-
question_answer1)
Three forces of magnitudes 1, 2, 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is [MNR 1987]
A)
114 dyne done
clear
B)
6 dyne done
clear
C)
5 dyne done
clear
D)
None of these done
clear
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question_answer2)
The vectors b and c are in the direction of north-east and north-west respectively and |b|=|c|= 4. The magnitude and direction of the vector d = c - b, are [Roorkee 2000]
A)
\[4\sqrt{2},\] towards north done
clear
B)
\[4\sqrt{2}\], towards west done
clear
C)
4, towards east done
clear
D)
4, towards south done
clear
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question_answer3)
If a, b and c are unit vectors, then \[|\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}}\] does not exceed [IIT Screening 2001]
A)
4 done
clear
B)
9 done
clear
C)
8 done
clear
D)
6 done
clear
View Solution play_arrow
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question_answer4)
The vectors \[\overrightarrow{AB}=3\mathbf{i}+5\mathbf{j}+4\mathbf{k}\] and \[\overrightarrow{AC}=5\mathbf{i}-5\mathbf{j}+2\mathbf{k}\] are the sides of a triangle ABC. The length of the median through A is [UPSEAT 2004]
A)
\[\sqrt{13}\] unit done
clear
B)
\[\theta \] unit done
clear
C)
5 unit done
clear
D)
10 unit done
clear
View Solution play_arrow
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question_answer5)
Let the value of \[\mathbf{p}=(x+4y)\,\mathbf{a}+(2x+y+1)\,\mathbf{b}\] and \[\mathbf{q}=(y-2x+2)\,\mathbf{a}+(2x-3y-1)\,\mathbf{b},\] where a and b are non-collinear vectors. If \[3\mathbf{p}=2\mathbf{q},\] then the value of x and y will be [RPET 1984; MNR 1984]
A)
- 1, 2 done
clear
B)
2, - 1 done
clear
C)
1, 2 done
clear
D)
2, 1 done
clear
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question_answer6)
The points D, E, F divide BC, CA and AB of the triangle ABC in the ratio 1 : 4, 3 : 2 and 3 : 7 respectively and the point K divides AB in the ratio \[1:3\], then \[(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF})\,\,:\,\,\overrightarrow{CK}\] is equal to [MNR 1987]
A)
1 : 1 done
clear
B)
2 : 5 done
clear
C)
5 : 2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer7)
If two vertices of a triangle are \[\mathbf{i}-\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\], then the third vertex can be [Roorkee 1995]
A)
\[\mathbf{i}+\mathbf{k}\] done
clear
B)
\[\mathbf{i}-2\mathbf{j}-\mathbf{k}\] done
clear
C)
\[\mathbf{i}-\mathbf{k}\] done
clear
D)
\[2\mathbf{i}-\mathbf{j}\] done
clear
E)
All the above done
clear
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question_answer8)
If a of magnitude 50 is collinear with the vector \[\mathbf{b}=6\,\mathbf{i}-8\,\mathbf{j}-\frac{15\,\mathbf{k}}{2},\] and makes an acute angle with the positive direction of z-axis, then the vector a is equal to [Pb. CET 2004]
A)
\[24\,\mathbf{i}-32\,\mathbf{j}+30\,\mathbf{k}\] done
clear
B)
\[-24\,\mathbf{i}+32\,\mathbf{j}+30\,\mathbf{k}\] done
clear
C)
\[16\,\mathbf{i}-16\,\mathbf{j}-15\,\mathbf{k}\] done
clear
D)
\[-12\,\mathbf{i}+16\,\mathbf{j}-30\,\mathbf{k}\] done
clear
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question_answer9)
If three non-zero vectors are \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k},\] \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\] and \[\mathbf{c}={{c}_{1}}\mathbf{i}+{{c}_{2}}\mathbf{j}+{{c}_{3}}\mathbf{k}.\] If c is the unit vector perpendicular to the vectors a and b and the angle between a and b is \[\frac{\pi }{6},\] then \[{{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix}\, \right|}^{2}}\] is equal to [IIT 1986]
A)
0 done
clear
B)
\[\frac{3\,(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})\,(\Sigma c_{1}^{2})}{4}\] done
clear
C)
1 done
clear
D)
