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Point Operations in Square and Asymptotic Limits

XAT 2010 · Quantitative Ability & Data Interpretation · Hard · Quantitative Ability

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Passage / data set

OABC is a square where O is the origin and AB = 1. Consider the set of points S=(xi,yi)S = (x_i, y_i) in the square such that xi+yi≤1x_i + y_i \le 1. Let P(x1,y1)P(x_1, y_1) and Q(x2,y2)Q(x_2, y_2) be two such points. Two operations addition (+) and multiplication (.) on S are defined as P+Q=(x1+x2−x1x2,y1y2)P + Q = (x_1 + x_2 - x_1 x_2, y_1 y_2) P.Q=(x1x2,y1+y2−y1y2)P . Q = (x_1 x_2, y_1 + y_2 - y_1 y_2)

Question 1 of 2

For a very large number n, Pn+QnP^n + Q^n is

  1. A.

    close to (0, 0)

  2. B.

    close to (1,0)

  3. C.

    close to (0, 1)

  4. D.

    any point in the region x + y < 1

  5. E.

    None of the above

Answer

C

Explanation

For P=(x1,y1)P = (x_1, y_1) with x1,y1<1x_1, y_1 < 1, as n→∞n \to \infty, Pn→(0,1)P^n \to (0, 1). Then (0,1)+(0,1)=(0+0−0,1⋅1)=(0,1)(0, 1) + (0, 1) = (0 + 0 - 0, 1 \cdot 1) = (0, 1).

Question 2 of 2

For a very large number n, nP+nQnP + nQ is

  1. A.

    close to (0, 0)

  2. B.

    close to (1,0)

  3. C.

    close to (0, 1)

  4. D.

    Any point in the region x + y < 1

  5. E.

    None of the above

Answer

B

Explanation

nPnP under repeated addition (+) gives xx-component approaching 1 and yy-component approaching 0. Thus nP→(1,0)nP \to (1,0). (1,0)+(1,0)=(1+1−1,0)=(1,0)(1,0) + (1,0) = (1+1-1, 0) = (1,0).

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