Question 1 :
If $A= (1, 2, 3), B = (2, 3, 4)$ and $AB$ is produced upto $C$ such that $2AB = BC$, then $C =$<br/>
Question 3 :
If the $zx$-plane divides the line segment joining $(1,-1,5)$ and $(2,3,4)$ in the ratio $p:1$, then $p+1=$
Question 4 :
In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are
Question 5 :
A point on XOZ-plane divides the join of $(5, -3, -2)$ and $(1, 2, -2)$ at
Question 6 :
Plane $ax + by + cz = 1$ intersect axes in $A, B, C$ respectively. If $G\left (\dfrac {1}{6}, -\dfrac {1}{3}, 1\right )$ is a centroid of $\triangle ABC$ then $a + b + 3c =$ _________.
Question 8 :
If $z = \cos \dfrac{\pi }{6} + i\sin \dfrac{\pi }{6}$, then
Question 9 : <div>State the following statement is True or False<br/></div>If two distinct lines are intersecting each other in a plane then they cannot have more than one point in common.<br/>
Question 10 :
The points (2, 5) and (5, 1) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line $y = 2x + k$, then the value of k is
Question 11 :
$A=\left(2,4,5\right)$ and $B=\left(3,5,-4\right)$ are two points. If the $xy$-plane, $yz$-plane divide $AB$ in the ratios $a:b,p:q$ respectively then $\dfrac{a}{b}+\dfrac{p}{q}$=
Question 13 :
The ratio in which $xy-$plane divides the line joining the points $(1, 0, -3)$ and $(1, -5, 7)$ is given by
Question 14 :
$A(3, 2, 0), B(5, 3, 2), C(-9, 6, -3)$ are three points forming a triangle. If $AD$, the bisector of $\angle BAC$ meets $BC$ in $D$ then coordinates of $D$ are
Question 16 :
The ratio in which the plane $2x+3y-2z+7=0$ divides the line segment joining the points $(-1, 1, 3)$, $(2, 3, 5)$ is
Question 18 :
The coordinate of any point, which lies in $xy$ plane , is
Question 19 :
The coordinates of any point, which lies in $yz$ plane, are
Question 20 :
The plane $ax+by+cz+(-3)=0$ meet the co-ordinate axes in A,B,C. Then centroid<br>of the triangle is<br>
Question 21 :
$A$ point $C$ with position vector $\frac{{3a + 4b - 5c}}{3}$ (where a,b and c are non co-planar vectors) divides the line joining $A$ and $B$ in the ratio $2:1$. If the position vector of $A$ is $a-2b+3c$, then the position vector of $B$ is
Question 22 :
The ratio in which the line joining $(3,4,-7)$ and $(4,2,1)$ is dividing the xy-plane
Question 23 :
The coordinates of a point which divides the line joining the points $P(2,3,1)$ and $Q(5,0,4)$ in the ratio $1:2$ are<br/>
Question 25 :
The equation of plane passing through $(-1,0,-1)$ parallel to $xz$ plane is
Question 26 :
If $P\left( x,y,z \right) $ is a point on the line segment joining $Q\left( 2,2,4 \right) $ and $R\left( 3,5,6 \right) $ such that the projection of $\overline { OP } $ on the axes are $\displaystyle\frac { 13 }{ 5 } ,\frac { 19 }{ 5 } ,\frac { 26 }{ 5 } ,$ respectively, then $P$ divides $QR$ in the ratio
Question 27 :
If the $zx$-plane divides the line segment joining $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $p : 1$, then $p + 1=$<br/>
Question 28 :