\[\frac{(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})}{4}\] done
clear
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question_answer10)
Let the unit vectors a and b be perpendicular and the unit vector c be inclined at an angle q to both a and b. If \[\mathbf{c}=\alpha \,\mathbf{a}+\beta \,\mathbf{b}+\gamma \,(\mathbf{a}\times \mathbf{b}),\] then [Orissa JEE 2003]
A)
\[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \] done
clear
B)
\[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=-\cos \,\,2\theta \] done
clear
C)
\[\alpha =\cos \theta ,\,\,\beta =\sin \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \] done
clear
D)
None of these done
clear
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question_answer11)
The vector \[\mathbf{a}+\mathbf{b}\] bisects the angle between the vectors a and b, if
A)
\[|\mathbf{a}|\,=\,|\mathbf{b}|\] done
clear
B)
\[|\mathbf{a}|\,=\,|\mathbf{b}|\] or angle between a and b is zero done
clear
C)
\[|\mathbf{a}|\,\,=m\,|\mathbf{b}|\] done
clear
D)
None of these done
clear
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question_answer12)
The points \[O,\,A,\,B,\,C,\,D\] are such that \[\overrightarrow{OA}=\mathbf{a},\] \[\overrightarrow{OB}=\mathbf{b},\,\] \[\overrightarrow{OC}=2\mathbf{a}+3\mathbf{b}\] and \[\overrightarrow{OD}=\mathbf{a}-2\mathbf{b}.\] If \[|\mathbf{a}|\,=3\,|\mathbf{b}|,\] then the angle between \[\overrightarrow{BD}\] and \[\overrightarrow{AC}\] is
A)
\[\frac{\pi }{3}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{6}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
If \[\overrightarrow{A}=\mathbf{i}+2\mathbf{j}+3\mathbf{k},\,\,\,\overrightarrow{B}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\overrightarrow{C}=3\mathbf{i}+\mathbf{j},\] then the value of t such that \[\overrightarrow{A}+t\overrightarrow{B}\] is at right angle to vector \[3\mathbf{i}+4\mathbf{j}\] is [RPET 2002]
A)
2 done
clear
B)
4 done
clear
C)
5 done
clear
D)
6 done
clear
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question_answer14)
Let \[\mathbf{b}=4\mathbf{i}+3\mathbf{j}\] and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by [IIT 1987]
A)
\[2\mathbf{i}-\mathbf{j},\,\,\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done
clear
B)
\[2\mathbf{i}+\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done
clear
C)
\[2\mathbf{i}+\mathbf{j},\,-\frac{2}{5}\mathbf{i}-\frac{11}{5}\mathbf{j}\] done
clear
D)
\[2\mathbf{i}-\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done
clear
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question_answer15)
Let \[\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\mathbf{j}-2\mathbf{k}\] be three vectors. A vector in the plane of b and c whose projection on a is of magnitude \[\sqrt{2/3}\] is [IIT 1993; Pb. CET 2004]
A)
\[2\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] done
clear
B)
\[2\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] done
clear
C)
\[-\,2\mathbf{i}-\mathbf{j}+5\mathbf{k}\] done
clear
D)
\[2\mathbf{i}+\mathbf{j}+5\mathbf{k}\] done
clear
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question_answer16)
A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then [IIT 1984]
A)
\[p=0\] done
clear
B)
\[p=1\] or \[-\frac{1}{3}\] done
clear
C)
\[p=-1\] or \[\frac{1}{3}\] done
clear
D)
\[p=1\] or \[-1\] done
clear
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question_answer17)
If \[\mathbf{u}=2\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and \[\mathbf{v}=6\,\mathbf{i}-3\,\mathbf{j}+2\,\mathbf{k},\] then a unit vector perpendicular to both u and v is [MP PET 1987]
A)
\[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\] done
clear
B)
\[\frac{1}{\sqrt{17}}\,\left( \frac{1}{5}\mathbf{i}-2\mathbf{j}-\frac{18}{5}\mathbf{k} \right)\] done
