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) = x^{2} then find \(\frac{f(2+h) – f(2)}{h}\)

b

Dry air is moving upward. If the ground temperature is 20^{0} and the temperature at a height of 1km is 10^{0} C, express the temperature T in ^{0}C 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 = x^{2} + 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) = 6t ^{2} + 1*. Its initial velocity is

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) = x^{2} – 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) = x ^{3} – x^{2} – 6x + 2* in [0, 3]

8

Find the third approximation x_{3} to the root of the equation *f(x) = x ^{3} – 2x – 7*, setting

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

15

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

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