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src/app/past-questions-archive/wdu/mth/2022_2023/200-level/1st-semester/mth203/page.jsx
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import Questions from "@/components/Questions"; | ||
import Answers from "@/components/Answers"; | ||
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||
export default function MTH202() { | ||
return ( | ||
<Questions | ||
school="western delta university" | ||
college="natural and applied sciences" | ||
department="mathematics" | ||
session="2022/2023" | ||
semester="first semester" | ||
courseCode="mth203" | ||
courseTitle="linear algebra I" | ||
allowedTime="2hrs: 30mins" | ||
instruction="answer any THREE (3) questions" | ||
> | ||
<li> | ||
<h4 className="mb-2 font-bold">Question 1</h4> | ||
<ol className="flex list-[lower-alpha] flex-col gap-2"> | ||
<li> | ||
What is a Vector Space?{" "} | ||
<strong className="whitespace-nowrap">(8 marks)</strong> | ||
<Answers questionNumber="1a" /> | ||
</li> | ||
<li> | ||
Give three (3) examples of a vector space{" "} | ||
<strong className="whitespace-nowrap">(6 marks)</strong> | ||
<Answers questionNumber="1b" /> | ||
</li> | ||
<li> | ||
Define the following and give one (1) example in each case: | ||
<ol className="mx-4 list-[lower-roman]"> | ||
<li>Echelon matrix</li> | ||
<li>Orthogonal matrix</li> | ||
<li>Skew-symmetric matrix</li> | ||
</ol> | ||
<strong className="whitespace-nowrap">(9 marks)</strong> | ||
<Answers questionNumber="1c" /> | ||
</li> | ||
</ol> | ||
</li> | ||
|
||
<li> | ||
<h4 className="mb-2 font-bold">Question 2</h4> | ||
<ol className="flex list-[lower-alpha] flex-col gap-2"> | ||
<li> | ||
Reduce the matrix below to echelon form: <br />A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-7xl">[</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span> -4 </span> | ||
<span>1</span> | ||
<span> -6 </span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>1</span> | ||
<span>2</span> | ||
<span> -5 </span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>6</span> | ||
<span>3</span> | ||
<span>4</span> | ||
</div> | ||
</div> | ||
<span className="text-7xl">]</span> | ||
</div>{" "} | ||
<strong className="whitespace-nowrap">(10 marks)</strong> | ||
<Answers questionNumber="2a" /> | ||
</li> | ||
<li> | ||
Let A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-5xl">(</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span>7</span> | ||
<span>3</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>3</span> | ||
<span> -1 </span> | ||
</div> | ||
</div> | ||
<span className="text-5xl">)</span> | ||
</div> | ||
, find an orthogonal matrix P such that D = P<sup>-1</sup>AP is | ||
diagonal <strong className="whitespace-nowrap">(13 marks)</strong> | ||
<Answers questionNumber="2b" /> | ||
</li> | ||
</ol> | ||
</li> | ||
|
||
<li> | ||
<h4 className="mb-2 font-bold">Question 3</h4> | ||
<ol className="flex list-[lower-alpha] flex-col gap-2"> | ||
<li> | ||
Given A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-7xl">[</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span>1</span> | ||
<span> -1 </span> | ||
<span>1</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>2</span> | ||
<span>4</span> | ||
<span>3</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>5</span> | ||
<span>6</span> | ||
<span> -2 </span> | ||
</div> | ||
</div> | ||
<span className="text-7xl">]</span> | ||
</div> | ||
, using elementary row operations, reduce A to row echelon form{" "} | ||
<strong className="whitespace-nowrap">(10 marks)</strong> | ||
<Answers questionNumber="3a" /> | ||
</li> | ||
<li> | ||
Given A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-7xl">[</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span>1</span> | ||
