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Questions › DSE › DSE 2010 › Q36
2 marksMathematics › Vector spaces, rank and eigenvalues › Linear maps, rank and projectionsModerateabout 3 min

DSE 2010 Question 36: Vector spaces, rank and eigenvalues

Setup, from Q35 of the paper:

Let A={(x,y)∈ℜ2∣x2+y2≤1}A=\{(x,y)\in\Re^{2}\mid x^{2}+y^{2}\le 1\}A={(x,y)∈ℜ2∣x2+y2≤1}. Given (a,b)≠(0,0)(a,b)\ne(0,0)(a,b)=(0,0) and c>1c>1c>1, let X={(x,y)+(a,b)∣(x,y)∈A}X=\{(x,y)+(a,b)\mid (x,y)\in A\}X={(x,y)+(a,b)∣(x,y)∈A}, Y={(2−1/2(x+y),2−1/2(x−y))∣(x,y)∈A}Y=\{(2^{-1/2}(x+y),2^{-1/2}(x-y))\mid (x,y)\in A\}Y={(2−1/2(x+y),2−1/2(x−y))∣(x,y)∈A}, Z={c(x,y)∣(x,y)∈A}Z=\{c(x,y)\mid (x,y)\in A\}Z={c(x,y)∣(x,y)∈A}. Which set is not a disc?

(Same definitions of A,X,Y,ZA,X,Y,ZA,X,Y,Z.) Which set has a larger area than AAA?

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Setup, from Q35 of the paper: Let A={(x,y)∈ℜ²| x²+y²≤ 1}. Given (a,b)≠(0,0) and c>1, let X={(x,y)+(a,b)| (x,y)∈ A}, Y={(2^(-1/2)(x+y),2^(-1/2)(x-y))| (x,y)∈ A}, Z={c(x,y)| (x,y)∈ A}. Which set is not a disc? (Same definitions of A,X,Y,Z .) Which set has a larger area than A? (A) X (B) Y (C) Z (D) None

Also written as: Setup, from Q35 of the paper: Let A={(x,y) in Re^2| x^2+y^2<= 1}. Given (a,b)!=(0,0) and c>1, let X={(x,y)+(a,b)| (x,y) in A}, Y={(2^(-1/2)(x+y),2^(-1/2)(x-y))| (x,y) in A}, Z={c(x,y)| (x,y) in A}. Which set is not a disc? (Same definitions of A,X,Y,Z .) Which set has a larger area than A?

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Linear maps, rank and projections, part of Vector spaces, rank and eigenvalues in Mathematics.

Linear transformations, their range and rank, and projection operators.

TestStop has 14 past-paper questions on this subtopic.

Practise Vector spaces, rank and eigenvaluesMore on Linear maps, rank and projections

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Question from the DSE 2010 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.