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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) \[a(x+y+z)+b(p+q+r)+c\]
B) 0
C) \[abc+xyz+pqr\]
D) None of these
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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
B) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
C) \[3abc\]
D) \[{{(a+b+c)}^{3}}\]
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question_answer3) \[\left| \,\begin{matrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \\ \end{matrix}\, \right|=\] [RPET 1996]
A) 1
B) 0
C) x
D) xy
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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) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
B) \[(a+b)\,(b+c)\,(c+a)\]
C) \[(a-b)(b-c)(c-a)\]
D) None of these
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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) \[-1,-2\]
B) \[-1,\,2\]
C) \[1,-2\]
D) \[1,\,2\]
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question_answer6) \[\left| \,\begin{matrix} 1 & 5 & \pi \\ {{\log }_{e}}e & 5 & \sqrt{5} \\ {{\log }_{10}}10 & 5 & e \\ \end{matrix}\, \right|=\]
A) \[\sqrt{\pi }\]
B) e
C) 1
D) 0
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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) \[x=0\]
B) \[x=a\]
C) \[x=b\]
D) \[x=c\]
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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) \[\left| \,\begin{matrix} 2 & 1 & 1 \\ 2 & 2 & 3 \\ 2 & 3 & 6 \\ \end{matrix}\, \right|\]
B) \[\left| \,\begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 3 \\ 4 & 3 & 6 \\ \end{matrix}\, \right|\]
C) \[\left| \begin{matrix} 1 & 2 & 1 \\ 1 & 5 & 3 \\ 1 & 9 & 6 \\ \end{matrix} \right|\]
D) \[\left| \,\begin{matrix} 3 & 1 & 1 \\ 6 & 2 & 3 \\ 10 & 3 & 6 \\ \end{matrix} \right|\,\]
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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
B) 0
C) \[\omega \]
D) \[{{\omega }^{2}}\]
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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
B) \[\pm \frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]
C) \[0,\,\pm \sqrt{\frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\]
D) \[0,\,\,\pm \sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
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question_answer11) \[\left| \,\begin{matrix} 1+i & 1-i & i \\ 1-i & i & 1+i \\ i & 1+i & 1-i \\ \end{matrix}\, \right|=\]
A) \[-4-7i\]
B) \[4+7i\]
C) \[3+7i\]
D) \[7+4i\]
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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
B) -1, 9
C) -1, -9
D) 1, -9
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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) \[{{(a+b+c)}^{2}}\]
B) \[{{(a+b+c)}^{3}}\]
C) \[(a+b+c)(ab+bc+ca)\]
D) None of these
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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) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-3abc\]
B) \[3ab\]
C) \[3a+5b\]
D) 0
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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) \[abc\]
B) \[2abc\]
C) \[3abc\]
D) \[4abc\]
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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
B) 0, 0, - 3
C) 0, 0, 0, - 3
D) None of these
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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) \[-(a+b)\]
B) \[-(b+c)\]
C) \[-a\]
D) \[-(a+b+c)\]
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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
B) - 2
C) \[{{x}^{2}}-2\]
D) None of these
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question_answer19) \[\left| \,\begin{matrix} 1 & a & b \\ -a & 1 & c \\ -b & -c & 1 \\ \end{matrix}\, \right|=\] [MP PET 1991]
A) \[1+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
B) \[1-{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
C) \[1+{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\]
D) \[1+{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\]
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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) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
B) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}+3abc\]
C) \[(a+b+c)(a-b)(b-c)(c-a)\]
D) None of these
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question_answer21) \[\left| \begin{matrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \\ \end{matrix} \right|=\] [MP PET 1992]
A) \[-2abc\]
B) \[abc\]
C) 0
D) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
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question_answer22) \[\left| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right|=\] [MP PET 1991]
A) \[3abc+{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
B) \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
C) \[abc-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
D) \[abc+{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
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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) \[abc(a+b+c)\]
B) \[3{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
C) 0
D) None of these
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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) \[abc\]
B) \[1/abc\]
C) \[ab+bc+ca\]
D) 0
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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) \[abc\]
