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Maths M.C.Q (1 Marks)

Introduction to Three Dimensional Geometry · Jharkhand Board (JAC) STD 11 Science (Jharkhand - English Medium). 159 questions.

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M.C.Q (1 Marks)

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159 questions · auto-graded multiple-choice test.

MCQ 11 Mark
If the distance between the points $(a, 0, 1)$ and $(0, 1, 2)$ is $\sqrt{27}$ then the value of $a$ is:
  • A
    $5$
  • ✓
    $\underline{+}5$
  • C
    $-5$
  • D
    None of these
Answer
Correct option: B.
$\underline{+}5$
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MCQ 21 Mark
If $A = (1, 2, 3), B = (2, 3, 4)$ and $AB$ is produced upto $C$ such that $\text{2AB = BC}$ then $C =$
  • A
    $(5, 4, 6)$
  • B
    $(6, 2, 4)$
  • ✓
    $(4, 5, 6)$
  • D
    $(6, 4, 5)$
Answer
Correct option: C.
$(4, 5, 6)$
Let the point $C$ be $(i, j, k)$
Since, $B$ divides $AC$ in the ratio $1 : 2$
Coordinates of $B$ should be $\Big(\frac{2+\text{i}}{3},\frac{4+\text{j}}{3},\frac{\text{k}+6}{3}\Big)$
Comparing the values given already for $B,$ we get, $i = 4, j = 5$ and $k = 6$
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MCQ 31 Mark
There is one and only one sphere through:
  • ✓
    $4$ points not in the same plane
  • B
    $4$ points not lie in the same straight line
  • C
    None of these
  • D
    $3$ points not lie in the same line
Answer
Correct option: A.
$4$ points not in the same plane
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MCQ 41 Mark
The vector equation of a sphere having centre at origin and radius $5$ is:
  • ✓
    $\mid{\text{r}}\mid = 5$
  • B
    $\mid{\text{r}}\mid = 25$
  • C
    $\mid{\text{r}}\mid = \sqrt{5}$
  • D
    None of these
Answer
Correct option: A.
$\mid{\text{r}}\mid = 5$
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MCQ 51 Mark
The perpendicular distance of the point $P(3, 3, 4)$ from the $x-$axis is
  • A
    $3\sqrt{2}$
  • ✓
    $5$
  • C
    $3$
  • D
    $4$
Answer
Correct option: B.
$5$
The perpendicular distance of the point $P(3, 3, 4)$ from the $x-$axis is given by
$\sqrt{3^2+4^2}$
$=\sqrt{25}$
$=5$
Hence, the correct answer is option $(b)$
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MCQ 61 Mark
The ratio in which $yz-$plane divides the line segment joining $(-3, 4, 2), (2, 1, 3)$ is:
  • A
    $-4 : 1$
  • ✓
    $3 : 2$
  • C
    $-2 : 3$
  • D
    $1: 4$
Answer
Correct option: B.
$3 : 2$
Let the plane divide the line in the ratio $p : 1$
A point that divides the line joining these $2$ points in the ratio $p : 1$
given by $\Big(\frac{2\text{p}-3}{\text{p}+1},\frac{\text{p}+4}{\text{p}+1},\frac{3\text{p}+2}{\text{p}+1}\Big)$
Since, this point has to lie on the $zy-$plane.
so, $2p - 3 = 0$
$\Rightarrow\text{p}=\frac{3}{2}$
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MCQ 71 Mark
Find the distance between $(12, 3, 4)$ and $(4, 5, 2):$
  • ✓
    $\sqrt{72}$
  • B
    $\sqrt{62}$
  • C
    $\sqrt{64}$
  • D
    None of these
Answer
Correct option: A.
$\sqrt{72}$
Consider the problem,
Let the given points
$A(12, 3, 4)$ and $B(4, 5, 2)$
So, distance between A and B by distance formula.
$\text{AB}=\sqrt{(4-12)^2+(5-3)^2+(2-4)^2}$
$=\sqrt{(-8)^2+2^2+(-2)^2}$
$=\sqrt{64+4+4}$
$=\sqrt{72}$
So, distance between the points $(12,3,4)$ and $(4,5,2)$ is $\sqrt{72}\text{ Sq. units.}$
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MCQ 81 Mark
Graph $x^2+ y^2= 4$ in $3D$ looks like:
  • A
    Circle
  • ✓
    Cylinder
  • C
    Hemisphere
  • D
    Sphere
Answer
Correct option: B.
Cylinder
The given curve is $x^2+ y^2= 4$
So $x$ coordinate and $y-$coordinate are connected by $x^2+ y^2= 4$ which is locus of a circle with radius $2$ But $z-$coordinate can be anything,
so in three dimension the circle $x^2+ y^2= 4$ will be stretched which will be a cylinder with radius same as the radius of the circle.
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MCQ 91 Mark
The coordinates of any point, which lies on $x$ axis are:
  • A
    $(0, x, 0)$
  • ✓
    $(x, 0, 0)$
  • C
    $(x, x, 0)$
  • D
    $(x, x, x)$
Answer
Correct option: B.
$(x, 0, 0)$
In $3-$dimensional plane, the point which lies on $x-$axis does not have any part in $y$ and $z$ axes.
At that point, the value of $y$ and $z$ will be $0.$
Hence, coordinate of any point which lies on $x$ axis are $(x, 0, 0)$.
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MCQ 101 Mark
If the plane $7x + 11y + 13z = 3003$ meets the axes in $\ce{A, B, C}$ then the centroid of $\Delta\text{ABC}$ is:
  • ✓
    $(143, 91, 77)$
  • B
    $(143, 77, 91)$
  • C
    $(91, 143, 77)$
  • D
    $(143, 66, 91)$
Answer
Correct option: A.
$(143, 91, 77)$
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MCQ 111 Mark
$G(1, 1, -2)$ is the centroid of the triangle $\text{ABC}$ and $D$ is the mid point of $\text{BC.}$ If $A = (-1, 1, -4) D=$
  • A
    $\Big(\frac{1}{2},1,\frac{-5}{2}\Big)$
  • B
    $(5, 1, 2)$
  • C
    $(-5, -1, -2)$
  • ✓
    $(2, 1, -1)$
Answer
Correct option: D.
$(2, 1, -1)$
Let the coordinates of $D$ be $(p, q, r)$
Since, the centroid divides the line joining $AD$ in the ratio $2 : 1$ the coordinates of centroid should be,
$\Big(\frac{2\text{p}-1}{3},\frac{2\text{q}+1}{3},\frac{2\text{r}-4}{3}\Big)$
Comparing it with the coordinates of the centroid given, $D (2, 1, -1).$
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MCQ 121 Mark
Point $A$ is $a + 2b,$ and $a$ divides $AB$ in the ratio $2 : 3.$ The position vector of $B$ is:
  • A
    $2a - b$
  • B
    $b - 2a$
  • ✓
    $a - 3b$
  • D
    $b$
Answer
Correct option: C.
$a - 3b$
Let us consider $x$ be the position vector of $B,$ then a divides $AB$ in the ratio $2 : 3$
$\text{a}=\frac{2\text{x}+3(\text{a+2b})}{2+3}$
$\Rightarrow\text{x}=\text{a - 3b}$
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MCQ 131 Mark
$\text{XOZ}$ plane divides the join of $(2, 3, 1)$ and $(6, 7, 1)$ in the ratio:
  • A
    $3 : 7$
  • B
    $2 : 7$
  • ✓
    $-3 : 7$
  • D
    $-2 : 7$
Answer
Correct option: C.
$-3 : 7$
Let the plane divide the line in the ratio $p : 1$
A point that divides the line joining these $2$ points in the ratio $p : 1$ is
given by $\Big(\frac{6\text{p}+2}{\text{p}+1},\frac{7\text{p}+3}{\text{p}+1},\frac{\text{p}+1}{\text{p}+1}\Big)$
Since, this point has to lie on the $zx$ plane,
so, $7p + 3 = 0$
$\Rightarrow\text{p}=\frac{-3}{7}$
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MCQ 141 Mark
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:
  • A
    $\Big(\frac{7}{3}, 1, \frac{5}{3}\Big)$
  • B
    (4, 1, 3)
  • ✓
    $(3, 2, 2)$
  • D
    $(1, -1, 1)$
Answer
Correct option: C.
$(3, 2, 2)$
Using section formula, Coordinate of the point which divides $P (2, 3, 1)$ and $Q (5, 0, 4)$ in ratio.
$1 : 2$ is $\Big(\frac{2.2+5}{2+1},\frac{2.3+0}{2+1},\frac{2.1+4}{2+1}\Big)=\Big(\frac{9}{6},\frac{6}{3},\frac{6}{3}\Big)=(3,2,2)$
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MCQ 151 Mark
What is the length of foot of perpendicular drawn from the point $P(3, 4, 5)$ on $y-$axis:
  • A
    $\sqrt{41}$
  • ✓
    $\sqrt{34}$
  • C
    $5$
  • D
    None of these
Answer
Correct option: B.
$\sqrt{34}$
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MCQ 161 Mark
Find the image of $(-2, 3, 4)$ in the $y z$ plane:
  • A
    $(-2, 3, 4)$
  • ✓
    $(2, 3, 4)$
  • C
    $(-2, -3, 4)$
  • D
    $(-2, -3, -4)$
Answer
Correct option: B.
$(2, 3, 4)$
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MCQ 171 Mark
If $\alpha,\beta,\text{y}$ are the angles made by a half ray of a line respectively with positive directions of $X-$axis, $Y-$axis and, $Z-$axis, then $\sin^2 \alpha + \sin^2 \beta + \sin^2 \text{y} =$
  • A
    $1$
  • B
    $0$
  • C
    $-1$
  • ✓
    None of these
Answer
Correct option: D.
None of these
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MCQ 181 Mark
The distance of the point $P(a, b, c)$ from the $x-$axis is:
  • A
    $\sqrt{(\text{a}^2 + \text{c}^2)}$
  • B
    $\sqrt{(\text{a}^2 + \text{b}^2)}$
  • ✓
    $\sqrt{(\text{b}^2 + \text{c}^2)}$
  • D
    None of these
Answer
Correct option: C.
$\sqrt{(\text{b}^2 + \text{c}^2)}$
The coordinate of the foot of the perpendicular from $P$ on $x-$axis are $(a, 0, 0)$.
So, the required distance $= \sqrt{{(\text{a – a})^2 + (\text{b – 0})^2 + (\text{c – 0})^2}}$
$=\sqrt{(\text{b}^2 + \text{c}^2)}$
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MCQ 191 Mark
What is the distance between the points $(2, -1, 3)$ and $(-2, 1, 3):$
  • ✓
    $2\sqrt{5}\text{ units}$
  • B
    $25\text{ units}$
  • C
    $4\sqrt{5}\text{ units}$
  • D
    $\sqrt{5}\text{ units}$
Answer
Correct option: A.
$2\sqrt{5}\text{ units}$
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MCQ 201 Mark
If $G$ is centroid of $\triangle\text{ABC}$ then:
  • A
    $\vec{G}=\vec{a}+\vec{b}+\vec{c}$
  • B
    $\vec{G}=\frac{\vec{a}+\vec{b}+\vec{c}}{2}$
  • ✓
    $3 \vec{ G }=\vec{a}+\vec{b}+\vec{c}$
  • D
    $3 \vec{G}=\frac{\vec{a}+\vec{b}+\vec{c}}{2}$
Answer
Correct option: C.
