Three-Dimensional Geometry - Class 12

Let $A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 2 & 1 \end{pmatrix}$. Let $M = \text{adj}(AB)$. Consider the system of linear equations $M X = \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$, where $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$. Let $P_1$ and $P_2$ be the planes represented by the first and second equations of this system, respectively. If $L$ is the line of intersection of the planes $P_1$ and $P_2$, and $L$ intersects the plane $2x - 3y + z = 3$ at the point $Q(\alpha, \beta, \gamma)$, then the value of $9(\alpha + \beta + \gamma)$ is:

A line $L$ passes through the origin $O$ and has direction cosines $(l, m, n)$ proportional to $(t^2-4, 4t, t^2+4)$ for a real parameter $t \notin \{0, 2\}$. It is given that the angle $\gamma$ made by $L$ with the positive z-axis is acute. The line $L$ intersects the plane $\Pi: x - y + z = 2\sqrt{2}$ at the point $Q$. If the sum of the $x$ and $y$ coordinates of $Q$ is at most $-\sqrt{2}$, then the complete set of values for $t$ is:

Let $L_1$ be the line given by the equations $x = y$ and $z = 0$. Let $L_2$ be a variable line parallel to the $z$-axis such that the shortest distance between $L_1$ and $L_2$ is $2$. Let $M$ be the point of intersection of $L_2$ with the plane $P: x + y + z = 1$. The coordinate axes are translated without rotation such that the new origin is at $O'(1, -1, 1)$. In the new coordinate system, the locus of $M$ consists of two parallel lines, $C_1$ and $C_2$. If $d_1$ and $d_2$ are the perpendicular distances from the new origin to $C_1$ and $C_2$ respectively, then the value of $d_1 d_2$ is:

Let $L'$ be the orthogonal projection of the line $L: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-2}{2}$ on the plane $\Pi: x + y - z = 0$. If the line $L'$ intersects the plane $4x - 2y + 2z = 33$ at the point $P(\alpha, \beta, \gamma)$, then the value of $2\alpha + 2\beta + \gamma$ is:

Let $P(14, 14, \sqrt{2})$ be a point in $\mathbb{R}^3$ and $\Pi$ be the plane $x + y + \sqrt{2}z = 4$. Let $P'$ be the image of $P$ in the plane $\Pi$. If $Q(6, 0, 0)$ is another point and $V$ is the volume of the tetrahedron $OPP'Q$ (where $O$ is the origin), then the value of $[V]$ is (where $[ \cdot ]$ denotes the greatest integer function):

Let a moving point $P$ trace a surface $S_1$ such that the difference of the squares of its distances from $A(1, 2, 3)$ and $B(3, 0, 1)$ is $-4$. Let another moving point $Q$ trace a surface $S_2$ such that the difference of the squares of its distances from $C(2, -1, 4)$ and $D(0, 1, 2)$ is $16$. A plane $\Pi$ passes through the curve of intersection of $S_1$ and $S_2$, and is perpendicular to the plane $2x + y + 3z = 5$. If $(\alpha, \beta, \gamma)$ is the image of the point $M(4, -1, 0)$ in the plane $\Pi$, find the value of $\alpha^2 + \beta^2 + \gamma^2$.

Let $P$ be the point of intersection of the line $L: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{2}$ and the plane $\Pi_1: x + y + z = 5$. Let $Q$ be the image of $P$ in the plane $\Pi_2: x - 2y + 2z = 6$. If $d$ is the length of the projection of the vector $\vec{PQ}$ on the line $L$, and $\theta$ is the acute angle between the line $L$ and the plane $\Pi_2$, find the value of $9d^2 + 81 \cos^2\theta$.

Let $y = f(x)$ be the general solution to the homogeneous differential equation $\frac{dy}{dx} = \frac{y^3 + x^2 y}{x^3 + x y^2}$. Consider the family of planes $\Pi_c$ in $\mathbb{R}^3$ given by the equation $y = f(x)$, where $c \in \mathbb{R}$ is the constant of integration. Let $I_c$ be the image of the point $A(3, 4, 0)$ in the plane $\Pi_c$. As $c$ varies over all real numbers, $I_c$ traces a curve $\Gamma$. Find the minimum distance of the curve $\Gamma$ from the point $P(8, 15, 12)$.

