-
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
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
View Solution play_arrow
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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
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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
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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
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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
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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
View Solution play_arrow
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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
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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
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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
View Solution play_arrow
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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
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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
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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
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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
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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
View Solution play_arrow
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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
View Solution play_arrow
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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