JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

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

1)

\[\left| \,\begin{matrix} a-b & b-c & c-a \\ x-y & y-z & z-x \\ p-q & q-r & r-p \\ \end{matrix}\, \right|=\] [MNR 1987]

A)
\[a(x+y+z)+b(p+q+r)+c\]

B)
0

C)
\[abc+xyz+pqr\]

D)
None of these
View Solution
2)

\[\left| \,\begin{matrix} 1 & a & {{a}^{2}}-bc \\ 1 & b & {{b}^{2}}-ac \\ 1 & c & {{c}^{2}}-ab \\ \end{matrix}\, \right|=\] [IIT 1988; MP PET 1990, 91; RPET 2002]

A)
0

B)
\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]

C)
\[3abc\]

D)
\[{{(a+b+c)}^{3}}\]
View Solution
3)

\[\left| \,\begin{matrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \\ \end{matrix}\, \right|=\] [RPET 1996]

A)
1

B)
0

C)
x

D)
xy
View Solution
4)

\[\left| \,\begin{matrix} 1 & a & {{a}^{2}} \\ 1 & b & {{b}^{2}} \\ 1 & c & {{c}^{2}} \\ \end{matrix}\, \right|=\] [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
View Solution
5)

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)
\[1,\,2\]
View Solution
6)

\[\left| \,\begin{matrix} 1 & 5 & \pi \\ {{\log }_{e}}e & 5 & \sqrt{5} \\ {{\log }_{10}}10 & 5 & e \\ \end{matrix}\, \right|=\]

A)
\[\sqrt{\pi }\]

B)
e

C)
1

D)
0
View Solution
7)

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)
\[x=c\]
View Solution
8)

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)
\[\left| \,\begin{matrix}    3 & 1 & 1 \\    6 & 2 & 3 \\    10 & 3 & 6 \\ \end{matrix} \right|\,\]
View Solution
9)

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)
\[{{\omega }^{2}}\]
View Solution
10)

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)
\[0,\,\,\pm \sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}\]
View Solution
11)

\[\left| \,\begin{matrix} 1+i & 1-i & i \\ 1-i & i & 1+i \\ i & 1+i & 1-i \\ \end{matrix}\, \right|=\]

A)
\[-4-7i\]

B)
\[4+7i\]

C)
\[3+7i\]

D)
\[7+4i\]
View Solution
12)

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
View Solution
13)

\[\left| \,\begin{matrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \\ \end{matrix}\, \right|=\] [RPET 1990, 95]

A)
\[{{(a+b+c)}^{2}}\]

B)
\[{{(a+b+c)}^{3}}\]

C)
\[(a+b+c)(ab+bc+ca)\]

D)
None of these
View Solution
14)

\[\left| \,\begin{matrix} a+b & a+2b & a+3b \\ a+2b & a+3b & a+4b \\ a+4b & a+5b & a+6b \\ \end{matrix}\, \right|=\] [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
View Solution
15)

\[\left| \,\begin{matrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \\ \end{matrix}\, \right|=\] [Roorkee 1980; RPET 1997, 99; KCET 1999; MP PET 2001]

A)
\[abc\]

B)
\[2abc\]

C)
\[3abc\]

D)
\[4abc\]
View Solution
16)

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
View Solution
17)

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)
\[-(a+b+c)\]
View Solution
18)

\[\left| \,\begin{matrix} x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14 \\ \end{matrix}\, \right|=\] [MNR 1985; UPSEAT 2000]

A)
2

B)
- 2

C)
\[{{x}^{2}}-2\]

D)
None of these
View Solution
19)

\[\left| \,\begin{matrix} 1 & a & b \\ -a & 1 & c \\ -b & -c & 1 \\ \end{matrix}\, \right|=\] [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)
\[1+{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\]
View Solution
20)

\[\left| \,\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ {{a}^{3}} & {{b}^{3}} & {{c}^{3}} \\ \end{matrix}\, \right|=\] [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
View Solution
21)

\[\left| \begin{matrix} 0 & a & -b \\ -a & 0 & c \\ b & -c & 0 \\ \end{matrix} \right|=\] [MP PET 1992]

A)
\[-2abc\]

B)
\[abc\]

