JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Expansion of determinants, Solution of equation in the form of determinants and properties of determinants Total Questions - 124

• question_answer1)
  \[\left| \,\begin{matrix}    a-b & b-c & c-a  \\    x-y & y-z & z-x  \\    p-q & q-r & r-p  \\ \end{matrix}\, \right|=\] [MNR 1987]
  A)
  \[a(x+y+z)+b(p+q+r)+c\] done clear
  B)
  0 done clear
  C)
  \[abc+xyz+pqr\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[\left| \,\begin{matrix}    1 & a & {{a}^{2}}-bc  \\    1 & b & {{b}^{2}}-ac  \\    1 & c & {{c}^{2}}-ab  \\ \end{matrix}\, \right|=\] [IIT 1988; MP PET 1990, 91; RPET 2002]
  A)
  0 done clear
  B)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\] done clear
  C)
  \[3abc\] done clear
  D)
  \[{{(a+b+c)}^{3}}\] done clear
  View Solution play_arrow

• question_answer3)
  \[\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & 1+x & 1  \\    1 & 1 & 1+y  \\ \end{matrix}\, \right|=\] [RPET 1996]
  A)
  1 done clear
  B)
  0 done clear
  C)
  x done clear
  D)
  xy done clear
  View Solution play_arrow

• question_answer4)
  \[\left| \,\begin{matrix}    1 & a & {{a}^{2}}  \\    1 & b & {{b}^{2}}  \\    1 & c & {{c}^{2}}  \\ \end{matrix}\, \right|=\]                             [Pb. CET 1997; DCE 2002]
  A)
  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  B)
  \[(a+b)\,(b+c)\,(c+a)\] done clear
  C)
  \[(a-b)(b-c)(c-a)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  The roots of the equation \[\left| \,\begin{matrix}    1 & 4 & 20  \\    1 & -2 & 5  \\    1 & 2x & 5{{x}^{2}}  \\ \end{matrix}\, \right|=0\]are [IIT 1987; MP PET 2002]
  A)
  \[-1,-2\] done clear
  B)
  \[-1,\,2\] done clear
  C)
  \[1,-2\] done clear
  D)
  \[1,\,2\] done clear
  View Solution play_arrow

• question_answer6)
  \[\left| \,\begin{matrix}    1 & 5 & \pi   \\    {{\log }_{e}}e & 5 & \sqrt{5}  \\    {{\log }_{10}}10 & 5 & e  \\ \end{matrix}\, \right|=\]
  A)
  \[\sqrt{\pi }\] done clear
  B)
  e done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer7)
  If \[a\ne b\ne c,\] the value of x which satisfies the equation \[\left| \,\begin{matrix}    0 & x-a & x-b  \\    x+a & 0 & x-c  \\    x+b & x+c & 0  \\ \end{matrix}\, \right|=0\], is [EAMCET 1988; Karnataka CET 1991; MNR 1980;                                      MP PET 1988, 99, 2001; DCE 2001]
  A)
  \[x=0\] done clear
  B)
  \[x=a\] done clear
  C)
  \[x=b\] done clear
  D)
  \[x=c\] done clear
  View Solution play_arrow

• question_answer8)
  The determinant \[\,\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & 2 & 3  \\    1 & 3 & 6  \\ \end{matrix}\, \right|\]is not equal to   [MP PET 1988]
  A)
  \[\left| \,\begin{matrix}    2 & 1 & 1  \\    2 & 2 & 3  \\    2 & 3 & 6  \\ \end{matrix}\, \right|\] done clear
  B)
  \[\left| \,\begin{matrix}    2 & 1 & 1  \\    3 & 2 & 3  \\    4 & 3 & 6  \\ \end{matrix}\, \right|\] done clear
  C)
  \[\left| \begin{matrix}    1 & 2 & 1  \\    1 & 5 & 3  \\    1 & 9 & 6  \\ \end{matrix} \right|\] done clear
  D)
  \[\left| \,\begin{matrix}    3 & 1 & 1  \\    6 & 2 & 3  \\    10 & 3 & 6  \\ \end{matrix} \right|\,\] done clear
  View Solution play_arrow

• question_answer9)
  If \[\omega \] is the cube root of unity, then \[\left| \begin{matrix}    1 & \omega  & {{\omega }^{2}}  \\    \omega  & {{\omega }^{2}} & 1  \\    {{\omega }^{2}} & 1 & \omega   \\ \end{matrix} \right|\]=   [RPET 1985, 93, 94; MP PET 1990, 2002;  Karnataka CET 1992; 93, 02, 05]
  A)
  1 done clear
  B)
  0 done clear
  C)
  \[\omega \] done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer10)
  If \[a+b+c=0\], then the solution of the equation \[\left| \,\begin{matrix}    a-x & c & b  \\    c & b-x & a  \\    b & a & c-x  \\ \end{matrix}\, \right|=0\] is [UPSEAT 2001]
  A)
  0 done clear
  B)
  \[\pm \frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done clear
  C)
  \[0,\,\pm \sqrt{\frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\] done clear
  D)
  \[0,\,\,\pm \sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\] done clear
  View Solution play_arrow

• question_answer11)
  \[\left| \,\begin{matrix}    1+i & 1-i & i  \\    1-i & i & 1+i  \\    i & 1+i & 1-i  \\ \end{matrix}\, \right|=\]
  A)
  \[-4-7i\] done clear
  B)
  \[4+7i\] done clear
  C)
  \[3+7i\] done clear
  D)
  \[7+4i\] done clear
  View Solution play_arrow

• question_answer12)
  If \[\left| \,\begin{matrix}    x+1 & 3 & 5  \\    2 & x+2 & 5  \\    2 & 3 & x+4  \\ \end{matrix}\, \right|=0\], then x =  [MP PET 1991]
  A)
  1, 9 done clear
  B)
  -1, 9 done clear
  C)
  -1, -9 done clear
  D)
  1, -9 done clear
  View Solution play_arrow

• question_answer13)
  \[\left| \,\begin{matrix}    a-b-c & 2a & 2a  \\    2b & b-c-a & 2b  \\    2c & 2c & c-a-b  \\ \end{matrix}\, \right|=\] [RPET 1990, 95]
  A)
  \[{{(a+b+c)}^{2}}\] done clear
  B)
  \[{{(a+b+c)}^{3}}\] done clear
  C)
  \[(a+b+c)(ab+bc+ca)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  \[\left| \,\begin{matrix}    a+b & a+2b & a+3b  \\    a+2b & a+3b & a+4b  \\    a+4b & a+5b & a+6b  \\ \end{matrix}\, \right|=\] [IIT 1986; MNR 1985; MP PET 1998; Pb. CET 2003]
  A)
  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-3abc\] done clear
  B)
  \[3ab\] done clear
  C)
  \[3a+5b\] done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer15)
  \[\left| \,\begin{matrix}    b+c & a & a  \\    b & c+a & b  \\    c & c & a+b  \\ \end{matrix}\, \right|=\]        [Roorkee 1980; RPET 1997, 99; KCET 1999; MP PET 2001]
  A)
  \[abc\] done clear
  B)
  \[2abc\] done clear
  C)
  \[3abc\] done clear
  D)
  \[4abc\] done clear
  View Solution play_arrow

