Similar Questions - CBSE SQP Class 12 2023-24 - Part 2 (from q26 to end).jpg

Slide42.JPG
Slide43.JPG
Slide44.JPG
Slide45.JPG Slide46.JPG Slide47.JPG Slide48.JPG

Go Ad-free

Transcript

Misc 7 Solve the system of the following equations 2/x + 3/y + 10/z = 4 4/x + 6/y + 5/z = 1 6/x + 9/y + 20/z = 2 The system of equations are 2/x + 3/y + 10/z = 4 4/x + 6/y + 5/z = 1 6/x + 9/y + 20/z = 2 Now let 𝟏/𝒙 = u , 𝟏/𝒚 = v , & 𝟏/𝒛 = w The system of equations become 2u + 3v + 10w = 4 4u – 6v + 5w = 1 6u + 9v – 20w = 2 Writing equation as AX = B [■8(2&3&10@4&−6&5@6&9&−20)] [■8(𝑢@𝑣@𝑤)] = [■8(4@1@2)] Hence A = [■8(2&3&10@4&−6&5@6&9&−20)] , X = [■8(𝑢@𝑣@𝑤)] & B = [■8(4@1@2)] Calculating |A| |A| = |■8(2&3&10@4&−6&5@6&9&−20)| = 2 |■8(−6&5@9&−20)| – 3 |■8(4&5@6&−20)| + 10 |■8(4&−6@6&9)| = 2 (120 – 45) –3 (–80 – 30) + 10 ( 36 + 36) = 2 (75) –3 (–110) + 10 (72) = 150 + 330 + 720 = 1200 ∴ |A|≠ 0 So, the system of equation is consistent & has a unique solution Now, AX = B X = A-1 B Calculating A-1 Now, A-1 = 1/(|A|) adj (A) adj (A) = [■8(A11&A12&A13@A21&A22&A23@A31&A32&A33)]^′ = [■8(A11&A21&A31@A12&A22&A32@A13&A23&A33)] A = [■8(2&3&10@4&−6&5@6&9&−20)] M11 = |■8(−6&5@9&−20)| = 120 – 45 = 75 M12 = |■8(4&5@6&−20)| = (–80 – 30) = –110 M13 = |■8(4&−6@6&9)| = 36 –36 = 72 M21 = |■8(3&10@9&−20)| = −60 – 90 = –150 M22 = |■8(2&10@6&−20)| = –40 – 60 = –100 M23 = |■8(2&3@6&9)| = 18 – 18 = 0 M31 = |■8(3&10@−6&5)| = 15 + 60 = 75 M32 = |■8(2&10@4&5)| = 10 – 40 = –30 M33 = |■8(2&3@4&−6)| = –12 – 12 = –24 Now, A11 = 〖"(–1)" 〗^(1+1) M11 = (–1)2 . 75 = 75 A12 = 〖"(–1)" 〗^"1+2" M12 = 〖"(–1)" 〗^3 . (–110) = 110 A13 = 〖(−1)〗^(1+3) M13 = 〖(−1)〗^4 . (72) = 72 A21 = 〖(−1)〗^(2+1) M21 = 〖(−1)〗^3 . (–150) = 150 A22 = 〖(−1)〗^(2+2) M22 = (–1)4 . (–100) = –100 A23 = 〖(−1)〗^(2+3). M23 = 〖(−1)〗^5. 0 = 0 A31 = 〖(−1)〗^(3+1). M31 = 〖(−1)〗^4 . 75 = 75 A32 = 〖(−1)〗^(3+2) . M32 = 〖(−1)〗^5. (–30) = 30 A33 = 〖(−1)〗^(3+3) . M33 = (–1)6 . –24 = –24 Thus, adj A = [■8(75&150&75@110&−110&30@72&0&−24)] Now, A-1 = 1/(|A|) adj A A-1 = 𝟏/𝟏𝟐𝟎𝟎 [■8(𝟕𝟓&𝟏𝟓𝟎&𝟕𝟓@𝟏𝟏𝟎&−𝟏𝟏𝟎&𝟑𝟎@𝟕𝟐&𝟎&−𝟐𝟒)] Also, X = A−1 B Putting Values [■8(𝑢@𝑣@𝑤)]= 1/1200 [■8(75&150&75@110&−110&30@72&0&−24)] [■8(4@1@2)] [■8(𝑢@𝑣@𝑤)]= 1/1200 [■8(75(4)+150(1)+75(4)@110(4)+(−110)(1)+30(1)@72(4)+0(1)+(−24)2)] [■8(𝑢@𝑣@𝑤)] = 1/1200 [■8(300+150+150@440−100+60@288+0−48)] = 1/1200 [■8(600@400@140)] [■8(𝒖@𝒗@𝒘)] = [■8(𝟏/𝟐@𝟏/𝟑@𝟏/𝟓)] Hence u = 1/2 , v = 1/3 , & w = 1/5 Thus, x = 2, y = 3 & z = 5 Putting u = 𝟏/𝒙 1/2 = 1/𝑥 x = 2 Putting v = 𝟏/𝒚 1/3 = 1/𝑦 y = 3 Putting w = 𝟏/𝒛 1/5 = 1/𝑧 z = 5

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo