11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank

done Critical Thinking Total Questions - 40

• question_answer1)
  If \[x=\sqrt{1+\sqrt{1+\sqrt{1+.......\text{to infinity}}}},\]then x =
  A)
  \[\frac{1+\sqrt{5}}{2}\] done clear
  B)
  \[\frac{1-\sqrt{5}}{2}\] done clear
  C)
  \[\frac{1\pm \sqrt{5}}{2}\] done clear
  D)
  None of these done clear
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• question_answer2)
  For the equation \[|{{x}^{2}}|+|x|-6=0\], the roots are [EAMCET 1988, 93]
  A)
  One and only one real number done clear
  B)
  Real with sum one done clear
  C)
  Real with sum zero done clear
  D)
  Real with product zero done clear
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• question_answer3)
  If\[a{{x}^{2}}+bx+c=0\], then x = [MP PET 1995]
  A)
  \[\frac{b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] done clear
  B)
  \[\frac{-b\pm \sqrt{{{b}^{2}}-ac}}{2a}\] done clear
  C)
  \[\frac{2c}{-b\pm \sqrt{{{b}^{2}}-4ac}}\] done clear
  D)
  None of these done clear
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• question_answer4)
  If the equations \[2{{x}^{2}}+3x+5\lambda =0\] and \[{{x}^{2}}+2x+3\lambda =0\] have a common root, then \[\lambda =\] [RPET 1989]
  A)
  0 done clear
  B)
  -1 done clear
  C)
  \[0,-1\] done clear
  D)
  2,-1 done clear
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• question_answer5)
  If the equation \[{{x}^{2}}+\lambda x+\mu =0\] has equal roots and one root of the equation \[{{x}^{2}}+\lambda x-12=0\]is 2, then \[(\lambda ,\mu )\]=
  A)
  (4, 4) done clear
  B)
  (-4,4) done clear
  C)
  \[(4,-4)\] done clear
  D)
  \[(-4,-4)\] done clear
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• question_answer6)
  If \[x\]is real and \[k=\frac{{{x}^{2}}-x+1}{{{x}^{2}}+x+1},\] then [MNR 1992; RPET 1997]
  A)
  \[\frac{1}{3}\le k\le 3\] done clear
  B)
  \[k\ge 5\] done clear
  C)
  \[k\le 0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If \[a<b<c<d\], then the roots of the equation \[(x-a)(x-c)+2(x-b)(x-d)=0\] are [IIT 1984]
  A)
  Real and distinct done clear
  B)
  Real and equal done clear
  C)
  Imaginary done clear
  D)
  None of these done clear
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• question_answer8)
  If the roots of the equation \[q{{x}^{2}}+px+q=0\]where p, q are real, be complex, then the roots of the equation \[{{x}^{2}}-4qx+{{p}^{2}}=0\] are
  A)
  Real and unequal done clear
  B)
  Real and equal done clear
  C)
  Imaginary done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  The values of \['a'\] for which \[({{a}^{2}}-1){{x}^{2}}+2(a-1)x+2\] is positive for any \[x\] are [UPSEAT 2001]
  A)
  \[a\ge 1\] done clear
  B)
  \[a\le 1\] done clear
  C)
  \[a>-3\] done clear
  D)
  \[a<-3\]or \[a>1\] done clear
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• question_answer10)
  If the roots of equation \[\frac{{{x}^{2}}-bx}{ax-c}=\frac{m-1}{m+1}\]are equal but opposite in sign, then the value of \[m\] will be [RPET 1988, 2001; MP PET 1996, 2002; Pb. CET 2000]
  A)
  \[\frac{a-b}{a+b}\] done clear
  B)
  \[\frac{b-a}{a+b}\] done clear
  C)
  \[\frac{a+b}{a-b}\] done clear
  D)
  \[\frac{b+a}{b-a}\] done clear
  View Solution play_arrow