• question_answer16)
  The roots of the equation \[\left| \,\begin{matrix}    1+x & 1 & 1  \\    1 & 1+x & 1  \\    1 & 1 & 1+x  \\ \end{matrix}\, \right|=0\]are  [MP PET 1989; Roorkee Qualifying 1998]
  A)
  0, - 3 done clear
  B)
  0, 0, - 3 done clear
  C)
  0, 0, 0, - 3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  One of the roots of the given equation \[\left| \,\begin{matrix}    x+a & b & c  \\    b & x+c & a  \\    c & a & x+b  \\ \end{matrix}\, \right|=0\] is  [MP PET 1988, 2002; RPET 1996]
  A)
  \[-(a+b)\] done clear
  B)
  \[-(b+c)\] done clear
  C)
  \[-a\] done clear
  D)
  \[-(a+b+c)\] done clear
  View Solution play_arrow

• question_answer18)
  \[\left| \,\begin{matrix}    x+1 & x+2 & x+4  \\    x+3 & x+5 & x+8  \\    x+7 & x+10 & x+14  \\ \end{matrix}\, \right|=\]    [MNR 1985; UPSEAT 2000]
  A)
  2 done clear
  B)
  - 2 done clear
  C)
  \[{{x}^{2}}-2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  \[\left| \,\begin{matrix}    1 & a & b  \\    -a & 1 & c  \\    -b & -c & 1  \\ \end{matrix}\, \right|=\] [MP PET 1991]
  A)
  \[1+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  B)
  \[1-{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  C)
  \[1+{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\] done clear
  D)
  \[1+{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer20)
  \[\left| \,\begin{matrix}    1 & 1 & 1  \\    a & b & c  \\    {{a}^{3}} & {{b}^{3}} & {{c}^{3}}  \\ \end{matrix}\, \right|=\]          [AMU 1979; RPET 1990; DCE 1999]
  A)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\] done clear
  B)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3abc\] done clear
  C)
  \[(a+b+c)(a-b)(b-c)(c-a)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  \[\left| \begin{matrix}    0 & a & -b  \\    -a & 0 & c  \\    b & -c & 0  \\ \end{matrix} \right|=\] [MP PET 1992]
  A)
  \[-2abc\] done clear
  B)
  \[abc\] done clear
  C)
  0 done clear
  D)
  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer22)
  \[\left| \,\begin{matrix}    a & b & c  \\    b & c & a  \\    c & a & b  \\ \end{matrix}\, \right|=\] [MP PET 1991]
  A)
  \[3abc+{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] done clear
  B)
  \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] done clear
  C)
  \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] done clear
  D)
  \[abc+{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] done clear
  View Solution play_arrow

• question_answer23)
  \[\left| \,\begin{matrix}    {{b}^{2}}-ab & b-c & bc-ac  \\    ab-{{a}^{2}} & a-b & {{b}^{2}}-ab  \\    bc-ac & c-a & ab-{{a}^{2}}  \\ \end{matrix}\, \right|=\] [MNR 1988]
  A)
  \[abc(a+b+c)\] done clear
  B)
  \[3{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  \[\left| \,\begin{matrix}    1/a & {{a}^{2}} & bc  \\    1/b & {{b}^{2}} & ca  \\    1/c & {{c}^{2}} & ab  \\ \end{matrix}\, \right|=\] [RPET 1990, 99]
  A)
  \[abc\] done clear
  B)
  \[1/abc\] done clear
  C)
  \[ab+bc+ca\] done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer25)
  \[\left| \,\begin{matrix}    {{b}^{2}}+{{c}^{2}} & {{a}^{2}} & {{a}^{2}}  \\    {{b}^{2}} & {{c}^{2}}+{{a}^{2}} & {{b}^{2}}  \\    {{c}^{2}} & {{c}^{2}} & {{a}^{2}}+{{b}^{2}}  \\ \end{matrix}\, \right|=\] [IIT 1980]
  A)
  \[abc\] done clear
  B)
  \[4abc\] done clear
  C)
  \[4{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  D)
  \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer26)
  \[\left| \,\begin{matrix}    1+x & 1 & 1  \\    1 & 1+y & 1  \\    1 & 1 & 1+z  \\ \end{matrix}\, \right|=\]    [RPET 1992; Kerala (Engg.) 2002]
  A)
  \[xyz\left( 1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right)\] done clear
  B)
  \[xyz\] done clear
  C)
  \[1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] done clear
  D)
  \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\] done clear
  View Solution play_arrow

• question_answer27)
  If \[\omega \]is a cube root of unity, then \[\left| \,\begin{matrix}    x+1 & \omega  & {{\omega }^{2}}  \\    \omega  & x+{{\omega }^{2}} & 1  \\    {{\omega }^{2}} & 1 & x+\omega   \\ \end{matrix}\, \right|=\] [MNR 1990; MP PET 1999]
  A)
  \[{{x}^{3}}+1\] done clear
  B)
  \[{{x}^{3}}+\omega \] done clear
  C)
  \[{{x}^{3}}+{{\omega }^{2}}\] done clear
  D)
  \[{{x}^{3}}\] done clear
  View Solution play_arrow

• question_answer28)
  If \[\left| \,\begin{matrix}    y+z & x & y  \\    z+x & z & x  \\    x+y & y & z  \\ \end{matrix}\, \right|=k(x+y+z){{(x-z)}^{2}}\], then \[k=\]
  A)
  \[2xyz\] done clear
  B)
  1 done clear
  C)
  \[xyz\] done clear
  D)
  \[{{x}^{2}}{{y}^{2}}{{z}^{2}}\] done clear
  View Solution play_arrow

• question_answer29)
  If - 9 is a root of the equation \[\left| \,\begin{matrix}    x & 3 & 7  \\    2 & x & 2  \\    7 & 6 & x  \\ \end{matrix}\, \right|=0\]then the other two roots are [IIT 1983; MNR 1992; MP PET 1995; DCE 1997; UPSEAT 2001]
  A)
  2, 7 done clear
  B)
  - 2, 7 done clear
  C)
  2, -7 done clear
  D)
  - 2, -7 done clear
  View Solution play_arrow

• question_answer30)
  If \[A=\left| \,\begin{matrix}    1 & 1 & 1  \\    a & b & c  \\    {{a}^{3}} & {{b}^{3}} & {{c}^{3}}  \\ \end{matrix}\, \right|,B=\left| \,\begin{matrix}    1 & 1 & 1  \\    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    {{a}^{3}} & {{b}^{3}} & {{c}^{3}}  \\ \end{matrix}\, \right|,C=\left| \,\begin{matrix}    a & b & c  \\    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    {{a}^{3}} & {{b}^{3}} & {{c}^{3}}  \\ \end{matrix}\, \right|,\] then which relation is correct 
  A)
  \[A=B\] done clear
  B)
  \[A=C\] done clear
  C)
  \[B=C\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer31)
  \[\left| \,\begin{matrix}    b+c & a-b & a  \\    c+a & b-c & b  \\    a+b & c-a & c  \\ \end{matrix}\, \right|=\] [MP PET 1990]
  A)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\] done clear
  B)
  \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] done clear
  C)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-{{a}^{2}}b-{{b}^{2}}c-{{c}^{2}}a\] done clear
  D)
  \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] done clear
  View Solution play_arrow

