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

done Solution of quadratic equations and Nature of roots Total Questions - 88

• question_answer1)
  The roots of the equation \[a({{x}^{2}}+1)-({{a}^{2}}+1)x=0\] are
  A)
  \[a,\frac{1}{a}\] done clear
  B)
  a, 2a done clear
  C)
  \[a,\frac{1}{2a}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  The roots of the equation \[{{x}^{4}}-8{{x}^{2}}-9=0\] are
  A)
  \[\pm 3,\ \pm 1\] done clear
  B)
  \[\pm 3,\ \pm i\] done clear
  C)
  \[\pm 2,\ \pm i\]   done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  The roots of the equation \[i{{x}^{2}}-4x-4i=0\] are
  A)
  \[-2i\] done clear
  B)
  \[2i\] done clear
  C)
  \[-2i,-2i\] done clear
  D)
  \[2i,2i\] done clear
  View Solution play_arrow

• question_answer4)
  The roots of the equation \[{{x}^{2/3}}+{{x}^{1/3}}-2=0\] are [UPSEAT 2004]
  A)
  1, 4 done clear
  B)
  \[1,-4\] done clear
  C)
  \[1,-8\] done clear
  D)
  1, 8 done clear
  View Solution play_arrow

• question_answer5)
  If \[x=2+{{2}^{2/3}}+{{2}^{1/3}},\]then \[{{x}^{3}}-6{{x}^{2}}+6x=\] [MNR 1985]
  A)
  3 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  The number of roots of the quadratic equation \[8{{\sec }^{2}}\theta -6\sec \theta +1=0\] is  [Pb. CET 1989, 94]
  A)
  Infinite done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer7)
  The roots of the equation \[\sqrt{3x+1}+1=\sqrt{x}\] are
  A)
  0 done clear
  B)
  1 done clear
  C)
  0, 1 done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer8)
  The number which exceeds its positive square root by 12 is
  A)
  9 done clear
  B)
  16 done clear
  C)
  25 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  The roots of the equation \[{{3}^{2x}}-{{10.3}^{x}}+9\]=0 are
  A)
  1, 2 done clear
  B)
  0, 2 done clear
  C)
  0, 1 done clear
  D)
  1, 3 done clear
  View Solution play_arrow

• question_answer10)
  If \[{{x}^{2/3}}-7{{x}^{1/3}}+10=0,\] then \[x=\] [BIT Ranchi 1992]
  A)
  {125} done clear
  B)
  {8} done clear
  C)
  \[\varphi \] done clear
  D)
  {125, 8} done clear
  View Solution play_arrow

• question_answer11)
  If \[{{x}^{2}}+{{y}^{2}}=25,\ xy=12\], then \[x=\] [BIT Ranchi 1992]
  A)
  {3, 4} done clear
  B)
  {3, -3} done clear
  C)
  {3, 4, -3, -4} done clear
  D)
  {-3, -3} done clear
  View Solution play_arrow

• question_answer12)
  The solution set of the equation \[{{x}^{{{\log }_{x}}{{(1-x)}^{2}}}}=9\] is [Pb. CET 2003]
  A)
  {- 2, 4} done clear
  B)
  {4} done clear
  C)
  {0, - 2, 4} done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  Let  one root of \[a{{x}^{2}}+bx+c=0\] where \[a,b,c\] are integers be \[3+\sqrt{5}\], then the other root is [MNR 1982]
  A)
  \[3-\sqrt{5}\] done clear
  B)
  3 done clear
  C)
  \[\sqrt{5}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  The number of real solutions of the equation\[|x{{|}^{2}}\]- \[3|x|+2=0\] are   [IIT 1982, 89; MP PET 1997; DCE 2002; AMU 2000; UPSEAT 1999; AIEEE 2003]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer15)
  The number of real roots of the equation \[{{e}^{\sin x}}-{{e}^{-\sin x}}-4\] \[=0\] are [IIT 1982; Pb. CET 2000]
  A)
  1 done clear
  B)
  2 done clear
  C)
  Infinite done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer16)
  The number of real solutions of the equation |\[{{x}^{2}}\] + 4x + 3| +  2x + 5 = 0  are [IIT 1988]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer17)
  The roots of the given equation \[(p-q){{x}^{2}}+(q-r)x+(r-p)=0\] are                    [RPET 1986; MP PET 1999; Pb. CET 2004]
  A)
  \[\frac{p-q}{r-p},1\] done clear
  B)
  \[\frac{q-r}{p-q},1\] done clear
  C)
  \[\frac{r-p}{p-q},1\] done clear
  D)
  \[1,\frac{q-r}{p-q}\] done clear
  View Solution play_arrow

