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

Relation between roots and coefficients Total Questions - 107

1)

If one root of \[5{{x}^{2}}+13x+k=0\] is reciprocal of the other, then \[k\]= [MNR 1980, 1983]

A)
0

B)
5

C)
1/6

D)
6
View Solution
2)

If \[\alpha \] and \[\beta \] are the roots of the equation \[4{{x}^{2}}+3x+7=0\], then \[\frac{1}{\alpha }+\frac{1}{\beta }\]= [MNR 1981; RPET 1990]

A)
\[-\frac{3}{7}\]

B)
\[\frac{3}{7}\]

C)
\[-\frac{3}{5}\]

D)
\[\frac{3}{5}\]
View Solution
3)

If the product of the roots of the equation \[(a+1){{x}^{2}}+(2a+3)x+(3a+4)=0\] be 2, then the sum of roots is

A)
1

B)
-1

C)
2

D)
-2
View Solution
4)

If the roots of the equation \[a{{x}^{2}}+bx+c=0\]be \[\alpha \]and \[\beta \], then the roots of the equation \[c{{x}^{2}}+bx+a=0\] are [MNR 1988; RPET 2003]

A)
\[-\alpha ,-\beta \]

B)
\[\alpha ,\frac{1}{\beta }\]

C)
\[\frac{1}{\alpha },\frac{1}{\beta }\]

D)
None of these
View Solution
5)

If \[\alpha ,\beta \] are the roots of the equation \[a{{x}^{2}}+bx+c=0\] then the equation whose roots are \[\alpha +\frac{1}{\beta }\]and \[\beta +\frac{1}{\alpha }\], is [RPET 1991]

A)
\[ac{{x}^{2}}+(a+c)bx+{{(a+c)}^{2}}=0\]

B)
\[ab{{x}^{2}}+(a+c)bx+{{(a+c)}^{2}}=0\]

C)
\[ac{{x}^{2}}+(a+b)cx+{{(a+c)}^{2}}=0\]

D)
None of these
View Solution
6)

If a root of the equation \[a{{x}^{2}}+bx+c=0\]be reciprocal of a root of the equation then\[{a}'{{x}^{2}}+{b}'x+{c}'=0\], then [IIT 1968]

A)
\[{{(c{c}'-a{a}')}^{2}}=(b{a}'-c{b}')(a{b}'-b{c}')\]

B)
\[{{(b{b}'-a{a}')}^{2}}=(c{a}'-b{c}')(a{b}'-b{c}')\]

C)
\[{{(c{c}'-a{a}')}^{2}}=(b{a}'+c{b}')(a{b}'+b{c}')\]

D)
None of these
View Solution
7)

If \[\alpha \]and \[\beta \] be the roots of the equation \[2{{x}^{2}}+2(a+b)x+{{a}^{2}}+{{b}^{2}}=0\], then the equation whose roots are \[{{(\alpha +\beta )}^{2}}\]and \[{{(\alpha -\beta )}^{2}}\] is

A)
\[{{x}^{2}}-2abx-{{({{a}^{2}}-{{b}^{2}})}^{2}}=0\]

B)
\[{{x}^{2}}-4abx-{{({{a}^{2}}-{{b}^{2}})}^{2}}=0\]

C)
\[{{x}^{2}}-4abx+{{({{a}^{2}}-{{b}^{2}})}^{2}}=0\]

D)
None of these
View Solution
8)

If \[2+i\sqrt{3}\] is a root of the equation \[{{x}^{2}}+px+q=0\], where p and q are real, then \[(p,q)\]= [IIT 1981; MP PET 1997, 2004]

A)
\[(-4,\,7)\]

B)
\[(4,\,-7)\]

C)
(4, 7)

D)
\[(-4,\,\,-7)\]
View Solution
9)

If the sum of the roots of the equation \[\lambda {{x}^{2}}+2x+3\lambda =0\] be equal to their product, then \[\lambda =\]

A)
4

B)
\[-4\]

C)
6

D)
None of these
View Solution
10)

If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}+6x+\lambda =0\] and \[3\alpha +2\beta =-20\], then \[\lambda =\]

A)
-8

B)
-16

C)
16

D)
8
View Solution
11)

If \[\alpha \] and \[\beta \] are the roots of the equation \[2{{x}^{2}}-3x+4=0\], then the equation whose roots are \[{{\alpha }^{2}}\] and \[{{\beta }^{2}}\] is

A)
\[4{{x}^{2}}+7x+16=0\]

B)
\[4{{x}^{2}}+7x+6=0\]

C)
\[4{{x}^{2}}+7x+1=0\]

D)
\[4{{x}^{2}}-7x+16=0\]
View Solution
12)

If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}-a(x+1)-b=0\] then \[(\alpha +1)(\beta +1)=\]

A)
b

B)
- b

C)
\[1-b\]

D)
\[b-1\]
View Solution
13)

If \[\alpha ,\beta \] be the roots of the equation \[2{{x}^{2}}-2({{m}^{2}}+1)x+{{m}^{4}}+{{m}^{2}}+1=0\], then \[{{\alpha }^{2}}+{{\beta }^{2}}\]=

A)
0

B)
1

C)
m

D)
\[{{m}^{2}}\]
View Solution
14)