C)
0

D)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
View Solution
22)

\[\left| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right|=\] [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)
\[abc+{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
View Solution
23)

\[\left| \,\begin{matrix} {{b}^{2}}-ab & b-c & bc-ac \\ ab-{{a}^{2}} & a-b & {{b}^{2}}-ab \\ bc-ac & c-a & ab-{{a}^{2}} \\ \end{matrix}\, \right|=\] [MNR 1988]

A)
\[abc(a+b+c)\]

B)
\[3{{a}^{2}}{{b}^{2}}{{c}^{2}}\]

C)
0

D)
None of these
View Solution
24)

\[\left| \,\begin{matrix} 1/a & {{a}^{2}} & bc \\ 1/b & {{b}^{2}} & ca \\ 1/c & {{c}^{2}} & ab \\ \end{matrix}\, \right|=\] [RPET 1990, 99]

A)
\[abc\]

B)
\[1/abc\]

C)
\[ab+bc+ca\]

D)
0
View Solution
25)

\[\left| \,\begin{matrix} {{b}^{2}}+{{c}^{2}} & {{a}^{2}} & {{a}^{2}} \\ {{b}^{2}} & {{c}^{2}}+{{a}^{2}} & {{b}^{2}} \\ {{c}^{2}} & {{c}^{2}} & {{a}^{2}}+{{b}^{2}} \\ \end{matrix}\, \right|=\] [IIT 1980]

A)
\[abc\]

B)
\[4abc\]

C)
\[4{{a}^{2}}{{b}^{2}}{{c}^{2}}\]

D)
\[{{a}^{2}}{{b}^{2}}{{c}^{2}}\]
View Solution
26)

\[\left| \,\begin{matrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \\ \end{matrix}\, \right|=\] [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)
\[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\]
View Solution
27)

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)
\[{{x}^{3}}\]
View Solution
28)

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)
\[{{x}^{2}}{{y}^{2}}{{z}^{2}}\]
View Solution
29)

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
View Solution
30)

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
View Solution
31)

\[\left| \,\begin{matrix} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \\ \end{matrix}\, \right|=\] [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)
\[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
View Solution
32)

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
View Solution
33)

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
View Solution
34)

\[\left| \,\begin{matrix} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \\ \end{matrix}\, \right|=\] [MP PET 1990]

A)
0

B)
187

C)
354

D)
54
View Solution
35)

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
View Solution
36)

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)
\[{{\omega }^{2}}\]
View Solution
37)

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
View Solution
38)

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
View Solution
39)

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)
\[1+a+b+c\]
View Solution
40)

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
View Solution
41)

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
View Solution
42)

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
View Solution
43)

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
View Solution
44)

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
View Solution
45)

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
View Solution
46)

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)
\[{{k}^{3}}\Delta \]
View Solution
47)

\[\left| \,\begin{matrix} a-1 & a & bc \\ b-1 & b & ca \\ c-1 & c & ab \\ \end{matrix}\, \right|=\] [RPET 1988]

A)
0

B)
\[(a-b)(b-c)(c-a)\]

C)
\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]

D)
None of these
View Solution
48)

\[\left| \,\begin{matrix} {{a}_{1}} & m{{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & m{{a}_{2}} & {{b}_{2}} \\ {{a}_{3}} & m{{a}_{3}} & {{b}_{3}} \\ \end{matrix}\, \right|=\] [RPET 1989]

A)
0

B)
\[m{{a}_{1}}{{a}_{2}}{{a}_{3}}\]

C)
\[m{{a}_{1}}{{a}_{2}}{{b}_{3}}\]

D)
\[m{{b}_{1}}{{a}_{2}}{{a}_{3}}\]
View Solution
49)

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
View Solution
50)

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
View Solution
51)

\[\Delta =\left| \,\begin{matrix} a & a+b & a+b+c \\ 3a & 4a+3b & 5a+4b+3c \\ 6a & 9a+6b & 11a+9b+6c \\ \end{matrix}\, \right|\] where \[a=i,b=\omega ,c={{\omega }^{2}}\], then \[\Delta \]is equal to

A)
i

B)
\[-{{\omega }^{2}}\]

C)
\[\omega \]

D)
\[-i\]
View Solution
52)

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
View Solution
53)

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)
\[x=1,y=3\]
View Solution
54)

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
View Solution
55)

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
View Solution
56)

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)
\[{D}'=D(1+pqr)\]
View Solution
57)

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
View Solution
58)

