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
2075
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 question
A function is defined by f(x) = |x| , calculate f(-3), f(4), and sketch the graph.
Prove that the \(\lim_{x \to 2} \frac{|x – 2|}{x – 2}\) doesn’t exist.
Find the domain and sketch the graph of the function \(f(x) = x^2 – 6x\).
Estimate the area between the curve y = x2 and the lines y = 1 and y = 2.
Find the Maclaurin series for cos x and prove that it represents cos x for all x.
Find the volume of a sphere of radius r
If f(x, y) = \(\frac{y}{x}\) does \(\lim_{(x,y) \to (0, 0)} f(x, y)\) exists? Justify
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 question
If f(x) = \(\sqrt{2-x}\) and g(x) = \(\sqrt{x}\), Find fog and fof
Define continuity on an interval. Show that the function \(f(x) = 1 – \sqrt{1-x^2}\) on the continuous on the interval [1,-1].
Verify Mean value theorem of f(x) = x3 – 3x + 2 for [-1, 2].
Stating with x1 = 2, find the third approximation x3 to the root of the equation x3 – 2x – 5 = 0
Evaluate \(\int_0^∞ x^3 \sqrt{1 – x^4}\) dx
Find the volume of the resulting solid which is enclosed by the curve y = x and y = x2 is rotated about the x-axis.
Find the solution of y” + 4y’ + 4 = 0.
Determine whether the series \(\sum_{n=1}^∞ \frac{n^2}{5n^2 + 4}\) converges or diverges.
test
Find \(\frac{∂z}{∂x} \enspace and \enspace \frac{∂z}{∂y}\) if z is defined as a function of x and y by the equation \(x^3 + y^3 + z^3 + 6xyz = 1\).
Find the extreme values of the function \(f(x, y) = x^2 + 2y^2\) on the circle \(x^2 + y^2 = 1\).