• question_answer32)
  If \[a,b,c\]are unequal what is the condition that the value of the following determinant is zero \[\Delta =\left| \,\begin{matrix}    a & {{a}^{2}} & {{a}^{3}}+1  \\    b & {{b}^{2}} & {{b}^{3}}+1  \\    c & {{c}^{2}} & {{c}^{3}}+1  \\ \end{matrix}\, \right|\] [IIT 1985; DCE 1999]
  A)
  \[1+abc=0\] done clear
  B)
  \[a+b+c+1=0\] done clear
  C)
  \[(a-b)(b-c)(c-a)=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[\omega \]is a complex cube root of unity, then the determinant \[\left| \,\begin{matrix}    2 & 2\omega  & -{{\omega }^{2}}  \\    1 & 1 & 1  \\    1 & -1 & 0  \\ \end{matrix}\, \right|=\]
  A)
  0 done clear
  B)
  1 done clear
  C)
  - 1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer34)
  \[\left| \,\begin{matrix}    19 & 17 & 15  \\    9 & 8 & 7  \\    1 & 1 & 1  \\ \end{matrix}\, \right|=\] [MP PET 1990]
  A)
  0 done clear
  B)
  187 done clear
  C)
  354 done clear
  D)
  54 done clear
  View Solution play_arrow

• question_answer35)
  If \[\left| \,\begin{matrix}    x+1 & x+2 & x+3  \\    x+2 & x+3 & x+4  \\    x+a & x+b & x+c  \\ \end{matrix}\, \right|=0\], then \[a,b,c\] are in [Pb. CET 1998]
  A)
  A. P. done clear
  B)
  G. P. done clear
  C)
  H. P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer36)
  If \[\omega \] be a complex cube root of unity, then                           \[\left| \,\begin{matrix}    1 & \omega  & -{{\omega }^{2}}/2  \\    1 & 1 & 1  \\    1 & -1 & 0  \\ \end{matrix}\, \right|=\]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\omega \] done clear
  D)
  \[{{\omega }^{2}}\] done clear
  View Solution play_arrow

• question_answer37)
  If \[p{{\lambda }^{4}}+q{{\lambda }^{3}}+r{{\lambda }^{2}}+s\lambda +t=\]\[\left| \,\begin{matrix}    {{\lambda }^{2}}+3\lambda  & \lambda -1 & \lambda +3  \\    \lambda +1 & 2-\lambda  & \lambda -4  \\    \lambda -3 & \lambda +4 & 3\lambda   \\ \end{matrix}\, \right|,\] the value of t is [IIT 1981]
  A)
  16 done clear
  B)
  18 done clear
  C)
  17 done clear
  D)
  19 done clear
  View Solution play_arrow

• question_answer38)
  The value of the determinant \[\left| \,\begin{matrix}    4 & -6 & 1  \\    -1 & -1 & 1  \\    -4 & 11 & -1\,  \\ \end{matrix} \right|\]is  [RPET 1992]
  A)
  - 75 done clear
  B)
  25 done clear
  C)
  0 done clear
  D)
  - 25 done clear
  View Solution play_arrow

• question_answer39)
  The value of the determinant \[\left| \,\begin{matrix}    1 & a & b+c  \\    1 & b & c+a  \\    1 & c & a+b  \\ \end{matrix}\, \right|\]is  [MP PET 1993; Karnataka CET 1994; Pb. CE 2004]
  A)
  \[a+b+c\] done clear
  B)
  \[{{(a+b+c)}^{2}}\] done clear
  C)
  0 done clear
  D)
  \[1+a+b+c\] done clear
  View Solution play_arrow

• question_answer40)
  If a, b and c are non zero numbers, then \[\Delta =\left| \,\begin{matrix}    {{b}^{2}}{{c}^{2}} & bc & b+c  \\    {{c}^{2}}{{a}^{2}} & ca & c+a  \\    {{a}^{2}}{{b}^{2}} & ab & a+b  \\ \end{matrix}\, \right|\] is equal to  [AMU 1992; Karnataka CET 2000; 03]
  A)
  \[abc\] done clear
  B)
  \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  C)
  \[ab+bc+ca\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer41)
  The determinant \[\left| \,\begin{matrix}    a & b & a\alpha +b  \\    b & c & b\alpha +c  \\    a\alpha +b & b\alpha +c & 0  \\ \end{matrix}\, \right|=0\], if \[a,b,c\]are in  [IIT 1986, 97; MNR 1992; DCE 2000, 01; UPSEAT 2002]
  A)
  A. P. done clear
  B)
  G. P. done clear
  C)
  H. P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer42)
  The value of the determinant \[\left| \,\begin{matrix}    31 & 37 & 92  \\    31 & 58 & 71  \\    31 & 105 & 24  \\ \end{matrix}\, \right|\]is [MP PET 1992]
  A)
  - 2 done clear
  B)
  0 done clear
  C)
  81 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  The value of the determinant \[\left| \,\begin{matrix}    1 & 2 & 3  \\    3 & 5 & 7  \\    8 & 14 & 20  \\ \end{matrix}\, \right|\]is  [MNR 1991]
  A)
  20 done clear
  B)
  10 done clear
  C)
  0 done clear
  D)
  250 done clear
  View Solution play_arrow

• question_answer44)
  If \[\left| \,\begin{matrix}    1 & k & 3  \\    3 & k & -2  \\    2 & 3 & -1  \\ \end{matrix}\, \right|=0\],then the value of k is  [IIT 1979]
  A)
  - 1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  The value of the determinant \[\left| \,\begin{matrix}    1 & 1 & 1  \\    b+c & c+a & a+b  \\    b+c-a & c+a-b & a+b-c  \\ \end{matrix}\, \right|\] is [RPET 1986]
  A)
  abc done clear
  B)
  \[a+b+c\] done clear
  C)
  \[ab+bc+ca\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer46)
  If \[\Delta =\left| \,\begin{matrix}    a & b & c  \\    x & y & z  \\    p & q & r  \\ \end{matrix}\, \right|\], then \[\left| \,\begin{matrix}    ka & kb & kc  \\    kx & ky & kz  \\    kp & kq & kr  \\ \end{matrix}\, \right|\]= [RPET 1986]
  A)
  \[\Delta \] done clear
  B)
  \[k\Delta \] done clear
  C)
  \[3k\Delta \] done clear
  D)
  \[{{k}^{3}}\Delta \] done clear
  View Solution play_arrow

• question_answer47)
  \[\left| \,\begin{matrix}    a-1 & a & bc  \\    b-1 & b & ca  \\    c-1 & c & ab  \\ \end{matrix}\, \right|=\] [RPET 1988]
  A)
  0 done clear
  B)
  \[(a-b)(b-c)(c-a)\] done clear
  C)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer48)
  \[\left| \,\begin{matrix}    {{a}_{1}} & m{{a}_{1}} & {{b}_{1}}  \\    {{a}_{2}} & m{{a}_{2}} & {{b}_{2}}  \\    {{a}_{3}} & m{{a}_{3}} & {{b}_{3}}  \\ \end{matrix}\, \right|=\] [RPET 1989]
  A)
  0 done clear
  B)
  \[m{{a}_{1}}{{a}_{2}}{{a}_{3}}\] done clear
  C)
  \[m{{a}_{1}}{{a}_{2}}{{b}_{3}}\] done clear
  D)
  \[m{{b}_{1}}{{a}_{2}}{{a}_{3}}\] done clear
  View Solution play_arrow

