University of Mumbai first-year engineering Semester I Applied Mathematics-I paper, December 2024, NEP 2020 scheme, with step-by-step solutions written with AI help and checked.
Mumbai University FE Sem 1 Applied Mathematics-I (BSC101) Question Paper December 2024
Written semester papers (NEP 2020 scheme): 60 marks and 2 hours for most first-year courses. Question 1 is compulsory, then any three of questions 2 to 6.
Sections
| Q.1 Attempt any five (compulsory) | 6 questions | |
| Q.2 | 3 questions | |
| Q.3 | 3 questions | |
| Q.4 | 3 questions | |
| Q.5 | 3 questions | |
| Q.6 | 3 questions |
Chapters covered
Instructions
Solve on paper with a timer, then check against the solutions.
Q.1 Attempt any five (compulsory)
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Q.1 (a) Find all values of \((1+i)^{1/3}\).
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Q.1 (b) Prove that: \(\tanh(\log\sqrt3)=0.5\)
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Q.1 (c) If \(u=(1-2xy+y^2)^{-1/2}\), prove that \(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=y^2u^3\).
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Q.1 (d) Find \(n^{th}\) derivative of \(y=e^{ax}\sin^2x\).
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Q.1 (e) Show that the matrix \(A=\frac{1}{\sqrt3}\begin{bmatrix}1 & 1+i\\ 1-i & -1\end{bmatrix}\) is unitary and hence find \(A^{-1}\).
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Q.1 (f) Using Newton Raphson's Method, find an iterative formula for \(\sqrt[3]{150}\).
Q.2
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Q.2 (a) If \(H=f(y-z,\,z-x,\,x-y)\), then prove that \(\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y}+\frac{\partial H}{\partial z}=0\).
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Q.2 (b) Prove that \(\log\left(e^{i\alpha}+e^{i\beta}\right)=\log\left[2\cos\left(\frac{\alpha-\beta}{2}\right)\right]+i\left(\frac{\alpha+\beta}{2}\right)\).
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Q.2 (c) Apply the Gauss Seidel method, to solve the following system of linear equations up to the two iterations. \(15x+3y-2z=85,\ x-2y+8z=5,\ 2x+10y+z=51\).
Q.3
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Q.3 (a) Discuss the maxima and minima of \((x^2+y^2+8x+6y+6)\).
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Q.3 (b) Solve the equation \(x^4+x^3+x^2+x+1=0\).
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Q.3 (c) If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), then prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial x^2}=-\frac{\sin u\cos2u}{4\cos^3u}\).
Q.4
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Q.4 (a) Prove that \(\mathrm{sech}^{-1}(\sin\theta)=\log\left(\cot\frac{\theta}{2}\right)\).
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Q.4 (b) Prove that \(\cos^6\theta=\frac{1}{32}(\cos6\theta+6\cos4\theta+15\cos2\theta+10)\).
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Q.4 (c) Find nonsingular matrices P & Q such that PAQ is in normal form and hence find rank of the matrix \(A\) for the following matrix \(A=\begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}\).
Q.5
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Q.5 (a) Find \(a,b,c\) & \(A^{-1}\) if \(A=\frac19\begin{bmatrix}8&-4&a\\1&4&b\\4&7&c\end{bmatrix}\) is orthogonal.
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Q.5 (b) State & prove Euler's theorem on homogeneous functions with two independent variables.
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Q.5 (c) If \(y=e^{\tan^{-1}x}\) then prove that \((1+x^2)y_{n+2}+[2(n+1)x-1]y_{n+1}+n(n+1)y_n=0\).
Q.6
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Q.6 (a) Find \(n^{th}\) derivative of \(y=\frac{x}{1+3x+2x^2}\).
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Q.6 (b) 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.
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Q.6 (c) Find the three sets of iterative solutions of the following equations by Gauss Jacobi method: \(5x-y+z=10;\ 2x+4y=12;\ x+y+5z=-1\).