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question_answer1)
The following system of equation\[3x-2y+z=0\], \[\lambda x-14y+15z=0\], \[x+2y-3z=0\]has a solution other than \[x=y=z=0\]for \[\lambda \] equal to [MP PET 1990]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
5 done
clear
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question_answer2)
If \[2x+3y-5z=7,x+y+z=6\], \[3x-4y+2z=1,\] then x = [MP PET 1987]
A)
\[\left| \,\begin{matrix} 2 & -5 & 7 \\ 1 & 1 & 6 \\ 3 & 2 & 1 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 7 & 3 & -5 \\ 6 & 1 & 1 \\ 1 & -4 & 2 \\ \end{matrix}\, \right|\] done
clear
B)
\[\left| \,\begin{matrix} -7 & 3 & -5 \\ -6 & 1 & 1 \\ -1 & -4 & 2 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 2 & 3 & -5 \\ 1 & 1 & 1 \\ 3 & -4 & 2 \\ \end{matrix}\, \right|\] done
clear
C)
\[\left| \,\begin{matrix} 7 & 3 & -5 \\ 6 & 1 & 1 \\ 1 & -4 & 2 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 2 & 3 & -5 \\ 1 & 1 & 1 \\ 3 & -4 & 2 \\ \end{matrix}\, \right|\] done
clear
D)
None of these done
clear
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question_answer3)
\[x+ky-z=0,3x-ky-z=0\]and \[x-3y+z=0\] has non-zero solution for k = [IIT 1988]
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_answer4)
The number of solutions of equations \[x+y-z=0\], \[3x-y-z=0,x-3y+z=0\] is [MP PET 1992]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
Infinite done
clear
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question_answer5)
If \[x+y-z=0,\,3x-\alpha y-3z=0,\,\,x-3y+z=0\] has non zero solution, then \[\alpha =\] [MP PET 1990]
A)
- 1 done
clear
B)
0 done
clear
C)
1 done
clear
D)
- 3 done
clear
View Solution play_arrow
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question_answer6)
The number of solutions of the equations \[x+4y-z=0,\] \[3x-4y-z=0,\,x-3y+z=0\] is [MP PET 1992]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
Infinite done
clear
View Solution play_arrow
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question_answer7)
If \[\Delta (x)=\left| \,\begin{matrix} {{x}^{n}} & \sin x & \cos x \\ n! & \sin \frac{n\pi }{2} & \cos \frac{n\pi }{2} \\ a & {{a}^{2}} & {{a}^{3}} \\ \end{matrix}\, \right|,\] then the value of \[\frac{{{d}^{n}}}{d{{x}^{n}}}[\Delta (x)]\] at \[x=0\]is
A)
- 1 done
clear
B)
0 done
clear
C)
1 done
clear
D)
Dependent of a done
clear
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question_answer8)
The value of a for which the system of equations \[{{a}^{3}}x+{{(a+1)}^{3}}y+{{(a+2)}^{3}}z=0,\]\[ax+(a+1)y+(a+2)z=0,\] \[x+y+z=0,\]has a nonzero solution is [Pb. CET 2000]
A)
- 1 done
clear
B)
0 done
clear
C)
1 done
clear
D)
None of these done
clear
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question_answer9)
If \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z=0,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z=0\] \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z=0\] and \[\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|=0,\]then the given system has [Roorkee 1990]
A)
One trivial and one non-trivial solution done
clear
B)
No solution done
clear
C)
One solution done
clear
D)
Infinite solution done
clear
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question_answer10)