• question_answer49)
  The value of \[\left| \,\begin{matrix}    265 & 240 & 219  \\    240 & 225 & 198  \\    219 & 198 & 181  \\ \end{matrix}\, \right|\] is equal to  [RPET 1989]
  A)
  0 done clear
  B)
  679 done clear
  C)
  779 done clear
  D)
  1000 done clear
  View Solution play_arrow

• question_answer50)
  If \[\left| \,\begin{matrix}    {{x}^{2}}+x & x+1 & x-2  \\    2{{x}^{2}}+3x-1 & 3x & 3x-3  \\    {{x}^{2}}+2x+3 & 2x-1 & 2x-1  \\ \end{matrix}\, \right|=Ax-12\], then the value of A is [IIT 1982]
  A)
  12 done clear
  B)
  24 done clear
  C)
  -12 done clear
  D)
  - 24 done clear
  View Solution play_arrow

• question_answer51)
  \[\Delta =\left| \,\begin{matrix}    a & a+b & a+b+c  \\    3a & 4a+3b & 5a+4b+3c  \\    6a & 9a+6b & 11a+9b+6c  \\ \end{matrix}\, \right|\] where \[a=i,b=\omega ,c={{\omega }^{2}}\], then \[\Delta \]is equal to
  A)
  i done clear
  B)
  \[-{{\omega }^{2}}\] done clear
  C)
  \[\omega \] done clear
  D)
  \[-i\] done clear
  View Solution play_arrow

• question_answer52)
  The value of the determinant \[\left| \,\begin{matrix}    2 & 8 & 4  \\    -5 & 6 & -10  \\    1 & 7 & 2  \\ \end{matrix}\, \right|\]is   [MP PET 1994]
  A)
  - 440 done clear
  B)
  0 done clear
  C)
  328 done clear
  D)
  488 done clear
  View Solution play_arrow

• question_answer53)
  Let \[\left| \,\begin{matrix}    6i & -3i & 1  \\    4 & 3i & -1  \\    20 & 3 & i  \\ \end{matrix}\, \right|=x+iy\], then [IIT 1998]
  A)
  \[x=3,y=1\] done clear
  B)
  \[x=0,y=0\] done clear
  C)
  \[x=0,y=3\] done clear
  D)
  \[x=1,y=3\] done clear
  View Solution play_arrow

• question_answer54)
  If \[a,b,c\] are positive integers, then the determinant \[\Delta =\left| \,\begin{matrix}    {{a}^{2}}+x & ab & ac  \\    ab & {{b}^{2}}+x & bc  \\    ac & bc & {{c}^{2}}+x  \\ \end{matrix}\, \right|\] is divisible by
  A)
  \[{{x}^{3}}\] done clear
  B)
  \[{{x}^{2}}\] done clear
  C)
  \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  If \[p+q+r=0=a+b+c\], then the value of the determinant \[\left| \,\begin{matrix}    pa & qb & rc  \\    qc & ra & pb  \\    rb & pc & qa  \\ \end{matrix}\, \right|\] is
  A)
  0 done clear
  B)
  \[pa+qb+rc\] done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer56)
  Suppose \[D=\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\]and     \[{D}'=\left| \,\begin{matrix}    {{a}_{1}}+p{{b}_{1}} & {{b}_{1}}+q{{c}_{1}} & {{c}_{1}}+r{{a}_{1}}  \\    {{a}_{2}}+p{{b}_{2}} & {{b}_{2}}+q{{c}_{2}} & {{c}_{2}}+r{{a}_{2}}  \\    {{a}_{3}}+p{{b}_{3}} & {{b}_{3}}+q{{c}_{3}} & {{c}_{3}}+r{{a}_{3}}  \\ \end{matrix}\, \right|\], then [Karnataka CET 1993; Pb. CET 1993]
  A)
  \[{D}'=D\] done clear
  B)
  \[{D}'=D(1-pqr)\] done clear
  C)
  \[{D}'=D(1+p+q+r)\] done clear
  D)
  \[{D}'=D(1+pqr)\] done clear
  View Solution play_arrow

• question_answer57)
  The roots of the equation \[\left| \,\begin{matrix}    0 & x & 16  \\    x & 5 & 7  \\    0 & 9 & x  \\ \end{matrix}\, \right|=0\]are [Pb. CET 2001; Karnataka CET 1994]
  A)
  \[0,\,\,12,\,\,12\] done clear
  B)
  0, 12, -12 done clear
  C)
  0, 12, 16 done clear
  D)
  0, 9, 16 done clear
  View Solution play_arrow

• question_answer58)
  If \[\left| \begin{matrix}    1 & 2 & 3  \\    2 & x & 3  \\    3 & 4 & 5  \\ \end{matrix}\, \right|=0,\]then x =       [Karnataka CET 1994]
  A)
  - 5/2 done clear
  B)
  -2/5 done clear
  C)
  5/2 done clear
  D)
  2/5 done clear
  View Solution play_arrow

• question_answer59)
  \[\left| \,\begin{matrix}    a+b & b+c & c+a  \\    b+c & c+a & a+b  \\    c+a & a+b & b+c  \\ \end{matrix}\, \right|=K\,\,\left| \,\begin{matrix}    a & b & c  \\    b & c & a  \\    c & a & b  \\ \end{matrix}\, \right|\,,\]then \[K=\] [EAMCET 1992; DCE 2000]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer60)
  \[\left| \,\begin{matrix}    0 & p-q & p-r  \\    q-p & 0 & q-r  \\    r-p & r-q & 0  \\ \end{matrix}\, \right|=\] [EAMCET 1993]
  A)
  0 done clear
  B)
  \[(p-q)(q-r)(r-p)\] done clear
  C)
  pqr done clear
  D)
  \[3pqr\] done clear
  View Solution play_arrow

• question_answer61)
  \[\left| \,\begin{matrix}    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    {{(a+1)}^{2}} & {{(b+1)}^{2}} & {{(c+1)}^{2}}  \\    {{(a-1)}^{2}} & {{(b-1)}^{2}} & {{(c-1)}^{2}}  \\ \end{matrix}\, \right|=\]
  A)
  \[4\,\left| \,\begin{matrix}    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    a & b & c  \\    1 & 1 & 1  \\ \end{matrix}\, \right|\] done clear
  B)
  \[3\,\,\left| \,\begin{matrix}    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    a & b & c  \\    1 & 1 & 1  \\ \end{matrix}\, \right|\] done clear
  C)
  \[2\,\,\left| \,\begin{matrix}    {{a}^{2}} & {{b}^{2}} & {{c}^{2}}  \\    a & b & c  \\    1 & 1 & 1  \\ \end{matrix}\, \right|\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer62)
  \[\left| \,\begin{matrix}    11 & 12 & 13  \\    12 & 13 & 14  \\    13 & 14 & 15  \\ \end{matrix}\, \right|=\] [Karnataka CET 1991]
  A)
  1 done clear
  B)
  0 done clear
  C)
  -1 done clear
  D)
  67 done clear
  View Solution play_arrow