If the ratio of the roots of the equation \[a{{x}^{2}}+bx+c=0\]be \[p:q\], then [Pb. CET 1994]

A)
\[pq{{b}^{2}}+{{(p+q)}^{2}}ac=0\]

B)
\[pq{{b}^{2}}-{{(p+q)}^{2}}ac=0\]

C)
\[pq{{a}^{2}}-{{(p+q)}^{2}}bc=0\]

D)
None of these
View Solution
15)

If \[\alpha ,\ \beta \] are the roots of the equation \[a{{x}^{2}}+bx+c=0\], then \[\frac{\alpha }{a\beta +b}+\frac{\beta }{a\alpha +b}=\]

A)
\[\frac{2}{a}\]

B)
\[\frac{2}{b}\]

C)
\[\frac{2}{c}\]

D)
\[-\frac{2}{a}\]
View Solution
16)

If the sum of the roots of the equation \[a{{x}^{2}}+bx+c=0\] be equal to the sum of their squares, then

A)
\[a(a+b)=2bc\]

B)
\[c(a+c)=2ab\]

C)
\[b(a+b)=2ac\]

D)
\[b(a+b)=ac\]
View Solution
17)

If the roots of the equation \[\frac{\alpha }{x-\alpha }+\frac{\beta }{x-\beta }=1\] be equal in magnitude but opposite in sign, then \[\alpha +\beta \]=

A)
0

B)
1

C)
2

D)
None of these
View Solution
18)

If \[\alpha ,\beta \] be the roots of the equation \[{{x}^{2}}-2x+3=0\], then the equation whose roots are \[\frac{1}{{{\alpha }^{2}}}\]and \[\frac{1}{{{\beta }^{2}}}\] is

A)
\[{{x}^{2}}+2x+1=0\]

B)
\[9{{x}^{2}}+2x+1=0\]

C)
\[9{{x}^{2}}-2x+1=0\]

D)
\[9{{x}^{2}}+2x-1=0\]
View Solution
19)

If \[\alpha ,\beta \] are the roots of \[{{x}^{2}}+px+1=0\] and \[\gamma ,\delta \]are the roots of \[{{x}^{2}}+qx+1=0\],then \[{{q}^{2}}-{{p}^{2}}\]= [IIT 1978; DCE 2000]

A)
\[(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )\]

B)
\[(\alpha +\gamma )(\beta +\gamma )(\alpha -\delta )(\beta +\delta )\]

C)
\[(\alpha +\gamma )(\beta +\gamma )(\alpha +\delta )(\beta +\delta )\]

D)
None of these
View Solution
20)

If \[\alpha ,\beta \] be the roots of \[{{x}^{2}}-px+q=0\]and \[{\alpha }',{\beta }'\] be the roots of \[{{x}^{2}}-{p}'x+{q}'=0\], then the value of \[{{(\alpha -\alpha ')}^{2}}+{{(\beta -{\alpha }')}^{2}}+{{(a-{\beta }')}^{2}}+{{(\beta -{\beta }')}^{2}}\] is

A)
\[2\{{{p}^{2}}-2q+{{{p}'}^{2}}-2{q}'-p{p}'\}\]

B)
\[2\{{{p}^{2}}-2q+{{{p}'}^{2}}-2{q}'-q{q}'\}\]

C)
\[2\{{{p}^{2}}-2q-{{{p}'}^{2}}-2{q}'-p{p}'\}\]

D)
\[2\{{{p}^{2}}-2q-{{{p}'}^{2}}-2{q}'-q{q}'\}\]
View Solution
21)

If one root of \[a{{x}^{2}}+bx+c=0\] be square of the other, then the value of \[{{b}^{3}}+a{{c}^{2}}+{{a}^{2}}c\]is

A)
\[3abc\]

B)
\[-3abc\]

C)
0

D)
None of these
View Solution
22)

The quadratic in \[t\], such that A.M. of its roots is \[A\] and G.M. is G, is [IIT 1968, 1974]

A)
\[{{t}^{2}}-2At+{{G}^{2}}=0\]

B)
\[{{t}^{2}}-2At-{{G}^{2}}=0\]

C)
\[{{t}^{2}}+2At+{{G}^{2}}=0\]

D)
None of these
View Solution
23)

If \[\alpha ,\beta \] are the roots of \[(x-a)(x-b)=c,\]\[c\ne 0,\] then the roots of \[(x-\alpha )(x-\beta )+c=0\] shall be [IIT 1992; MP PET 2000; DCE 2000]

A)
a, c

B)
\[b,c\]

C)
\[a,b\]

D)
\[a+c,b+c\]
View Solution
24)

If the difference of the roots of \[{{x}^{2}}-px+8=0\] be 2, then the value of p is [Roorkee 1992]

A)
\[\pm 2\]

B)
\[\pm 4\]

C)
\[\pm 6\]

D)
\[\pm 8\]
View Solution
25)

If the sum of the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\] is equal to the sum of the squares of their reciprocals, then \[a/c,\,b/a,\,c/b\]are in [AIEEE 2003; DCE 2000]

A)
A.P.

B)
G.P.

C)
H.P.

