-
question_answer1)
The locus of a point equidistant from two given points a and b is given by
A)
\[[\mathbf{r}-\frac{1}{2}(\mathbf{a}+\mathbf{b})]\,.\,\,(\mathbf{a}-\mathbf{b})=0\] done
clear
B)
\[[\mathbf{r}-\frac{1}{2}(\mathbf{a}-\mathbf{b})]\,.\,\,(\mathbf{a}+\mathbf{b})=0\] done
clear
C)
\[[\mathbf{r}-\frac{1}{2}(\mathbf{a}+\mathbf{b})].(\mathbf{a}+\mathbf{b})=0\] done
clear
D)
\[[\mathbf{r}-\frac{1}{2}(\mathbf{a}-\mathbf{b})]\,.\,\,(\mathbf{a}-\mathbf{b})=0\] done
clear
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question_answer2)
If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\] is given by
A)
\[\mathbf{r}=x\mathbf{a}+\frac{1}{\mathbf{a}\,.\,\,\mathbf{a}}(\mathbf{a}\times \mathbf{b})\] done
clear
B)
\[\mathbf{r}=x\mathbf{b}-\frac{1}{\mathbf{b}\,.\,\,\mathbf{b}}(\mathbf{a}\times \mathbf{b})\] done
clear
C)
\[\mathbf{r}=x\mathbf{a}\times \mathbf{b}\] done
clear
D)
\[\mathbf{r}=x\mathbf{b}\times \mathbf{a}\] done
clear
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question_answer3)
If r be position vector of any point on a sphere and a, b are respectively position vectors of the extremities of a diameter, then [MP PET 1994]
A)
\[\mathbf{r}\,.\,(\mathbf{a}-\mathbf{b})=0\] done
clear
B)
\[\mathbf{r}\,.\,(\mathbf{r}-\mathbf{a})=0\] done
clear
C)
\[(\mathbf{r}+\mathbf{a})\,.\,(\mathbf{r}+\mathbf{b})=0\] done
clear
D)
\[(\mathbf{r}-\mathbf{a})\,.\,(\mathbf{r}-\mathbf{b})=0\] done
clear
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question_answer4)
Angle between the line \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (\mathbf{i}-\mathbf{j}+\mathbf{k})\] and the normal to the plane \[\mathbf{r}\,.\,(2\mathbf{i}-\mathbf{j}+\mathbf{k})=4\] is [MP PET 1997]
A)
\[{{\sin }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done
clear
B)
\[{{\cos }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done
clear
C)
\[{{\tan }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done
clear
D)
\[{{\cot }^{-1}}\,\left( \frac{2\sqrt{2}}{3} \right)\] done
clear
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question_answer5)
If the equation of a line through a point a and parallel to vector b is \[\mathbf{r}=\mathbf{a}+t\,\mathbf{b},\] where t is a parameter, then its perpendicular distance from the point c is [MP PET 1998]
A)
\[|(\mathbf{c}-\mathbf{b})\times \mathbf{a}|\div |\mathbf{a}|\] done
clear
B)
\[|(\mathbf{c}-\mathbf{a})\times \mathbf{b}|\div |\mathbf{b}|\] done
clear
C)
\[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{c}|\] done
clear
D)
\[|(\mathbf{a}-\mathbf{b})\times \mathbf{c}|\div |\mathbf{a}+\mathbf{c}|\] done
clear
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question_answer6)
If \[a=i+j\] and \[b=2i-k\] are two vectors, then the point of intersection of two lines \[r\times a=b\times a\] and \[r\times b=a\times b\] is [RPET 2000]
A)
i + j ? k done
clear
B)
i ? j + k done
clear
C)
3i + j ? k done
clear
D)
3i ? j + k done
clear
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question_answer7)
If a, b, c are three non-coplanar vectors, then the vector equation \[\mathbf{r}=(1-\mathbf{p}-\mathbf{q})\,\mathbf{a}+p\mathbf{b}+q\mathbf{c}\] represents a [EAMCET 2003]
A)
Straight line done
clear
B)
Plane done
clear
C)
Plane passing through the origin done
clear
D)
Sphere done
clear
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question_answer8)
The vector equation of the line joining the points \[i-2j+k\] and \[-2j+3k\] is [MP PET 2003]
A)
\[r=t(i+j+k)\] done
clear
B)
\[r={{t}_{1}}(i-2j+k)+{{t}_{2}}(3k-2j)\] done
clear
C)
\[r=(i-2j+k)+t(2k-i)\] done
clear
D)
\[r=t(2k-i)\] done
clear
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question_answer9)
