| Class XI |
Mathematics |
Mathematics |
4 |
3i and 2 2 2 z z = 2 +(−3) =13 Therefore, the multiplicative inverse2 −3iis given by z 2+3i 2 3 z–1= 2 = = + i z 13 13 13 The above working can be ... |
|
| Class XI |
Mathematics |
Mathematics |
4 |
the form of a + ib : (3+i 5 )(3−i 5 ) ( 3 + 2i −) ( 3 −i 2 ) 4.5 ArgandPlaneandPolarRepresentation We already know that corresponding to each order... |
|
| Class XI |
Mathematics |
Mathematics |
4 |
4.3 Reprint 2026-27 COMPLEX NUMBERS AND QUADRATIC EQUATIONS 85 Miscellaneous Examples (3−2i)(2+3i) Example 7 Find the conjugate of . (1+ 2i)(2−i) (... |
|
| Class XI |
Mathematics |
Mathematics |
4 |
a + ib = 2x +1 , prove that a + b = (2x +1 ) . 7. Let z1= 2 – i, z2= –2 + i. Find z1 2 1 (i)Re , (ii) Im . z1 z1 1 8. Find th... |
|
| Class XI |
Mathematics |
Mathematics |
4 |
any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists the a −b complex number +i , denoted by 1 or z , called the a +b 2 a +b 2 z a −... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
Chapter 5 LINEARINEQUALITIES v Mathematics is the art of saying many things in many different ways. – MAXWELL v 5.1 Introduction Inearlierclasses,w... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
such as (1), (2) and (3) above are inequalities. 3 < 5; 7 > 5 are the examples of numerical inequalities while x < 5; y > 2; x ≥ 3, y ≤ 4 are some ... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
is true. For x = 1, L.H.S. = 30 (1) = 30 < 200 (R.H.S.), which is true. For x = 2, L.H.S. = 30 (2) = 60 < 200, which is true. For x = 3, L.H.S. = 3... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
withoutaffectingthesignofinequality. Rule 2 Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both s... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
the set of real numbers. Henceforth, unless stated otherwise, we shall solve the inequalities in this Chapter in the set of real numbers. Reprint 2... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
first and second terminal examinationare62and48,respectively.Findtheminimummarksheshouldgetinthe annual examination to have an average of at least ... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
5(2−x) 1 x 1 11. 5 ≤ 3 12. 2 +4 (3−6) 13. 2 (2x + 3) – 10 < 6 (x – 2) 14. 37 – (3x + 5) > 9x – 8 (x – 3) 15. x<(5x−2) (−x−3) 16. (2x−1) (≥x ... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
Miscellaneous Examples Example 9 Solve – 8 ≤ 5x – 3 < 7. Solution In this case, we have two inequalities, – 8 ≤ 5x – 3 and 5x – 3 < 7, which we wil... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
– 32) < 35, 9 9 or × (30) < (F – 32) < × (35) 5 5 or 54 < (F – 32) < 63 or 86 < F < 95. Thus, the required range of temperature is between 86° F an... |
|
| Class XI |
Mathematics |
Mathematics |
5 |
1 > – 24, 5x – 1 < 24 8. 2 (x – 1) < x + 5, 3 (x + 2) > 2 – x 9. 3x – 7 > 2 (x – 6) , 6 – x > 11 – 2x 10. 5 (2x – 7) – 3 (2x + 3) ≤ 0 , 2x + 19 ≤ 6... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
100 MATHEMATICS Chapter 6 PERMUTATIONSANDCOMBINATIONS v Every body of discovery is mathematical in form because there is no other guidance we can h... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
digits 1, 2, 3, 4, 5 if the digits can be repeated? Solution There will be as many ways as there are ways of filling 2 vacant places insuccessionby... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
Howmany3-digitevennumberscanbeformedfromthedigits1,2,3,4,5,6ifthe digits can be repeated? 3. How many 4-letter code can be formed using the first 1... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
objects do not repeat is n ( n – 1) ( n – 2). . .( n – r + 1), which is denoted by P . r Proof There will be as many permutations as there are ways... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
2 × 1! Clearly, for a natural number n n ! = n (n – 1) ! = n (n – 1) (n – 2) ! [provided (n ≥ 2)] = n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)] ... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
n = 9, r = 5. 6.3.3 Derivation of the formula for Pn r nP = n! r (n−r !) , 0 ≤ r ≤ n Let us now go back to the stage where we had determined the fo... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
r 4 In Example 1, the required number of words = P = 4! = 44. Here repetition is notallowed. Ifrepetitionisallowed,therequirednumberofwordswouldbe4... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
O1R 2 TO O R T O O R 2 1 Reprint 2026-27 PERMUTATIONS AND COMBINATIONS 109 RO T1O 2 R O T O RO T 2 1 T O 1 O 2 T O R O T O R O 2 1 RT... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
U2T 2 Hen3e, total number of differentpermutationswillbe 9! 2!3! We can state (without proof) the following theorems: Theorem 3 The number of permu... |
|
| Class XI |
Mathematics |
Mathematics |
6 |
taking 2 at a time. This number is P . So 6! 5! The required number = P −3P = 2 − 3! 3! = 4 × 5 × 6 – 4 ×5 = 100 Example 12 Find the value of n suc... |
|