clear
C)
\[\frac{1}{\sqrt{473}}\,(7\mathbf{i}-10\mathbf{j}-18\mathbf{k})\] done
clear
D)
None of these done
clear
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question_answer18)
If \[\mathbf{a}=2\mathbf{i}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=4\mathbf{i}-3\mathbf{j}+7\mathbf{k}.\] If \[\mathbf{d}\times \mathbf{b}=\mathbf{c}\times \mathbf{b}\] and \[\mathbf{d}\,.\,\mathbf{a}=0,\] then d will be [IIT 1990]
A)
\[\mathbf{i}+8\mathbf{j}+2\mathbf{k}\] done
clear
B)
\[\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] done
clear
C)
\[-\mathbf{i}+8\mathbf{j}-\mathbf{k}\] done
clear
D)
\[-\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] done
clear
View Solution play_arrow
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question_answer19)
If \[\mathbf{a}\times \mathbf{r}=\mathbf{b}+\lambda \mathbf{a}\] and \[\mathbf{a}\,\,.\,\,\mathbf{r}=3,\] where \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=-\mathbf{i}-2\mathbf{j}+\mathbf{k},\] then r and l are equal to
A)
\[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] done
clear
B)
\[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{5}{6}\] done
clear
C)
\[\mathbf{r}=\frac{6}{7}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] done
clear
D)
None of these done
clear
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question_answer20)
Let the vectors a, b, c and d be such that\[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=0\]. Let \[{{P}_{1}}\] and \[{{P}_{2}}\] be planes determined by pair of vectors a, b and c, d respectively. Then the angle between \[{{P}_{1}}\] and \[{{P}_{2}}\] is [IIT Screening 2000; MP PET 2004]
A)
\[{{0}^{o}}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{3}\] done
clear
D)
\[\frac{\pi }{2}\] done
clear
View Solution play_arrow
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question_answer21)
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{a}\,.\,\mathbf{b}=1\] and \[\mathbf{a}\times \mathbf{b}=\mathbf{j}-\mathbf{k},\] then \[\mathbf{b}=\] [IIT Screening 2004]
A)
\[\mathbf{i}\] done
clear
B)
\[\mathbf{i}-\mathbf{j}+\mathbf{k}\] done
clear
C)
\[2\mathbf{j}-\mathbf{k}\] done
clear
D)
\[2\mathbf{i}\] done
clear
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question_answer22)
The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]
A)
\[\frac{1}{4}\,|a\times b+b\times d+d\times a|\] done
clear
B)
\[\frac{1}{4}\,\left| b\times c+c\times d+a\times d+b\times a \right|\] done
clear
C)
\[\frac{1}{4}\,\left| a\times b+b\times c+c\times d+d\times a \right|\] done
clear
D)
\[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\]. done
clear
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question_answer23)
The moment about the point \[M(-2,\,4,\,-6)\] of the force represented in magnitude and position by \[\overrightarrow{AB}\] where the points A and B have the co-ordinates \[(1,\,2,\,-3)\] and \[(3,\,-4,\,2)\] respectively, is [MP PET 2000]
A)
8i - 9j - 14k done
clear
B)
2i - 6j + 5k done
clear
C)
- 3i + 2j - 3k done
clear
D)
- 5i + 8j - 8k done
clear
View Solution play_arrow
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question_answer24)
If the vectors \[a\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}+b\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}+c\mathbf{k}\] \[(a\ne b\ne c\ne 1)\] are coplanar, then the value of \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\] [BIT Ranchi 1988; RPET 1987; IIT 1987; DCE 2001; MP PET 2004; Orissa JEE 2005]
A)
- 1 done
clear
B)
\[-\frac{1}{2}\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer25)
If \[\alpha \,(\mathbf{a}\times \mathbf{b})+\beta \,(\mathbf{b}\times \mathbf{c})+\gamma \,(\mathbf{c}\times \mathbf{a})=\mathbf{0}\] and at least one of the numbers \[\alpha ,\,\,\beta \] and \[\gamma \] is non-zero, then the vectors a, b and c are