$A = (1, -2, 3)$ , $B =$ (2, 1, 3), $C =$ (4, 2, 1) and $G= (-1,3, 5)$ is the centroid of the tetrahedron $ABCD$. Then the fourth coordinate is <br/>
Question 29 :
Assertion(A): If centroid and circumcentre of a triangle are known its orthocentre can be found.<br/>Reason (R) : Centriod, orthocentre and circumcentre of a triangle are collinear<br/>
Question 30 :
In the tetrahedron $ABCD,\ A= (1, 2, -3)$ and $G(-3,4, 5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta BCD$, then $AP=$<br>
Question 32 :
If $\mathrm{A}= (-1,6, 6)$ , $\mathrm{B}=(-4,9, 6)$ , $\displaystyle \mathrm{G}=\frac{1}{3}(-5,22,22)$ and $\mathrm{G}$ is the centroid of the $\Delta \mathrm{A}\mathrm{B}\mathrm{C}$ then the name of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is<br/>
Question 33 :
$G(1, 1, -2)$ is the centroid of the triangle $ABC$ and $D$ is the mid point of $BC$. If $A = (-1, 1, -4)$, then $D =$<br/>
Question 34 :
If P $(3, 2, -4)$ , Q $(5, 4, -6)$ and R $(9, 8, -10)$ are collinear, then R divides PQ in the ratio
Question 35 :
If the centroid of triangle whose vertices are $(a, 1, 3), (-2, b, - 5)$ and $(4, 7, c)$ be the origin, then the values of $a, b$ and $c$ are
Question 37 :
$\mathrm{If} \mathrm{A}=(1, 2, 3)$ , $\mathrm{B}=(2,3, 4)$ and $\mathrm{C}$ is a point of trisection of$\mathrm{A}\mathrm{B}$ such that $\displaystyle \mathrm{C}_{\mathrm{x}}+\mathrm{C}_{\mathrm{y}}=\frac{13}{3}$ then $\mathrm{C}_{\mathrm{z}}=$<br/>
Question 38 :
Three vertices of a tetrahedron are $(0, 0, 0), (6, -5, -1) $ and $(-4, 1, 3)$. If the centroid of the tetrahedron be $(1, -2, 5) $ then the fourth vertex is<br/>
Question 39 :
A plane intersects the co ordinate axes at $A, B, C$. If $O= (0, 0, 0)$ and $(1, 1, 1)$ is the centroid of the tetrahedron $O ABC$, then the sum of the reciprocals of the intercepts of the plane<br/>
Question 40 :
The points $(3,\ 2,\ 0),\ (5,\ 3,\ 2)$ and $(-9,\ 6,\ -3)$, are the vertices of a triangle $ABC.AD$ is the internal bisector of $\angle\ BAC$ which meets $BC$ at $D$. Then the co-ordinates of $D$, are
Question 41 :
The plane $XOZ$ divides the join of $(1, - 1, 5)$ and $(2, 3, 4)$ in the ratio $\lambda : 1$, then $\lambda$ is -
Question 42 :
The ratio in which $yz$-plane divides the line segment joining $(-3, 4, 2), (2, 1, 3)$ is<br/>
Question 43 :
Four vertices of a tetrahedron are $(0, 0, 0), (4, 0, 0), (0, -8, 0)$ and $(0, 0, 12)$. Its centroid has the coordinates<br/>
Question 44 :
If the orthocentre, circumcentre of a triangle are $(-3, 5, 2), (6, 2, 5)$ respectively then the centroid of the triangle is<br>
Question 45 :
$XOZ$ plane divides the join of $(2,3,1)$ and $(6,7,1)$ in the ratio<br/>
Question 46 :
The planes $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$ are
Question 47 :
If the centroid of tetrahedron $OABC$ where $A,B,C$ are given by $(a,2,3), (1,b,2)$ and $(2,1,c)$ respectively is $(1,2,-2)$, then distance of $P(a,b,c)$ from origin is<br/>
Question 48 :
If $P(x,y,z)$ is a point on the line segment joining $A (2,2,4)$ and $B(3,5,6)$ such that projection of $\vec{OP}$ on axes are $\displaystyle \frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then P divide AB in the ratio
