#mu engineering
- If \(u=f\left(e^{x-y},e^{y-z},e^{z-x}\right)\), then prove that \(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0\). Let X=e^(x-y),\ Y=e^(y-z),\ Z=e^(z-x) , so u=f(X,Y,Z) . ∂ X/∂ x=X,\ ∂ X/∂ y=-X,\ ∂ X/∂ z=0 ; ∂ Y/∂ x=0,\ ∂ Y/∂ y=Y,\ ∂ Y/∂ z=-Y ; ∂ Z/∂ x=-Z... 1
- Reduce the following matrix to normal form and find the rank of matrix \(A=\begin{bmatrix}1&1&1\\1&-1&-1\\3&1&1\end{bmatrix}\). R₂→ R₂-R₁,\ R₃→ R₃-3R₁ : beginbmatrix1&1&1\\0&-2&-2\\0&-2&-2endbmatrix R₃→ R₃-R₂ : beginbmatrix1&1&1\\0&-2&-2\\0&0&0endbmatrix C₂→ C₂-C₁,\ C... 1
- Find \(n^{th}\) derivative of \(y=\sin2x\cos6x\). Using 2sin Acos B=sin(A+B)+sin(A-B) : y=1/2[sin8x+sin(-4x)]=1/2[sin8x-sin4x] . Using Dⁿsin(ax)=aⁿsin(ax+fracnπ2) : y_n=1/2[8ⁿsin(8x+fracnπ2)... 1
- If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\). u(tx,ty)=sqrt t(sqrt x+sqrt y)/t(x+y)=t^(-1/2)u , so u is homogeneous of degree n=-1/2 . By Euler's theorem xu_x+yu_y=nu : x∂ u/∂ x+y∂ u/∂ y... 1
- If \(z=\frac1{\sqrt2}+\frac i{\sqrt2}\) and \(\bar z\) is the conjugate of z, then prove that \((z)^{10}+(\bar z)^{10}=0\). z=cosfracπ4+isinfracπ4, bar z=cosfracπ4-isinfracπ4 . By De Moivre: z¹⁰=cosfrac10π4+isinfrac10π4 , bar z¹⁰=cosfrac10π4-isinfrac10π4 . z¹⁰+bar... 1
- Express the matrix \(A=\begin{bmatrix}2+3i&2&3i\\-2i&0&1+2i\\4&2+5i&-i\end{bmatrix}\) as the sum of a Hermitian and a skew-Hermitian matrix. A=P+Q with P=1/2(A+A^θ) (Hermitian) and Q=1/2(A-A^θ) (skew-Hermitian), where A^θ=(bar A)^T . A^θ=beginbmatrix2-3i&2i&4\\2&0&2-5i\\-3i&1-2i&i... 1
- Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\). Let x+iy=re^iθ where r=√(x²+y²),\ θ=tan⁻¹y/x . Then x-iy=re^(-iθ) . log(x+iy/x-iy)=logfracre^iθre^(-iθ)=log e^2iθ=2iθ (principal value) =2it... 1
- Solve \(10x+y+z=12,\ x+10y+z=12,\ x+y+10z=12\) using the Gauss-Jacobi's method by taking two iterations. Diagonally dominant. x=12-y-z/10,\ y=12-x-z/10,\ z=12-x-y/10 . Start x₀=y₀=z₀=0 . Iteration 1: x₁=y₁=z₁=12/10=1.2 Iteration 2: x₂=12-1.2-1.2... 1
- If \(z=f(x,y),\ x=r\cos\theta,\ y=r\sin\theta\) then prove that \(\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2=\left(\frac{\partial z}{\partial r}\right)^2+\frac1{r^2}\left(\frac{\partial z}{\partial\theta… ∂ x/∂ r=cosθ,\ ∂ y/∂ r=sinθ,\ ∂ x/∂θ=-rsinθ,\ ∂ y/∂θ=rcosθ . Chain rule: ∂ z/∂ r=∂ z/∂ xcosθ+∂ z/∂ ysinθ ...(1) ∂ z/∂θ=-rsinθ∂ z/∂ x+rcosθ∂... 1
- Using the Newton-Raphson method find the root of the equation \(2x^3-3x+4=0\) lying between -2 and -1, correct to four places of decimals. f(x)=2x³-3x+4,\ f'(x)=6x²-3 . f(-2)=-6,\ f(-1)=5 → root in (−2, −1); f(-1.5)=1.75 , so take x₀=-1.5 . x_(n+1)=x_n-2x_n³-3x_n+4/6x_n²-3=4x_n³... 1
- If \(\tan(\alpha+i\beta)=e^{i\theta}\), prove that \(\alpha=\frac{n\pi}2+\frac\pi4\) and \(\beta=\frac12\log\left[\tan\left(\frac\pi4+\frac\theta2\right)\right]\). tan(α+iβ)=cosθ+isinθ⇒tan(α-iβ)=cosθ-isinθ . For α : tan2α=tan[(α+iβ)+(α-iβ)]=2cosθ/1-(cos²θ+sin²θ)=2cosθ/0=∞ . ⇒2α=nπ+fracπ2⇒ α=fracnπ2+frac... 1
- Find the rank of the matrix by reducing it to normal form \(A=\begin{bmatrix}1&-1&-2&-3\\4&1&0&2\\0&3&1&4\\0&1&0&2\end{bmatrix}\). C₂→ C₂+C₁,\ C₃→ C₃+2C₁,\ C₄→ C₄+3C₁ : beginbmatrix1&0&0&0\\4&5&8&14\\0&3&1&4\\0&1&0&2endbmatrix R₂→ R₂-4R₁ , then R₂rightarrow R₄ : beginbma... 1