• question_answer18)
  If a root of the equation \[{{x}^{2}}+px+12=0\] is 4, while the roots of the equation \[{{x}^{2}}+px+q=0\] are same, then the value of \[q\]will be      [RPET 1991; AIEEE 2004]
  A)
  4 done clear
  B)
  4/49 done clear
  C)
  49/4 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  How many roots the equation \[x-\frac{2}{x-1}=1-\frac{2}{x-1}\]have  [IIT 1984; UPSEAT 1999; Pb. CET 2003]
  A)
  One done clear
  B)
  Two done clear
  C)
  Infinite done clear
  D)
   None done clear
  View Solution play_arrow

• question_answer20)
  The solution of the equation \[x+\frac{1}{x}=2\] will be [MNR 1983]
  A)
  2, -1 done clear
  B)
  0, -1, \[-\frac{1}{5}\] done clear
  C)
  \[-1,-\frac{1}{5}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  If the product of the roots of the equation \[2{{x}^{2}}+6x+\] \[{{\alpha }^{2}}+1=0\] is \[-\alpha \], then the value of \[\alpha \] will be
  A)
  -1 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  -2 done clear
  View Solution play_arrow

• question_answer22)
  If \[\sqrt{3{{x}^{2}}-7x-30}+\sqrt{2{{x}^{2}}-7x-5}=x+5\],then x is equal to
  A)
  2 done clear
  B)
  3 done clear
  C)
  6 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer23)
  The value of  \[2+\frac{1}{2+\frac{1}{2+...........\infty }}\] is
  A)
  \[1-\sqrt{2}\] done clear
  B)
  \[1+\sqrt{2}\] done clear
  C)
  \[1\pm \sqrt{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  The roots of the equation \[{{2}^{x+2}}{{27}^{x/(x-1)}}=9\] are given by
  A)
  \[1-{{\log }_{2}}3,\,2\] done clear
  B)
  \[{{\log }_{2}}\left( \frac{2}{3} \right)\,,\,\ 1\] done clear
  C)
  \[2,-2\] done clear
  D)
  \[-2,\ 1-\frac{\log 3}{\log 2}\] done clear
  View Solution play_arrow

• question_answer25)
  Let \[\alpha \]and \[\beta \] be the roots of the equation \[{{x}^{2}}+x+1=0\] The equation whose roots are \[{{\alpha }^{19}},{{\beta }^{7}}\] is [IIT Screening 1994]
  A)
  \[{{x}^{2}}-x-1=0\] done clear
  B)
  \[{{x}^{2}}-x+1=0\] done clear
  C)
  \[{{x}^{2}}+x-1=0\] done clear
  D)
  \[{{x}^{2}}+x+1=0\] done clear
  View Solution play_arrow

• question_answer26)
  If \[|x-2|+|x-3|=7\], then x =
  A)
  6 done clear
  B)
  -1 done clear
  C)
  6 or -1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  If \[{{x}_{1}},{{x}_{2}},{{x}_{3}}\] are distinct roots of the equation \[a{{x}^{2}}+bx+c=0\] then
  A)
  \[a=b=0,c\in R\] done clear
  B)
  \[a=c=0,b\in R\] done clear
  C)
  \[{{b}^{2}}-4ac\ge 0\] done clear
  D)
  \[a=b=c=0\] done clear
  View Solution play_arrow