D)
None of these
View Solution
26)

If \[\alpha \] and \[\beta \] are roots of \[a{{x}^{2}}+2bx+c=0\], then \[\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}\]is equal to [BIT Ranchi 1990]

A)
\[\frac{2b}{ac}\]

B)
\[\frac{2b}{\sqrt{ac}}\]

C)
\[-\frac{2b}{\sqrt{ac}}\]

D)
\[\frac{-b}{\sqrt{2}}\]
View Solution
27)

The quadratic equation with real coefficients whose one root is\[7+5i\], will be [RPET 1992]

A)
\[{{x}^{2}}-14x+74=0\]

B)
\[{{x}^{2}}+14x+74=0\]

C)
\[{{x}^{2}}-14x-74=0\]

D)
\[{{x}^{2}}+14x-74=0\]
View Solution
28)

If the roots of the equation \[\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}\] are equal in magnitude but opposite in sign, then the product of the roots will be [IIT 1967; RPET 1999]

A)
\[\frac{{{p}^{2}}+{{q}^{2}}}{2}\]

B)
-\[\frac{({{p}^{2}}+{{q}^{2}})}{2}\]

C)
\[\frac{{{p}^{2}}-{{q}^{2}}}{2}\]

D)
-\[\frac{({{p}^{2}}-{{q}^{2}})}{2}\]
View Solution
29)

If the roots of the equation \[a{{x}^{2}}+bx+c=0\] are reciprocal to each other, then [RPET 1985]

A)
\[a-c=0\]

B)
\[b-c=0\]

C)
\[a+c=0\]

D)
\[b+c=0\]
View Solution
30)

The quadratic equation whose one root is \[2-\sqrt{3}\]will be [RPET 1985]

A)
\[{{x}^{2}}-4x-1=0\]

B)
\[{{x}^{2}}-4x+1=0\]

C)
\[{{x}^{2}}+4x-1=0\]

D)
\[{{x}^{2}}+4x+1=0\]
View Solution
31)

If the roots of the equation \[A{{x}^{2}}+Bx+C=0\] are \[\alpha ,\beta \] and the roots of the equation \[{{x}^{2}}+px+q=0\] are \[{{\alpha }^{2}},\ {{\beta }^{2}}\], then value of p will be [RPET 1986]

A)
\[\frac{{{B}^{2}}-2AC}{{{A}^{2}}}\]

B)
\[\frac{2AC-{{B}^{2}}}{{{A}^{2}}}\]

C)
\[\frac{{{B}^{2}}-4AC}{{{A}^{2}}}\]

D)
None of these
View Solution
32)

The quadratic equation whose one root is \[\frac{1}{2+\sqrt{5}}\] will be [RPET 1987]

A)
\[{{x}^{2}}+4x-1=0\]

B)
\[{{x}^{2}}+4x+1=0\]

C)
\[{{x}^{2}}-4x-1=0\]

D)
\[\sqrt{2}{{x}^{2}}-4x+1=0\]
View Solution
33)

If the roots of the equation \[{{x}^{2}}+x+1=0\] are \[\alpha ,\beta \] and the roots of the equation \[{{x}^{2}}+px+q=0\] are \[\frac{\alpha }{\beta },\frac{\beta }{\alpha }\] then \[p\] is equal to [RPET 1987]

A)
-2

B)
-1

C)
1

D)
2
View Solution
34)

If \[\alpha ,\beta \]are the roots of the equation \[{{x}^{2}}+ax+b=0\]then the value of \[{{\alpha }^{3}}+{{\beta }^{3}}\]is equal to [RPET 1989; Pb. CET 1991]

A)
\[-({{a}^{3}}+3ab)\]

B)
\[{{a}^{3}}+3ab\]

C)
\[-{{a}^{3}}+3ab\]

D)
\[{{a}^{3}}-3ab\]
View Solution
35)

If the sum of the roots of the equation \[{{x}^{2}}+px+q=0\] is three times their difference, then which one of the following is true [Dhanbad Engg. 1968]

A)
\[9{{p}^{2}}=2q\]

B)
\[2{{q}^{2}}=9p\]

C)
\[2{{p}^{2}}=9q\]

D)
\[9{{q}^{2}}=2p\]
View Solution
36)

If the roots of the equation \[{{x}^{2}}+2mx+{{m}^{2}}-2m+6=0\] are same, then the value of m will be [MP PET 1986]

A)
3

B)
0

C)
2

D)
-1
View Solution
37)

If the roots of the given equation\[(2k+1){{x}^{2}}-(7k+3)x+k+2=0\]are reciprocal to each other, then the value of k will be [MP PET 1986]

A)
0

B)
1

C)
2

D)
3
View Solution
38)

If the roots of the equation \[a{{x}^{2}}+bx+c=0\]are \[l\] and\[2l\], then [MP PET 1986; MP PET 2002]

A)
\[{{b}^{2}}=9ac\]

B)
\[2{{b}^{2}}=9ac\]

C)
\[{{b}^{2}}=-4ac\]

D)
\[{{a}^{2}}={{c}^{2}}\]
View Solution
39)

The sum of the roots of a equation is 2 and sum of their cubes is 98, then the equation is [MP PET 1986]

A)
\[{{x}^{2}}+2x+15=0\]

B)
\[{{x}^{2}}+15x+2=0\]

C)
\[2{{x}^{2}}-2x+15=0\]

