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 = xand 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 – 6x2y 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) = x3 – 3x + 2 for [-1, 2].

8

Stating with x1 = 2, find the third approximation x3 to the root of the equation x3 – 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 = xis 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 $$x^2 + y^2 = 1$$.