- For \(u=\sin^{-1}(x^2+y^2)^{1/5}\) deduce \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=\frac2{25}\tan u\,(2\tan^2u-3)\). z=sin u=(x²+y²)^(1/5) is homogeneous of degree n=2/5 . Euler: xz_x+yz_y=nz⇒cos u (xu_x+yu_y)=2/5sin u⇒ xu_x+yu_y=2/5tan u=g(u) . Deduction:... 1
- If \(\sin^4\theta\cos^3\theta=a_1\cos\theta+a_3\cos3\theta+a_5\cos5\theta+a_7\cos7\theta\), then prove that \(a_1+9a_3+25a_5+49a_7=0\). Let x=cosθ+isinθ : x-1/x=2isinθ,\ x+1/x=2cosθ,\ xⁿ+frac1xⁿ=2cos nθ . (2isinθ)⁴(2cosθ)³=(x-1/x)⁴(x+1/x)³=(x-1/x)(x²-frac1x²)³ =(x-1/x)(x⁶-3x²... 1
- Test the consistency and hence solve the following system of linear equations, \(-2y+3t=2,\ 2x+y+z+t=-4,\ 4x-3y+z+7t=8\). [As printed; the first equation is very likely \(x-2y+3t=2\), with the \(x\) dropped in print.] The first equation is printed as -2y+3t=2 , with no x-term visible. The standard textbook problem is x-2y+3t=2 , so that is solved first; th... 1
- Show that \(\log[\sin(x+iy)]=\frac12\log\left(\frac{\cosh2y-\cos2x}2\right)+i\tan^{-1}(\cot x\tanh y)\). sin(x+iy)=sin xcos iy+cos xsin iy=sin xcosh y+icos xsinh y . For log(a+ib)=1/2log(a²+b²)+itan⁻¹b/a : a²+b²=sin²xcosh²y+cos²xsinh²y=sin²x(1+s... 1
- If \(u=\log(x^3+y^3+z^3-3xyz)\), prove that \(\left(\frac\partial{\partial x}+\frac\partial{\partial y}+\frac\partial{\partial z}\right)^2u=-\frac9{(x+y+z)^2}\). u_x=3x²-3yz/x³+y³+z³-3xyz,\ u_y=3y²-3zx/·s,\ u_z=3z²-3xy/·s u_x+u_y+u_z=3(x²+y²+z²-xy-yz-zx)/(x+y+z)(x²+y²+z²-xy-yz-zx)=frac3x+y+z (using x³... 1
- For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank. A=beginbmatrix2&3&4\\3&4&5\\4&5&6endbmatrix R₂→ R₂-R₁,\ R₃→ R₃-R₁ : beginbmatrix2&3&4\\1&1&1\\2&2&2endbmatrix ; R₃→ R₃-2R₂ : beginbmatrix2&3... 1
- Find all the values of \(\left(\frac12+i\frac{\sqrt3}2\right)^{3/4}\) and show that their continued product is 1. Step 1 (polar form): 1/2+isqrt3/2=cosπ/3+isinπ/3 . Including the general angle, 1/2+isqrt3/2=cos(2kπ+π/3)+isin(2kπ+π/3) . Step 2 (De Moivre)... 1
- Discuss the maxima and minima of \(x^3y^2(1-x-y)\). f=x³y²-x⁴y²-x³y³ . f_x=3x²y²-4x³y²-3x²y³=x²y²(3-4x-3y) ; f_y=2x³y-2x⁴y-3x³y²=x³y(2-2x-3y) . Stationary points (x, y ≠ 0): 4x+3y=3,\ 2x+3y=2⇒... 1
- Solve the following equations by Gauss-Seidel method by taking two iterations. \(28x+4y-z=32,\ 2x+17y+4z=35,\ x+3y+10z=24\). Diagonally dominant. x=frac128(32-4y+z),\ y=frac117(35-2x-4z),\ z=frac110(24-x-3y) . Start y=z=0 , using latest values. Iteration 1: x₁=32/2... 1
- If \(y=e^{m\sin^{-1}x}\), prove that \((1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2+m^2)y_n=0\) y₁=e^msin⁻¹x·frac m√(1-x²)⇒√(1-x²) y₁=my⇒(1-x²)y₁²=m²y² . Differentiate: (1-x²)2y₁y₂-2xy₁²=2m²yy₁ ; divide by 2y₁ : (1-x²)y₂-xy₁-m²y=0 . Lei... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the system of equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No Solution, ii) Unique Solution iii) Many 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
- If \(u=(1-2xy+y^2)^{-1/2}\), prove that \(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=y^2u^3\). ∂ u/∂ x=-1/2(1-2xy+y²)^(-3/2)(-2y)=y u³ ∂ u/∂ y=-1/2(1-2xy+y²)^(-3/2)(-2x+2y)=(x-y) u³ x∂ u/∂ x-y∂ u/∂ y=xy u³-(xy-y²)u³ =y²u³ (proved). 1