-
question_answer1)
If a, b, c are three non-coplanar vector, then \[\frac{\mathbf{a}\,.\,\mathbf{b}\times \mathbf{c}}{\mathbf{c}\times \mathbf{a}\,.\,\mathbf{b}}+\frac{\mathbf{b}\,.\,\mathbf{a}\times \mathbf{c}}{\mathbf{c}\,.\,\mathbf{a}\times \mathbf{b}}\]= [IIT 1985, 86; UPSEAT 2003]
A)
0 done
clear
B)
2 done
clear
C)
? 2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer2)
If a, b, c be any three non-coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\,\mathbf{b}+\mathbf{c}\,\,\,\mathbf{c}+\mathbf{a}]=\] [RPET 1988; MP PET 1990, 02; Kerala (Engg.) 2002]
A)
\[|\mathbf{a}\,\mathbf{b}\,\mathbf{c}|\] done
clear
B)
2\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done
clear
C)
\[{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] done
clear
D)
\[2\,{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] done
clear
View Solution play_arrow
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question_answer3)
If the vectors \[2\mathbf{i}-3\mathbf{j},\,\,\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[3\mathbf{i}-\mathbf{k}\] form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is [IIT 1983; RPET 1995; DCE 2001; Kurukshetra CEE 1998; MP PET 2001]
A)
8 done
clear
B)
10 done
clear
C)
4 done
clear
D)
14 done
clear
View Solution play_arrow
-
question_answer4)
If a, b, c are any three coplanar unit vectors, then
A)
\[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=1\] done
clear
B)
\[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=3\] done
clear
C)
\[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}=0\] done
clear
D)
\[(\mathbf{c}\times \mathbf{a})\,.\,\mathbf{b}=1\] done
clear
View Solution play_arrow
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question_answer5)
If a and b be parallel vectors, then [a c b] =
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer6)
If the vectors \[2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-3\mathbf{k}\] and \[3\mathbf{i}+\lambda \mathbf{j}+5\mathbf{k}\] be coplanar, then \[\lambda =\] [Roorkee 1986; RPET 1999, 02; Kurukshetra CEE 2002]
A)
? 1 done
clear
B)
? 2 done
clear
C)
? 3 done
clear
D)
? 4 done
clear
View Solution play_arrow
-
question_answer7)
If a, b, c are the three non-coplanar vectors and p, q, r are defined by the relations \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] then (a+b) . p +(b+c) . q +(c+a) . r = [IIT 1988; BIT Mesra 1996; AMU 2002]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
-
question_answer8)
If the points whose position, vectors are \[3\mathbf{i}-2\mathbf{j}-\mathbf{k},\] \[2\mathbf{i}+3\mathbf{j}-4\mathbf{k},\] \[-\mathbf{i}+\mathbf{j}+2\mathbf{k}\]and \[4\mathbf{i}+5\mathbf{j}+\lambda \mathbf{k}\] lie on a plane, then \[\lambda =\] [IIT 1986; Pb. CET 2003]
A)
\[-\frac{146}{17}\] done
clear
B)
\[\frac{146}{17}\] done
clear
C)
\[-\frac{17}{146}\] done
clear
D)
\[\frac{17}{146}\] done
clear
View Solution play_arrow
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question_answer9)
If \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\]where a, b, c are three non-coplanar vectors, then the value of \[(\mathbf{a}+\mathbf{b}+\mathbf{c})\,.\,(\mathbf{p}+\mathbf{q}+\mathbf{r})\] is given by [MNR 1992; UPSEAT 2000]
A)
3 done
clear
B)
2 done
clear
C)
1 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer10)
The volume of the parallelopiped whose edges are represented by \[-12\mathbf{i}+\alpha \mathbf{k},\,\,3\mathbf{j}-\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-15\mathbf{k}\] is 546. Then \[\alpha =\] [IIT Screening 1989; MNR 1987]
A)
3 done
clear
