Applied Mathematics-I
- Find \(a,b,c\) & \(A^{-1}\) if \(A=\frac19\begin{bmatrix}8&-4&a\\1&4&b\\4&7&c\end{bmatrix}\) is orthogonal. A is orthogonal ⇒ its columns are orthonormal. Check: C₁· C₁=64+1+16/81=1,\ C₂· C₂=16+16+49/81=1,\ C₁· C₂=-32+4+28/81=0 . ✓ C₃⊥ C₁:\ 8a+b+4c... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the system of linear equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No solution, (ii) Unique solution, (iii) Infinite number of solutions. Augmented matrix: [Amid B]=[beginarrayccc|c1&1&1&6\\1&2&3&10\\1&2&λ&μendarray] R₂→ R₂-R₁, R₃→ R₃-R₁ : [beginarrayccc|c1&1&1&6\\0&1&2&4\\0&1&... 1
- Apply the Gauss Seidel method, to solve the following system of linear equations up to the two iterations. \(15x+3y-2z=85,\ x-2y+8z=5,\ 2x+10y+z=51\). Rearrange for diagonal dominance: 15x+3y-2z=85 ; 2x+10y+z=51 ; x-2y+8z=5 . x=1/15(85-3y+2z), y=1/10(51-2x-z), z=1/8(5-x+2y) Start with y=z=0... 1
- Find the three sets of iterative solutions of the following equations by Gauss Jacobi method: \(5x-y+z=10;\ 2x+4y=12;\ x+y+5z=-1\). The system is diagonally dominant ( 5>1+1,\ 4>2,\ 5>1+1 ). x=1/5(10+y-z), y=1/4(12-2x), z=1/5(-1-x-y) Start with x₀=y₀=z₀=0 ; in Jacobi's me... 1
- If \(H=f(y-z,\,z-x,\,x-y)\), then prove that \(\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y}+\frac{\partial H}{\partial z}=0\). Let u=y-z,\ v=z-x,\ w=x-y , so H=f(u,v,w) . ∂ u/∂ x=0,\ ∂ v/∂ x=-1,\ ∂ w/∂ x=1 ; ∂ u/∂ y=1,\ ∂ v/∂ y=0,\ ∂ w/∂ y=-1 ; ∂ u/∂ z=-1,\ ∂ v/∂ z=1... 1
- If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), then prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial x^2}=-\frac{\sin u\cos2u}{4\cos^3u}\). (Note: the last term on the left is y²∂²u/∂ y² ; it is misprinted as ∂ x² in the paper.) Given: u=sin⁻¹(x+y/sqrt x+sqrt y)⇒ z=sin u=x+y/sqrt... 1
- State & prove Euler's theorem on homogeneous functions with two independent variables. Statement: If u=f(x,y) is a homogeneous function of degree n in x and y, then x∂ u/∂ x+y∂ u/∂ y=nu . Proof: Since u is homogeneous of degree... 1
- Prove that: \(\tanh(\log\sqrt3)=0.5\) tanh x=frace^x-e^(-x)e^x+e^(-x) . With x=logsqrt3 : e^x=sqrt3,\ e^(-x)=1/sqrt3 . tanh(logsqrt3)=fracsqrt3-frac1sqrt3sqrt3+frac1sqrt3=3-1/3+1... 1
- Find all values of \((1+i)^{1/3}\). Polar form: 1+i=sqrt2(cosπ/4+isinπ/4)=sqrt2[cos(2kπ+π/4)+isin(2kπ+π/4)] . De Moivre: (1+i)^(1/3)=2^(1/6)[cos(8k+1)π/12+isin(8k+1)π/12],\ k=0... 1