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3D Geometry

436 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2005 JEE Mains MCQ
AIEEE 2005
The plane $x+2y-z=4$ cuts the sphere ${x^2} + {y^2} + {z^2} - x + z - 2 = 0$ in a circle of radius
A.
$3$
B.
$1$
C.
$2$
D.
${\sqrt 2 }$
2005 JEE Mains MCQ
AIEEE 2005
The angle between the lines $2x=3y=-z$ and $6x=-y=-4z$ is :
A.
${0^ \circ }$
B.
${90^ \circ }$
C.
${45^ \circ }$
D.
${30^ \circ }$
2005 JEE Mains MCQ
AIEEE 2005
If the plane $2ax-3ay+4az+6=0$ passes through the midpoint of the line joining the centres of the spheres

${x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13$ and

${x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8$ then a equals :
A.
$-1$
B.
$1$
C.
$-2$
D.
$2$
2005 JEE Mains MCQ
AIEEE 2005
The distance between the line

$\overrightarrow r = 2\widehat i - 2\widehat j + 3\widehat k + \lambda \left( {i - j + 4k} \right),$ and the plane

$\overrightarrow r .\left( {\widehat i + 5\widehat j + \widehat k} \right) = 5$ is
A.
${{10} \over 9}$
B.
${{10} \over {3\sqrt 3 }}$
C.
${{3} \over 10}$
D.
${{10} \over 3}$
2005 JEE Mains MCQ
AIEEE 2005
If the angel $\theta $ between the line ${{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}$ and

the plane $2x - y + \sqrt \lambda \,\,z + 4 = 0$ is such that $\sin \,\,\theta = {1 \over 3}$ then value of $\lambda $ is :
A.
${5 \over 3}$
B.
${-3 \over 5}$
C.
${3 \over 4}$
D.
${-4 \over 3}$
2005 JEE Advanced MCQ
IIT-JEE 2005 Screening
A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $A,$ $B$ and $C.$ If the centroid $D$ $(x, y, z)$ of triangle $ABC$ satisfies the relation ${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = k,$ then the value $k$ is
A.
$3$
B.
$1$
C.
${1 \over 3}$
D.
$9$
2005 JEE Advanced MCQ
IIT-JEE 2005 Mains

Find the equation of the plane containing the line $2 x-y+z-3=0,3 x+y+z=5$ and at a distance of $\frac{1}{\sqrt{6}}$ from the point $(2,1,-1)$.

A.
$62x+19y+29z-105=0$
B.
$62x+29y+z-105=0$
C.
$29x+62y+19z-105=0$
D.
$62x+29y+19z-105=0$
2005 JEE Advanced Numerical
IIT-JEE 2005
Find the equation of the plane containing the line $2x-y+z-3=0,3x+y+z=5$ and at a distance of ${1 \over {\sqrt 6 }}$ from the point $(2, 1, -1).$
2004 JEE Mains MCQ
AIEEE 2004
A line makes the same angle $\theta $, with each of the $x$ and $z$ axis.

If the angle $\beta \,$, which it makes with y-axis, is such that $\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$ then ${\cos ^2}\theta $ equals :
A.
${2 \over 5}$
B.
${1 \over 5}$
C.
${3 \over 5}$
D.
${2 \over 3}$
2004 JEE Mains MCQ
AIEEE 2004
The intersection of the spheres
${x^2} + {y^2} + {z^2} + 7x - 2y - z = 13$ and
${x^2} + {y^2} + {z^2} - 3x + 3y + 4z = 8$
is the same as the intersection of one of the sphere and the plane
A.
$2x-y-z=1$
B.
$x-2y-z=1$
C.
$x-y-2z=1$
D.
$x-y-z=1$
2004 JEE Mains MCQ
AIEEE 2004
Distance between two parallel planes