Let $P_1: 2x + y + 3z = 6$ and $P_2: x + 3y + 2z = 4$ be two planes. Let $\Pi$ be a plane passing through the line of intersection of $P_1$ and $P_2$, and bisecting one of the angles between them. The normal vector $\vec{n} = a\hat{i} + b\hat{j} + c\hat{k}$ to the plane $\Pi$ is chosen such that $a, b, c$ are coprime integers with $a > 0$. It is given that the projection of $\vec{n}$ onto the $xy$-plane is a vector whose tip lies on the director circle of the ellipse $16x^2 + 9y^2 = 144$ (assuming its tail is at the origin). If $\Pi$ intersects the coordinate axes at points $A, B,$ and $C$, then the area of $\triangle ABC$ is:

Let $P_1$ and $P_2$ be two planes given by the equations $\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 4$ and $\vec{r} \cdot (\hat{i} + 2\hat{j} + 2\hat{k}) = 3$, respectively. A plane $\Pi$ is chosen from the family of planes passing through the line of intersection of $P_1$ and $P_2$. A fair coin is tossed. If it shows Heads, $\Pi$ is chosen to be $\Pi_1$, which is parallel to the vector $\vec{v}_1 = 2\hat{i} + \hat{j} - \hat{k}$. If it shows Tails, $\Pi$ is chosen to be $\Pi_2$, which is parallel to the vector $\vec{v}_2 = \hat{i} - 3\hat{j}$.

A point $Q$ is randomly generated by a computer. The probability that $Q$ lies on $\Pi_1$ is $0.175$, and the probability that $Q$ lies on $\Pi_2$ is $0.186$. Given that the generated point $Q$ lies on the chosen plane $\Pi$, let $p$ be the probability that the chosen plane was $\Pi_1$.

Let $\vec{n}_1$ be the normal vector to $\Pi_1$, scaled such that its $\hat{i}$ component is $1$. A vector $\vec{a}$ has magnitude $\sqrt{26}$ and satisfies $\vec{a} \cdot \vec{n}_1 = -10$. Find the value of: $361 p + |\vec{n}_1 \times \vec{a}|^2$

Let $A(1, \lambda, (\lambda-2)^2)$, $B(\lambda, 1, (\lambda-2)^2)$, $C((\lambda-2)^2, \lambda, 1)$ and $D(1, 1, 1)$ be four distinct points in $\mathbb{R}^3$. Find the sum of all real values of $\lambda$ for which these points are coplanar.

Let $z = x + iy$ be a complex number satisfying $|z|^2 = 56$. A line $L$ has direction vector $\vec{v} = x\hat{i} + y\hat{j} + c\hat{k}$ (where $c > 0$). Consider a plane $P$ with normal vector $\vec{n} = \hat{i} + \hat{j} + \hat{k}$ and a vector $\vec{u} = 2\hat{i} - \hat{j} + \hat{k}$. If the vectors satisfy the relation $(\vec{v} \cdot \vec{n}) (\vec{v} \times \vec{u}) = \vec{v} \times (\vec{n} \times \vec{v}) + (\vec{v} \cdot \vec{n})\vec{v} - 244 \vec{n}$ then the value of $c^2$ is:

Let $\vec{a}(x) = \hat{i} + x\hat{j} + x^2\hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$, and $\vec{c}(x) = \hat{i} + f(x)\hat{j} + g(x)\hat{k}$, where $f(x) = \int_{-x}^{x} \ln\left(t + \sqrt{t^2+1}\right) dt$ $g(x) = \int_{-x}^{x} \sqrt{t^2} dt$ If $V(x)$ represents the volume of the parallelepiped formed by the vectors $\vec{a}(x)$, $\vec{b}$, and $\vec{c}(x)$ for $x \in [-3, 3]$, what is the maximum value of $V(x)$?

Let $T_m$ be the numerically greatest term in the expansion of $(3 + 2x)^{10}$ at $x = 1$. Let $m$ arithmetic means $A_1, A_2, \dots, A_m$ be inserted between $5$ and $23$. The points $P(A_1, A_2, A_1+A_2)$, $Q(A_3, \beta, A_3+\beta)$, and $R(A_5, \alpha, A_5+\alpha)$ are collinear in 3D space.