• question_answer11)
  The coefficient of \[x\] in the equation \[{{x}^{2}}+px+q=0\]was taken as 17 in place of 13, its roots were found to be -2 and -15, The roots of the original equation are      [IIT 1977, 79]
  A)
   3, 10 done clear
  B)
  - 3, - 10 done clear
  C)
  - 5, - 18 done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer12)
  If one root of the equation \[a{{x}^{2}}+bx+c=0\]be \[n\] times the other root, then
  A)
  \[n{{a}^{2}}=bc{{(n+1)}^{2}}\] done clear
  B)
  \[n{{b}^{2}}=ac{{(n+1)}^{2}}\] done clear
  C)
  \[n{{c}^{2}}=ab{{(n+1)}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  If one root of the quadratic equation \[a{{x}^{2}}+bx+c=0\] is equal to the nth power of the other root, then the value of \[{{(a{{c}^{n}})}^{\frac{1}{n+1}}}+{{({{a}^{n}}c)}^{\frac{1}{n+1}}}=\] [IIT 1983]
  A)
  \[b\] done clear
  B)
  - b done clear
  C)
  \[{{b}^{\frac{1}{n+1}}}\] done clear
  D)
  \[-{{b}^{\frac{1}{n+1}}}\] done clear
  View Solution play_arrow

• question_answer14)
  If \[\sin \alpha ,\cos \alpha \] are the roots of the equation \[a{{x}^{2}}+bx+c=0\], then [MP PET 1993]
  A)
  \[{{a}^{2}}-{{b}^{2}}+2ac=0\] done clear
  B)
  \[{{(a-c)}^{2}}={{b}^{2}}+{{c}^{2}}\] done clear
  C)
  \[{{a}^{2}}+{{b}^{2}}-2ac=0\] done clear
  D)
  \[{{a}^{2}}+{{b}^{2}}+2ac=0\] done clear
  View Solution play_arrow

• question_answer15)
  If both the roots of the quadratic equation\[{{x}^{2}}-2kx+{{k}^{2}}+k-5=0\]are less than 5, then \[k\] lies in the interval [AIEEE 2005]
  A)
  \[(-\infty ,\,4)\] done clear
  B)
  [4, 5] done clear
  C)
  (5, 6] done clear
  D)
  (6, \[\infty \]) done clear
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• question_answer16)
  If the roots of the equations \[{{x}^{2}}-bx+c=0\] and \[{{x}^{2}}-cx+b=0\] differ by the same quantity, then  \[b+c\] is equal to                            [BIT Ranchi 1969; MP PET 1993]
  A)
  4 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  -4 done clear
  View Solution play_arrow

• question_answer17)
  If the product of roots of the equation\[{{x}^{2}}-3kx+2{{e}^{2\log k}}-1=0\]is 7, then its roots will real when [IIT 1984]
  A)
  \[k=1\] done clear
  B)
  \[k=2\] done clear
  C)
  \[k=3\] done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer18)
  If a root of the given equation \[a(b-c){{x}^{2}}+b(c-a)x+c(a-b)=0\]is 1, then the other will be    [RPET 1986]
  A)
  \[\frac{a(b-c)}{b(c-a)}\] done clear
  B)
  \[\frac{b(c-a)}{a(b-c)}\] done clear
  C)
  \[\frac{c(a-b)}{a(b-c)}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  In a triangle \[ABC\] the value of \[\angle A\] is given by \[5\cos A+3=0\], then the equation  whose roots are \[\sin A\] and \[\tan A\] will be [Roorkee 1972]
  A)
  \[15{{x}^{2}}-8x+16=0\] done clear
  B)
  \[15{{x}^{2}}+8x-16=0\] done clear
  C)
  \[15{{x}^{2}}-8\sqrt{2}x+16=0\] done clear
  D)
  \[15{{x}^{2}}-8x-16=0\] done clear
  View Solution play_arrow

• question_answer20)
  If one root of the equation \[a{{x}^{2}}+bx+c=0\]the square of the other, then\[a{{(c-b)}^{3}}=cX\], where X is
  A)
  \[{{a}^{3}}+{{b}^{3}}\] done clear
  B)
  \[{{(a-b)}^{3}}\] done clear
  C)
  \[{{a}^{3}}-{{b}^{3}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  If 8, 2 are the roots of \[{{x}^{2}}+ax+\beta =0\] and 3, 3 are the roots of \[{{x}^{2}}+\alpha \,x+b=0\], then the roots of \[{{x}^{2}}+ax+b=0\] are [EAMCET 1987]
  A)
  \[8,\,-1\] done clear
  B)
  - 9, 2 done clear
  C)
  \[-8,-2\] done clear
  D)
  9, 1 done clear
  View Solution play_arrow