• question_answer28)
  The number of roots of the equation \[|x{{|}^{2}}-7|x|+12=0\] is [MNR 1995]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer29)
  The number of solutions of \[\frac{\log 5+\log ({{x}^{2}}+1)}{\log (x-2)}=2\] is
  A)
  2 done clear
  B)
  3 done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer30)
  If \[x=\sqrt{7+4\sqrt{3}},\] then \[x+\frac{1}{x}=\] [EAMCET 1994]
  A)
  4 done clear
  B)
  6 done clear
  C)
  3 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer31)
  If \[{{\log }_{2}}x+{{\log }_{x}}2=\frac{10}{3}={{\log }_{2}}y+{{\log }_{y}}2\] and \[x\ne y,\] then \[x+y=\] [EAMCET 1994]
  A)
  2 done clear
  B)
  65/8 done clear
  C)
  37/6 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  The value of  \[x=\sqrt{2+\sqrt{2+\sqrt{2+.....}}}\]is  [Karnataka CET 2001]
  A)
  -1 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer33)
  The value of x in the given equation\[{{4}^{x}}-{{3}^{x\,\ -\ \frac{1}{2}}}={{3}^{x+\frac{1}{2}}}-{{2}^{2x-1}}\]is
  A)
  \[\frac{4}{3}\] done clear
  B)
  \[\frac{3}{2}\] done clear
  C)
  \[\frac{2}{1}\] done clear
  D)
  \[\frac{5}{3}\] done clear
  View Solution play_arrow

• question_answer34)
  The equation\[{{e}^{x}}-x-1=0\] has      [Kurukshetra CEE 1998]
  A)
  Only one real root \[x=0\] done clear
  B)
  At least two real roots done clear
  C)
  Exactly two real roots done clear
  D)
  Infinitely many real roots done clear
  View Solution play_arrow

• question_answer35)
  The equation \[{{x}^{2}}+qx+rp=0,\] has [IIT 1997 Cancelled]
  A)
  No solution done clear
  B)
  One solution done clear
  C)
  Two solutions done clear
  D)
  More than two solutions done clear
  View Solution play_arrow

• question_answer36)
  The equation \[{{\log }_{e}}x+{{\log }_{e}}(1+x)=0\] can be written as [Kurukshetra CEE 1998; MP PET 1989]
  A)
  \[{{x}^{2}}+x-e=0\] done clear
  B)
  \[{{x}^{2}}+x-1=0\] done clear
  C)
  \[{{x}^{2}}+x+1=0\] done clear
  D)
  \[{{x}^{2}}+xe-e=0\] done clear
  View Solution play_arrow

• question_answer37)
  If \[x=\sqrt{6+\sqrt{6+\sqrt{6+....\text{to}\,\,\infty }},}\] then  [Pb. CET 1999]
  A)
  x is an irrational number done clear
  B)
  \[2<x<3\] done clear
  C)
  \[x=3\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer38)
  The real roots of the equation \[{{x}^{2}}+5|x|+\,\,4=0\] are  [UPSEAT 1993, 99; Orissa JEE 2004]
  A)
  - 1, 4 done clear
  B)
  1, 4 done clear
  C)
  - 4, 4 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  A real root of the equation \[{{\log }_{4}}\{{{\log }_{2}}(\sqrt{x+8}-\sqrt{x})\}=0\] is [AMU 1999]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer40)
  \[\{x\in R:|x-2|\,\,={{x}^{2}}\}=\] [EAMCET 2000]
  A)
  { - 1, 2} done clear
  B)
  {1, 2} done clear
  C)
  { - 1, - 2} done clear
  D)
  {1, - 2} done clear
  View Solution play_arrow

• question_answer41)
  The number of solutions of \[{{\log }_{4}}(x-1)={{\log }_{2}}(x-3)\] [IIT Screening 2001]
  A)
  3 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer42)
  The roots of \[|x-2{{|}^{2}}+|x-2|-6=0\]are  [UPSEAT 2003]
  A)
  0, 4 done clear
  B)
  -1, 3 done clear
  C)
  4, 2 done clear
  D)
  5, 1 done clear
  View Solution play_arrow

