $x^2-92y^2=1$ is a mathematician” — Brahmagupta
Niccolo Fontana Tartaglia was an Italian mathematician, engineer, surveyor and bookkeeper from then Republic of Venice (now Italy). He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs; his work was later validated by Galileo’s studies on falling bodies.
Tartaglia along with Cardano were credited for finding methods to solve any third degree polynomials called cubic equations. He also provided a nice formula for calculating volume of any tetrahedron using distance between pairs of its four vertices.
Algebra can be thought of as the next level of study of numbers. If we need to determine anything subject to certain specific conditions, then we need Algebra. In that sense, the study of Algebra is considered as “Science of determining unknowns”. During third century AD(CE) Diophantus of Alexandria wrote a monumental book titled “Arithmetica” in thirteen volumes of which only six has survived. This book is the first source where the conditions of the problems are stated as equations and they are eventually solved. Diophantus realized that for many real life situation problems, the variables considered are usually positive integers.
The term “Algebra” has evolved as a misspelling of the word ‘al-jabr’ from one of the important work titled Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala (“The Compendious Book on Calculation by Completion and Balancing”) written by Persian Mathematician Al-Khwarizmi of 9th Century AD(CE). Since Al-Khwarizmi’s Al-Jabr book provided the most appropriate methods of solving equations, he is hailed as “Father of Algebra”.
In the earlier classes, we had studied several important concepts in Algebra. In this class, we will continue our journey to understand other important concepts which will be of much help in solving problems of greater scope. Real understanding of these ideas will benefit much in learning higher mathematics in future classes.
Let us recall solving a pair of linear equations in two variables.
Definition — Linear Equation in two variables
Any first degree equation containing two variables $x$ and $y$ is called a linear equation in two variables. The general form of linear equation in two variables $x$ and $y$ is $ax+by+c=0$, where atleast one of $a,b$ is non-zero and $a,b,c$ are real numbers.
Note that linear equations are first degree equations in the given variables.
Note
$xy-7=3$ is not a linear equation in two variables since the term $xy$ is of degree 2.
A linear equation in two variables represent a straight line in $xy$ plane.
The father’s age is six times his son’s age. Six years hence, the age of father will be four times his son’s age. Find the present ages (in years) of the son and father.
Solution Let the present age of father be $x$ years and the present age of son be $y$ years.
Substituting $y=-\dfrac45$ in (2), $x-\dfrac45=1$, we get $x=\dfrac95$.
Therefore, $x=\dfrac95$, $y=-\dfrac45$.
Fig. 3.1
3.2 Simultaneous Linear Equations in Three Variables#
Right from the primitive needs of calculating amount spent for various items in a super market, finding ages of people under specific conditions, finding path of an object when it is thrown upwards at an angle, Algebra plays a vital role in our daily life.
Any point in the space can be determined uniquely by knowing its latitude, longitude and altitude. Hence to locate the position of an object at a particular place situated on the Earth, three satellites are positioned to arrive three equations. Among these three equations, we get two linear equations and one quadratic (second degree) equation. Hence we can solve for the variables latitude, longitude and altitude to uniquely fix the position of any object at a given point of time. This is the basis of Global Positioning System (GPS). Hence the concept of linear equations in three variables is used in GPS systems.
Fig. 3.2
3.2.1 System of Linear Equations in Three Variables#
In earlier classes, we have learnt different methods of solving Simultaneous Linear Equations in two variables. Here we shall learn to solve the system of linear equations in three variables namely, $x,y$ and $z$. The general form of a linear equation in three variables $x,y$ and $z$ is $ax+by+cz+d=0$ where $a,b,c,d$ are real numbers, and atleast one of $a,b,c$ is non-zero.
Note
A linear equation in two variables of the form $ax+by+c=0$, represents a straight line.
A linear equation in three variables of the form $ax+by+cz+d=0$, represents a plane.
Fig. 3.3(i)
Fig. 3.3(ii)
General Form: A system of linear equations in three variables $x,y,z$ has the general form
Each equation in the system represents a plane in three dimensional space and solution of the system of equations is precisely the point of intersection of the three planes defined by the three linear equations of the system. The system may have only one solution, infinitely many solutions or no solution depending on how the planes intersect one another.
The figures presented below illustrate each of these possibilities.
Fig. 3.4 — Only one solution, infinitely many solutions, no solution
Procedure for solving system of linear equations in three variables#
By taking any two equations from the given three, first multiply by some suitable non-zero constant to make the co-efficient of one variable (either $x$ or $y$ or $z$) numerically equal.
Eliminate one of the variables whose co-efficients are numerically equal from the equations.
Eliminate the same variable from another pair.
Now we have two equations in two variables.
Solve them using any method studied in earlier classes.
The remaining variable is then found by substituting in any one of the given equations.
Note
If you obtain a false equation such as $0=1$, in any of the steps then the system has no solution.
If you do not obtain a false solution, but obtain an identity, such as $0=0$ then the system has infinitely many solutions.
In an interschool athletic meet, with total of 24 individual prizes, securing a total of 56 points, a first place secures 5 points, a second place secures 3 points, and a third place secures 1 point. Having as many third place finishers as first and second place finishers, find how many athletes finished in each place.
Solution Let the number of I, II and III place finishers be $x,y$ and $z$ respectively.
Total number of prizes $=24$; Total number of points $=56$.
Hence, the linear equations in three variables are
The sum of thrice the first number, second number and twice the third number is 5. If thrice the second number is subtracted from the sum of first number and thrice the third we get 2. If the third number is subtracted from the sum of twice the first, thrice the second, we get 1. Find the numbers.
Solution Let the three numbers be $x,y,z$.
From the given data we get the following equations,
Vani, her father and her grand father have an average age of 53. One-half of her grand father’s age plus one-third of her father’s age plus one fourth of Vani’s age is 65. Four years ago if Vani’s grandfather was four times as old as Vani then how old are they all now?
The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three digit number?
There are 12 pieces of five, ten and twenty rupee currencies whose total value is ₹105. When first 2 sorts are interchanged in their numbers its value will be increased by ₹20. Find the number of currencies in each sort.
3.3.1 Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of Polynomials#
In our previous class we have learnt how to find the GCD (HCF) of second degree and third degree expressions by the method of factorization. Now we shall learn how to find the GCD of the given polynomials by the method of long division.
As discussed in Chapter 2, (Numbers and Sequences) to find GCD of two positive integers using Euclidean Algorithm, similar techniques can be employed for two given polynomials also.
The following procedure gives a systematic way of finding Greatest Common Divisor of two given polynomials $f(x)$ and $g(x)$.
Step 1: First, divide $f(x)$ by $g(x)$ to obtain $f(x)=g(x)q(x)+r(x)$ where $q(x)$ is the quotient and $r(x)$ is the remainder. Then, $\deg[r(x)]<\deg[g(x)]$.
Step 2: If the remainder $r(x)$ is non-zero, divide $g(x)$ by $r(x)$ to obtain $g(x)=r(x)q_1(x)+r_1(x)$ where $r_1(x)$ is the new remainder. Then $\deg[r_1(x)]<\deg[r(x)]$. If the remainder $r_1(x)$ is zero, then $r(x)$ is the required GCD.
Step 3: If $r_1(x)$ is non-zero, then continue the process till we get zero as remainder. The divisor at this stage will be the required GCD.
We write $\operatorname{GCD}[f(x),g(x)]$ to denote the GCD of the polynomials $f(x),g(x)$.
Note
If $f(x)$ and $g(x)$ are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend. In case, if both have the same coefficient then compare the next least degree’s coefficient and proceed with the division.
Progress Check
When two polynomials of same degree has to be divided, __________ should be considered to fix the dividend and divisor.
If $r(x)=0$ when $f(x)$ is divided by $g(x)$ then $g(x)$ is called ________ of the polynomials.
If $f(x)=g(x)q(x)+r(x)$, _________ must be added to $f(x)$ to make $f(x)$ completely divisible by $g(x)$.
If $f(x)=g(x)q(x)+r(x)$, _________ must be subtracted to $f(x)$ to make $f(x)$ completely divisible by $g(x)$.
The Least Common Multiple of two or more algebraic expressions is the expression of highest degree (or power) such that the expressions exactly divide it.
Consider the following simple expressions $a^3b^2$, $a^2b^3$.
For these expressions $\operatorname{LCM}=a^3b^3$.
To find LCM by factorization method:
(i) Each expression is first resolved into its factors.
(ii) The highest power of the factors will be the LCM.
(iii) If the expressions have numerical coefficients, find their LCM.
(iv) The product of the LCM of factors and coefficient is the required LCM.
An expression is called a rational expression if it can be written in the form $\dfrac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials and $q(x)\ne0$. A rational expression is the ratio of two polynomials.