B) \[4abc\]
C) \[4{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
D) \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
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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) \[xyz\left( 1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right)\]
B) \[xyz\]
C) \[1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\]
D) \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\]
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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) \[{{x}^{3}}+1\]
B) \[{{x}^{3}}+\omega \]
C) \[{{x}^{3}}+{{\omega }^{2}}\]
D) \[{{x}^{3}}\]
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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) \[2xyz\]
B) 1
C) \[xyz\]
D) \[{{x}^{2}}{{y}^{2}}{{z}^{2}}\]
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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
B) - 2, 7
C) 2, -7
D) - 2, -7
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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) \[A=B\]
B) \[A=C\]
C) \[B=C\]
D) None of these
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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) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
B) \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
C) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-{{a}^{2}}b-{{b}^{2}}c-{{c}^{2}}a\]
D) \[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
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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) \[1+abc=0\]
B) \[a+b+c+1=0\]
C) \[(a-b)(b-c)(c-a)=0\]
D) None of these
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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
B) 1
C) - 1
D) None of these
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question_answer34) \[\left| \,\begin{matrix} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \\ \end{matrix}\, \right|=\] [MP PET 1990]
A) 0
B) 187
C) 354
D) 54
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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.
B) G. P.
C) H. P.
D) None of these
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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
B) 1
C) \[\omega \]
D) \[{{\omega }^{2}}\]
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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
B) 18
C) 17
D) 19
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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
B) 25
C) 0
D) - 25
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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) \[a+b+c\]
B) \[{{(a+b+c)}^{2}}\]
C) 0
D) \[1+a+b+c\]
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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) \[abc\]
B) \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
C) \[ab+bc+ca\]
D) None of these
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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.
B) G. P.
C) H. P.
D) None of these
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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
B) 0
C) 81
D) None of these
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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
B) 10
C) 0
D) 250
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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
B) 0
C) 1
D) None of these
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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
B) \[a+b+c\]
C) \[ab+bc+ca\]
D) None of these
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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) \[\Delta \]
B) \[k\Delta \]
C) \[3k\Delta \]
D) \[{{k}^{3}}\Delta \]
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question_answer47) \[\left| \,\begin{matrix} a-1 & a & bc \\ b-1 & b & ca \\ c-1 & c & ab \\ \end{matrix}\, \right|=\] [RPET 1988]
A) 0
B) \[(a-b)(b-c)(c-a)\]
C) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
D) None of these
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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
B) \[m{{a}_{1}}{{a}_{2}}{{a}_{3}}\]
C) \[m{{a}_{1}}{{a}_{2}}{{b}_{3}}\]
D) \[m{{b}_{1}}{{a}_{2}}{{a}_{3}}\]
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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
B) 679
C) 779
D) 1000
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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
B) 24
C) -12
D) - 24
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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
B) \[-{{\omega }^{2}}\]
C) \[\omega \]
D) \[-i\]
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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
B) 0
C) 328
D) 488
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question_answer53) Let \[\left| \,\begin{matrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \\ \end{matrix}\, \right|=x+iy\], then [IIT 1998]
A) \[x=3,y=1\]
B) \[x=0,y=0\]
C) \[x=0,y=3\]
D) \[x=1,y=3\]
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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) \[{{x}^{3}}\]
B) \[{{x}^{2}}\]
C) \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]
D) None of these
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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
B) \[pa+qb+rc\]
C) 1
D) None of these
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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) \[{D}'=D\]