• question_answer43)
  The solution of equation\[\frac{p+q-x}{r}+\frac{q+r-x}{p}\]+\[\frac{r+p-x}{q}\]+\[\frac{4x}{p+q+r}=0\] is  [MP PET 2004]
  A)
  \[x=p+q+r\] done clear
  B)
  \[x=p-q+r\] done clear
  C)
  \[x=\frac{p+q}{q+r}\] done clear
  D)
  \[x=\frac{p}{q}+r\] done clear
  View Solution play_arrow

• question_answer44)
  The number of solutions for the equation \[{{x}^{2}}-5|x|+\,6=0\] is [Karnataka CET 2004]
  A)
  4 done clear
  B)
  3 done clear
  C)
  2 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer45)
  If the roots of the given equation \[({{m}^{2}}+1){{x}^{2}}+2amx+{{a}^{2}}-{{b}^{2}}=0\] be equal, then
  A)
  \[{{a}^{2}}+{{b}^{2}}({{m}^{2}}+1)=0\] done clear
  B)
  \[{{b}^{2}}+{{a}^{2}}({{m}^{2}}+1)=0\] done clear
  C)
  \[{{a}^{2}}-{{b}^{2}}({{m}^{2}}+1)=0\] done clear
  D)
  \[{{b}^{2}}-{{a}^{2}}({{m}^{2}}+1)=0\] done clear
  View Solution play_arrow

• question_answer46)
  If \[P(x)=a{{x}^{2}}+bx+c\] and \[Q(x)=-a{{x}^{2}}+dx+c\]where \[ac\ne 0\], then \[P(x).Q(x)=0\]has at least [IIT 1985; Pb. CET 2003; AMU 2005]
  A)
  Four real roots done clear
  B)
  Two real roots done clear
  C)
    Four imaginary roots done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  Both the roots of the given equation\[(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0\] are always [MNR 1986; IIT 1980; Kurukshetra CEE 1998; RPET 2002]
  A)
  Positive done clear
  B)
  Negative done clear
  C)
  Real done clear
  D)
  Imaginary done clear
  View Solution play_arrow

• question_answer48)
  If the roots of the given equation \[2{{x}^{2}}+3(\lambda -2)x+\lambda +4=0\] be equal in magnitude but opposite in sign, then \[\lambda \]=
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  2/3 done clear
  View Solution play_arrow

• question_answer49)
  If the roots of the equation \[({{p}^{2}}+{{q}^{2}}){{x}^{2}}\]\[-2q(p+r)x\]+ \[({{q}^{2}}+{{r}^{2}})=0\] be real and equal, then \[p,q,r\]will be  in
  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_answer50)
  If \[a+b+c=0\], then the roots of the equation \[4a{{x}^{2}}+3bx+2c=0\] are
  A)
  Equal done clear
  B)
  Imaginary done clear
  C)
  Real done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer51)
  The roots of the given equation \[2({{a}^{2}}+{{b}^{2}}){{x}^{2}}+2(a+b)x+1=0\] are
  A)
  Rational done clear
  B)
  Irrational done clear
  C)
  Real done clear
  D)
  Imaginary done clear
  View Solution play_arrow

• question_answer52)
  If the roots of the equations \[p{{x}^{2}}+2qx+r=0\]and \[q{{x}^{2}}-2\sqrt{pr}x+q=0\] be real, then
  A)
  \[p=q\] done clear
  B)
  \[{{q}^{2}}=pr\] done clear
  C)
  \[{{p}^{2}}=qr\] done clear
  D)
  \[{{r}^{2}}=pq\] done clear
  View Solution play_arrow

• question_answer53)
  If the roots of the equation \[a{{x}^{2}}+x+b=0\] be real, then the roots of the equation \[{{x}^{2}}-4\sqrt{ab}x+1=0\] will be
  A)
  Rational done clear
  B)
  Irrational done clear
  C)
  Real done clear
  D)
  Imaginary done clear
  View Solution play_arrow

