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Data Sufficiency: Geometry & Equations

XAT 2011 · Quantitative Ability & Data Interpretation · Hard · Data Sufficiency

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Passage / data set

Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below: A. Statement I alone is sufficient to answer the question. B. Statement II alone is sufficient to answer the question. C. Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question. D. Either Statement I or Statement II alone is sufficient to answer the question. E. Neither Statement I nor Statement II is necessary to answer the question.

Question 1 of 2

Let PQRS be a quadrilateral. Two circles O1 and O2 are inscribed in triangles PQR and PSR respectively. Circle O1 touches PR at M and circle O2 touches PR at N. Find the length of MN. I. A circle is inscribed in the quadrilateral PQRS. II. The radii of the circles O1 and O2 are 5 and 6 units respectively.

  1. A.

    Statement I alone is sufficient to answer the question.

  2. B.

    Statement II alone is sufficient to answer the question.

  3. C.

    Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.

  4. D.

    Either Statement I or Statement II alone is sufficient to answer the question.

  5. E.

    Neither Statement I nor Statement II is necessary to answer the question.

Answer

A

Explanation

Using properties of tangents in inscribed circles, Statement I yields MN=0MN = 0 uniquely. Thus Statement I alone is sufficient.

Question 2 of 2

Given below is an equation where the letters represent digits. (PQ)⋅(RQ)=XXX(PQ) \cdot (RQ) = XXX. Determine the sum of P+Q+R+XP + Q + R + X. I. X=9X = 9. II. The digits are unique.

  1. A.

    Statement I alone is sufficient to answer the question.

  2. B.

    Statement II alone is sufficient to answer the question.

  3. C.

    Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.

  4. D.

    Either Statement I or Statement II alone is sufficient to answer the question.

  5. E.

    Neither Statement I nor Statement II is necessary to answer the question.

Answer

E

Explanation

XXX=X×111=X×3×37XXX = X \times 111 = X \times 3 \times 37. For PQ×RQ=XXXPQ \times RQ = XXX where PQPQ and RQRQ share unit digit QQ, the only two-digit factorization with matching unit digits is 27×37=99927 \times 37 = 999. Thus P=2,Q=7,R=3,X=9P=2, Q=7, R=3, X=9 uniquely without needing either statement.

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