8666 + Mcqs in Mathematics Page 39 McqOptions

1901.	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.	\[{{x}^{2}}-6x+9=0\]
Answer» D. \[{{x}^{2}}-6x+9=0\]

Discussion

1902.	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.	\[(-4,\,\,-7)\]
Answer» B. \[(4,\,-7)\]

Discussion

1903.	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
Answer» D. None of these

Discussion

1904.	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
Answer» D. None of these

Discussion

1905.	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
Answer» C. 125 and 215

Discussion

1906.	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
Answer» C. 9

Discussion

1907.	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.	\[2{{n}^{2}}\]
Answer» D. \[2{{n}^{2}}\]

Discussion

1908.	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.	\[\frac{{{B}^{3}}-3ABC}{{{B}^{3}}}\]
Answer» B. \[\frac{3ABC+{{B}^{3}}}{{{A}^{3}}}\]

Discussion

1909.	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
Answer» B. 2

Discussion

1910.	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
Answer» C. \[{{p}^{2}}=4q-1\]

Discussion

1911.	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
Answer» C. 6

Discussion

1912.	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
Answer» C. \[{{x}^{2}}-4abx+{{({{a}^{2}}-{{b}^{2}})}^{2}}=0\]

Discussion

1913.	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
Answer» C. -16

Discussion

1914.	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.	\[p=-6,q=25\]
Answer» E.

Discussion

1915.	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
Answer» B. -2

Discussion

1916.	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.	\[5\pm \sqrt{2}\]
Answer» D. \[5\pm \sqrt{2}\]

Discussion

1917.	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.	\[r{{c}^{2}}=b{{q}^{2}}\]
Answer» D. \[r{{c}^{2}}=b{{q}^{2}}\]

Discussion

1918.	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.	\[a{{x}^{2}}+x(b-4a)+4a-2b+c=0\]
Answer» E.

Discussion

1919.	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
Answer» E.

Discussion

1920.	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
Answer» C. 64

Discussion

1921.	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
Answer» B. \[{{(b{b}'-a{a}')}^{2}}=(c{a}'-b{c}')(a{b}'-b{c}')\]

Discussion

1922.	If roots of \[{{x}^{2}}-7x+6=0\] are \[\alpha ,\beta \], then \[\frac{1}{\alpha }+\frac{1}{\beta }\]= [RPET 1995]
A.	44383
B.	44354
C.	44476
D.	44447
Answer» C. 44476

Discussion

1923.	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.
B.	i
C.	\[2+i\]
D.	\[2-i\]
Answer» B. i

Discussion

1924.	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.	\[p=-2,q=0\]
Answer» B. \[p=-2,q=1\]

Discussion

1925.	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.	\[{{\left( \frac{a}{A} \right)}^{2}}\]
Answer» C. \[{{\left( \frac{A}{a} \right)}^{2}}\]

Discussion

1926.	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
Answer» C. 2

Discussion

1927.	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
Answer» C. \[\frac{2b}{ac}=\frac{{{q}^{2}}}{pr}\]

Discussion

1928.	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
Answer» B. \[{{b}^{2}}-2ac\]

Discussion

1929.	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
Answer» C. Negative

Discussion

1930.	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.	\[{{x}^{2}}+[\alpha \beta +(\alpha +\beta )]x-\alpha \beta (\alpha +\beta )=0\]
Answer» D. \[{{x}^{2}}+[\alpha \beta +(\alpha +\beta )]x-\alpha \beta (\alpha +\beta )=0\]

Discussion

1931.	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
Answer» B. \[ab{{x}^{2}}+(a+c)bx+{{(a+c)}^{2}}=0\]

Discussion

1932.	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.	\[-\frac{3}{2}\]
Answer» C. \[\frac{3}{2}\]

Discussion

1933.	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.	\[{{x}^{60}}+{{x}^{30}}+1=0\]
Answer» D. \[{{x}^{60}}+{{x}^{30}}+1=0\]

Discussion

1934.	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
Answer» C. 8

Discussion

1935.	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
Answer» C. -52

Discussion

1936.	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.	\[{{x}^{2}}-7x+6=0\]
Answer» B. \[6{{x}^{2}}-7x+1=0\]

Discussion

1937.	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.	\[{{x}^{2}}+ax+ab=0\]
Answer» B. \[{{x}^{2}}-abx+{{b}^{3}}=0\]

Discussion

1938.	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.	\[-\frac{1}{3}\]
Answer» D. \[-\frac{1}{3}\]

Discussion

1939.	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
Answer» B. 0

Discussion

1940.	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
Answer» D. None of these

Discussion

1941.	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.	\[{{x}^{2}}-2x-15=0\]
Answer» E.

Discussion

1942.	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.	\[{{a}^{2}}={{c}^{2}}\]
Answer» C. \[{{b}^{2}}=-4ac\]

Discussion

1943.	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
Answer» C. 2

Discussion

1944.	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.	\[9{{q}^{2}}=2p\]
Answer» D. \[9{{q}^{2}}=2p\]

Discussion

1945.	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
Answer» B. 0

Discussion

1946.	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.	\[{{a}^{3}}-3ab\]
Answer» D. \[{{a}^{3}}-3ab\]

Discussion

1947.	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
Answer» D. 2

Discussion

1948.	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.	\[\sqrt{2}{{x}^{2}}-4x+1=0\]
Answer» B. \[{{x}^{2}}+4x+1=0\]

Discussion

1949.	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.	\[{{x}^{2}}+4x+1=0\]
Answer» C. \[{{x}^{2}}+4x-1=0\]

Discussion

1950.	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
Answer» C. \[\frac{{{B}^{2}}-4AC}{{{A}^{2}}}\]

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

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]

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]

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]

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]

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]

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

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]

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]

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]

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

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]

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

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

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

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]

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

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

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]

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

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

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]

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

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

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

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]

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

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]

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]

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]

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]

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]

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

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]

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]

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]

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

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]

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]

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]

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]

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

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

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]

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]

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]

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]

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]

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

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

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]