JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank

done Condition for common roots, Quadratic expressions and Position of roots Total Questions - 37

• question_answer1)
  If both the roots of \[k(6{{x}^{2}}+3)+rx+2{{x}^{2}}-1=0\] and \[6k(2{{x}^{2}}+1)+px+4{{x}^{2}}-2=0\] are common, then \[2r-p\] is equal to [MNR 1983]
  A)
  -1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  2 done clear
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• question_answer2)
  If a root of the equations \[{{x}^{2}}+px+q=0\] and \[{{x}^{2}}+\alpha x+\beta =0\] is common, then its value will be (where \[p\ne \alpha \] and \[q\ne \beta \]) [IIT 1974, 1976; RPET 1997]
  A)
  \[\frac{q-\beta }{\alpha -p}\] done clear
  B)
  \[\frac{p\beta -\alpha q}{q-\beta }\] done clear
  C)
   \[\frac{q-\beta }{\alpha -p}\]or \[\frac{p\beta -\alpha q}{q-\beta }\] done clear
  D)
  None of these done clear
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• question_answer3)
  If  the two equations \[{{x}^{2}}-cx+d=0\] and \[{{x}^{2}}-ax+b=0\] have one common root and the second has equal roots, then \[2(b+d)=\]
  A)
  0 done clear
  B)
  \[a+c\] done clear
  C)
  \[ac\] done clear
  D)
  \[-ac\] done clear
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• question_answer4)
  If \[{{x}^{2}}-hx-21=0,{{x}^{2}}-3hx+35=0\]\[(h>0)\]has a common root, then the value of \[h\] is equal to [EAMCET 1986]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer5)
  If every pair of the equations\[{{x}^{2}}+px+qr=0\], \[{{x}^{2}}+qx+rp=0,\] \[{{x}^{2}}+rx+pq=0\] have a common root, then the sum of three common roots is
  A)
  \[\frac{-(p+q+r)}{2}\] done clear
  B)
  \[\frac{-p+q+r}{2}\] done clear
  C)
  \[-(p+q+r)\] done clear
  D)
  \[-p+q+r\] done clear
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• question_answer6)
  If \[a{{x}^{2}}+bx+c=0\] and \[b{{x}^{2}}+cx+a=0\] have a common root \[a\ne 0\], then \[\frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{abc}=\] [IIT 1982; Kurukshetra CEE 1983]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  None of these done clear
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• question_answer7)
  If the equation \[{{x}^{2}}+px+q=0\] and \[{{x}^{2}}+qx+p=0\], have a common root, then \[p+q+1=\] [Orissa JEE 2002]
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  - 1 done clear
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• question_answer8)
  If \[{{x}^{2}}+ax+10=0\] and \[{{x}^{2}}+bx-10=0\] have a common root, then \[{{a}^{2}}-{{b}^{2}}\] is equal to  [Kerala (Engg.) 2002]
  A)
  10 done clear
  B)
  20 done clear
  C)
  30 done clear
  D)
  40 done clear
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• question_answer9)
  \[{{x}^{2}}-11x+a\] and \[{{x}^{2}}-14x+2a\]will have a common factor, if \[a=\] [Roorkee 1981]
  A)
  24 done clear
  B)
  0, 24 done clear
  C)
  3, 24 done clear
  D)
  0, 3 done clear
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• question_answer10)
  If  \[{{x}^{2}}-3x+2\]be a factor of \[{{x}^{4}}-p{{x}^{2}}+q,\]then \[(p,q)=\]   [IIT 1974; MP PET 1995; Pb. CET 2001]
  A)
  (3, 4) done clear
  B)
  (4, 5) done clear
  C)
  (4, 3) done clear
  D)
  (5, 4) done clear
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• question_answer11)
  If \[x\] be real, then the minimum value of \[{{x}^{2}}-8x+17\] is [MNR 1980]
  A)
  -1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer12)
  If x is real, then the maximum and minimum values of expression \[\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}\] will be [Dhanbad Engg. 1968]
  A)
  4, - 5 done clear
  B)
  5,  - 4 done clear
  C)
  - 4, 5 done clear
  D)
  - 4, - 5 done clear
  View Solution play_arrow