• question_answer63)
  \[\left| \,\begin{matrix}    x & 4 & y+z  \\    y & 4 & z+x  \\    z & 4 & x+y  \\ \end{matrix}\, \right|=\] [Karnataka CET 1991]
  A)
  4 done clear
  B)
  \[x+y+z\] done clear
  C)
  xyz done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer64)
  The value of the determinant \[\left| \,\begin{matrix}    -1 & 1 & 1  \\    1 & -1 & 1  \\    1 & 1 & -1  \\ \end{matrix}\, \right|\]is equal to [Roorkee 1992]
  A)
  - 4 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer65)
  A root of the equation \[\left| \,\begin{matrix}    3-x & -6 & 3  \\    -6 & 3-x & 3  \\    3 & 3 & -6-x  \\ \end{matrix}\, \right|=0\]is  [Roorkee 1991; RPET 2001; J & K 2005]
  A)
  6 done clear
  B)
  3 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  \[\left| \,\begin{matrix}    {{\sin }^{2}}x & {{\cos }^{2}}x & 1  \\    {{\cos }^{2}}x & {{\sin }^{2}}x & 1  \\    -10 & 12 & 2  \\ \end{matrix}\, \right|=\] [EAMCET 1994]
  A)
  0 done clear
  B)
  \[12{{\cos }^{2}}x-10{{\sin }^{2}}x\] done clear
  C)
  \[12{{\sin }^{2}}x-10{{\cos }^{2}}x-2\] done clear
  D)
  \[10\sin 2x\] done clear
  View Solution play_arrow

• question_answer67)
  The roots of the equation \[\left| \,\begin{matrix}    x-1 & 1 & 1  \\    1 & x-1 & 1  \\    1 & 1 & x-1  \\ \end{matrix}\, \right|=0\]are [Karnataka CET 1992]
  A)
  1, 2 done clear
  B)
  - 1, 2 done clear
  C)
  1, - 2 done clear
  D)
  -1, - 2 done clear
  View Solution play_arrow

• question_answer68)
  \[\left| \,\begin{matrix}    bc & b{c}'+{b}'c & {b}'{c}'  \\    ca & c{a}'+{c}'a & {c}'{a}'  \\    ab & a{b}'+{a}'b & {a}'{b}'  \\ \end{matrix}\, \right|\] is equal to
  A)
  \[(ab-{a}'{b}')(bc-{b}'{c}')(ca-{c}'{a}')\] done clear
  B)
  \[(ab+{a}'{b}')(bc+{b}'{c}')(ca+{c}'{a}')\] done clear
  C)
  \[(a{b}'-{a}'b)(b{c}'-{b}'c)(c{a}'-{c}'a)\] done clear
  D)
  \[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\] done clear
  View Solution play_arrow

• question_answer69)
  The roots of the determinant equation (in x)  \[\left| \,\begin{matrix}    a & a & x  \\    m & m & m  \\    b & x & b  \\ \end{matrix}\, \right|=0\] [EAMCET 1993]
  A)
  \[x=a,b\] done clear
  B)
  \[x=-a,-b\] done clear
  C)
  \[x=-a,b\] done clear
  D)
  \[x=a,-b\] done clear
  View Solution play_arrow

• question_answer70)
  \[2\,\,\left| \,\begin{matrix}    1 & 1 & 1  \\    a & b & c  \\    {{a}^{2}}-bc & {{b}^{2}}-ac & {{c}^{2}}-ab  \\ \end{matrix}\, \right|=\] [EAMCET 1991; UPSEAT 1999]
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  \[3abc\] done clear
  View Solution play_arrow

• question_answer71)
  If \[{{D}_{p}}=\left| \,\begin{matrix}    p & 15 & 8  \\    {{p}^{2}} & 35 & 9  \\    {{p}^{3}} & 25 & 10  \\ \end{matrix}\, \right|\], then \[{{D}_{1}}+{{D}_{2}}+{{D}_{3}}+{{D}_{4}}+{{D}_{5}}=\] [Kurukshetra CEE 1998]
  A)
  0 done clear
  B)
  25 done clear
  C)
  625 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer72)
  The value of \[\left| \,\begin{matrix}    a & a+b & a+2b  \\    a+2b & a & a+b  \\    a+b & a+2b & a  \\ \end{matrix}\, \right|\]is equal to   [Kerala (Engg.) 2001]
  A)
  \[9{{a}^{2}}(a+b)\] done clear
  B)
  \[9{{b}^{2}}(a+b)\] done clear
  C)
  \[{{a}^{2}}(a+b)\] done clear
  D)
  \[{{b}^{2}}(a+b)\] done clear
  View Solution play_arrow

• question_answer73)
  If  \[a,b,c\] are different and \[\left| \,\begin{matrix}    a & {{a}^{2}} & {{a}^{3}}-1  \\    b & {{b}^{2}} & {{b}^{3}}-1  \\    c & {{c}^{2}} & {{c}^{3}}-1  \\ \end{matrix}\, \right|=0\], then [EAMCET 1989]
  A)
   \[a+b+c=0\] done clear
  B)
  \[abc=1\] done clear
  C)
  \[a+b+c=1\] done clear
  D)
  \[ab+bc+ca=0\] done clear
  View Solution play_arrow

• question_answer74)
  If \[\left| \,\begin{matrix}    -{{a}^{2}} & ab & ac  \\    ab & -{{b}^{2}} & bc  \\    ac & bc & -{{c}^{2}}  \\ \end{matrix}\, \right|=K{{a}^{2}}{{b}^{2}}{{c}^{2}},\]then \[K=\] [Kurukshetra CEE 1996, 98, 2002; RPET 1997; MP PET 1998, 99; Tamilnadu (Engg.) 2002]
  A)
  - 4 done clear
  B)
  2 done clear
  C)
  4 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer75)
  \[\left| \,\begin{matrix}    1 & 1+ac & 1+bc  \\    1 & 1+ad & 1+bd  \\    1 & 1+ae & 1+be  \\ \end{matrix}\, \right|=\] [MP PET 1996]
  A)
  1 done clear
  B)
  0 done clear
  C)
  3 done clear
  D)
  \[a+b+c\] done clear
  View Solution play_arrow

• question_answer76)
  The value of the determinant \[\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & 1-x & 1  \\    1 & 1 & 1+y  \\ \end{matrix}\, \right|\]is  [Pb. CET 2003]
  A)
  \[3-x+y\] done clear
  B)
  \[(1-x)(1+y)\] done clear
  C)
  \[xy\] done clear
  D)
  \[-xy\] done clear
  View Solution play_arrow

• question_answer77)
  \[\left| \,\begin{matrix}    13 & 16 & 19  \\    14 & 17 & 20  \\    15 & 18 & 21  \\ \end{matrix}\, \right|=\] [MP PET 1996]
  A)
  0 done clear
  B)
  - 39 done clear
  C)
  96 done clear
  D)
  57 done clear
  View Solution play_arrow

