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In pursuit of excellence Department of Applied Informatics \& Mathematics Mathematics II Assignment 1 Assessor: Mr. Y. Tshazibana Marks: 50 Due Date: 29 April 2024 Instructions: Show all steps and give answers correct to 2 decimal places, unless otherwise stated. OUESTION 1 [16] 1.1 The $ 2 \times 2 $ Matrices A, B and C are given below in terms of the scalar $ x $ : \[ A=\left[\begin{array}{ll} 2 & x \\ 3 & 1 \end{array}\right], B=\left[\begin{array}{ll} 2 & 1 \\ 1 & 4 \end{array}\right] \text { and } C=\left[\begin{array}{cc} 3 x+2 & 7 \\ 7-x & 7 \end{array}\right] \] Determine the value of $ x $, given $ B^{T} A^{T}=C $ 1.2 Find the value(s) of $ a $ for which the matrix: $ \left[\begin{array}{ccc}a^{2} & 0 & 3 \\ 5 & a & 2 \\ 3 & 0 & 1\end{array}\right] $ is a singular. 1.3 Use the inverse matrix method to solve the following system of linear equations. \[ \left\{\begin{array}{c} x+y+z=6 \\ 2 y+5 z=-4 \\ 2 x+5 y-z=27 \end{array}\right. \] QUESTION $ 2[24] $ 2.1 Write the following trigonometric identity to its corresponding hyperbolic identity: $ 1+\cot ^{2} x=\csc ^{2} x $, Hence use hyperbolic definitions to prove the corresponding hyperbolic. (5)

          In pursuit of excellence

Department of Applied Informatics \& Mathematics
Mathematics II
Assignment 1
Assessor: Mr. Y. Tshazibana
Marks: 50

Due Date: 29 April 2024

Instructions:
Show all steps and give answers correct to 2 decimal places, unless otherwise stated.
OUESTION 1 [16]
1.1 The \( 2 \times 2 \) Matrices A, B and C are given below in terms of the scalar \( x \) :
\[
A=\left[\begin{array}{ll}
2 & x \\
3 & 1
\end{array}\right], B=\left[\begin{array}{ll}
2 & 1 \\
1 & 4
\end{array}\right] \text { and } C=\left[\begin{array}{cc}
3 x+2 & 7 \\
7-x & 7
\end{array}\right]
\]

Determine the value of \( x \), given \( B^{T} A^{T}=C \)
1.2 Find the value(s) of \( a \) for which the matrix: \( \left[\begin{array}{ccc}a^{2} & 0 & 3 \\ 5 & a & 2 \\ 3 & 0 & 1\end{array}\right] \) is a singular.
1.3 Use the inverse matrix method to solve the following system of linear equations.
\[
\left\{\begin{array}{c}
x+y+z=6 \\
2 y+5 z=-4 \\
2 x+5 y-z=27
\end{array}\right.
\]

QUESTION \( 2[24] \)
2.1 Write the following trigonometric identity to its corresponding hyperbolic identity: \( 1+\cot ^{2} x=\csc ^{2} x \),
Hence use hyperbolic definitions to prove the corresponding hyperbolic.
(5)

$In pursuit of excellence Department of Applied Informatics & Mathematics Mathematics II Assignment 1 Assessor: Mr. Y. Tshazibana Marks: 50 Due Date: 29 April 2024 Instructions: Show all steps and give answers correct to 2 decimal places, unless otherwise stated. OUESTION 1 [16] 1.1 The 2 × 2 Matrices A, B and C are given below in terms of the scalar x : A=[ 2 x 3 1 ], B=[ 2 1 1 4 ] and C=[ 3 x+2 7 7-x 7 ] Determine the value of x, given B^T A^T=C 1.2 Find the value(s) of a for which the matrix: [ a^2 0 3 5 a 2 3 0 1 ] is a singular. 1.3 Use the inverse matrix method to solve the following system of linear equations. { x+y+z=6 2 y+5 z=-4 2 x+5 y-z=27 . QUESTION 2[24] 2.1 Write the following trigonometric identity to its corresponding hyperbolic identity: 1+cot ^2 x=csc ^2 x, Hence use hyperbolic definitions to prove the corresponding hyperbolic. (5)$

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