$3 \vec{ G }=\vec{a}+\vec{b}+\vec{c}$
We have,
In a $\triangle\text{ABC}$
$\overrightarrow {\text{A}}=\overrightarrow{\text{a}}$
$\overrightarrow {\text{B}}=\overrightarrow{\text{b}}$
$\overrightarrow {\text{C}}=\overrightarrow{\text{c}}$
then,
we know that,
$\overrightarrow {\text{G}} = \frac {\overrightarrow{\text{a}} + \overrightarrow {\text{b}} +\overrightarrow{\text{c}}}{3}$
$3\overrightarrow {\text{G}}=\overrightarrow {\text{a}}+ \overrightarrow {\text{b}}+\overrightarrow {\text{c}}$
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MCQ 211 Mark
The point $(0, -2, 5)$ lies on the:
  • A
    $z$ axis
  • B
    $x$ axis
  • C
    $xy$ plane
  • ✓
    $yz$ plane
Answer
Correct option: D.
$yz$ plane
Given, point is $(0, -2, 5)$
The $X-$coordinate in the given point is zero.
so, the point lies on $yz$ plane.
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MCQ 221 Mark
Three vertices of a parallelogram $\text{ABCD}$ are $A(1, 2, 3), B(-1, -2, -1)$ and $C(2, 3, 2).$ Find the fourth vertex $D:$
  • A
    $(-4, -7, -6)$
  • ✓
    $(4, 7, 6)$
  • C
    $(4, 7, -6)$
  • D
    None of these
Answer
Correct option: B.
$(4, 7, 6)$
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MCQ 231 Mark
The mid$-$points of the sides of a triangle are $(5, 7, 11), (0, 8, 5)$ and $(2, 3, -1)$. Then, the vertices are:
  • ✓
    $(7, 2, 5), (3, 12, 17), (-3, 4, -7)$
  • B
    $(7, 2, 5), (3, 12, 17), ( 3, 4, 7)$
  • C
    $(7, 2, 5), (-3, 12, 17), (-3, -4, -7)$
  • D
    None of the above
Answer
Correct option: A.
$(7, 2, 5), (3, 12, 17), (-3, 4, -7)$
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MCQ 241 Mark
The points on the $y-$axis which are at a distance of $3$ units from the point $(2, 3, -1)$ is:
  • A
    Either $(0, -1, 0)$ or $(0, -7, 0)$
  • B
    Either $(0, 1, 0)$ or $(0, 7, 0)$
  • C
    Either $(0, 1, 0)$ or $(0, -7, 0)$
  • ✓
    Either $(0, -1, 0)$ or $(0, 7, 0)$
Answer
Correct option: D.
Either $(0, -1, 0)$ or $(0, 7, 0)$
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MCQ 251 Mark
Choose the correct answer. What is the length of foot of perpendicular drawn from the point $P(3, 4, 5)$ on $y-$axis.
  • A
    $\sqrt{41}$
  • ✓
    $\sqrt{34}$
  • C
    $5$
  • D
    $\text{None of these.}$
Answer
Correct option: B.
$\sqrt{34}$
We know that, on the $y-$axis $x = 0$ and $z = 0.$
$\therefore$ Point $\text{A}\equiv(0,4,0)$
$\therefore\text{PA}=\sqrt{(0-3)^2+(4-4)^2+(0-5)^2}$
$=\sqrt{9+0+25}$
$=\sqrt{34}$
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MCQ 261 Mark
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 projections of $OP$ on the axes are $\frac{13}{5}, \frac{19}{5}, \frac{26}{5} $ respectively, then $P$ divides $QR$ in the ration:
  • A
    $1 : 2$
  • ✓
    $3 : 2$
  • C
    $2 : 3$
  • D
    $1 : 3$
Answer
Correct option: B.
$3 : 2$
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MCQ 271 Mark
Find the ratio in which $2x + 3y + 5z = 1$ divides the line joining the points $(1, 0, -3)$ and $(1, -5, 7):$
  • A
    $1 : 2$
  • B
    $2 : 1$
  • C
    $3 : 2$
  • ✓
    $2 : 3$
Answer
Correct option: D.
$2 : 3$
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MCQ 281 Mark
Choose the correct answer. $L$ is the foot of the perpendicular drawn from a point $P(3, 4, 5)$ on the $xy-$plane. The coordinates of point $L$ are:
  • A
    $(3, 0, 0).$
  • B
    $(0, 4, 5).$
  • C
    $(3, 0, 5).$
  • ✓
    None of these.
Answer
Correct option: D.
None of these.
We know that on the $xy-$plane, $z = 0.$
Hence, the coordinates of the points $L$ are $(3, 4, 0).$
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MCQ 291 Mark
$L$ is the foot of the perpendicular drawn from a point $P(3, 4, 5)$ on the $xy-$plane. The coordinates of point $L$ are:
  • A
    $(3, 0, 0)$
  • B
    $(0, 4, 5)$
  • C
    $(3, 0, 5)$
  • ✓
    None of these
Answer
Correct option: D.
None of these
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MCQ 301 Mark
Choose the correct answer. $x-$axis is the intersection of two planes:
  • ✓
    $xy$ and $xz.$
  • B
    $yz$ and $zx.$
  • C
    $xy$ and $yz.$
  • D
    None of these.
Answer
Correct option: A.
$xy$ and $xz.$
We know that on the $xy$ and $xz-$planes, the line of intersection is $x-$axis.
Hence, the correct option is $(a).$
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MCQ 311 Mark
The locus represented by $xy + yz = 0$ is:
  • A
    A pair of perpendicular lines
  • B
    A pair of parallel lines
  • C
    A pair of parallel planes
  • ✓
    A pair of perpendicular planes
Answer
Correct option: D.
A pair of perpendicular planes
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MCQ 321 Mark
The three planes divides the space into:
  • A
    Four parts.
  • B
    Six parts.
  • ✓
    Eight parts.
  • D
    Sixteen parts.
Answer
Correct option: C.
Eight parts.
Three planes divides the space into eight regions.
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MCQ 331 Mark
If the $zx-$plane divides the line segment joining $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $p : 1$ then $p + 1 =$
  • A
    $\frac{1}{3}$
  • B
    $1:3$
  • C
    $\frac{3}{4}$
  • ✓
    $\frac{4}{3}$
Answer
Correct option: D.
$\frac{4}{3}$
Given, points are $(1, -1, 5)$ and $(2, 3, 4)$ since $ZX-$plane divides the line segment in the ratio $p : 1, y-$coordinate will be $0$ the $y-$coordinate of the point dividing the line segment will be.
$=\frac{3\text{p}-1}{\text{p} + 1}=0,\text{ p}=\frac{1}{3}\text{ p}+1=\frac{1}{3}+1=\frac{4}{3}$
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MCQ 341 Mark
Choose the correct answer. The locus of a point for which $x = 0$ is:
  • A
    $xy-$plane.
  • ✓
    $yz-$plane.
  • C
    $zx-$plane.
  • D
    None of these
Answer
Correct option: B.
$yz-$plane.
On the $yz-$plane, $x = 0$
Hence, the locus of the point is $yz-$plane.
So, the correct option is $(b).$
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MCQ 351 Mark
Choose the correct answer.
If a parallelopiped is formed by planes drawn through the points $(5, 8, 10)$ and $(3, 6, 8)$ parallel to the coordinate planes, then the length of diagonal of the parallelopiped is:
  • ✓
    $2\sqrt{3}$
  • B
    $3\sqrt{2}$
  • C
    $\sqrt{2}$
  • D
    $\sqrt{3}$
Answer
Correct option: A.
$2\sqrt{3}$
Given parallelepiped passes through $A(5, 8, 10)$ and $B(3, 6, 8)$
$\therefore$ Length of the diagonal,
$\text{AB}=\sqrt{(5-3)^2+(8-6)^2+(10-8)^2}$
$=\sqrt{4+4+4}$
$=2\sqrt{3}$
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MCQ 361 Mark
The equation of the set of point $P,$ the sum of whose distance from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to $10$ is:
  • A
    $9 x^2+25 y^2+25 z^2+225=0$
  • ✓
    $9 x^2+25 y^2+25 z^2-225=0$
  • C
    $9 x^2+25 y^2-25 z^2-225=0$
  • D
    $9 x^2-25 y^2-25 z^2-225=0$
Answer
Correct option: B.
$9 x^2+25 y^2+25 z^2-225=0$
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MCQ 371 Mark
The plane $x = 0$ divides the joinning of $(-2, 3, 4)$ and $(1, -2, 3)$ in the ratio:
  • ✓
    $2 : 1$
  • B
    $1 : 2$
  • C
    $3 : 2$
  • D
    $-4 : 3$
Answer
Correct option: A.
$2 : 1$
$\text{R.E.F}$ image
Given, place $x = 0$ and two points
$\Rightarrow (-2, 3, 4)$ and $(1, -2, 3)$
let say a point $(x, y, z)$ in $x = 0$ place
$\text{x}=\frac{\text{m+n}(-2)}{\text{m+n}}=\frac{\text{m - 2n}}{\text{m+n}}$
$0=\frac{\text{m-2n}}{\text{m+n}}$
$\Rightarrow{\text{m}=2{\text{n}}}$
so $\frac{\text{m}}{\text{n}}=\frac{2}{1}$
$\Rightarrow2:1$
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MCQ 381 Mark
$\text{XOZ-}$plane divides the join of $(2, 3, 1)$ and $(6, 7, 1)$ in the ratio
  • A
    $3 : 7$
  • B
    $2 : 7$
  • ✓
    $-3 : 7$
  • D
    $-2 : 7$
Answer
Correct option: C.
$-3 : 7$
Let $A ≡ (2, 3, 1)$ and $B ≡ (6, 7, 1)$
Let the line joining $A$ and $B$ be divided by the $xz-$plane at point $P$ in the ratio $\lambda:1.$
Then, we have,
$\text{P}\equiv\Big(\frac{6\lambda+2}{\lambda+1},\ \frac{7\lambda+3}{\lambda+1},\ \frac{\lambda+1}{\lambda+1}\Big)$
Since $P$ lies on the $xz-$plane, the $y-$coordinate of $P$ will be zero.
$\therefore\frac{7\lambda+3}{\lambda+1}=0$
$\Rightarrow7\lambda+3=0$
$\Rightarrow\lambda=\frac{-3}{7}$
Hence, the $xz-$plane divides $AB$ in the ratio $-3 : 7$
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MCQ 391 Mark
The points $(5, 2, 4), (6, -1, 2)$ and $(8, -7, k)$ are collinear, if $k$ is equal to:
  • ✓
    $-2$
  • B
    $2$
  • C
    $3$
  • D
    $-1$
Answer
Correct option: A.
$-2$
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MCQ 401 Mark
Distance between $A(4, 5 ,6)$ from origin $O$ is:
  • A
    $25\sqrt3$
  • ✓
    $\sqrt{77}$
  • C
    $3\sqrt{5}$
  • D
    Data Insufficient
Answer
Correct option: B.
$\sqrt{77}$
Origin is $O(0, 0, 0)$ and given point is $A(4, 5, 6)$
So, distance $=\sqrt{(4-0)^2+(5-0)^2+(6-0)^2}$
$=\sqrt{4^2+5^2+6^2}$
$=\sqrt{77}$
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MCQ 411 Mark
The locus of a first$-$degree equation in $x, y, z$ is $a:$
  • A
    Sphere
  • B
    Straight line
  • ✓
    Plane
  • D
    None of these
Answer
Correct option: C.
Plane
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MCQ 421 Mark
The coordinates of any point, which lies in $yz$ plane, are:
  • A
    $(x, y, y)$
  • ✓
    $(0, y, y)$
  • C
    $(0, y, x)$
  • D
    $(x, y, z)$
Answer
Correct option: B.
$(0, y, y)$
In a $y-z$ plane, $x$ co$-$ordinate is always $0$
So $(0, y, y)$ and $(0, y, x)$ are point in a $y-z$ plane.
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MCQ 431 Mark
Which octant do the point $(-5, 4, 3)$ lie:
  • A
    Octant $I$
  • ✓
    Octant $II$
  • C
    Octant $III$
  • D
    Octant $IV$
Answer
Correct option: B.