B) \[{D}'=D(1-pqr)\]
C) \[{D}'=D(1+p+q+r)\]
D) \[{D}'=D(1+pqr)\]
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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) \[0,\,\,12,\,\,12\]
B) 0, 12, -12
C) 0, 12, 16
D) 0, 9, 16
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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
B) -2/5
C) 5/2
D) 2/5
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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
B) 2
C) 3
D) 4
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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
B) \[(p-q)(q-r)(r-p)\]
C) pqr
D) \[3pqr\]
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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) \[4\,\left| \,\begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix}\, \right|\]
B) \[3\,\,\left| \,\begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix}\, \right|\]
C) \[2\,\,\left| \,\begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix}\, \right|\]
D) None of these
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question_answer62) \[\left| \,\begin{matrix} 11 & 12 & 13 \\ 12 & 13 & 14 \\ 13 & 14 & 15 \\ \end{matrix}\, \right|=\] [Karnataka CET 1991]
A) 1
B) 0
C) -1
D) 67
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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
B) \[x+y+z\]
C) xyz
D) 0
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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
B) 0
C) 1
D) 4
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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
B) 3
C) 0
D) None of these
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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
B) \[12{{\cos }^{2}}x-10{{\sin }^{2}}x\]
C) \[12{{\sin }^{2}}x-10{{\cos }^{2}}x-2\]
D) \[10\sin 2x\]
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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
B) - 1, 2
C) 1, - 2
D) -1, - 2
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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) \[(ab-{a}'{b}')(bc-{b}'{c}')(ca-{c}'{a}')\]
B) \[(ab+{a}'{b}')(bc+{b}'{c}')(ca+{c}'{a}')\]
C) \[(a{b}'-{a}'b)(b{c}'-{b}'c)(c{a}'-{c}'a)\]
D) \[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\]
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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) \[x=a,b\]
B) \[x=-a,-b\]
C) \[x=-a,b\]
D) \[x=a,-b\]
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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
B) 1
C) 2
D) \[3abc\]
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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
B) 25
C) 625
D) None of these
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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) \[9{{a}^{2}}(a+b)\]
B) \[9{{b}^{2}}(a+b)\]
C) \[{{a}^{2}}(a+b)\]
D) \[{{b}^{2}}(a+b)\]
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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) \[a+b+c=0\]
B) \[abc=1\]
C) \[a+b+c=1\]
D) \[ab+bc+ca=0\]
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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
B) 2
C) 4
D) 8
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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
B) 0
C) 3
D) \[a+b+c\]
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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) \[3-x+y\]
B) \[(1-x)(1+y)\]
C) \[xy\]
D) \[-xy\]
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question_answer77) \[\left| \,\begin{matrix} 13 & 16 & 19 \\ 14 & 17 & 20 \\ 15 & 18 & 21 \\ \end{matrix}\, \right|=\] [MP PET 1996]
A) 0
B) - 39
C) 96
D) 57
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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.
B) \[a,b,c\]are in G. P.
C) \[a,b,c\]are in H. P.
D) None of these
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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
B) 2
C) 6
D) 7
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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
B) 2/3, 11/3
C) 1/2, 1
D) 11/3, 1
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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) \[x-(a+b+c)\]
B) \[9{{x}^{2}}+a+b+c\]
C) \[a+b+c\]
D) 0
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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) \[{{\Delta }^{2}}\]
B) \[4\Delta \]
C) \[3\Delta \]
D) None of these
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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) \[a+b+c\]
B) 0
C) \[{{b}^{3}}\]
D) \[ab+bc\]
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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
B) abc
C) - abc
D) None of these
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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) \[{{a}^{2}}\]
B) \[{{b}^{2}}\]
C) \[{{c}^{2}}\]
D) \[{{x}^{2}}\]
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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
B) On n
C) Both on x and n
D) None of these
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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
B) 0
C) |A|
D) \[\frac{1}{2}|A|\]
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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
B) 10
C) 0
D) 5
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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.
B) G. P.
C) H. P.
