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It's a quiz document, Cheat Sheet of Mathematical Methods

University of ZambiaMathematical Methods
Prof. Trevor Chimpande

You are going to enjoy how the quizzes look like

Typology: Cheat Sheet

2023/2024

Uploaded on 02/11/2025

racheal-hantukumane-1
racheal-hantukumane-1 ๐Ÿ‡ฟ๐Ÿ‡ฒ

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The University of Zambia
Department of Mathematics and Statistics
Academic Year 2022/23
MAT 3110 Quiz 2
6th August, 2023 12.00 - 13.00
Instructions:
โ€ขThis paper has 2 pages and 4 questions. Answer all the questions.
โ€ขDuration is 60 minutes.
โ€ขTotal marks is 30 marks
First Name: ......................................................................................
Last Name: .......................................................................................
Computer Number: ..........................................................................
Department : .......................................................................................
Question: 1 2 3 4 Total
Points: 8 7 8 7 30
Score:
1. (a) Find the coordinates of the points where the line x= 1 + 2t, y =โˆ’1โˆ’t, z = 3t
meets the coordinate planes in 3D-space. (The coordinates planes are the xy-plane,
the xz-plane and the yz -plane) [3]
(b) i. Show that the planes 3x+ 6z= 1 and 2x+ 2yโˆ’z= 3 are othorgonal (perpen-
dicular). [2]
ii. Find the parametric equations of the line of intersection of the planes 3x+6z= 1
and 2x+ 2yโˆ’z= 3. [3]
Total marks : 8
1
pf2

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The University of Zambia

Department of Mathematics and Statistics

Academic Year 2022/

MAT 3110 Quiz 2

6th August, 2023 12.00 - 13.

Instructions:

  • This paper has 2 pages and 4 questions. Answer all the questions.
  • Duration is 60 minutes.
  • Total marks is 30 marks

First Name: ......................................................................................

Last Name: .......................................................................................

Computer Number: ..........................................................................

Department : .......................................................................................

Question: 1 2 3 4 Total

Points: 8 7 8 7 30

Score:

  1. (a) Find the coordinates of the points where the line x = 1 + 2t, y = โˆ’ 1 โˆ’ t, z = 3t meets the coordinate planes in 3D-space. (The coordinates planes are the xy-plane, the xz-plane and the yz-plane) [3] (b) i. Show that the planes 3x + 6z = 1 and 2x + 2y โˆ’ z = 3 are othorgonal (perpen- dicular). [2] ii. Find the parametric equations of the line of intersection of the planes 3x+6z = 1 and 2x + 2y โˆ’ z = 3. [3]

Total marks : 8

1

MAT 3110 Quiz 2 - Page 2 of 2 6th August, 2023

  1. The following is a diagram of a parallelogram in 3D-space with vertices A(2, โˆ’ 1 , 4), B(1, 0 , โˆ’1), C(1, 2 , 3) and D.

x

y

z

A(2, โˆ’ 1 , 4)

B(1, 0 , โˆ’1)

C(1, 2 , 3)

D

(a) Find the coordinates of D. [3] (b) Find the equation of the plane on which this parallelogram lies. [4]

Total marks : 7

  1. (a) Sketch the region in 3D-space that is enclosed by the surfaces z = 12 โˆ’ 3 x^2 โˆ’ 3 y^2 and z = x^2 + y^2 โˆ’ 8. [4] (b) Sketch the region in 3D-space that is enclosed by the surfaces z =

p 16 โˆ’ x^2 โˆ’ y^2 and z =

p 3 x^2 + 3y^2. [4]

Total marks : 8

  1. (a) The equation z = |r| is a representation of a surface in 3D-space using cylindrical coordinates. i. Translate the equation into rectangular coordinates. [1] ii. Translate the equation into spherical coordinates. [1] iii. Identify the surface. [1] (b) The equation ฯ cos ฯ• + ฯ^2 sin^2 ฯ• = 1 is a representation of a surface in 3D-space using spherical coordinates. i. Translate the equation into rectangular coordinates. [2] ii. Translate the equation into cylindrical coordinates. [1] iii. Identify the surface. [1]

Total marks : 7

End of Quiz!