Octant $II$
Given, $(-5, 4, 3)$ is the point
Here, the x-coordinate is negative but $y$ and $z$ coordinates are positive
$\therefore (-5, 4, 3)$ lie in octant $II.$
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MCQ 441 Mark
The ratio of $yz-$plane divide the line joining the points $A (3, 1, -5), B (1, 4, -6)$ is:
  • A
    $3 : 1$
  • ✓
    $-1 : 3$
  • C
    $1 : 3$
  • D
    $-3 : 1$
Answer
Correct option: B.
$-1 : 3$
Let $yz-$plane divide the line segment joining the points $A (3, 1, -5), B (1, 4, -6)$ in the ratio $m : n$
Then, $(0, y, z) =\Big(\frac{3\text{m+n}}{\text{m+n}},\frac{\text{m+4n}}{\text{m+n}},\frac{\text{-5m-6n}}{\text{m+n}}\Big)$
$\Rightarrow\frac{3\text{m+n}}{\text{m+n}}=0$
$\Rightarrow\text{m}:\text{n}=-1:3$
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MCQ 451 Mark
A cube of side $5$ has one vertex at the point $(1, 0, -1)$ 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:
  • A
    $(1, 0, 1)$
  • ✓
    $(0, -1, 0)$
  • C
    $(0, 0, -1)$
  • D
    $(1, 0, 0)$
Answer
Correct option: B.
$(0, -1, 0)$
Consider the problem Below, are four complete cube face on $XZ-$plane, $(y = 0)$
Given, point $(1, 0, -1)$
End of the edge parallel to negative $x-$axis $(0, 0 - 1)$ Origin $(0, 0, 0)$
End of the edge parallel to positive $z-$axis $(1, 0, 0)$
And, below Four point complete the opposite face of cube.
consider $P,$ end of edge parallel to negative $y-$axis $(1, -1, -1)$
Edge from $P$ parallel to positive $z-$axis $(0, -1, 0)$
Edge from $P$ parallel to negative $x-$axis $(0, -1, -1)$ And $(0, -1, 0)$
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MCQ 461 Mark
Coordinate planes divide the space into octants:
  • A
    $4$
  • B
    $6$
  • ✓
    $8$
  • D
    $10$
Answer
Correct option: C.
$8$
The coordinate planes divide the three dimensional space into eight octants.
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MCQ 471 Mark
The graph of the equation $y^2+ z^2= 0$ in three dimensional space is:
  • ✓
    $x-$axis
  • B
    $y-$axis
  • C
    $z-$axis
  • D
    $yz-$plane
Answer
Correct option: A.
$x-$axis
Consider the problem
$y^2 + z^2= 0$
$x = 0$ and $z = 0$
$\therefore$ The graph of the equation
$y^2+ z^2= 0$ is $x-$axis
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MCQ 481 Mark
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:
  • ✓
    $9, 18$
  • B
    $-9, -18$
  • C
    $3, -18$
  • D
    $-3, 18$
Answer
Correct option: A.
$9, 18$
Since $a, b$ are the roots of the equation.
$x^2+ px + q = 0$
$\Rightarrow a + b = -p$ and $ab = q$ Centroid of triangle is $\Big(\frac{\text{a}+6}{3},-3,0\Big)$
Given, centroid $(0, b, 0)$ Comparing, we get $b = -3$ and
$\frac{\text{a}+6}{3}=0\Rightarrow\text{a}=-6$
$p = 9, q = 18$
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MCQ 491 Mark
Name three undefined terms:
  • A
    Point
  • B
    Line
  • C
    Plane
  • ✓
    All of the above
Answer
Correct option: D.
All of the above
The basic undefined term is point. Line is formed from points and plane is formed from many lines. Undefined terms are point, line and plane.
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MCQ 501 Mark
The coordinate of any point, which lies in $xy$ plane, is:
  • A
    $(x, 0, y)$
  • ✓
    $(x, x, 0)$
  • C
    $(x, 0, x)$
  • D
    $(y, 0, x)$
Answer
Correct option: B.
$(x, x, 0)$
Given, that the point lies in $xy$ plane In $xy$ plane, the coordinate of $z$ will be zero so $(x, x, 0)$ represents a point which lies in $xy$ plane.
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MCQ 511 Mark
The equation of plane passing through $(-1, 0, -1)$ parallel to $xz$ plane is:
  • A
    $y = -2$
  • ✓
    $y = 0$
  • C
    $-x - z = 0$
  • D
    None of the above
Answer
Correct option: B.
$y = 0$
Given, that the plane is parallel to $xz$ plane and the plane passes through $(-1, 0, -1)$ since the plane is parallel to $xz$ plane, the $y-$coordinate should be constant
Given, that it passes through point $(-1, 0, -1)$
$\therefore$ The plane lies on $xz$ plane
$\therefore$ The equation of plane is $y = 0$
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MCQ 521 Mark
Arrange the points. $A(1, 2, -3), B(-1, -2, -3), C(-1, -2, -3)$ and $D(1, -2, -3)$ in the increasing order of their octant numbers:
  • ✓
    $\text{A, B, C, D}$
  • B
    $\text{B, C, D, A}$
  • C
    $\text{C, D, A, B}$
  • D
    $\text{D, C, B, A}$
Answer
Correct option: A.
$\text{A, B, C, D}$
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MCQ 531 Mark
The image of the point $P(1, 3, 4)$ in the plane $2x - y + z = 0$ is:
  • ✓
    $(-3, 5, 2)$
  • B
    $(3, 5, 2)$
  • C
    $(3, -5, 2)$
  • D
    $(3, 5, -2)$
Answer
Correct option: A.
$(-3, 5, 2)$
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MCQ 541 Mark
The ratio in which $xy-$plane divides the line joining the points $(1, 0, -3)$ and $(1, -5, 7)$ is given by:
  • A
    $7 : 3$
  • ✓
    $3 : 7$
  • C
    $3 : 4$
  • D
    $4 : 7$
Answer
Correct option: B.
$3 : 7$
Let $xy-$planexy divide the line joining the given points in the ratio $k : 1$ and the point of intersection is $x, y, 0$
$=\frac{\text{7k} - 3}{\text{k} + 1}=0$
$=7\text{k}-3=0$
$\text{k}=\frac{3}{7}$
$\therefore$ The ratio is $3 : 7$
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MCQ 551 Mark
The plane $\text{XOZ}$ divides the join of $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $\lambda:1$ then $\lambda$ is:
  • A
    $-3$
  • B
    $\frac{1}{4}$
  • C
    $3$
  • ✓
    $\frac{1}{3}$
Answer
Correct option: D.
$\frac{1}{3}$
Given points are $(1, -1, 5)$ and $(2, 3, 4)$ Using section formula the desired points is
$=\Big(\frac{2\lambda+1}{\lambda+1},\frac{3\lambda-1}{\lambda+1},\frac{4\lambda+5}{\lambda+1}\Big)$
Since, this point lies in $\text{XOZ}$ plane then its $yy-$co$-$ordinate should be zero.
$\Rightarrow\frac{3\lambda-1}{\lambda+1}=0$
$\Rightarrow\lambda=\frac{1}{3}$
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MCQ 561 Mark
$D(2, 1, 0), E(2, 0, 0), F(0, 1, 0)$ are mid point of the sides $\text{BC, CA, AB}$ of $\Delta\text{ABC}$ respectively, The the centroid of $\Delta\text{ABC}$ is:
  • A
    $\Big(\frac{1}{3},\frac{1}{3},\frac{1}{3}\Big)$
  • ✓
    $\Big(\frac{4}{3},\frac{2}{3},0\Big)$
  • C
    $\Big(-\frac{1}{3},\frac{1}{3},\frac{1}{3}\Big)$
  • D
    $\Big(\frac{2}{3},\frac{1}{3},\frac{1}{3}\Big)$
Answer
Correct option: B.
$\Big(\frac{4}{3},\frac{2}{3},0\Big)$
Centroid of triangle coincide with the centroid of triangle formed by joing the mid$-$point of sides of triangle.
So, centroid of $\triangle\text{ABC} =$ centroid of.
$\triangle\text{DEF}=\Big(\frac{2+2+0}{3},\frac{1+0+1}{3},\frac{0+0+0}{3}\Big) = \Big(\frac{4}{3},\frac{2}{3},0\Big)$
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MCQ 571 Mark
If the vertices of a triangle are $A(0, 4, 1), B(2, 3, -1)$ and $C(4, 5, 0),$ then orthocentre of a $\triangle\text{ABC}$ is:
  • A
    $(4, 5, 0)$
  • ✓
    $(2, 3, -1)$
  • C
    $(-2, 3, -1)$
  • D
    None of the above
Answer
Correct option: B.
$(2, 3, -1)$
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MCQ 581 Mark
If the $zx-$plane divides the line segment joining $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $p : 1$ then $p + 1 =$
  • A
    $\frac{1}{3}$
  • B
    $1$
  • C
    $\frac{3}{4}$
  • ✓
    $\frac{4}{3}$
Answer
Correct option: D.
$\frac{4}{3}$
Let the points be given by $(1, -1, 5)$ and $(2, 3, 4)$
A point that divides the line joining these $2$ points in the ratio $p : 1$ is
given by $\Big(\frac{2\text{p}+1}{\text{p}+1},\frac{3\text{p}-1}{\text{p}+1},\frac{4\text{p}+5}{\text{p}+1}\Big)$
Since, this point has to lie on the $zx-$plane.
so, $3p - 1 = 0$
$\Rightarrow\text{p}=\frac{1}{3}$
$\Rightarrow1+\text{p}=\frac{4}{3}$
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MCQ 591 Mark
An ordered triplet corresponds to in three dimensional space:
  • A
    Three points
  • ✓
    A unique point
  • C
    A point in each octant
  • D
    Infinite number of points
Answer
Correct option: B.
A unique point
It is fundamental fact that, The ordered triplet $(x, y, z)$ represents an unique point in three dimensional space.
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MCQ 601 Mark
Two opposite vertices of a rectangle are $(1, 3)$ and $(5, 1).$ If the rest two vertices lie on the line $y - x + l = 0,$ then $l$ is equal to:
  • ✓
    $1$
  • B
    $-1$
  • C
    $2$
  • D
    $3$
Answer
Correct option: A.
$1$
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MCQ 611 Mark
Choose the correct answer. The distance of point $P(3, 4, 5)$ from the $yz-$plane is:
  • ✓
    $3$ units.
  • B
    $4$ units.
  • C
    $5$ units.
  • D
    $550.$
Answer
Correct option: A.
$3$ units.
Given point is $P(3, 4, 5)$
$\therefore$ Distance of from yz-plane
$=\sqrt{(0-3)^2+(4-4)^2+(5-5)^2}$
$=\sqrt{9}$
$=3$ units
Hence, the correct option is $(a).$
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MCQ 621 Mark
Choose the correct answer. Distance of the point $(3, 4, 5)$ from the origin $(0, 0, 0)$ is:
  • ✓
    $\sqrt{50}$
  • B
    $3$
  • C
    $4$
  • D
    $5$
Answer
Correct option: A.
$\sqrt{50}$
Given point $A(3, 4, 5)$ and the given $O(0, 0, 0)$
$\therefore\sqrt{(3-0)^2+(4-0)^2+(5-0)^2}$
$=\sqrt{9+16+25}$
$=\sqrt{50}$
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MCQ 631 Mark
Distance between the points $(12, 4, 7)$ and $(10, 5, 3)$ is:
  • A
    $\sqrt{27}$
  • B
    $\sqrt{5}$
  • C
    $\sqrt{17}$
  • ✓
    None of these
Answer
Correct option: D.
None of these
Consider the problem,
Let the given points
$A(12, 4, 7)$ and $B(10, 5, 3)$
So, distance between A and B by distance formula.
