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
2078
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 \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{3 – x}\) then find f0g and its domain and range.
A rectangular storage container with an open top has a volume of 20m3. The length of its base is twice its width. Material for the base costs Rs.10 per square meter material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base.
Using rectangular, estimate the area under the parabola y = x2 from 0 to 1.
A particle moves along a line so that its velocity v at time t is
v = t2 – t + 6
- Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
- Find the distance travelled during this time period.
Find the area of the region bounded by y = x2 and y = 2x – x2
Using trapezoidal rule, approximate \(\int_{1}^{2}\frac{1}{x}dx\) with n = 5
Solve y’ = x2/y2, y(0) = 2
Solve the initial value problem: y” + y’ – 6y = 0, y(0) = 1, y'(0) = 0
Group B
Attempt any TEN Questions
Recent studies indicates that the average surface temperature of the earth has been rising rapidly. Some scientists have modeled the temperature by the linear function T = 0.03t + 8.50, where T is temperature in degree centigrade and t represents years since 1900.
- What do the slope and T-intercept represent?
- Use the equation to predict the average global surface temperature in 2100
Find the equation of tangent at (1, 2) to the curve y = 2x2
State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x2 – 3x + 2 in [0, 3]
Use Newton’s method to find 6√2 correct five decimal places.
Find the derivatives of r(t) = (1 + t2)i – te-tj + sin 2tk and find the unit tangent vector at t=0.
Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.
Solve: y’ + 2xy – 1 = 0
What is sequence? Is the sequence
\(a_{n} = \frac{n}{\sqrt{5 + n}}\)
convergent?
Find a vector perpendicular to the plane that passes through the points:p(1, 4, 6), Q(-2, 5, -1) and R(1. -1, 1)
Find the partial derivative of f(x, y) = x2 + 2x3y2 – 3y2 + x + y at (1. 2)
Find the local maximum and minimum values, saddle points of f(x,y) = x4 + y4 – 4xy + 1