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Introduction to Computer Science I.

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Introduction to Computer Science I.

Repeated First Midterm Test 2017. December 11.

1. Determine all the four-digit integers which give a remainder of 3 when divided by 51, furthermore if we multiply them by 17, then the last two digits of the product are 15.

2. Determine the remainder we get if we divide 7337+ 3773 by 108.

3. Use the algorithm we learnt to determine the remainder we get if we divide 585 by 155.

4. Does the plane through the pointsA(−1,−2,1), B(3,1,3) andC(7,6,3) contain a point which is on the y axis? If yes, then which is it?

5. Let u = (0,0,1,2)T, v = (0,1,2,5)T and w = (1,2,4,11)T be vectors in R4. Determine hu, v, wi, the subspace generated by them. (That is, give a (system of) equation(s), satisfied by the vectors in hu, v, wi.) 6. Suppose that for the vectorsv1, v2, ..., v10, winRnit holds thatv1, v2, ..., v10

are linearly independent, but v1, v2, ..., v10, w are linearly dependent, and w6= 0. Show that there is an index 1≤i≤10 and a scalar α6= 0, such that the vectorsv1, v2, . . . , vi−1, vi+α·w, vi+1, . . . , v10 are linearly dependent.

The full solution of each problem is worth 10 points. Show all your work!

Results without proper justification or work shown deserve no credit.

Calculators (or other devices) are not allowed to use.

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