$\text{AB}=\sqrt{(10-12)^2+(5-4)^2+(3-7)^2}$
$=\sqrt{(-2)^2+1^2+(-4)^2}$
$=\sqrt{4+1+6}$
$=\sqrt{21}$
So, distance between the points $(12, 4, 7)$ and $(10, 5, 3)$ is $\sqrt{21}\text{ Sq. units}$
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MCQ 641 Mark
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:
  • A
    $4$
  • B
    $3$
  • ✓
    $-4$
  • D
    $-3$
Answer
Correct option: C.
$-4$
Points $(2, 5)$ and $(5, 1)$ form a diagonal of the rectangle the mid point of these points will lie on the other diagonal. Mid Point $\Big(\frac{2+5}{2},\frac{5+1}{2}\Big)=\Big(\frac{7}{2},3\Big)$
Equation of the other diagonal is $y = 2x + k$
$\therefore3=2\times\frac{7}{2}+\text{k}$
$\Rightarrow\text{k}=-4$
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MCQ 651 Mark
The plane. $\text{ax + by + cz + (-3) = }0$ meet the co$-$ordinate axes in $\text{A, B, C.}$ The centroid of the triangle is:
  • A
    $\big(3\text{a, 3b, 3c}\big)$
  • B
    $\Big(\frac{3}{\text{a}},\frac{3}{\text{b}},\frac{3}{\text{c}}\Big)$
  • C
    $\Big(\frac{\text{a}}{3},\frac{\text{b}}{3},\frac{\text{c}}{3}\Big)$
  • ✓
    $\Big(\frac{1}{\text{a}},\frac{1}{\text{b}},\frac{1}{\text{c}}\Big)$
Answer
Correct option: D.
$\Big(\frac{1}{\text{a}},\frac{1}{\text{b}},\frac{1}{\text{c}}\Big)$
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MCQ 661 Mark
If the orthocentre, circumcentre of a triangle are $(-3, 5, 2), (6, 2, 5)$ respectively then the centroid of the triangle is:
  • ✓
    $(3, 3, 4)$
  • B
    $\Big(\frac{3}{2},\frac{7}{2},\frac{9}{2}\Big)$
  • C
    $(9, 9, 12)$
  • D
    $\Big(\frac{9}{2}\frac{-3}{2},\frac{3}{2}\Big)$
Answer
Correct option: A.
$(3, 3, 4)$
Since, the centroid divides the line joining the orthocentre and circumcentre in the ratio $2 : 1$
The coordinates of the centroid will be, $\Big(\frac{9}{3},\frac{9}{3},\frac{12}{3}\Big)$
$=(3, 3, 4)$
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MCQ 671 Mark
The ratio in which the line joining the points $(a, b, c)$ and $(-a, -c, -b)$ is divided by the $xy-$plane is
  • A
    $a : b$
  • B
    $b : c$
  • C
    $a : c$
  • ✓
    $c : d$
Answer
Correct option: D.
$c : d$
Let $A ≡ (a, b, c)$ and $B ≡ (-a, -c, -b)$
Let the line joining $A$ and $B$ be divided by the $xy-$plane at point $P$ in the ratio $\lambda:1.$
Then, we have,
$\text{P}\equiv\Big(\frac{-\text{a}\lambda+\text{a}}{\lambda+1},\ \frac{-\text{c}\lambda+\text{b}}{\lambda+1},\ \frac{-\text{b}\lambda+\text{a}}{\lambda+1}\Big)$
Since $P$ lies on the $xy-$plane, the $z-$coordinate of $P$ will be zero
$\therefore\frac{-\text{b}\lambda+\text{c}}{\lambda+1}=0$
$\Rightarrow-\text{b}\lambda+\text{c}=0$
$\Rightarrow\lambda=\frac{\text{c}}{\text{b}}$
Hence, the $xz-$plane divides $AB$ in the ratio $c : b$
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MCQ 681 Mark
Plane $ax + by + cz = 1$ intersect axes in $\text{A, B, C}$ respectively. If $\text{G}\Big(\frac{1}{6},-\frac{1}{3},{1}\Big)$ is a centroid of $\triangle\text{ABC}$ then $\text{a + b + 3c:}$
  • A
    $\frac{4}{3}$
  • B
    $4$
  • ✓
    $2$
  • D
    $\frac{5}{6}$
Answer
Correct option: C.
$2$
$\text{A}\Big(\frac{1}{\text{a}},0,0\Big)\text{ B}\Big(0,\frac{1}{\text{b}},0\Big)\text{ C}\Big(0,0,\frac{1}{\text{c}}\Big)$
centroid $\Rightarrow\Big(\frac{1}{3\text{a}},\frac{1}{3\text{b}},\frac{1}{3\text{c}}\Big)=\Big(\frac{1}{6},\frac{-1}{3},1\Big)$ On comparing we get,
$3a = 6$
$\Rightarrow a = 2$
$3b = -3$
$\Rightarrow b = -1$
$3c = 1$
$\Rightarrow\text{c}=\frac{1}{3}$
$\therefore a = 2, b = -1,$
$\Rightarrow\text{c}=\frac{1}{3}$
$\therefore a + b + 3c = 2$
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MCQ 691 Mark
The distance from the origin to the centroid of the tetrahedron formed by the points $(0, 0, 0), (3, 0, 0), (0, 4, 0), (0, 0, 5)$ is:
  • A
    $\frac{\sqrt{3+4+5}}{4}$
  • B
    $\frac{\sqrt{3+4+5}}{3}$
  • C
    $\frac{\sqrt{3^2+4^2+5^2}}{16}$
  • ✓
    $\frac{\sqrt{3^2+4^2+5^2}}{4}$
Answer
Correct option: D.
$\frac{\sqrt{3^2+4^2+5^2}}{4}$
$\text{G}=\Big(\frac{\text{a}}{4},\frac{\text{b}}{4},\frac{\text{c}}{4}\Big)\therefore\sqrt{\Big({\frac{\text{a}^2}{16}+\frac{\text{b}^2}{16}+\frac{\text{c}^2}{16}\Big)}}$
here, $a = 3, b = 4, c = 5$ substituting in above equation we get.
$\text{OG}=\sqrt{\frac{3^2+4^2+5^2}{4}}$
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MCQ 701 Mark
A point $C$ with position vector $\frac{\text{3a}+4\text{b}-5\text{c}}{3} ($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:
  • A
    $2a + 3b - 4c$
  • B
    $2a - 3b + 4c$
  • C
    $2a + 3b + 4c$
  • ✓
    $a + 3b - 4c$
Answer
Correct option: D.
$a + 3b - 4c$
$\frac{\text{3a}+4\text{b}-5\text{c}}{3}$
$\overrightarrow{\text{c}}=\frac{2\overrightarrow{\text{b}}+\overrightarrow{\text{a}}}{3}$
$\overrightarrow{\text{b}}=\frac{3\overrightarrow{\text{c}}-\overrightarrow{\text{a}}}{2}$
$=\frac{\big(3\overrightarrow{a}+4\overrightarrow{b}-5\overrightarrow{c}\big)-\big(\overrightarrow{a}+2\overrightarrow{b}-3\overrightarrow{c}\big)}{2}$
$=\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}$
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MCQ 711 Mark
The plane $\text{XOZ}$ divides the join of $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $\lambda:1$ then $\lambda$ is:
  • A
    $-3$
  • B
    $\frac{-1}{3}$
  • C
    $3$
  • ✓
    $\frac{1}{3}$
Answer
Correct option: D.
$\frac{1}{3}$
The plane $\text{XOZ}$ divides the join of $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio $\lambda:1$
i.e. $y = 0$ divide the join of $(1, -1, 5)$ and $(2, 3, 4)$ in the ratio.
$\lambda:1$
$\therefore\frac{3\lambda−1}{\lambda+1}=0$
$\Rightarrow\lambda=\frac{1}{3}$
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MCQ 721 Mark
If the points $A(3, -2, 4), B(1, 1, 1)$ and $C(-1, 4, -2)$ are collinear, then the ratio in which $C$ divides $AB$ is:
  • A
    $1 : 2$
  • ✓
    $-2 : 1$
  • C
    $-1 : 2$
  • D
    $4 : 0$
Answer
Correct option: B.
$-2 : 1$
Let $C$ divide $AB$ in the ratio $x : y$
Let us compare the $x-$coordinate of $C$ by using section formula.
$-1=\frac{\text{x}\times1+\text{y}\times3}{\text{x+y}}$
$\Rightarrow x + 3y = -x - y$
$\Rightarrow x = -2y$
point $C$ divides $AB$ in the ratio $-2 : 1.$
As the ratio is negative,
it means $C$ divides $AB$ externally.
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MCQ 731 Mark
If $A = (2, -3, 1), B = (3, -4, 6)$ and $C$ is a point of trisection of $AB,$ then $C_y​=$
  • A
    $\frac{11}{3}$
  • B
    $-11$
  • C
    $\frac{10}{3}$
  • ✓
    $\frac{-11}{3}$
Answer
Correct option: D.
$\frac{-11}{3}$
Given, $C$ is a point of trisection of $AB.$
$C$ either divides $AB$ in the ratio $2 : 1$ or $1 : 2$
Case $1: C$ divides in the ratio $2 : 1$
The coordinates of $C$ will be $\Big(\frac{8}{3},-\frac{11}{3},\frac{13}{3}\Big)$
Case $2: C$ divides in the ratio $1 : 2$
The coordinates of $C$ will be $\Big(\frac{7}{3},-\frac{10}{3},\frac{8}{3}\Big)$
either $C_y=$ $\frac{-11}{3}$ or $-\frac{10}{3}$
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MCQ 741 Mark
In geometry, we take a point, a line and a plane as undefined terms:
  • ✓
    True
  • B
    False
  • C
    Ambiguous
  • D
    Data Insufficient
Answer
Correct option: A.
True
In Geometry, we define a point as a location and no size. A line is defined as something that extends infinitely in either direction but has no width and is one dimensional while a plane extends infinitely in two dimensions.
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MCQ 751 Mark
The perpendicular distance of the point $P(6, 7, 8)$ from the $XY-$Plane is:
  • ✓
    $8$
  • B
    $7$
  • C
    $6$
  • D
    None of the above
Answer
Correct option: A.
$8$
Let $Q$ be the foot of perpendicular drawn from the point $P (6, 7, 8)$ to the $XY-$plane.
Thus, the distance of this foot $Q$ from $P$ is $z-$coordinate of $P,$
i.e. $8$ units.
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MCQ 761 Mark
Choose the correct answer. The point $(-2, -3, -4)$ lies in the:
  • A
    First octant.
  • ✓
    Seventh octant.
  • C
    Second octant.
  • D
    Eighth octant.
Answer
Correct option: B.
Seventh octant.
The point $(-2, -3, -4)$ lies in seventh octant.
Hence the correct option is $(b).$
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MCQ 771 Mark
If $x-$coordinate of a point $P$ of line joining the points $Q(2, 2, 1)$ and $R(5, 2, -2)$ is $4,$ then the $z-$coordinate of $P$ is:
  • A
    $-2$
  • ✓
    $-1$
  • C
    $1$
  • D
    $2$
Answer
Correct option: B.
$-1$
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MCQ 781 Mark
$(-1, -5, -7)$ lies in Octant:
  • A
    $I$
  • ✓
    $VII$
  • C
    $V$
  • D
    $III$
Answer
Correct option: B.
$VII$
Here all the three $x, y, z$ coordinate are negative of the given point.
Therefore, it will lie in the seventh Octant.
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MCQ 791 Mark
There are three points with position vectors $-2a + 3b + 5c, a + 2b + 3c$ and $7a - c.$ What is the relation between the three points:
  • ✓
    Collinear
  • B
    Forms a triangle
  • C
    In different plane
  • D
    None of the above
Answer
Correct option: A.
Collinear
The relation between the three points are collinear.