Consider the polynomial $g(x) = \frac{x^4}{4} - \frac{13 x^3}{3} + \frac{c x^2}{2} - q x + r$. If $g(x)$ has a point of inflection at $x = \alpha$ (where $g'(\alpha)=0$) and a local minimum at $x = \beta$, find the value of $c + q$.

A line $L$ passes through the origin $O$ and has direction ratios $(1, -1, \sqrt{2})$. A point $P(x, y, z)$ lies on the line $L$ at a distance $r$ ($r > 0$) from $O$. The point $P$ satisfies the condition $x \sin\left(\frac{\pi r}{2}\right) + y \cos\left(\frac{\pi r}{2}\right) + z \ge \frac{r}{\sqrt{2}}$ and its distance $r$ satisfies $\frac{r^2 - 11r + 28}{r - 3} \le 0$. Find the sum of all possible integral values of $r$.

Let $L_1$ and $L_2$ be two lines in $\mathbb{R}^3$ passing through the origin, having direction ratios $(1, 2, 2)$ and $(2, 3, 6)$ respectively. Let $P_1$ and $P_2$ be the $3 \times 3$ matrices representing the orthogonal projections onto $L_1$ and $L_2$ respectively. If $M = P_1 + P_2 - \lambda I$, where $I$ is the $3 \times 3$ identity matrix, then the set of all real values of $\lambda$ for which $M$ is a singular matrix is:

Let $S_t$ be the family of quadric surfaces defined by $(t^2 - t)x^2 + (t-1)y^2 + (z-3)^2 = t-1$, where $t \in \mathbb{R}$. Let $\Pi_t$ be the polar plane of the point $P(0, 0, 5)$ with respect to $S_t$. Let $C_t$ be the intersection of $\Pi_t$ and $S_t$. Let $Z$ be the set of all $z$-coordinates of the points where $C_t$ intersects the $z$-axis, considering all $t \in \mathbb{R}$ for which such an intersection exists. Find the sum of the elements of $Z$.

Let $P_n$ be a family of planes given by the equation $x + y + z = d_n$, where $d_n$ is the coefficient of $t^2$ in the polynomial $(1+t)^{n+3} - (1+t)^{n+1}$. A line $L$ is given by the symmetric equations $\frac{x - 2023}{1} = \frac{y - 2024}{1} = \frac{z - 2025}{1}$. Let $Q_n(x_n, y_n, z_n)$ be the unique point of intersection of the plane $P_n$ and the line $L$. Find the value of $\sum_{n=1}^{8} (x_n + y_n + z_n)$.

Let $\vec{r}(t) = (2t+1)\hat{i} + (t-2)\hat{j} + (3t+4)\hat{k}$ be the parametric equation of a line. Let $\vec{u} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{v} = \hat{i} - \hat{j} + \hat{k}$. A function $f: \mathbb{R} \to \mathbb{R}$ is defined as $f(x) = \int_0^x |y| e^{-y^2} dy$. Let $g(t) = f([\vec{r}(t) \ \vec{u} \ \vec{v}])$. Let $t_0$ be the value of $t$ where $\vec{r}(t)$ becomes linearly dependent on $\vec{u}$ and $\vec{v}$. If $L = \lim_{h \to 0^+} \frac{g'(t_0+h) - g'(t_0)}{h}$ and $R = \lim_{h \to 0^-} \frac{g'(t_0+h) - g'(t_0)}{h}$, find the value of $|L - R|$.

Let $A = \begin{pmatrix} 0 & 2 & 0 \\ 4 & -2 & 4 \\ 2 & -3 & 2 \end{pmatrix}$. A plane $P$ has the equation $ax + by + cz = d$. The system of linear equations $A \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is known to be consistent. If the plane $P$ passes through the points $(1, 1, 1)$ and $(2, 0, 1)$, find the square of the perpendicular distance from the point $(10, 10, -8)$ to the plane $P$.