The following are examples of rational expressions.
The rational expressions are applied for describing distance-time, modeling multi-task problems, to combine workers or machines to complete a job schedule and much more.
We have studied the concepts of addition, subtraction, multiplication and division of rational numbers in previous classes. Now, let us generalize the above for rational expressions.
In other words, the product of two rational expression is the product of their numerators divided by the product of their denominators and the resulting expression is then reduced to its lowest form.
Thus division of one rational expression by other is equivalent to the product of first and reciprocal of the second expression. If the resulting expression is not in its lowest form then reduce to its lowest form.
Progress Check
Find the unknown expression in the following figures.
If $x=\dfrac{a^2+3a-4}{3a^2-3}$ and $y=\dfrac{a^2+2a-8}{2a^2-2a-4}$ find the value of $x^2y^{-2}$.
If a polynomial $p(x)=x^2-5x-14$ is divided by another polynomial $q(x)$ we get $\dfrac{x-7}{x+2}$, find $q(x)$.
Activity 1
(i) The length of a rectangular garden is the sum of a number and its reciprocal. The breadth is the difference of the square of the same number and its reciprocal. Find the length, breadth and the ratio of the length to the breadth of the rectangle.
Rectangular garden
(ii) Find the ratio of the perimeter to the area of the given triangle.
Addition and Subtraction of Rational Expressions with unlike Denominators#
(i) Determine the Least Common Multiple of the denominator.
(ii) Rewrite each fraction as an equivalent fraction with the LCM obtained in step (i). This is done by multiplying both the numerators and denominator of each expression by any factors needed to obtain the LCM.
(iii) Follow the same steps given for doing addition or subtraction of the rational expression with like denominators.
Progress Check
Write an expression that represents the perimeter of the figure and simplify.
Triangle for perimeter calculation
Find the base of the given parallelogram whose perimeter is $\dfrac{4x^2+10x-50}{(x-3)(x+5)}$.
Simplify (i) $\dfrac{x(x+1)}{x-2}+\dfrac{x(1-x)}{x-2}$ (ii) $\dfrac{x+2}{x+3}+\dfrac{x-1}{x-2}$ (iii) $\dfrac{x^3}{x-y}+\dfrac{y^3}{y-x}$.
Simplify (i) $\dfrac{(2x+1)(x-2)}{x-4}-\dfrac{2x^2-5x+2}{x-4}$ (ii) $\dfrac{4x}{x^2-1}-\dfrac{x+1}{x-1}$.
Subtract $\dfrac1{x^2+2}$ from $\dfrac{2x^3+x^2+3}{(x^2+2)^2}$.
Which rational expression should be subtracted from $\dfrac{x^2+6x+8}{x^3+8}$ to get $\dfrac3{x^2-2x+4}$?
If $A=\dfrac{2x+1}{2x-1}$, $B=\dfrac{2x-1}{2x+1}$ find $\dfrac1{A-B}-\dfrac{2B}{A^2-B^2}$.
If $A=\dfrac{x}{x+1}$, $B=\dfrac1{x+1}$, prove that $\dfrac{(A+B)^2+(A-B)^2}{A\div B}=\dfrac{2(x^2+1)}{x(x+1)^2}$.
Pari needs 4 hours to complete a work. His friend Yuvan needs 6 hours to complete the same work. How long will it take to complete if they work together?
Iniya bought 50 kg of fruits consisting of apples and bananas. She paid twice as much per kg for the apple as she did for the banana. If Iniya bought ₹ 1800 worth of apples and ₹ 600 worth bananas, then how many kgs of each fruit did she buy?
The square root of a given positive real number is another number which when multiplied with itself is the given number.
Similarly, the square root of a given expression $p(x)$ is another expression $q(x)$ which when multiplied by itself gives $p(x)$, that is, $q(x)\cdot q(x)=p(x)$.
So, $|q(x)|=\sqrt{p(x)}$ where $|q(x)|$ is the absolute value of $q(x)$.
The following two methods are used to find the square root of a given expression:
(i) Factorization method (ii) Division method.
Progress Check
Is $x^2+4x+4$ a perfect square?
What is the value of $x$ in $3\sqrt{x}=9$?
The square root of $361x^4y^2$ is _______.
$\sqrt{a^2x^2+2abx+b^2}=$ _______.
If a polynomial is a perfect square then, its factors will be repeated _______ number of times (odd / even).
3.5.1 Find the Square Root by Factorization Method#
Arab mathematician Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book ‘Liber Embadorum’ published in 1145 AD(CE) which is the first book published in Europe to give the complete solution of a quadratic equation.
For a period of more than three thousand years beginning from early civilizations to current times, humanity knew how to solve a general quadratic equation in terms of its co-efficients by using four arithmetical operations and extraction of roots. This process is called “Solving by Radicals”. Huge amount of research has been carried to this day in solving various types of equations.
An expression of degree $n$ in variable $x$ is $a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n$ where $a_0\ne0$ and $a_1,a_2,\ldots,a_n$ are real numbers. $a_0,a_1,a_2,\ldots,a_n$ are called coefficients of the expression.
In particular an expression of degree 2 is called a Quadratic Expression which is expressed as $p(x)=ax^2+bx+c$, $a\ne0$ and $a,b,c$ are real numbers.
Let $ax^2+bx+c=0$, $(a\ne0)$ be a quadratic equation. The values of $x$ such that the expression $ax^2+bx+c$ becomes zero are called roots of the quadratic equation $ax^2+bx+c=0$.
$$
\alpha\beta=\left(\frac{-b+\sqrt{b^2-4ac}}{2a}\right)\times\left(\frac{-b-\sqrt{b^2-4ac}}{2a}\right)=\frac ca.
$$
Since, $(x-\alpha)$ and $(x-\beta)$ are factors of $ax^2+bx+c=0$,
We have $(x-\alpha)(x-\beta)=0$.
Hence, $x^2-(\alpha+\beta)x+\alpha\beta=0$.
That is, $x^2-(\text{sum of roots})x+\text{product of roots}=0$ is the general form of the quadratic equation when the roots are given.
Note
$ax^2+bx+c=0$ can equivalently be expressed as $x^2+\dfrac bax+\dfrac ca=0$, since $a\ne0$.
Activity 2
Consider a rectangular garden in front of a house, whose dimensions are $(2k+6)$ metre and $k$ metre. A smaller rectangular portion of the garden of dimensions $k$ metre and 3 metres is leveled. Find the area of the garden, not leveled.
Rectangular garden with a smaller leveled rectangular portion
We have already learnt how to solve linear equations in one, two and three variable(s). Recall that the values of the variables which satisfies a given equation are called its solution(s). In this section, we are going to study three methods of solving quadratic equation, namely factorization method, completing the square method and using formula.
Solving a quadratic equation by factorization method.#
We follow the steps provided below to solve a quadratic equation through factorization method.
Step 1: Write the equation in general form $ax^2+bx+c=0$.
Step 2: By splitting the middle term, factorize the given equation.
Step 3: After factoring, the given quadratic equation can be written as product of two linear factors.
Step 4: Equate each linear factor to zero and solve for $x$.
These values of $x$ gives the roots of the equation.
$m+2=0\Rightarrow m=-2$ or $2m+15=0$ we get, $m=\dfrac{-15}2$.
Therefore, the roots are $-2,\dfrac{-15}2$.
Some equations which are not quadratic can be solved by reducing them to quadratic equations by suitable substitutions. Such examples are illustrated below.
Solve the following quadratic equations by factorization method.
(i) $4x^2-7x-2=0$
(ii) $3(p^2-6)=p(p+5)$
(iii) $\sqrt{a(a-7)}=3\sqrt2$
(iv) $\sqrt2x^2+7x+5\sqrt2=0$
(v) $2x^2-x+\dfrac18=0$
The number of volleyball games that must be scheduled in a league with $n$ teams is given by $G(n)=\dfrac{n^2-n}2$ where each team plays with every other team exactly once. A league schedules 15 games. How many teams are in the league?
Solving a Quadratic Equation by Completing the Square Method#
In deriving the formula for the roots of a quadratic equation we used completing the squares method. The same technique can be applied in solving any given quadratic equation through the following steps.
Step 1: Write the quadratic equation in general form $ax^2+bx+c=0$.
Step 2: Divide both sides of the equation by the coefficient of $x^2$ if it is not 1.
Step 3: Shift the constant term to the right hand side.
Step 4: Add the square of one-half of the coefficient of $x$ to both sides.
Step 5: Write the left hand side as a square and simplify the right hand side.
