Guessed Subjective Questions on Mathematics-II for End Terminal Examination
Tips for preparing end Terminal Examination of Mathematics-II:
When preparing for an end-of-semester exam in Mathematics II, it can be helpful to make an educated guess about the types of questions that might appear on the exam. Here are some tips that can help you prepare:
- Review your notes: Review the notes you took in class and familiarize yourself with the topics covered in the course.
- Practice: Solve practice problems and review any material that you’re not confident about.
- Make connections: Think about how different topics are related and how they might be applied to solve a problem.
- Think logically: Make sure you understand the logic behind the material and how it can be applied.
- Make predictions: Make an educated guess about the types of questions that might appear on the exam.
- Review examples: If possible, look at examples of questions that have been asked in previous exams.
- Ask questions: If you’re unsure about something, don’t be afraid to ask questions with me.
- Take breaks: Make sure to take breaks and give your brain a rest when studying for the exam.
Suggested Questions for Practice: (Please practice the following problems as much as you can do.)
- If $ f(x)=\dfrac{x^{2}-1}{x^{2}+1} $, then prove that $ f(x)+f(\frac{1}{x})=0 $.
- Let $ f:A\to B $ be defined by $ f(x)=\dfrac{x-2}{x+3} $, where $ A=\mathbb{R}-\left\lbrace 3\right\rbrace $ and $ A=\mathbb{R}-\left\lbrace 1\right\rbrace $. Show that $ f $ is one-one and onto.
- Evaluate $\lim\limits_{x\rightarrow 0} \dfrac{\sin (\alpha + \beta)x + \sin (\alpha – \beta)x +\sin 2\alpha x}{\cos 2\beta x-\cos 2\alpha x} $.
- Evaluate $\lim\limits_{n\rightarrow \infty}\left[ \dfrac{1^{2}}{n^{3}}+\dfrac{2^{2}}{n^{3}}+\dfrac{3^{2}}{n^{3}}+\cdots+\dfrac{n^{2}}{n^{3}}\right] $.
- Find differential co-efficient of $ \tan x $ with respect to x by first principle.
- Differentiate $ \sin\left[\cos\left\lbrace \tan (\cot x)\right\rbrace\right] $ with respect to $ x $.
- Find $ \dfrac{dy}{dx} $, when $ y=x^{x}+\sin x^{\cos x} $.
- If $ y=A\left(x+\sqrt{x^{2}-1}\right)^{n} + B \left(x-\sqrt{x^{2}-1}\right)^{n} $, then prove that \[ (x^{2}-1)y_{2} +xy_{1}-n^{2}y=0.\]
- If $ x^{m}y^{n}=(x+y)^{m+n} $, then find $ \dfrac{dy}{dx} $.
- Integrate $ \displaystyle \int \tan^{4}x ~dx $.
- Integrate $ \displaystyle \int \dfrac{dx}{\sqrt{x+a}-\sqrt{x-a}} $.
- Solve $ x \cos^{2} y dx -y \cos^{2}x dy=0 $.
- Find the sin of the angle between the two vectors $ 3 \hat{i} + \hat{j} +2\hat{k} $ and $ 2\hat{i}-2\hat{j}+4\hat{k} $.
- Show that the three points $ -2\hat{i} + 3\hat{j} + 5\hat{k}, \hat{i} + 2\hat{j} + 3\hat{k} $ and $ 7\hat{i}-\hat{k} $ are collinear.
- Evaluate $\lim\limits_{n\rightarrow \infty}\left[ \dfrac{1^{2}+1}{n^{3}}+\dfrac{2^{2}+2}{n^{3}}+\dfrac{3^{2}+3}{n^{3}}+\cdots+\dfrac{n^{2}+n}{n^{3}}\right] $.
- Evaluate $\lim\limits_{x\rightarrow \alpha} \dfrac{x\sin\alpha -\alpha\sin x}{x-\alpha} $.
- The function $ f(x) $ is defined in the following form: \[ f(x) =\begin{cases} \dfrac{x^{2}-a^{2}}{x-a}, ~\text{when}~ 0\leq x\leq a,\\ 2a, ~~~~\text{when} ~ x \geq a,\end{cases}\] Examine the continuity at $ x=a $.
- Find $ \dfrac{dy}{dx} $, when $ \sin x^{y} + \sin y^{x}=2 $.
- Integrate $ \displaystyle \int_{0}^{\frac{\pi}{2}} \log \tan x~ dx=0 $.
- Solve the differential equation $ (y+x)^{2} \dfrac{dy}{dx} -2(y+x)^{2}+3=0 $.
- Solve the differential equation $ xdx + ydy = \dfrac{a^{2}(xdy – ydx)}{x^{2}+y^{2}} $.
- Two forces $ -\hat{i} + 2\hat{j} – \hat{k} $ and $ 2\hat{i} – 5\hat{j} + 6\hat{k} $ acts on a particle whose position vector is $ 4\hat{i} – 3\hat{j} + 2\hat{k} $ and displaces it to another point whose position vector is $ 6\hat{i}+\hat{j} -3\hat{k} $. Find the work done by the forces.
- The force represented by $ 5\hat{i}-\hat{k} $ is acting through the point $ 9\hat{i}-\hat{j}+2\hat{k} $. find its vector moment about the point $ 3\hat{i}+2\hat{j}+\hat{k} $. Also find magnitude of the moment.