• question_answer78)
  If \[\left| \,\begin{matrix}    a & b & a\alpha -b  \\    b & c & b\alpha -c  \\    2 & 1 & 0  \\ \end{matrix}\, \right|=0\]and \[\alpha \ne \frac{1}{2},\]then [MP PET 1998]
  A)
  \[a,b,c\] are in A. P. done clear
  B)
  \[a,b,c\]are in G. P. done clear
  C)
  \[a,b,c\]are in H. P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer79)
  If \[{{\left| \,\begin{matrix}    4 & 1  \\    2 & 1  \\ \end{matrix}\, \right|}^{2}}=\left| \,\begin{matrix}    3 & 2  \\    1 & x  \\ \end{matrix}\, \right|-\left| \,\begin{matrix}    x & 3  \\    -2 & 1  \\ \end{matrix}\, \right|\], then x = [RPET 1996]
  A)
  - 14 done clear
  B)
  2 done clear
  C)
  6 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer80)
  If \[\left| \,\begin{matrix}    3x-8 & 3 & 3  \\    3 & 3x-8 & 3  \\    3 & 3 & 3x-8  \\ \end{matrix}\, \right|=0,\]then the values of x are [RPET 1997]
  A)
  0, 2/3 done clear
  B)
  2/3, 11/3 done clear
  C)
  1/2, 1 done clear
  D)
  11/3, 1 done clear
  View Solution play_arrow

• question_answer81)
  If \[a,b,c\] are in A.P., then the value of \[\left| \,\begin{matrix}    x+2 & x+3 & x+a  \\    x+4 & x+5 & x+b  \\    x+6 & x+7 & x+c  \\ \end{matrix}\, \right|\] is [RPET 1999]
  A)
  \[x-(a+b+c)\] done clear
  B)
  \[9{{x}^{2}}+a+b+c\] done clear
  C)
  \[a+b+c\] done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer82)
  If \[\Delta =\left| \,\begin{matrix}    x & y & z  \\    p & q & r  \\    a & b & c  \\ \end{matrix}\, \right|,\]then \[\left| \,\begin{matrix}    x & 2y & z  \\    2p & 4q & 2r  \\    a & 2b & c  \\ \end{matrix}\, \right|\]equals [RPET 1999]
  A)
  \[{{\Delta }^{2}}\] done clear
  B)
  \[4\Delta \] done clear
  C)
  \[3\Delta \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer83)
  If \[a\ne 6,b,c\]satisfy \[\left| \,\begin{matrix}    a & 2b & 2c  \\    3 & b & c  \\    4 & a & b  \\ \end{matrix}\, \right|=0,\]then \[abc=\] [EAMCET 2000]
  A)
  \[a+b+c\] done clear
  B)
  0 done clear
  C)
  \[{{b}^{3}}\] done clear
  D)
  \[ab+bc\] done clear
  View Solution play_arrow

• question_answer84)
  If \[{{a}^{-1}}+{{b}^{-1}}+{{c}^{-1}}=0\] such that \[\left| \,\begin{matrix}    1+a & 1 & 1  \\    1 & 1+b & 1  \\    1 & 1 & 1+c  \\ \end{matrix}\, \right|=\lambda \], then the value of \[\lambda \]is [RPET 2000]
  A)
  0 done clear
  B)
  abc done clear
  C)
  - abc done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer85)
  \[\left| \,\begin{matrix}    {{a}^{2}}+{{x}^{2}} & ab & ca  \\    ab & {{b}^{2}}+{{x}^{2}} & bc  \\    ca & bc & {{c}^{2}}+{{x}^{2}}  \\ \end{matrix}\, \right|\] is divisor of  [RPET 2000]
  A)
  \[{{a}^{2}}\] done clear
  B)
  \[{{b}^{2}}\] done clear
  C)
  \[{{c}^{2}}\] done clear
  D)
  \[{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer86)
  \[\left| \,\begin{matrix}    1 & 1 & 1  \\    \cos (nx) & \cos (n+1)x & \cos (n+2)x  \\    \sin (nx) & \sin (n+1)x & \sin (n+2)x  \\ \end{matrix}\, \right|\] is not depend  [RPET 2000]
  A)
  On x done clear
  B)
  On n done clear
  C)
  Both on x and n done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer87)
  The sum of the products of the elements of any row of a determinant A with the same row is always equal to [Karnataka CET 2000]
  A)
  1 done clear
  B)
  0 done clear
  C)
  |A| done clear
  D)
  \[\frac{1}{2}|A|\] done clear
  View Solution play_arrow

• question_answer88)
  The value of the determinant given below \[\left| \text{ }\begin{matrix}    1 & 2 & 3  \\    3 & 5 & 7  \\    8 & 14 & 20  \\ \end{matrix} \right|\] is  [UPSEAT 2000]
  A)
  20 done clear
  B)
  10 done clear
  C)
  0 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer89)
  If \[\left| \,\begin{matrix}    a & b & a+b  \\    b & c & b+c  \\    a+b & b+c & 0  \\ \end{matrix}\, \right|=0\]; then \[a,b,c\] are in   [AMU 2000]
  A)
  A. P. done clear
  B)
  G. P. done clear
  C)
  H. P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer90)
  If \[ab+bc+ca=0\] and \[\left| \,\begin{matrix}    a-x & c & b  \\    c & b-x & a  \\    b & a & c-x  \\ \end{matrix}\, \right|=0\], then one of the value of x is  [AMU 2000]
  A)
  \[{{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}^{\frac{1}{2}}}\] done clear
  B)
  \[{{\left[ \frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}}) \right]}^{\frac{1}{2}}}\] done clear
  C)
  \[{{\left[ \frac{1}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}}) \right]}^{\frac{1}{2}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer91)
  If \[\left| \,\begin{matrix}    a & b & c  \\    m & n & p  \\    x & y & z  \\ \end{matrix}\, \right|=k\], then \[\left| \,\begin{matrix}    6a & 2b & 2c  \\    3m & n & p  \\    3x & y & z  \\ \end{matrix}\, \right|=\] [Tamilnadu (Engg.) 2002]
  A)
  \[k/6\] done clear
  B)
  \[2k\] done clear
  C)
  \[3k\] done clear
  D)
  \[6k\] done clear
  View Solution play_arrow

• question_answer92)
  If \[A=\left| \,\begin{matrix}    -1 & 2 & 4  \\    3 & 1 & 0  \\    -2 & 4 & 2  \\ \end{matrix}\, \right|\]and \[B=\left| \,\begin{matrix}    -2 & 4 & 2  \\    6 & 2 & 0  \\    -2 & 4 & 8  \\ \end{matrix}\, \right|\], then B is given by [Tamilnadu (Engg.) 2002]
  A)
  \[B=4A\] done clear
  B)
  \[B=-4A\] done clear
  C)
  \[B=-A\] done clear
  D)
  \[B=6A\] done clear
  View Solution play_arrow

• question_answer93)
  If \[\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|=5\]; then the value of  \[\left| \,\begin{matrix}    {{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}} & {{c}_{2}}{{a}_{3}}-{{c}_{3}}{{a}_{2}} & {{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}}  \\    {{b}_{3}}{{c}_{1}}-{{b}_{1}}{{c}_{3}} & {{c}_{3}}{{a}_{1}}-{{c}_{1}}{{a}_{3}} & {{a}_{3}}{{b}_{1}}-{{a}_{1}}{{b}_{3}}  \\    {{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}} & {{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}} & {{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}  \\ \end{matrix}\, \right|\] is [Tamilnadu (Engg.) 2002]
  A)
  5 done clear
  B)
  25 done clear
  C)
  125 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer94)
  \[\Delta =\left| \,\begin{matrix}    a+x & b & c  \\    b & x+c & a  \\    c & a & x+b  \\ \end{matrix}\, \right|\],which of the following is a factor for the above determinant         [Tamilnadu (Engg.) 2002]
  A)
  \[x-(a+b+c)\] done clear
  B)
  \[x+(a+b+c)\] done clear
  C)
  \[a+b+c\] done clear
  D)
  \[-(a+b+c)\] done clear
  View Solution play_arrow