Step 6: Take the square root on both sides and solve for $x$.
$$
\begin{aligned}
x^2-3x-2&=0\\
x^2-3x&=2\quad(\text{Shifting the Constant to RHS})\\
x^2-3x+\left(\frac32\right)^2&=2+\left(\frac32\right)^2\quad\left(\text{Add }\left[\frac12(\text{co-efficient of }x)\right]^2\text{ to both sides}\right)\\
\left(x-\frac32\right)^2&=\frac{17}4\quad(\text{writing the LHS as complete square})\\
x-\frac32&=\pm\frac{\sqrt{17}}2\quad(\text{Taking the square root on both sides})\\
x&=\frac32+\frac{\sqrt{17}}2\quad\text{or}\quad x=\frac32-\frac{\sqrt{17}}2.
\end{aligned}
$$
The formula for finding roots of a quadratic equation $ax^2+bx+c=0$ (derivation given in section 3.6.2) is $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Do You Know?
The formula for finding roots of a quadratic equation was known to Ancient Babylonians, though not in a form as we derived. They found the roots by creating the steps as a verse, which is a common practice at their times. Babylonians used quadratic equations for deciding to choose the dimensions of their land for agriculture.
Solve the following quadratic equations by completing the square method.
(i) $9x^2-12x+4=0$
(ii) $\dfrac{5x+7}{x-1}=3x+2$
Solve the following quadratic equations by formula method.
(i) $2x^2-5x+2=0$
(ii) $\sqrt2f^2-6f+3\sqrt2=0$
(iii) $3y^2-20y-23=0$
(iv) $36y^2-12ay+(a^2-b^2)=0$
A ball rolls down a slope and travels a distance $d=t^2-0.75t$ feet in $t$ seconds. Find the time when the distance travelled by the ball is 11.25 feet.
A ladder 17 feet long is leaning against a wall. If the ladder, vertical wall and the floor from the bottom of the wall to the ladder form a right triangle, find the height of the wall where the top of the ladder meets if the distance between bottom of the wall to bottom of the ladder is 7 feet less than the height of the wall?
Solution
Let the height of the wall $AB=x$ feet.
As per the given data $BC=(x-7)$ feet.
In the right triangle $ABC$, $AC=17$ ft, $BC=(x-7)$ feet.
A flock of swans contained $x^2$ members. As the clouds gathered, $10x$ went to a lake and one-eighth of the members flew away to a garden. The remaining three pairs played about in the water. How many swans were there in total?
Solution
As given there are $x^2$ swans.
As per the given data $x^2-10x-\dfrac18x^2=6$ we get, $7x^2-80x-48=0$.
A passenger train takes 1 hr more than an express train to travel a distance of 240 km from Chennai to Virudhachalam. The speed of the express train is more than that of the passenger train by 20 km per hour. Find the average speed of both the trains.
Solution
Let the average speed of passenger train be $x$ km/hr.
Then the average speed of express train will be $(x+20)$ km/hr.
Time taken by the passenger train to cover distance of 240 km $=\dfrac{240}x$ hr.
Time taken by express train to cover distance of 240 km $=\dfrac{240}{x+20}$ hr.
If the difference between a number and its reciprocal is $\dfrac{24}5$, find the number.
A garden measuring 12 m by 16 m is to have a pedestrian pathway that is ‘$w$’ meters wide installed all the way around so that it increases the total area to 285 m². What is the width of the pathway?
A bus covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more it would have taken 30 minutes less for the journey. Find the original speed of the bus.
A girl is twice as old as her sister. Five years hence, the product of their ages (in years) will be 375. Find their present ages.
A pole has to be erected at a point on the boundary of a circular ground of diameter 20 m in such a way that the difference of its distances from two diametrically opposite fixed gates P and Q on the boundary is 4 m. Is it possible to do so? If answer is yes at what distance from the two gates should the pole be erected?
From a group of $2x^2$ black bees, square root of half of the group went to a tree. Again eight-ninth of the bees went to the same tree. The remaining two got caught up in a fragrant lotus. How many bees were there in total?
Music is been played in two opposite galleries with certain group of people. In the first gallery a group of 4 singers were singing and in the second gallery 9 singers were singing. The two galleries are separated by the distance of 70 m. Where should a person stand for hearing the same intensity of the singers voice? (Hint: The ratio of the sound intensity is equal to the square of the ratio of their corresponding distances).
There is a square field whose side is 10 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹3 and ₹4 per square metre respectively is ₹364. Find the width of the gravel path.
The hypotenuse of a right angled triangle is 25 cm and its perimeter 56 cm. Find the length of the smallest side.
The roots of the quadratic equation $ax^2+bx+c=0$, $a\ne0$ are found using the formula $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here, $b^2-4ac$ called as the discriminant (which is denoted by $\Delta$) of the quadratic equation, decides the nature of roots as follows.
$$
\alpha+\beta=\frac{-b}a=\frac{-\text{Co-efficient of }x}{\text{Co-efficient of }x^2}
$$$$
\alpha\beta=\frac ca=\frac{\text{Constant term}}{\text{Co-efficient of }x^2}.
$$
If $\alpha,\beta$ are the roots of the equation $3x^2+7x-2=0$, find the values of (i) $\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}$ (ii) $\dfrac{\alpha^2}{\beta}+\dfrac{\beta^2}{\alpha}$.
If $\alpha,\beta$ are the roots of the equation $2x^2-x-1=0$, then form the equation whose roots are (i) $\dfrac1\alpha,\dfrac1\beta$ (ii) $\alpha^2\beta,\beta^2\alpha$ (iii) $2\alpha+\beta,2\beta+\alpha$.
$$
\text{Sum of the roots}=\frac1\alpha+\frac1\beta=\frac{\alpha+\beta}{\alpha\beta}=\frac{\frac12}{-\frac12}=-1.
$$$$
\text{Product of the roots}=\frac1\alpha\times\frac1\beta=\frac1{\alpha\beta}=\frac1{-\frac12}=-2.
$$
The required equation is $x^2-(\text{Sum of the roots})x+(\text{Product of the roots})=0$.
$$
x^2-(-1)x-2=0\Rightarrow x^2+x-2=0.
$$
(ii) Given roots are $\alpha^2\beta,\beta^2\alpha$.
$$
\text{Sum of the roots }\alpha^2\beta+\beta^2\alpha=\alpha\beta(\alpha+\beta)=-\frac12\left(\frac12\right)=-\frac14.
$$$$
\text{Product of the roots }(\alpha^2\beta)\times(\beta^2\alpha)=\alpha^3\beta^3=(\alpha\beta)^3=\left(-\frac12\right)^3=-\frac18.
$$
The required equation is $x^2-(\text{Sum of the roots})x+(\text{Product of the roots})=0$.
$$
\text{Sum of the roots }2\alpha+\beta+2\beta+\alpha=3(\alpha+\beta)=3\left(\frac12\right)=\frac32.
$$$$
\begin{aligned}\text{Product of the roots}&=(2\alpha+\beta)(2\beta+\alpha)=4\alpha\beta+2\alpha^2+2\beta^2+\alpha\beta\\&=5\alpha\beta+2(\alpha^2+\beta^2)=5\alpha\beta+2[(\alpha+\beta)^2-2\alpha\beta]\\&=5\left(-\frac12\right)+2\left[\frac14-2\times-\frac12\right]=0.\end{aligned}
$$
The required equation is $x^2-(\text{Sum of the roots})x+(\text{Product of the roots})=0$.
Every day, Harini travels from her home, cycling at a uniform speed, to reach her school. You can state this mathematically by an equation $d=rt$, where $d$ stands for distance travelled at any time $t$ and $r$ is the uniform rate of speed.
Suppose you want to find the distance covered by her at a speed of 20 km per hour when she has cycled for fifteen minutes.
$r=20$ and $t=\dfrac14$ (how?) and we find $d$ to be $rt=20\times\dfrac14=5$ km.
Here, we say that $d$ is a dependent variable and $r$ and $t$ are independent variables. As the distance $d$ travelled depends upon the rate $r$ and time used $t$.
Thus, an independent variable represents a quantity that is manipulated in a given situation where as a dependent variable represents a quantity whose value depends on how the independent variable is manipulated.
Equations that describe the relationship between two variables in a sentence express the variation between those variables.
Consider the monthly income of Server Suresh who works in a hotel where he is paid ₹50 per hour.
There are two variables here. One is the monthly income and the other is the number of hours he works. Which among the two is the independent variable?
You know how to calculate the area of a circle when the length of its radius is given. If the area required is $A$ and the length of radius is $r$, then the formula
$$
A=\pi r^2
$$
gives the required result. Here, the area $A$ depends upon the length $r$ of radius; thus $A$ is a dependent variable and $r$ is the independent variable. But what can we say about $\pi$? It is a number that remains the same in all the situations. It is constant.
A constant is a quantity that assumes a fixed value throughout in a specific mathematical context.