D)
\[{{x}^{2}}-2x-15=0\]
View Solution
40)

If the roots of the equation \[a{{x}^{2}}+bx+c=0\] are \[\alpha ,\beta \], then the value of \[\alpha {{\beta }^{2}}+{{\alpha }^{2}}\beta +\alpha \beta \] will be [EAMCET 1980; AMU 1984]

A)
\[\frac{c(a-b)}{{{a}^{2}}}\]

B)
0

C)
\[-\frac{bc}{{{a}^{2}}}\]

D)
None of these
View Solution
41)

If the product of roots of the equation, \[m{{x}^{2}}+6x+(2m-1)=0\] is -1, then the value of m will be [Pb. CET 1990]

A)
1

B)
- 1

C)
\[\frac{1}{3}\]

D)
\[-\frac{1}{3}\]
View Solution
42)

The roots of the equation \[{{x}^{2}}+ax+b=0\]are p, and q, then the equation whose roots are \[{{p}^{2}}q\] and \[p{{q}^{2}}\] will be [MP PET 1980]

A)
\[{{x}^{2}}+abx+{{b}^{3}}=0\]

B)
\[{{x}^{2}}-abx+{{b}^{3}}=0\]

C)
\[b{{x}^{2}}+x+a=0\]

D)
\[{{x}^{2}}+ax+ab=0\]
View Solution
43)

The equation whose roots are \[\frac{1}{3+\sqrt{2}}\]and \[\frac{1}{3-\sqrt{2}}\] is [MP PET 1994]

A)
\[7{{x}^{2}}-6x+1=0\]

B)
\[6{{x}^{2}}-7x+1=0\]

C)
\[{{x}^{2}}-6x+7=0\]

D)
\[{{x}^{2}}-7x+6=0\]
View Solution
44)

If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}-4x+1=0\] the value of \[{{\alpha }^{3}}+{{\beta }^{3}}\]is [MP PET 1994]

A)
76

B)
52

C)
-52

D)
-76
View Solution
45)

A two digit number is four times the sum and three times the product of its digits. The number is [MP PET 1994]

A)
42

B)
24

C)
12

D)
21
View Solution
46)

If \[\alpha ,\beta \]be the roots of the equation \[2{{x}^{2}}-35x+2=0\] then the value of \[{{(2\alpha -35)}^{3}}.{{(2\beta -35)}^{3}}\] is equal to [Bihar CEE 1994]

A)
1

B)
64

C)
8

D)
None of these
View Solution
47)

Let \[\alpha ,{{\alpha }^{2}}\]be the roots of \[{{x}^{2}}+x+1=0\], then the equation whose roots are \[{{\alpha }^{31}},{{\alpha }^{62}}\]is [AMU 1999]

A)
\[{{x}^{2}}-x+1=0\]

B)
\[{{x}^{2}}+x-1=0\]

C)
\[{{x}^{2}}+x+1=0\]

D)
\[{{x}^{60}}+{{x}^{30}}+1=0\]
View Solution
48)

If \[3{{p}^{2}}=5p+2\] and \[3{{q}^{2}}=5q+2\], where , then pq is equal to

A)
\[\frac{2}{3}\]

B)
\[-\frac{2}{3}\]

C)
\[\frac{3}{2}\]

D)
\[-\frac{3}{2}\]
View Solution
49)

If \[\alpha ,\beta \] are the roots of the quadratic equation \[{{x}^{2}}+bx-c=0\], then the equation whose roots are \[b\]and \[c\] is [Pb. CET 1989]

A)
\[{{x}^{2}}+\alpha x-\beta =0\]

B)
\[{{x}^{2}}-[(\alpha +\beta )+\alpha \beta ]x-\alpha \beta (\alpha +\beta )=0\]

C)
\[{{x}^{2}}+[(\alpha +\beta )+\alpha \beta ]x+\alpha \beta (\alpha +\beta )=0\]

D)
\[{{x}^{2}}+[\alpha \beta +(\alpha +\beta )]x-\alpha \beta (\alpha +\beta )=0\]
View Solution
50)

If \[\alpha \] and \[\beta \] are the roots of the equation \[a{{x}^{2}}+bx+c=0\] \[(a\ne 0;\]\[a,b,c\] being different), then \[(1+\alpha +{{\alpha }^{2}})\] \[(1+\beta +{{\beta }^{2}})\] = [DCE 2000]

A)
Zero

B)
Positive

C)
Negative

D)
None of these
View Solution
51)

If the roots of the equation \[a{{x}^{2}}+bx+c=0\] are real and of the form \[\frac{\alpha }{\alpha -1}\]and \[\frac{\alpha +1}{\alpha }\], then the value of \[{{(a+b+c)}^{2}}\]is [AMU 2000]

A)
\[{{b}^{2}}-4ac\]

B)
\[{{b}^{2}}-2ac\]

C)
\[2{{b}^{2}}-ac\]

D)
None of these
View Solution
52)

If the ratio of the roots of \[a{{x}^{2}}+2bx+c=0\] is same as the ratio of the roots of \[p{{x}^{2}}+2qx+r=0\], then [Pb. CET 1991]

A)
\[\frac{b}{ac}=\frac{q}{pr}\]

B)
\[\frac{{{b}^{2}}}{ac}=\frac{{{q}^{2}}}{pr}\]