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MCQ 801 Mark
The length of the perpendicular drawn from the point $P(3, 4, 5)$ on $y-$axis is
  • A
    $10$
  • ✓
    $\sqrt{34}$
  • C
    $\sqrt{113}$
  • D
    $512$
Answer
Correct option: B.
$\sqrt{34}$
The length of the perpendicular drawn from the point $P (3, 4, 5)$ on $y-$axis is given by
$\sqrt{3^2+5^2}$
$=\sqrt{34}$
Hence, the correct answer is option $(b)$
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MCQ 811 Mark
The point $(3, 0, -4)$ lies on the:
  • A
    $Y-$axis
  • B
    $Z-$axis
  • C
    $XY-$plane
  • ✓
    $XZ-$plane
Answer
Correct option: D.
$XZ-$plane
$(3, 0, -4)$ Given pointClearly, $y = 0$ and $x$ and $z$ have non$-$zero value.
If the point lies on $x - z$ plane, this condition is possible. the answer is $XZ-$plane.
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MCQ 821 Mark
Which of the following is true for a plane:
  • ✓
    A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus
  • B
    Value of $y$ in a $zx$ plane is non$-$zero
  • C
    Value of $z$ in a $xy$ plane is zero
  • D
    None of the above
Answer
Correct option: A.
A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus
Option $A$ and $C$ are correct $A$ locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus. and also Value of $z$ in a $xy$ plane is zero.
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MCQ 831 Mark
Points $\text{A}\big(3,2,4),\text{B}\Big(\frac{33}{5},\frac{28}{5},\frac{38}{5}\Big),\text{C}\big(9,8,10\big)$ are given The ratio in which $B$ divides $\overline{\text{AC}}$ is:
  • A
    $5 : 3$
  • B
    $2 : 1$
  • C
    $1 : 3$
  • ✓
    $3 : 2$
Answer
Correct option: D.
$3 : 2$
$B$ divides $AC$ in the ratio is $x_1 - x_2​: x_2​- x_3$
$3-\frac{33}{5}:\frac{33}{5}-9$
$3:2$
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MCQ 841 Mark
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:
  • A
    $3 : 5$
  • B
    $7 : 5$
  • C
    $9 : 11$
  • ✓
    $1 : 5$ externally
Answer
Correct option: D.
$1 : 5$ externally
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MCQ 851 Mark
The point $A(1, -1, 3), B(2, -4, 5)$ and $C(5, -13, 11)$ are:
  • ✓
    Collinear
  • B
    Non$-$collinear
  • C
    Do not say anything
  • D
    None of these
Answer
Correct option: A.
Collinear
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MCQ 861 Mark
The ordinate of the point which divides the lines joining the origin and the point $(1, 2)$ externally in the ratio of $3 : 2$ is:
  • A
    $-2$
  • B
    $\frac{3}{5}$
  • C
    $\frac{2}{5}$
  • ✓
    $6$
Answer
Correct option: D.
$6$
Co$-$ordinates of the required point will be
$\text{y}=\frac{\text{m}_{1}\text{y}_{2}-\text{m}_{2}\text{y}_{1}}{\text{m}_{1}-\text{m}_{2}}$
$=\frac{3\times2-2\times0}{3-2}$
$=6$
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MCQ 871 Mark
A point at which all the three perpendicular coordinate axes meets is known as:
  • A
    Meeting point
  • ✓
    Origin
  • C
    Triple point
  • D
    None of these
Answer
Correct option: B.
Origin
The three perpendicular coordinate axes meets at one of the point which divides the plane into eight quadrant.
The $1^{st}$ quadrant has all positive points, $2^{nd}$ has $x-ve$ and remaining $2\ +ve$ points and so on.
Only $(0, 0, 0)$ is not included in any quadrant and is the intersection point.
Thus, the three axes meet at $(0, 0,0)$ from where the eight quadrants originate.
Hence, the point is known as origin.
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MCQ 881 Mark
In the $\Delta \text{ABC} A = (1, 3, -2)$ and $G (-1, 4, 2)$ is the centroid of the triangle. If $D$ is the mid point of $BC$ then $AD =$
  • A
    $\frac{\sqrt21}{2}$
  • ✓
    $\frac{3\sqrt21}{2}$
  • C
    $\sqrt{21}$
  • D
    $\frac{63}{2}$
Answer
Correct option: B.
$\frac{3\sqrt21}{2}$
First, we calculate the distance $AG,$
It is $(4 + 1 + 16)^{0.5}= 21^{0.5}$
From the property of the centroid that it divides the line joining $AD$ in the ratio $2 : 1.$
The distance $=\text{AD}=\frac{3}{2}\text{AG}$
$\text{AD}=\frac{3}{2}{ 21}^{0.5}$
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MCQ 891 Mark
Area of quadrilateral whose vertices are $(2, 3), (3, 4), (4, 5)$ and $(5, 6),$ is equal to:
  • ✓
    $0$
  • B
    $4$
  • C
    $6$
  • D
    None of these
Answer
Correct option: A.
$0$
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MCQ 901 Mark
The perpendicular distance of the point $P(6, 7, 8)$ from $xy-$plane is
  • ✓
    $8$
  • B
    $7$
  • C
    $6$
  • D
    $10$
Answer
Correct option: A.
$8$
The distance of the point $P(6, 7, 8)$ from the $xy-$plane is equal to the $z-$coordinate of the point.
Here, the value of $z-$coordinate is $8.$
Hence, the correct answer is option $(a)$.
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MCQ 911 Mark
if $P(0, 1, 2), Q (4, -2, 1)$ and $O(0, 0, 0)$ are three points, then $\angle\text{POQ}=$
  • A
    $\frac{\pi}{6}$
  • B
    $\frac{\pi}{4}$
  • C
    $\frac{\pi}{3}$
  • ✓
    $\frac{\pi}{2}$
Answer
Correct option: D.
$\frac{\pi}{2}$
$\mathrm{PQ}^2=(4-0)^2+(-2-1)^2+(1-2)^2=16+9+1=26$
$\mathrm{OP}^2=(0-0)^2+(1-0)^2+(2-0)^2=0+1+4=5$
$\mathrm{QO}^2=(0-4)^2+(0+2)^2+(0-1)^2=16+1+4=21$
Since, $\mathrm{PQ}^2=\mathrm{OP}^2+\mathrm{QO}^2$
Hence, $\angle\text{POQ}=\frac{π}{2}$
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MCQ 921 Mark
The position vectors of the four angular point of a tetrahedron $\text{OABC}$ are $(0, 0, 0), (0, 0, 2), (0, 4, 0)$ and $(6, 0, 0)$ respectively. Find the coordinates of cenroid:
  • A
    $\Big(2,\frac{4}{3},\frac{2}{3}\Big)$
  • ✓
    $\Big(\frac{6}{4},1,\frac{2}{4}\Big)$
  • C
    $(0, 0, 0)$
  • D
    None of these
Answer
Correct option: B.
$\Big(\frac{6}{4},1,\frac{2}{4}\Big)$
Angular points of tetrahedron $\text{OABC}$ are.
$(0, 0, 0), (0, 0, 2), (0, 4, 0), (6, 0, 0)$ To find the coordinates of the centroid of the tetrahedron whose vertices are
$\left(x_1, y_1, z_1\right),\left(x_2, y_2, z_2\right),\left(x_3, y_3, z_3\right)$ and $\left(x_4, y_4, z_4\right)$ the centroid is
$\Big(\frac{\text{x}_{1}+\text{x}_{2}+\text{x}_{3}+\text{x}_{4}}{4}\Big),\Big(\frac{\text{y}_{1}+\text{y}_{2}+\text{y}_{3}+\text{y}_{4}}{4}\Big),\Big(\frac{\text{z}_{1}+\text{z}_{2}+\text{z}_{3}+\text{z}_{4}}{4}\Big)$
Now, substituting the values we get
$\Big(\frac{0+0+0+6}{4}\Big),\Big(\frac{0+0+4+0}{4}\Big),\Big(\frac{0+2+0+0}{4}\Big)$
$\therefore$ The coordinates of the centroid are $\Big(\frac{6}{4},1,\frac{2}{4}\Big)$
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MCQ 931 Mark
Under what condition does the equation $x^2+ y^2+ z^2+ 2ux + 2vy + 2wz + d$ represent a real sphere:
  • A
    $ u^2+v^2+w^2=d^2 $
  • ✓
    $ u^2+v^2+w^2>d $
  • C
    $ u^2+v^2+w^2$
  • D
    $ u^2+v^2+w^2$
Answer
Correct option: B.
$ u^2+v^2+w^2>d $
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MCQ 941 Mark
Choose the correct answer. If the distance between the points $(a, 0, 1)$ and $(0, 1, 2)$ is $27$, then the value of a is:
  • A
    $5$
  • ✓
    $\pm5$
  • C
    $-5$
  • D
    None of these.
Answer
Correct option: B.
$\pm5$
Given points are $A(a, 0, 1)$ and $B(0, 1, 2).$
$\therefore\text{AB}=\sqrt{(\text{a}-0)^2+(0-1)^2+(1-2)^2}$
$=\sqrt{27} ($Given$)$
$\Rightarrow27=\text{a}^2+2$
$\Rightarrow\text{a}^2=25$
$\Rightarrow\text{a}=\pm5$
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MCQ 951 Mark
An equation of sphere with centre at origin and radius $r$ can be represented as:
  • A
    $x^2+y^2+z^2=r$
  • ✓
    $x^2+y^2+z^2=r^2$
  • C
    $x^2+y^2+z^2=2 r^2$
  • D
    None of the above
Answer
Correct option: B.
$x^2+y^2+z^2=r^2$
Sphere is locus of a point in $3D$ whose distance from a fixed point $($center$)$ is constant $($radius$)$
$\Rightarrow\sqrt{({\text{x}-0})^2+(\text{y}-0)^2+(\text{z}-0)^2}$
$=\mid\text{r}\mid\Rightarrow\text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2$ square both sides.
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MCQ 961 Mark
A point is on the $x-$axis. Which of the following represent the point:
  • A
    $(0, x, 0)$
  • B
    $(0, 0, x)$
  • ✓
    $(x, 0, 0)$
  • D
    None of the above
Answer
Correct option: C.
$(x, 0, 0)$
At $x-$axis, $y$ and $z$ coordinates are zero.
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MCQ 971 Mark
In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are:
  • A
    Three mutually parallel lines
  • ✓
    Three mutually perpendicular lines
  • C
    Two mutually perpendicular lines and any two parallel
  • D
    None of these
Answer
Correct option: B.
Three mutually perpendicular lines
In three dimensions, the coordinate axes, i.e. $x, y$ and $z$ axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. The word rectangular is used to indicate perpendicularity among the axes.
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MCQ 981 Mark
Assertion $(A):$ If centroid and circumcentre of a triangle are known its orthocentre can be found
Reason $(R):$ Centriod, orthocentre and circumcentre of a triangle are collinear
  • ✓
    Both $A$ and $R$ are individually true and $R$ is the correct explanation of $A.$
  • B
    Both $A$ and $R$ individually true but $R$ is not the correct explanation of $A.$
  • C
    $A$ is true but $R$ is false.
  • D
    $A$ is false but $R$ is true.
Answer
Correct option: A.
Both $A$ and $R$ are individually true and $R$ is the correct explanation of $A.$
Centroid, orthocentre and circumcentre are collinear and centroid dividesthe line joining orthocentre and circumcentre in the ratio $2 : 1$ so if any two points are given then this one can be found.
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MCQ 991 Mark
If the extremities of the diagonal of a square are $(1, -2, 3)$ and $(2, -3, 5),$ then the length of the side is
  • A
    $\sqrt{6}$
  • ✓
    $\sqrt{3}$
  • C
    $\sqrt{5}$
  • D
    $\sqrt{7}$
Answer
Correct option: B.
