Mathematics II Exam Prep
Full explanation, worked examples & memory tricks in the app
🎯 Exam Preparation Kit
Mathematics II Exam Prep
English 19 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
Verify Green's theorem for $\int_C [(xy+y^2)dx + x^2dy]$ Where C is bounded by $y=x$ and $y = x^2$.
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Show that line integral equals double integral over region.
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8 marks Long answer
Evaluate $\int_0^2 \int_y^2 rac{xdy dx}{x^2+y^2}$ By changing cartesian to polar form.
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Convert to polar coordinates and integrate.
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8 marks Long answer
State Gauss Divergence theorem and evaluate $\int_S F \cdot ds$ with the help of Gauss Divergence theorem, where $F = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k}$ And S is the surface bounding the region. $x^2 + y^2 = 4, z = 0$ and $z=3$.
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Apply divergence theorem to calculate surface integral.
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8 marks Long answer
Solve the Differential Equation $ rac{dy}{dx} + x \sin^2 y = x^2 \cos^2 y$.
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Use integrating factor or substitution to solve.
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8 marks Long answer
Solve the Differential equation $y(xy+2x^2y^2)dx + x(xy - x^2y^2) dy = 0$.
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Separate variables and integrate.
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8 marks Long answer
Solve the differential equation $p^2 + 2py \cot x = y^2$.
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Rearrange and solve for p.
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8 marks Long answer
Solve the Differential Equation $(x^2D^2-xD+1)y=\log x$.
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Use complementary and particular solutions.
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8 marks Long answer
Solve using the variation of parameters $ rac{d^2y}{dx^2}+a^2y = \sec ax$.
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Apply variation of parameters to find solution.
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8 marks Long answer
If $i^{i^{... \infty}} = \alpha + i\beta$ then prove that $ an \left(\frac{\pi\alpha}{2}\right) = \frac{B}{A}$ and $A^2 + B^2 = e^{-\pi\beta}$.
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Use properties of complex exponentiation.
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8 marks Long answer
If $w = \phi + i\psi$ represent the complex potential for an electric field and $\psi = x^2 - y^2 + \frac{x}{x^2 - y^2}$ Determine the function $\phi$.
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Use Cauchy-Riemann equations to find $\phi$.
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8 marks Long answer
If $ an(\theta+i\phi) = e^{i\alpha}$ show that $ heta = \left(n+\frac{1}{2}\right)\frac{\pi}{2}$ $\phi = \frac{1}{2}\log \tan\left(\frac{\pi}{4}+\frac{\alpha}{2}\right)$.
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Use properties of logarithms and trigonometric identities.
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8 marks Long answer
Apply indenting method to prove that $\int_0^\infty \frac{\sin x}{x} dx = \frac{\pi}{2}$.
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Use integration by parts and contour integration.
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8 marks Long answer
Find the Taylor's or Laurent's series expansion of the function $ rac{1}{(z-1)(z-2)}$ When (i) $|z|<1$ (ii) $1<|z|<2$ (iii) $|z|>2$.
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Use partial fractions and series expansion techniques.
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8 marks Long answer
Find the sum of residue of function $f(z) = \frac{\sin z}{z \cos z}$ At its poles inside the circle $|z|=2$.
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Identify poles and apply the residue theorem.
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5 marks Long answer
Find the volume bounded by the cylinder $x^2 + y^2 = 4$ and the plane $y + z = 4$ and $z=0$.
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Use triple integrals to find the volume.
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2 marks Short answer
Explain method of variation of parameter.
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A method for finding particular solutions of non-homogeneous ODEs.
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2 marks Short answer
Write the real and imaginary part of $(\alpha+i\beta)^{x+iy}$.
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Use Euler's formula to express in terms of sine and cosine.
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2 marks Short answer
Define Singularity and its types.
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A point where a function ceases to be well-defined.
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1 marks Short answer
Define Exact Differential Equation.
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An equation where a total differential equals zero.
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