Engineering Mathematics II Problems: Practice Questions

Here, you can find a list of engineering mathematics questions that can be treated as practice problems in Mid-Sem exam, Feb-Mar 2026.

For the study material of Engineering Mathematics II, please visit this LINK.

Engineering Mathematics II Mid Term Practice Problems

Q1: Define a convergent, divergent and oscillatory sequence with an example.

Q2: Write down the definition of a bounded sequence with examples.

Q3: What is a monotonically increasing/decreasing sequence? Show that the sequence $\left\{ \dfrac{n+1}{2n+3} \right\}$ is monotonically increasing.

Q4: State ratio and root test for a series. Test the convergence of the following series:

1. $\sum \sin \left( \dfrac{1}{n} \right)$ [Full Solution]
2. $\sum \dfrac{1}{n} \sin \left( \dfrac{1}{n} \right)$ [Full Solution]
3. $\sum_{n=1}^\infty \dfrac{\sin nx}{n^2} \, (x>0)$
4. $\sum_{n=2}^\infty \dfrac{n^{n^2}}{(n-1)^{n^2}}$ [Full Solution]
5. $\sum \left( 1-\dfrac{1}{n}\right)^{n^2}$
6. $\sum_{n=1}^\infty \dfrac{2n+4}{(2n-1)(2n+1)(2n+3)}$ [Apply limit comparison test with $\sum \dfrac{1}{n^2}$.]

Q5: Test the convergence of the series $\dfrac{2}{1 \cdot 3 \cdot 5}+\dfrac{4}{3 \cdot 5 \cdot 7}+\dfrac{6}{5 \cdot 7 \cdot 9}+\cdots$

Q6: What is an alternating series? State Leibnitz’s test for an alternating series. Using this test or otherwise, find the values of $x$ for which the series

$x-\dfrac{x^2}{2^2}+\dfrac{x^3}{3^2}-\dfrac{x^4}{4^2}+\cdots$

converges.

Q7: Find the radius of convergence of the series: $1+2x+2^2 x^2+2^3x^3+\cdots$.

Q8: Write down the orders and degrees of the differential equations listed below:

1. $\frac{dy}{dx}=\sin^2(x+y)$
2. $\frac{d^2y}{dx^2}+\frac{dy}{dx}=\sqrt{x+y}$.

Q9: Eliminate A and B to form an ODE: [Full Solution]

1. y=Ax+A³
2. y=e^-x(Acosx + Bsinx).

Q10: Find the value of $m$ for which the differential equation

$\left(xy^2+mx^2y\right)dx + \left(x+y\right)x^2dy=0$

is exact and solve it for this value of $m$.

Q11: What is an integrating factor of a ODE? For the inexact differential equation xdy-ydx=0, write down all possible integrating factors. [Note: Both 1/x² and 1/y² are integrating factors. Verify it.]

Q12: Solve:

1. $\dfrac{dy}{dx}=\dfrac{y}{x}+\tan\left(\dfrac{y}{x}\right)$
2. $(x^2-y^2)dx-xydy=0$.
3. $\dfrac{dy}{dx}=\sin(x+y)$.
4. $x\dfrac{dy}{dx}=y-\sqrt{x^2+y^2}$ (homogeneous differential equation)
5. $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$

Q13: Solve the differential equation $\frac{dy}{dx}+y \cot x=\text{cosec}~ x$. [View Solution]

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