The spheres\[{{r}^{2}}+2{{u}_{1}}\,.\,r+2{{d}_{1}}=0\] and \[{{r}^{2}}+2{{u}_{2}}\,.\,r+2{{d}_{2}}=0\] cut orthogonally, if [AMU 1999]
A)
\[{{u}_{1}}\,.\,{{u}_{2}}=0\] done
clear
B)
\[{{u}_{1}}+{{u}_{2}}=0\] done
clear
C)
\[{{u}_{1}}\,.\,{{u}_{2}}={{d}_{1}}+{{d}_{2}}\] done
clear
D)
\[({{u}_{1}}-{{u}_{2}})\,.\,({{u}_{1}}+{{u}_{2}})=d_{1}^{2}+d_{2}^{2}\] done
clear
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question_answer10)
A tetrahedron has vertices at \[O(0,\,0,\,0)\], \[A(1,\,2,\,1),B(2,\,1,\,3)\] and \[C(-1,\,1,\,2)\]. Then the angle between the faces \[OAB\]and \[ABC\]will be [MNR 1994; UPSEAT 2000; AIEEE 2003]
A)
\[{{\cos }^{-1}}\left( \frac{19}{35} \right)\] done
clear
B)
\[{{\cos }^{-1}}\left( \frac{17}{31} \right)\] done
clear
C)
\[30{}^\circ \] done
clear
D)
\[90{}^\circ \] done
clear
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question_answer11)
A vector \[\mathbf{n}\] of magnitude 8 units is inclined to x-axis at \[{{45}^{o}}\], y-axis at \[{{60}^{o}}\] and an acute angle with z-axis. If a plane passes through a point \[(\sqrt{2},\,-1,\,1)\] and is normal to \[\mathbf{n}\], then its equation in vector form is
A)
\[\mathbf{r}.(\sqrt{2}\mathbf{i}+\mathbf{j}+\mathbf{k})=4\] done
clear
B)
\[\mathbf{r}.(\sqrt{2}\mathbf{i}+\mathbf{j}+\mathbf{k})=2\] done
clear
C)
\[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=4\] done
clear
D)
None of these done
clear
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question_answer12)
The vector equation of a plane, which is at a distance of 8 unit from the origin and which is normal to the vector \[2\mathbf{i}+\mathbf{j}+2\mathbf{k},\] is
A)
\[\mathbf{r}.(2\mathbf{i}+\mathbf{j}+\mathbf{k})=24\] done
clear
B)
\[\mathbf{r}.(2\mathbf{i}+\mathbf{j}+2\mathbf{k})=24\] done
clear
C)
\[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=24\] done
clear
D)
None of these done
clear
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question_answer13)
The distance of the point \[2\mathbf{i}+\mathbf{j}-\mathbf{k}\] from the plane \[\mathbf{r}.(\mathbf{i}-2\mathbf{j}+4\mathbf{k})=9\] is
A)
\[\frac{13}{\sqrt{21}}\] done
clear
B)
\[\frac{13}{21}\] done
clear
C)
\[\frac{13}{3\sqrt{21}}\] done
clear
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question_answer14)
The centre of the circle given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}+2\mathbf{k})=15\] and \[|\mathbf{r}-(\mathbf{j}+2\mathbf{k})|=4\]is
A)
(0, 1, 2) done
clear
B)
(1, 3, 4) done
clear
C)
(?1, 3, 4) done
clear
D)
None of these done
clear
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question_answer15)
A vector r is equally inclined with the co-ordinate axes. If the tip of r is in the positive octant and |r| = 6, then \[\mathbf{r}\] is
A)
\[2\sqrt{3}(\mathbf{i}-\mathbf{j}+\mathbf{k})\] done
clear
B)
\[2\sqrt{3}(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
C)
\[2\sqrt{3}(\mathbf{i}+\mathbf{j}-\mathbf{k})\] done
clear
D)
\[2\sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
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question_answer16)
The position vectors of two points P and Q are \[3\mathbf{i}+\mathbf{j}+2\mathbf{k}\] and \[\mathbf{i}-2\mathbf{j}-4\mathbf{k}\] respectively. The equation of the plane through Q and perpendicular to PQ is
A)
\[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})=28\] done
clear
B)
\[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})=32\] done
clear
C)
\[\mathbf{r}.(2\mathbf{i}+3\mathbf{j}+6\mathbf{k})+28=0\] done
clear
D)
None of these done
clear
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question_answer17)
The vector equation of the plane passing through the origin and the line of intersection of the plane \[\mathbf{r}.\mathbf{a}=\lambda \] and \[\mathbf{r}.\mathbf{b}=\mu \] is
A)
