### Exam Year

Tribhuvan University

Institute of Science and Technology

2078

Bachelor Level / first-semester / Science

Computer Science and Information Technology( MTH112 )

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

1 (a)

If $$f(x) = \sqrt{x}$$ and $$g(x) = \sqrt{3 – x}$$ then find f0g and its domain and range.

b

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.

2 (a)

Using rectangular, estimate the area under the parabola y = x2 from 0 to 1.

b

A particle moves along a line so that its velocity v at time t is

v = t2 – t + 6

1. Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
2. Find the distance travelled during this time period.
3 (a)

Find the area of the region bounded by y = x2 and y = 2x – x2

b

Using trapezoidal rule, approximate $$\int_{1}^{2}\frac{1}{x}dx$$ with n = 5

4 (a)

Solve y’ = x2/y2, y(0) = 2

b

Solve the initial value problem: y” + y’ – 6y = 0,  y(0) = 1, y'(0) = 0

Group B

Attempt any TEN Questions

5

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.

1. What do the slope and T-intercept represent?
2. Use the equation to predict the average global surface temperature in 2100
6

Find the equation of tangent at (1, 2) to the curve y = 2x2

7

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x2 – 3x + 2 in [0, 3]

8

Use Newton’s method to find 6√2 correct five decimal places.

9

Find the derivatives of r(t) = (1 + t2)i – te-tj + sin 2tk and find the unit tangent vector at t=0.

10

Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.

11

Solve: y’ + 2xy – 1 = 0

12

What is sequence? Is the sequence

$$a_{n} = \frac{n}{\sqrt{5 + n}}$$

convergent?

13

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)

14

Find the partial derivative of f(x, y) = x2 + 2x3y2 – 3y2 + x + y at (1. 2)

15

Find the local maximum and minimum values, saddle points of f(x,y) = x4 + y4 – 4xy + 1

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