C)
\[\frac{2b}{ac}=\frac{{{q}^{2}}}{pr}\]

D)
None of these
View Solution
53)

Roots of the equation \[{{x}^{2}}+bx-c=0(b,c>0)\]are

A)
Both positive

B)
Both negative

C)
Of opposite sign

D)
None of these
View Solution
54)

If p and q are the roots of the equation \[{{x}^{2}}+pq=(p+1)x\], then q=

A)
-1

B)
1

C)
2

D)
None of these
View Solution
55)

If the roots of \[a{{x}^{2}}+bx+c=0\] are \[\alpha ,\beta \] and the roots of \[A{{x}^{2}}+Bx+C=0\]are \[\alpha -k,\beta -k,\]then \[\frac{{{B}^{2}}-4AC}{{{b}^{2}}-4ac}\] is equal to [RPET 1999]

A)
0

B)
1

C)
\[{{\left( \frac{A}{a} \right)}^{2}}\]

D)
\[{{\left( \frac{a}{A} \right)}^{2}}\]
View Solution
56)

If p and q are the roots of \[{{x}^{2}}+px+q=0,\] then [IIT 1995;AIEEE 2002; UPSEAT 2003;RPET 2001]

A)
\[p=1,q=-2\]

B)
\[p=-2,q=1\]

C)
\[p=1,q=0\]

D)
\[p=-2,q=0\]
View Solution
57)

If one root of the quadratic equation, \[i{{x}^{2}}-2(i+1)x+(2-i)=0\]is \[2-i\], then the other root is

A)
- i

B)
i

C)
\[2+i\]

D)
\[2-i\]
View Solution
58)

If the roots of equation \[5{{x}^{2}}-7x+k=0\] are reciprocal to each other, then value of \[k\] is [RPET 1995; MP PET 2002]

A)
5

B)
2

C)
1/5

D)
1
View Solution
59)

If roots of \[{{x}^{2}}-7x+6=0\] are \[\alpha ,\beta \], then \[\frac{1}{\alpha }+\frac{1}{\beta }\]= [RPET 1995]

A)
6/7

B)
7/6

C)
7/10

D)
8/9
View Solution
60)

If \[\alpha ,\beta \] are the roots of\[{{x}^{2}}-2x+4=0\], then \[{{\alpha }^{5}}+{{\beta }^{5}}\] is equal to [EAMCET 1990]

A)
16

B)
32

C)
64

D)
None of these
View Solution
61)

If \[a{{(p+q)}^{2}}+2bpq+c=0\] and \[a{{(p+r)}^{2}}+2bpr+c=0\], then \[qr\]=

A)
\[{{p}^{2}}+\frac{c}{a}\]

B)
\[{{p}^{2}}+\frac{a}{c}\]

C)
\[{{p}^{2}}+\frac{a}{b}\]

D)
\[{{p}^{2}}+\frac{b}{a}\]
View Solution
62)

The roots of the quadratic equation \[(a+b-2c){{x}^{2}}-(2a-b-c)x+(a-2b+c)=0\] are

A)
\[a+b+c\]and \[a-b+c\]

B)
\[\frac{1}{2}\]and \[a-2b+c\]

C)
\[a-2b+c\]and \[\frac{1}{a+b-x}\]

D)
None of these
View Solution
63)

If \[\alpha ,\beta \]are the roots of \[a{{x}^{2}}+bx+c=0\], then the equation whose roots are \[2+\alpha ,\,2+\beta \]is [EAMCET 1994]

A)
\[a{{x}^{2}}+x(4a-b)+4a-2b+c=0\]

B)
\[a{{x}^{2}}+x(4a-b)+4a+2b+c=0\]

C)
\[a{{x}^{2}}+x(b-4a)+4a+2b+c=0\]

D)
\[a{{x}^{2}}+x(b-4a)+4a-2b+c=0\]
View Solution
64)

If the ratio of the roots of \[{{x}^{2}}+bx+c=0\] and \[{{x}^{2}}+qx+r=0\] be the same, then [EAMCET 1994]

A)
\[{{r}^{2}}c={{b}^{2}}q\]

B)
\[{{r}^{2}}b={{c}^{2}}q\]

C)
\[r{{b}^{2}}=c{{q}^{2}}\]

D)
\[r{{c}^{2}}=b{{q}^{2}}\]
View Solution
65)

If one root of \[{{x}^{2}}-x-k=0\] is square of the other, then k = [EAMCET 1986, 1987]

A)
\[2\pm \sqrt{3}\]

B)
\[3\pm \sqrt{2}\]

C)
\[2\pm \sqrt{5}\]

D)
\[5\pm \sqrt{2}\]
View Solution
66)

If \[3+4i\] is a root of the equation \[{{x}^{2}}+px+q=0\] (p, q are real numbers), then [EAMCET 1985]

A)
\[p=6,q=25\]

B)
\[p=6,q=1\]

C)
\[p=-6,q=-7\]

D)
\[p=-6,q=25\]
View Solution
67)

If the sum of the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\]is equal to the sum of the squares of their reciprocals, then \[\frac{{{b}^{2}}}{ac}+\frac{bc}{{{a}^{2}}}=\] [BIT Ranchi 1996]