$\sqrt{3}$
Length of the diagonal $=\sqrt{(2 − 1)^2 + (−3 + 2)^2 + (5 − 3)^2}$
$=\sqrt{1 + 1 + 4}$
$=\sqrt{6}$
$\therefore$ Length of the side $=\frac{\text{Length of diagonal}}{\sqrt{2}}$
$=\frac{\sqrt{6}}{\sqrt{2}}$
$=\sqrt{3}$
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MCQ 1001 Mark
The coordinates of the foot of the perpendicular drawn from the point $P(3, 4, 5)$ on the $yz-$ plane are
  • A
    $(3, 4, 0)$
  • ✓
    $(0, 4, 5)$
  • C
    $(3, 0, 5)$
  • D
    $(3, 0, 0)$
Answer
Correct option: B.
$(0, 4, 5)$
We know that the x-coordinate on $yz-$plane is $0.$
The coordinates of the foot of the perpendicular drawn from the point $P(3, 4, 5)$ on the $yz-$plane are $(0, 4, 5).$
Hence, the correct answer is option $(b).$
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MCQ 1011 Mark
The points $(3, 2, 0), (5, 3, 2)$ and $(-9, 6, -3),$ are the vertices of a triangle $\text{ABC.AD}$ is the internal bisector of $\angle\text{BAC}$ which meets $BC$ at $D.$ Then the co$-$ordinates of $D,$ are:
  • A
    $\Big[\frac{17}{16},\frac{57}{16},\frac{19}{8}\Big]$
  • ✓
    $\Big[\frac{19}{8},\frac{57}{16},\frac{17}{16}\Big]$
  • C
    $\Big[0,0,\frac{17}{16}\Big]$
  • D
    $\Big[\frac{17}{16},0,0\Big]$
Answer
Correct option: B.
$\Big[\frac{19}{8},\frac{57}{16},\frac{17}{16}\Big]$
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MCQ 1021 Mark
Find the coordinates of the point which divides the line segment joining the points $(-2, 3, 5)$ and $(1, -4, 6)$ in the ratio $2 : 3$ externally:
  • A
    $(-8, -17, 3)$
  • ✓
    $(-8, 17, 3)$
  • C
    $(8, -17, 3)$
  • D
    None of these
Answer
Correct option: B.
$(-8, 17, 3)$
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MCQ 1031 Mark
$A = (1, -1, 2)$ and $B = (2, 3, 7)$ are two points. lf $P, O$ divide $AB$ in the ratios $2 : 3, -2 : 3$ respectively then $P_x + Q_y​=$
  • ✓
    $\frac{-38}{5}$
  • B
    $\frac{38}{5}$
  • C
    $\frac{-2}{5}$
  • D
    $\frac{-47}{6}$
Answer
Correct option: A.
$\frac{-38}{5}$
$P$ divides line joining $A (1, -1, 2)$ and $B (2, 3, 7)$ in the ratio $2 : 3$
$\therefore\text{P}_{\text{x}}=\frac{2\times2+3\times1}{2+3}=\frac{7}{5}$
Similarly, $Q$ divides line joining $A (1, -1, 2)$ and $B (2, 3, 7)$ in the ratio $-2 : 3$
$\therefore\text{Q}_{\text{y}}=\frac{-2\times3+3\times-1}{-2+3}=-9$
$\Rightarrow\text{P}_{\text{x}}+\text{Q}_{\text{y}}$
$=-9+\frac{7}{5}$
$=\frac{-38}{5}$
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MCQ 1041 Mark
Choose the correct answer. $L$ is the foot of the perpendicular drawn from a point $(3, 4, 5)$ on $x-$axis. The coordinates of $L$ are:
  • ✓
    $(3, 0, 0).$
  • B
    $(0, 4, 0).$
  • C
    $(0, 0, 5).$
  • D
    None of these.
Answer
Correct option: A.
$(3, 0, 0).$
On the $x-$axis, $y = 0$ and $z = 0.$
Hence, the required coordinates are $(3, 0, 0).$
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MCQ 1051 Mark
The coordinates of the foot of the perpendicular from a point $P(6, 7, 8)$ on $x-$axis are
  • ✓
    $(6, 0, 0)$
  • B
    $(0, 7, 0)$
  • C
    $(0, 0, 8)$
  • D
    $(0, 7, 8)$
Answer
Correct option: A.
$(6, 0, 0)$
We know that the $y$ and $z$ coordinates on $x-$axis are $0$
The coordinates of the foot of the perpendicular from a point $P(6, 7, 8)$ on $x-$axis are $(6, 0, 0)$
Hence, the correct answer is option $(a).$
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MCQ 1061 Mark
If point $p$ lies in first octant, then the sign of $x-$ coordinate will always be:
  • ✓
    $+$
  • B
    $−$
  • C
    $x$ coordinate is always $0$
  • D
    $x$ coordinate can be $+$ or $-$
Answer
Correct option: A.
$+$
In the first octant, the values of $x, y$ and $z$ axes are positive.
Any point which lies in first octant has all their coordinate values as positive.
Since point $p$ lies in first octant,
so, $p$ will have all its coordinates as positive.
Hence, sign of $x-$coordinate will always be $+.$
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MCQ 1071 Mark
The $xy-$plane divides the line joining the points $(-1, 3, 4)$ and $(2, -5, 6):$
  • A
    Internally in the ratio $2 : 3$
  • ✓
    Externally in the ratio $-2 : 3$
  • C
    Internally in the ratio $3 : 2$
  • D
    Externally in the ratio $3 : 2$
Answer
Correct option: B.
Externally in the ratio $-2 : 3$
The ratio that $xy-$plane divides the line joining the points $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)=-z_1: z_2$
IF the result is positive, it divides internally otherwise externally.
The ratio that $xy-$plane divides the line joining the points $(-1, 3, 4)$ and $(2, -5, 6) = -4 : 6 = -2 : 3$
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MCQ 1081 Mark
Choose the correct answer. A plane is parallel to $yz-$plane so it is perpendicular to:
  • ✓
    $x-$axis.
  • B
    $y-$axis.
  • C
    $z-$axis.
  • D
    None of these.
Answer
Correct option: A.
$x-$axis.
Any plane parallel to $yz-$plane,
so it is perpendicular to $x-$axis.
Hence, the correct option is $(a)$
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MCQ 1091 Mark
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:
  • ✓
    $(-11, 3, 8)$
  • B
    $(-11, 3, -8)$
  • C
    $(-8, 12, 20)$
  • D
    $(13, 6, -13)$
Answer
Correct option: A.
$(-11, 3, 8)$
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MCQ 1101 Mark
The points $A(5, -1, 1), B(7, -4, 7), C(1, -6, 10)$ and $D(-1, -3, 4)$ are vertices of a:
  • A
    Square
  • ✓
    Rhombus
  • C
    Rectangle
  • D
    None of these
Answer
Correct option: B.
Rhombus
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MCQ 1111 Mark
$A = (2, 4, 5)$ 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 $\frac{\text{a}}{\text{b}}+\frac{\text{p}}{\text{q}}=$
  • A
    $\frac{23}{12}$
  • B
    $\frac{-7}{12}$
  • ✓
    $\frac{7}{12}$
  • D
    $\frac{-22}{15}$
Answer
Correct option: C.
$\frac{7}{12}$
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MCQ 1121 Mark
$(-1, -5, -7)$ lies in Octant:
  • A
    $I$
  • ✓
    $VII$
  • C
    $V$
  • D
    $III$
Answer
Correct option: B.
$VII$
Here all the three $x, y, z$ coordinate are negative of the given point.
$\therefore$ it will lie in the seventh Octant.
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MCQ 1131 Mark
The planes $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$ are:
  • ✓
    Parallel
  • B
    Perpendicular
  • C
    Intersect
  • D
    Intersect x-axis
Answer
Correct option: A.
Parallel
Planes are $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$
Multiply both sides by $2$ to the second equation
$\Rightarrow 10x - 5y + 20 = 12$ Now divide both sides by $2$
$\Rightarrow2\text{x - y + 4z}=\frac{12}{5}$
Clearly both planes are parallel.
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MCQ 1141 Mark
The distance of point $P(3, 4, 5)$ from the $yz-$plane is:
  • ✓
    $3$ units
  • B
    $4$ units
  • C
    $5$ units
  • D
    $550$ units
Answer
Correct option: A.
$3$ units
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MCQ 1151 Mark
Four vertices of a tetrahedron are $(0, 0, 0), (4, 0, 0), (0, -8, 0)$ and $(0, 0, 12)$ Its centroid has the coordinates:
  • A
    $\Big(\frac{4}{3},-\frac{8}{3},4\Big)$
  • B
    $(2, -4, 6)$
  • ✓
    $(1, -2, 3)$
  • D
    None of these
Answer
Correct option: C.
$(1, -2, 3)$
The centroid of the coordinates is
$\Big(\frac{\text{x}_{1}+\text{x}_{2}+\text{x}_{3}+\text{x}_{4}}{4},\frac{\text{y}_{1}+\text{y}_{2}+\text{y}_{3}+\text{y}_{4}}{4},\frac{\text{z}_{1}+\text{z}_{2}+\text{z}_{3}+\text{z}_{4}}{4}\Big)$
Thus by substituting the vertices we get
$=\Big(\frac{0+4+0+0}{4},\frac{0+0-8+0}{4},\frac{0+0+0+12}{4}\Big)$
$=\Big(\frac{4}{4},\frac{-8}{4},\frac{12}{4}\Big)$
$\therefore$ The centroid of the coordinates is $(1, -2, 3)$
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MCQ 1161 Mark
Find the distance between the points whose position vectors are given as follows : $4\hat{\text{i}} + 3\hat{\text{j}} - 6\hat{\text{k}},-2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$
  • ✓
    $\sqrt{65}$
  • B
    $\sqrt{69}$
  • C
    $1$
  • D
    None of these
Answer
Correct option: A.
$\sqrt{65}$
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MCQ 1171 Mark
Let $(3, 4, -1)$ and $(-1, 2, 3)$ be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
  • A
    $2$
  • ✓
    $3$
  • C
    $6$
  • D
    $7$
Answer
Correct option: B.
$3$
$\text{d}^2=(−1−3)^2+(2−4)^2+(3+1)^2$
$\Rightarrow\text{d}^2=(−4)^2+(−2)^2+(4)^2$
$\Rightarrow\text{d}^2=16+4+16$
$\Rightarrow\text{d}^2=36$
$\Rightarrow\text{d}^2=6$
Hence, radius of the sphere is $3$ units.
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MCQ 1181 Mark
Find the value of $x$ for which the points $(x, -1), (2, 1)$ and $(4, 5)$ are collinear:
  • A
    $-1$
  • ✓
    $1$
  • C
    $2$
  • D
    $-2$
Answer
Correct option: B.
$1$
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MCQ 1191 Mark
A plane intersects the co ordinate axes at $\text{A, B, C.}$ If $O = (0, 0, 0)$ and $(1, 1, 1)$ is the centroid of the tetrahedron $\text{OABC,}$ then the sum of the reciprocals of the intercepts of the plane:
  • A
    $12$
  • B
    $\frac{4}{3}$
  • C
    $1$
  • ✓
    $\frac{3}{4}$
Answer
Correct option: D.
$\frac{3}{4}$
Let the point of intersections be,
$A (a, 0, 0), B (0, b, 0)$ and $C (0, 0, c)$
The coordinates of the centroid are $\Big(\frac{\text{a}}{4},\frac{\text{b}}{4},\frac{\text{c}}{4}\Big)$
Comparing it with the coordinates given, we get
$a = 4, b = 4, c = 4$
$\Big(\frac{1}{\text{a}}, \frac{1}{\text{b}},\frac{1}{\text{c}}\Big)=\frac{3}{4}$
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MCQ 1201 Mark
In a three dimensional space, the equation $3x - 4y = 0$ represents:
  • A
    A plane containing $y$ axis
  • B
    A plane containing $x$ axis
  • ✓
    A plane containing $z$ axis
  • D
    None  of these
Answer
Correct option: C.