Question 49 :
If xy -plane and yz-plane divides the line segment joining A(2,4,5) and B(3,5,-4) in the ratio a:b and p:q respectively then value of $\left( {{a \over b},{p \over q}} \right)$ may be<br/>
Question 50 :
The ratio in which the plane $\displaystyle \bar {r} .(\bar {i} - 2 \bar {j} + 3 \bar {k}) = 17$ divides the line joining the points $\displaystyle -2 \bar {i} + 4 \bar {j} + 7 \bar {k} $ and $\displaystyle 3 \bar {i} - 5 \bar {j} + 8 \bar {k}$ is
Question 51 :
<br/>$A=(2, 4, 5)$ and $B=(3,5, -4)$ are two points. lf the $XY$-plane, $YZ$-plane divide $AB$ in the ratio $a:b$ and $ p:q$ respectively, then $\dfrac {a}{b}+\dfrac {p}{q}=$
Question 52 :
Three vertices of a tetrahedron are $(0,0,0),(6,-5,-1)$and $(-4,1,3)$. If the centroid of the tetrahedron be$(1,-2,5)$ then the fourth vertex is<br>
Question 53 :
In the $\Delta $ ABC , A $=$ (1, 3, -2) and G (-1, 4, 2) is the centroid of the triangle. If D is the mid point of BC then AD $=$<br>
Question 54 :
If two vertices of a triangle $ABC$ are $A(-1,2,4)$and $B(2,-3,0)$,and the centroid is $(2,0,2)$ then the vertex $C$ has the coordinates<br>
Question 55 :
If $P(x,y,z)$ is a point on the line segment joining $Q(2,2,4)$ and $R(3,5,6)$ such that the projection of $\overrightarrow { OP } $ on the axes are $\displaystyle \frac { 13 }{ 5 } ,\frac { 19 }{ 5 } ,\frac { 26 }{ 5 } $ respectively, then $P$ divides $QR$ in ratio
Question 56 :
Arrange the points: $\mathrm{A}(1,2-3), \mathrm{B}(-1,2,-3), \mathrm{C}(-1,-2-3)$ and $\mathrm{D}(1,-2, -3)$ in the increasing order of their octant numbers:<br/>
Question 57 :
A cube of side <b>5</b> has one vertex at the point <b>(1,0,-1)</b>, and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Question 58 :
The plane $ax+by + cz + d = 0$ divides the line joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the ratio
Question 59 :
If $xy-$plane and $yz-$plane divides the line segment joining $A(2,4,5)$ and $B(3,5,-4)$ in the ratio $a:b$ and $p:q$ respectively then value of $\left(\dfrac {a}{b}+\dfrac {p}{q}\right)$ may be
Question 60 :
If the plane a $2x-3y+5_{Z}-2=0$ divides the line segment joining $(1, 2, 3)$ and $(2, 1, k)$ in the ratio $9 : 11$, then $k$ is<br/>
Question 61 :
An equation of sphere with centre at origin and radius $r$ can be represented as
Question 62 :
If $A = (2, -3, 1), B = (3, -4, 6)$ and $C$ is a point of trisection of $AB$, then ${C}_{{y}}=$<br/>
Question 63 :
The graph of the equation $y^{2}+z^{2}=0$ in three dimensional space is
Question 65 :
The ratio of $yz$-plane divide the line joining the points $A(3, 1,- 5), B(1, 4, -6)$ is
Question 66 :
If the line joining $A(1, 3, 4)$ and $B$ is divided by the point $(-2, 3, 5)$ in the ratio $1 : 3$, then $B$ is<br/>
Question 67 :
A tangent to the curve $y = f(x)$ at $p(x, y)$ meets $x - axis$ at $A$ and $y-axis$ at $B$. If $\overline {AP} : \overline {BP} = 1 : 3$ and $f(1) = 1$ then the curve also passes through the point.