B)
2 done
clear
C)
? 3 done
clear
D)
? 2 done
clear
View Solution play_arrow
-
question_answer11)
Let a, b, c be distinct non-negative numbers. If the vectors \[a\mathbf{i}+a\mathbf{j}+c\mathbf{k},\,\,\mathbf{i}+\mathbf{k}\] and \[c\mathbf{i}+c\mathbf{j}+b\mathbf{k}\] lie in a plane, then c is [IIT 1993; AIEEE 2005]
A)
The arithmetic mean of a and b done
clear
B)
The geometric mean of a and b done
clear
C)
The harmonic mean of a and b done
clear
D)
Equal to zero done
clear
View Solution play_arrow
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question_answer12)
If a, b, c are any three vectors and their inverse are \[{{\mathbf{a}}^{-1}},\,{{\mathbf{b}}^{-1}},\,{{\mathbf{c}}^{-1}}\]and \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\ne 0,\] then \[[{{\mathbf{a}}^{-1}}\,{{\mathbf{b}}^{-1}}\,{{\mathbf{c}}^{-1}}]\] will be [Roorkee 1989]
A)
Zero done
clear
B)
One done
clear
C)
Non-zero done
clear
D)
[a b c] done
clear
View Solution play_arrow
-
question_answer13)
If \[\mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=3\mathbf{i}+p\mathbf{j}+5\mathbf{k}\] are coplanar then the value of p will be [RPET 1985, 86, 88, 91]
A)
? 6 done
clear
B)
? 2 done
clear
C)
2 done
clear
D)
6 done
clear
View Solution play_arrow
-
question_answer14)
If \[\mathbf{i},\,\mathbf{j},\,\mathbf{k}\] are the unit vectors and mutually perpendicular, then \[[\mathbf{i}\,\mathbf{k}\,\mathbf{j}]\] is equal to [RPET 1986]
A)
0 done
clear
B)
? 1 done
clear
C)
1 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer15)
If three vectors \[\mathbf{a}=12\mathbf{i}+4\mathbf{j}+3\mathbf{k},\,\,\]\[\mathbf{b}=8\mathbf{i}-12\mathbf{j}-9\mathbf{k}\] and \[\mathbf{c}=33\mathbf{i}-4\mathbf{j}-24\mathbf{k}\] represents a cube, then its volume will be [Roorkee 1988]
A)
616 done
clear
B)
308 done
clear
C)
154 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer16)
If \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-\mathbf{j}+2\mathbf{k},\] then \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\] [RPET 1989, 2001]
A)
6 done
clear
B)
10 done
clear
C)
12 done
clear
D)
24 done
clear
View Solution play_arrow
-
question_answer17)
Three concurrent edges OA, OB, OC of a parallelopiped are represented by three vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-3\mathbf{i}-\mathbf{j}+\mathbf{k},\] the volume of the solid so formed in cubic unit is [Kurukshetra CEE 1998]
A)
5 done
clear
B)
6 done
clear
C)
7 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer18)
If \[\mathbf{x}\,.\,\mathbf{a}=0,\,\,\mathbf{x}\,.\,\mathbf{b}=0\] and \[\mathbf{x}\,.\,\mathbf{c}=0\] for some non-zero vector x, then the ture statement is [IIT 1983; Karnataka CET 2002]
A)
\[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]=0\] done
clear
B)
\[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]\ne 0\] done
clear
C)
\[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]=1\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
If the given vectors \[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] \[({{a}^{2}}+ac,\,-ac,\,{{c}^{2}}+ac)\] and \[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] are coplanar, where none of a, b and c is zero, then
A)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] done
clear
B)
\[bc+ca+ab=0\] done
clear
C)
\[a+b+c=0\] done
clear
D)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\] done
clear
View Solution play_arrow
-
question_answer20)
If a,b,c are three coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{c}+\mathbf{a}]=\] [MP PET 1995]
A)
[a b c] done
clear
B)
2 [a b c] done