When two things are in proportion, there is a relation between them, due to which, if the value of one of them changes, the value of the other also changes. We look into two types of variations here:
When you go to the market, to buy more apples, you’ll have to spend more amount of money. If the cost of one kg of apples is ₹200, you pay as follows:
Weight (Kg)
1
2
3
4
5
Cost (₹)
200
400
600
800
1000
Fig. 3.8
You find that $\dfrac1{200}=\dfrac2{400}=\dfrac3{600}=\dfrac4{800}=\dfrac5{1000}=\cdots$.
This kind of proportionate variation is known as Direct variation. Here to find the cost, the weight is multiplied by the constant 200.
If we denote the variable weight as $x$ and the variable cost as $y$ we can express this algebraically as $y=200x$. The multiplying constant here is 200.
If $\dfrac yx=k$ where $k$ is a positive number (a constant), then $x$ and $y$ are said to vary directly. Here, $k$ is known as the constant of proportionality.
Do You Know — Mathematics in real life: This figure shows that doubling the force doubles the displacement. This is a consequence of what is known as Hooke’s law. It states $F=kx$ where $F$ is the force needed to produce a displacement of $x$ in the position of a spring. To double the displacement, you double the force on the spring; the constant of proportionality $k$ depends on the stiffness of the spring. So this is an example of a direct proportionality.
To identify direct variation is to look at the equation and determine if it is of the form $y=kx$, where $k$ is the constant of proportionality. Thus, an equation like $y=5x$ will always indicate direct proportion among variables.
Thinking Corner
What can you say if the variables $x$ and $y$ are related by the equation $3y-7x=0$? It also indicates direct variation. How? Think about it. In that case, what is the constant of proportionality?
Observe this graph:
The distance travelled and the time taken are proportional, but how do we know that?
Note that
(i) The graph is a straight line.
(ii) The line passes through the origin. When both of these features are present we know that the two quantities on the graph must be directly proportional.
Do you see this in the graph?
Time (in minutes)
4
8
12
16
Distance (in km)
8
16
24
32
Fig. 3.9
If one variable doubles, the other also doubles. From this you can see the relation $d=rt$ and it is easy to guess the constant of proportionality.
Varshika drew 6 circles with different sizes. Draw a graph for the relationship between the diameter and circumference (approximately related) of each circle as shown in the table and use it to find the circumference of a circle when its diameter is 6 cm.
Diameter ($x$) cm
1
2
3
4
5
Circumference ($y$) cm
3.1
6.2
9.3
12.4
15.5
Solution:
From the table, we found that as $x$ increases, $y$ also increases. Thus, the variation is a direct variation.
Let $y=kx$, where $k$ is a constant of proportionality.
The distance between Chennai and Madurai is (nearly) 480 km. Think of a train that starts from Chennai and travels towards Madurai. As it increases speed more and more, the time taken for travel will decrease. In the following table speed $v$ is given in km and time $t$ is given in hours:
Speed ($v$) (km/hr)
30
40
60
80
Time ($t$) (hours)
16
12
8
6
From the table it is clear that if you travel at a slower speed, the time increases and if the train is faster, the time decreases. You find, $30\times16=40\times12=60\times8=80\times6$, which tells that $vt$ is a constant. Here, $vt=480$. In such a case, we say the variables $v$ and $t$ are inversely proportional. Observe that the graph of equation like $vt=480$ will not be a straight line. Inverse variation implies that as one variable increases, the other variable decreases.
A company initially started with 40 workers to complete the work by 150 days. Later, it decided to fasten up the work increasing the number of workers as shown below.
Number of workers ($x$)
40
50
60
75
Number of days ($y$)
150
120
100
80
(i) Graph the above data and identify the type of variation.
(ii) From the graph, find the number of days required to complete the work if the company decides to opt for 120 workers?
(iii) If the work has to be completed by 200 days, how many workers are required?
(i)
Fig. 3.13
From the given table, we observe that as $x$ increases, $y$ decreases. Thus, the variation is an inverse variation.
Let $y=\dfrac kx$.
$\Rightarrow xy=k$, $k>0$ is called the constant of variation.
From the table, $k=40\times150=50\times120=\cdots=75\times80=6000$.
Therefore, $xy=6000$.
Plot the points $(40,150)$, $(50,120)$, $(60,100)$ of $(75,80)$ and join to get a free hand smooth curve (Rectangular Hyperbola).
(ii) From the graph, the required number of days to complete the work when the company decides to work with 120 workers is 50 days.
Also, from $xy=6000$ if $x=120$, then $y=\dfrac{6000}{120}=50$.
(iii) From the graph, if the work has to be completed by 200 days, the number of workers required is 30.
Also, from $xy=6000$ if $y=200$, then $x=\dfrac{6000}{200}=30$.
Nishanth is the winner in a Marathon race of 12 km distance. He ran at the uniform speed of 12 km/hr and reached the destination in 1 hour. He was followed by Aradhana, Jeyanth, Sathya and Swetha with their respective speed of 6 km/hr, 4 km/hr, 3 km/hr and 2 km/hr. And, they covered the distance in 2 hrs, 3 hrs, 4 hrs and 6 hours respectively.
Draw the speed-time graph and use it to find the time taken to Kaushik with his speed of 2.4 km/hr.
Solution: Let us form the table with the given details.
Speed $x$ (km/hr)
12
6
4
3
2
Time $y$ (hours)
1
2
3
4
6
From the table, we observe that as $x$ decreases, $y$ increases. Hence, the type is inverse variation.
Let $y=\dfrac kx$.
$\Rightarrow xy=k$, $k>0$ is called the constant of variation.
From the table $k=12\times1=6\times2=\cdots=2\times6=12$.
Therefore, $xy=12$.
Plot the points $(12,1)$, $(6,2)$, $(4,3)$, $(3,4)$, $(2,6)$ and join these points by a smooth curve (Rectangular Hyperbola).
Fig. 3.14
From the graph, we observe that Kaushik takes 5 hrs with a speed of 2.4 km/hr.
Note
Already we learned that, the linear equation of straight line is $y=mx+c$, where $m$ is the slope of the straight line and $c$ is the $y$-intercept. Also, the equation reduces to $y=mx$ when the straight line passes through origin. As the graph of direct variation refer to straight line and its general form is $y=kx$, we can conclude that ‘constant of proportionality’ is nothing but ‘slope’ of its straight line.
A garment shop announces a flat 50% discount on every purchase of items for their customers. Draw the graph for the relation between the Marked Price and the Discount. Hence find
(i) the marked price when a customer gets a discount of ₹3250 (from graph)
(ii) the discount when the marked price is ₹2500.
Draw the graph of $xy=24$, $x,y>0$. Using the graph find, (i) $y$ when $x=3$ and (ii) $x$ when $y=6$.
Graph the following linear function $y=\dfrac12x$. Identify the constant of variation and verify it with the graph. Also (i) find $y$ when $x=9$ (ii) find $x$ when $y=7.5$.
The following table shows the data about the number of pipes and the time taken to fill the same tank.
No. of pipes ($x$)
2
3
6
9
Time Taken (in min) ($y$)
45
30
15
10
Draw the graph for the above data and hence
(i) find the time taken to fill the tank when five pipes are used
(ii) Find the number of pipes when the time is 9 minutes.
A school announces that for a certain competitions, the cash price will be distributed for all the participants equally as show below
No. of participants ($x$)
2
4
6
8
10
Amount for each participant in ₹ ($y$)
180
90
60
45
36
(i) Find the constant of variation.
(ii) Graph the above data and hence, find how much will each participant get if the number of participants are 12.
A two wheeler parking zone near bus stand charges as below.
Time (in hours) ($x$)
4
8
12
24
Amount ₹ ($y$)
60
120
180
360
Check if the amount charged are in direct variation or in inverse variation to the parking time. Graph the data. Also (i) find the amount to be paid when parking time is 6 hr; (ii) find the parking duration when the amount paid is ₹150.
The trajectory followed by an object (say, a ball) thrown upward at an angle gives a curve known as a parabola. Trajectory of water jets in a fountain or of a bouncing ball results in a parabolic path. A parabola represents a Quadratic function.
A quadratic function has the form $f(x)=ax^2+bx+c$, where $a,b,c$ are constants, and $a\ne0$.
Fig. 3.15
Many quadratic functions can be graphed easily by hand using the techniques of stretching/shrinking and shifting the parabola $y=x^2$ (We can easily sketch the curve $y=x^2$ by preparing a table of values and plotting the ordered pairs).
The “basic” parabola, $y=x^2$, looks like this Fig. 3.16.
Fig. 3.16
The coefficient $a$ in the general equation is responsible for parabolas to open upward or downward and vary in “width” (“wider” or “skinnier”), but they all have the same basic “∪” shape.
