8666 + Mcqs in Mathematics Page 105 McqOptions

5201.	\[\left\| \,\begin{matrix} 13 & 16 & 19 \\ 14 & 17 & 20 \\ 15 & 18 & 21 \\ \end{matrix}\, \right\|=\] [MP PET 1996]
A.	0
B.	-39
C.	96
D.	57
Answer» B. -39

Discussion

5202.	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.	\[-xy\]
Answer» E.

Discussion

5203.	\[\left\| \,\begin{matrix} 1 & 1+ac & 1+bc \\ 1 & 1+ad & 1+bd \\ 1 & 1+ae & 1+be \\ \end{matrix}\, \right\|=\] [MP PET 1996]
A.	1
B.	0
C.	3
D.	\[a+b+c\]
Answer» C. 3

Discussion

5204.	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
Answer» D. 8

Discussion

5205.	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.	\[ab+bc+ca=0\]
Answer» C. \[a+b+c=1\]

Discussion

5206.	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.	\[{{b}^{2}}(a+b)\]
Answer» C. \[{{a}^{2}}(a+b)\]

Discussion

5207.	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
Answer» E.

Discussion

5208.	\[2\,\,\left\| \,\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ {{a}^{2}}-bc & {{b}^{2}}-ac & {{c}^{2}}-ab \\ \end{matrix}\, \right\|=\] [EAMCET 1991; UPSEAT 1999]
A.	0
B.	1
C.	2
D.	\[3abc\]
Answer» B. 1

Discussion

5209.	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.	\[x=c\]
Answer» B. \[x=a\]

Discussion

5210.	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.	\[x=a,-b\]
Answer» B. \[x=-a,-b\]

Discussion

5211.	\[\left\| \,\begin{matrix} bc & b{c}'+{b}'c & {b}'{c}' \\ ca & c{a}'+{c}'a & {c}'{a}' \\ ab & a{b}'+{a}'b & {a}'{b}' \\ \end{matrix}\, \right\|\] 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.	\[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\]
Answer» D. \[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\]

Discussion

5212.	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
Answer» C. 1, - 2

Discussion

5213.	\[\left\| \,\begin{matrix} {{\sin }^{2}}x & {{\cos }^{2}}x & 1 \\ {{\cos }^{2}}x & {{\sin }^{2}}x & 1 \\ -10 & 12 & 2 \\ \end{matrix}\, \right\|=\] [EAMCET 1994]
A.	0
B.	\[12{{\cos }^{2}}x-10{{\sin }^{2}}x\]
C.	\[12{{\sin }^{2}}x-10{{\cos }^{2}}x-2\]
D.	\[10\sin 2x\]
Answer» B. \[12{{\cos }^{2}}x-10{{\sin }^{2}}x\]

Discussion

5214.	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
Answer» D. None of these

Discussion

5215.	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
Answer» E.

Discussion

5216.	\[\left\| \,\begin{matrix} x & 4 & y+z \\ y & 4 & z+x \\ z & 4 & x+y \\ \end{matrix}\, \right\|=\] [Karnataka CET 1991]
A.	4
B.	\[x+y+z\]
C.	xyz
D.	0
Answer» E.

Discussion

5217.	\[\left\| \,\begin{matrix} 11 & 12 & 13 \\ 12 & 13 & 14 \\ 13 & 14 & 15 \\ \end{matrix}\, \right\|=\] [Karnataka CET 1991]
A.	1
B.	0
C.	-1
D.	67
Answer» C. -1

Discussion

5218.	\[\left\| \,\begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ {{(a+1)}^{2}} & {{(b+1)}^{2}} & {{(c+1)}^{2}} \\ {{(a-1)}^{2}} & {{(b-1)}^{2}} & {{(c-1)}^{2}} \\ \end{matrix}\, \right\|=\]
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
Answer» B. \[3\,\,\left\| \,\begin{matrix} {{a}^{2}} & {{b}^{2}} & {{c}^{2}} \\ a & b & c \\ 1 & 1 & 1 \\ \end{matrix}\, \right\|\]

Discussion

5219.	\[\left\| \,\begin{matrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \\ \end{matrix}\, \right\|=\] [EAMCET 1993]
A.	0
B.	\[(p-q)(q-r)(r-p)\]
C.	pqr
D.	\[3pqr\]
Answer» B. \[(p-q)(q-r)(r-p)\]

Discussion

5220.	\[\left\| \,\begin{matrix} 1 & 5 & \pi \\ {{\log }_{e}}e & 5 & \sqrt{5} \\ {{\log }_{10}}10 & 5 & e \\ \end{matrix}\, \right\|=\]
A.	\[\sqrt{\pi }\]
B.	e
C.	1
D.	0
Answer» E.

Discussion

5221.	\[\left\| \,\begin{matrix} a+b & b+c & c+a \\ b+c & c+a & a+b \\ c+a & a+b & b+c \\ \end{matrix}\, \right\|=K\,\,\left\| \,\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix}\, \right\|\,,\]then \[K=\] [EAMCET 1992; DCE 2000]
A.	1
B.	2
C.	3
D.	4
Answer» C. 3

Discussion

5222.	If \[\left\| \begin{matrix} 1 & 2 & 3 \\ 2 & x & 3 \\ 3 & 4 & 5 \\ \end{matrix}\, \right\|=0,\]then x = [Karnataka CET 1994]
A.	-2.5
B.	-0.4
C.	44232
D.	44318
Answer» D. 44318

Discussion

5223.	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
Answer» C. 0, 12, 16

Discussion

5224.	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.	\[{D}'=D(1+pqr)\]
Answer» E.

