Mathematics I
This course makes students able to understand and formulate real-world problems into mathematical statements and also develop solutions to mathematical problems at the level appropriate to the course.
Tribhuvan University
Institute of Science and Technology
2074
Bachelor Level / first-semester / Science
Computer Science and Information Technology( MTH117 )
Mathematics I
Full Marks: 80 + 20
Pass Marks: 32 + 8
Time: 3 Hours
Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.
Group A
Attempt any three questions
A function is defined by f(x) = \(\left\{\begin{matrix}
x + 2 \enspace \enspace if \enspace x < 0\\
1 – x \enspace \enspace if \enspace x > 0
\end{matrix}\right.\), Concluate f(-1), f(3) and sketch graph.
Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.
Estimate the area between the curve y2 = x and the lines x = 0 and x = 2.
Find the Maclaurin series for ex and prove that it represents ex for all x.
Define Initial Value Problem. Solve that value problem of \(y^2 + 5y = 1\), y(0) = 2
Find the volume of a sphere of radius r
For What values of x does the series \(\sum_{n=1}^{∞} \frac{(x – 3)^n}{x}\) converge?
Calculate \(\int_R\int f(x,y) dA\) for f(x,y) = 100 – 6x2y and \(R: 0 \leq x \leq 2, -1 \leq y \leq 1 \)
Group B
Attempt any ten questions
If \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{3-x}\), find gof and gog.
Use Continuity to evaluate the limit, \(\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )\)
Verify Mean Value Theorem by f(x) = x3 – 3x + 3 for [-1, 2]
Sketch the curve y = x^3 + x
Determine whether the integral \(\int_{1}^{∞}\left ( \frac{1}{x} \right )dx\) is convergent or divergent
Find the length of the arc of the semi cubical parabola y2 = x2 between the points (1,1) and (4,8)
Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1
Test the convergence of the series \(\sum_{n=1}^{∞} \left ( \frac{n^n}{n!} \right )\)
Define cross product of two vectors. If \(\vec{a} = \vec{i} + 3\vec{j} + 4\vec{k}\) and \(\vec{b} = 2\vec{i} + 7\vec{j} – 5\vec{k}\) find the vector \(\vec{b} \times \vec{a} \enspace and \enspace \vec{a} \times \vec{b}\)
Define limit of a function. Find limit \(\lim_{x \to ∞} (x – \sqrt{x}) \)
Find the extremes values of f(x, y) = y2 – x2