clear
C)
3 [a b c] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer21)
\[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{a}\times \mathbf{b}]\] is equal to
A)
\[|\,\,\mathbf{a}\times \mathbf{b}|\] done
clear
B)
\[|\,\,\mathbf{a}\times \mathbf{b}{{|}^{2}}\] done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer22)
If \[\mathbf{a}\,.\,\mathbf{i}=4,\] then \[(\mathbf{a}\times \mathbf{j})\,.\,(2\mathbf{j}-3\mathbf{k})=\] [EAMCET 1994]
A)
12 done
clear
B)
2 done
clear
C)
0 done
clear
D)
? 12 done
clear
View Solution play_arrow
-
question_answer23)
If the vectors \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[x\mathbf{i}-\mathbf{j}+2\mathbf{k}\] are coplanar, then \[x=\] [EAMCET 1994]
A)
\[\frac{8}{5}\] done
clear
B)
\[\frac{5}{8}\] done
clear
C)
0 done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer24)
Volume of the parallelopiped whose coterminous edges are \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,3\mathbf{i}-\mathbf{j}+\mathbf{k},\] is [EAMCET 1993]
A)
5 cubic unit done
clear
B)
6 cubic unit done
clear
C)
7 cubic unit done
clear
D)
8 cubic unit done
clear
View Solution play_arrow
-
question_answer25)
If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{a}\,.\,\mathbf{b})=\] [Karnataka CET 1994]
A)
3a done
clear
B)
\[3\sqrt{14}\] done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer26)
\[\mathbf{i}\,.\,(\mathbf{j}\times \mathbf{k})+\mathbf{j}\,.\,(\mathbf{k}\times \mathbf{i})+\mathbf{k}\,.\,(\mathbf{i}\times \mathbf{j})=\] [Karnataka CET 1994]
A)
1 done
clear
B)
3 done
clear
C)
? 3 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer27)
If \[\mathbf{a}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[\mathbf{b}=3\mathbf{i}+\mathbf{j}+2\mathbf{k},\] then the unit vector perpendicular to a and b is [MP PET 1996]
A)
\[\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
B)
\[\frac{\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
C)
\[\frac{-\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
D)
\[\frac{\mathbf{i}-\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] done
clear
View Solution play_arrow
-
question_answer28)
If \[\mathbf{a}=-3\mathbf{i}+7\mathbf{j}+5\mathbf{k},\] \[\mathbf{b}=-3\mathbf{i}+7\mathbf{j}-3\mathbf{k}\], \[\mathbf{c}=7\mathbf{i}-5\mathbf{j}-3\mathbf{k}\] are the three coterminous edges of a parallelopiped, then its volume is [MP PET 1996]
A)
108 done
clear
B)
210 done
clear
C)
272 done
clear
D)
308 done
clear
View Solution play_arrow
-
question_answer29)
\[\mathbf{a}\,.\,(\mathbf{a}\times \mathbf{b})=\] [MP PET 1996]
A)
b . b done
clear
B)
\[{{a}^{2}}b\] done
clear
C)
0 done
clear
D)
\[{{a}^{2}}+ab\] done
clear
View Solution play_arrow
-
question_answer30)
If three conterminous edges of a parallelopiped are represented by \[\mathbf{a}-\mathbf{b},\,\,\mathbf{b}-\mathbf{c}\] and \[\mathbf{c}-\mathbf{a}\], then its volume is [MP PET 1999; Pb. CET 2003]
A)
[a b c] done
clear
B)
2 [a b c] done
clear
C)
\[\,{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer31)
For three vectors u, v, w which of the following expressions is not equal to any of the remaining three [IIT 1998]
A)
\[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] done
clear
B)
\[\,(\mathbf{v}\times \mathbf{w})\,.\,\mathbf{u}\,\] done
clear
C)
\[\mathbf{v}\,.\,(\mathbf{u}\times \mathbf{w})\] done
clear
D)
\[(\mathbf{u}\times \mathbf{v})\,.\,\mathbf{w}\] done
clear
View Solution play_arrow
-
question_answer32)
Which of the following expressions are meaningful [IIT 1998; RPET 2001]
A)
\[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] done
