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 20m^{3}. 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 = x^{2} from 0 to 1.

b

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

v = t^{2} – t + 6

- Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
- Find the distance travelled during this time period.

3 (a)

Find the area of the region bounded by y = x^{2} and y = 2x – x^{2}

b

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

4 (a)

Solve y’ = x^{2}/y^{2}, 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.

- What do the slope and T-intercept represent?
- 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 = 2x^{2}

7

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x^{2} – 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 + t ^{2})i – te^{-t}j +*

10

Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^{2}.

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) = x^{2} + 2x^{3}y^{2} – 3y^{2} + x + y at (1. 2)

15

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

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