List of top Mathematics Questions on Three Dimensional Geometry asked in AP EAPCET

Let \[ O(0,0,0) \] and \[ A(2,1,-3) \] be vertices of a triangle \(OAB\). If \[ (-1,2,1) \] is the midpoint of side \(AB\), and the perimeter of the triangle is \[ \sqrt2\,(k+l\sqrt7+m\sqrt{13}), \] then the value of \[ k+l+m \] is:

If the feet of the perpendiculars drawn from the point \[ (3,4,5) \] to the \(X\)-, \(Y\)- and \(Z\)-coordinate axes are \(A,B,C\) respectively and the angle between \(AB\) and \(AC\) is \[ \cos^{-1}\left(\frac{9}{a}\right), \] then the value of \(a\) is:

\( A(1,2,3) \), \( B(3,4,k) \), \( C(2,1,4) \) form an isosceles triangle. If \( AB=BC \), then the area of \( \triangle ABC \) is

If \( A(1,0,1) \), \( B(0,1,-1) \), \( C(-1,1,0) \) are the vertices of a triangle ABC, then \( \cos^{2}A+\cos^{2}B = \)

If the angle between the planes \( \lambda x-2y+3z+1=0 \) and \( 2x+3y-\lambda z+\lambda=0 \) is \( \cos^{-1}(\frac{12}{49}) \) and \( \lambda\in\mathbb{Z} \), then the sum of the perpendicular distances from the origin to these planes is

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 \( 4\overline{i}+7\overline{j}-4\overline{k} \) then the length of PM is

If a line makes angles $45^\circ$ and $60^\circ$ with the positive $x$ and $y$ axes respectively, then the acute angle it makes with the $z$-axis is:

A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\)-axis and \(Z\)-axis respectively. Then the value of the sine of the angle made by the ray with \(X\)-axis is

A plane meets the \(X, Y, Z\)-axes in \(A, B, C\) respectively. If the centroid of the triangle \(ABC\) is \((2,-3,5)\), then the perpendicular distance from origin to the given plane is

If \((l_1,m_1,n_1)\) and \((l_2,m_2,n_2)\) are the direction cosines of two lines satisfying the relations \[ l^2+mn-6n^2=0 \] and \[ 2l-m+3n=0, \] then \[ |l_1l_2|+|m_1m_2|= \]

If the centroid of triangle whose vertices are \[ (a,1,3),\ (-2,b,-5)\ \text{and}\ (4,7,c) \] be the origin, then \[ a^2+b^2+c^2= \] is:

If \(\alpha,2\alpha,3\alpha\) are angles made by a ray with \(OX\), \(OY\), \(OZ\) axes respectively, then all the possible values of \(\alpha\) are:

If \(P\) divides the line segment joining the points \(A(1,2,-1)\) and \(B(-1,0,1)\) externally in the ratio \(1:2\) and \(Q=(1,3,-1)\), then \(PQ=\)

If the direction cosines of a line are \[ \left( \frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}} \right) \] and \(c-a=4\), then \(ca=\)