The value of k for which the set of equations\[x+ky+3z=0,\]\[3x+ky-2z=0,\]\[2x+3y-4z=0\]has a nontrivial solution over the set of rationals is [Kurukshetra CEE 1996]
A)
15 done
clear
B)
31/2 done
clear
C)
16 done
clear
D)
33/2 done
clear
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question_answer11)
If the system of equations, \[x+2y-3z=1\], \[(k+3)z=3,\] \[(2k+1)x+z=0\]is inconsistent, then the value of k is [Roorkee 2000]
A)
- 3 done
clear
B)
1/2 done
clear
C)
0 done
clear
D)
2 done
clear
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question_answer12)
If the system of equations \[x-ky-z=0\], \[kx-y-z=0\] and \[x+y-z=0\] has a non zero solution, then the possible value of k are [IIT Screening 2000]
A)
- 1, 2 done
clear
B)
1, 2 done
clear
C)
0, 1 done
clear
D)
- 1, 1 done
clear
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question_answer13)
Set of equations \[a+b-2c=0,\]\[2a-3b+c=0\] and \[a-5b+4c=\alpha \] is consistent for \[\alpha \]equal to [Orissa JEE 2004]
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_answer14)
\[{{x}_{1}}+2{{x}_{2}}+3{{x}_{3}}=a2{{x}_{1}}+3{{x}_{2}}+{{x}_{3}}=\] \[b3{{x}_{1}}+{{x}_{2}}+2{{x}_{3}}=c\] this system of equations has [Orissa JEE 2004]
A)
Infinite solution done
clear
B)
No solution done
clear
C)
Unique solution done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer15)
The system of equations \[\lambda x+y+z=0,\] \[-x+\lambda y+z=0,\] \[-x-y+\lambda z=0\], will have a non zero solution if real values of \[\lambda \]are given by [IIT 1984]
A)
0 done
clear
B)
1 done
clear
C)
3 done
clear
D)
\[\sqrt{3}\] done
clear
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question_answer16)
The value of \[\sum\limits_{n=1}^{N}{{{U}_{n}},}\] if \[{{U}_{n}}=\left| \,\begin{matrix} n & 1 & 5 \\ {{n}^{2}} & 2N+1 & 2N+1 \\ {{n}^{3}} & 3{{N}^{2}} & 3N \\ \end{matrix}\, \right|\] is [MNR 1994]
A)
0 done
clear
B)
1 done
clear
C)
- 1 done
clear
D)
None of these done
clear
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question_answer17)
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},........,{{a}_{n}},......\] are in G.P. and \[{{a}_{i}}>0\]for each i, then the value of the determinant \[\Delta =\left| \,\begin{matrix} \log {{a}_{n}} & \log {{a}_{n+2}} & \log {{a}_{n+4}} \\ \log {{a}_{n+6}} & \log {{a}_{n+8}} & \log {{a}_{n+10}} \\ \log {{a}_{n+12}} & \log {{a}_{n+14}} & \log {{a}_{n+16}} \\ \end{matrix} \right|\] is equal to
A)
1 done
clear
B)
2 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer18)
If \[{{D}_{r}}=\left| \begin{matrix} {{2}^{r-1}} & {{2.3}^{r-1}} & {{4.5}^{r-1}} \\ x & y & z \\ {{2}^{n}}-1 & {{3}^{n}}-1 & {{5}^{n}}-1 \\ \end{matrix} \right|\], then the value of \[\sum\limits_{r=1}^{n}{{{D}_{r}}=}\]
A)
1 done
clear
B)
- 1 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
If \[a_{i}^{2}+b_{i}^{2}+c_{i}^{2}=1,\,\,(i=1,2,3)\] and \[{{a}_{i}}{{a}_{j}}+{{b}_{i}}{{b}_{j}}+{{c}_{i}}{{c}_{j}}=0\] \[(i\ne j,i,j=1,2,3)\] then the value of \[{{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix}\, \right|}^{2}}\] is [AMU 1994; DCE 2001]
A)
0 done
clear
B)
1/2 done
clear
C)
1 done
clear
D)
2 done
clear
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question_answer20)
If the system of equation\[3x-2y+z=0\], \[\lambda x-14y+15z=0\], \[x+2y+3z=0\]have a non-trivial solution, then \[\lambda =\] [EAMCET 1993]