• question_answer22)
  The set of values of \[x\] which satisfy \[5x+2<3x+8\] and \[\frac{x+2}{x-1}<4,\] is [EAMCET 1989]
  A)
  \[(2,\,3)\] done clear
  B)
  \[(-\infty ,\,1)\cup (2,\,3)\] done clear
  C)
  \[(-\infty ,\,1)\] done clear
  D)
  \[(1,\,3)\] done clear
  View Solution play_arrow

• question_answer23)
  If \[\alpha ,\beta \]are the roots of \[{{x}^{2}}-ax+b=0\] and if \[{{\alpha }^{n}}+{{\beta }^{n}}={{V}_{n}}\], then      [RPET 1995; Karnataka CET 2000; Pb. CET 2002]
  A)
  \[{{V}_{n+1}}=a{{V}_{n}}+b{{V}_{n-1}}\] done clear
  B)
  \[{{V}_{n+1}}=a{{V}_{n}}+a{{V}_{n-1}}\] done clear
  C)
  \[{{V}_{n+1}}=a{{V}_{n}}-b{{V}_{n-1}}\] done clear
  D)
  \[{{V}_{n+1}}=a{{V}_{n-1}}-b{{V}_{n}}\] done clear
  View Solution play_arrow

• question_answer24)
  The value of ?\[c\]?for which \[|{{\alpha }^{2}}-{{\beta }^{2}}|=\frac{7}{4}\], where \[\alpha \] and \[\beta \] are the roots of \[2{{x}^{2}}+7x+c=0\],  is
  A)
  4 done clear
  B)
  0 done clear
  C)
  6 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer25)
  For what value of \[\lambda \]the sum of the squares of the roots of \[{{x}^{2}}+(2+\lambda )\,x-\frac{1}{2}(1+\lambda )=0\] is minimum   [AMU 1999]
  A)
  3/2 done clear
  B)
  1 done clear
  C)
  1/2 done clear
  D)
  11/4 done clear
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• question_answer26)
  The product of all real roots of the equation \[{{x}^{2}}-|x|-\,6=0\] is [Roorkee 2000]
  A)
  - 9 done clear
  B)
  6 done clear
  C)
  9 done clear
  D)
  36 done clear
  View Solution play_arrow

• question_answer27)
  For the equation \[3{{x}^{2}}+px+3=0,\,p>0\] if one of the root is square of the other, then p is equal to  [IIT Screening 2000]
  A)
  \[\frac{1}{3}\] done clear
  B)
  1 done clear
  C)
  3 done clear
  D)
  \[\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer28)
  If a, b be the roots of \[{{x}^{2}}+px+q=0\] and \[\alpha +h,\,\beta +h\] are the roots of \[{{x}^{2}}+rx+s=0\], then [AMU 2001]
  A)
  \[\frac{p}{r}=\frac{q}{s}\] done clear
  B)
  \[2h=\left[ \frac{p}{q}+\frac{r}{s} \right]\] done clear
  C)
  \[{{p}^{2}}-4q={{r}^{2}}-4s\] done clear
  D)
  \[p{{r}^{2}}=q{{s}^{2}}\] done clear
  View Solution play_arrow

