CDS 2026 (I) Elementary Mathematics, question 76

Topic: Algebra

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Q76

A Question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question : Is rnr^{n} less than 11, where rr is a real number and nn is a natural number ?

Statement-I : 0<r2<10<r^{2}<1

Statement-II : 0<r3<20<r^{3}<2

Which one of the following is correct in respect of the above Question and the Statements ?

  1. (a) The Question can be answered by using one of the statements alone, but cannot be answered using the other statement aloneCorrect answer
  2. (b) The Question can be answered by using either Statement alone
  3. (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
  4. (d) The Question cannot be answered even by using both the Statements together

Solution.

Answer: (a) The Question can be answered by using one of the statements alone, but cannot be answered using the other statement alone

  1. I alone implies 0<∣r∣<10 < |r| < 1.
  2. For every positive integer n,∣rn∣<1n,\left| r^{n} \right| < 1, so certainly rn<1r^{n} < 1, whether r is positive or negative.
  3. Sufficient.
  4. II alone allows r=1/2r = 1/2 and r=1r = 1; both satisfy 0<r3<20 < r^{3} < 2, but rn<1r^{n} < 1 is true in the first case and false in the second.
  5. Insufficient.