JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done System of linear equations, Some special determinants, differentiation and integration of determinants Total Questions - 38

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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

• 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
  View Solution play_arrow

• 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

• 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

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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

• 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

• 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
  View Solution play_arrow

• 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
  View Solution play_arrow

• 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

• 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

• 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
  View Solution play_arrow

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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
  View Solution play_arrow

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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
  View Solution play_arrow

• 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

• 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