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
View Solution
59)

\[\left| \,\begin{matrix} a+b & b+c & c+a \\ b+c & c+a & a+b \\ c+a & a+b & b+c \\ \end{matrix}\, \right|=K\,\,\left| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right|\,,\]then \[K=\] [EAMCET 1992; DCE 2000]

A)
1

B)
2

C)
3

D)
4
View Solution
60)

\[\left| \,\begin{matrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \\ \end{matrix}\, \right|=\] [EAMCET 1993]

A)
0

B)
\[(p-q)(q-r)(r-p)\]

C)
pqr

D)
\[3pqr\]
View Solution
61)

\[\left| \,\begin{matrix}    {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\    {{(a+1)}^{2}} & {{(b+1)}^{2}} & {{(c+1)}^{2}} \\    {{(a-1)}^{2}} & {{(b-1)}^{2}} & {{(c-1)}^{2}} \\ \end{matrix}\, \right|=\]

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
View Solution
62)

\[\left| \,\begin{matrix} 11 & 12 & 13 \\ 12 & 13 & 14 \\ 13 & 14 & 15 \\ \end{matrix}\, \right|=\] [Karnataka CET 1991]

A)
1

B)
0

C)
-1

D)
67
View Solution
63)

\[\left| \,\begin{matrix} x & 4 & y+z \\ y & 4 & z+x \\ z & 4 & x+y \\ \end{matrix}\, \right|=\] [Karnataka CET 1991]

A)
4

B)
\[x+y+z\]

C)
xyz

D)
0
View Solution
64)

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
View Solution
65)

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
View Solution
66)

\[\left| \,\begin{matrix} {{\sin }^{2}}x & {{\cos }^{2}}x & 1 \\ {{\cos }^{2}}x & {{\sin }^{2}}x & 1 \\ -10 & 12 & 2 \\ \end{matrix}\, \right|=\] [EAMCET 1994]

A)
0

B)
\[12{{\cos }^{2}}x-10{{\sin }^{2}}x\]

C)
\[12{{\sin }^{2}}x-10{{\cos }^{2}}x-2\]

D)
\[10\sin 2x\]
View Solution
67)

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
View Solution
68)

\[\left| \,\begin{matrix} bc & b{c}'+{b}'c & {b}'{c}' \\ ca & c{a}'+{c}'a & {c}'{a}' \\ ab & a{b}'+{a}'b & {a}'{b}' \\ \end{matrix}\, \right|\] 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)
\[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\]
View Solution
69)

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)
\[x=a,-b\]
View Solution
70)

\[2\,\,\left| \,\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ {{a}^{2}}-bc & {{b}^{2}}-ac & {{c}^{2}}-ab \\ \end{matrix}\, \right|=\] [EAMCET 1991; UPSEAT 1999]

A)
0

B)
1

C)
2

D)
\[3abc\]
View Solution
71)

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
View Solution
72)

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)
\[{{b}^{2}}(a+b)\]
View Solution
73)

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)
\[ab+bc+ca=0\]
View Solution
74)

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
View Solution
75)

\[\left| \,\begin{matrix} 1 & 1+ac & 1+bc \\ 1 & 1+ad & 1+bd \\ 1 & 1+ae & 1+be \\ \end{matrix}\, \right|=\] [MP PET 1996]

A)
1

B)
0

C)
3

D)
\[a+b+c\]
View Solution
76)

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)
\[-xy\]
View Solution
77)

\[\left| \,\begin{matrix} 13 & 16 & 19 \\ 14 & 17 & 20 \\ 15 & 18 & 21 \\ \end{matrix}\, \right|=\] [MP PET 1996]

A)
0

B)
- 39

C)
96

D)
57
View Solution
78)

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
View Solution
79)

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
View Solution
80)

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
View Solution
81)

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
View Solution
82)

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
View Solution
83)

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)
\[ab+bc\]
View Solution
84)

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
View Solution
85)

\[\left| \,\begin{matrix} {{a}^{2}}+{{x}^{2}} & ab & ca \\ ab & {{b}^{2}}+{{x}^{2}} & bc \\ ca & bc & {{c}^{2}}+{{x}^{2}} \\ \end{matrix}\, \right|\] is divisor of [RPET 2000]

A)
\[{{a}^{2}}\]

B)
\[{{b}^{2}}\]

C)
\[{{c}^{2}}\]