Question 68 :
In geometry, we take a point, a line and a plane as undefined terms.<br/>
Question 69 :
A triangle $ABC$ is placed so that the midpoints of its sides are on the $x, y$ and $z$ axes respectively. Lengths of the intercepts made by the plane containing the triangle on these axes are respectively $\displaystyle \alpha ,\beta ,\gamma$, then the coordinates of the centroid of the triangle $ABC$ are
Question 71 :
If $(0, b, 0)$ is the centroid of the triangle formed by the points $(4, 2, -3)$ , $({a}, -5, 1)$ and $(2, -6, 2)$ . If $a ,b$ are the roots of the quadratic equation $ x^2+px+q = 0 $, then $p,q$ are <br/>
Question 72 :
The plane $\displaystyle ax + by + cz + (-3) = 0$ meet the co-ordinate axes in $A, B, C$. The centroid of the triangle is
Question 73 :
The coordinates of the point where the line segment joining $A(5,1,6)$ and $B (3,4,1)$ crosses the yz plane are
Question 74 :
Point A is $\displaystyle a+2b,$ and a divides AB in the ratio 2 : 3. The position vector of B is
Question 75 :
If the vertices of a triangle are $(-1,6,-4),(2,1,1)$ and $(5,-1,0)$ then the centroid of the triangle is<br>
Question 76 :
The plane $x = 0$ divides the joinning of $( - 2, 3, 4)$ and $(1, - 2, 3)$ in the ratio :
Question 77 :
$D(2, 1, 0), E(2, 0, 0), F(0, 1, 0)$ are mid point of the sides $BC, CA, AB$ of $\Delta$ $ABC$ respectively, The the centroid of $\Delta$ABC is
Question 78 :
In the triangle with vertices $A(1, -1, 2), B(5, -6, 2)$ and $C(1,3,-1)$ find the altitude $n=|BD|$.<br/>
Question 79 :
The chord of contact of tangents from a point $P$ to a circle passes through $q$<i>. </i>If $l_1$ and $l_2$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $PQ$ is equal to
Question 80 :
There are three points with position vectors $ -2a+3b+5c, a+2b+3c $ and$ 7a-c$. What is the relation between the three points?
Question 81 :
Find the ratio in which $2x + 3y + 5z = 1$ divides the line joining the points $(1,\ 0,\ -3)$ and $(1,\ -5,\ 7)$.
Question 82 :
Four vertices of a tetrahedron are $(0,0,0),(4,0,0),(0,-8,0)$ and $(0,0,12)$,Its centroid has the coordinates<br>
Question 83 :
If the plane $7x + 11y+ 13z= 3003$ meets the axes in $A, B, C$, then the centroid of $\Delta ABC$ is
Question 85 :
If $P(x,y,x)$ is a point on the line segment joining $Q(2,2,4)$ and $R(3,5,6)$ such that the projection of $\overline { OP } $ on the axis are $\displaystyle \frac { 13 }{ 5 } ,\frac { 19 }{ 5 } ,\frac { 26 }{ 5 } ,$ respectively, then $P$ divides $QR$ in the ratio
Question 86 :
If $A(2, 2, -3), B(5, 6, 9), C(2, 7, 9)$ be the vertices of a triangle. The internal bisector of the angle $A$ meets $BC$ at the point $D$, then find the coordinates of $D$.
Question 87 :
The plane $ax+by+cz+d=0$ divides the line joining the points $\left( { x }_{ 1 },{ y }_{ 1 },{ z }_{ 1 } \right) $ and $\left( { x }_{ 2 },{ y }_{ 2 },{ z }_{ 2 } \right) $ in the ratio
Question 88 :
A tetrahedron is a three dimensional figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points A,B,C,D as its vertices which have coordinates $(x1, y1, z_{1}) (x_{2}, y_{2}, z_{2})$ , $(x_{3}, y_{3}, z_{3})$ and $(x_{4}, y_{4}, z_{4})$, respectively in a rectangular three dimensional space. Then, the coordinates of its centroid are $[\dfrac{x_{1}+x_{2}+x_{3}+x_{4}}{4},\dfrac{y_{1}+y_{2}+y_{3}+y_{4}}{4},\dfrac{z_{1}+z_{2}+z_{3}+z_{4}}{4}]$.<br/>Let a tetrahedron have three of its vertices represented by the points $(0,0,0) ,(6,5,1)$ and $(4,1,3)$ and its centroid lies at the point $(1,2,5)$. Now, answer the following question. The coordinate of the fourth vertex of the tetrahedron is:<br/>
