Set Theory (Class 11th CBSE)

1. we have given the introduction on sets and how to represent them in set builder form and roaster form.

2. We have covered the topic " Types of Sets" which includes-Empty set, Singleton set, Equal sets and Equivalent sets.



3. we have introduced the concept of subsets, definition and its types by giving examples.



4. we have explained how to apply operations on sets such as union , intersection, subtraction.



5. we have explained the laws of operations on sets namely Idempotent law, commutative law, Identity law, Associative law, Distributive law and demorgans law.




6. we have explained the concept of Venn diagrams, definition and how to read them


List of Indian States and Union territory with Capital

STATES AND CAPITALS

Andhra Pradesh – Hyderabad ( newly proposed Amaravati )
Arunachal Pradesh – Itanagar
Assam – Dispur
Bihar – Patna
Chhattisgarh – Raipur
Goa – Panaji
Gujarat – Gandhinagar
Haryana – Chandigarh
Himachal Pradesh – Shimla
Jammu & Kashmir – Jammu (Winter Capital) & Srinagar (Summer Capital)
Jharkhand – Ranchi
Karnataka – Bengaluru
Kerala – Thiruvananthapuram
Madhya Pradesh – Bhopal
Maharashtra – Mumbai
Manipur – Imphal
Meghalaya – Shillong
Mizoram – Aizawl
Nagaland – Kohima
Orissa – Bhubaneshwar
Punjab – Chandigarh
Rajasthan – Jaipur
Sikkim – Gangtok
Tamil Nadu – Chennai
Telangana – Hyderabad
Tripura – Agartala
Uttar Pradesh – Lucknow
Uttarakhand – Dehradun
West Bengal – Kolkata

UNION TERRITORIES AND CAPITALS


Andaman & Nicobar Islands – Port Blair
Dadra and Nagar Haveli – Silvassa
Daman and Diu – Daman
Lakshadweep – Kavaratti
Puducherry – Puducherry/ Pondicherry
Delhi (National Capital Territory) – New Delhi
Chandigarh – Chandigarh


Dynamic Programming

Dynamic Programming with proper indented code and example.
Covers following topics

1.Optimal Parenthesization of Matrix Chain Product
2.Least Common Subsequence
3.Optimal Binary Search Tree
4.0-1 Knapsack Problem
5.Binomial Coefficient Computation
6.Floyd-Warshall Algorithm

https://drive.google.com/open?id=0B6JEvMco0pOEUVBXNUdwQUdBMGc

APTITUDE PREPARATION-5

APTITUDE PREPARATION-5

SQUARE ROOTS AND CUBE ROOTS

IMPORTANT FACTS AND FORMULAE


Square Root: If x2 = y, we say that the square root of y is x and we write, √y = x.
Thus, √4 = 2, √9 = 3, √196 = 14.
Cube Root: The cube root of a given number x is the number whose cube is x. We denote the cube root of x by 3√x.
Thus, 3√8  = 3√2 x 2 x 2 = 2, 3√343 = 3√7 x 7 x 7 = 7 etc.
Note:
1.√xy = √x * √y                     2. √(x/y) = √x / √y  = (√x / √y) * (√y / √y) = √xy / y

SOLVED EXAMPLES


Ex. 1. Evaluate √6084 by factorization method .
Sol.     Method: Express the given number as the product of prime factors.          2    6084  
            Now, take the product of these prime factors choosing one out of              2    3042
            every pair of the same primes. This product gives the square root              3    1521  
            of the given number.                                                                                     3    507
Thus, resolving 6084 into prime factors, we get:                                        13   169
6084 = 22 x 32 x 132                                                                                                         13                \6084 = (2 x 3 x 13) = 78.       
                                                                     

