MTH301 Midterm Solved Paper

1. In the form of interval, real line can be denoted by (-∞ , +∞) 2. To each point on a co-ordinate line or real line, there is associated i. A real number ii. An integer iii. A natural number iv. A rational number. 3. How many real numbers lie in the interval [1, 5] i. 5 ii. 3 iii. 10 iv. Infinite 4. All the three axes are positive in i. Third octant ii. First octant iii. Second octant iv. Eigth octant

5. Let w = f(x, y, z) such that 1) 2) 4) x y z w a b c where a,

b and c are real numbers, not equal to zero. Is the function defined at origin? If yes, what is its value and if not, give the reason.

Solution:

Origin means (x, y, z) = (0, 0, 0). So putting these values in given equation 1) 2) 4)

(0 1) (0 2) (0 4) x y z w a b c

a b c

a b c Thus the given function is defined at origin and its value is

a b c 6 .

6. Find the domain of the following functions. i. f x y x y Domain of f consists of region in xy-plane where − x y ≥ 0 x y x y Now is an equation of surface of elliptic cylinder in three dimensional space which has infinite length along z-axis. x y = 4

Thus domain of f is inside and on the surface of elliptic cylinder.

ii. f ( , ) x y x y Domain of f is entire xy-plane except the origin because at origin function is not defined. iii. cos( ) f x y x y xy xy Domain of f is entire xy-plane.

7. Which of the following statement is true? Correct the false statement. i. A straight line is a special kind of curve. TRUE Generally, a curve is considered to be any one-dimensional collection of points. For your convenience, think a curve as a thread which we use in our daily life. The straight line is a special kind of curve.

ii. Plane is an example of surface. TRUE Surface is a two-dimensional geometric figure (a collection of points) in three-dimensional space. The simplest example is a plane—a flat surface. Some other common surfaces are spheres, cylinders, and cones.

iii. The intersection of two surfaces in three dimensional space gives a surface. FALSE Correct statement is “Intersection of two surfaces in three dimensional space gives a curve”.

8. Identify the curve or surface in three dimensional space for each of the following equation. i. y2 + z2 = 100 This is the equation of surface of circular cylinder which has infinite length along x-axis.