A plane containing $z$ axis
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MCQ 1211 Mark
The ratio in which the line joining $(3, 4, -7)$ and $(4, 2, 1)$ is dividing the $xy-$plane:
  • A
    $3 : 4$
  • B
    $2 : 1$
  • ✓
    $7 : 1$
  • D
    $4 : 3$
Answer
Correct option: C.
$7 : 1$
Let the given points be
$A (3, 4, -7), B (4, 2, 1)$
Let a point on $XY-$plane be $P (x, y, 0)$ and the line $AB$ in the ratio $k : 1$
then by section formula
$0=\frac{\text{k}\times1+1\times-7}{\text{k}+1}$
$k - 7 = 0$
$k = 7$
$\therefore$ ratio is $7 : 1$
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MCQ 1221 Mark
If $(1, -1, 0), (-2, 1, 8)$ and $(-1, 2, 7)$ are three consecutive vertices of a parallelogram then the fourth vertex is:
  • ✓
    $(2, 0, -1)$
  • B
    $(1, 0, -1)$
  • C
    $(1, -2, 0)$
  • D
    $(0, -2, 1)$
Answer
Correct option: A.
$(2, 0, -1)$
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MCQ 1231 Mark
Locus of a point $P$ which such that $\text{PA = PB}$ where $A = (0, 3, 2)$ and $B = (2, 4, 1)$ is:
  • ✓
    $2x + y - z = 4$
  • B
    $x - 2y + z + 1 = 0$
  • C
    $9x - 2y + 4z - 5 = 0$
  • D
    None of these
Answer
Correct option: A.
$2x + y - z = 4$
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MCQ 1241 Mark
The distance of origin from the image of $(1, 2, 3)$ in plane $x - y + z = 5$ is:
  • A
    $\sqrt{17}$
  • B
    $\sqrt{29}$
  • ✓
    $\sqrt{34}$
  • D
    $\sqrt{41}$
Answer
Correct option: C.
$\sqrt{34}$
$P(1, 2, 3),$ Plane: $x - y + z = 5$
$F$ is foot of perpendicular form $P$ to plane and $I$ is image, then $\text{PF = FI}$
$\therefore$ If $(x, y, z) = (r + 1, -r + 2, r + 3)$ are foot of perpendicular.
$\Rightarrow (r + 1) - (-r + 2) + r + 3 = 5$
$\Rightarrow r = 1$
$\therefore F = (2, 1, 4)$
$\therefore I = (3, 0, 5)$
$\therefore$ Distance of $I$ from origin $=\sqrt{3^2+0^2+5^2}=\sqrt{34}$
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MCQ 1251 Mark
If the plane a $2x - 3y + 5Z ​- 2 = 0$ divides the line segment joining $(1, 2, 3)$ and $(2, 1, k)$ in the ratio $9 : 11,$ then $k$ is:
  • A
    $11$
  • ✓
    $-2$
  • C
    $-10$
  • D
    $-\frac{1}{2}$
Answer
Correct option: B.
$-2$
Coordinate of the point which divides the line segment joining the points
$(1, 2, 3)$ and $(2, 1, k)$ in the ratio $9 : 11$ are $\Big(\frac{29}{20},\frac{31}{20},\frac{9\text{k}+33}{20}\Big)$
Also, this point will lie on the given plane
$\Rightarrow2\times\frac{29}{20}-3\times\frac{31}{20}+5\times\frac{9\text{k}+33}{20}-2=0$
$\Rightarrow\text{k}=-2$
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MCQ 1261 Mark
A point on $\text{XOZ-}$plane divides the join of $(5, -3, -2)$ and $(1, 2, -2)$ at:
  • ✓
    $\Big(\frac{13}{5},0,-2\Big)$
  • B
    $\Big(\frac{13}{5},0,2\Big)$
  • C
    $\Big(5, 0, 2\Big)$
  • D
    $\Big(5, 0, -2\Big)$
Answer
Correct option: A.
$\Big(\frac{13}{5},0,-2\Big)$
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MCQ 1271 Mark
If $A = (1, 2, 3), B = (2, 3, 4)$ and $C$ is a point of trisection of $AB$ such that $\text{C}_{\text{x}} + \text{C}_{\text{y}} = \frac{13}{3}$ then $\text{C}_\text{z}=$
  • A
    $\frac{10}{3}$
  • ✓
    $\frac{11}{3}$
  • C
    $\frac{11}{2}$
  • D
    $11$
Answer
Correct option: B.
$\frac{11}{3}$
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MCQ 1281 Mark
The points $(5, -4, 2), (4, -3, 1), (7, -6, 4)$ and $(8, -7, 5)$ are the vertices of:
  • A
    A rectangle
  • B
    A square
  • ✓
    A parallelogram
  • D
    None of these
Answer
Correct option: C.
A parallelogram
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MCQ 1291 Mark
In the tetrahedron $\text{ABCD, A} = (1, 2, -3)$ and $G (-3, 4, 5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta\text{BCD}$ then $AP =$
  • ✓
    $\frac{8\sqrt{21}}{3}$
  • B
    $ \frac{4\sqrt{21}}{3}$
  • C
    $4\sqrt{21}$
  • D
    $\frac{\sqrt{21}}{3}$
Answer
Correct option: A.
$\frac{8\sqrt{21}}{3}$
Given, $A = (1, 2, -3), G (-3, 4, 5)$
$\therefore\text{AG}=\sqrt{(-3-1)^2+(4-2)^2+(5-(-3))^2}$
and $AG=\sqrt { 84 } =2\sqrt { 21 }$
$P$ is the centroid of $\triangle\text{BCD}$
So, $G$ divides $AP$ in $3 : 1$
Let $AG = 3x,$ then, $GP = x$
$\text{3x}=2\sqrt{21}$
$\text{x}=\frac{2\sqrt2}{3}$
Now $\text{AP = AG + GP}$
$\Rightarrow AP = 3x + x$
$\Rightarrow AP = 4x$
$\Rightarrow\text{AP}=4\Big(\frac{2\sqrt2}{3}\Big)=\frac{8\sqrt21}{3}$
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MCQ 1301 Mark
Find the ratio in which the $YZ-$plane divides the line segment formed by joining the points $(-2, 4, 7)$ and $(3, -5, 8):$
  • A
    Externally $2 : 3$
  • ✓
    Internally $2 : 3$
  • C
    Internally $3 : 2$
  • D
    Externally $3 : 2$
Answer
Correct option: B.
Internally $2 : 3$
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MCQ 1311 Mark
Find the coordinates of the points which trisect the line segment joining the points $P(4, 2, -6)$ and $Q(10, -16, 6):$
  • A
    $(6, -4, -2), (8, -10, 2)$
  • ✓
    $(6, 4, -2), (8, -10, 2)$
  • C
    $(6, -4, -2), (8, 10, 2)$
  • D
    None of these
Answer
Correct option: B.
$(6, 4, -2), (8, -10, 2)$
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MCQ 1321 Mark
The centroid of triangle $A(3, 4, 5), B(6, 7, 2), C(0, -5, 2)$ is:
  • ✓
    $(3, 2, 3)$
  • B
    $(5, 2, 1)$
  • C
    $(2, 5, 1)$
  • D
    $(3, 4, 1)$
Answer
Correct option: A.
$(3, 2, 3)$
$A (3, 4, 5), B (6, 7, 2), C (0, -5, 2)$ Centroid is given as.
$\Big(\frac{3+6+0}{3},\frac{4+7-5}{3},\frac{5+2+2}{3}\Big)$
$=\Big(\frac{9}{3},\frac{6}{3},\frac{9}{3}\Big)$
$=(3, 2, 3)$
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MCQ 1331 Mark
The cartesian equation of the line is $3x + 1 = 6y - 2 = 1 - z$ then its direction ratio are:
  • ✓
    $\frac{1}{3},\frac{1}{6},1$
  • B
    $\frac{-1}{3},\frac{1}{6},1$
  • C
    $\frac{1}{3},\frac{-1}{6},1$
  • D
    $\frac{1}{3},\frac{1}{6},-1$
Answer
Correct option: A.
$\frac{1}{3},\frac{1}{6},1$
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MCQ 1341 Mark
What is the locus of a point for which $y = 0, z = 0?$
  • ✓
    $x-$axis
  • B
    $y-$axis
  • C
    $z-$axis
  • D
    $yz-$plane
Answer
Correct option: A.
$x-$axis
We know that on $x-$axis both $y = 0, z = 0.$
Hence, the correct answer is option $(a)$
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MCQ 1351 Mark
If the vertices of a triangle are $(-1, 6, -4), (2, 1, 1)$ and $(5, -1, 0)$ then the centroid of the triangle is:
  • A
    $(6, 6, -3)$
  • ✓
    $(2, 2, -1)$
  • C
    $\Big(3,3,-\frac{3}{2}\Big)$
  • D
    None of these
Answer
Correct option: B.
$(2, 2, -1)$
The centroid of the triangle is
$\Big(\frac{\text{x}_{1}+\text{x}_{2}+\text{x}_{3}}{3},\frac{\text{y}_{1}+\text{y}_{2}+\text{y}_{3}}{3},\frac{\text{z}_{1}+\text{z}_{2}+\text{z}_{3}}{3}\Big)$
Thus by substituting the vertices we get
$=\Big(\frac{-1+2+5}{6},\frac{6+1-1}{3},\frac{-4+1+0}{3}\Big)=\Big(\frac{6}{3},\frac{6}{3},\frac{-3}{3}\Big)$
$\therefore$ The centroid of the triangle is $(2, 2, -1)$
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MCQ 1361 Mark
The ratio in which the line joining the points $(1, 2, 3)$ and $(-3, 4, -5)$ is divided by the xy-plane is:
  • A
    $2 : 5$
  • ✓
    $3 : 5$
  • C
    $5 : 2$
  • D
    $5 : 3$
Answer
Correct option: B.
$3 : 5$
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MCQ 1371 Mark
If $x^2+ y^2= 1$, then the distance from the point $(x, y, 1 - x^2- y^2)$ to the origin is:
  • ✓
    $1$
  • B
    $-1$
  • C
    $0$
  • D
    $2$
Answer
Correct option: A.
$1$
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MCQ 1381 Mark
The points $(5, –4, 2), (4, –3, 1), (7, 6, 4)$ and $(8, –7, 5)$ are the vertices of
  • A
    A rectangle.
  • B
    A square.
  • C
    A parallelogram.
  • ✓
    None of these
Answer
Correct option: D.
None of these
Suppose:
$A(5, -4, 2)$
$B(4, -3, 1)$
$C(7, 6, 4)$
$D(8, -7, 5)$
$\text{AB}=\sqrt{(4 − 5)^2 + (−3 + 4)^2 + (1 − 2)^2}$
$=\sqrt{(−1)^2 + (1)^2 + (−1)^2}$
$=\sqrt{1 + 1 + 1}$
$=\sqrt{3}$
$\text{BC}=\sqrt{(7 − 4)^2 + (6 + 3)^2 + (4 − 1)^2}$
$=\sqrt{(3)^2 + (9)^2 + (3)^2}$
$=\sqrt{9 + 81 + 9}$
$=\sqrt{99}$
$=3\sqrt{11}$
$\text{CD}=\sqrt{(8 − 7)^2 + (−7 − 6)^2 + (5 − 4)^2}$
$=\sqrt{(1)^2 + (-13)^2 + (1)^2}$
$=\sqrt{1 + 169 + 1}$
$=\sqrt{171}$
$\text{DA}=\sqrt{(8 − 5)^2 + (−7 + 4)^2 + (5 − 2)^2}$
$=\sqrt{(3)^2 + (-3)^2 + (3)^2}$
$=\sqrt{9 + 9 + 9}$
$=\sqrt{27}$
$=3\sqrt{3}$
We see that none of the sides are equal.
