Tribhuvan University

Institute of Science and Technology

2075

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 question**

1 (a)

A function is defined by f(x) = |x| , calculate f(-3), f(4), and sketch the graph.

b

Prove that the \(\lim_{x \to 2} \frac{|x – 2|}{x – 2}\) doesn’t exist.

2 (a)

Find the domain and sketch the graph of the function \(f(x) = x^2 – 6x\).

b

Estimate the area between the curve y = x^{2 }and the lines y = 1 and y = 2.

3 (a)

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

b

Find the volume of a sphere of radius r

4 (a)

If f(x, y) = \(\frac{y}{x}\) does \(\lim_{(x,y) \to (0, 0)} f(x, y)\) exists? Justify

b

Calculate \(\int_R\int f(x,y) dA\) for f(x,y) = 100 – 6x^{2}y and \(R: 0 \leq x \leq 2, -1 \leq y \leq 1 \)

**Group B**

**Attempt any ten question**

5

If f(x) = \(\sqrt{2-x}\) and g(x) = \(\sqrt{x}\), Find fog and fof

6

Define continuity on an interval. Show that the function \(f(x) = 1 – \sqrt{1-x^2}\) on the continuous on the interval [1,-1].

7

Verify Mean value theorem of f(x) = x^{3} – 3x + 2 for [-1, 2].

8

Stating with x_{1} = 2, find the third approximation x_{3} to the root of the equation x^{3} – 2x – 5 = 0

9

Evaluate \(\int_0^∞ x^3 \sqrt{1 – x^4}\) dx

10

Find the volume of the resulting solid which is enclosed by the curve y = x and y = x^{2 }is rotated about the x-axis.

11

Find the solution of y” + 4y’ + 4 = 0.

12

Determine whether the series \(\sum_{n=1}^∞ \frac{n^2}{5n^2 + 4}\) converges or diverges.

13

If a = (4, 0, 3) and b = (-2, 1, 5) find |a|, the vector a – b and 2a + b

14

Find \(\frac{∂z}{∂x} \enspace and \enspace \frac{∂z}{∂y}\) if z is defined as a function of x and y by the equation \(x^3 + y^3 + z^3 + 6xyz = 1\).

15

Find the extreme values of the function \(f(x, y) = x^2 + 2y^2\) on the circle \(x^2 + y^2 = 1\).

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