\[\mathbf{r}.(\lambda \mathbf{a}-\mu \mathbf{b})=0\] done
clear
B)
\[\mathbf{r}.\,(\lambda \mathbf{b}-\mu \mathbf{a})=0\] done
clear
C)
\[\mathbf{r}.(\lambda \mathbf{a}+\mu \mathbf{b})=0\] done
clear
D)
\[\mathbf{r}.(\lambda \mathbf{b}+\mu \mathbf{a})=0\] done
clear
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question_answer18)
The position vectors of points A and B are \[\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] respectively. The equation of a plane is \[\mathbf{r}.(5\mathbf{i}+2\mathbf{j}-7\mathbf{k})+9=0\]. The points A and B
A)
Lie on the plane done
clear
B)
Are on the same side of the plane done
clear
C)
Are on the opposite side of the plane done
clear
D)
None of these done
clear
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question_answer19)
The vector equation of the plane through the point \[2\mathbf{i}-\mathbf{j}-4\mathbf{k}\] and parallel to the plane \[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})-7=0\] is
A)
\[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=0\] done
clear
B)
\[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=32\] done
clear
C)
\[\mathbf{r}.(4\mathbf{i}-12\mathbf{j}-3\mathbf{k})=12\] done
clear
D)
None of these done
clear
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question_answer20)
The vector equation of the plane through the point (2, 1, ?1) and passing through the line of intersection of the plane \[\mathbf{r}.(\mathbf{i}+3\mathbf{j}-\mathbf{k})=0\] and \[\mathbf{r}.(\mathbf{j}+2\mathbf{k})=0\] is
A)
\[\mathbf{r}.(\mathbf{i}+9\mathbf{j}+11\mathbf{k})=0\] done
clear
B)
\[\mathbf{r}.(\mathbf{i}+9\mathbf{j}+11\mathbf{k})=6\] done
clear
C)
\[\mathbf{r}.(\mathbf{i}-3\mathbf{j}-13\mathbf{k})=0\] done
clear
D)
None of these done
clear
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question_answer21)
The vector equation of the plane through the point \[\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and perpendicular to the line of intersection of the planes \[\mathbf{r}.(3\mathbf{i}-\mathbf{j}+\mathbf{k})=1\] and \[\mathbf{i}+4\mathbf{j}-2\mathbf{k}=2\] is
A)
\[\mathbf{r}.(2\mathbf{i}+7\mathbf{j}-13\mathbf{k})=1\] done
clear
B)
\[\mathbf{r}.(2\mathbf{i}-7\mathbf{j}-13\mathbf{k})=1\] done
clear
C)
\[\mathbf{r}.(2\mathbf{i}+7\mathbf{j}+13\mathbf{k})=0\] done
clear
D)
None of these done
clear
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question_answer22)
The equation of the plane containing the lines \[\mathbf{r}={{\mathbf{a}}_{1}}+\lambda {{\mathbf{a}}_{2}}\] and \[\mathbf{r}={{\mathbf{a}}_{2}}+\lambda {{\mathbf{a}}_{1}}\] is
A)
\[[\mathbf{r}\,\ {{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]=0\] done
clear
B)
\[[\mathbf{r}\ \,{{\mathbf{a}}_{1}}\ \,{{\mathbf{a}}_{2}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] done
clear
C)
\[[\mathbf{r}\ \,{{\mathbf{a}}_{2}}\ \,{{\mathbf{a}}_{1}}]={{\mathbf{a}}_{1}}.\ {{\mathbf{a}}_{2}}\] done
clear
D)
None of these done
clear
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question_answer23)
The vector equation of the plane containing the lines \[\mathbf{r}=(\mathbf{i}+\mathbf{j})+\lambda (\mathbf{i}+2\mathbf{j}-\mathbf{k})\] and \[\mathbf{r}=(\mathbf{i}+\mathbf{j})+\mu (-\mathbf{i}+\mathbf{j}-2\mathbf{k})\] is
A)
\[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=0\] done
clear
B)
\[\mathbf{r}.(\mathbf{i}-\mathbf{j}-\mathbf{k})=0\] done
clear
C)
\[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=3\] done
clear
D)
None of these done
clear
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question_answer24)
The cartesian equation of the plane \[\mathbf{r}=(1+\lambda -\mu )\mathbf{i}+(2-\lambda )\mathbf{j}+(3-2\lambda +2\mu )\mathbf{k}\] is
A)
\[2x+y=5\] done
clear
B)
\[2x-y=5\] done
clear
