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JEE Advanced 2022 Paper 2 · Mathematics · Quadratic Equations · medium

For a real number k, let p(x)=x^3-3x+k with roots a,b,c (counted with multiplicity).

One or more options correct

For a real number kk, let p(x)=x3−3x+kp(x)=x^{3}-3x+k with roots a,b,ca,b,c (counted with multiplicity). Which of the following statements is/are correct?
  1. A. pp has three distinct real roots if and only if −2<k<2-2<k<2
  2. B. For k=2k=2, pp has a repeated root and the remaining root is −2-2
  3. C. a2+b2+c2=6a^{2}+b^{2}+c^{2}=6 for every real kk
  4. D. For k=3k=3, the only real root of pp lies in the interval (−2,−1)(-2,-1)
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Answer: A, B, C

Local extrema of pp are at x=±1x=\pm1 with values k+2k+2 (at x=−1x=-1) and k−2k-2 (at x=1x=1). Three distinct real roots occur exactly when k+2>0>k−2k+2>0>k-2, i.e. −2<k<2-2<k<2, so A is true. For k=2k=2, p(x)=(x−1)2(x+2)p(x)=(x-1)^2(x+2), so B is true. From Vieta's relations a+b+c=0a+b+c=0 and ab+bc+ca=−3ab+bc+ca=-3, hence a2+b2+c2=0−2(−3)=6a^2+b^2+c^2=0-2(-3)=6 for every kk, so C is true. For k=3k=3, p(−2)=1>0p(-2)=1>0 and p(−3)=−15<0p(-3)=-15<0, so the unique real root lies in (−3,−2)(-3,-2), hence D is false (numerically it is about −2.10-2.10).
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