The greater the quadratic coefficient of $x^2$, the narrower is the parabola.
The lesser the quadratic coefficient of $x^2$, the wider is the parabola.
Fig. 3.17
Graph $y=x^2$ is broader than graph $y=4x^2$.
Fig. 3.18
Graph $y=x^2$ is narrower than graph $y=\dfrac14x^2$.
A parabola is symmetric with respect to a line called the axis of symmetry. The point of intersection of the parabola and the axis of symmetry is called the vertex of the parabola. The graph of any second degree polynomial gives a curve called “parabola”.
Hint: For a quadratic equation, the axis is given by $x=\dfrac{-b}{2a}$ and the vertex is given by $\left(\dfrac{-b}{2a},\dfrac{-\Delta}{4a}\right)$ where $\Delta=b^2-4ac$ is the discriminant of the quadratic equation $ax^2+bx+c=0$. Where $a\ne0$.
We have already studied how to find the roots of any quadratic equation $ax^2+bx+c=0$ where $a,b,c\in\mathbb R$ and $a\ne0$ theoretically. In this section, we will learn how to solve a quadratic equation and obtain its roots graphically.
3.8.1 Finding the Nature of Solution of Quadratic Equations Graphically#
To obtain the roots of the quadratic equation $ax^2+bx+c=0$ graphically, we first draw the graph of $y=ax^2+bx+c$.
The solutions of the quadratic equation are the $x$ coordinates of the points of intersection of the curve with $X$ axis.
To determine the nature of solutions of a quadratic equation, we can use the following procedure.
(i) If the graph of the given quadratic equation intersect the $X$ axis at two distinct points, then the given equation has two real and unequal roots.
(ii) If the graph of the given quadratic equation touch the $X$ axis at only one point, then the given equation has only one root which is same as saying two real and equal roots.
(iii) If the graph of the given equation does not intersect the $X$ axis at any point then the given equation has no real root.
Discuss the nature of solutions of the following quadratic equations.
(i) $x^2+x-12=0$ (ii) $x^2-8x+16=0$ (iii) $x^2+2x+5=0$.
Solution
(i) $x^2+x-12=0$.
Step 1: Prepare the table of values for the equation $y=x^2+x-12$.
$x$
$-5$
$-4$
$-3$
$-2$
$-1$
0
1
2
3
4
$y$
8
0
$-6$
$-10$
$-12$
$-12$
$-10$
$-6$
0
8
Step 2: Plot the points for the above ordered pairs $(x,y)$ on the graph using suitable scale.
Step 3: Draw the parabola and mark the co-ordinates of the parabola which intersect the $X$ axis.
Step 4: The roots of the equation are the $x$ coordinates of the intersecting points $(-4,0)$ and $(3,0)$ of the parabola with the $X$ axis which are $-4$ and 3 respectively.
Since there are two points of intersection with the $X$ axis, the quadratic equation $x^2+x-12=0$ has real and unequal roots.
Fig. 3.19
(ii) $x^2-8x+16=0$.
Step 1: Prepare the table of values for the equation $y=x^2-8x+16$.
$x$
$-1$
0
1
2
3
4
5
6
7
8
$y$
25
16
9
4
1
0
1
4
9
16
Step 2: Plot the points for the above ordered pairs $(x,y)$ on the graph using suitable scale.
Step 3: Draw the parabola and mark the coordinates of the parabola which intersect with the $X$ axis.
Step 4: The roots of the equation are the $x$ coordinates of the intersecting points of the parabola with the $X$ axis $(4,0)$ which is 4.
Since there is only one point of intersection with $X$ axis, the quadratic equation $x^2-8x+16=0$ has real and equal roots.
Fig. 3.20
(iii) $x^2+2x+5=0$.
Let $y=x^2+2x+5$.
Step 1: Prepare a table of values for the equation $y=x^2+2x+5$.
$x$
$-3$
$-2$
$-1$
0
1
2
3
$y$
8
5
4
5
8
13
20
Step 2: Plot the above ordered pairs $(x,y)$ on the graph using suitable scale.
Step 3: Join the points by a free-hand smooth curve this smooth curve is the graph of $y=x^2+2x+5$.
Step 4: The solutions of the given quadratic equation are the $x$ coordinates of the intersecting points of the parabola the $X$ axis.
Here, the parabola doesn’t intersect or touch the $X$ axis.
So, we conclude that there is no real root for the given quadratic equation.
Fig. 3.21
Progress Check
Connect the graphs to its respective number of points of intersection with $X$ axis and to its corresponding nature of solutions which is given in the following table.
S. No.
Graphs
Number of points of Intersection with $X$ axis
Nature of solutions
1.
Graph 1
2
Real and equal roots
2.
Graph 2
1
No real roots
3.
Graph 3
2
No real roots
4.
Graph 4
0
Real and equal roots
5.
Graph 5
0
Real and unequal roots
6.
Graph 6
1
Real and unequal roots
The illustrated connection is Graph 1 → 0 points of intersection → No real roots.
3.8.2 Solving quadratic equations through intersection of lines#
We can determine roots of a quadratic equation graphically by choosing appropriate parabola and intersecting it with a desired straight line.
(i) If the straight line intersects the parabola at two distinct points, then the $x$ coordinates of those points will be the roots of the given quadratic equation.
(ii) If the straight line just touch the parabola at only one point, then the $x$ coordinate of the common point will be the single root of the quadratic equation.
(iii) If the straight line doesn’t intersect or touch the parabola then the quadratic equation will have no real roots.
Let us consider the following information. Vanitha has 12 story books, 20 notebooks and 4 pencils. Radha has 27 story books, 17 notebooks and 6 pencils. Gokul has 7 story books, 11 notebooks and 4 pencils. Geetha has 10 story books, 12 notebooks and 5 pencils.
Details
Story Books
Note Books
Pencils
Vanitha
12
20
4
Radha
27
17
6
Gokul
7
11
4
Geetha
10
12
5
Now, we arrange this information in the tabular form as follows.
The three columns are labelled First Column, Second Column and Third Column respectively.
Here, the items possessed by four people are aligned or positioned in a rectangular array containing four horizontal and three vertical arrangements. The horizontal arrangements are called “rows” and the vertical arrangements are called “columns”. The whole rectangular arrangement is called a “Matrix”. Generally, if we arrange things in a rectangular array, we call it as “Matrix”.
Applications of matrices are found in several scientific fields. In Physics, matrices are applied in the calculations of battery power outputs, resistor conversion of electrical energy into other forms of energy. In computer based applications, matrices play a vital role in the projection of three dimensional image into a two dimensional screen, creating a realistic seeming motions. In graphic software, Matrix Algebra is used to process linear transformations to render images. One of the most important usage of matrices are encryption of message codes. The encryption and decryption processes are carried out using matrix multiplication and inverse operations. The concept of matrices is used in transmission of codes when the messages are lengthy. In Geology, matrices are used for taking seismic surveys. In Robotics, matrices are used to identify the robot movements.
Definition
A matrix is a rectangular array of elements. The horizontal arrangements are called rows and vertical arrangements are called columns.
For example, $\begin{pmatrix}4&8&0\\1&9&-2\end{pmatrix}$ is a matrix.
Usually capital letters such as $A,B,C,X,Y,\ldots$ etc., are used to represent the matrices and small letters such as $a,b,c,l,m,n,a_{12},a_{13},\ldots$ to indicate the entries or elements of the matrices.
The following are some examples of matrices
(i) $\begin{pmatrix}8&4&-1\\\frac12&5&4\\9&0&1\end{pmatrix}$ (ii) $\begin{pmatrix}1+x&x^3&\sin x\\\cos x&2&\tan x\end{pmatrix}$ (iii) $\begin{pmatrix}3+1&\sqrt2&-1\\1.5&8&9\\\frac13&13&-\frac79\end{pmatrix}$.
If a matrix $A$ has $m$ number of rows and $n$ number of columns, then the order of the matrix $A$ is (Number of rows) × (Number of columns) that is, $m\times n$. We read $m\times n$ as $m$ cross $n$ or $m$ by $n$. It may be noted that $m\times n$ is not a product of $m$ and $n$.
General form of a matrix $A$ with $m$ rows and $n$ columns (order $m\times n$) can be written in the form
where, $a_{11},a_{12},\ldots$ denote entries of the matrix. $a_{11}$ is the element in first row, first column, $a_{12}$ is the element in the first row, second column, and so on.
Progress Check
Find is the element in the second row and third column of the matrix $\begin{pmatrix}1&-2&3\\2&1&5\end{pmatrix}$.
Find is the order of the matrix $\begin{pmatrix}\sin\theta\\\cos\theta\\\tan\theta\end{pmatrix}$.
