List of top Mathematics Questions on 3D Geometry asked in AP EAPCET

If the direction ratios of two lines are given by $ l + m + n = 0 $ and $ mn - 2ln + lm = 0 $, then the angle between the lines is:

The direction cosines of the line making angles $$ \frac{\pi}{4}, \frac{\pi}{3} $$ and $ \theta $ (where $ 0<\theta<\frac{\pi}{2} $) with X, Y, and Z axes respectively are:

Let vectors: $ \vec{A} = \hat{i} - 2\hat{j} + \hat{k},\ \vec{B} = \hat{i} + \hat{j} - 2\hat{k},\ \vec{C} = 2\hat{i} - \hat{j},\ \vec{D} = \hat{i} + \hat{j} + \hat{k} $ If $ P $ divides $ AB $ in ratio 2:1 internally, and $ Q $ divides $ CD $ in ratio 1:2 externally, find the ratio in which the point $ 5\hat{i} - 6\hat{j} - 5\hat{k} $ divides line $ PQ $

If A(1, 2, 3), B(2, 3, -1), C(3, -1, -2) are the vertices of a triangle ABC, then the direction ratios of the bisector of $\angle$ABC are

Let A = (2, 0, -1), B = (1, -2, 0), C = (1, 2, -1), and D = (0, -1, -2) be four points. If $\theta$ is the acute angle between the plane determined by A, B, C and the plane determined by A, C, D, then $\tan\theta =$

The points $ A(-1, 2, 3), B(2, -3, 1), C(3, 1, -2) $ are:

If the equation of the plane passing through point $ (3, 2, 5) $ and perpendicular to the planes $$ 2x - 3y + 5z = 7, \quad 5x + 2y - 3z = 11 $$ is $$ x + by + cz + d = 0, $$ then find $ 2b + 3c + d $.

If the line joining points $ \vec{r}_1 = \hat{i} + 2\hat{j} $ and $ \vec{r}_2 = \hat{j} - 2\hat{k} $ intersects the plane through the points $ \vec{A} = 2\hat{i} - \hat{j},\ \vec{B} = -2\hat{j} + 3\hat{k},\ \vec{C} = \hat{k} - 2\hat{i} $ at $ T $, then find $ \vec{r}_T \cdot (\hat{i} + \hat{j} + \hat{k}) $

The vector equation of a plane passing through the line of intersection of the planes $$ \vec{r} \cdot (\hat{i} - 2\hat{k}) = 3,\quad \vec{r} \cdot (\hat{j} + \hat{k}) = 5 $$ and also passing through the point $ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} $ is:

If $ \vec{a} = \hat{i} + \hat{j}, \vec{b} = 2\hat{j} - \hat{k} $ are two vectors such that $ \vec{r} \times \vec{a} = \vec{b} \times \vec{a}, \vec{r} \times \vec{b} = \vec{a} \times \vec{b} $, then the unit vector in the direction of $ \vec{r} $ is:

If $ y = x - x^2 $, then the rate of change of $ y^2 $ with respect to $ x^2 $ at $ x = 2 $ is:

If $ A(1,0,2) $, $ B(2,1,0) $, $ C(2,-5,3) $, and $ D(0,3,2) $ are four points and the point of intersection of the lines $ AB $ and $ CD $ is $ P(a,b,c) $, then $ a + b + c = ? $

If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then $ (\alpha, \beta) $ is:

The equation of the plane which bisects the angle between the planes $$ 3x - 6y + 2z + 5 = 0 \quad \text{and} \quad 4x - 12y + 3z - 3 = 0 \quad \text{which contains the origin is:} $$

The distance of the point $ O(0,0,0) $ from the plane $ \overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 5 $ measured parallel to $ 2\hat{i} + 3\hat{j} - 6\hat{k} $ is?

The distance from a point $ (1,1,1) $ to a variable plane $\pi$ is 12 units and the points of intersections of the plane with X, Y, Z-axes are $ A, B, C $ respectively. If the point of intersection of the planes through the points $ A, B, C $ and parallel to the coordinate planes is $ P $, then the equation of the locus of $ P $ is:

If $(l_1, m_1, n_1), (l_2, m_2, n_2)$ are the direction cosines of two lines, then \[ (l_2m_1 - l_1m_2)^2 + (m_1n_2 - m_2n_1)^2 + (n_1l_2 - n_2l_1)^2 + (l_1l_2 + m_1m_2 + n_1n_2)^2 = \]

If a straight line is equally inclined at an angle $ \theta $ with all the three coordinate axes, then $ \tan \theta =$

If the points $ (k, 1, 5), (1, 0, 3), (7, -2, m) $ are collinear, then $ (k, m) = $

If $A = (2, 3, 4)$ and $B = (-2, 3, 4)$, then the locus of a point $P$ such that $PA + PB = 4$ is:

The transformed equation of $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$ when the axes are translated to the point $(2,-3,1)$ is

If a point R$(4,y,z)$ lies on the line joining $P(2,-3,4)$ and $Q(8,0,10)$, then distan of R from origin is

If the foot of the perpendicular drawn from $(0,0,0)$ to a plane is $(1,2,3)$, then the equation of the plane is

A point on the plane passing through the points $(\sqrt{2},1,4), (0,-1,0)$ and $(0,0,1)$ is:

In $ \triangle ABC $, if the midpoints of the sides $ AB, BC, CA $ are respectively $ (l, 0, 0), (0, m, 0), (0, 0, n) $, then: \[ \frac{AB^2 + BC^2 + CA^2}{l^2 + m^2 + n^2} = ? \]