A)
2

B)
-2

C)
1

D)
-1
View Solution
68)

If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}-6x+a=0\] and satisfy the relation \[3\alpha +2\beta =16,\]then the value of a is

A)
- 8

B)
8

C)
- 16

D)
9
View Solution
69)

If \[\alpha ,\beta \] are the roots of the equation \[l{{x}^{2}}+mx+n=0\], then the equation whose roots are \[{{\alpha }^{3}}\beta \] and \[\alpha {{\beta }^{3}}\] is [MP PET 1997]

A)
\[{{l}^{4}}{{x}^{2}}-nl({{m}^{2}}-2nl)x+{{n}^{4}}=0\]

B)
\[{{l}^{4}}{{x}^{2}}+nl({{m}^{2}}-2nl)x+{{n}^{4}}=0\]

C)
\[{{l}^{4}}{{x}^{2}}+nl({{m}^{2}}-2nl)x-{{n}^{4}}=0\]

D)
\[{{l}^{4}}{{x}^{2}}-nl({{m}^{2}}+2nl)x+{{n}^{4}}=0\]
View Solution
70)

If the roots of equation \[{{x}^{2}}+px+q=0\] differ by 1, then [MP PET 1999]

A)
\[{{p}^{2}}=4q\]

B)
\[{{p}^{2}}=4q+1\]

C)
\[{{p}^{2}}=4q-1\]

D)
None of these
View Solution
71)

The harmonic mean of the roots of the equation \[(5+\sqrt{2}){{x}^{2}}-(4+\sqrt{5})x+8+2\sqrt{5}=0\] is [IIT 1999; MP PET 2000]

A)
2

B)
4

C)
6

D)
8
View Solution
72)

If the roots of \[{{x}^{2}}-bx+c=0\] are two consecutive integers, then \[{{b}^{2}}-4c\] is [RPET 1991; Kurukshetra CEE 1998; AIEEE 2005]

A)
1

B)
2

C)
3

D)
4
View Solution
73)

If \[\alpha \]and \[\beta \] are roots of the equation \[A{{x}^{2}}+Bx+C=0\], then value of \[{{\alpha }^{3}}+{{\beta }^{3}}\] is [RPET 1996; DCE 2005]

A)
\[\frac{3ABC-{{B}^{3}}}{{{A}^{3}}}\]

B)
\[\frac{3ABC+{{B}^{3}}}{{{A}^{3}}}\]

C)
\[\frac{{{B}^{3}}-3ABC}{{{A}^{3}}}\]

D)
\[\frac{{{B}^{3}}-3ABC}{{{B}^{3}}}\]
View Solution
74)

If \[\alpha ,\beta \] are the roots of the equation \[{{x}^{2}}-(1+{{n}^{2}})x+\frac{1}{2}(1+{{n}^{2}}+{{n}^{4}})=0\]then the value of \[{{\alpha }^{2}}+{{\beta }^{2}}\] is [RPET 1996]

A)
\[2n\]

B)
\[{{n}^{3}}\]

C)
\[{{n}^{2}}\]

D)
\[2{{n}^{2}}\]
View Solution
75)

The value of p for which one root of the equation \[{{x}^{2}}-30x+p=0\]is the square of the other, are [Roorkee Qualifying 1998]

A)
125 only

B)
125 and \[-216\]

C)
125 and 215

D)
216 only
View Solution
76)

What is the sum of the squares of roots of \[{{x}^{2}}-3x+1=0\] [Karnataka CET 1998]

A)
5

B)
7

C)
9

D)
10
View Solution
77)

Sum of roots is \[-1\] and sum of their reciprocals is \[\frac{1}{6}\], then equation is [Karnataka CET 1998]

A)
\[{{x}^{2}}+x-6=0\]

B)
\[{{x}^{2}}-x+6=0\]

C)
\[6{{x}^{2}}+x+1=0\]

D)
\[{{x}^{2}}-6x+1=0\]
View Solution
78)

If the sum of the roots of the equation \[{{x}^{2}}+px+q=0\] is equal to the sum of their squares, then [Pb. CET 1999]

A)
\[{{p}^{2}}-{{q}^{2}}=0\]

B)
\[{{p}^{2}}+{{q}^{2}}=2q\]

C)
\[{{p}^{2}}+p=2q\]

D)
None of these
View Solution
79)

If a, b are roots of \[{{x}^{2}}-3x+1=0,\] then the equation whose roots are \[\frac{1}{\alpha -2},\frac{1}{\beta -2}\] is [RPET 1999]

A)
\[{{x}^{2}}+x-1=0\]

B)
\[{{x}^{2}}+x+1=0\]

C)
\[{{x}^{2}}-x-1=0\]

D)
None of these
View Solution
80)

The equation formed by decreasing each root of \[a{{x}^{2}}+bx+c=0\] by 1 is \[2{{x}^{2}}+8x+2=0,\] then [EAMCET 2000]

A)
a = - b

B)
b = - c

C)
c = - a

D)
b = a + c
View Solution
81)

If a, b are the roots of \[9{{x}^{2}}+6x+1=0,\] then the equation with the roots \[\frac{1}{\alpha },\,\frac{1}{\beta }\] is [EAMCET 2000]

A)
\[2{{x}^{2}}+3x+18=0\]