C)
\[2x+z=5\] done
clear
D)
\[2x-z=5\] done
clear
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question_answer25)
The length of the perpendicular from the origin to the plane passing through three non-collinear points \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] is
A)
\[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{c}\times \mathbf{a}+\mathbf{b}\times \mathbf{c}|}\] done
clear
B)
\[\frac{2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done
clear
C)
\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done
clear
D)
None of these done
clear
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question_answer26)
The length of the perpendicular from the origin to the plane passing through the point a and containing the line \[\mathbf{r}=\mathbf{b}+\lambda \mathbf{c}\] is
A)
\[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done
clear
B)
\[\frac{\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}|}\] done
clear
C)
\[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\] done
clear
D)
\[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{c}\times \mathbf{a}+\mathbf{a}\times \mathbf{b}|}\] done
clear
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question_answer27)
The position vector of a point at a distance of \[3\sqrt{11}\] units from \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] on a line passing through the points \[\mathbf{i}-\mathbf{j}+2\mathbf{k}\] and \[3\mathbf{i}+\mathbf{j}+\mathbf{k}\] is
A)
\[10\mathbf{i}+2\mathbf{j}-5\mathbf{k}\] done
clear
B)
\[-8\mathbf{i}-4\mathbf{j}-\mathbf{k}\] done
clear
C)
\[8\mathbf{i}+4\mathbf{j}+\mathbf{k}\] done
clear
D)
\[-10\mathbf{i}-2\mathbf{j}-5\mathbf{k}\] done
clear
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question_answer28)
The line joining the points \[6\mathbf{a}-4\mathbf{b}+4\mathbf{c},\,-4\mathbf{c}\] and the line joining the points \[-\mathbf{a}-2\mathbf{b}-3\mathbf{c},\,\mathbf{a}+2\mathbf{b}-5\mathbf{c}\] intersect at
A)
\[-4\mathbf{a}\] done
clear
B)
\[4\mathbf{a}-\mathbf{b}-\mathbf{c}\] done
clear
C)
\[4\mathbf{c}\] done
clear
D)
None of these done
clear
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question_answer29)
Angle between the line \[\mathbf{r}=(2\mathbf{i}-\mathbf{j}+\mathbf{k})+\lambda (-\mathbf{i}+\mathbf{j}+\mathbf{k})\] and the plane \[\mathbf{r}.(3\mathbf{i}+2\mathbf{j}-\mathbf{k})=4\] is
A)
\[{{\cos }^{-1}}\left( \frac{2}{\sqrt{42}} \right)\] done
clear
B)
\[{{\cos }^{-1}}\left( \frac{-2}{\sqrt{42}} \right)\] done
clear
C)
\[{{\sin }^{-1}}\left( \frac{2}{\sqrt{42}} \right)\] done
clear
D)
\[{{\sin }^{-1}}\left( \frac{-2}{\sqrt{42}} \right)\] done
clear
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question_answer30)
The line through \[\mathbf{i}+3\mathbf{j}+2\mathbf{k}\] and perpendicular to the lines \[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})\] and \[\mathbf{r}=(2\mathbf{i}+6\mathbf{j}+\mathbf{k})+\mu (\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is
A)
\[\mathbf{r}=(\mathbf{i}+2\mathbf{j}-\mathbf{k})+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] done
clear
B)
\[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}-5\mathbf{j}+3\mathbf{k})\] done
clear
C)
\[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (\mathbf{i}+5\mathbf{j}+3\mathbf{k})\] done
clear
D)
\[\mathbf{r}=\mathbf{i}+3\mathbf{j}+2\mathbf{k}+\lambda (-\mathbf{i}+5\mathbf{j}-3\mathbf{k})\] done
clear
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question_answer31)
The distance from the point \[-\mathbf{i}+2\mathbf{j}+6\mathbf{k}\] to the straight line through the point (2, 3, ?4) and parallel to the vector \[6\mathbf{i}+3\mathbf{j}-4\mathbf{k}\] is
A)
7 done
clear
B)
10 done
clear
C)
9 done
clear
D)