clear
B)
\[(\mathbf{u}\,.\,\mathbf{v})\,.\,\mathbf{w}\] done
clear
C)
\[(\mathbf{u}\,.\,\mathbf{v})\,\mathbf{w}\] done
clear
D)
\[\mathbf{u}\times (\mathbf{v}\,.\,\mathbf{w})\] done
clear
View Solution play_arrow
-
question_answer33)
If \[\mathbf{a,}\,\mathbf{b,}\,\mathbf{c}\] are non-coplanar vectors and \[\mathbf{d}=\lambda \mathbf{a}+\mu \,\mathbf{b}+\nu \mathbf{c},\] then \[\lambda \] is equal to [Roorkee 1999]
A)
\[\frac{[\mathbf{d}\,\mathbf{b}\,\mathbf{c}]}{[\mathbf{b}\,\mathbf{a}\,\mathbf{c}]}\] done
clear
B)
\[\frac{[\mathbf{b}\,\mathbf{c}\,\mathbf{d}]}{[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]}\] done
clear
C)
\[\frac{[\mathbf{b}\,\mathbf{d}\,\mathbf{c}]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] done
clear
D)
\[\frac{[\mathbf{c}\,\mathbf{b}\,\mathbf{d}\,]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] done
clear
View Solution play_arrow
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question_answer34)
If vectors \[\vec{A}=2\mathbf{i}+3\mathbf{j}+4\mathbf{k}\], \[\vec{B}=\mathbf{i}+\mathbf{j}+5\mathbf{k}\], and \[\vec{C}\] form a left handed system, then \[\vec{C}\] is [Roorkee 1999]
A)
11i ? 6j ? k done
clear
B)
? 11i + 6j + k done
clear
C)
11i ? 6j + k done
clear
D)
? 11i + 6j ? k done
clear
View Solution play_arrow
-
question_answer35)
What will be the volume of that parallelopiped whose sides are a = i ? j + k, b = i ? 3j + 4k and c = 2i ? 5j + 3k [UPSEAT 1999]
A)
5 unit done
clear
B)
6 unit done
clear
C)
7 unit done
clear
D)
8 unit done
clear
View Solution play_arrow
-
question_answer36)
Given vectors a, b, c such that \[\mathbf{a}\,.(\mathbf{b}\times \mathbf{c})\]\[=\lambda \ne 0,\,\] the value of \[(\mathbf{b}\times \mathbf{c})\,.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})/\lambda \] is [AMU 1999]
A)
3 done
clear
B)
1 done
clear
C)
\[-3\lambda \] done
clear
D)
\[3/\lambda \] done
clear
View Solution play_arrow
-
question_answer37)
If \[a,\,b\] and c are unit coplanar vectors then the scalar triple product \[[2a-b\,\,2b-c\,\,2c-a]\] is equal to [IIT Screening 2000; Kerala (Engg.) 2005]
A)
0 done
clear
B)
1 done
clear
C)
\[-\sqrt{3}\] done
clear
D)
\[\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer38)
If the vectors \[i+3j-2k\], \[2i-j+4k\] and \[3i+2j+xk\] are coplanar, then the value of x is [Karnataka CET 2000]
A)
? 2 done
clear
B)
2 done
clear
C)
1 done
clear
D)
3 done
clear
View Solution play_arrow
-
question_answer39)
The value of [a ? b b ? c c ? a], where \[|a|\,=1\], \[|b|\,=5\] and \[|c|\,=3\] is [RPET 2000]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
4 done
clear
View Solution play_arrow
-
question_answer40)
\[a=i+j+k,\,\mathbf{b}=2i-4k,\,c=i+\lambda \,j+3k\] are coplanar, then the value of \[\lambda \] is [MP PET 2000]
A)
5/2 done
clear
B)
3/5 done
clear
C)
7/3 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer41)
Let \[\overrightarrow{A}=i+j+k\], \[\overrightarrow{B}=i,\,\overrightarrow{C}={{C}_{1}}i+{{C}_{2}}j+{{C}_{3}}k\]. If \[{{C}_{2}}=-1\], and \[{{C}_{3}}=1\], then to make three vectors coplanar [AMU 2000]
A)
\[{{C}_{1}}=0\] done
clear
B)
\[{{C}_{1}}=1\] done
clear
C)
\[{{C}_{1}}=2\] done
clear
D)
No value of \[{{C}_{1}}\] can be found done
clear
View Solution play_arrow
-
question_answer42)
Let \[a=i-k,\,\,\,b=xi+j+(1-x)\,k\],\[c=yi+xj+(1+x-y)k\]. Then \[[a\,\,b\,\,c]\] depends on [IIT Screening 2001; AIEEE 2005]
A)
Only x done
clear
B)
Only y done
clear
C)
Neither x nor y done
clear
D)
Both x and y done
clear
View Solution play_arrow
-
question_answer43)