$\,2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is :
A.
${9 \over 2}$
B.
${5 \over 2}$
C.
${7 \over 2}$
D.
${3 \over 2}$
2004 JEE Mains MCQ
AIEEE 2004
A line with direction cosines proportional to $2,1,2$ meets each of the lines $x=y+a=z$ and $x+a=2y=2z$ . The co-ordinates of each of the points of intersection are given by :
A.
$\left( {2a,3a,3a} \right),\left( {2a,a,a} \right)$
B.
$\left( {3a,2a,3a} \right),\left( {a,a,a} \right)$
C.
$\left( {3a,2a,3a} \right),\left( {a,a,2a} \right)$
D.
$\left( {3a,3a,3a} \right),\left( {a,a,a} \right)$
2004 JEE Mains MCQ
AIEEE 2004
If the straight lines
$x=1+s,y=-3$$ - \lambda s,$ $z = 1 + \lambda s$ and $x = {t \over 2},y = 1 + t,z = 2 - t,$ with parameters $s$ and $t$ respectively, are co-planar, then $\lambda $ equals :
A.
$0$
B.
$-1$
C.
$ - {1 \over 2}$
D.
$-2$
2004 JEE Advanced MCQ
IIT-JEE 2004 Screening
If the lines ${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$ and $\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$ intersect, then the value of $k$ is
A.
$3/2$
B.
$9/2$
C.
$-2/9$
D.
$-3/2$
2004 JEE Advanced Numerical
IIT-JEE 2004
A parallelopiped $'S'$ has base points $A, B, C$ and $D$ and upper face points $A',$ $B',$ $C'$ and $D'.$ This parallelopiped is compressed by upper face $A'B'C'D'$ to form a new parallelopiped $'T'$ having upper face points $A'',B'',C''$ and $D''.$ Volume of parallelopiped $T$ is $90$ percent of the volume of parallelopiped $S.$ Prove that the locus of $'A''',$ is a plane.
2004 JEE Advanced Numerical
IIT-JEE 2004
${P_1}$ and ${P_2}$ are planes passing through origin. ${L_1}$ and ${L_2}$ are two line on ${P_1}$ and ${P_2}$ respectively such that their intersection is origin. Show that there exists points $A, B, C,$ whose permutation $A',B',C'$ can be chosen such that (i) $A$ is on ${L_1},$ $B$ on ${P_1}$ but not on ${L_1}$ and $C$ not on ${P_1}$ (ii) $A'$ is on ${L_2},$ $B'$ on ${P_2}$ but not on ${L_2}$ and $C'$ not on ${P_2}$
2004 JEE Advanced Numerical
IIT-JEE 2004
Find the equation of plane passing through $(1, 1, 1)$ & parallel to the lines ${L_1},{L_2}$ having direction ratios $(1,0,-1),(1,-1,0).$ Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes.
2003 JEE Mains MCQ
AIEEE 2003
The shortest distance from the plane $12x+4y+3z=327$ to the sphere

${x^2} + {y^2} + {z^2} + 4x - 2y - 6z = 155$ is
A.
$39$
B.
$26$
C.
$11{4 \over {13}}$
D.
$13$
2003 JEE Mains MCQ
AIEEE 2003
Two systems of rectangular axes have the same origin. If a plane cuts then at distances $a,b,c$ and $a', b', c'$ from the origin then
A.
${1 \over {{a^2}}} + {1 \over {{b^2}}} + {1 \over {{c^2}}} - {1 \over {a{'^2}}} - {1 \over {b{'^2}}} - {1 \over {c{'^2}}} = 0$
B.
$\,{1 \over {{a^2}}} + {1 \over {{b^2}}} + {1 \over {{c^2}}} + {1 \over {a{'^2}}} + {1 \over {b{'^2}}} + {1 \over {c{'^2}}} = 0$
C.
${1 \over {{a^2}}} + {1 \over {{b^2}}} - {1 \over {{c^2}}} + {1 \over {a{'^2}}} - {1 \over {b{'^2}}} - {1 \over {c{'^2}}} = 0$
D.
${1 \over {{a^2}}} - {1 \over {{b^2}}} - {1 \over {{c^2}}} + {1 \over {a{'^2}}} - {1 \over {b{'^2}}} - {1 \over {c{'^2}}} = 0$
2003 JEE Mains MCQ
AIEEE 2003
The radius of the circle in which the sphere

