#mu engineering
- 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
- 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
- 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 \(H=f(y-z,\,z-x,\,x-y)\), 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Write meaning of the following idioms: IdiomsMeaning Strike a blow Game over All comers Ahead of your its time Under the weather IdiomMeaning Strike a blow (for/against)To do something that helps (or harms) a cause or principle, e.g. "The new law strikes a blow for wom... 1
- Give Diagrammatic representation and explanation of the Complete Block style. Diagram (Complete / Full Block style): Sender's address City – PIN code Date Receiver's name / designation Organisation Address, City – PIN... 1
- Write an email to your class teacher for requesting a two days leave. To: priya.deshmukh@abccoe.edu.in Cc: (none) Subject: Request for leave on 19 and 20 June 2025 – Rahul Verma, FE Comps A, Roll No. 42 Dear Pr... 1
- Read the following Paragraph and answer the questions given there below: India, with its large and youthful population, holds immense potential for economic growth. However, a significant challenge the country faces is the skills gap — the mismatch b… Answer: B. The mismatch between workforce skills and job requirements Answer: (B) The mismatch between workforce skills and job requirements... 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