Three Dimensional Geometry · Mathematics · TS EAMCET
MCQ (Single Correct Answer)
1
A plane $\pi$ passing through the points $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}, 3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ is parallel to the vector $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$. If a line joining the points $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ intersects the plane $\pi$ at the point $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, then $a+b+2 c=$
TG EAPCET 2024 (Online) 11th May Morning Shift
2
$\hat{\mathbf{r}} .(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ and $\hat{\mathbf{r}} .(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})=3$ are two planes. A plane $\pi$ passing through the line of intersection of these two planes, passes through the point $(0,1,2)$. If the equation of $\pi$ is $\hat{\mathbf{r}} .(a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}})=m$, then $\frac{b c}{a^{2}}=$
TG EAPCET 2024 (Online) 11th May Morning Shift
3
If $A(-2,4, a), B(1, b, 3), C(c, 0,4)$ and $D(-5,6,1)$ are collinear points, then $a+b+c=$
TG EAPCET 2024 (Online) 11th May Morning Shift
4
$A(1,-2,1)$ and $B(2,-1,2)$ are the end points of a line segment. If $D(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from $C(1,2,3)$ to $A B$, then $\alpha^{2}+\beta^{2}+\gamma^{2}=$
TG EAPCET 2024 (Online) 11th May Morning Shift
5
The foot of the perpendicular drawn from the point $(-2,-1,3)$ to a plane $\pi$ is $(1,0,-2)$. If $a, b, c$ are the intercepts made by the plane $\pi$ on $X, Y, Z$-axis respectively, then $3 a+b+5 c=$
TG EAPCET 2024 (Online) 11th May Morning Shift
6
$\mathbf{n}$ is a unit vector normal to the plane $\pi$ containing the vectors $\hat{\mathbf{i}}+3 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$. If this plane $\pi$ passes through the point $(-3,7,1)$ and $p$ is the perpendicular distance from the origin to this plane $\pi$, then $\sqrt{p^{2}+5}=$
TG EAPCET 2024 (Online) 10th May Evening Shift
7
If the harmonic conjugate of $P(2,3,4)$ with respect to the line segment joining the points $A(3,-2,2)$ and $B(6,-17,-4)$ is $Q(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$
TG EAPCET 2024 (Online) 10th May Evening Shift
8
If $L$ is the line of intersection of two planes $x+2 y+2 z=15$ and $x-y+z=4$ and the direction ratio of the line $L$ are $(a, b, c)$, then $\frac{\left(a^{2}+b^{2}+c^{2}\right)}{b^{2}}=$
TG EAPCET 2024 (Online) 10th May Evening Shift
9
The foot of the perpendicular drawn from $A(1,2,2)$ oril the the plane $x+2 y+2 z-5=0$ is $B(\alpha, \beta, \gamma)$. If $\pi(x, y, z)$ $=x+2 y+2 z+5=0$ is a plane, then $-\pi(A): \pi(B)=$
TG EAPCET 2024 (Online) 10th May Evening Shift
10
A plane $\pi_1$ passing through the point $3 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is perpendicular to the vector $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and another plane $\pi_2$ passing through the point $2 \hat{\mathbf{i}}+7 \hat{\mathbf{k}}-8 \hat{\mathbf{k}}$ is perpendicular to the vector $3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$. If $p_1$ and $p_2$ are the perpendicular distances from the origin to the planes $\pi_1$ and $\pi_2$ respectively, then $p_1-p_2=$
TG EAPCET 2024 (Online) 10th May Morning Shift
11
$A(2,3, k), B(-1, k,-1)$ and $C(4,-3,2)$ are the vertices of $\triangle A B C$. If $A B=A C$ and $k>0$, then $\triangle A B C$ is
TG EAPCET 2024 (Online) 10th May Morning Shift
12
If $a, b$ and $c$ are the intercepts made on $X, Y, Z$-axes respectively by the plane passing through the points $(1,0,-2),(3,-1,2)$ and $(0,-3,4)$, then $3 a+4 b+7 c=$
TG EAPCET 2024 (Online) 10th May Morning Shift
13
If $\hat{\mathbf{i}}+\hat{\mathbf{j}}, \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{k}}+\hat{\mathbf{i}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are the position vectors of the points $A, B, C, D, E$ respectively, then the point of intersection of the line $A B$ and the plane passing through $C, D, E$ is.
TG EAPCET 2024 (Online) 9th May Evening Shift
14
A plane $(\pi)$ passing through the point $(1,2,-3)$ is perpendicular to the planes $x+y-z+4=0$ and $2 x-y+z+1=0$. If the equation of the plane $(\pi)$ is $a x+b y+c z+1=0$, then $a^2+b^2+c^2=$
TG EAPCET 2024 (Online) 9th May Evening Shift
15
If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $Y Z$ - plane is $2: 3$, then the equation of the locus of $P$ is
TG EAPCET 2024 (Online) 9th May Morning Shift
16
The direction cosines of two lines are connected by the relations $l-m+n=0$ and $2 l m-3 m n+n l=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$
TG EAPCET 2024 (Online) 9th May Morning Shift
17
A plane $\pi$ passes through the points $(5,1,2),(3,-4,6)$ and $(7,0,-1)$. If $p$ is the perpendicular distance from the origin to the plane $\pi$ and $l, m$ and $n$ are the direction cosines of a normal to the plane $\pi$, the $|3 l+2 m+5 n|=$
TG EAPCET 2024 (Online) 9th May Morning Shift
18
If $M$ is the foot of the perpendicular drawn from $P($ -1,2,-1 ) to the plane passing through the point $A(3,-2,1)$ and perpendicular to the vector $4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$, then the length of $P M$ is
TS EAMCET 2023 (Online) 12th May Evening Shift
19
If $A=(1,-1,2), B=(3,4,-2), C=(0,3,2)$ and $D=(3$, $5,6)$, then the angle between the lines $\mathbf{A B}$ and $\mathbf{C D}$ is
TS EAMCET 2023 (Online) 12th May Evening Shift
20
Consider the following statements:
Assertion (A) : The direction ratios of a line $L_1$ are 2,5, 7 and the direction ratios of another line $L_2$ are $\frac{4}{\sqrt{19}}$, $\frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}$. Then, the lines $L_1, L_2$ are parallel.
Reason : ( $\mathbf{R}$ ) If the direction ratios of a line $L_1$ are $a_1, b_1, c_1$ the direction ratios of a line $L_2$ are $a_2, b_2, c_2$ and $a_1 a_2+b_1 b_2+c_1 c_2=0$, then the lines of $L_1, L_2$ are parallel.
TS EAMCET 2023 (Online) 12th May Evening Shift
21
A line $L$ is parallel to both the planes $2 x+3 y+z=1$ and $x+3 y+2 z=2$. If line $L$ makes an angle $\alpha$ with the positive direction of $X$-axis, then $\cos \alpha=$
TS EAMCET 2023 (Online) 12th May Evening Shift
22
If the direction cosines $(l, m, n)$ of two lines are connected by the relations $l+m+n=0$ and $l m=0$, then the angle between those lines is
TS EAMCET 2023 (Online) 12th May Morning Shift
23
The sum of the squares of the perpendicular distances of a point $(x, y, z)$ from the coordinate axes is $k$ times the square of the distance of the point from the origin Then, $k=$
TS EAMCET 2023 (Online) 12th May Morning Shift
24
Equation of the plane through the mid-point of the line segment joining the points $A(4,5,-10)$ and $B(-1,2,1)$ and perpendicular to $A B$ is
TS EAMCET 2023 (Online) 12th May Morning Shift