A)
5 done
clear
B)
- 5 done
clear
C)
- 29 done
clear
D)
29 done
clear
View Solution play_arrow
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question_answer21)
The system of linear equations \[x+y+z=2\], \[2x+y-z=3,\] \[3x+2y+kz=4\]has a unique solution if [EAMCET 1994; DCE 2000]
A)
\[k\ne 0\] done
clear
B)
\[-1<k<1\] done
clear
C)
\[-2<k<2\] done
clear
D)
\[k=0\] done
clear
View Solution play_arrow
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question_answer22)
The system of equations \[{{x}_{1}}-{{x}_{2}}+{{x}_{3}}=2,\] \[\,3{{x}_{1}}-{{x}_{2}}+2{{x}_{3}}=-6\]and \[3{{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-18\] has [AMU 2001]
A)
No solution done
clear
B)
Exactly one solution done
clear
C)
Infinite solutions done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
The number of values of k for which the system of equations \[(k+1)x+8y=4k,\] \[kx+(k+3)y=3k-1\] has infinitely many solutions, is [IIT Screening 2002]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
Infinite done
clear
View Solution play_arrow
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question_answer24)
If \[\left| \,\begin{matrix} 1+ax & 1+bx & 1+cx \\ 1+{{a}_{1}}x & 1+{{b}_{1}}x & 1+{{c}_{1}}x \\ 1+{{a}_{2}}x & 1+{{b}_{2}}x & 1+{{c}_{2}}x \\ \end{matrix}\, \right|,\] \[={{A}_{0}}+{{A}_{1}}x+{{A}_{2}}{{x}^{2}}+{{A}_{3}}{{x}^{3}}\] then \[{{A}_{1}}\] is equal to [AMU 2002]
A)
abc done
clear
B)
0 done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
The existence of the unique solution of the system \[x+y+z=\lambda ,\] \[5x-y+\mu z=10\], \[2x+3y-z=6\] depends on [Kurukshetra CEE 2002]
A)
\[\mu \]only done
clear
B)
\[\lambda \]only done
clear
C)
\[\lambda \]and \[\mu \] both done
clear
D)
Neither \[\lambda \]nor \[\mu \] done
clear
View Solution play_arrow
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question_answer26)
The system of equations \[x+y+z=2\],\[3x-y+2z=6\] and \[3x+y+z=-18\] has [Kurukshetra CEE 2002]
A)
A unique solution done
clear
B)
No solutions done
clear
C)
An infinite number of solutions done
clear
D)
Zero solution as the only solution done
clear
View Solution play_arrow
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question_answer27)
If \[a>0\]and discriminant of \[a{{x}^{2}}+2bx+c\]is negative, then \[\left| \,\begin{matrix} a & b & ax+b \\ b & c & bx+c \\ ax+b & bx+c & 0 \\ \end{matrix}\, \right|\] is [AIEEE 2002]
A)
Positive done
clear
B)
\[(ac-{{b}^{2}})(a{{x}^{2}}+2bx+c)\] done
clear
C)
Negative done
clear
D)
0 done
clear
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question_answer28)
For what value of \[\lambda \], the system of equations \[x+y+z=6,x+2y+3z=10,\]\[x+2y+\lambda z=12\]is inconsistent [AIEEE 2002]
A)
\[\lambda =1\] done
clear
B)
\[\lambda =2\] done
clear
C)
\[\lambda =-2\] done
clear
D)
\[\lambda =3\] done
clear
View Solution play_arrow
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question_answer29)
If x is a positive integer, then \[\Delta =\left| \,\begin{matrix} x! & (x+1)! & (x+2)! \\ (x+1)! & (x+2)! & (x+3)! \\ (x+2)! & (x+3)! & (x+4)! \\ \end{matrix}\, \right|\] is equal to
A)
\[2(x!)(x+1)!\] done
clear
B)
\[2(x!)(x+1)!(x+2)!\] done
clear
C)
\[2(x!)(x+3)!\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer30)