• question_answer54)
  If one of the roots of the equation \[{{x}^{2}}+ax+b=0\] and \[{{x}^{2}}+bx+a=0\] is coincident, then the numerical value of \[(a+b)\] is                  [IIT 1986; RPET 1992; EAMCET 2002]
  A)
  0 done clear
  B)
  - 1 done clear
  C)
  2 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer55)
  The equation \[{{x}^{(3/4){{({{\log }_{2}}x)}^{2}}+({{\log }_{2}}x)-5/4}}=\sqrt{2}\] has [IIT 1989]
  A)
  At least one real solution done clear
  B)
  Exactly three real solutions done clear
  C)
  Exactly one irrational solution done clear
  D)
  All the above done clear
  View Solution play_arrow

• question_answer56)
  If \[a>0,b>0,c>0\] then both the roots of the equation \[a{{x}^{2}}+bx+c=0\]
  A)
  Are real and negative done clear
  B)
  Have negative real parts done clear
  C)
  Are rational numbers done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  The value of \[k\]for which \[2{{x}^{2}}-kx+x+8=0\]has equal and real roots are [BIT Ranchi  1990]
  A)
  -9 and -7 done clear
  B)
  9 and 7 done clear
  C)
  -9 and 7 done clear
  D)
  9 and -7 done clear
  View Solution play_arrow

• question_answer58)
  The  roots of the quadratic equation \[2{{x}^{2}}+3x+1=0\], are  [IIT 1983]
  A)
  Irrational done clear
  B)
  Rational done clear
  C)
  Imaginary done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer59)
  If  \[l,m,n\] are real and \[l\ne m\], then the roots of the equation \[(l-m){{x}^{2}}-5(l+m)x-2(l-m)=0\]are [IIT 1979; RPET 1983]
  A)
  Complex done clear
  B)
  Real and distinct done clear
  C)
  Real and equal   done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer60)
  If the roots of the equation \[{{x}^{2}}-8x+({{a}^{2}}-6a)=0\] are real, then                                    [RPET 1987, 97; MP PET 1999]
  A)
  \[-2<a<8\] done clear
  B)
  \[2<a<8\] done clear
  C)
  \[-2\le a\le 8\] done clear
  D)
  \[2\le a\le 8\] done clear
  View Solution play_arrow

• question_answer61)
  The roots of the equation \[{{x}^{2}}+2\sqrt{3}x+3=0\]are  [RPET 1986]
  A)
  Real and unequal   done clear
  B)
  Rational and equal done clear
  C)
  Irrational and equal done clear
  D)
  Irrational and unequal done clear
  View Solution play_arrow

• question_answer62)
  If the roots of the given equation \[(\cos p-1){{x}^{2}}+(\cos p)x+\sin p=0\] are real, then  [IIT 1990; RPET 1995]
  A)
  \[p\in (-\pi ,0)\] done clear
  B)
  \[p\in \left( -\frac{\pi }{2},\frac{\pi }{2} \right)\] done clear
  C)
  \[p\in (0,\pi )\] done clear
  D)
  \[p\in (0,2\pi )\] done clear
  View Solution play_arrow

• question_answer63)
  If \[{{x}^{2}}+2x+2xy+my-3\] has two rational factors, then the value of m will be     [RPET 1990]
  A)
  \[-6,-2\] done clear
  B)
  \[-6,2\] done clear
  C)
  \[6,-2\] done clear
  D)
  6, 2 done clear
  View Solution play_arrow

• question_answer64)
  If a and b are the odd integers, then the roots of the equation \[2a{{x}^{2}}+(2a+b)x+b=0,\ a\ne 0,\] will be [Pb. CET 1988]
  A)
  Rational done clear
  B)
  Irrational done clear
  C)
  Non-real done clear
  D)
  Equal done clear
  View Solution play_arrow

• question_answer65)
  Roots of \[a{{x}^{2}}+b=0\] are real and distinct if
  A)
  \[ab>0\] done clear
  B)
  \[ab<0\] done clear
  C)
  \[a,b>0\] done clear
  D)
  \[a,b<0\] done clear
  View Solution play_arrow