Determine the entries denoted by $a_{11},a_{22},a_{33},a_{44}$ from the matrix $\begin{pmatrix}2&1&3&4\\5&9&-4&\sqrt7\\3&\frac52&8&9\\7&0&1&4\end{pmatrix}$.
In general, $a_{ij}$ is the element in the $i$th row and $j$th column and is referred as $(i,j)$th element.
With this notation, we can express the matrix $A$ as $A=(a_{ij})_{m\times n}$ where $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$.
The total number of entries in the matrix $A=(a_{ij})_{m\times n}$ is $mn$.
Note
When giving the order of a matrix, you should always mention the number of rows first, followed by the number of columns.
A matrix is said to be a row matrix if it has only one row and any number of columns. A row matrix is also called as a row vector.
For example, $A=\begin{pmatrix}8&9&4&3\end{pmatrix}$, $B=\begin{pmatrix}-\dfrac{\sqrt3}2&1&\sqrt3\end{pmatrix}$ are row matrices of order $1\times4$ and $1\times3$ respectively.
In general $A=\begin{pmatrix}a_{11}&a_{12}&a_{13}&\ldots&a_{1n}\end{pmatrix}$ is a row matrix of order $1\times n$.
A matrix is said to be a column matrix if it has only one column and any number of rows. It is also called as a column vector.
For example, $A=\begin{pmatrix}\sin x\\\cos x\\1\end{pmatrix}$, $B=\begin{pmatrix}\sqrt5\\7\end{pmatrix}$ and $C=\begin{pmatrix}8\\-3\\23\\17\end{pmatrix}$ are column matrices of order $3\times1$, $2\times1$ and $4\times1$ respectively.
In general, $A=\begin{pmatrix}a_{11}\\a_{21}\\a_{31}\\\vdots\\a_{m1}\end{pmatrix}$ is a column matrix of order $m\times1$.
A matrix in which the number of rows is equal to the number of columns is called a square matrix. Thus a matrix $A=(a_{ij})_{m\times n}$ will be a square matrix if $m=n$.
For example, $\begin{pmatrix}1&3\\4&5\end{pmatrix}_{2\times2}$, $\begin{pmatrix}-1&0&2\\3&6&8\\2&3&5\end{pmatrix}_{3\times3}$ are square matrices.
In general, $\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}_{2\times2}$, $\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{pmatrix}$ are square matrices of orders $2\times2$ and $3\times3$ respectively.
$A=(a_{ij})_{m\times m}$ is a square matrix of order $m$.
Definition: In a square matrix, the elements of the form $a_{11},a_{22},a_{33},\ldots$ (i.e) $a_{ii}$ are called leading diagonal elements. For example in the matrix $\begin{pmatrix}1&3\\4&5\end{pmatrix}$, 1 and 5 are leading diagonal elements.
A square matrix, all of whose elements, except those in the leading diagonal are zero is called a diagonal matrix.
(ie) A square matrix $A=(a_{ij})$ is said to be diagonal matrix if $a_{ij}=0$ for $i\ne j$. Note that some elements of the leading diagonal may be zero but not all.
For example, $\begin{pmatrix}8&0&0\\0&-3&0\\0&0&11\end{pmatrix}$, $\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}$ are diagonal matrices.
A matrix is said to be a zero matrix or null matrix if all its elements are zero.
For example, $(0)$, $\begin{pmatrix}0&0\\0&0\end{pmatrix}$, $\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}$ are all zero matrices of order $1\times1$, $2\times2$ and $3\times3$ but of different orders. We denote zero matrix of order $n\times n$ by $O_n$.
$\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}$ is a zero matrix of the order $2\times3$.
The matrix which is obtained by interchanging the elements in rows and columns of the given matrix $A$ is called transpose of $A$ and is denoted by $A^T$.
For example,
(a) If $A=\begin{pmatrix}5&3&-1\\2&8&9\\-4&7&5\end{pmatrix}_{3\times3}$ then $A^T=\begin{pmatrix}5&2&-4\\3&8&7\\-1&9&5\end{pmatrix}_{3\times3}$.
(b) If $B=\begin{pmatrix}1&5\\8&9\\4&3\end{pmatrix}_{3\times2}$ then $B^T=\begin{pmatrix}1&8&4\\5&9&3\end{pmatrix}_{2\times3}$.
If order of $A$ is $m\times n$ then order of $A^T$ is $n\times m$.
A square matrix in which all the entries above the leading diagonal are zero is called a lower triangular matrix.
If all the entries below the leading diagonal are zero, then it is called an upper triangular matrix.
Definition: A square matrix $A=(a_{ij})_{n\times n}$ is called upper triangular matrix if $a_{ij}=0$ for $i>j$ and is called lower triangular matrix if $a_{ij}=0$, $i
For example, $A=\begin{pmatrix}1&7&-3\\0&2&4\\0&0&7\end{pmatrix}$ is an upper triangular matrix and $B=\begin{pmatrix}8&0&0\\4&5&0\\-11&3&1\end{pmatrix}$ is a lower triangular matrix.
Two matrices $A$ and $B$ are said to be equal if and only if they have the same order and each element of matrix $A$ is equal to the corresponding element of matrix $B$. That is, $a_{ij}=b_{ij}$ for all $i,j$.
For example, if $A=\begin{pmatrix}5&1\\0&3\end{pmatrix}$,
The negative of a matrix $A_{m\times n}$ denoted by $-A_{m\times n}$ is the matrix formed by replacing each element in the matrix $A_{m\times n}$ with its additive inverse.
Additive inverse of an element $k$ is $-k$. That is, every element of $-A$ is the negative of the corresponding element of $A$.
For example, if $A=\begin{pmatrix}2&-4&9\\5&-3&-1\end{pmatrix}_{2\times3}$ then $-A=\begin{pmatrix}-2&4&-9\\-5&3&1\end{pmatrix}_{2\times3}$.
If a matrix has 16 elements, what are the possible orders it can have?
Solution We know that a matrix of order $m\times n$, has $mn$ elements. Thus to find all possible orders of a matrix with 16 elements, we will find all ordered pairs of natural numbers whose product is 16.
Such ordered pairs are $(1,16)$, $(16,1)$, $(4,4)$, $(8,2)$, $(2,8)$.
Hence, possible orders are $1\times16$, $16\times1$, $4\times4$, $2\times8$, $8\times2$.
In the matrix $A=\begin{pmatrix}8&9&4&3\\-1&\sqrt7&\frac{\sqrt3}{2}&5\\1&4&3&0\\6&8&-11&1\end{pmatrix}$, write (i) The number of elements (ii) The order of the matrix (iii) Write the elements $a_{22},a_{23},a_{24},a_{34},a_{43},a_{44}$.
If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?
Construct a $3\times3$ matrix whose elements are given by (i) $a_{ij}=|i-2j|$ (ii) $a_{ij}=\frac{(i+j)^3}{3}$.
If $A=\begin{pmatrix}5&4&3\\1&-7&9\\3&8&2\end{pmatrix}$ then find the transpose of $A$.
If $A=\begin{pmatrix}\sqrt7&-3\\-\sqrt5&2\\\sqrt3&-5\end{pmatrix}$ then find the transpose of $-A$.
If $A=\begin{pmatrix}5&2&2\\-\sqrt{17}&0.7&\frac52\\8&3&1\end{pmatrix}$ then verify $(A^T)^T=A$.
Find the values of $x,y$ and $z$ from the following equations: (i) $\begin{pmatrix}12&3\\x&5\end{pmatrix}=\begin{pmatrix}y&z\\3&5\end{pmatrix}$ (ii) $\begin{pmatrix}x+y&2\\5+z&xy\end{pmatrix}=\begin{pmatrix}6&2\\5&8\end{pmatrix}$ (iii) $\begin{pmatrix}x+y+z\\x+z\\y+z\end{pmatrix}=\begin{pmatrix}9\\5\\7\end{pmatrix}$.
Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements.
If $A=(a_{ij})$, $B=(b_{ij})$, $i=1,2,\ldots,m$, $j=1,2,\ldots,n$ then $C=A+B$ is such that $C=(c_{ij})$ where $c_{ij}=a_{ij}+b_{ij}$ for all $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$.
Two examinations were conducted for three groups of students namely group 1, group 2, group 3 and their data on average of marks for the subjects Tamil, English, Science and Mathematics are given below in the form of matrices $A$ and $B$. Find the total marks of both the examinations for all the three groups.
The columns are Tamil, English, Science and Mathematics respectively; the rows are Group 1, Group 2 and Group 3.
We can multiply the elements of the given matrix $A$ by a non-zero number $k$ to obtain a new matrix $kA$ whose elements are multiplied by $k$. The matrix $kA$ is called scalar multiplication of $A$.
