First Year (FE)
- 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
- 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
- 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
- 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
- Write antonyms of the following words WordSynonym Abundant Arrive Borrow Conceal Enter [The question asks for antonyms, though the table's second column is printed as "Synonym".] The instruction asks for antonyms (words of opposite meaning): WordAntonym AbundantScarce (insufficient) ArriveDepart (leave) BorrowLend Con... 1
- Write one sentence summary of the following extracts: 1. World Book Day, celebrated on April 23rd, is an international event organized by UNESCO to promote reading, publishing, and the love of literature. The date was chosen to honour the memory of g… 1. World Book Day, organised by UNESCO on 23 April in honour of great literary figures, promotes reading, publishing and the love of books,... 1
- Punctuate the following sentences You may seat wherevr you laik i had seen a movi yesterdays i believe on him sahu did not understood what you say its amazing Corrected sentences (punctuation, capitalisation, spelling and grammar): You may sit wherever you like. ("seat" is a noun/transitive verb; "... 1
- Do as Directed: The correct option should be clicked (Rewrite as an imperative sentence). Actual fact. (Identify the redundant word). End result (identify the redundant phrase). Yours Faithfully, ( Rewrite the complimentary close in open punctuation… Click the correct option. Redundant word: "Actual" – a fact is by definition actual, so "fact" alone is enough. Redundant word: "End" – a re... 1
- You ordered a cellphone from Click & Buy online shopping portal. On opening the parcel, you noticed that the cell phone was damaged. Write a complaint cum claim letter to the Customer Relations Manager. (Complete block format, open punctuation – every line begins at the left margin.) B-204, Sai Darshan Society L.B.S. Marg, Ghatkopar (West) M... 1
- Write a set of instruction for making an UPI payment. Instructions for Making a UPI Payment Using a Mobile App Purpose: Unified Payments Interface (UPI) lets you transfer money instantly from yo... 1
- Write a short paragraph on 'Responsible Use of Social Media'. Responsible Use of Social Media Social media has become a powerful tool for communication, learning and networking, but it must be used resp... 1
- Describe a Scientific Calculator Definition and function: A scientific calculator is a hand-held electronic device that performs arithmetic operations as well as scientific... 1