• question_answer95)
  The value of \[\left| \,\begin{matrix}    {{5}^{2}} & {{5}^{3}} & {{5}^{4}}  \\    {{5}^{3}} & {{5}^{4}} & {{5}^{5}}  \\    {{5}^{4}} & {{5}^{5}} & {{5}^{7}}  \\ \end{matrix}\, \right|\]is
  A)
  \[{{5}^{2}}\] done clear
  B)
  0 done clear
  C)
  \[{{5}^{13}}\] done clear
  D)
  \[{{5}^{9}}\] done clear
  View Solution play_arrow

• question_answer96)
  At what value of \[x,\]will \[\left| \,\begin{matrix}    x+{{\omega }^{2}} & \omega  & 1  \\    \omega  & {{\omega }^{2}} & 1+x  \\    1 & x+\omega  & {{\omega }^{2}}  \\ \end{matrix}\, \right|=0\] [DCE 2000, 01]
  A)
  \[x=0\] done clear
  B)
  \[x=1\] done clear
  C)
  \[x=-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer97)
  Let \[\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}\]. Then the value of the determinant \[\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & -1-{{\omega }^{2}} & {{\omega }^{2}}  \\    1 & {{\omega }^{2}} & {{\omega }^{4}}  \\ \end{matrix}\, \right|\]is [IIT Screening 2002]
  A)
  \[3\omega \] done clear
  B)
  \[3\omega (\omega -1)\] done clear
  C)
  \[3{{\omega }^{2}}\] done clear
  D)
  \[3\omega (1-\omega )\] done clear
  View Solution play_arrow

• question_answer98)
  If \[\left| \,\begin{matrix}    {{(b+c)}^{2}} & {{a}^{2}} & {{a}^{2}}  \\    {{b}^{2}} & {{(c+a)}^{2}} & {{b}^{2}}  \\    {{c}^{2}} & {{c}^{2}} & {{(a+b)}^{2}}  \\ \end{matrix}\, \right|=k\,abc{{(a+b+c)}^{3}}\], then the value of k is       [Tamilnadu (Engg.) 2001]
  A)
  - 1 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  -2 done clear
  View Solution play_arrow

• question_answer99)
  The value of \[\left| \,\begin{matrix}    41 & 42 & 43  \\    44 & 45 & 46  \\    47 & 48 & 49  \\ \end{matrix}\, \right|=\] [Karnataka CET 2001]
  A)
  2 done clear
  B)
  4 done clear
  C)
  0 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer100)
  If A, B, C be the angles of a triangle, then \[\left| \,\begin{matrix}    -1 & \cos C & \cos B  \\    \cos C & -1 & \cos A  \\    \cos B & \cos A & -1  \\ \end{matrix}\, \right|=\] [Karnataka CET 2002]
  A)
  1 done clear
  B)
  0 done clear
  C)
  \[\cos A\cos B\cos C\] done clear
  D)
  \[\cos A+\cos B\cos C\] done clear
  View Solution play_arrow

• question_answer101)
  \[\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & {{\omega }^{2}} & \omega   \\    1 & \omega  & {{\omega }^{2}}  \\ \end{matrix}\, \right|=\] [RPET 2002]
  A)
  \[3\sqrt{3}i\] done clear
  B)
  \[-3\sqrt{3}i\] done clear
  C)
  \[i\sqrt{3}\] done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer102)
  \[\left| \,\begin{matrix}    1/a & 1 & bc  \\    1/b & 1 & ca  \\    1/c & 1 & ab  \\ \end{matrix}\, \right|=\] [RPET 2002]
  A)
  0 done clear
  B)
  abc done clear
  C)
  1/abc done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer103)
  \[\left| \,\begin{matrix}    {{({{a}^{x}}+{{a}^{-x}})}^{2}} & {{({{a}^{x}}-{{a}^{-x}})}^{2}} & 1  \\    {{({{b}^{x}}+{{b}^{-x}})}^{2}} & {{({{b}^{x}}-{{b}^{-x}})}^{2}} & 1  \\    {{({{c}^{x}}+{{c}^{-x}})}^{2}} & {{({{c}^{x}}-{{c}^{-x}})}^{2}} & 1  \\ \end{matrix}\, \right|=\][UPSEAT 2002; AMU 2005]
  A)
  0 done clear
  B)
  \[2abc\] done clear
  C)
  \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer104)
  The determinant \[\left| \,\begin{matrix}    a & b & a-b  \\    b & c & b-c  \\    2 & 1 & 0  \\ \end{matrix}\, \right|\] is equal to zero if \[a,b,c\]are in [UPSEAT 2002]
  A)
  G. P. done clear
  B)
  A. P. done clear
  C)
  H. P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer105)
  If \[\left| \,\begin{matrix}    x+1 & 1 & 1  \\    2 & x+2 & 2  \\    3 & 3 & x+3  \\ \end{matrix}\, \right|=0,\]then x is  [Kerala (Engg.) 2002]
  A)
  0, - 6 done clear
  B)
  0, 6 done clear
  C)
  6 done clear
  D)
  - 6 done clear
  View Solution play_arrow

• question_answer106)
  Solution of the equation \[\left| \,\begin{matrix}    1 & 1 & x  \\    p+1 & p+1 & p+x  \\    3 & x+1 & x+2  \\ \end{matrix}\, \right|=0\]are [AMU 2002]
  A)
  \[x=1,\,2\] done clear
  B)
  \[x=2,\,3\] done clear
  C)
  \[x=1,\,p,\,2\] done clear
  D)
  \[x=1,\,2,\,-p\] done clear
  View Solution play_arrow

• question_answer107)
  The values of the determinant \[\left| \,\begin{matrix}    1 & \cos (\alpha -\beta ) & \cos \alpha   \\    \cos (\alpha -\beta ) & 1 & \cos \beta   \\    \cos \alpha  & \cos \beta  & 1  \\ \end{matrix}\, \right|\] is [UPSEAT 2003]
  A)
  \[{{\alpha }^{2}}+{{\beta }^{2}}\] done clear
  B)
  \[{{\alpha }^{2}}-{{\beta }^{2}}\] done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer108)
  The value of  \[\left| \,\begin{matrix}    {{1}^{2}} & {{2}^{2}} & {{3}^{2}}  \\    {{2}^{2}} & {{3}^{2}} & {{4}^{2}}  \\    {{3}^{2}} & {{4}^{2}} & {{5}^{2}}  \\ \end{matrix}\, \right|\]is [Kerala (Engg.) 2001]
  A)
  8 done clear
  B)
  - 8 done clear
  C)
  400 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer109)
  The values of x in the following determinant equation, \[\left| \,\begin{matrix}    a+x & a-x & a-x  \\    a-x & a+x & a-x  \\    a-x & a-x & a+x  \\ \end{matrix}\, \right|=0\] are [MP PET 2003]
  A)
  \[x=0,x=4a\] done clear
  B)
  \[x=0,x=a\] done clear
  C)
  \[x=0,x=2a\] done clear
  D)
  \[x=0,x=3a\] done clear
  View Solution play_arrow

