Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Pair of Linear Equations in Two Variables Class 10 Chapter 3

Pair of Linear Equations in Two Variables
Basic points and complete explanations of all methods of solving pair of linear equations in two variables, for class X, chapter 3 Mathematics. 
Pair of linear equations in two variables

General pair of linear equations in two variables
\[a_{1}x+b_{1}y=c_{1}\]\[a_{2}x+b_{2}y=c_{2}\]
There are four methods of solving pair of linear equations in two variables
1) Graphical Method :
In this method the point of intersection of two lines on the graph is the solution of two equations.
2) Substitution Method
3) Elimination Method
4) Cross Multiplication Method

Types of graphs, nature of solutions, consistency/ inconsistency all are discussed in the following table.

Conditions
Nature of Solution
Type of Graph
Consistent /Inconsistent
\[\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\]
Unique Solution
Two intersecting lines
Consistent
\[\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\]
Unique Solution
Two intersecting lines
Consistent
\[\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}\]
Infinitely many solutions
Coincident lines
Consistent
\[\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\]
No Solution
Two Parallel Lines
Inconsistent



Explanation of Algebraic Methods of solving Pair of Linear Equations in two Variables

Substitution Method of solving

Pair of linear equations in two variables


Let Given Pair of linear equations in two variables
        x + y = 3………………(1)
2x + 3y = 10…………..(2)
From equation (1)
x = 3 – y………….(3)
Putting eqn. (3) in eqn. (2) we get
2(3 - y) + 3y = 10
  6 - 2y + 3y = 10
           6 + y = 10
                      y = 10 - 6
             y = 4
Putting y = 4 in eqn. (3) we get
   x = 3 - y
    x = 3 - 4
x = -1
x = -1 and y = 4 is the required solution.
*******************************************
Elimination Method of solving

Pair of linear equations in two variables

x + y = 3…………(1)
2x + 3y = 10…………..(2)

Eqn. (1) x 2 – eqn. (2) x 1 we get
y = 4
Putting y = 4 in equation (1)
x + 4 = 3
                   x = 3 - 4 = -1
x = -1 and y = 4 is the required solution.
********************************************
Cross Multiplication Method of solving

Pair of linear equations in two variables


There are two cases to explain this method

Case I :  Let general pair of linear equations in two variables :
\[a_{1}x+b_{1}y=c_{1}\]\[a_{2}x+b_{2}y=c_{2}\]
If constant terms on the right hand side then place the coefficients as shown below
\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}=\frac{y}{c_{1}a_{2}-c_{2}a_{1}}=\frac{-1}{a_{1}b_{2}-a_{2}b_{1}}\]
Case II :  If constant terms are on right hand side then we must follow the following steps
General pair of linear equations in two variables
\[a_{1}x+b_{1}y+c_{1}=0\]\[a_{2}x+b_{2}y+c_{2}=0\]
\[\frac{x}{b_{1}c_{2}-b_{2}c_{1}}=\frac{y}{c_{1}a_{2}-c_{2}a_{1}}=\frac{1}{a_{1}b_{2}-a_{2}b_{1}}\]

Explanation of cross multiplication method by taking an example
Let given a pair of equations in two variables
x + y = 3…………(1)
2x + 3y = 10…………..(2)

Exercise 3.7
Q1 The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani  and Biju is twice as old as his sister cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
Sol. :         Let age of Ani = x
                Let age of Biju = y
According to the question
                                  x - y = 3 .....................(1)
Age of Dharam = 2x
    Age of cathy = y/2
A.T.Q                 2x - y/2 = 30
Multiplying this equation by 2 we get
                               4x - y = 60 ....................(2)
Solving equation (1) and eqn(2) by elimination method we get
                    x = 19 years, y=16 years.

Q2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their capital?
Sol. :           Let first friend have = Rs x
                     Second friend have = Rs y
A.T.Q. 
                      x + 100 = 2(y - 100)
                      x + 100 = 2y - 200
                         x - 2y = - 200 - 100
                         x - 2y = - 300 ...............(1)
Again ATQ
                  6(x - 10) = y + 10
                    6x - 60 = y + 10
                      6x - y = 10 + 60
                      6x - y = 70 ......................(2)
Solving equation (1) and (2) by elimination method we get
           x = 40 and y = 170

Q3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And if the train were slower by 10 km/h, it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
Sol. :            Let speed of train = x km/h
                 Time taken by train = y Km/h
                                    Distance = Speed x Time
                                                   = xy
Case 1 
                 New speed  = (x + 10) km/h
         New time taken = y - 2) hours
                     Distance = Speed x Time
                                    = (x + 10) x (y - 2)
                                    = xy - 2x + 10y - 20
A.T.O.             
            xy - 2x + 10y - 20 =  xy
                  -2x + 10y = 20 .......................(1)
Case II
             New speed = (x - 10) km/h
    New time taken  = (y + 3) hours
Distance = (x - 10) x (y + 3)
                 = xy + 3x - 10y - 30
A.T.Q.
          xy + 3x - 10y - 30 = xy
                  3x - 10y - 30 = 0
                         3x - 10y = 30 .....................(2)
By elimination method solve equation (1) and (2) we get
       x = 50, y = 12
Distance = xy = 50 x 12 = 600km