Thus if $A=(a_{ij})_{m\times n}$ then, $kA=(ka_{ij})_{m\times n}$ for all $i=1,2,\ldots,m$ and $\forall j=1,2,\ldots,n$.
If $A=\begin{pmatrix}5&4&-2\\\frac12&\frac34&\sqrt2\\1&9&4\end{pmatrix}$, $B=\begin{pmatrix}-7&4&-3\\\frac14&\frac72&3\\5&-6&9\end{pmatrix}$, find $4A-3B$.
Solution Since $A,B$ are of the same order $3\times3$, subtraction of $4A$ and $3B$ is defined.
If $A=\begin{pmatrix}1&8&3\\3&5&0\\8&7&6\end{pmatrix}$, $B=\begin{pmatrix}8&-6&-4\\2&11&-3\\0&1&5\end{pmatrix}$, $C=\begin{pmatrix}5&3&0\\-1&-7&2\\1&4&3\end{pmatrix}$ compute the following: (i) $3A+2B-C$ (ii) $\frac12A-\frac32B$.
If $A=\begin{pmatrix}1&9\\3&4\\8&-3\end{pmatrix}$, $B=\begin{pmatrix}5&7\\3&3\\1&0\end{pmatrix}$ then verify that (i) $A+B=B+A$ (ii) $A+(-A)=(-A)+A=O$.
If $A=\begin{pmatrix}4&3&1\\2&3&-8\\1&0&-4\end{pmatrix}$, $B=\begin{pmatrix}2&3&4\\1&9&2\\-7&1&-1\end{pmatrix}$ and $C=\begin{pmatrix}8&3&4\\1&-2&3\\2&4&-1\end{pmatrix}$ then verify that $A+(B+C)=(A+B)+C$.
Find $X$ and $Y$ if $X+Y=\begin{pmatrix}7&0\\3&5\end{pmatrix}$ and $X-Y=\begin{pmatrix}3&0\\0&4\end{pmatrix}$.
If $A=\begin{pmatrix}0&4&9\\8&3&7\end{pmatrix}$, $B=\begin{pmatrix}7&3&8\\1&4&9\end{pmatrix}$ find the value of (i) $B-5A$ (ii) $3A-9B$.
Find the values of $x,y,z$ if (i) $\begin{pmatrix}x-3&3x-z\\x+y+7&x+y+z\end{pmatrix}=\begin{pmatrix}1&0\\1&6\end{pmatrix}$ (ii) $\begin{pmatrix}x&y-z&z+3\end{pmatrix}+\begin{pmatrix}y&4&3\end{pmatrix}=\begin{pmatrix}4&8&16\end{pmatrix}$.
Find $x$ and $y$ if $x\begin{pmatrix}4\\-3\end{pmatrix}+y\begin{pmatrix}-2\\3\end{pmatrix}=\begin{pmatrix}4\\6\end{pmatrix}$.
Find the non-zero values of $x$ satisfying the matrix equation $x\begin{pmatrix}2x&2\\3&x\end{pmatrix}+2\begin{pmatrix}8&5x\\4&4x\end{pmatrix}=2\begin{pmatrix}x^2+8&24\\10&6x\end{pmatrix}$.
Solve for $x,y$: $\begin{pmatrix}x^2\\y^2\end{pmatrix}+2\begin{pmatrix}-2x\\-y\end{pmatrix}=\begin{pmatrix}5\\8\end{pmatrix}$.
To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Consider the multiplications of $3\times3$ and $3\times2$ matrices.
The inner dimensions must be equal; the outer dimensions determine the order of the product matrix.
(Order of left hand matrix) $\times$ (order of right hand matrix) $\to$ (order of product matrix).
Matrices are multiplied by multiplying the elements in a row of the first matrix by the elements in a column of the second matrix, and adding the results.
The product $AB$ can be found if the number of columns of matrix $A$ is equal to the number of rows of matrix $B$. If the order of matrix $A$ is $m\times n$ and $B$ is $n\times p$ then the order of $AB$ is $m\times p$.
(a) Matrix multiplication is not commutative in general
If $A$ is of order $m\times n$ and $B$ of the order $n\times p$ then $AB$ is defined but $BA$ is not defined. Even if $AB$ and $BA$ are both defined, it is not necessary that they are equal. In general $AB\ne BA$.
(b) Matrix multiplication is distributive over matrix addition
(i) If $A,B,C$ are $m\times n$, $n\times p$ and $n\times p$ matrices respectively then $A(B+C)=AB+AC$ (Right Distributive Property).
(ii) If $A,B,C$ are $m\times n$, $m\times n$ and $n\times p$ matrices respectively then $(A+B)C=AC+BC$ (Left Distributive Property).
(c) Matrix multiplication is always associative
If $A,B,C$ are $m\times n$, $n\times p$ and $p\times q$ matrices respectively then $(AB)C=A(BC)$.
(d) Multiplication of a matrix by a unit matrix
If $A$ is a square matrix of order $n\times n$ and $I$ is the unit matrix of same order then $AI=IA=A$.
Note
If $x$ and $y$ are two real numbers such that $xy=0$ then either $x=0$ or $y=0$. But this condition may not be true with respect to two matrices.
$AB=0$ does not necessarily imply that $A=0$ or $B=0$ or both $A,B=0$.
If $A=\begin{pmatrix}1&-1&2\end{pmatrix}$, $B=\begin{pmatrix}1&-1\\2&1\\1&3\end{pmatrix}$ and $C=\begin{pmatrix}1&2\\2&-1\end{pmatrix}$ show that $(AB)C=A(BC)$.
If $A=\begin{pmatrix}1&1\\-1&3\end{pmatrix}$, $B=\begin{pmatrix}1&2\\-4&2\end{pmatrix}$, $C=\begin{pmatrix}-7&6\\3&2\end{pmatrix}$ verify that $A(B+C)=AB+AC$.
If $A$ is of order $p\times q$ and $B$ is of order $q\times r$ what is the order of $AB$ and $BA$?
$A$ has ‘$a$’ rows and ‘$a+3$’ columns. $B$ has ‘$b$’ rows and ‘$17-b$’ columns, and if both products $AB$ and $BA$ exist, find $a,b$?
If $A=\begin{pmatrix}2&5\\4&3\end{pmatrix}$, $B=\begin{pmatrix}1&-3\\2&5\end{pmatrix}$ find $AB,BA$ and verify $AB=BA$?
Given that $A=\begin{pmatrix}1&3\\5&-1\end{pmatrix}$, $B=\begin{pmatrix}1&-1&2\\3&5&2\end{pmatrix}$, $C=\begin{pmatrix}1&3&2\\-4&1&3\end{pmatrix}$ verify that $A(B+C)=AB+AC$.
Show that the matrices $A=\begin{pmatrix}1&2\\3&1\end{pmatrix}$, $B=\begin{pmatrix}1&-2\\-3&1\end{pmatrix}$ satisfy commutative property $AB=BA$.
Let $A=\begin{pmatrix}1&2\\1&3\end{pmatrix}$, $B=\begin{pmatrix}4&0\\1&5\end{pmatrix}$, $C=\begin{pmatrix}2&0\\1&2\end{pmatrix}$. Show that (i) $A(BC)=(AB)C$ (ii) $(A-B)C=AC-BC$ (iii) $(A-B)^T=A^T-B^T$.
If $A=\begin{pmatrix}\cos\theta&0\\0&\cos\theta\end{pmatrix}$, $B=\begin{pmatrix}\sin\theta&0\\0&\sin\theta\end{pmatrix}$ then show that $A^2+B^2=I$.
If $A=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$ prove that $AA^T=I$.
Verify that $A^2=I$ when $A=\begin{pmatrix}5&-4\\6&-5\end{pmatrix}$.
If $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ and $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ show that $A^2-(a+d)A=(bc-ad)I_2$.
If $A=\begin{pmatrix}5&2&9\\1&2&8\end{pmatrix}$, $B=\begin{pmatrix}1&7\\1&2\\5&-1\end{pmatrix}$ verify that $(AB)^T=B^TA^T$.
If $A=\begin{pmatrix}3&1\\-1&2\end{pmatrix}$ show that $A^2-5A+7I_2=0$.
Which of the following can be calculated from the given matrices $A=\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}$, $B=\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}$, (i) $A^2$ (ii) $B^2$ (iii) $AB$ (iv) $BA$.
(A) (i) and (ii) only (B) (ii) and (iii) only (C) (ii) and (iv) only (D) all of these.
If $A=\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}$, $B=\begin{pmatrix}1&0\\2&-1\\0&2\end{pmatrix}$ and $C=\begin{pmatrix}0&1\\-2&5\end{pmatrix}$. Which of the following statements are correct? (i) $AB+C=\begin{pmatrix}5&5\\5&5\end{pmatrix}$ (ii) $BC=\begin{pmatrix}0&1\\2&-3\\-4&10\end{pmatrix}$ (iii) $BA+C=\begin{pmatrix}2&5\\3&0\end{pmatrix}$ (iv) $(AB)C=\begin{pmatrix}-8&20\\-8&13\end{pmatrix}$.