• question_answer110)
  If \[\left| \,\begin{matrix}    x-1 & 3 & 0  \\    2 & x-3 & 4  \\    3 & 5 & 6  \\ \end{matrix}\, \right|=0\], then x = [RPET 2003]
  A)
  0 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer111)
  The roots of the equation \[\left| \,\begin{matrix}    x & 0 & 8  \\    4 & 1 & 3  \\    2 & 0 & x  \\ \end{matrix}\, \right|=0\]are equal to  [Pb. CET 2000]
  A)
  \[(-4,\,4)\] done clear
  B)
  \[(2,\,-4)\] done clear
  C)
  \[(2,\,4)\] done clear
  D)
  \[(2,\,8)\] done clear
  View Solution play_arrow

• question_answer112)
  The value of \[x,\]if \[\left| \,\begin{matrix}    -x & 1 & 0  \\    1 & -x & 1  \\    0 & 1 & -x  \\ \end{matrix}\, \right|=0\]is equal to  [Pb. CET 2002]
  A)
  \[\pm \sqrt{6}\] done clear
  B)
  \[\pm \sqrt{2}\] done clear
  C)
  \[\pm \sqrt{3}\] done clear
  D)
  \[\sqrt{2},\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer113)
  \[\left| \,\begin{matrix}    5 & 3 & -1  \\    -7 & x & -3  \\    9 & 6 & -2  \\ \end{matrix}\, \right|=0\], then x is equal to [Pb. CET 2002]
  A)
  3 done clear
  B)
  5 done clear
  C)
  7 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer114)
  If \[\omega \] is an imaginary root of unity, then the value of \[\left| \,\begin{matrix}    a & b{{\omega }^{2}} & a\omega   \\    b\omega  & c & b{{\omega }^{2}}  \\    c{{\omega }^{2}} & a\omega  & c  \\ \end{matrix}\, \right|\] is [MP PET 2004]
  A)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\] done clear
  B)
  \[{{a}^{2}}b-{{b}^{2}}c\] done clear
  C)
  0 done clear
  D)
  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer115)
  The value of \[\left| \,\begin{matrix}    1 & 1 & 1  \\    bc & ca & ab  \\    b+c & c+a & a+b  \\ \end{matrix}\, \right|\]is [Karnataka CET 2004]
  A)
  1 done clear
  B)
  0 done clear
  C)
  \[(a-b)(b-c)(c-a)\] done clear
  D)
  \[(a+b)(b+c)(c+a)\] done clear
  View Solution play_arrow

• question_answer116)
  The value of \[\left| \,\begin{matrix}    441 & 442 & 443  \\    445 & 446 & 447  \\    449 & 450 & 451  \\ \end{matrix}\, \right|\] is [Karnataka CET 2004]
  A)
  \[441\times 446\times 451\] done clear
  B)
  0 done clear
  C)
  - 1 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer117)
  If a, b, c are all different and \[\left| \,\begin{matrix}    a & {{a}^{3}} & {{a}^{4}}-1  \\    b & {{b}^{3}} & {{b}^{4}}-1  \\    c & {{c}^{3}} & {{c}^{4}}-1  \\ \end{matrix}\, \right|\] = 0 , then the value of \[abc(ab+bc+ca)\]is  [Kurukshetra CEE 2002]
  A)
  \[a+b+c\] done clear
  B)
  0 done clear
  C)
  \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done clear
  D)
  \[{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer118)
  If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=-2\]and \[f(x)=\left| \begin{matrix}    1+{{a}^{2}}x & (1+{{b}^{2}})x & (1+{{c}^{2}})x  \\    (1+{{a}^{2}})x & 1+{{b}^{2}}x & (1+{{c}^{2}})x  \\    (1+{{a}^{2}})x & (1+{{b}^{2}})x & 1+{{c}^{2}}x  \\ \end{matrix} \right|\] then f(x) is a polynomial of degree [AIEEE 2005]
  A)
  3 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer119)
  The determinant \[\left| \,\begin{matrix}    4+{{x}^{2}} & -6 & -2  \\    -6 & 9+{{x}^{2}} & 3  \\    -2 & 3 & 1+{{x}^{2}}  \\ \end{matrix}\, \right|\] is not divisible by [J & K 2005]
  A)
  x done clear
  B)
  \[{{x}^{3}}\] done clear
  C)
  \[14+{{x}^{2}}\] done clear
  D)
  \[{{x}^{5}}\] done clear
  View Solution play_arrow

• question_answer120)
  The value of the determinant  \[\left| \,\begin{matrix}    0 & {{b}^{3}}-{{a}^{3}} & {{c}^{3}}-{{a}^{3}}  \\    {{a}^{3}}-{{b}^{3}} & 0 & {{c}^{3}}-{{b}^{3}}  \\    {{a}^{3}}-{{c}^{3}} & {{b}^{3}}-{{c}^{3}} & 0  \\ \end{matrix}\, \right|\] is equal to  is equal to [J & K 2005]
  A)
  \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] done clear
  B)
  \[{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\] done clear
  C)
  0 done clear
  D)
  \[-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] done clear
  View Solution play_arrow

• question_answer121)
  The solutions of the equation \[\left| \,\begin{matrix}    x & 2 & -1  \\    2 & 5 & x  \\    -1 & 2 & x  \\ \end{matrix}\, \right|=0\] are  [Karnataka CET 2005]
  A)
  \[3,\,\,-1\] done clear
  B)
  \[-3,\,\,1\] done clear
  C)
  3, 1 done clear
  D)
  \[-3,\,\,-1\] done clear
  View Solution play_arrow

• question_answer122)
  \[\left| \begin{matrix}    1+{{\sin }^{2}}\theta  & {{\sin }^{2}}\theta  & {{\sin }^{2}}\theta   \\    {{\cos }^{2}}\theta  & 1+{{\cos }^{2}}\theta  & {{\cos }^{2}}\theta   \\    4\sin 4\theta  & 4\sin 4\theta  & 1+4\sin 4\theta   \\ \end{matrix} \right|=0\] then \[\sin 4\theta \]equal to  [Orissa JEE 2005]
  A)
  1/2 done clear
  B)
  1 done clear
  C)
  -1/2 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer123)
  If \[f(x)=\left| \begin{matrix}    x-3 & 2{{x}^{2}}-18 & 3{{x}^{3}}-81  \\    x-5 & 2{{x}^{2}}-50 & 4{{x}^{3}}-500  \\    1 & 2 & 3  \\ \end{matrix} \right|\] then \[f(1).f(3)+f(3).f(5)+f(5).f(1)\]= [Kerala (Engg.) 2005]
  A)
  \[f(1)\] done clear
  B)
  f (3) done clear
  C)
  \[f(1)+f(3)\] done clear
  D)
  \[f(1)+f(5)\] done clear
  E)
  (e) \[f(1)+f(3)+f(5)\] done clear
  View Solution play_arrow

• question_answer124)
  If \[\left| \,\begin{matrix}    y+z & x-z & x-y  \\    y-z & z-x & y-x  \\    z-y & z-x & x+y  \\ \end{matrix}\, \right|=k\,xyz\], then the value of k is [AMU 2005]
  A)
  2 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
   8 done clear
  View Solution play_arrow