Basic Mathematics 2078

Section A

Long Answer Questions.

Attempt any two questions.        (2x10=20)

1.  a) What do you mean by asymptotes? How many types of asymptotes define each? (1+3)

    b) Find horizontal and vertical asymptotes of the following functions

            Does there other asymptotes exist?                (5+1)

2. Define area between two curves

    a) Find area of the region enclosed by the parabola y=2 - x² and the line y = -x.                       

    b) Define volume integral. Find the volume of solid generated by revolving the region bounded by the curve y² = x and the line y = 1, x = 4 about the line y = 1.   (1+5)

3.  a) Define Newton's Raphson method with their formula.    (2)

    b) An open top box is to be made by cutting small congruent squares from the corners of square sheet of tin having length 12 inch and is bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?        (8)

Section B

Short Answer Questions.

Attempt any eight questions.        (8x5=40)

4. Graph the following functions. Write their symmetricity and specify the interval over which the function is increasing and decreasing.

    a. y = -x³

    b. y = x²                        (2.5+2.5)

5. Find the equations of tangent and normal to the curve x³ + y³ - 9xy = 0 at the point (2, 4).   (3+2)

6. What is L' Hospital rule? Using this rule evaluate the following

        a. 

        b.                 (1+2+2)

7. Define integration. Evaluate the following integral.

        a. 

        b.                     (1+2+2)

8. Define integral test and determine the convergence or divergence of the series.  (1+4)

9. Solve the following differential equation.

10. Find the derivative of , at point P(1,1,0) in the direction of .

11. State Mean value theorem. Verify the mean value theorem if f(x) = x² + 2x -1 on [0, 1].    (1+4)

12.  a. Find  if .

        b. Find the slope of circle  at the point (3,4).