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• Arithmetics of complex numbers III

Inverse and division of complex numbers.  Four parts.

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• Truth tables 4 (v2)

Create a truth table for a logical expression of the form $((a \operatorname{op1} b) \operatorname{op2}(c \operatorname{op3} d))\operatorname{op4}e$ where each of $a, \;b,\;c,\;d,\;e$ can be one the Boolean variables $p,\;q,\;r,\;\neg p,\;\neg q,\;\neg r$ and each of $\operatorname{op1},\;\operatorname{op2},\;\operatorname{op3},\;\operatorname{op4}$ one of $\lor,\;\land,\;\to$.

For example: $((q \lor \neg r) \to (p \land \neg q)) \land \neg r$

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• Transportation problem - stepping stones

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• Simplex method - 4 or 5 random inequalities

Abstract simplex method question. Given optimal tableau, student must identify optimal solution and objective value.

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• Simplex algorithm - bus company

Linear program described in words. Student must write out constraints as equations in standard form, then identify the optimal solution in a finished simplex tableau.

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• Hungarian algorithm

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• Finding bases for the null space and row space of a matrix

Given a matrix in row reduced form use this to find bases for the null, column and row spaces of the matrix.

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• Find n such that $\sigma(n)$ is the given number

Given $m \in \mathbb{N}$, find values of $n\in \mathbb{N}$ such that $\sigma(n)=m$.

There are at most two such solutions in this question.

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• Find determinant of a 3x3 matrix with a free variable by row reduction

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

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• Find determinant of a 3x3 matrix by row reduction

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.

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