Let $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$, and $\vec{c} = \hat{i} + \hat{j} + 2\hat{k}$. Let $\hat{n} = l\hat{i} + m\hat{j} + n\hat{k}$ be a unit vector parallel to $\vec{v} = \vec{a} \times (\vec{b} \times \vec{c}) - 4\hat{i} - 3\hat{j} + \hat{k}$ such that $m > 0$. A plane $\Pi$ passes through the point $P(3, 5, -2)$ and has $\hat{n}$ as its normal vector. The perpendicular distance from the point $Q(l, m, n)$ to the plane $\Pi$ is:

Let $S_1, S_2,$ and $S_3$ be the sets of all real roots of the following equations, respectively:

(i) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{\sqrt{1-t^2}} \right) = 0$

(ii) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{\sqrt{t^2-1}} \right) = 0$

(iii) $\frac{d}{dx} \left( \int_{x}^{x^2} \frac{dt}{t\sqrt{t^2-1}} \right) = 0$

Let $P(\alpha, \beta, \gamma)$ be a point in 3D space where $\alpha = \sum_{x \in S_1} \frac{1}{x^2}$, $\beta = \sum_{x \in S_2} x^2$, and $\gamma = \sum_{x \in S_3} x^2$. (If any set is empty, its corresponding sum is $0$).

If $\Pi$ is the plane whose $x, y,$ and $z$ intercepts are $3, -6,$ and $3$ respectively, then the perpendicular distance from the point $P$ to the plane $\Pi$ is:

Let $E$ be the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ in the $xy$-plane ($z=0$). A point $P$ moves along the portion of $E$ that lies within the region defined by $y^2 \le \frac{9}{4}x$. Find the maximum distance from the point $P$ to the plane $\Pi: x - 2\sqrt{3}y + 3z = 5$.

Let $P(\alpha, \beta, \gamma)$ be a point in 3D space whose coordinates are given by: $\alpha = 35 \int_{-\pi/2}^{\pi/2} \text{sgn}(x) \sin^3 x \cos^4 x \, dx$ $\beta = \frac{32}{\pi} \int_{-\pi/2}^{\pi/2} \text{sgn}(x) \cos^6 x \, dx$ $\gamma = \frac{32}{\pi} \int_{-\pi/2}^{\pi/2} (1 + \text{sgn}(x)) \sin^6 x \, dx$ where $\text{sgn}(x)$ denotes the signum function. The perpendicular distance from the point $P$ to the plane $x + 2y + 2z - 6 = 0$ is:

Let $M = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & -1 & 2 \end{pmatrix}$ and let $\lambda_1 < \lambda_2 < \lambda_3$ be the eigenvalues of $M$. A plane $\Pi$ is defined by the equation $\lambda_1 x + \lambda_2 y + \lambda_3 z = 12$. Let $A = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$. A line $L$ passes through the point $P(3, 3, 1)$ and has a direction vector $\vec{v} = AB\vec{u}$, where $\vec{u} = \hat{i} + \hat{j} - \hat{k}$. If $S$ is the set of all points of intersection of the line $L$ and the plane $\Pi$, let $d$ be the shortest distance from the origin to the set $S$. Find the value of $d^2$.

A line $L$ lies in the plane $P: x + y - z = 0$ and is tangent to the surface $S: z = xy$ at a point in the first octant. Let $L$ intersect the planes $x = 0$ and $y = 0$ at points $A$ and $B$ respectively. Find the minimum value of $OA^2 + OB^2$, where $O$ is the origin.

Let a curve $L$ be the locus of points $(x, y, z)$ satisfying the equations: $y = 2x - 1 - x \ln x + \int_1^x \ln(t) dt$ $z = x + \frac{11}{4} - \frac{x^2}{2} \ln x + \frac{x^2}{4} + \int_1^x t \ln(t) dt$ Find the length of the segment of $L$ that lies inside the sphere $S: x^2 + y^2 + z^2 - 2x - 4y - 6z = 0$.