Ex. 2. Find the square root of 1471369.                                                   
Sol.     Explanation: In the given number, mark off the digits              1  1471369 (1213
in pairs starting from the unit's digit. Each pair and                        1             
the remaining one digit is called a period.                                22     47
Now, 12 = 1. On subtracting, we get 0 as remainder.                        44
Now, bring down the next period i.e., 47.                               241      313
Now, trial divisor is 1 x 2 = 2 and trial dividend is 47.                       241
So, we take 22 as divisor and put 2 as quotient.                    2423        7269
The remainder is 3.                                                                                7269
Next, we bring down the next period which is 13.                                  x
Now, trial divisor is 12 x 2 = 24 and trial dividend is
313. So, we take 241 as dividend and 1 as quotient.
The remainder is 72. ­
Bring down the next period i.e., 69.
Now, the trial divisor is 121 x 2 = 242 and the trial
dividend is 7269. So, we take 3as quotient and 2423
as divisor. The remainder is then zero.
Hence, √1471369 = 1213.

Ex. 3. Evaluate: 248 + 51 + √ 169 .
Sol.     Given expression = 248 + √51 + 13 = 248 + √64    = √ 248 + 8 = √256 = 16.

Ex. 4. If a * b * c = √(a + 2)(b + 3) / (c + 1), find the value of 6 * 15 * 3.
Sol.      6 * 15 * 3 = √(6 + 2)(15 + 3) / (3 + 1) = √8 * 18 / 4 = √144 / 4 = 12 / 4 = 3.
Ex. 5. Find the value of √25/16.
Sol.    √ 25 / 16   = √ 25 / √ 16 = 5 / 4


Ex. 6. What is the square root of 0.0009?
Sol.      √0.0009= √ 9 / 1000  = 3 / 100 = 0.03.

Ex. 7. Evaluate √175.2976.
Sol.      Method: We make even number of decimal places              1   175.2976 (13.24  
by affixing a zero, if necessary. Now, we mark off                   1
periods and extract the square root as shown.                     23     75            
                                                                                                      69
\√175.2976 = 13.24                                                         262       629
                                                                                                                    524
                                                                                                     2644       10576
                                                                                                                    10576
                                                                                                                        x


Ex. 8. What will come in place of question mark in each of the following questions?
(i) √32.4 / ?  = 2                       (ii) √86.49 + √ 5 + ( ? )2 = 12.3.
            Sol.      (i) Let √32.4 / x = 2. Then, 32.4/x = 4 <=> 4x = 32.4 <=> x = 8.1.
                     
(ii) Let √86.49 + √5 + x2 = 12.3.
      Then, 9.3 + √5+x2 = 12.3 <=> √5+x= 12.3 - 9.3 = 3
      <=> 5 + x2 = 9   <=> x2 = 9 - 5= 4   <=>   x = √4 = 2.


Ex.9. Find the value of √ 0.289 / 0.00121.
 


Sol.      √0.289 / 0.00121 = √0.28900/0.00121 = √28900/121 = 170 / 11.

 


Ex.10. If √1 + (x / 144) = 13 / 12, the find the value of x.

Sol.      √1 + (x / 144) = 13 / 12 Þ ( 1 + (x / 144)) = (13 / 12 )2 = 169 / 144
  Þx / 144 = (169 / 144) - 1
  Þx / 144 = 25/144 Þ x = 25.

Ex. 11. Find the value of √3 up to three places of decimal.
Sol.                
  1    3.000000   (1.732
                                1
27    200
189
                      343      1100
                                  1029
                    3462          7100
6924                          \√3 = 1.732.



Ex. 12. If √3 = 1.732, find the value of √192 - 1 √48 - √75 correct to 3 places
                                                                            2                        
of decimal.                                                                                     (S.S.C. 2004)
Sol.     192 - (1 / 2)√48 - √75 = √64 * 3 - (1/2) √ 16 * 3  - √ 25 * 3
                                                =8√3 - (1/2) * 4√3 - 5√3
                                                =3√3 - 2√3 = √3 = 1.732
 


Ex. 13. Evaluate: √(9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021)
Sol.      Given exp. = √(9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021)
            Now, since the sum of decimal places in the numerator and denominator under the            radical sign is the same, we remove the decimal.
\        Given exp = √(95 * 85 * 18900) / (17 * 19 * 21) = √ 5 * 5 * 900  = 5 * 30 = 150.