• question_answer29)
  If \[{{x}^{2}}+px+q=0\] is the quadratic equation whose roots are a - 2 and b - 2 where a and b are the roots of \[{{x}^{2}}-3x+1=0\], then [Kerala (Engg.) 2002]
  A)
  \[p=1,\,q=5\]  done clear
  B)
  \[p=1,\,q=-5\] done clear
  C)
  \[p=-1,\,\,q=1\] done clear
  D)
  None of these done clear
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• question_answer30)
  The value of 'a' for which one root of the quadratic equation \[({{a}^{2}}-5a+3){{x}^{2}}+(3a-1)x+2=0\] is twice as large as the other, is [AIEEE 2003]
  A)
  \[\frac{2}{3}\] done clear
  B)
  \[-\frac{2}{3}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  \[-\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer31)
  If \[a,b,c\] are in G.P., then the equations \[a{{x}^{2}}+2bx+c=0\] and \[d{{x}^{2}}+2ex+f=0\] have a common root if \[\frac{d}{a},\frac{e}{b},\frac{f}{c}\] are in [IIT 1985; Pb. CET 2000; DCE 2000]
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  The value of 'a' for which the equations \[{{x}^{2}}-3x+a=0\] and \[{{x}^{2}}+ax-3=0\] have a common root is  [Pb. CET 1999]
  A)
  3 done clear
  B)
  1 done clear
  C)
  - 2 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer33)
  If \[(x+1)\] is a factor of\[{{x}^{4}}-(p-3){{x}^{3}}-(3p-5){{x}^{2}}\] \[+(2p-7)x+6\], then \[p=\] [IIT 1975]
  A)
  4 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  The roots of the equation\[4{{x}^{4}}-24{{x}^{3}}+57{{x}^{2}}+18x-45=0\], If one of them is\[3+i\sqrt{6}\], are
  A)
  \[3-i\sqrt{6},\pm \sqrt{\frac{3}{2}}\] done clear
  B)
  \[3-i\sqrt{6},\pm \frac{3}{\sqrt{2}}\] done clear
  C)
  \[3-i\sqrt{6},\pm \frac{\sqrt{3}}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  The values of a for which \[2{{x}^{2}}-2\,(2a+1)\,\,x+a(a+1)=0\] may have one root less than a and other root greater than a are given by [UPSEAT 2001]
  A)
  \[1>a>0\] done clear
  B)
  \[-1<a<0\] done clear
  C)
  \[a\ge 0\] done clear
  D)
  \[a>0\,\,\text{or  }a<-1\] done clear
  View Solution play_arrow

• question_answer36)
  Let \[a,b,c\] be real numbers \[a\ne 0\]. If \[\alpha \]is a root \[{{a}^{2}}{{x}^{2}}+bx+c=0\], \[\beta \] is a root of \[{{a}^{2}}{{x}^{2}}-bx-c=0\] and \[0<\alpha <\beta \], then the equation \[{{a}^{2}}{{x}^{2}}+2bx+2c=0\]has a root \[\gamma \]that always satisfies [IIT 1989]
  A)
  \[\gamma =\frac{\alpha +\beta }{2}\] done clear
  B)
  \[\gamma =\alpha +\frac{\beta }{2}\] done clear
  C)
  \[\gamma =\alpha \] done clear
  D)
  \[\alpha <\gamma <\beta \] done clear
  View Solution play_arrow

• question_answer37)
  If a, b, g  are roots of equation \[{{x}^{3}}+a{{x}^{2}}+bx+c=0\], then \[{{\alpha }^{-1}}+{{\beta }^{-1}}+{{\gamma }^{-1}}=\] [EAMCET 2002]
  A)
  a/c done clear
  B)
  - b/c done clear
  C)
  b/a done clear
  D)
  c/a done clear
  View Solution play_arrow

• question_answer38)
  If \[\frac{2x}{2{{x}^{2}}+5x+2}\]>\[\frac{1}{x+1}\], then [IIT 1987]
  A)
  \[-2>x>-1\] done clear
  B)
  \[-2\ge x\ge -1\] done clear
  C)
  \[-2<x<-1\] done clear
  D)
  \[-2<x\le -1\] done clear
  View Solution play_arrow

• question_answer39)
  If \[a<0\] then the inequality \[a{{x}^{2}}-2x+4>0\] has the solution represented by [AMU 2001]
  A)
  \[\frac{1+\sqrt{1-4a}}{a}>x>\frac{1-\sqrt{1-4a}}{a}\] done clear
  B)
  \[x<\frac{1-\sqrt{1-4a}}{a}\] done clear
  C)
  x < 2 done clear
  D)
  \[2>x>\frac{1+\sqrt{1-4a}}{a}\] done clear
  View Solution play_arrow

• question_answer40)
  The two roots of an equation \[{{x}^{3}}-9{{x}^{2}}+14x+24=0\] are in the ratio 3 : 2.  The roots will be [UPSEAT 1999]
  A)
  6, 4, - 1 done clear
  B)
  6, 4, 1 done clear
  C)
  - 6, 4, 1 done clear
  D)
  - 6, - 4, 1 done clear
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