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MCQ 1391 Mark
The maximum distance between points $ (3\sin \theta, 0, 0)$ and $(4\cos \theta, 0, 0)$ is:
  • A
    $3$
  • B
    $4$
  • ✓
    $5$
  • D
    Can not be find
Answer
Correct option: C.
$5$
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MCQ 1401 Mark
Choose the correct answer. Equation of $y-$axis is considered as:
  • A
    $x = 0, y = 0.$
  • B
    $y = 0, z = 0.$
  • ✓
    $z = 0, x = 0.$
  • D
    None of these.
Answer
Correct option: C.
$z = 0, x = 0.$
On $y-$axis, $x = 0$ and $z = 0$
Hence, the correct option is $(c).$
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MCQ 1411 Mark
The locus of a point for which $y = 0, z = 0$ is:
  • ✓
    Equation of $x-$axis
  • B
    Equation of $y-$axis
  • C
    Equation of $z-$axis
  • D
    None of these
Answer
Correct option: A.
Equation of $x-$axis
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MCQ 1421 Mark
The image of the point $P(1,3,4)$ in the plane $2x - y + z = 0$ is:
  • ✓
    $(-3, 5, 2)$
  • B
    $(3, 5, 2)$
  • C
    $(3, -5, 2)$
  • D
    $(3, 5, -2)$
Answer
Correct option: A.
$(-3, 5, 2)$
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MCQ 1431 Mark
The triangle formed by the points $(0, 7, 10), (-1, 6, 6), (-4, 9, 6)$ is:
  • A
    Equilateral
  • B
    Isosceles
  • C
    Right angled
  • ✓
    Right angled isosceles
Answer
Correct option: D.
Right angled isosceles
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MCQ 1441 Mark
Find the centroid of a triangle, the mid$-$point of whose sides are $D(1, 2, -3), E(3, 0, 1)$ and $F(-1, 1, -4):$
  • A
    $(1, 1, 2)$
  • ✓
    $(1, 1, -2)$
  • C
    $(-1, -1, -2)$
  • D
    $(1, -1, -2)$
Answer
Correct option: B.
$(1, 1, -2)$
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MCQ 1451 Mark
If a parallelopiped is formed by planes drawn through the points $(5, 8, 10)$ and $(3, 6, 8)$ parallel to the coordinate planes, then the length of diagonal of the parallelopiped is:
  • ✓
    $2\sqrt{3}$
  • B
    $3\sqrt{2}$
  • C
    $\sqrt{2}$
  • D
    $\sqrt{3}$
Answer
Correct option: A.
$2\sqrt{3}$
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MCQ 1461 Mark
The points $A(2a, 4a), B(2a, 6a)$ and $C(2a + 3a, 5a), a > 0$ are the vertices of:
  • A
    An isosceles triangle
  • B
    A right angled triangle
  • ✓
    An acute angled triangle
  • D
    None of the above
Answer
Correct option: C.
An acute angled triangle
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MCQ 1471 Mark
A plane is parallel $xy-$plane, so it is perpendicular to:
  • ✓
    $z-$axis
  • B
    $y-$axis
  • C
    $x-$axis
  • D
    None of these
Answer
Correct option: A.
$z-$axis
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MCQ 1481 Mark
The length of the perpendicular drawn from the point $P(a, b, c)$ from $z-$axis is
  • ✓
    $\sqrt{\text{a}^2+\text{b}^2}$
  • B
    $\sqrt{\text{b}^2+\text{c}^2}$
  • C
    $\sqrt{\text{a}^2+\text{c}^2}$
  • D
    $\sqrt{\text{a}^2+\text{b}^2+\text{c}^2}$
Answer
Correct option: A.
$\sqrt{\text{a}^2+\text{b}^2}$
The length of the perpendicular drawn from the point $P(x, y, z)$ from $z-$axis is given by $\sqrt{\text{y}^2+\text{x}^2}$
Thus, the length of the perpendicular drawn from the point $P(a, b, c)$ from $z-$axis is $\sqrt{\text{a}^2+\text{b}^2}$
Hence, the correct answer is option $(a)$
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MCQ 1491 Mark
The plane $ax + by + cz + (-3) = 0$ meet the co$-$ordinate axes in $\text{A, B, C.}$ Then centroid of the triangle is:
  • A
    $\big(3\text{a},3\text{b},3\text{c}\big)$
  • B
    $\Big( \frac{3}{\text{a}}\frac{3}{\text{b}},\frac{3}{\text{c}}\Big)$
  • C
    $\Big(\frac{\text{a}}{3},\frac{\text{b}}{3},\frac{\text{c}}{3}\Big)$
  • ✓
    $\Big(\frac{1}{\text{a}}, \frac{1}{\text{b}},\frac{1}{\text{c}}\Big)$
Answer
Correct option: D.
$\Big(\frac{1}{\text{a}}, \frac{1}{\text{b}},\frac{1}{\text{c}}\Big)$
For finding the coordinates of the point where the plane $ax + by + cz - 3 = 0$ cuts the $x-$axis, we equate $y$ and $z$ to zero.
The point becomes $\Big(\frac{3}{\text{a}},0,0\Big)$ Similarly, the point on $y-$axis becomes $\Big(0,\frac{3}{\text{b}},0\Big)$ And that on $z$ axis becomes $\Big(0,0,\frac{3}{\text{c}}\Big)$ The centroid of the triangle.
formed by these points would be $\Bigg(\frac{\frac{3}{\text{a}}+0+0}{3},\frac{0+\frac{3}{\text{b}}+0}{3},\frac{0+0+\frac{3}{\text{c}}}{3}\Bigg)=\bigg(\frac{1}{\text{a}},\frac{1}{\text{b}},\frac{1}{\text{c}}\bigg)$
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MCQ 1501 Mark
$A(3, 2, 0), B(5, 3, 2), C(-9, 6, -3)$ are three points forming a triangle. If $AD,$ the bisector of $\angle\text{BAC}$ meets $BC$ in $D$ then coordinates of $D$ are:
  • ✓
    $\Big(-\frac {19}{8}, \frac {57}{16}, \frac {17}{16}\Big)$
  • B
    $\Big( \frac {19}{8}, -\frac {57}{16}, \frac {17}{16}\Big)$
  • C
    $\Big( \frac {19}{8}, \frac {57}{16}, \frac {17}{16}\Big)$
  • D
    None of these
Answer
Correct option: A.
$\Big(-\frac {19}{8}, \frac {57}{16}, \frac {17}{16}\Big)$
According to question,
$\text{AB}=\sqrt{4+1+4}=3$
$\text{AC}=\sqrt{144+16+9}=13$
$\text{BD}:\text{DC}=\text{AB}:\text{AC}=3:13$
$\text{D}=\Big(\frac{3(-9)+13(15)}{3+13},\frac{3(6)+13(3)}{3+13},\frac{3(-3)+13(2)}{3+13}\Big)$
$=\Big(\frac{-38}{16},\frac{57}{16},\frac{17}{16}\Big)$
$\therefore\Big(-\frac {19}{8}, \frac {57}{16}, \frac {17}{16}\Big)$
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MCQ 1511 Mark
Solve the following differential equation. $\frac{\text{dy}}{\text{dx}}=\text{x}-1$
  • A
    $ y=x^2+x $
  • B
    $ y=x^2 $
  • ✓
    $ y=x^2-x $
  • D
    None of the above
Answer
Correct option: C.
$ y=x^2-x $
Given, $\frac{\text{dy}}{\text{dx}}=\text{x}-1$
Integrating on both sides
$\int\frac{\text{dy}}{\text{dx}}=\int\text{x}-1\text{ dx}$
$y = x^2- x + c$
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MCQ 1521 Mark
If $P(3, 2, -4), Q(5, 4, -6)$ and $R(9, 8, -10)$ are collinear, then $R$ divides $PQ$ in the ratio:
  • A
    $3 : 2$ internally
  • ✓
    $3 : 2$ externally
  • C
    $2 : 1$ internally
  • D
    $2 : 1$ externally
Answer
Correct option: B.
$3 : 2$ externally
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MCQ 1531 Mark
Choose the correct answer. The locus of a point for which $y = 0, z = 0$ is:
  • ✓
    Equation of $x-$axis.
  • B
    Equation of $y-$axis.
  • C
    Equation at $z-$axis.
  • D
    None of these.
Answer
Correct option: A.
Equation of $x-$axis.
We know that one equation of $x-$axis, $y = 0, z = 0$
Hence, the locus of the point is equation of $x-$axis.
So, the correct option is $(a).$
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MCQ 1541 Mark
In a three dimensional space the equation $x^2- 5x + 6 = 0$ represents
  • A
    Points.
  • B
    Planes.
  • ✓
    Curves.
  • D
    Pair of straight lines.
Answer
Correct option: C.
Curves.
Since, there is only one variable in the given equation.
Also, it is quadratic equation.
Hence, It represents curves in $yz$ plane.
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MCQ 1551 Mark
The points $(-5, 12), (-2, -3), (9, -10), (6, 5)$ taken in order, form:
  • ✓
    Parallelogram
  • B
    Rectangle
  • C
    Rhombus
  • D
    Square
Answer
Correct option: A.
Parallelogram
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MCQ 1561 Mark
The ratio in which the line joining $(2, 4, 5)$ and $(3, 5, -9)$ is divided by the $yz-$plane is
  • A
    $2 : 3$
  • B
    $3 : 2$
  • ✓
    $-2 : 3$
  • D
    $4 : -3$
Answer
Correct option: C.
$-2 : 3$
Let $A ≡ (2, 4, 5)$ and $B ≡ (3, 5, 9)$
Let the line joining $A$ and $B$ be divided by the $yz-$plane at point $P$ in the ratio $\lambda:1.$
Then, we have,
$\text{P}\equiv\Big(\frac{3\lambda+2}{\lambda+1},\ \frac{5\lambda+4}{\lambda+1},\ \frac{-9\lambda+5}{\lambda+1}\Big)$
Since $P$ lies on the $yz-$plane, the $x-$coordinate of $P$ will be zero
$\therefore\frac{3\lambda+2}{\lambda+1}=0$
$\Rightarrow3\lambda+2=0$
$\Rightarrow\lambda=\frac{-2}{3}$
Hence, the $yz-$plane divides $AB$ in the ratio $-2 : 3$
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MCQ 1571 Mark
$A = (1, 1, 4)$ and $B = (5, -3, 4)$ are two points. If the points $P, Q$ are on the line $AB$ such that $\text{AP = PQ = QB}$ then $\text{PQ} =$
  • A
    $2\sqrt{2}$
  • B
    $4$
  • ✓
    $\sqrt{\frac{32}{9}}$
  • D
    $\sqrt{2}$
Answer
Correct option: C.
$\sqrt{\frac{32}{9}}$
$\text{AB}=\sqrt{(1-5)^2+(1+3)^2+(4+4)^2}$
$\text{AB}=\sqrt{(-4)^2+4^2}$
$\text{AB}=\sqrt{32}$
$\text{AB}=3\times\text{PQ}$
$=\frac{\sqrt{132}}{3}$
$=\sqrt{\frac{32}{9}}$
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MCQ 1581 Mark
The point $(-2, -3, -4)$ lies in the:
  • A
    First octant
  • ✓
    Seventh octant
  • C
    Second octant
  • D
    Eight octant
Answer
Correct option: B.
Seventh octant
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MCQ 1591 Mark
The ratio in which the line joining the points $(1, 2, 3)$ and $(-3, 4, -5)$ is divided by the $xy-$plane is:
  • A
    $2 : 5$
  • ✓
    $3 : 5$
  • C
    $5 : 2$
  • D
    $5 : 3$
Answer
Correct option: B.
$3 : 5$
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