None of these done
clear
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question_answer32)
The position vector of the point in which the line joining the points \[\mathbf{i}-2\mathbf{j}+\mathbf{k}\] and \[3\mathbf{k}-2\mathbf{j}\] cuts the plane through the origin and the points \[4\mathbf{j}\] and \[2\mathbf{i}+\mathbf{k}\], is
A)
\[6\mathbf{i}-10\mathbf{j}+3\mathbf{k}\] done
clear
B)
\[\frac{1}{5}(6\mathbf{i}-10\mathbf{j}+3\mathbf{k})\] done
clear
C)
\[-6\mathbf{i}+10\mathbf{j}-3\mathbf{k}\] done
clear
D)
None of these done
clear
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question_answer33)
The distance between the planes given by \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-2\mathbf{k})+5=0\] and \[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-2\mathbf{k})-8=0\] is
A)
1 unit done
clear
B)
\[\frac{13}{3}\] unit done
clear
C)
13 unit done
clear
D)
None of these done
clear
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question_answer34)
The equation of the plane containing the line \[\mathbf{r}=\mathbf{i}+\mathbf{j}+\lambda (2\mathbf{i}+\mathbf{j}+4\mathbf{k})\] is
A)
\[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-\mathbf{k})=3\] done
clear
B)
\[\mathbf{r}.(\mathbf{i}+2\mathbf{j}-\mathbf{k})=6\] done
clear
C)
\[\mathbf{r}.(-\mathbf{i}-2\mathbf{j}+\mathbf{k})=3\] done
clear
D)
None of these done
clear
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question_answer35)
The equation \[|\mathbf{r}{{|}^{2}}-\mathbf{r}.(2\mathbf{i}+4\mathbf{j}-2\mathbf{k})-10=0\] represents a
A)
Circle done
clear
B)
Plane done
clear
C)
Sphere of radius 4 done
clear
D)
Sphere of radius 3 done
clear
E)
None of these done
clear
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question_answer36)
The centre of the sphere \[\alpha \,\mathbf{r}-2\mathbf{u}.\mathbf{r}=\beta ,(\alpha \ne 0)\] is [AMU 1999]
A)
\[-\mathbf{u}/\alpha \] done
clear
B)
\[\mathbf{u}/\alpha \] done
clear
C)
\[\alpha \mathbf{u}/\beta \] done
clear
D)
\[\frac{\alpha +\beta }{\alpha }\mathbf{u}\] done
clear
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question_answer37)
The shortest distance between the lines \[\mathbf{r}=(3\mathbf{i}-2\mathbf{j}-2\mathbf{k})+\mathbf{i}t\] and \[\mathbf{r}=\mathbf{i}-\mathbf{j}+2\mathbf{k}+\mathbf{j}s\] (t and s being parameters) is [AMU 1999]
A)
\[\sqrt{21}\] done
clear
B)
\[\sqrt{102}\] done
clear
C)
4 done
clear
D)
3 done
clear
View Solution play_arrow
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question_answer38)
The equation of the line passing through the points \[{{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\] and \[{{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\]is [RPET 2002]
A)
\[({{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k})+t({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})\] done
clear
B)
\[({{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k})-t({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})\] done
clear
C)
\[{{a}_{1}}(1-t)\mathbf{i}+{{a}_{2}}(1-t)\mathbf{j}+{{a}_{3}}(1-t)\mathbf{k}+({{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k})t\] done
clear
D)
None of these done
clear
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question_answer39)
The distance between the line \[\mathbf{r}=2\mathbf{i}-2\mathbf{j}+3\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}+4\mathbf{k})\] and the plane \[\mathbf{r}.(\mathbf{i}+5\mathbf{j}+\mathbf{k})=5\] is [AIEEE 2005]
A)
\[\frac{3}{10}\] done
clear
B)
\[\frac{10}{3}\] done
clear
C)
\[\frac{10}{9}\] done
clear
D)
\[\frac{10}{3\sqrt{3}}\] done
clear
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question_answer40)