<span>0</span> | ||
<span>2</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>2</span> | ||
<span> -1 </span> | ||
<span>3</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>4</span> | ||
<span>1</span> | ||
<span>8</span> | ||
</div> | ||
</div> | ||
<span className="text-7xl">]</span> | ||
</div> | ||
, find A<sup>-1</sup> using elementary row operations{" "} | ||
<strong className="whitespace-nowrap">(13 marks)</strong> | ||
<Answers questionNumber="3b" /> | ||
</li> | ||
</ol> | ||
</li> | ||
|
||
<li> | ||
<h4 className="mb-2 font-bold">Question 4</h4> | ||
<ol className="flex list-[lower-alpha] flex-col gap-2"> | ||
<li> | ||
Solve the system, using Gaussian elimination method: | ||
<ol className="mx-4 list-[lower-roman]"> | ||
<li>x + 2y + z = 3</li> | ||
<li>2x + 5y - z = -4</li> | ||
<li>3x + 2y - z = 5</li> | ||
</ol>{" "} | ||
<strong className="whitespace-nowrap">(13 marks)</strong> | ||
<Answers questionNumber="4a" /> | ||
</li> | ||
<li> | ||
Given Y ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-5xl">[</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span> | ||
Y<sub>1</sub> | ||
</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span> | ||
Y<sub>2</sub> | ||
</span> | ||
</div> | ||
</div> | ||
<span className="text-5xl">]</span> | ||
</div> | ||
and A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-5xl">[</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span>2</span> | ||
<span>2</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>2</span> | ||
<span>0</span> | ||
</div> | ||
</div> | ||
<span className="text-5xl">]</span> | ||
</div> | ||
, find Y<sup>1</sup>AY{" "} | ||
<strong className="whitespace-nowrap">(10 marks)</strong> | ||
<Answers questionNumber="4b" /> | ||
</li> | ||
</ol> | ||
</li> | ||
|
||
<li> | ||
<h4 className="mb-2 font-bold">Question 5</h4> | ||
If A ={" "} | ||
<div className="inline-flex"> | ||
<span className="text-5xl">(</span>{" "} | ||
<div className="flex flex-col"> | ||
<div className="flex gap-6"> | ||
<span>2</span> | ||
<span>2</span> | ||
</div> | ||
<div className="flex gap-6"> | ||
<span>1</span> | ||
<span>3</span> | ||
</div> | ||
</div> | ||
<span className="text-5xl">)</span> | ||
</div> | ||
<ol className="mx-4 list-[lower-roman]"> | ||
<li> | ||
Find all eigen values and corresponding eigenvectors.{" "} | ||
<strong className="whitespace-nowrap">(5 marks)</strong> | ||
<Answers questionNumber="5i" /> | ||
</li> | ||
<li> | ||
Find a non singular matrix P such that D = P<sup>-1</sup>AP is | ||
diagonal and P<sup>-1</sup>{" "} | ||
<strong className="whitespace-nowrap">(10 marks)</strong> | ||
<Answers questionNumber="5ii" /> | ||
</li> | ||
<li> | ||
Find A<sup>6</sup> and F(A), where t<sup>3</sup> - 3t<sup>2</sup> + | ||
7t + 3 <strong className="whitespace-nowrap">(8 marks)</strong> | ||
<Answers questionNumber="5iii" /> | ||
</li> | ||
</ol> | ||
</li> | ||
|
||
<li> | ||
<h4 className="mb-2 font-bold">Question 1</h4> | ||
<ol className="flex list-[lower-alpha] flex-col gap-2"> | ||
<li> | ||
When are vectors linearly dependent?{" "} | ||
<strong className="whitespace-nowrap">(3 marks)</strong> | ||
<Answers questionNumber="6a" /> | ||
</li> | ||
<li> | ||
Determine whether or not U and V are linearly dependent. | ||
<ol className="mx-4 list-[lower-roman]"> | ||
<li>U = (1, -3), V = (-2, 6)</li> | ||
<li>U = (1, 2, -3), V = (4, 5, -6)</li> | ||
<li>U = (2, 4, -8), V = (3, 6, -2)</li> | ||
</ol> | ||
<strong className="whitespace-nowrap">(15 marks)</strong> | ||
<Answers questionNumber="6b" /> | ||
</li> | ||
</ol> | ||
</li> | ||
</Questions> | ||
); | ||
} |
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