MTH301 Midterm Past Paper 2010 Session 2

Q1. Which of the following is the interval notation of real line? (1 marks)

1. (-∞ , 0)
2. (0 , +∞)

Q2. What is the general equation of parabola whose axis of symmetry is parallel to y-axis? (1 marks)

0) y ax b a

0) x ay b a

0) y ax bx c a

0) x ay by c a

Q3. Which of the following is geometrical representation of the equation (1 marks)

y = , in three

dimensional space?

1. A point on y-axis
2. Plane parallel to xy-plane
3. Plane parallel to yz-axis
4. Plane parallel to xz-plane

Q4. Suppose (1 marks)

xy f x y x e . Which one of the statements is correct?

xy f x e y

xy f x e y

xy f x e y xy f x ye y

Q5. If (1 marks)

ln f x y x y y x

then f

x

2xy x

2 y x

2xy x

2 y x

Q6. Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule (1 marks)

w

r

w x w y w z

x r y r z r

w x w y w z

r r r r r r

w x x w y y w z z

x r s y r s z r s

w r w r w r

r x r y r z

Q7. Is the function (1 marks)

f x y continuous at origin? If not, why?

x y if x y f x y x y

if x y

f x y is continuous at origin (0, 0) f is not defined

(0, 0) f is defined but (0, 0) lim x y f x y does not exist

(0, 0) f is defined and (0, 0) lim x y f x y exists but these two numbers are not

equal.

Q8. Let R be a closed region in two dimensional space. What does the double integral over R (1 marks)

calculates?

1. Area of R.
2. Radius of inscribed circle in R.
3. Distance between two endpoints of R.
4. None of these

Q9. Two surfaces are said to be orthogonal at a point of their intersection if their normals at (1 marks)

that point are ---------

1. Parallel
2. Perpendicular
3. In opposite direction

Q10. Two surfaces are said to intersect orthogonally if their normals at every point common to (1 marks)

them are ----------

1. perpendicular
2. parallel
3. in opposite direction

Q11. f x y (1 marks)

xx yy xy f f and f x y and let

xx yy xy D f x y f x y f x y

If D > and xx f x y then f has ---------------

1. Relative maximum at

x y

1. Relative minimum at

x y

1. Saddle point at

x y