University of Mumbai first-year engineering Semester I Applied Mathematics-I paper, December 2025, 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 2025
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) If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\).
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Q.1 (b) If \(z=\frac1{\sqrt2}+\frac i{\sqrt2}\) and \(\bar z\) is the conjugate of z, then prove that \((z)^{10}+(\bar z)^{10}=0\).
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Q.1 (c) Find \(n^{th}\) derivative of \(y=\sin2x\cos6x\).
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Q.1 (d) 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.
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Q.1 (e) Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\).
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Q.1 (f) Solve \(10x+y+z=12,\ x+10y+z=12,\ x+y+10z=12\) using the Gauss-Jacobi's method by taking two iterations.
Q.2
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Q.2 (a) If \(z=f(x,y),\ x=r\cos\theta,\ y=r\sin\theta\) then prove that \(\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2=\left(\frac{\partial z}{\partial r}\right)^2+\frac1{r^2}\left(\frac{\partial z}{\partial\theta}\right)^2\).
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Q.2 (b) 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.
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Q.2 (c) If \(\tan(\alpha+i\beta)=e^{i\theta}\), prove that \(\alpha=\frac{n\pi}2+\frac\pi4\) and \(\beta=\frac12\log\left[\tan\left(\frac\pi4+\frac\theta2\right)\right]\).
Q.3
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Q.3 (a) Find the rank of the matrix by reducing it to normal form \(A=\begin{bmatrix}1&-1&-2&-3\\4&1&0&2\\0&3&1&4\\0&1&0&2\end{bmatrix}\).
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Q.3 (b) For \(u=\sin^{-1}(x^2+y^2)^{1/5}\) deduce \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=\frac2{25}\tan u\,(2\tan^2u-3)\).
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Q.3 (c) If \(\sin^4\theta\cos^3\theta=a_1\cos\theta+a_3\cos3\theta+a_5\cos5\theta+a_7\cos7\theta\), then prove that \(a_1+9a_3+25a_5+49a_7=0\).
Q.4
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Q.4 (a) 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.]
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Q.4 (b) Show that \(\log[\sin(x+iy)]=\frac12\log\left(\frac{\cosh2y-\cos2x}2\right)+i\tan^{-1}(\cot x\tanh y)\).
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Q.4 (c) If \(u=\log(x^3+y^3+z^3-3xyz)\), prove that \(\left(\frac\partial{\partial x}+\frac\partial{\partial y}+\frac\partial{\partial z}\right)^2u=-\frac9{(x+y+z)^2}\).
Q.5
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Q.5 (a) For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank.
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Q.5 (b) Find all the values of \(\left(\frac12+i\frac{\sqrt3}2\right)^{3/4}\) and show that their continued product is 1.
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Q.5 (c) Discuss the maxima and minima of \(x^3y^2(1-x-y)\).
Q.6
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Q.6 (a) Solve the following equations by Gauss-Seidel method by taking two iterations. \(28x+4y-z=32,\ 2x+17y+4z=35,\ x+3y+10z=24\).
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Q.6 (b) If \(y=e^{m\sin^{-1}x}\), prove that \((1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2+m^2)y_n=0\)
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Q.6 (c) 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.