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
2079
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
If a function is defined by
f(x)={1+x, if x<=-1
{x2, if x> -1,
evaluate f(-3), f(-1) and f(0) and sketch the graph.
Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.
Sketch the curve y=x2 +1 with the guidelines of sketching.
If z=xy2 + y3 , x= sint, y=cost, find dz/dt at t=0
Estimate the area between the curve y=x2 and the lines x=0 and x=1, using rectangle method, with four sub intervals.
A particle moves a line so that its velocity v at time t is
(1) Find the displacement of the particle during the fine period 1 ≤ t ≤ 4
(2) Find the distance travelled during this time period.
Define initial value problem. Solve:
yH+y‘ -6y =0, y(0)=1, y‘(0)=0
Find the Taylor’s series expansion for cosx at x=0.
Group B
Attempt any TEN questions
Dry air is moving upward. If the ground temperature is 20° and the temperature at a height of 2km is 10° c, express the temperature T in ° c as a function of the height h(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km?
Find the equation of the tangent at (1,3) to the curve y=2x2 + 1.
State Rolle’s theorem and verify the theorem for f(x) = x2 – 9, x ε[-3,3]
Starting with x1= 1, find the third approximate x3 to the root of the equation x³ – x – 5 = 0
Show the integral coverages
∫0³ dx/x-1
Use Trapezoidal rule to approximate the integral 1 ∫² dx/x, with n=5.
Find the derivative of (r(t)) = t^2i – te(-t)j + sin(2t)k and find the unit tangent vector at t = 0
What is sequence? Is the sequence
\(a_{n} = \frac{n}{\sqrt{5 + n}}\)
convergent?
Find the angle between the vectors a = (2, 2, -1) and b = (1, 3, 2)
Find the partial derivative fxx and fyy of f(x,y)= x2 + x3y2 – y2 + xy, at (1,2).
Evaluate
0∫3 1∫2 x2y dxdy