List of top Mathematics Questions on Three Dimensional Geometry asked in KEAM

A straight line passing through (6,1,3) meets the line \( \frac{x-1}{2} = \frac{y}{1} = \frac{z-2}{3} \) at Q. If the lines are perpendicular to each other, then the coordinates of Q are:

The plane passing through the points(2,1,0), (5,0,1)and (4,1,1)intersects the x axis at

The angle between the lines, whose direction cosines are proportional to 4,√3-1,-√3-1 and 4,-√3-1,√3-1, is

Given the points \(A(6,-7,0),B(16,-19,-4),C(0,3,-6)\) and \(D(2,-5,10)\),the point of intersection of the lines \(AB\) and \(CD\) is 

Let L₁ be the line joining (0, 0, 0) and (1, 2, 3) and L₂ be the line joining (2, 3, 4) and (3, 4, 5). The point of intersection of L₁ and L₂ is

The distance from the point (2, 2, 2) to the plane 2x-y+3z = 5 is equal to

If the points (1, 0, 0), (0, 3, 0) and (0, 0, 2) lie on a plane, then the unit normal vector \(\hat n\) to the plane is

The equation of the line through the point (1, 1, 1) and parallel to the line joining the points (-2, 2, 0) and (-1, 1, 1) is

The equation of the plane passing through the point (-1,-2,-3) and perpendicular to the x-axis is

The Cartesian equation of the line passing through (7, 5, 3) and perpendicular to the plane 3x + 2y + z = 6 is

If the point P(x, 1, 4) lies on the line \(\overrightarrow{r}=\hat{i}+3\hat{j}+4\hat{k}+\lambda(2\hat{i}-\hat{j})\), then the value of x is equal to

If a is the angle made by the vector \(\overrightarrow{a}=5\hat{i}+3\hat{j}+4\hat{k}\) with the positive x-axis, then cosa =

If the equation of the sphere through the circle \( x^2 + y^2 + z^2 = 9; \; 2x + 3y + 4z = 5 \) and through the point \( (1,2,3) \) is \( 3(x^2 + y^2 + z^2) - 2x - 3y - 4z = C \), then the value of \( C \) is

If a straight line makes the angles \(60^\circ,45^\circ,\alpha\) with x, y and z axes respectively, then \( \sin^2\alpha \) is

If a straight line makes angles \( \alpha, \beta, \gamma \) with the coordinate axes, then evaluate \[ 1 - \frac{\tan^2\alpha}{1+\tan^2\alpha} + \frac{1}{\sec^2\beta} - 2\sin^2\gamma \]

The equation of the plane which bisects the line segment joining the points $(3, 2, 6)$ and $(5,4, 8)$ and is perpendicular to the same line segment, is

The equation of the plane containing the lines $ \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{3} $ and $ \frac{x}{2}=\frac{y-2}{-1}=\frac{z+1}{3} $ is

A plane makes intercepts $a, b, c$ at $A, B, C$ on the coordinate axes respectively. If the centroid of the $ \Delta ABC $ is at $(3, 2, 1)$, then the equation of the plane is

If the plane $ 3x+y+2z+6=0 $ is parallel to the line $ \frac{3x-1}{2b}=3-y=\frac{z-1}{a}, $ then the value of $ 3a+3b $ is

The angle between the straight lines $x-1=\frac{2y+3}{3}=\frac{z+5}{2}$ and $x-3r+2; y=-2r-1; z=2,$ where $r$ is a parameter, is

The distance between the line $ \overrightarrow{r}=(2\hat{i}+2\hat{j}-\hat{k})+\lambda (2\hat{i}+\hat{j}-2\hat{k}) $ and the plane $ \overrightarrow{r}.(\hat{i}+2\hat{j}+2\hat{k})=10 $ is equal to

Equation of the plane passing through the intersection of the planes $ x+y+z=6 $ and $ 2x+3y+4z+5=0 $ and the point $(1, 1, 1)$ is

The plane $ \overrightarrow{r}=s(\hat{i}+2\hat{j}-4\hat{k})+t(3\hat{i}+4\hat{j}-4\hat{k}) $ $ +(1-t)(2\hat{i}-7\hat{j}-3\hat{k}) $ is parallel to the line

If a straight line makes angles $\alpha , \beta , \gamma $ with the coordinate axes, then $\frac{1-\tan^{2} \alpha }{1+tan^{2} \alpha} +\frac{1}{sec 2 \beta} -2\sin^{2} \gamma =$