Misc 2 - Chapter 4 Class 12 Determinants

Transcript

Misc 2 (Method 1) Evaluate |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| Expanding Determinant along C1 = cos α cos β |■8(cos⁡𝛽&0@sin⁡〖𝛼 sin⁡𝛽 〗&cos⁡𝛼 )| – cos α sin β |■8(〖−sin〗⁡𝛽&0@sin⁡〖𝛼 𝑐𝑜𝑠⁡𝛽 〗&cos⁡𝛼 )| – sin α |■8(sin⁡𝛽&cos⁡𝛽@sin⁡〖𝛼 𝑐𝑜𝑠⁡𝛽 〗&sin⁡〖𝛼 sin⁡𝛽 〗 )| = cos α cos β (cos β cos α – 0) – cos α sin β (– sin β cos α – 0) – sin α (– sin2 β sin α – cos2 β sin α) = cos α cos β ( cos β cos α) – cos α sin β ( – sin β cos α) + sin2 α sin2 β + cos2 β sin2 α = cos2 α cos2 β + cos2 α sin2 β + sin2 α ( sin2 β + cos2 β) = cos2 α (1) + sin2 α (1) = cos 2 α + sin2α = 1 Misc 2 (Method 2) Evaluate |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| Let ∆ = |■8(cos⁡〖α cosβ〗&cos⁡〖α sinβ〗&−sin⁡α@−sinβ&cosβ&0@sin⁡α cosβ&sin⁡〖α sin⁡β 〗&cos⁡α )| = |■8(𝐜𝐨𝐬⁡〖𝛂 〗 cosβ&𝐜𝐨𝐬⁡〖𝛂 〗 sin β&(−𝒄𝒐𝒔⁡𝜶)/cos⁡𝛼 sin⁡α@−sin β&cos β&0@𝐬𝐢𝐧⁡𝛂 cosβ&𝐬𝐢𝐧⁡𝛂 sin⁡β&𝒔𝒊𝒏⁡𝜶/sin⁡𝛼 〖.cos〗⁡α )| Taking common cos α from R1 & sin α from R3 = cos α.sin β |■8(cos⁡𝛽&sin⁡𝛽&−tan⁡𝛼@−sinβ&cosβ&0@cosβ&sin⁡𝛽&co𝑡⁡α )| Applying R1→ R1 – R3 = cos α.sin β |■8(cos⁡〖𝛽 〗−𝐜𝒐𝒔⁡𝜷&sin⁡〖𝛽−𝒔𝒊𝒏⁡𝜷 〗&−tan⁡〖𝛼−𝒄𝒐𝒕⁡𝜶 〗@−sinβ&cosβ&0@cosβ&sin⁡𝛽&co𝑡⁡α )| Expanding determinant along R1 = cos α . sin α(0|■8(cos⁡𝛽&0@sin⁡𝛽&cot⁡𝛼 )|−0|■8(〖−sin〗⁡𝛽&0@cos⁡𝛽&cot⁡𝛼 )|−〖(tan〗⁡〖𝛼−cot⁡〖𝛼)|■8(sin⁡𝛽&cos⁡𝛽@cos⁡𝛽&sin⁡𝛽 )|〗 〗 ) = cos α . sin α(0−0−〖(tan〗⁡〖𝛼−cot⁡〖𝛼)|■8(sin⁡𝛽&cos⁡𝛽@cos⁡𝛽&sin⁡𝛽 )|〗 〗 ) = cos α sin α (– (tan α + cotα ) ( – sin2 β – cos2 β )) = cos α sin α ( tan + cot α ) (sin2 β + cos2 β ) = cos α sin α (tan α + cot α ) (1) = cos α sin α (sin⁡𝛼/cos⁡𝛼 + cos⁡𝛼/sin⁡𝛼 ) = cos α sin α ((𝒔𝒊𝒏𝟐 𝜶 + 𝒄𝒐𝒔𝟐 𝜶)/cos⁡〖𝛼 sin⁡𝛼 〗 ) = cos α sin α (𝟏/cos⁡〖𝛼 sin⁡𝛼 〗 ) = 1