If the system of equations \[x+ay=0,\]\[az+y=0\] and \[ax+z=0\] has infinite solutions, then the value of a is [IIT Screening 2003]
A)
-1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
No real values done
clear
View Solution play_arrow
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question_answer31)
The values of \[x,y,z\] in order of the system of equations \[3x+y+2z=3,\] \[2x-3y-z=-3\], \[x+2y+z=4,\] are [MP PET 2003]
A)
2, 1, 5 done
clear
B)
1, 1, 1 done
clear
C)
1, -2, -1 done
clear
D)
1, 2, -1 done
clear
View Solution play_arrow
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question_answer32)
The value of \[\lambda \] for which the system of equations \[2x-y-z=12,\]\[x-2y+z=-4,\]\[x+y+\lambda z=4\] has no solution is [IIT Screening 2004]
A)
3 done
clear
B)
- 3 done
clear
C)
2 done
clear
D)
- 2 done
clear
View Solution play_arrow
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question_answer33)
\[2x+3y+4z=9\],\[4x+9y+3z=10,\]\[5x+10y+5z=1\]then the value of x is [UPSEAT 2002]
A)
\[\left| \,\begin{matrix} 9 & 3 & 4 \\ 10 & 9 & 3 \\ 11 & 10 & 5 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 2 & 3 & 4 \\ 4 & 9 & 3 \\ 5 & 10 & 5 \\ \end{matrix}\, \right|\] done
clear
B)
\[\left| \,\begin{matrix} 9 & 4 & 3 \\ 10 & 3 & 9 \\ 11 & 5 & 10 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 2 & 3 & 4 \\ 4 & 9 & 3 \\ 5 & 10 & 5 \\ \end{matrix}\, \right|\] done
clear
C)
\[\left| \,\begin{matrix} 9 & 4 & 9 \\ 10 & 3 & 3 \\ 11 & 5 & 10 \\ \end{matrix}\, \right|\div \left| \,\begin{matrix} 3 & 2 & 4 \\ 9 & 4 & 3 \\ 10 & 5 & 5 \\ \end{matrix}\, \right|\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer34)
The system of equations\[x+y+z=6\], \[x+2y+3z=10,x+2y+\lambda z=\mu \], has no solution for [Orissa JEE 2003]
A)
\[\lambda \ne 3,\mu =10\] done
clear
B)
\[\lambda =3,\mu \ne 10\] done
clear
C)
\[\lambda \ne 3,\mu \ne 10\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer35)
If \[a,b,c\] are respectively the \[{{p}^{th}},{{q}^{th}}{{r}^{th}}\]terms of an \[A.P.,\] the \[\left| \,\begin{matrix} a & p & 1 \\ b & q & 1 \\ c & r & 1 \\ \end{matrix}\, \right|=\] [Kerala (Engg.) 2002]
A)
1 done
clear
B)
-1 done
clear
C)
0 done
clear
D)
pqr done
clear
View Solution play_arrow
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question_answer36)
If the system of linear equation \[x+2ay+az=0,\] \[x+3by+bz=0,\] \[x+4cy+cz=0\]has a non zero solution, then \[a,b,c\] [AIEEE 2003]
A)
Are in A.P. done
clear
B)
Are in G. P. done
clear
C)
Are in H. P. done
clear
D)
Satisfy \[a+2b+3c=0\] done
clear
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question_answer37)
The system of equations\[\begin{align} & \alpha x+y+z=\alpha -1 \\ & x+\alpha y+z=\alpha -1 \\ & x+y+\alpha z=\alpha -1 \\ \end{align}\] has no solution, if \[\alpha \] is [AIEEE 2005]
A)
Not - 2 done
clear
B)
1 done
clear
C)
- 2 done
clear
D)
Either - 2 or 1 done
clear
View Solution play_arrow
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question_answer38)
If a system of the equation \[{{(\alpha +1)}^{3}}x+{{(\alpha +2)}^{3}}y-{{(\alpha +3)}^{3}}=0\] and \[(\alpha +1)x+(\alpha +2)y-(\alpha +3)=0,x+y-1=0\]is constant. what is the value of \[\alpha \] [Orissa JEE 2005]
A)
1 done
clear
B)
0 done
clear
C)
- 3 done
clear
D)
- 2 done
clear
View Solution play_arrow