B)
\[{{x}^{2}}+6x-9=0\]

C)
\[{{x}^{2}}+6x+9=0\]

D)
\[{{x}^{2}}-6x+9=0\]
View Solution
82)

If a and b are the roots of \[6{{x}^{2}}-6x+1=0,\] then the value of \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]\] \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]+\frac{1}{2}\left[ \,a+b\beta +c{{\beta }^{2}}+d{{\beta }^{3}}\, \right]\] is [RPET 2000]

A)
\[\frac{1}{4}(a+b+c+d)\]

B)
\[\frac{a}{1}+\frac{b}{2}+\frac{c}{3}+\frac{d}{4}\]

C)
\[\frac{a}{2}-\frac{b}{2}+\frac{c}{3}-\frac{d}{4}\]

D)
None of these
View Solution
83)

Given that \[\tan \alpha \] and \[\tan \beta \] are the roots of \[{{x}^{2}}-px+q=0,\] then the value of \[{{\sin }^{2}}(\alpha +\beta )=\][RPET 2000]

A)
\[\frac{{{p}^{2}}}{{{p}^{2}}+{{(1-q)}^{2}}}\]

B)
\[\frac{{{p}^{2}}}{{{p}^{2}}+{{q}^{2}}}\]

C)
\[\frac{{{q}^{2}}}{{{p}^{2}}+{{(1-q)}^{2}}}\]

D)
\[\frac{{{p}^{2}}}{{{(p+q)}^{2}}}\]
View Solution
84)

If the roots of the quadratic equation \[\frac{x-m}{mx+1}=\frac{x+n}{nx+1}\] are reciprocal to each other, then [MP PET 2001]

A)
\[n=0\]

B)
\[m=n\]

C)
\[m+n=1\]

D)
\[{{m}^{2}}+{{n}^{2}}=1\]
View Solution
85)

If the roots of the equation \[{{x}^{2}}-5x+16=0\] are \[\alpha ,\beta \] and the roots of equation \[{{x}^{2}}+px+q=0\] are \[{{\alpha }^{2}}+{{\beta }^{2}},\] \[\frac{\alpha \beta }{2},\] then [MP PET 2001]

A)
p = 1, q = - 56

B)
p = - 1, q = - 56

C)
p = 1, q = 56

D)
p = - 1, q = 56
View Solution
86)

The value of \[k\] for which one of the roots of \[{{x}^{2}}-x+3k=0\] is double of one of the roots of \[{{x}^{2}}-x+k=0\] is [UPSEAT 2001]

A)
1

B)
- 2

C)
2

D)
None of these
View Solution
87)

Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}-x+p=0\] and \[\gamma ,\delta \] be the roots of \[{{x}^{2}}-4x+q=0\]. If \[\alpha ,\beta ,\gamma ,\delta \] are in G.P., then integral values of \[p,\,q\] are respectively [IIT Screening 2001]

A)
- 2, - 32

B)
- 2, 3

C)
- 6, 3

D)
- 6, - 32
View Solution
88)

If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7, then the equation is [AMU 2001]

A)
\[5{{x}^{2}}-16x+7\]= 0

B)
\[7{{x}^{2}}-16x+5=0\]

C)
\[7{{x}^{2}}-16x+8=0\]

D)
\[3{{x}^{2}}-12x+7=0\]
View Solution
89)

If \[1-i\] is a root of the equation \[{{x}^{2}}-ax+b=0\], then \[b=\] [EAMCET 2002]

A)
- 2

B)
- 1

C)
1

D)
2
View Solution
90)

If 3 is a root of \[{{x}^{2}}+kx-24=0,\] it is also a root of [EAMCET 2002]

A)
\[{{x}^{2}}+5x+k=0\]

B)
\[{{x}^{2}}-5x+k=0\]

C)
\[{{x}^{2}}-kx+6=0\]

D)
\[{{x}^{2}}+kx+24=0\]
View Solution
91)

If \[\alpha \ne \beta \] but \[{{\alpha }^{2}}=5\alpha -3\] and \[{{\beta }^{2}}=5\beta -3\], then the equation whose roots are \[\alpha /\beta \] and \[\beta /\alpha \] is [AIEEE 2002]

A)
\[3{{x}^{2}}-25x+3=0\]

B)
\[{{x}^{2}}+5x-3=0\]

C)
\[{{x}^{2}}-5x+3=0\]

D)
\[3{{x}^{2}}-19x+3=0\]
View Solution
92)

Difference between the corresponding roots of \[{{x}^{2}}+ax+b=0\] and \[{{x}^{2}}+bx+a=0\] is same and \[a\ne b\], then [AIEEE 2002]

A)
\[a+b+4=0\]

B)
\[a+b-4=0\]

C)
\[a-b-4=0\]

D)
\[a-b+4=0\]
View Solution
93)

Product of real roots of the equation \[{{t}^{2}}{{x}^{2}}+|x|+\,9=0\] [AIEEE 2002]

A)
Is always positive

B)
Is always negative

C)
Does not exist

D)
None of these
View Solution
94)

If the roots of the equation \[12{{x}^{2}}-mx+5=0\] are in the ratio 2 : 3, then m = [RPET 2002]

A)
\[5\sqrt{10}\]

