Applied Mathematics-I
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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 \(z=f(x,y),\ x=r\cos\theta,\ y=r\sin\theta\) 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
- 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 \(u=\log(x^3+y^3+z^3-3xyz)\), 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
- Find \(n^{th}\) derivative of \(y=e^{ax}\sin^2x\). y=e^ax·1-cos2x/2=1/2e^ax-1/2e^axcos2x . Standard results: Dⁿ(e^ax)=aⁿe^ax ; Dⁿ[e^axcos(bx+c)]=rⁿe^axcos(bx+c+nφ) where r=√(a²+b²),\ φ=tan⁻¹b... 1
- Discuss the maxima and minima of \((x^2+y^2+8x+6y+6)\). f=x²+y²+8x+6y+6 . Stationary points: f_x=2x+8=0⇒ x=-4 ; f_y=2y+6=0⇒ y=-3 . Stationary point (-4,-3) . Second derivatives: r=f_xx=2,\ s=f_xy=... 1
- If \(y=e^{\tan^{-1}x}\) then prove that \((1+x^2)y_{n+2}+[2(n+1)x-1]y_{n+1}+n(n+1)y_n=0\). y₁=e^tan⁻¹x·frac11+x²=frac y1+x²⇒(1+x²)y₁=y . Differentiate again: (1+x²)y₂+2xy₁=y₁⇒(1+x²)y₂+(2x-1)y₁=0 . Leibnitz's theorem (differentiate... 1
- Find \(n^{th}\) derivative of \(y=\frac{x}{1+3x+2x^2}\). 1+3x+2x²=(1+x)(1+2x) . Partial fractions: x/(1+x)(1+2x)=frac A1+x+frac B1+2x⇒ x=A(1+2x)+B(1+x) . x=-1:\ -1=-A⇒ A=1 ; x=-1/2:\ -1/2=B/2⇒ B=-1... 1
- Solve the equation \(x^4+x^3+x^2+x+1=0\). Multiply by (x-1) : (x-1)(x⁴+x³+x²+x+1)=x⁵-1=0 , with x≠1 . So the required roots are the fifth roots of unity other than 1: x⁵=1=cos2kπ+isi... 1
- Prove that \(\mathrm{sech}^{-1}(\sin\theta)=\log\left(\cot\frac{\theta}{2}\right)\). Let sech⁻¹(sinθ)=u . Then sech u=sinθ⇒cosh u=1/sinθ (take 0<θ<π ). sinh u=√(cosh²u-1)=√(1-sin²θ/sin²θ)=cosθ/sinθ . e^u=cosh u+sinh u=1+cosθ/... 1
- Using Newton Raphson's Method, find an iterative formula for \(\sqrt[3]{150}\). Let x=sqrt[3]150⇒ f(x)=x³-150=0,\ f'(x)=3x² . Newton–Raphson: x_(n+1)=x_n-f(x_n)/f'(x_n)=x_n-x_n³-150/3x_n² Iterative formula: x_(n+1)=2x_n³... 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
- Prove that \(\cos^6\theta=\frac{1}{32}(\cos6\theta+6\cos4\theta+15\cos2\theta+10)\). Let x=cosθ+isinθ . Then x+1/x=2cosθ and xⁿ+frac1xⁿ=2cos nθ . (2cosθ)⁶=(x+1/x)⁶=x⁶+6x⁴+15x²+20+15/x²+frac6x⁴+frac1x⁶ (binomial coefficients 1... 1
- Prove that \(\log\left(e^{i\alpha}+e^{i\beta}\right)=\log\left[2\cos\left(\frac{\alpha-\beta}{2}\right)\right]+i\left(\frac{\alpha+\beta}{2}\right)\). Take out e^(i(α+β)/2) : e^iα+e^iβ=e^(iα+β/2)(e^(iα-β/2)+e^(-iα-β/2))=e^(iα+β/2)·2cosα-β/2 (using e^iθ+e^(-iθ)=2cosθ ). Taking logarithm (pri... 1
- Show that the matrix \(A=\frac{1}{\sqrt3}\begin{bmatrix}1 & 1+i\\ 1-i & -1\end{bmatrix}\) is unitary and hence find \(A^{-1}\). Given: A=1/sqrt3beginbmatrix1 & 1+i\\ 1-i & -1endbmatrix . A is unitary if AA^θ=I , where A^θ=(bar A)^T . Step 1: bar A=1/sqrt3beginbmatrix1... 1
- Find nonsingular matrices P & Q such that PAQ is in normal form and hence find rank of the matrix \(A\) for the following matrix \(A=\begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}\). Write A=I₃ A I₃ . Row operations act on A and the pre-factor; column operations act on A and the post-factor. R₁→ R₁-R₂ : beginbmatrix1&0&0\... 1