${x^2} + {y^2} + {z^2} + 2x - 2y - 4z - 19 = 0$ is cut by the plane

$x+2y+2z+7=0$ is
A.
$4$
B.
$1$
C.
$2$
D.
$3$
2003 JEE Mains MCQ
AIEEE 2003
The lines ${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$ and ${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$ are coplanar if :
A.
$k=3$ or $-2$
B.
$k=0$ or $-1$
C.
$k=1$ or $-1$
D.
$k=0$ or $-3$
2003 JEE Mains MCQ
AIEEE 2003
The two lines $x=ay+b,z=cy+d$ and $x = a'y + b',z = c'y + d'$ will be perpendicular, if and only if :
A.
$aa' + cc' + 1 = 0$
B.
$aa' + bb'cc' + 1 = 0$
C.
$aa' + bb'cc' = 0$
D.
$\left( {a + a'} \right)\left( {b + b'} \right) + \left( {c + c'} \right) = 0$
2003 JEE Advanced MCQ
IIT-JEE 2003 Screening
The value of $k$ such that ${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$ lies in the plane $2x -4y +z = 7,$ is
A.
$7$
B.
$-7$
C.
no real value
D.
$4$
2003 JEE Advanced Numerical
IIT-JEE 2003
(i) Find the equation of the plane passing through the points $(2, 1, 0), (5, 0, 1)$ and $(4, 1, 1).$
(ii) If $P$ is the point $(2, 1, 6)$ then find the point $Q$ such that $PQ$ is perpendicular to the plane in (i) and the mid point of $PQ$ lies on it.
2002 JEE Mains MCQ
AIEEE 2002
A plane which passes through the point $(3,2,0)$ and the line

${{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \over 4}$ is :
A.
$x-y+z=1$
B.
$x+y+z=5$
C.
$x+2y-z=1$
D.
$2x-y+z=5$
2002 JEE Mains MCQ
AIEEE 2002
The $d.r.$ of normal to the plane through $(1, 0, 0), (0, 1, 0)$ which makes an angle $\pi /4$ with plane $x+y=3$ are :
A.
$1,\sqrt 2 ,1$
B.
$1,1,\sqrt 2 $
C.
$1, 1, 2$
D.
$\sqrt 2 ,1,1$
1996 JEE Advanced Numerical
IIT-JEE 1996
The position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ are $\widehat i + \widehat j + \widehat k,\,\widehat i$ and $3\widehat i\,,$ respectively. The altitude from vertex $D$ to the opposite face $ABC$ meets the median line through $A$ of the triangle $ABC$ at a point $E.$ If the length of the side $AD$ is $4$ and the volume of the tetrahedron is ${{2\sqrt 2 } \over 3},$ find the position vector of the point $E$ for all its possible positions.
1994 JEE Advanced MCQ
IIT-JEE 1994
Let $\overrightarrow p $ and $\overrightarrow q $ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2:3$ respectively. If $OR$ and $OS$ are perpendicular then
A.
$9{q^2} = 4{q^2}$
B.
$4{p^2} = 9{q^2}$
C.
$9p = 4q$
D.
$4p = 9q$
1994 JEE Advanced MCQ
IIT-JEE 1994
Let $\alpha ,\beta ,\gamma $ be distinct real numbers. The points with position
vectors $\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$
A.
are collinear
B.
form an equilateral triangle
C.
form a scalene triangle
D.
form a right-angled triangle
1994 JEE Advanced Numerical
IIT-JEE 1994
A unit vector perpendicular to the plane determined by the points $P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)$ and $R\left( {0,2,1} \right)$ is ............
1983 JEE Advanced MCQ
IIT-JEE 1983
The points with position vectors $60i+3j,$ $40i-8j,$ $ai-52j$ are collinear if
A.
$a=-40$
B.
$a=40$
C.
$a=20$
D.
none of these
1983 JEE Advanced MCQ
IIT-JEE 1983
The volume of the parallelopiped whose sides are given by
$\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,$ is
A.
${4 \over {13}}$
B.
$4$
C.
${2 \over 7}$
D.
none of these
1983 JEE Advanced Numerical
IIT-JEE 1983
A vector $\overrightarrow A $ has components ${A_1},{A_2},{A_3}$ in a right -handed rectangular Cartesian coordinate system $oxyz.$ The coordinate system is rotated about the $x$-axis through an angle ${\pi \over 2}.$ Find the components of $A$ in the new coordinate system in terms of ${A_1},{A_2},{A_3}.$
1983 JEE Advanced Numerical
IIT-JEE 1983
The unit vector perpendicular to the plane determined by $P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)$ and $R\left( {0,2,1} \right)$ is ...........
1983 JEE Advanced Numerical
IIT-JEE 1983
The area of the triangle whose vertices are $A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)$ is ..........
1978 JEE Advanced Numerical
IIT-JEE 1978
From a point $O$ inside a triangle $ABC,$ perpendiculars $OD$, $OE, OF$ are drawn to the sides $BC, CA, AB$ respectively. Prove that the perpendiculars from $A, B, C$ to the sides $EF, FD, DE$ are concurrent.