The image of the point with position vector \[\mathbf{i}+3\mathbf{k}\]in the plane \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=1\]is [J & K 2005]
A)
\[\mathbf{i}+2\mathbf{j}+\mathbf{k}\] done
clear
B)
\[\mathbf{i}-2\mathbf{i}+\mathbf{k}\] done
clear
C)
\[-\mathbf{i}-2\mathbf{j}+\mathbf{k}\] done
clear
D)
\[\mathbf{i}+2\mathbf{j}-\mathbf{k}\] done
clear
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question_answer41)
The equation of the plane passing through the points \[(-1,-2,\,0),(2,\,3,\,5)\] and parallel to the line \[\mathbf{r}=-3\mathbf{j}+\mathbf{k}+\mathbf{\lambda }(2\mathbf{i}+5\mathbf{j}-\mathbf{k})\] is [J & K 2005]
A)
\[\mathbf{r}.(-30\mathbf{i}+13\mathbf{j}+5\mathbf{k})=4\] done
clear
B)
\[\mathbf{r}.(30\mathbf{i}+13\mathbf{j}+5\mathbf{k})=4\] done
clear
C)
\[\mathbf{r}.(30\mathbf{i}+13\mathbf{j}-5\mathbf{k})=4\] done
clear
D)
\[\mathbf{r}.(30\mathbf{i}-13\mathbf{j}-5\mathbf{k})=4\] done
clear
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question_answer42)
The shortest distance between the lines \[{{\mathbf{r}}_{1}}=4\mathbf{i}-3\mathbf{j}-\mathbf{k}+\lambda (\mathbf{i}-4\mathbf{j}+7\mathbf{k})\] and \[{{\mathbf{r}}_{2}}=\mathbf{i}-\mathbf{j}-10\mathbf{k}+\lambda (2\mathbf{i}-3\mathbf{j}+8\mathbf{k})\]is [J & K 2005]
A)
3 done
clear
B)
1 done
clear
C)
2 done
clear
D)
0 done
clear
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question_answer43)
The position vector of the point where the line \[\mathbf{r}=\mathbf{i}-\mathbf{j}+\mathbf{k}+t(\mathbf{i}+\mathbf{j}+\mathbf{k})\]meets the plane \[\mathbf{r}.(\mathbf{i}+\mathbf{j}+\mathbf{k})=5\]is [Kerala (Engg.) 2005]
A)
\[5\mathbf{i}+\mathbf{j}-\mathbf{k}\] done
clear
B)
\[5\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] done
clear
C)
\[2\mathbf{i}+\mathbf{j}+2\mathbf{k}\] done
clear
D)
\[5\mathbf{i}+\mathbf{j}+\mathbf{k}\] done
clear
E)
\[4\mathbf{i}+2\mathbf{j}-2\mathbf{k}\] done
clear
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question_answer44)
A Plane meets the co-ordinate axes at P, Q and R such that the position vector of the centroid of \[\Delta PQR\] is \[2\mathbf{i}-5\mathbf{j}+8\mathbf{k}\]. Then the equation of the plane is [J & K 2005]
A)
\[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=120\] done
clear
B)
\[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=1\] done
clear
C)
\[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=2\] done
clear
D)
\[\mathbf{r}.(20\mathbf{i}-8\mathbf{j}+5\mathbf{k})=20\] done
clear
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question_answer45)
The line of intersection of the planes \[\mathbf{r}.(\mathbf{i}-3\mathbf{j}+\mathbf{k})=1\] and \[\mathbf{r}.(2\mathbf{i}+5\mathbf{j}-3\mathbf{k})=2\] is parallel to the vector
A)
\[-4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\] done
clear
B)
\[4\mathbf{i}+5\mathbf{j}+11\mathbf{k}\] done
clear
C)
\[4\mathbf{i}-5\mathbf{j}+11\mathbf{k}\] done
clear
D)
\[4\mathbf{i}-5\mathbf{j}-11\mathbf{k}\] done
clear
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question_answer46)
The equation of plane passing through a point \[A(2,-1,\,3)\] and parallel to the vectors \[\mathbf{a}=(3,\,0,-1)\] and \[\mathbf{b}=(-3,\,\,2,\,2)\] is [Orissa JEE 2005]
A)
\[2x-3y+6z-25=0\] done
clear
B)
\[2x-3y+6z+25=0\] done
clear
C)
\[3x-2y+6z-25=0\] done
clear
D)
\[3x-2y+6z+25=0\] done
clear
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question_answer47)
If the position vectors of two point P and Q are respectively \[9\mathbf{i}-\mathbf{j}+5\mathbf{k}\] and \[\mathbf{i}+3\mathbf{j}+5\mathbf{k}\], and the line segment PQ intersects the YOZ plane at a point R, the PR : RQ is equal to [J & K 2005]
A)
9 : 1 done
clear
B)
1 : 9 done
clear
C)
?1 : 9 done
clear
D)
? 9 : 1 done
clear
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