Tribhuvan University

Institute of Science and Technology

2079

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 a function is defined by

f(x)={1+x, if x<=-1

{x^{2}, if x> -1,

evaluate f(-3), f(-1) and f(0) and sketch the graph.

b

Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.

2 (a)

Sketch the curve y=x^{2} +1 with the guidelines of sketching.

b

If z=xy^{2 }+ y^{3} , x= sint, y=cost, find dz/dt at t=0

3 (a)

Estimate the area between the curve y=x^{2 }and the lines x=0 and x=1, using rectangle method, with four sub intervals.

b

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

(1) Find the displacement of the particle during the fine period 1 ≤ t ≤ 4

(2) Find the distance travelled during this time period.

4 (a)

Define initial value problem. Solve:

y^{H}+y^{‘} -6y =0, y(0)=1, y^{‘}(0)=0

b

Find the Taylor’s series expansion for cosx at x=0.

**Group B**

**Attempt any TEN questions**

5

Dry air is moving upward. If the ground temperature is 20^{°} and the temperature at a height of 2km is 10^{°} c, express the temperature T in ^{°} c as a function of the height h(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km?

6

Find the equation of the tangent at (1,3) to the curve y=^{2}x^{2} + 1.

7

State Rolle’s theorem and verify the theorem for f(x) = x^{2} – 9, x ε[-3,3]

8

Starting with x_{1}= 1, find the third approximate x_{3} to the root of the equation x³ – x – 5 = 0

9

Show the integral coverages

∫_{0}^{³ }dx/x-1

10

Use Trapezoidal rule to approximate the integral _{1} ∫^{² }dx/x, with n=5.

11

Find the derivative of (r(t)) = t^2i – te^{(-t)j} + sin(2t)k and find the unit tangent vector at t = 0

12

What is sequence? Is the sequence

\(a_{n} = \frac{n}{\sqrt{5 + n}}\)

convergent?

13

Find the angle between the vectors a = (2, 2, -1) and b = (1, 3, 2)

14

Find the partial derivative f_{xx} and f_{yy} of f(x,y)= x^{2} + x^{3}y^{2} – y^{2 }+ xy, at (1,2).

15

Evaluate

_{0}∫^{3} _{1}∫^{2} x^{2}y dxdy

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