University of Mumbai first-year engineering Semester I Applied Mathematics-I paper, June 2025 (KT), 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 June 2025 (KT)
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) Prove that \(\sqrt{1+\mathrm{cosec}(\theta/2)}=\left(1-e^{i\theta}\right)^{-1/2}+\left(1-e^{-i\theta}\right)^{-1/2}\).
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Q.1 (b) Find \(n^{th}\) derivative of \(\frac{x^2}{(x-1)(x-2)(x-3)}\). [The numerator is faint in the source scan; it appears to be \(x^2\).]
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Q.1 (c) If \(5\sinh x-\cosh x=5\), find \(\tanh x\).
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Q.1 (d) 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 (e) If \(u=2(ax+by)^2-k(x^2+y^2)\) and \(a^2+b^2=k\), find \(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\).
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Q.1 (f) Using Newton-Raphson Method, find the real root of \(x^3-2x-5=0\) correct to three decimal places.
Q.2
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Q.2 (a) If \(\tan[\log(x+iy)]=a+ib\), prove that \(\tan[\log(x^2+y^2)]=\frac{2a}{1-a^2-b^2}\) when \(a^2+b^2\ne1\).
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Q.2 (b) Examine the function \(f(x,y)=y^2+4xy+3x^2+x^3\) for extreme values.
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Q.2 (c) If \(u=f\left(e^{x-y},e^{y-z},e^{z-x}\right)\), then prove that \(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0\).
Q.3
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Q.3 (a) Reduce the following matrix to normal form and find the rank of matrix \(A=\begin{bmatrix}1&1&1\\1&-1&-1\\3&1&1\end{bmatrix}\).
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Q.3 (b) Solve \(x^6+1=0\).
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Q.3 (c) Apply Gauss-Seidel iteration method to solve the following linear equations up to the three iterations. \(20x+y-2z=17,\ 3x+20y-z=-18,\ 2x-3y+20z=25\).
Q.4
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Q.4 (a) Expand \(\sin^5\theta\cos^3\theta\) in a series of sines of multiples of \(\theta\).
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Q.4 (b) Investigate for what values of \(\lambda\) and \(\mu\) the equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No Solution (ii) Unique Solution (iii) Infinitely Many Solutions.
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Q.4 (c) If \(y=a\cos(\log x)+b\sin(\log x)\), prove that \(x^2y_{n+2}+(2n+1)xy_{n+1}+(n^2+1)y_n=0\).
Q.5
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Q.5 (a) If \(\cos\alpha+\cos\beta+\cos\gamma=0,\ \sin\alpha+\sin\beta+\sin\gamma=0\), prove that (i) \(\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)\) (ii) \(\sin3\alpha+\sin3\beta+\sin3\gamma=3\sin(\alpha+\beta+\gamma)\)
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Q.5 (b) If \(\alpha+i\beta=\tanh\left(x+\frac{i\pi}{4}\right)\), prove that \(\alpha^2+\beta^2=1\).
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Q.5 (c) If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=-\frac{\sin u\cos2u}{4\cos^3u}\)
Q.6
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Q.6 (a) Find all the roots of \(\left(\frac12+i\frac{\sqrt3}{2}\right)^{3/4}\), hence find the continued product of the roots.
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Q.6 (b) Show that the minimum value of \(u=xy+a^3\left(\frac1x+\frac1y\right)\) is \(3a^2\).
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Q.6 (c) Find non-singular matrices P & Q such that PAQ is in normal form & hence find rank of the matrix A and obtain \(A^{-1}\). \(A=\begin{bmatrix}1&2&-2\\-1&3&0\\0&-2&1\end{bmatrix}\)