MTH301 Practice Questions with Answers Lectures 1-22

Practice Questions of Lectures 1-22

Lecture 1

Q1. What is the difference between quadrant and octant?

Q2. Describe two applications of Differential Calculus and two applications of Integral Calculus.

Q3. In two dimensions, in which quadrant, are the following points located?

1. ( 2, 5 )
2. ( − 3, 7 )
3. ( 9, − 2 )
4. ( − 8, −1)

Q4. In three dimensions, in which octant, are the following points located?

1. (1, 2, 3)
2. ( − 5, 7,1)
3. ( 9, − 2 , 6)
4. ( − 9, − 3, 9 )

e) ( − 4, − 2, − 6 ) f) ( −1, 2, − 3 ) g) ( 2, 2, − 2 )

Q5. Write down the equations of the following:

1. x – axis
2. y – axis
3. z – axis
4. xy – plane

e) yz – plane f) xz – plane

Q6. Identify the functions of single variable, two and three variables.

1. y = 2x + 1
2. z = 5x + 6 y + 10
3. w = 3x y z + 5
4. y = x2 + 5x + 6

Answers

Q1. In two dimension plane, say xy- plane, we divide the plane into four parts. Each part is called quadrant.

In three dimension space, we divide the whole space in eight parts. Each part is called Octant.

Q2. See handouts lecture No. 1. The answer is there.

Q3. (a) First quadrant

1. Second quadrant
2. 4th quadrant
3. 3rd quadrant

Q4. (a) First octant

1. 2nd octant
2. 4th octant
3. 3rd octant

(e) 7th octant (f) 6th octant

(g) 5th octant

Q5. See the page 14 Lecture No. 3

Q6. (a) single variable

1. two variables
2. three variables
3. single variable

(e) single variable (f) two variables

Lecture 2,3

Q1. Let f (x, y) = x2 + 9x3 y − 2y . Is the function defined at the point (4 , 2)? If yes, what is its value and if not, give the reason.

Sol:

f (x, y) = x2 + 9x3 y − 2 y f (4, 2) = (4)2 + 9(4)3 (2) − 2(2) = 16 +1152 − 4 = 1164

Q2. Let f (x, y, z) = x2 yexyz . Is the function defined at point (1, 3, 6) ? If yes, what is its value and if not, give the reason.

Sol:

f (x, y, z) = x2 yexyz f (1, 3, 6) = (1)2 (3) e(1)(3)(6) = 196979907.4

Q3. Let f (x, y, z) such that f (x, y, z) = ln

. Is the function defined at origin? If yes, what is its value and if not, give the reason.

Sol: f is not defined at origin.

Q4. Find the points where the parabola, represented by the equation 𝑦 = 𝑥2 − 4𝑥 , meets x-axis.

Sol: See solution of Q#5.

Q6. Find the mid point between (6,2,4) and (2, 5, 8).

Sol: 6+2 , 2+5 , 4+8 2 , 12 2 ) = (4, 2 , 6 )

Q7. Find the distance between the points A(4, 5, 9 ) and B(1, 3, 7 ) .

Solution: √(4 − 1)2 + (5 − 3)2 + (9 − 7)2 = √17

Q8. For a line joining the points A(3, 2, 6) and B(0, 3, 4) , find

x2 + y2 + z2