pyq 2024 apr-may
Full explanation, worked examples & memory tricks in the app
🎯 Exam Preparation Kit
Mathematics-II Exam Prep
English 27 leveled MCQs Free
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1
What you'll study, topic by topic
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Sample questions with model answers
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8 marks Long answer
Uses stoke's theorem evaluate $\oint_C [(x+y)dx+(2x-z)dy+(y+z)dz]$ where C is the boundary for the triangle with vertices (2,0,0), (0,3,0) and (0,0,6).
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Apply Stoke's theorem to evaluate the line integral.
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8 marks Long answer
Find the volume bounded by the cylinder $x^2 + y^2 = 4$ and the planes $y + z = 4$ and $z = 0$.
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Use cylindrical coordinates to compute the volume.
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8 marks Long answer
Solve $y = 2px + y^2 p^3$.
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Use the method of solving first-order PDEs.
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8 marks Long answer
Solve $y - 2px = \tan^{-1}(xp^2)$.
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Apply the method of characteristics to solve the equation.
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8 marks Long answer
Solve $p^2 + 2py \cot x = y^2$.
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Use the method of solving first-order PDEs.
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8 marks Long answer
Solve $\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = xe^x + \sin 2x$.
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Use the method of undetermined coefficients to find the solution.
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8 marks Long answer
Solve, by the method of variation of parameters $\frac{d^2y}{dx^2} - y = \frac{2}{(1+e^x)}$.
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Apply the variation of parameters method to find the solution.
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8 marks Long answer
Express $f(x) = x^4 + 3x^3 - x^2 + 5x - 2$ in terms of legendre polynomials.
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Use the orthogonality of Legendre polynomials.
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8 marks Long answer
If $(a+ib)^x = m^{x+iy}$, prove that one of the values of $y/x$ is $2 \tan^{-1}(b/a) + \log(a^2+b^2)$.
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Use properties of logarithms and complex numbers.
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8 marks Long answer
If $\tan(\theta+i\phi) = e^{i\alpha}$, show that $\theta = (n+\frac{1}{2})\frac{\pi}{2}$ and $\phi = \frac{1}{2} \log \tan(\frac{\pi}{4}+\frac{\alpha}{2})$.
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Use the properties of the tangent function in complex analysis.
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8 marks Long answer
Find the bilinear transformation which maps the points $z = 1, i, -1$ onto the points $w = i, 0, -i$. Hence find the image of $|z|<1$.
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Construct the transformation and find the image region.
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8 marks Long answer
Evaluate $\oint_C \frac{e^z}{\cos \pi z} dz$, where C is the unit circle $|z|=1$.
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Apply the residue theorem to evaluate the integral.
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8 marks Long answer
Show that $\int_{0}^{2\pi} \frac{\cos 2\theta}{1-2a\cos\theta+a^2} d\theta = \frac{2\pi a^2}{1-a^2} (a^2 < 1)$.
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Use contour integration and residue theorem.
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5 marks Long answer
Evaluate $\int_{-1}^{1} \int_{0}^{z} \int_{x-z}^{x+z} (x + y + z) dx dy dz$.
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Evaluate the triple integral using appropriate limits.
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4 marks Medium answer
Change the order of integration in the integral: $I = \int_{0}^{a} \int_{y^{2}/a}^{y} f(x, y) dx dy$.
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Rearrange the limits of integration based on the region of integration.
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4 marks Medium answer
(i) Solve $p = \sin(y - xp)$.
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Use the method of characteristics to find the solution.
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4 marks Medium answer
(i) Explain method of variation of parameters.
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A method to find particular solutions of linear differential equations.
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4 marks Medium answer
Expand $f(z) = \frac{1}{(z-1)(z-2)}$ in the region. (i) $|z|<1$.
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Use partial fraction decomposition.
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4 marks Medium answer
(ii) $1<|z|<2$.
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Use partial fraction decomposition.
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4 marks Medium answer
(iii) $|z|>2$.
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Use partial fraction decomposition.
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4 marks Medium answer
(iv) $0<|z-1|<1$.
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Use partial fraction decomposition.
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2 marks Short answer
(ii) Define Euler's equation.
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Euler's equation is $\frac{dy}{dx} = f(x,y)$.
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2 marks Short answer
(ii) Define Legendre polynomials.
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Legendre polynomials are solutions to Legendre's differential equation.
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2 marks Short answer
(i) Write Cauchy-Riemann equations.
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The Cauchy-Riemann equations are $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$.
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2 marks Short answer
(ii) Define Analytic function.
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A function is analytic if it is differentiable in a neighborhood of every point in its domain.
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2 marks Short answer
(i) Define Cauchy integral formula.
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The Cauchy integral formula states $f(z) = \frac{1}{2\pi i} \oint_C \frac{f(\xi)}{\xi - z} d\xi$.
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2 marks Short answer
(ii) Define Cauchy Residue theorem (Without proof).
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The residue theorem states that $\oint_C f(z) dz = 2\pi i \sum \text{Res}(f, a_k)$ for isolated singularities.
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