Ex. 14. Simplify: √ [( 12.1 )2 - (8.1)2] / [(0.25)2 + (0.25)(19.95)]
Sol.      Given exp. = √ [(12.1 + 8.1)(12.1 - 8.1)] / [(0.25)(0.25 + 19.95)]
                       
                              =√ (20.2 * 4) /( 0.25 * 20.2)   = √ 4 / 0.25 = √400 / 25 = √16 = 4.
Ex. 15. If x = 1 + √2 and y = 1 - √2, find the value of (x2 + y2).
Sol.      x2 + y2 = (1 + √2)2 + (1 - √2)2 = 2[(1)2 + (√2)2] = 2 * 3 = 6.

Ex. 16. Evaluate: √0.9 up to 3 places of decimal.
Sol.                 
9    0.900000(0.948
         81
184         900
736
                  1888         16400
                                   15104                             \√0.9 = 0.948
                         

Ex.17. If √15 = 3.88, find the value of √ (5/3).
Sol.      √ (5/3) = √(5 * 3) / (3 * 3)  = √15 / 3 = 3.88 / 3 = 1.2933…. = 1.293.
    
Ex. 18. Find the least square number which is exactly divisible by 10,12,15 and 18.
Sol.      L.C.M. of 10, 12, 15, 18 = 180. Now, 180 = 2 * 2 * 3 * 3 *5 = 22 * 32 * 5.
            To make it a perfect square, it must be multiplied by 5.
\         Required number = (22 * 32 * 52) = 900.

Ex. 19. Find the greatest number of five digits which is a perfect square.
(R.R.B. 1998)
Sol.      Greatest number of 5 digits is 99999.
                                                    3    99999(316
      9 
                                                  61      99
                                                            61
626      3899
           3756
                                                             143
\                  Required number == (99999 - 143) = 99856.

Ex. 20. Find the smallest number that must be added to 1780 to make it a perfect
square.
Sol.
                                    4    1780 (42
                                          16
82      180         
164
            16
 


      \               Number to be added = (43)2 - 1780 = 1849 - 1780 = 69.

Ex. 21. √2 = 1.4142, find the value of √2 / (2 + √2).
Sol.      √2 / (2 + √2) = √2 / (2 + √2) * (2 - √2) / (2 - √2) = (2√2 – 2) / (4 – 2)
                                 = 2(√2 – 1) / 2 = √2 – 1 = 0.4142.


22. If x = (√5 + √3) / (√5 - √3) and y = (√5 - √3) / (√5 + √3), find the value of (x+ y2).
Sol.     
x = [(√5 + √3) / (√5 - √3)] * [(√5 + √3) / (√5 + √3)] = (√5 + √3)2 / (5 - 3)
   =(5 + 3 + 2√15) / 2 = 4 + √15.        
            y = [(√5 - √3) / (√5 + √3)] * [(√5 - √3) / (√5 - √3)] = (√5 - √3)2 / (5 - 3)
               =(5 + 3 - 2√15) / 2 = 4 - √15.
\         x2 + y2 = (4 + √15)2 + (4 - √15)2 = 2[(4)2 + (√15)2] = 2 * 31 = 62.


Ex. 23. Find the cube root of 2744.
Sol.    Method: Resolve the given number as the product                 2    2744
          of prime factors and take the product of prime                        2    1372
           factors, choosing one out of three of the same                        2      686         
           prime factors. Resolving 2744 as the product of                     7      343
           prime factors, we get:                                                              7  ­      49
                                                                                                                       7
           2744 = 23 x 73.                                                                            
\      3√2744= 2 x 7 = 14.

                                                                                                    
Ex. 24. By what least number 4320 be multiplied to obtain a number which is a perfect cube?
Sol.      Clearly, 4320 = 23 * 33 * 22 * 5.

            To make it a perfect cube, it must be multiplied by 2 * 52 i.e,50.
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