• question_answer13)
  If x is real, the expression  \[\frac{x+2}{2{{x}^{2}}+3x+6}\] takes all value in the interval [IIT 1969]
  A)
  \[\left( \frac{1}{13},\frac{1}{3} \right)\] done clear
  B)
  \[\left[ -\frac{1}{13},\frac{1}{3} \right]\] done clear
  C)
  \[\left( -\frac{1}{3},\frac{1}{13} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  If \[{{x}^{2}}+px+1\] is a factor of the expression \[a{{x}^{3}}+bx+c\], then [IIT 1980]
  A)
  \[{{a}^{2}}+{{c}^{2}}=-ab\] done clear
  B)
  \[{{a}^{2}}-{{c}^{2}}=-ab\] done clear
  C)
  \[{{a}^{2}}-{{c}^{2}}=ab\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer15)
  If \[x,\ y,\ z\] are real and distinct, then\[u={{x}^{2}}+4{{y}^{2}}+9{{z}^{2}}-6yz-3zx-zxy\] is always [IIT 1979]
  A)
  Non-negative done clear
  B)
  Non-positive done clear
  C)
  Zero done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  If \[x\] be real, then the maximum value of \[5+4x-4{{x}^{2}}\] will be equal to [MNR 1979]
  A)
  5 done clear
  B)
  6 done clear
  C)
  1 done clear
  D)
    2 done clear
  View Solution play_arrow

• question_answer17)
  If \[x\] is real, the function \[\frac{(x-a)(x-b)}{(x-c)}\] will assume all real values, provided                [IIT 1984; Karnataka CET 2002]
  A)
  \[a>b>c\] done clear
  B)
  \[a<b<c\] done clear
  C)
  \[a>c<b\] done clear
  D)
  \[a<c<b\] done clear
  View Solution play_arrow

• question_answer18)
  If \[x\] is real, then the maximum and minimum values of the expression \[\frac{{{x}^{2}}-3x+4}{{{x}^{2}}+3x+4}\] will be [IIT 1984]
  A)
  2, 1 done clear
  B)
  \[5,\frac{1}{5}\] done clear
  C)
  \[7,\frac{1}{7}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[x\] is real, then the value of \[\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\] does not lie between [Roorkee 1983]
  A)
  -9 and -5 done clear
  B)
  -5 and  9 done clear
  C)
  0 and 9   done clear
  D)
  5 and 9 done clear
  View Solution play_arrow

• question_answer20)
  . If \[x\] is real, then the value of \[{{x}^{2}}-6x+13\] will not be less than [RPET 1986]
  A)
  4 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer21)
   The smallest value of \[{{x}^{2}}-3x+3\] in the interval \[(-3,\,3/2)\] is [EAMCET 1991; 93]
  A)
  3/4 done clear
  B)
  5 done clear
  C)
  -15 done clear
  D)
  -20 done clear
  View Solution play_arrow

• question_answer22)
  If the roots of \[{{x}^{2}}+x+a=0\]exceed a, then [EAMCET 1994]
  A)
  \[2<a<3\] done clear
  B)
  \[a>3\] done clear
  C)
  \[-3<a<3\] done clear
  D)
  \[a<-2\] done clear
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• question_answer23)
  If the roots of the equation \[{{x}^{2}}-2ax+{{a}^{2}}+a-3=0\]are real and less than 3, then [IIT 1999; MP PET 2000]
  A)
  \[a<2\] done clear
  B)
  \[2\le a\le 3\] done clear
  C)
  \[3<a\le 4\] done clear
  D)
  \[a>4\] done clear
  View Solution play_arrow

• question_answer24)
  If \[x\] be real, the least value of \[{{x}^{2}}-6x+10\] is [Kurukshetra CEE 1998]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer25)
  Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}+(3-\lambda )x-\lambda =0.\]  The value of \[\lambda \] for which \[{{\alpha }^{2}}+{{\beta }^{2}}\] is minimum, is [AMU 2002]
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer26)
  Let  \[f(x)={{x}^{2}}+4x+1\]. Then
  A)
  \[f(x)>0\] for all x done clear
  B)
  \[f(x)>1\] when \[x\ge 0\] done clear
  C)
  \[f(x)\ge 1\] when \[x\le -4\] done clear
  D)
  \[f(x)=f(-x)\] for all \[x\] done clear
  View Solution play_arrow

