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

Linear Inequality Chapter 6 Class XI : Basics


Linear Inequalities
Basic concepts based on linear inequalities chapter 6 class XI. Important points and revision notes on linear inequalities.
Definition :
 If any two numbers or algebraic expressions are related with four symbols like   < , > ,  ≤ and    form an inequality.

If degree of the algebraic expression is 1 then it is called a linear inequality.

For example  2x - 3  5,     2x+ 3 > 5  are the linear inequalities.

Solutions of linear inequility :-
Real solutions of inequality are represented in the form of  intervals. Intervals  are of 4 types

1. Open interval written as       (    )

2. Close Interval written as      [   ]

3.  Semi open or semi close interval written as   (    ]

4.  Semi open or semi close interval  written as  [     )

* Small square bracket always denote the  open interval.

* Capital square bracket represents the close interval.

For example  
 (2, 8) it means all real numbers between 2 and 8, except  2 and 8

[2, 8] it means all real numbers from 2  to  8

[2,8) it means  that  number 8 is not included in this interval.

(2,8] it means  that  number 2 is not included in this interval.

Solution of the Linear inequality
Solutions of linear inequality are represented in the form of  a set.

Eg.   {a, b, c, d, ……..} depending upon the nature of the solution.

All real solutions are represented in the form of intervals.

All the intervals can be expressed on the number line.

Real Number Line
Case 1: Graph of the line given below is called real number line


Real numbers start from -∞ and goes upto +

Set of real numbers are represented by  (-∞, )

Case II

In the above graph O represent the open interval  at 1 and this solution is written as  (1, )

Case III

In the above graph the dark spot represent the close interval at 1 and this solution is written as [1. )

Case IV

Above solution can be written as  (-2, 3) it means  all terms between -2 and 3

Case V

Above solution can be written as    [-2, 3] it means all numbers from -2 to 3
If an inequality is multiplied or divided by negative sign then sign of the inequality changes.
Eg.     2x < 5   ⇒    -2x > - 5

Graphical solution of linear equations in two variables
Graph of the line divide the Cartesian plane into two half planes.

A Vertical line divides the plane into two half planes called left half plane and right half plane as shown in the figure.


Non-vertical line divides the plane into two half planes called upper half plane and lower half plane as shown in the figure below.


Note :-
  • A point in the Cartesian plane either lie on the line or lie on either of the half plane.
  • The region containing all the solutions of an inequality is called the solution region or feasible region.
  • In order to identify the half plane represented by an inequality, it is sufficient to take any point (a, b) (not on the line ) and check whether it satisfies the inequality or not.
  • If it satisfies the given inequality then the half plane in which the point lie is called the solution region.
  • If the point does not satisfy the inequality then the region in which the point does not lie is called the solution region. For convenience  the point (0,0) is preferred.
  • We should shade the solution region identified in the above steps.
  • For the inequality with the sign ≤ or  ≥ , the points on the line are also included in the solution region or feasible region. In this case the graph line is the full line.
  • For the inequality with the sign < or >, the points on the line are not included in the feasible region or solution region. In this case the graph line is a dotted line as shown in the figure.


Solution of system of linear inequalities in two variables
  • Here we may be given two or three or four equations.
  • We find the solution region for all the equations as discussed above.
  • The common solution region of all the equations in the given system is called the solution region or feasible region of the system of equations.
NCERT Miscellaneous Exercise
Q 12  A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the  2% solution will have to be added ?

Solution 
 Amount of 8% boric acid solution = 640 litre
Let 2% boric acid solution added = x litre
Total solution becomes = (640 + x) litre
According to the question
4% (640 + x)  8% (640) + 2% of x  6% (640 + x)
\[\frac{4}{100}(640+x)\leq \frac{8}{100}\times 640+\frac{2}{100}\times x\leq \frac{6}{100}\left ( 640+x \right )\]
\[4(640+x)\leq 8\times 640+2\times x\leq 6\left ( 640+x \right )\]
\[2560+4x\leq 5120+2x\leq 3840+6x\]

Case I

\[2560+4x\leq 5120+2x\]

\[4x-2x\leq 5120-2560\]

\[2x\leq 2560\]

\[x\leq 1280\]

Case II

\[5120+2x\leq 3840+6x\]

\[5120-3840\leq 6x-2x\]

\[1280\leq 4x\]

\[1280\leq 4x\]

\[320\leq x\]

\[320\leq x\leq 1280\]

Hence amount of 2% acid solution added is ≥ 320 litre but ≤ 1280 litre

 

Case Study Based Questions

Ans

1

Marks obtained by Radhika in quarterly and half yearly examinations of Mathematics are 60 and 70 respectively.

Based on the above information, answer the following questions

 

(i)

Minimum marks she should get in the annual exam to have an average of atleast 70 marks is

a)    80               b) 85                  c) 75                  d)  90

a

(ii)

Maximum marks, she should get in the annual exam to have an average of atmost 75 marks is

a)    85              b) 90                  c) 95                  d) 80

c

(iii)

Range of marks in annual exam, so that the average marks is atleast 60 and atmost 70 is

a)    [60, 70]            b) [50, 80]                 c) [50, 70]             d) [60, 80]

b

(iv)

If the average of atleast 60 marks is considered pass, then minimum marks she need to score in annual exam to pass is

a)    60                   b) 65                         c) 70                        d) 50

d

(v)

If she scored atleast 20 and atmost 80 marks in annual exam, then the range of average marks is

a)    [50, 70]            b) [60, 70]                c) 50, 60]                 d [50, 80]

a



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