• question_answer66)
  Roots of the equations \[2{{x}^{2}}-5x+1=0\], \[{{x}^{2}}+5x+2=0\] are
  A)
  Reciprocal and of same sign done clear
  B)
  Reciprocal and of opposite sign done clear
  C)
  Equal in product done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer67)
  If \[a+b+c=0,\]\[a\ne 0,a,b,c\in Q\], then both the roots of the equation \[a{{x}^{2}}+bx+c=0\] are
  A)
  Rational done clear
  B)
  Non-real done clear
  C)
  Irrational done clear
  D)
  Zero done clear
  View Solution play_arrow

• question_answer68)
  If \[a,b,c\in Q\], then roots of the equation\[(b+c-2a){{x}^{2}}+\] \[(c+a-2b)x+(a+b-2c)=0\] are
  A)
  Rational done clear
  B)
  Non-real done clear
  C)
  Irrational done clear
  D)
  Equal done clear
  View Solution play_arrow

• question_answer69)
  The expression \[{{x}^{2}}+2bx+c\] has the positive value if [Roorkee 1995]
  A)
  \[{{b}^{2}}-4c>0\] done clear
  B)
  \[{{b}^{2}}-4c<0\] done clear
  C)
  \[{{c}^{2}}<b\] done clear
  D)
  \[{{b}^{2}}<c\] done clear
  View Solution play_arrow

• question_answer70)
  If the roots of \[4{{x}^{2}}+px+9=0\] are equal, then absolute value of p is [MP PET 1995]
  A)
  144 done clear
  B)
  12 done clear
  C)
  \[-12\] done clear
  D)
  \[\pm 12\] done clear
  View Solution play_arrow