Discussion

5225.	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
Answer» B. \[pa+qb+rc\]

Discussion

5226.	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
Answer» C. \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]

Discussion

5227.	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.	\[x=1,y=3\]
Answer» C. \[x=0,y=3\]

Discussion

5228.	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
Answer» C. 328

Discussion

5229.	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
Answer» C. -12

Discussion

5230.	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.	\[1,\,2\]
Answer» C. \[1,-2\]

Discussion

5231.	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
Answer» B. 679

Discussion

5232.	\[\left\| \,\begin{matrix} {{a}_{1}} & m{{a}_{1}} & {{b}_{1}} \\ {{a}_{2}} & m{{a}_{2}} & {{b}_{2}} \\ {{a}_{3}} & m{{a}_{3}} & {{b}_{3}} \\ \end{matrix}\, \right\|=\] [RPET 1989]
A.	0
B.	\[m{{a}_{1}}{{a}_{2}}{{a}_{3}}\]
C.	\[m{{a}_{1}}{{a}_{2}}{{b}_{3}}\]
D.	\[m{{b}_{1}}{{a}_{2}}{{a}_{3}}\]
Answer» B. \[m{{a}_{1}}{{a}_{2}}{{a}_{3}}\]

Discussion

5233.	\[\left\| \,\begin{matrix} a-1 & a & bc \\ b-1 & b & ca \\ c-1 & c & ab \\ \end{matrix}\, \right\|=\] [RPET 1988]
A.	0
B.	\[(a-b)(b-c)(c-a)\]
C.	\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc\]
D.	None of these
Answer» E.

Discussion

5234.	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.	\[{{k}^{3}}\Delta \]
Answer» E.

Discussion

5235.	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
Answer» E.

Discussion

5236.	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
Answer» E.

Discussion

5237.	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
Answer» C. 81

Discussion

5238.	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
Answer» C. H. P.

Discussion

5239.	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
Answer» E.

Discussion

5240.	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.	\[1+a+b+c\]
Answer» D. \[1+a+b+c\]

Discussion

5241.	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
Answer» E.

Discussion

5242.	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
Answer» C. 17

Discussion

5243.	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.	\[{{\omega }^{2}}\]
Answer» B. 1

Discussion

5244.	\[\left\| \,\begin{matrix} 19 & 17 & 15 \\ 9 & 8 & 7 \\ 1 & 1 & 1 \\ \end{matrix}\, \right\|=\] [MP PET 1990]
A.	0
B.	187
C.	354
D.	54
Answer» B. 187

Discussion

5245.	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
Answer» B. 1

Discussion

5246.	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
Answer» B. \[a+b+c+1=0\]

Discussion

5247.	\[\left\| \,\begin{matrix} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \\ \end{matrix}\, \right\|=\] [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.	\[3abc-{{a}^{3}}-{{b}^{3}}-{{c}^{3}}\]
Answer» C. \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-{{a}^{2}}b-{{b}^{2}}c-{{c}^{2}}a\]

Discussion

5248.	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
Answer» E.

Discussion

5249.	\[\left\| \,\begin{matrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \\ \end{matrix}\, \right\|=\] [RPET 1996]
A.	1
B.	0
C.	x
D.	xy
Answer» E.

Discussion

5250.	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
Answer» B. - 2, 7

Discussion

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

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

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]

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

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]

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]

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]

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]

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

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]

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]

\[\left| \,\begin{matrix} bc & b{c}'+{b}'c & {b}'{c}' \\ ca & c{a}'+{c}'a & {c}'{a}' \\ ab & a{b}'+{a}'b & {a}'{b}' \\ \end{matrix}\, \right|\] is equal to

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]

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

A 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]

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]

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

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

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

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

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

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

If \[\left| \begin{matrix} 1 & 2 & 3 \\ 2 & x & 3 \\ 3 & 4 & 5 \\ \end{matrix}\, \right|=0,\]then x = [Karnataka CET 1994]

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]

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

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

Let \[\left| \,\begin{matrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \\ \end{matrix}\, \right|=x+iy\], then [IIT 1998]

The value of the determinant \[\left| \,\begin{matrix} 2 & 8 & 4 \\ -5 & 6 & -10 \\ 1 & 7 & 2 \\ \end{matrix}\, \right|\]is [MP PET 1994]

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]

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]

The value of \[\left| \,\begin{matrix} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \\ \end{matrix}\, \right|\] is equal to [RPET 1989]

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

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

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]

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]

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]

The value of the determinant \[\left| \,\begin{matrix} 31 & 37 & 92 \\ 31 & 58 & 71 \\ 31 & 105 & 24 \\ \end{matrix}\, \right|\]is [MP PET 1992]

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]

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]

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]

The value of the determinant \[\left| \,\begin{matrix} 4 & -6 & 1 \\ -1 & -1 & 1 \\ -4 & 11 & -1\, \\ \end{matrix} \right|\]is [RPET 1992]

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]

If \[\omega \] be a complex cube root of unity, then \[\left| \,\begin{matrix} 1 & \omega & -{{\omega }^{2}}/2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \\ \end{matrix}\, \right|=\]

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

If \[\omega \]is a complex cube root of unity, then the determinant \[\left| \,\begin{matrix} 2 & 2\omega & -{{\omega }^{2}} \\ 1 & 1 & 1 \\ 1 & -1 & 0 \\ \end{matrix}\, \right|=\]

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]

\[\left| \,\begin{matrix} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \\ \end{matrix}\, \right|=\] [MP PET 1990]

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

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]