(A) (i) and (ii) only (B) (ii) and (iii) only (C) (iii) and (iv) only (D) all of these.
One hundred and fifty students are admitted to a school. They are distributed over three sections $A,B$ and $C$. If 6 students are shifted from section A to section $C$, the sections will have equal number of students. If 4 times of students of section $C$ exceeds the number of students of section $A$ by the number of students in section $B$, find the number of students in the three sections.
In a three-digit number, when the tens and the hundreds digit are interchanged the new number is 54 more than three times the original number. If 198 is added to the number, the digits are reversed. The tens digit exceeds the hundreds digit by twice as that of the tens digit exceeds the unit digit. Find the original number.
Find the least common multiple of $xy(k^2+1)+k(x^2+y^2)$ and $xy(k^2-1)+k(x^2-y^2)$.
Find the GCD of the following by division algorithm $2x^4+13x^3+27x^2+23x+7$, $x^3+3x^2+3x+1$, $x^2+2x+1$.
Reduce the given Rational expressions to its lowest form (i) $\frac{x^{3a}-8}{x^{2a}+2x^a+4}$ (ii) $\frac{10x^3-25x^2+4x-10}{-4-10x^2}$.
Arul, Madan and Ram working together can clean a store in 6 hours. Working alone, Madan takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?
Find the square root of $289x^4-612x^3+970x^2-684x+361$.
Solve $\sqrt{y+1}+\sqrt{2y-5}=3$.
A boat takes 1.6 hours longer to go 36 kms up a river than down the river. If the speed of the water current is 4 km per hr, what is the speed of boat in still water?
Is it possible to design a rectangular park of perimeter 320 m and area $4800\text{ m}^2$? If so find its length and breadth.
At $t$ minutes past 2 pm, the time needed to 3 pm is 3 minutes less than $\frac{t^2}{4}$. Find $t$.
The number of seats in a row is equal to the total number of rows in a hall. The total number of seats in the hall will increase by 375 if the number of rows is doubled and the number of seats in each row is reduced by 5. Find the number of rows in the hall at the beginning.
If $\alpha$ and $\beta$ are the roots of the polynomial $f(x)=x^2-2x+3$, find the polynomial whose roots are (i) $\alpha+2,\beta+2$ (ii) $\frac{\alpha-1}{\alpha+1},\frac{\beta-1}{\beta+1}$.
If $-4$ is a root of the equation $x^2+px-4=0$ and if the equation $x^2+px+q=0$ has equal roots, find the values of $p$ and $q$.
Two farmers Thilagan and Kausigan cultivates three varieties of grains namely rice, wheat and ragi. If the sale (in ₹) of three varieties of grains by both the farmers in the month of April is given by the matrix.
April sale in ₹; columns rice, wheat, ragi; rows Thilagan, Kausigan:
and the May month sale (in ₹) is exactly twice as that of the April month sale for each variety.
(i) What is the average sales of the months April and May.
(ii) If the sales continues to increase in the same way in the successive months, what will be sales in the month of August?
If $\cos\theta\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}+\sin\theta\begin{pmatrix}x&-\cos\theta\\\cos\theta&x\end{pmatrix}=I_2$, find $x$.
Given $A=\begin{pmatrix}p&0\\0&2\end{pmatrix}$, $B=\begin{pmatrix}0&-q\\1&0\end{pmatrix}$, $C=\begin{pmatrix}2&-2\\2&2\end{pmatrix}$ and if $BA=C^2$, find $p$ and $q$.
$A=\begin{pmatrix}3&0\\4&5\end{pmatrix}$, $B=\begin{pmatrix}6&3\\8&5\end{pmatrix}$, $C=\begin{pmatrix}3&6\\1&1\end{pmatrix}$ find the matrix $D$, such that $CD-AB=0$.
A system of linear equations in three variables will be according to one of the following cases.
(i) Unique solution (ii) Infinitely many solutions (iii) No solution.
The least common multiple of two or more algebraic expressions is the expression of lowest degree (or power) such that the expressions exactly divides it.
A polynomial of degree two in variable $x$ is called a quadratic polynomial in $x$. Every quadratic polynomial can have atmost two zeroes. Also the zeroes of a quadratic polynomial intersects the $x$-axis.
The roots of the quadratic equation $ax^2+bx+c=0$, $(a\ne0)$ are given by $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
For a quadratic equation $ax^2+bx+c=0$, $a\ne0$
Sum of the roots $\alpha+\beta=\frac{-b}{a}=\frac{-\text{Co-efficient of }x}{\text{Co-efficient of }x^2}$.
Product of the roots $\alpha\beta=\frac ca=\frac{\text{Constant term}}{\text{Co-efficient of }x^2}$.
If the roots of a quadratic equation are $\alpha$ and $\beta$, then the equation is given by $x^2-(\alpha+\beta)x+\alpha\beta=0$.
The value of the discriminant $(\Delta=b^2-4ac)$ decides the nature of roots as follows
(i) When $\Delta>0$, the roots are real and unequal.
(ii) When $\Delta=0$, the roots are real and equal.
(iii) When $\Delta<0$, there are no real roots.
Solving quadratic equation graphically.
A matrix is a rectangular array of elements arranged in rows and columns.
Order of a matrix
If a matrix $A$ has $m$ number of rows and $n$ number of columns, then the order of the matrix $A$ is (Number of rows) $\times$ (Number of columns) that is, $m\times n$. We read $m\times n$ as $m$ cross $n$ or $m$ by $n$. It may be noted that $m\times n$ is not a product of $m$ and $n$.
Types of matrices
(i) A matrix is said to be a row matrix if it has only one row and any number of columns. A row matrix is also called as a row vector.
(ii) A matrix is said to be a column matrix if it has only one column and any number of rows. It is also called as a column vector.
(iii) A matrix in which the number of rows is equal to the number of columns is called a square matrix.
(iv) A matrix is said to be a zero matrix or null matrix if all its elements are zero.
(v) If $A$ is a matrix, the matrix obtained by interchanging the rows and columns of $A$ is called its transpose and is denoted by $A^T$.
(vi) A square matrix, all of whose elements, except those in the leading diagonal are zero is called a diagonal matrix.
(vii) A diagonal matrix in which all the leading diagonal elements are same is called a scalar matrix.
(viii) A square matrix in which elements in the leading diagonal are all “1” and rest are all zero is called an identity matrix (or) unit matrix.
(ix) A square matrix in which all the entries above the leading diagonal are zero is called a lower triangular matrix.
If all the entries below the leading diagonal are zero, then it is called an upper triangular matrix.
(x) Two matrices $A$ and $B$ are said to be equal if and only if they have the same order and each element of matrix $A$ is equal to the corresponding element of matrix $B$. That is, $a_{ij}=b_{ij}$ for all $i,j$.
The negative of a matrix $A_{m\times n}$ denoted by $-A_{m\times n}$ is the matrix formed by replacing each element in the matrix $A_{m\times n}$ with its additive inverse.
Addition and subtraction of matrices
Two matrices can be added or subtracted if they have the same order. To add or subtract two matrices, simply add or subtract the corresponding elements.
Multiplication of matrix by a scalar
We can multiply the elements of the given matrix $A$ by a non-zero number $k$ to obtain a new matrix $kA$ whose elements are multiplied by $k$. The matrix $kA$ is called scalar multiplication of $A$.
Thus if $A=(a_{ij})_{m\times n}$ then, $kA=(ka_{ij})_{m\times n}$ for all $i=1,2,\ldots,m$ and for all $j=1,2,\ldots,n$.
Step 1: Open the Browser type the URL Link given below (or) Scan the QR Code. Chapter named “Algebra” will open. Select the work sheet “Simultaneous equations”.
Step 2: In the given worksheet you can see three linear equations and you can change the equations by typing new values for a, b and c for each equation. You can move the 3-D graph to observe. Observe the nature of solutions by changing the equations.
Step - 1: Open the Browser type the URL Link given below (or) Scan the QR Code. GeoGebra work book named “ALGEBRA” will open. Click on the worksheet named “Nature of Quadratic Equation”.
Step - 2: In the given worksheet you can change the co-efficient by moving the sliders given. Click on “New position” and move the sliders to fix the boundary for throwing the shell. Then click on “Get Ball” and click “fire” to hit the target. Here you can learn what happen to the curve when each co-efficient is changed.
ICT 3.2: Step 1, Step 2 and Expected results.
You can repeat the same steps for other activities
