507 results in Content created by Newcastle University - search across all projects.
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Given a set of vectors, find a basis which generates their span as a subspace of $\mathbb{Z}_n$.
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Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.
The correct solution is given, however the accuracy of 0.001 is not enough as in some cases answers near to the correct solution are also marked as correct.
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Given a set of codewords generating a code, give a generating matrix, encode three data vectors, and decode one codeword.
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Given vectors $\boldsymbol{v,\;w}$, find the angle between them.
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Given a set of codewords generating a code, write down a generator matrix, encode three data vectors, and decode one codeword.
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Calculation of the length and alternative form of the parameteric representation of a curve.
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Parametric form of a curve, cartesian points, tangent vector, and speed.
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Parametric form of a curve, cartesian points, tangent vector, and speed.
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Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.
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When are vectors $\boldsymbol{v,\;w}$ orthogonal?
Part b) is not answered in Advice, the given solution is for a different question.
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Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.
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Find all words with given Hamming distance from a given codeword.
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Given a set of codewords generating a code, give a generating matrix, encode three data vectors, and decode one codeword.
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Gradient of $f(x,y,z)$.
Should include a warning to insert * between multiplied terms
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A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.
There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.
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Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\boldsymbol{r=r_0+\lambda a+\mu b}$.
The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.
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(Green’s theorem). $\Gamma$ a rectangle, find: $\displaystyle \oint_{\Gamma} \left(ax^2-by \right)\;dx+\left(cy^2+px\right)\;dy$.
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Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$
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Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.
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Given a pair of 3D position vectors, find the vector equation of the line through both. Find two such lines and their point of intersection.
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Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.
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Given a matrix in the field $\mathbb{Z}_n$. By reducing it to row-echelon form (or otherwise), find a basis for the row space of the matrix, as a list of vectors.
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Compute the minimum distance between codewords of a code, given a parity check matrix.
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3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.
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Calculate a repeated integral of the form $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$
The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.
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Two double integrals with numerical limits
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Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.
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Three 3 dim vectors, one with a parameter $\lambda$ in the third coordinate. Find value of $\lambda$ ensuring vectors coplanar. Scalar triple product.
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Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.
Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.
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Determine if various combinations of vectors are defined or not.