If \[\mathbf{a}=3\mathbf{i}-2\mathbf{j}+2\mathbf{k},\,\,\,\mathbf{b}=6\mathbf{i}+4\mathbf{j}-2\mathbf{k}\] and \[\mathbf{c}=3\mathbf{i}-2\mathbf{j}-4\mathbf{k}\], then \[\mathbf{a}\,.\,\,(\mathbf{b}\times \mathbf{c})\] is [Karnataka CET 2001]
A)
122 done
clear
B)
? 144 done
clear
C)
120 done
clear
D)
? 120 done
clear
View Solution play_arrow
-
question_answer44)
\[(\mathbf{a}+\mathbf{b})\,.\,(\mathbf{b}+\mathbf{c})\times (\mathbf{a}+\mathbf{b}+\mathbf{c})=\] [EAMCET 2002]
A)
? [a b c] done
clear
B)
[a b c] done
clear
C)
0 done
clear
D)
2[a b c] done
clear
View Solution play_arrow
-
question_answer45)
a.(b × c) is equal to [RPET 2001]
A)
b.(a × c) done
clear
B)
c.(b × a) done
clear
C)
It is obvious. done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer46)
If \[a,\,b,\,c\] are vectors such that \[[a\,b\,c\,]=4\], then \[[a\times b\,\,b\times c\,\,c\times a]\] = [AIEEE 2002]
A)
16 done
clear
B)
64 done
clear
C)
4 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer47)
The volume of the parallelopiped whose conterminous edges are \[i-j+k,\,\,2i-4j+5k\]and \[3i-5j+2k\] is [Kerala (Engg.) 2002]
A)
4 done
clear
B)
3 done
clear
C)
2 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer48)
\[[i\,\,k\,\,j]+[k\,\,j\,\,i]+[j\,\,k\,\,i]\] [UPSEAT 2002]
A)
1 done
clear
B)
3 done
clear
C)
? 3 done
clear
D)
? 1 done
clear
View Solution play_arrow
-
question_answer49)
If u, v and w are three non-coplanar vectors, then \[(u+v-w)\,.\,[(u-v)\times (v-w)]\] equals [AIEEE 2003; DCE 2005]
A)
0 done
clear
B)
\[u\,.\,(v\times w)\] done
clear
C)
\[u\,.\,(w\times v)\] done
clear
D)
\[3u\,.\,(v\times w)\] done
clear
View Solution play_arrow
-
question_answer50)
\[a\,.\,[(b+c)\times (a+b+c)]\] is equal to [IIT 1981; UPSEAT 2003; RPET 1988, 2002; MP PET 2004]
A)
[a b c] done
clear
B)
2[a b c] done
clear
C)
3[a b c] done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer51)
If the vectors \[4i+11j+mk,\,7i+2j+6k\] and \[i+5j+4k\] are coplanar, then m is [Karnataka CET 2003]
A)
38 done
clear
B)
0 done
clear
C)
10 done
clear
D)
? 10 done
clear
View Solution play_arrow
-
question_answer52)
Vector coplanar with vectors i + j and j + k and parallel to the vector 2i ? 2j ? 4k, is [Roorkee 2000]
A)
i ? k done
clear
B)
i ? j ? 2k done
clear
C)
i + j ? k done
clear
D)
3i + 3j ? 6k done
clear
View Solution play_arrow
-
question_answer53)
The value of l for which the four points \[2\mathbf{i}+3\mathbf{j}-\mathbf{k},\] \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], \[3\mathbf{i}+4\mathbf{j}-2\mathbf{k},\,\,\mathbf{i}-\lambda \mathbf{j}+6\mathbf{k}\] are coplanar [MP PET 2004]
A)
8 done
clear
B)
0 done
clear
C)
? 2 done
clear
D)
6 done
clear
View Solution play_arrow
-
question_answer54)
If \[\mathbf{a},\,\,\mathbf{b},\,\,\mathbf{c}\] are non-coplanar vectors and l is a real number, then the vectors \[\mathbf{a}+2\mathbf{b}+3\mathbf{c},\,\lambda \,\mathbf{b}+4\mathbf{c}\] and \[(2\lambda -1)\mathbf{c}\] are non-coplanar for [AIEEE 2004]
A)
No value of l done
clear
B)
All except one value of l done
clear
C)
All except two values of l done
clear
D)
All values of l done
clear
View Solution play_arrow
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question_answer55)
Let \[\mathbf{a},\,\mathbf{b}\] and \[\mathbf{c}\] be three vectors. Then scalar triple product x
is equal to [UPSEAT 2004]
A)
B)
C)
D)
View Solution play_arrow
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question_answer56)
. If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] then the value of [a b c] is equal to [Pb. CET 2000]
A)
1 done
clear
B)
? 1 done
clear
C)
