### Exam Year

Tribhuvan University

Institute of Science and Technology

2077

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) = x2 then find $$\frac{f(2+h) – f(2)}{h}$$

b

Dry air is moving upward. If the ground temperature is 200 and the temperature at a height of 1km is 100 C, express the temperature T in 0C as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b)Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2km?

c

Find the equation of the tangent to the parabola y = x2 + x + 1 at (0, 1)

2 (a)

A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?

b

Sketch the curve $$y = \frac{1}{x – 3}$$

3 (a)

Show that the converges $$\int_{∞}^{1}\frac{1}{x^2} \,dx$$ and diverges $$\int_{∞}^{1}\frac{1}{x}\,dx$$

b

If $$f(x,y) = \frac{xy}{x^2 + y^2}$$, does f(x, y) exist, as (x, y) → (0, 0)?

c

A particle moves in a straight line and has acceleration given by a(t) = 6t2 + 1. Its initial velocity is 4m/sec and its initial displacement is s(0) = 5cm. Find its position function s(t).

4 (a)

Evaluate $$\int_{3}^{2} \int_{0}^{\frac{\pi}{2}} \enspace (y + y^{2}cosx ) \enspace dx dy$$

b

Find the Maclaurin series for cos x and prove that it represents cos x for all x.

Group B

Attempt ten questions

5

If f(x) = x2 – 1, g(x) = 2x + 1, find fog and gof and domain of fog.

6

Define continuity of a function at a point x = a. Show that the function $$f(x) = \sqrt{1-x^2}$$ is   continuous on the interval[1, -1].

7

State Rolle’s theorem and verify the Rolle’s theorem for f(x) = x3 – x2 – 6x + 2 in [0, 3]

8

Find the third approximation x3 to the root of the equation f(x) = x3 – 2x – 7, setting x1 = 2.

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” + y’ = 0, y(0) = 5, y(π/4) = 3

12

Show that the series $$\sum_{n=0}^{∞} \frac{1}{1 + n^2}$$ converges.

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) = x3 + 2x3y3 – 3y2 + x + y, at (2,1)

15

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