B)
\[3\sqrt{10}\]

C)
\[2\sqrt{10}\]

D)
None of these
View Solution
95)

If one root of the equation \[{{x}^{2}}+px+q=0\] is \[2+\sqrt{3}\], then values of p and q are [UPSEAT 2002]

A)
- 4, 1

B)
4, - 1

C)
2, \[\sqrt{3}\]

D)
\[-2,\,\,-\sqrt{3}\]
View Solution
96)

The condition that one root of the equation \[a{{x}^{2}}+bx+c=0\]is three times the other is [DCE 2002]

A)
\[{{b}^{2}}=8ac\]

B)
\[3{{b}^{2}}+16ac=0\]

C)
\[3{{b}^{2}}=16ac\]

D)
\[{{b}^{2}}+3ac=0\]
View Solution
97)

The equation whose roots are reciprocal of the roots of the equation \[3{{x}^{2}}-20x+17=0\] is [DCE 2002]

A)
\[3{{x}^{2}}+20x-17=0\]

B)
\[17{{x}^{2}}-20x+3=0\]

C)
\[17{{x}^{2}}+20x+3=0\]

D)
None of these
View Solution
98)

If \[\alpha ,\beta \] are the roots of the equation \[{{x}^{2}}+2x+4=0,\] then \[\frac{1}{{{\alpha }^{3}}}+\frac{1}{{{\beta }^{3}}}\] is equal to [Kerala (Engg.) 2002]

A)
\[-\frac{1}{2}\]

B)
\[\frac{1}{2}\]

C)
32

D)
\[\frac{1}{4}\]
View Solution
99)

The equation of the smallest degree with real coefficients having \[1+i\] as one of the root is [Kerala (Engg.) 2002]

A)
\[{{x}^{2}}+x+1=0\]

B)
\[{{x}^{2}}-2x+2=0\]

C)
\[{{x}^{2}}+2x+2=0\]

D)
\[{{x}^{2}}+2x-2=0\]
View Solution
100)

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation [AIEEE 2004]

A)
\[{{x}^{2}}-18x-16=0\]

B)
\[{{x}^{2}}-18x+16=0\]

C)
\[{{x}^{2}}+18x-16=0\]

D)
\[{{x}^{2}}+18x+16=0\]
View Solution
101)

If \[\alpha ,\beta \] are the roots of the equation \[6{{x}^{2}}-5x+1=0\]. Then the value of \[{{\tan }^{-1}}\alpha +{{\tan }^{-1}}\beta \]is [MP PET 2004]

A)
\[\pi /4\]

B)
1

C)
0

D)
\[\pi /2\]
View Solution
102)

If a and b are roots of \[{{x}^{2}}-px+q=0\], then \[\frac{1}{a}+\frac{1}{b}=\] [Orissa JEE 2004]

A)
\[\frac{1}{p}\]

B)
\[\frac{1}{q}\]

C)
\[\frac{1}{2p}\]

D)
\[\frac{p}{q}\]
View Solution
103)

If one root of the equation \[{{x}^{2}}+px+q=0\]is the square of the other, then [IIT Screening 2004]

A)
\[{{p}^{3}}+{{q}^{2}}-q(3p+1)=0\]

B)
\[{{p}^{3}}+{{q}^{2}}+q(1+3p)=0\]

C)
\[{{p}^{3}}+{{q}^{2}}+q(3p-1)=0\]

D)
\[{{p}^{3}}+{{q}^{2}}+q(1-3p)=0\]
View Solution
104)

If one of the roots of equation \[{{x}^{2}}+ax+3=0\] is 3 and one of the roots of the equation \[{{x}^{2}}+ax+b=0\] is three times the other root, then the value of b is equal to [J & K 2005]

A)
3

B)
4

C)
2

D)
1
View Solution
105)

If \[\alpha ,\beta \] are the roots of \[a{{x}^{2}}+bx+c=0\] and \[\alpha +\beta ,\] \[\,{{\alpha }^{2}}+{{\beta }^{2}},\] \[\,{{\alpha }^{3}}+{{\beta }^{3}}\] are in G.P., where \[\Delta ={{b}^{2}}-4ac\], then [IIT Screening 2005]

A)
\[\Delta \ne 0\]

B)
\[b\Delta =0\]

C)
\[cb\ne 0\]

D)
\[c\Delta =0\]
View Solution
106)

\[2{{x}^{2}}-(p+1)x+(p-1)=0\]. If \[\alpha -\beta =\alpha \beta \], then what is the value of p [Orissa JEE 2005]

A)
1

B)
2

C)
3

D)
- 2
View Solution
107)

If \[3{{p}^{2}}=5p+2\] and \[3{{q}^{2}}=5q+2\] where \[p\ne q\], then the equation whose roots are \[3p-2q\] and \[3q-2p\] is [Kerala (Engg.) 2005]

A)
\[3{{x}^{2}}-5x-100=0\]

B)
\[5{{x}^{2}}+3x+100=0\]

C)
\[3{{x}^{2}}-5x+100=0\]

D)
\[5{{x}^{2}}-3x-100=0\] \[5{{x}^{2}}-3x-100=0\]
View Solution

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

done Relation between roots and coefficients Total Questions - 107

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Relation between roots and coefficients Total Questions - 107