• question_answer71)
  The condition for the roots of the equation,\[({{c}^{2}}-ab){{x}^{2}}-\]\[2({{a}^{2}}-bc)x+({{b}^{2}}-ac)=0\] to be equal is   [TS Rajendra 1982]
  A)
  \[a=0\] done clear
  B)
  \[b=0\] done clear
  C)
  \[c=0\] done clear
  D)
  None of these done clear
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• question_answer72)
  If \[{{b}_{1}}{{b}_{2}}=2\]\[({{c}_{1}}+{{c}_{2}})\], then at least one of the equations \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\]  and \[{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0\] has
  A)
  Real roots done clear
  B)
  Purely imaginary roots done clear
  C)
  Imaginary roots done clear
  D)
  None of these done clear
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• question_answer73)
  The value of \[k\] for which the quadratic equation,\[k{{x}^{2}}+1=\]\[kx+3x-11{{x}^{2}}\] has real and equal roots are [BIT Ranchi 1993]
  A)
  \[-11,-3\] done clear
  B)
  \[5,\,7\] done clear
  C)
  \[5,-7\] done clear
  D)
  None of these done clear
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• question_answer74)
  The expression \[y=a{{x}^{2}}+bx+c\] has always the same sign as c if
  A)
  \[4ac<{{b}^{2}}\] done clear
  B)
  \[4ac>{{b}^{2}}\] done clear
  C)
  \[ac<{{b}^{2}}\] done clear
  D)
  \[ac>{{b}^{2}}\] done clear
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• question_answer75)
  The value of m  for which the equation\[\frac{a}{x+a+m}+\frac{b}{x+b+m}=1\]has roots equal in magnitude but opposite in sign is
  A)
  \[\frac{a+b}{a-b}\] done clear
  B)
  0 done clear
  C)
    \[\frac{a-b}{a+b}\] done clear
  D)
  \[\frac{2(a-b)}{a+b}\] done clear
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• question_answer76)
  The roots of the equation\[({{a}^{2}}+{{b}^{2}}){{t}^{2}}-2(ac+bd)t+({{c}^{2}}+{{d}^{2}})=0\] are equal, then  [MP PET 1996]
  A)
  \[ab=dc\] done clear
  B)
  \[ac=bd\] done clear
  C)
  \[ad+bc=0\] done clear
  D)
  \[\frac{a}{b}=\frac{c}{d}\] done clear
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• question_answer77)
  For what values of k will the equation\[{{x}^{2}}-2(1+3k)x+\]\[7(3+2k)=0\] have equal roots   [MP PET 1997]
  A)
  \[1,-\frac{10}{9}\] done clear
  B)
  \[2,-\frac{10}{9}\] done clear
  C)
  \[3,-\frac{10}{9}\] done clear
  D)
  \[4,-\frac{10}{9}\] done clear
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• question_answer78)
  If the roots of equation \[{{x}^{2}}+{{a}^{2}}=8x+6a\] are real, then [MP PET 1999]
  A)
  \[a\in [2,\,8]\] done clear
  B)
  \[a\in [-2,\,8]\] done clear
  C)
  \[a\in (2,\,8)\] done clear
  D)
  \[a\in (-2,\,8)\] done clear
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• question_answer79)
  Let\[p,q\in \{1,\,2,\,3,\,4\}\]. The number of equations of the form \[p{{x}^{2}}+qx+1=0\] having real roots is   [IIT Screening 1994]
  A)
  15 done clear
  B)
  9 done clear
  C)
  7 done clear
  D)
  8 done clear
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• question_answer80)
  For what value of k will the equation\[{{x}^{2}}-(3k-1)x+\]\[2{{k}^{2}}+2k\] have equal roots  [Karnataka CET 1998]
  A)
  5 done clear
  B)
  9 done clear
  C)
  Both (a) and (b) done clear
  D)
  0 done clear
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• question_answer81)
  The value of k for which the equation\[(k-2){{x}^{2}}+8x+k+4=0\] has both real, distinct and negative is[Orissa JEE 2002]
  A)
  0 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  - 4 done clear
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• question_answer82)
  If \[k\in (-\infty ,\,-2)\cup (2,\infty ),\] then the roots of the equation \[{{x}^{2}}+2kx+4=0\] are [DCE 2002]
  A)
  Complex done clear
  B)
  Real and unequal done clear
  C)
  Real and equal done clear
  D)
  One real and one imaginary done clear
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• question_answer83)
  If the equation \[(m-n){{x}^{2}}+(n-l)x+l-m=0\] has equal roots, then l, m and n satisfy [DCE 2002]
  A)
  \[2l=m+n\] done clear
  B)
  \[2m=n+l\] done clear
  C)
  \[m=n+l\] done clear
  D)
  \[l=m+n\] done clear
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• question_answer84)
  The least integer k which makes the roots of the equation \[{{x}^{2}}+5x+k=0\] imaginary is  [Kerala (Engg.) 2002]
  A)
  4 done clear
  B)
  5 done clear
  C)
  6 done clear
  D)
  7 done clear
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• question_answer85)
  The roots of \[4{{x}^{2}}+6px+1=0\] are equal, then the value of p is [MP PET 2003]
  A)
  \[\frac{4}{5}\] done clear
  B)
  \[\frac{1}{3}\] done clear
  C)
  \[\frac{2}{3}\] done clear
  D)
  \[\frac{4}{3}\] done clear
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• question_answer86)
  \[{{x}^{2}}+x+1+2k\,({{x}^{2}}-x-1)=0\]is a perfect square for how many values of k [Orissa JEE 2004]
  A)
  2 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  3 done clear
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• question_answer87)
  If \[\sin A,\sin B,\cos A\] are in G.P., then roots of \[{{x}^{2}}+2x\cot B+1=0\] are always [Orissa JEE 2005]
  A)
  Real done clear
  B)
  Imaginary done clear
  C)
  Greater than 1 done clear
  D)
  Equal done clear
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• question_answer88)
  The values of 'a' and 'b' for which equation \[{{x}^{4}}-4{{x}^{3}}+a{{x}^{2}}+bx+1=0\] have four real roots [DCE 2005]
  A)
  - 6, - 4 done clear
  B)
  - 6, 5 done clear
  C)
  - 6, 4 done clear
  D)
  6, - 4 done clear
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