TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2080 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) $\vec{a}=(4,0,3)$ and $\vec{b}=(-2,1,5)$, find $|\vec{a}|$, $|\vec{b}|$, $\vec{a}+\vec{b}$ and $2\vec{a}+5\vec{b}$. (b) Estimate the value of $\lim_{x\to 0} \frac{\sqrt{x^2+9}-3}{x^2}$. (5+5)
- 2(a) The area of the parabola $y=x^2$ from $(1,1)$ to $(2,4)$ is rotated about the y-axis. Find the area of the resulting surface. (b) Find the solution of the equation $y^2 dy = x^2 dx$ that satisfies the initial condition $y(0)=2$. (5+5)
- 3(a) As dry air moves upward, it expands and cools. If ground temp is $20$°C and temp at height $1$ km is $10$°C, express temperature $T$ as function of height $h$. (b) Draw graph and what does slope represent? (c) Temperature at $2.5$ km? (5+3+2)
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4Evaluate $\int_{0}^{1} x^2 \sqrt{x^3+1} \, dx$. (5)
- 5Find the Maclaurin series expansion of $f(x) = e^x$ at $x=0$. (5)
- 6Find where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing and where decreasing. (5)
- 7Find $y'$ if $x^3 + y^3 = 6xy$. (5)
- 8Show that $y = x - \frac{1}{x}$ is a solution of differential equation $xy' + y = 2x$. (5)
- 9Sketch the graph and find the domain and range of $f(x) = 2x - 1$. (5)
- 10Determine whether the series $\sum \frac{n^2}{5n^2 + 4}$ converges or diverges. (5)
- 11If $f(x,y) = x^3 + x^2 y^3 - 2y^2$, find $f_x(2,1)$ and $f_y(2,1)$. (5)
- 12Show that the function $f(x) = x^2 + \sqrt{7-x}$ is continuous at $x=4$. (5)