UPSC Prelims 2006 · General Studies

Question 101: Basic Numeracy

There are three parallel straight lines. Two points A and B are marked on the first line, points C and D are marked on the second line, and points E and F are marked on the third line. Each of these six points can move to any position on its respective straight line. Consider the following statements:

1.The minimum number of triangles that can be formed by joining these points is zero.
2.The maximum number of triangles that can be formed by joining these points is twenty.

Which of the statements given above is/are correct?

a1 only

b2 only

cBoth 1 and 2

dNeither 1 nor 2

AnswerThe answer is (c).

For statement 2: six points give C(6,3) = 20 triples, and if no three points are collinear every triple is a triangle, so the maximum is 20. For statement 1: if the two points on each line are moved to coincide, and the three coincident spots are placed on one line cutting all three parallels, all six points become collinear and no triangle can be formed, so the minimum is zero. Both statements hold.

1. All six points can be brought onto one transversal, giving zero triangles.
2. C(6,3) = 20, reached when no three points are collinear.

How you could have got this

If you were sure statement 1 is correct, (b) and (d) go: each of them leaves 1 out.
What you needed to know: All six points can be brought onto one transversal, giving zero triangles.
Then, if you were sure statement 2 is correct, (a) goes: it leaves 2 out.
What you needed to know: C(6,3) = 20, reached when no three points are collinear.

Only (c) is left. This one needs every statement.

This follows an unofficial answer: UPSC has not published its key for this year, and if that answer is wrong, so is this path.

Which options go is worked out from UPSC’s key and the options, not by a model.

Why each option is right or wrong

(a) Statement 2 is also correct (maximum 20).
(b) Statement 1 is also correct, since the points can be made collinear.
(c) Correct: minimum zero and maximum twenty.
(d) Both statements hold.

Worth knowing: Statement 1 holds only if two points on a line may coincide; if they must stay distinct, statement 1 fails and the answer would be (b).

Explanation: written by UPSC Grade, not by UPSC.

Practise this paper timedPrevious question Next question

Source: UPSC, Civil Services (Preliminary) Examination 2006, General Studies. UPSC's own paper isn't online; the words are from a published compilation. The paper on upsc.gov.in.