• question_answer27)
  The adjoining figure shows the graph of \[y=a{{x}^{2}}+bx+c\]. Then
  A)
  \[a<0\] done clear
  B)
  \[{{b}^{2}}<4ac\] done clear
  C)
  \[c>0\] done clear
  D)
    a and b are of opposite signs done clear
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• question_answer28)
  If \[a,b,c\] are real numbers such that \[a+b+c=0,\] then the quadratic equation \[3a{{x}^{2}}+2bx+c=0\]has [MNR 1992; DCE 1999]
  A)
  At least one root in [0, 1] done clear
  B)
  At least one root in [1, 2] done clear
  C)
  At least one root in \[[-1,\,0]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  If a, b  be the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\] and \[k\] be a real number, then the condition so that  \[\alpha <k<\beta \] is given by
  A)
  \[ac>0\] done clear
  B)
  \[a{{k}^{2}}+bk+c=0\] done clear
  C)
  \[ac<0\] done clear
  D)
  \[{{a}^{2}}{{k}^{2}}+abk+ac<0\] done clear
  View Solution play_arrow

• question_answer30)
  The value of p for which both the roots of the equation \[4{{x}^{2}}-20px+(25{{p}^{2}}+15p-66)=0\]are less than 2, lies in
  A)
  \[(4/5,\ 2)\] done clear
  B)
  \[(2,\,\,\infty )\] done clear
  C)
  \[(-1,\,-4/5)\] done clear
  D)
  \[(-\infty ,\,-1)\] done clear
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• question_answer31)
  If a and b (a < b) are the roots of the equation \[{{x}^{2}}+bx+c=0,\] where \[c<0<b,\] then [IIT Screening 2000; Pb. CET 2000]
  A)
  \[0<\alpha <\beta \] done clear
  B)
  \[\alpha <0<\beta <\,|\alpha |\] done clear
  C)
  \[\alpha <\beta <0\] done clear
  D)
  \[\alpha <0<\,|\alpha |\,<\beta \] done clear
  View Solution play_arrow

• question_answer32)
  If \[b>a\], then the equation \[(x-a)\,(x-b)=1\] has [IIT Screening 2000]
  A)
  Both roots in \[[a,\,b]\] done clear
  B)
  Both roots in \[(-\infty ,\,a)\] done clear
  C)
  Both roots in \[(b,\,+\infty )\] done clear
  D)
  One root in \[(-\infty ,\,a)\] and the other in \[(b,\,+\infty )\] done clear
  View Solution play_arrow

• question_answer33)
  The maximum possible number of real roots of equation \[{{x}^{5}}-6{{x}^{2}}-4x+5=0\] is [EAMCET 2002]
  A)
  0 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer34)
  If \[2a+3b+6c=0\] then at least one root of the equation \[a{{x}^{2}}+bx+c=0\]lies in the interval [Kurukshetra CEE 2002; AIEEE 2002, 04]
  A)
  (0, 1) done clear
  B)
  (1, 2) done clear
  C)
  (2, 3) done clear
  D)
  (3, 4) done clear
  View Solution play_arrow

• question_answer35)
  If the equation \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+....+{{a}_{1}}x=0\], \[{{a}_{1}}\ne 0\], \[\,n\ge 2\], has a positive root \[x=\alpha \], then the equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+....+{{a}_{1}}=0\] has a positive root, which is [AIEEE 2005]
  A)
  Greater than or equal to a done clear
  B)
  Equal to \[\alpha \] done clear
  C)
  Greater than \[\alpha \] done clear
  D)
  Smaller than \[\alpha \] done clear
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• question_answer36)
  If S is a set of \[P(x)\] is polynomial of degree \[\le 2\] such that \[P(0)=0,\]\[P(1)=1\],\[P'(x)>0\text{ }\forall x\in (0,\,1)\], then [IIT Screening 2005]
  A)
  \[S=0\] done clear
  B)
  \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in (0,\infty )\] done clear
  C)
  \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in R\] done clear
  D)
  \[S=ax+(1-a){{x}^{2}}\text{ }\forall a\in (0,2)\] done clear
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• question_answer37)
  If \[\alpha \] and \[\beta \], \[\alpha \] and \[\gamma \], \[\alpha \] and \[\delta \] are the roots of the equations \[a{{x}^{2}}+2bx+c=0\], \[2b{{x}^{2}}+cx+a=0\] and \[c{{x}^{2}}+ax+2b=0\] respectively, where \[a,b\] and \[c\] are positive real numbers, then \[\alpha +{{\alpha }^{2}}\]= [Kerala (Engg.) 2005]
  A)
  - 1 done clear
  B)
  0 done clear
  C)
  abc done clear
  D)
  \[a+2b+c\]  done clear
  E)
   abc done clear
  View Solution play_arrow