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
2080
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.
Section A
Attempt any two questions.
a) If \( \vec{a} = (4,0,3) \) and \( \vec{b} = (-2,1,5) \), find \( |\vec{a}| \), \( 3\vec{b} \), \( \vec{a} + \vec{b} \) and \( \overrightarrow{2a} + 5\vec{b} \)
Estimate the value of limx→0 ( (√(x2 + 9) ) – 3 )/ x2
(a)The area of the parabola y = x2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulitng surface.
(b) Find the solution of the equation y2dy = x2dx that satisfies the initial condition y(0) = 2.
As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at height of 1 km is 10C, express the temeperature T(in °C) as a function of height h(in kilometer), assuming that the linear model is appropriate.
(b) Draw a graph of the function in part(a). What does the slope represent?
(c) What is the temperature at a height of 2.5 km?
Section B
Attempt any eight questions.
Integrate 0∫1 x2√(x3 + 1) dx.
Find the Maclaurin series expansion of f(x) = ex at x =0.
Find where the function f(x) = 3×4 – 4×3 – 12 x2 + 5 is increasing and where it is decreasing.
Find y’ if x3 + y3 = 6xy.
Show that y = x – 1/x is a solution of the differential equation xy’ + y = 2x.
Sketch the graph and find the domain and range of the function f(x) =2x-1.
Determine whether the series n=1∑∞ n2 / (5n2 + 4) converges or diverges.
If f(x,y) = x3 + x2y3 – 2y2, find fx(2,1) and fy(2,1).
Show that the function f(x) = x2 + √(7-x) is continuous at x = 4.