REAL NUMBERS(LECTURE 8) & STATISTICS (LECTURE 1)

Real Numbers (Worksheet Queries) & STATISTICS (Introduction)_CLASS 10

LESSON 8, Day-2, 

Good Morning  !

Hope you have completed the worksheet (Real numbers) that was given to you on 1st April 2020! 😀

Let us spend some time in clarifying any doubts !!

take your SET-A Mathematics Note Book

Our handwriting reflects a lot about us. It will be awesome if you use good presentation and cursive hand writing. Hope you are at it !!

Make a column on the right hand side, if you need to do any rough work

Leave two lines where you finished yesterday’s work and draw a horizontal line. Recall SDG-12 (turn the leaves of your notebooks to read that!) Thank you😄

Write today's date on the line after that. 

Now Queries (if any...)

Re posting the worksheet Questions and Answers from the worksheet for you 👇

REAL NUMBERS {assignment based on Board papers)

Q1) Prove that √2 is irrational                                                                       [CBSE 2008,2019]

A1) Refer to pg 12 (thm 1.4)

Q2) Prove that the following are irrationals:-

  (i) 2- √3          (ii) 5-2√3                                                                                          [CBSE 2008]

   (iii) 3-5√2        (iv) 4-5√3         (v) 7+3√2                                                              [CBSE 2010]
A2i)   Let 2 - √3   =  p/q , {where p & q are coprime Z,q ≠ 0}
          ⟹  √3   =  2 - p/q
          ⟹  √3   =  (2q - p) /q
           Now p & q are Q no.s , ⟹ (2q - p) /q   ⟹  √3  is also a Q no.
           BUT , WE KNOW THAT √3  is an irrational no.
            ⟹  Contradiction ⟹  (2- √3 )  is an irrational no.

A2iv)  Let 4 - 5√3   =  p/q , {where p & q are coprime Z,q ≠ 0}
         ⟹   5√3   =  4 - p/q
         ⟹   5√3   =  (4q - p) /q
         ⟹   √3   =  (4q - p) / 5q
         ⟹    Now p & q are Q no.s , ⟹ (4q - p) / 5q   ⟹  √3  is also a Q no.
           BUT , WE KNOW THAT √3  is an irrational no.
            ⟹  Contradiction ⟹  (4 - 5√3 )  is an irrational no.

If you have any query in any other part of Q2, please ask in the comment below !

Q3)  Find the HCF of 1260 and 7344 using Euclid’s algorithm.                            [CBSE 2019]

A3)  Let a = 7344 , b = 1260
        Applying Euclid's Division Algorithm 👇
        7344 = 1260 ⨯ 5 + 1044
        1260 = 1044 ⨯ 1 + 216
        1044 = 216 ⨯ 4 +  180
        216 = 180 ⨯ 1 + 36
        180 = 36 ⨯ 3  + 0
        ⟹ HCF (7344, 1260) = 36

Q4) Use Euclid’s  division Lemma to show that the cube of any positive integer is of the form 9q or 9q +1 or 9q+8 for some integer q.                                                                     [CBSE 2009 c]
A4)  Refer to Solution covered in Lesson 6

5) Show that one and only one out of n, n+2,n+4 is divisible by 3, where n is any positive integer.                                                                                                                      [CBSE 2008C]

Q6)  Use Euclid’s  division Lemma to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m.                                                                   [CBSE 2008]

A6)  Refer to Solution covered in Lesson 6

Q7)  Show that every positive odd integer is of the form (4q+ 1) or (4q+ 3), where  q is some integer.                                                                                                                       [CBSE 2019]

A7)  Refer to Eg 3 on pg 6 of NCERT

Q8)  Show that (3 +√7)/5 is an irrational number, given that √7 is irrational.      [CBSE 2019]
A8)  Similar to Q2 explained above

Q9)   Prove that(n² + n) is divisible by 2 for any positive integer n.                       [CBSE 2019C]
A9)   (n² + n)  = n(n + 1)
         "n" and "(n+1)" are two consecutive numbers
           Case 1 : Let n be an even number , then (n + 1) is an odd number.
                         Product of even and odd is always even .
                          ⟹ (n² + n) is divisible by 2
            Case 2 : Let n be an odd number , then (n + 1) is an even number.
                         Product of even and odd is always even .

                          ⟹ (n² + n) is divisible by 2
       ⟹ There were only two possibilities and in both cases the                 result has been verified.  Hence Proved

Q10)  Prove that (2 +√3)/5 is an irrational number, given that √3 is irrational.    [CBSE 2019]
A10)   Similar to Q2 explained above

Just reminding you about the important life skill that we learnt in lesson 4: "Assume & Examine and only then make your Perception"  



HOME WORK: Revisit all the words that you have written in the diary

After you have, completed your incomplete home work,
Please turn to fresh page and write "STATISTICS"

Learning Outcomes

I will be able to  :                                               

• Recall the different ways in which data be written.         
• Recall the 3 measures of central tendencies.

Click on the link given below to initiate talk about "DATA"

Intro to Stats-1


Q1)  State a few examples of Raw data.

Q2)  State a few examples of Processed data.  

Click on the link given below to discuss what you have written in your answers

Discussion for Q1 and Q2


Example of Ungrouped data : 👇

Example of Grouped data : 👇

That will be all for today !

Good Morning and Thank you boys !  😊

See you in the next class tomorrow

Take a good care of yourselves !