Q4. The students of a class are made to stand in rows. If 3 students are extra in each row, there would be 1 row less. If 3 students are less in each row, there would be 2 rows more. Find the number of students in the class.
Sol.                   Let number of rows = x
     Number of students in each row = y
                Total students in the class = xy
A.T.Q.
              (x + 3)(y - 1) = xy
            xy - x + 3y - 3 = xy
                 - x + 3y - 3 = 0
                      - x + 3y = 3 .................(1)
Again ATQ
                (x - 3)(y + 2) = xy
           xy + 2x - 3y - 6 = xy
                         2x - 3y = 6 ....................(2)
Solving equations (1) and (2) by elimination method we get
x = 9, y = 4
Number of students in the class = 9 x 4 = 36

Q7. Solve the following equations
i) Solve                               
 px + qy = p – q
  qx - py = p + q
By cross multiplication method


\[\frac{x}{q(p+q)+p(p-q)}=\frac{y}{q(p-q)-p(p+q)}=\frac{-1}{-p^{2}-q^{2}}\]\[\frac{x}{pq+q^{2}+p^{2}-pq}=\frac{y}{pq-q^{2}-p^{2}-pq}=\frac{-1}{-\left (p^{2}+q^{2} \right )}\]\[\frac{x}{p^{2}+q^{2}}=\frac{y}{-\left (p^{2}+q^{2} \right )}=\frac{1}{\left (p^{2}+q^{2} \right )}\]\[x=1, y=-1\]
ii) Solve : 
ax + by = c
bx + ay = 1+c
By Cross multiplication method

\[\frac{x}{b+bc-ac}=\frac{y}{bc-a-ac}=\frac{-1}{a^{2}-b^{2}}\]\[x=\frac{-\left ( b+bc-ac \right )}{a^{2}-b^{2}},\; \; y=\frac{-\left ( bc-a-ac \right )}{a^{2}-b^{2}}\]\[x=\frac{ac-b-bc}{a^{2}-b^{2}},\; \; y=\frac{a+ac-bc}{a^{2}-b^{2}}\]

iii) Solve : \[\frac{x}{a}-\frac{y}{b}=0 ........(1)\]\[ax+by=a^{2}+b^{2} .......(2)\]\[From \; \; equation\; \; (1),\; \; \frac{x}{a}=\frac{y}{b}\]\[x=\frac{a}{b}y .........(3)\]Putting equation (3) in equation (2) we get\[a\times \frac{ay}{b}+by=a^{2}+b^{2}\]\[\frac{a^{2}y+b^{2}y}{b}=a^{2}+b^{2}\] \[\frac{\left (a^{2}+b^{2} \right )y}{b}=a^{2}+b^{2}\]\[y=b\]Putting y = b in equation (3) we get\[x=\frac{a}{b}\times b\Rightarrow x=a\]

Q7 iv)            (a - b)x + (a + b)y = a2 - 2ab - b2  …..(1)


                (a + b)(x + y) = a2 + b2

                    (a + b)x + (a + b)y = a2 + b2 …….(2)

Eqn. (1) – Eqn.(2)
2bx = - 2b(a – b)

x  = a – b

Putting x = a - b in equation  (2)

(a + b)(a - b) + (a + b)y = a2 + b2

         a2 - b2  + (a + b)y = a2 + b2

\[(a+b)y=b^{2}+b^{2}\]\[y = \frac{2b^{2}}{a+b}\]

Q7(iv)            152x – 378y = -74 …….(1)

                     -378x + 152y = - 604 ……(2)

Adding eqn.(1) and eqn.(2)

Dividing by -226 we get

                          x + y = 3 …………..(3)

Subtracting eqn.(1) and eqn.(2) we get 


Dividing by 530 we get

                         x – y =1 ……………(4)

By elimination method solving eqn. (3) and (4) we get

                           x = 2, y = 1

Q 8)

ABCD is a cyclic 
quadrilateral, find the measure of angles of the cyclic quadrilateral ?

Sol. Since opposite angles of a cyclic quadrilateral are supplementary.
Therefore
               ∠A + ∠C = 180o
            4y + 20 - 4x = 180
                 - 4x + 4y = 180 - 20
                 - 4x + 4y = 160
Dividing by 4 we get
                     -x + y = 40..........(1)
               ∠B + ∠D = 180o
       3y - 5 - 7x + 5 = 180
                -7x + 3y = 180  ..................(2)
Solving these equations by elimination method:-
Eqn.(1) x 3 - eqn. (2) x 1 we get
x = -60/4 ⇒ x = -15
Putting x = -15 in eqn. (1) we get
                                    -(-15) + y = 40
       y = 40 - 15
 y = 25
                   x = -15, y = 25
Now we find the angles of quadrilateral
              ∠A = 4y + 20 = 4 x 25 + 20 = 120o
              ∠B = 3y - 5    = 3 x 25 - 5    = 70o
              ∠C = - 4x       = -4 x -15       = 60o
              ∠D = -7x + 5 = -7 x - 15 + 5 =  105 + 5 = 110o


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