List of top Mathematics Questions on Three Dimensional Geometry asked in TS EAMCET

If \(P(1,2,5)\), \(Q(3,0,7)\), \(R(6,-3,10)\) are collinear and \((\alpha,\beta,\gamma)\) is a point at distance 3 from \(P\) on the same line, then the value of \(\alpha+\beta+\gamma\) lying between 6 and 7 is:

If the direction cosines of the line common to the planes \[ x+2y-z-1=0 \] and \[ 3x-4y+z-5=0 \] are \((l,m,n)\), then \(|l+m-n|=\)

If the area of the triangle formed by the points \((0,0,0)\), \((1,1,1)\) and \((t,2t,3t)\) is \(\sqrt6\), then the sum of squares of all possible values of \(t\) is:

\(\overline{a}, \overline{b}, \overline{c}\) are non-coplanar vectors. If \(\overline{x}=2\overline{a}+3\overline{b}+4\overline{c}\), \(\overline{y}=3\overline{a}+4\overline{b}+5\overline{c}\), \(\overline{z}=4\overline{a}+5\overline{b}+6\overline{c}\), then \([\overline{x}\ \overline{y}\ \overline{z}]=\):

Let \(A(4,3,-2)\), \(B(0,-4,2)\), and \(C(-4,7,6)\) be the vertices of a triangle \(ABC\). If \(D(p,q,r)\) is the point of intersection of the bisector of angle \(A\) and the side \(BC\), then \(2p+q+r=\)

Let \(OA\), \(OB\), \(OC\) lying along the \(X\)-, \(Y\)- and \(Z\)-axes respectively represent the coterminous edges of a rectangular parallelepiped. If \(OA=1\), \(OB=2\), \(OC=3\), then the angle between a pair of diagonals of the parallelepiped drawn through the vertices \(O\) and \(A\) is

A line passing through the points \((9,7,5)\) and \((2,10,0)\) is perpendicular to a plane \(\pi\) passing through the point \((200,30,116)\). If the plane \(\pi\) cuts the \(X\)-, \(Y\)-, \(Z\)-axes at the points \(A\), \(B\), \(C\) respectively, then the centroid of \(\triangle ABC\) is

If \[ \vec a=2\hat i+3\mu\hat j-\hat k, \qquad \vec b=\mu\hat i-2\hat j+3\hat k, \qquad \vec c=\hat i+3\hat j-2\mu\hat k \] are three vectors such that \[ \alpha\vec a+\beta\vec b+\gamma\vec c=\vec 0 \] only when \[ \alpha=\beta=\gamma=0, \] then the set of all real values of \(\mu\) is

Let $\vec{a}=\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$, $\vec{c}=6\hat{i}+3\hat{j}-2\hat{k}$ be three vectors. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$, $\vec{b}$ and $|\vec{d}\times\vec{c}|=14$, then $|\vec{d}\cdot\vec{c}|=$

If the plane $-4x-2y+2z+\alpha=0$ is at a distance of two units from the plane $2x+y-z+1=0$, then the product of all the possible values of $\alpha$ is

The direction ratios of the line bisecting the angle between the x-axis and the line having direction ratios (3, -1, 5) are

Let A($\alpha$,4,7) and B(3,$\beta$,8) be two points in space. If YZ plane and ZX plane respectively divide the line segment joining the points A and B in the ratio 2:3 and 4:5, then the point C which divides AB in the ratio $\alpha:\beta$ externally is

Consider the following
Assertion (A): The two lines $\vec{r} = \vec{a}+t(\vec{b})$ and $\vec{r}=\vec{b}+s(\vec{a})$ intersect each other.
Reason (R): The shortest distance between the lines $\vec{r}=\vec{p}+t(\vec{q})$ and $\vec{r}=\vec{c}+s(\vec{d})$ is equal to the length of projection of the vector $(\vec{p}-\vec{c})$ on $(\vec{q}\times\vec{d})$.
The correct answer is

If the foot of the perpendicular drawn from the point (2,0,-3) to the plane $\pi$ is (1,-2,0) and the equation of the plane is $ax+by-3z+d=0$ then $a+b+d=$

A line makes angles 60$^\circ$, 45$^\circ$, $\theta$ with positive X, Y, Z-axes respectively. If $\theta$ is an acute angle, then $\tan\theta =$

The number of values of 'k' for which the points (-4,9,k), (-1,6,k), (0,7,10) form a right-angled isosceles triangle is

If A(2,1,-1), B(6,-3,2), C(-3,12,4) are the vertices of a triangle ABC and the equation of the plane containing the triangle ABC is $53x+by+cz+d=0$, then $\frac{d}{b+c}=$

If $(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from a point $(-1,2,-1)$ to the line joining the points $(2,-1,1)$ and $(1,1,-2)$, then $\alpha+\beta+\gamma=$

If m:n is the ratio in which the point $\left(\frac{8}{5}, \frac{1}{5}, \frac{8}{5}\right)$ divides the line segment joining the points (2,p,2) and (p,-2,p) where p is an integer then $\frac{3m+n}{3n}=$

Let $\pi_1$ be the plane determined by the vectors $\hat{i}+\hat{j}, \hat{i}+\hat{k}$ and $\pi_2$ be the plane determined by the vectors $\hat{j}-\hat{k}, \hat{k}-\hat{i}$. Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the planes $\pi_1$ and $\pi_2$. If $\vec{b} = \hat{i}+\hat{j}-\hat{k}$ then the angle between the vectors $\vec{a}$ and $\vec{b}$ is