A)
Perpendicular done
clear
B)
Parallel done
clear
C)
Coplanar done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
The volume of the tetrahedron, whose vertices are given by the vectors \[-\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}-\mathbf{k}\]with reference to the fourth vertex as origin, is
A)
\[\frac{5}{3}\]cubic unit done
clear
B)
\[\frac{2}{3}\] cubic unit done
clear
C)
\[\frac{3}{5}\]cubic unit done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer27)
Let \[\mathbf{a}=\mathbf{i}-\mathbf{j},\,\,\mathbf{b}=\mathbf{j}-\mathbf{k},\,\,\mathbf{c}=\mathbf{k}-\mathbf{i}.\] If \[\mathbf{\hat{d}}\] is a unit vector such that \[\mathbf{a}\,.\,\mathbf{\hat{d}}=0=[\mathbf{b}\,\,\mathbf{c}\,\,\mathbf{\hat{d}}],\] then \[\mathbf{\hat{d}}\] is equal to [IIT 1995]
A)
\[\pm \frac{\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] done
clear
B)
\[\pm \frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
C)
\[\pm \frac{\mathbf{i}+\mathbf{j}-2\mathbf{k}}{\sqrt{6}}\] done
clear
D)
\[\pm \,\,\mathbf{k}\] done
clear
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question_answer28)
The value of 'a' so that the volume of parallelopiped formed by \[\mathbf{i}+a\mathbf{j}+\mathbf{k},\mathbf{j}+a\,\mathbf{k}\] and \[a\,\mathbf{i}+\mathbf{k}\] becomes minimum is [IIT Screening 2003]
A)
- 3 done
clear
B)
3 done
clear
C)
\[\frac{1}{\sqrt{3}}\] done
clear
D)
\[\sqrt{3}\] done
clear
View Solution play_arrow
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question_answer29)
If b and c are any two non-collinear unit vectors and a is any vector, then \[(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}+(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{c}+\frac{\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})}{|\mathbf{b}\times \mathbf{c}|}\,(\mathbf{b}\times \mathbf{c})=\] [IIT 1996]
A)
a done
clear
B)
b done
clear
C)
c done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer30)
If a, b, c are non-coplanar unit vectors such that \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\], then the angle between a and b is [IIT 1995]
A)
\[\frac{\pi }{4}\] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{3\pi }{4}\] done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
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question_answer31)
\[[(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c})\,(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})\,(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})]=\,\]
A)
\[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] done
clear
B)
\[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{3}}\] done
clear
C)
\[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{4}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer32)
Unit vectors a, b and c are coplanar. A unit vector d is perpendicular to them. If \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\frac{1}{6}\mathbf{i}-\frac{1}{3}\mathbf{j}+\frac{1}{3}\mathbf{k}\] and the angle between a and b is \[{{30}^{o}}\], then c is [Roorkee Qualifying 1998]
A)
\[\frac{(\mathbf{i}-2\mathbf{j}+2\mathbf{k})}{3}\] done
clear
B)
\[\frac{(2\mathbf{i}+\mathbf{j}-\mathbf{k})}{3}\] done
clear
C)
\[\frac{(-\mathbf{i}+2\mathbf{j}-2\mathbf{k})}{3}\] done
clear
D)
\[\frac{(-\mathbf{i}+2\mathbf{j}+\mathbf{k})}{3}\] done
clear
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question_answer33)
The radius of the circular section of the sphere \[|\mathbf{r}|\,=5\]by the plane \[\mathbf{r}\,.\,(\mathbf{i}+\mathbf{j}+\mathbf{k})=3\sqrt{3}\] is [DCE 1999]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer34)
