TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2081 B.S.
Candidates are required to give their answers in their own words as far as practicable.
Group A
Long Answer QuestionsAttempt any Two questions.
- 1(a) Sketch the graph of $f(x)=x^2$. Find its domain and range. (b) Evaluate: $\lim_{x\rightarrow 1}\sin^{-1}\left(\frac{1-\sqrt{x}}{1-x}\right)$
- 2(a) Where is $f(x)=|x|$ differentiable? (b) Find dimensions of a rectangular field (1200m fencing) for largest area bordering a river.
- 3(a) Solve $x^2y'+xy=1, y(1)=2, x>0$. (b) Find area enclosed by $y=x-1$ and $y^2=2x+6$.
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4Evaluate: $\int_{0}^{\sqrt{3}}\sqrt{1+x^2}x^3dx.$
- 5Find Maclaurin series expansion of $f(x)=\sin x$.
- 6Find unit normal and binormal vectors for $\vec{r}(t)=\cos t\vec{i}+\sin t\vec{j}+t\vec{k}$.
- 7Does $\lim_{(x,y)\rightarrow(0,0)}\frac{xy}{x^2+y^2}$ exist? Justify.
- 8Determine if sequence $a_n=(-1)^n$ is convergent or divergent.
- 9Find velocity, speed, and acceleration for $\vec{r}(t)=t^3\vec{i}+t^2\vec{j}$ at $t=1$.
- 10Show $y=\frac{1+ce^t}{1-ce^t}$ is a solution of $y'=\frac{1}{2}(y^2-1)$.
- 11If $f(x,y)=2x^3+x^2y^3-y^4$, find $f_x(1,-2), f_y(1,-1)$ and $f_{yx}(1,-1)$.
- 12Use cylindrical shells to find volume rotating about x-axis for $y=\sqrt{x}$ from $x=0$ to $x=1$.