\[|\mathbf{a}||\mathbf{b}||\mathbf{c}|\] done
clear
D)
0 done
clear
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question_answer57)
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\alpha \mathbf{j}\] are coplanar vectors, the value of \[\alpha \] is [UPSEAT 2004]
A)
\[-\frac{4}{3}\] done
clear
B)
\[\frac{3}{4}\] done
clear
C)
\[\frac{4}{3}\] done
clear
D)
2 done
clear
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question_answer58)
Out of the following which one is not true [Orissa JEE 2004]
A)
\[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})\] done
clear
B)
\[(\mathbf{b}\times \mathbf{c})\,.\,\mathbf{a}\] done
clear
C)
\[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}\] done
clear
D)
\[(\mathbf{a}.\mathbf{c})\,\times \,\mathbf{b}\] done
clear
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question_answer59)
If \[\mathbf{a}\] is perpendicular to \[\mathbf{b}\]and \[\mathbf{c},|\mathbf{a}|=2,|\mathbf{b}|=3\], \[|\mathbf{c}|=4\] and the angle between \[\mathbf{b}\] and \[\mathbf{c}\]is \[\frac{2\pi }{3}\], then \[[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]\] is equal to [Kerala (Engg.) 2005]
A)
\[4\sqrt{3}\] done
clear
B)
\[6\sqrt{3}\] done
clear
C)
\[12\sqrt{3}\] done
clear
D)
\[18\sqrt{3}\] done
clear
E)
\[8\sqrt{3}\] done
clear
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question_answer60)
If \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] are non-coplanar vectors and \[\lambda \] is a real number then \[[\lambda (\mathbf{a}+\mathbf{b})\,\,\,\,{{\lambda }^{2}}\mathbf{b}\,\,\,\,\,\lambda \mathbf{c}]=\left[ \mathbf{a}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{b} \right]\] for [AIEEE 2005]
A)
Exactly three values of \[\lambda \] done
clear
B)
Exactly two values of \[\lambda \] done
clear
C)
Exactly one value of \[\lambda \] done
clear
D)
No value of \[\lambda \] done
clear
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question_answer61)
If the vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,-\mathbf{i}+2\mathbf{j}+\lambda \mathbf{k}\] and \[-5\mathbf{i}+2\mathbf{j}-\mathbf{k}\] are coplanar, then the value of \[\lambda \] is equal [J & K 2005]
A)
? 13 done
clear
B)
13/9 done
clear
C)
? 13/9 done
clear
D)
? 9/13 done
clear
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question_answer62)
If a,b,c are three non-zero, non-coplanar vectors and \[{{b}_{1}}=b-\frac{b.a}{|a{{|}^{2}}}a,\,{{b}_{2}}=b+\frac{b.a}{|a{{|}^{2}}}a\],\[{{c}_{1}}=c-\frac{c.a}{|a{{|}^{2}}}a-\frac{c.b}{|b{{|}^{2}}}b\], \[{{c}_{2}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{1}}}{|{{b}_{1}}{{|}^{2}}}{{b}_{1}}\], \[{{c}_{3}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{2}}}{|{{b}_{2}}{{|}^{2}}}{{b}_{2}}\], \[{{c}_{4}}=a-\frac{c.a}{|a{{|}^{2}}}a\]. Then which of the following is a set of mutually orthogonal vectors is [IIT Screening 2005]
A)
\[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{1}}\}\] done
clear
B)
\[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{2}}\}\] done
clear
C)
\[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{3}}\}\] done
clear
D)
\[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{4}}\}\] done
clear
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question_answer63)
If a vector a lie in the plane b and g then which is correct [Orissa JEE 2005]
A)
\[[\alpha \,\,\beta \,\,\gamma ]=0\] done
clear
B)
\[[\alpha \,\,\beta \,\,\gamma ]=1\] done
clear
C)
\[[\alpha \,\,\beta \,\,\gamma ]=3\] done
clear
D)
\[[\beta \,\,\gamma \,\,\alpha ]=1\] done
clear
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