If \[\mathbf{x}\] is parallel to \[\mathbf{y}\] and \[\mathbf{z}\] where \[\mathbf{x}=2\mathbf{i}+\mathbf{j}+\alpha \mathbf{k}\], \[\mathbf{y}=\alpha \mathbf{i}+\mathbf{k}\] and \[\mathbf{z}=5\mathbf{i}-\mathbf{j}\], then \[\alpha \] is equal to [J & K 2005]
A)
\[\pm \sqrt{5}\] done
clear
B)
\[\pm \sqrt{6}\] done
clear
C)
\[\pm \sqrt{7}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer35)
The vector c directed along the internal bisector of the angle between the vectors \[\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}\] and \[\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}\] with \[|\mathbf{c}|\,=5\sqrt{6},\] is
A)
\[\frac{5}{3}\,(\mathbf{i}-7\mathbf{j}+2\mathbf{k})\] done
clear
B)
\[\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] done
clear
C)
\[\frac{5}{3}\,(\mathbf{i}+7\mathbf{j}+2\mathbf{k})\] done
clear
D)
\[\frac{5}{3}\,(-5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] done
clear
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question_answer36)
The distance of the point \[B\,(\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] from the line which is passing through \[A\,(4\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] and which is parallel to the vector \[\overrightarrow{C}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}\] is [Roorkee 1993]
A)
10 done
clear
B)
\[\sqrt{10}\] done
clear
C)
100 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer37)
Let a, b, c are three non-coplanar vectors such that
\[{{\mathbf{r}}_{1}}=\mathbf{a}-\mathbf{b}+\mathbf{c},\,\,{{\mathbf{r}}_{2}}=\mathbf{b}+\mathbf{c}-\mathbf{a},\,\,{{\mathbf{r}}_{3}}=\mathbf{c}+\mathbf{a}+\mathbf{b},\] |
\[\mathbf{r}=2\mathbf{a}-3\mathbf{b}+4\mathbf{c}.\] If \[\mathbf{r}={{\lambda }_{1}}{{\mathbf{r}}_{1}}+{{\lambda }_{2}}{{\mathbf{r}}_{2}}+{{\lambda }_{3}}{{\mathbf{r}}_{3}},\] then |
A)
\[{{\lambda }_{1}}=7\] done
clear
B)
\[{{\lambda }_{1}}+{{\lambda }_{3}}=3\] done
clear
C)
\[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4\] done
clear
D)
\[{{\lambda }_{3}}+{{\lambda }_{2}}=2\] done
clear
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question_answer38)
Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and a unit vector c be coplanar. If c is perpendicular to a, then c = [IIT 1999; Pb. CET 2003; DCE 2005]
A)
\[\frac{1}{\sqrt{2}}(-\mathbf{j}+\mathbf{k})\] done
clear
B)
\[\frac{1}{\sqrt{3}}(-\mathbf{i}-\mathbf{j}-\mathbf{k})\] done
clear
C)
\[\frac{1}{\sqrt{5}}\,(\mathbf{i}-2\mathbf{j})\] done
clear
D)
\[\frac{1}{\sqrt{3}}(\mathbf{i}-\mathbf{j}-\mathbf{k})\] done
clear
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question_answer39)
Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation \[\mathbf{p}\times \{(\mathbf{x}-\mathbf{q})\times \mathbf{p}\}+\mathbf{q}\times \{(\mathbf{x}-\mathbf{r})\times \mathbf{q}\}+\mathbf{r}\times \{(\mathbf{x}-\mathbf{p})\times \mathbf{r}\}=0,\] then x is given by [IIT 1997 Cancelled]
A)
\[\frac{1}{2}\,(\mathbf{p}+\mathbf{q}-2\mathbf{r})\] done
clear
B)
\[\frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] done
clear
C)
\[\frac{1}{3}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] done
clear
D)
\[\frac{1}{3}(2\mathbf{p}+\mathbf{q}-\mathbf{r})\] done
clear
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question_answer40)
The point of intersection of \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\] and \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\], where \[\mathbf{a}=\mathbf{i}+\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}-\mathbf{k}\] is [Orissa JEE 2004]
A)
\[3\mathbf{i}+\mathbf{j}-\mathbf{k}\] done
clear
B)
\[3\mathbf{i}-\mathbf{k}\] done
clear
C)
\[3\mathbf{i}+2\mathbf{j}+\mathbf{k}\] done
clear
D)
None of these done
clear
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