D) None of these
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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) \[{{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}^{\frac{1}{2}}}\]
B) \[{{\left[ \frac{3}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}}) \right]}^{\frac{1}{2}}}\]
C) \[{{\left[ \frac{1}{2}({{a}^{2}}+{{b}^{2}}+{{c}^{2}}) \right]}^{\frac{1}{2}}}\]
D) None of these
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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) \[k/6\]
B) \[2k\]
C) \[3k\]
D) \[6k\]
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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) \[B=4A\]
B) \[B=-4A\]
C) \[B=-A\]
D) \[B=6A\]
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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
B) 25
C) 125
D) 0
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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) \[x-(a+b+c)\]
B) \[x+(a+b+c)\]
C) \[a+b+c\]
D) \[-(a+b+c)\]
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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) \[{{5}^{2}}\]
B) 0
C) \[{{5}^{13}}\]
D) \[{{5}^{9}}\]
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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) \[x=0\]
B) \[x=1\]
C) \[x=-1\]
D) None of these
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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) \[3\omega \]
B) \[3\omega (\omega -1)\]
C) \[3{{\omega }^{2}}\]
D) \[3\omega (1-\omega )\]
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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
B) 1
C) 2
D) -2
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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
B) 4
C) 0
D) 1
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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
B) 0
C) \[\cos A\cos B\cos C\]
D) \[\cos A+\cos B\cos C\]
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question_answer101) \[\left| \,\begin{matrix} 1 & 1 & 1 \\ 1 & {{\omega }^{2}} & \omega \\ 1 & \omega & {{\omega }^{2}} \\ \end{matrix}\, \right|=\] [RPET 2002]
A) \[3\sqrt{3}i\]
B) \[-3\sqrt{3}i\]
C) \[i\sqrt{3}\]
D) 3
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question_answer102) \[\left| \,\begin{matrix} 1/a & 1 & bc \\ 1/b & 1 & ca \\ 1/c & 1 & ab \\ \end{matrix}\, \right|=\] [RPET 2002]
A) 0
B) abc
C) 1/abc
D) None of these
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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
B) \[2abc\]
C) \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
D) None of these
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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.
B) A. P.
C) H. P.
D) None of these
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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
B) 0, 6
C) 6
D) - 6
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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) \[x=1,\,2\]
B) \[x=2,\,3\]
C) \[x=1,\,p,\,2\]
D) \[x=1,\,2,\,-p\]
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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) \[{{\alpha }^{2}}+{{\beta }^{2}}\]
B) \[{{\alpha }^{2}}-{{\beta }^{2}}\]
C) 1
D) 0
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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
B) - 8
C) 400
D) 1
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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) \[x=0,x=4a\]
B) \[x=0,x=a\]
C) \[x=0,x=2a\]
D) \[x=0,x=3a\]
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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
B) 2
C) 3
D) 1
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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) \[(-4,\,4)\]
B) \[(2,\,-4)\]
C) \[(2,\,4)\]
D) \[(2,\,8)\]
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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) \[\pm \sqrt{6}\]
B) \[\pm \sqrt{2}\]
C) \[\pm \sqrt{3}\]
D) \[\sqrt{2},\sqrt{3}\]
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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
B) 5
C) 7
D) 9
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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) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
B) \[{{a}^{2}}b-{{b}^{2}}c\]
C) 0
D) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
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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
B) 0
C) \[(a-b)(b-c)(c-a)\]
D) \[(a+b)(b+c)(c+a)\]
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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) \[441\times 446\times 451\]
B) 0
C) - 1
D) 1
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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) \[a+b+c\]
B) 0
C) \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
D) \[{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\]
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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
B) 2
C) 1
D) 0
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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
B) \[{{x}^{3}}\]
C) \[14+{{x}^{2}}\]
D) \[{{x}^{5}}\]
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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) \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
B) \[{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
C) 0
D) \[-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
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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) \[3,\,\,-1\]
B) \[-3,\,\,1\]
C) 3, 1
D) \[-3,\,\,-1\]
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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
B) 1
C) -1/2
D) -1
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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) \[f(1)\]
B) f (3)
C) \[f(1)+f(3)\]
D) \[f(1)+f(5)\]
E) (e) \[f(1)+f(3)+f(5)\]
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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
B) 4
C) 6
D) 8
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