Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 4 4.3.2 Difference of two complex numbers Given any two complex numbers z and 1 z2, the difference z1– z 2s defined as follows: z1– z 2 z +1(– z ).2 ...
Class XI Mathematics Mathematics 4 (vi) The distributive law For any three complex numbers z , z , z , 1 2 3 (a) z 1z 2 z )3= z z1+ 2 z 1 3 (b) (z + z ) z = z z + z z 1 2 3 1 3 2 3 4...
Class XI Mathematics Mathematics 4 4.3.6 The square roots of a negative real number Note that i = –1 and ( – i) = i = – 1 Therefore, the square roots of – 1 are i, – i. However, by t...
Class XI Mathematics Mathematics 4 + z ) = (z + z ) (z + z ), 1 2 1 2 1 2 = (z 1 z )2z +1(z + 1 ) z2 2 (Distributivelaw) = z1+ z 2 1 z z1 2z 2 (Distributivelaw) = z + z z + z z + z 2...
Class XI Mathematics Mathematics 4 – 198i. Example 4 Express ( − 3 + −2 2 3)(i )in the form of a + ib Solution We have, (− 3 + −2 )( 2 3 −i ) (= − 3 + 2i )(2 3 −i ) 2 −6+ 2 + 3 1+2 2...
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. – MAXWELLv 5.1 Introduction Inearlierclasses,we...
Class XI Mathematics Mathematics 5 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 examp...
Class XI Mathematics Mathematics 5 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. = 30 (...
Class XI Mathematics Mathematics 5 Rule 2 Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a n...
Class XI Mathematics Mathematics 5 set of real numbers. Henceforth, unless stated otherwise, we shall solve the inequalities in this Chapter in the set of real numbers. Reprint 2026-...
Class XI Mathematics Mathematics 5 and second terminal examinationare62and48,respectively.Findtheminimummarksheshouldgetinthe annual examination to have an average of at least 60 mar...
Class XI Mathematics Mathematics 5 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 − 2) (2...
Class XI Mathematics Mathematics 5 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 will solve simult...
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 and ...
Class XI Mathematics Mathematics 5 > – 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 ≤ 6x ...
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 box can be chosen in 3 different ways. Hence, there are 2×3=6pairsofschoolbagandatiffin box. For each of these pairs a water bottlecanbechosenin2di...
Class XI Mathematics Mathematics 6 the occurence of the events in this chosen order. Example 1 Find the number of 4 letter words, with or without meaning, which can be formed out of ...
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...