Revision: Complex Numbers Applied Mathematics 1 BE Civil Engineering Semester 1 (FE First Year) University of Mumbai
- If U = X 2 + Y 2 + Z 2 Where X = E T , Y = E T Sin T , Z = E T Cos T Prove that D U D T = 4 E 2 T
- If Cos α Cos β = X 2 , Sin α Sin β = Y 2 , Prove That: Sec ( α − I β ) + Sec ( α − I β ) = 4 X X 2 − Y 2
- Find the Nth Derivative of Cos 5x.Cos 3x.Cos X.
- Evaluate : Lim X → 0 ( X ) 1 1 − X
- If X = Uv, Y = U + V U − V . Find ∂ ( U , V ) ∂ ( X , Y ) .
- If Y = 2 X Sin 2 X Cos X Find Y N
- If Z = Log ( E X + E Y ) Show that Rt − S 2 = 0 Where R = ∂ 2 Z ∂ X 2 , T = ∂ 2 Z ∂ Y 2 S = ∂ 2 Z ∂ X ∂ Y
- If Z=Tan^1 (X/Y), Where X = 2 T , Y = 1 − T 2 , Prove that D Z D T = 2 1 + T 2 .
- By Using De MoiréS Theorem Obtain Tan 5θ in Terms of Tan θ and Show that 1 − 10 Tan 2 ( π 10 ) + 5 Tan 4 ( π 10 ) = 0 .
- Using De Moivre’S Theorem Prove That] Cos 6 θ − Sin 6 θ = 1 16 ( Cos 6 θ + 15 Cos 2 θ )
- Solve X 4 − X 3 + X 2 − X + 1 = 0 .
- Find All Values of ( 1 + I ) 1 3 and Show that Their Continued Product is (1+I).
- Show that the Roots of X5 =1 Can Be Written as 1, α 1 , α 2 , α 3 , α 4 .Hence Show that ( 1 − α 1 ) ( 1 − α 2 ) ( 1 − α 3 ) ( 1 − α 4 ) = 5 .
- Show that All Roots of ( X + 1 ) 6 + ( X − 1 ) 6 = 0 Are Given by -icot ( 2 K + 1 ) N 12 Where K=0,1,2,3,4,5.
- Find All Values of `(1 + I)^(1/3` and Show that Their Continued Product is (1+ 𝒊 ).
- Expand 2 X 3 + 7 X 2 + X − 1 in Powers of X - 2
- Prove that Sin − 1 ( Cos E C θ ) = π 2 + I . Log ( Cot θ 2 )
- Expand 2 X 3 + 7 X 2 + X − 6 in Powers of (X-2)
- Prove that sin 5 θ = 1 16 [ sin 5 θ − 5 sin 3 θ + 10 sin θ ]
- If U= F ( Y − X X Y , Z − X X Z ) , Show that X 2 ∂ U ∂ X + Y 2 ∂ U ∂ Y + X 2 ∂ U ∂ Z = 0
- If X = U V and Y = U V Prove that J J 1 = 1
- Prove that 𝒕𝒂𝒏𝒉−𝟏(𝒔𝒊𝒏 𝜽) = 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄 𝜽)
- Prove that the Matrix 1 √ 3 [ 1 1 + I 1 1 − I − 1 ] is Unitary.
- Hence Proved.
- Review of Complex Numbers‐Algebra of Complex Number
- Different Representations of a Complex Number and Other Definitions
- D’Moivre’S Theorem
- Powers and Roots of Exponential Function
- Powers and Roots of Trigonometric Functions
- Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
- Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
- .Circular Functions of Complex Number
- Hyperbolic functions of complex number
- Inverse Circular Functions
- Inverse Hyperbolic Functions
- Separation of Real and Imaginary Parts of All Types of Functions