D)
\[{{x}^{2}}\]
View Solution
86)

\[\left| \,\begin{matrix} 1 & 1 & 1 \\ \cos (nx) & \cos (n+1)x & \cos (n+2)x \\ \sin (nx) & \sin (n+1)x & \sin (n+2)x \\ \end{matrix}\, \right|\] is not depend [RPET 2000]

A)
On x

B)
On n

C)
Both on x and n

D)
None of these
View Solution
87)

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)
\[\frac{1}{2}|A|\]
View Solution
88)

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
View Solution
89)

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
View Solution
90)

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
View Solution
91)

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)
\[6k\]
View Solution
92)

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)
\[B=6A\]
View Solution
93)

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
View Solution
94)

\[\Delta =\left| \,\begin{matrix} a+x & b & c \\ b & x+c & a \\ c & a & x+b \\ \end{matrix}\, \right|\],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)
\[-(a+b+c)\]
View Solution
95)

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)
\[{{5}^{9}}\]
View Solution
96)

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
View Solution
97)

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)
\[3\omega (1-\omega )\]
View Solution
98)

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
View Solution
99)

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
View Solution
100)

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)
\[\cos A+\cos B\cos C\]
View Solution
101)

\[\left| \,\begin{matrix} 1 & 1 & 1 \\ 1 & {{\omega }^{2}} & \omega \\ 1 & \omega & {{\omega }^{2}} \\ \end{matrix}\, \right|=\] [RPET 2002]

A)
\[3\sqrt{3}i\]

B)
\[-3\sqrt{3}i\]

C)
\[i\sqrt{3}\]

D)
3
View Solution
102)

\[\left| \,\begin{matrix} 1/a & 1 & bc \\ 1/b & 1 & ca \\ 1/c & 1 & ab \\ \end{matrix}\, \right|=\] [RPET 2002]

A)
0

B)
abc

C)
1/abc

D)
None of these
View Solution
103)

\[\left| \,\begin{matrix} {{({{a}^{x}}+{{a}^{-x}})}^{2}} & {{({{a}^{x}}-{{a}^{-x}})}^{2}} & 1 \\ {{({{b}^{x}}+{{b}^{-x}})}^{2}} & {{({{b}^{x}}-{{b}^{-x}})}^{2}} & 1 \\ {{({{c}^{x}}+{{c}^{-x}})}^{2}} & {{({{c}^{x}}-{{c}^{-x}})}^{2}} & 1 \\ \end{matrix}\, \right|=\][UPSEAT 2002; AMU 2005]

A)
0

B)
\[2abc\]

C)
\[{{a}^{2}}{{b}^{2}}{{c}^{2}}\]

D)
None of these
View Solution
104)

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
View Solution
105)

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
View Solution
106)

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)
\[x=1,\,2,\,-p\]
View Solution
107)

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
View Solution
108)

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
View Solution
109)

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)
\[x=0,x=3a\]
View Solution
110)

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
View Solution
111)

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)
\[(2,\,8)\]
View Solution
112)

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)
\[\sqrt{2},\sqrt{3}\]
View Solution
113)

\[\left| \,\begin{matrix} 5 & 3 & -1 \\ -7 & x & -3 \\ 9 & 6 & -2 \\ \end{matrix}\, \right|=0\], then x is equal to [Pb. CET 2002]

A)
3

B)
5

C)
7

D)
9
View Solution
114)

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)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
View Solution
115)

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)
\[(a+b)(b+c)(c+a)\]
View Solution
116)

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
View Solution
117)

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)
\[{{a}^{2}}-{{b}^{2}}+{{c}^{2}}\]
View Solution
118)

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
View Solution
119)

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)
\[{{x}^{5}}\]
View Solution
120)

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)
\[-{{a}^{3}}+{{b}^{3}}+{{c}^{3}}\]
View Solution
121)

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)
\[-3,\,\,-1\]
View Solution
122)

\[\left| \begin{matrix} 1+{{\sin }^{2}}\theta & {{\sin }^{2}}\theta & {{\sin }^{2}}\theta \\ {{\cos }^{2}}\theta & 1+{{\cos }^{2}}\theta & {{\cos }^{2}}\theta \\ 4\sin 4\theta & 4\sin 4\theta & 1+4\sin 4\theta \\ \end{matrix} \right|=0\] then \[\sin 4\theta \]equal to [Orissa JEE 2005]

A)
1/2

B)
1

C)
-1/2

D)
-1
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123)

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) \[f(1)+f(3)+f(5)\]
View Solution
124)

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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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

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