Let $L_1$ be the line of intersection of the planes $P_1: 2x + y - 2z = 5$ and $P_2: x - 2y + 2z = 6$. Let $L_2$ be the line given by the symmetric form $\frac{x-2}{0} = \frac{y+1}{1} = \frac{z-3}{1}$. A plane $\Pi$ contains the line $L_1$ and is parallel to the line $L_2$. The perpendicular distance from the origin to the plane $\Pi$ is:

Let $L$ be the line of intersection of the planes $x - 2y + z = 1$ and $2x + y - z = 2$. Let $Q$ be the image of the point $P(-1, -1, 8)$ in the line $L$. If $R(\alpha, \beta, \gamma)$ is the image of $Q$ in the plane $x + y + z = 2$, then the value of $\alpha^2 + \beta^2 + \gamma^2$ is

Let $ABCD$ be a tetrahedron such that its two opposite edges $AB$ and $CD$ lie on the lines $L_1$ and $L_2$ respectively, where: $L_1: x+y+z-1 = 0 = x-y+z-1$ $L_2: x+y-z+2 = 0 = -x+y+z+2$ The vertices $A$ and $B$ are at a distance of $1$ unit from the origin, while the vertices $C$ and $D$ are at a distance of $\sqrt{6}$ units from the origin. Let $G(\alpha, \beta, \gamma)$ be the centroid of the tetrahedron $ABCD$. If $G$ lies in the $k^{\text{th}}$ octant (using the standard I-VIII convention), then the value of $k^2 + 8(\alpha + \beta + \gamma)$ is:

Let $y = y(x)$ be a solution to the differential equation $x \ln x \frac{dy}{dx} + y = 2 \ln x, \quad x > 1$ A line $L$ has direction ratios proportional to $(2, y(e), e y'(e))$. It is given that $L$ intersects the $y$-axis and lies entirely in the plane $P_1: x + 2y + z = 10$. If $L$ intersects the plane $P_2: 3x - y + 2z = 17$ at the point $Q(\alpha, \beta, \gamma)$, then the value of $\alpha + \beta + \gamma$ is:

Let $S$ be the locus of a complex number $z$ in the Argand plane satisfying $|z - 2| = |z - 2i|$. Let $P$ be the plane in $\mathbb{R}^3$ containing the $z$-axis and the locus $S$ (where the Argand plane is identified with the $xy$-plane). Let $L$ be the line of intersection of the planes $2x - y + z = 2$ and $x + 2y - z = 1$. If $L'$ is the orthogonal projection of the line $L$ onto the plane $P$, then which of the following points lies on $L'$?

Let $\Delta(t) = \begin{vmatrix} x & y & z \\ e^{-t} & 1-\cos t & \ln(1-t) \\ \sin t & e^{-2t} & -t \end{vmatrix}$ for $t < 1$. A plane $P$ is defined by the equation $\left[ \frac{d}{dt} \Delta(t) \right]_{t=0} = 0$. Let $L_1$ be the line $\frac{x-5}{1} = \frac{y-6}{2} = \frac{z}{1}$ and $L_1'$ be its orthogonal projection onto the plane $P$. If $d$ is the shortest distance between $L_1'$ and the line $L_2: \frac{x}{2} = \frac{y-2}{-1} = \frac{z-7}{0}$, then the value of $d^2$ is:

Let $L$ be the line given by $x - 1 = 2 - y = \frac{z - 3}{\sqrt{2}}$. Let $A(1, 2, 3)$ be a point on $L$, and $B$ be another point on $L$ such that the distance $AB = 2$. For each positive integer $n$, let $\Pi_n$ be the plane defined by the equation $\sqrt{n^2+n-1}(x+y) + \sqrt{2}z = n^3$. Let $p_n$ denote the length of the orthogonal projection of the line segment $AB$ onto the plane $\Pi_n$. Find the value of $\sum_{n=1}^{99} p_n^2$.

Let $S$ be the set of all real values of $\lambda$ such that the orthogonal projection of the line $L: \frac{x - 1}{\lambda} = \frac{y - 2}{\lambda^2 - 4} = \frac{z - 1}{\lambda - 1}$ on the plane $\Pi: x + 3y + 2z = 0$ is coplanar with the z-axis. If $z \in \mathbb{C}$ satisfies $|z - \lambda| \le 4$ for all $\lambda \in S$, then the maximum value of $|z - 2|$ is:

Let $A$ be a $3 \times 3$ matrix with real entries such that: $A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad A \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \\ 2 \end{pmatrix}$ Let $P_1$ and $P_2$ be two planes passing through the origin. The normal vector to $P_1$ is parallel to the first row vector of $A$, and the normal vector to $P_2$ is parallel to the first column vector of $\text{adj}(A)$. Let $\vec{n}_1$ and $\vec{n}_2$ be the normal vectors of $P_1$ and $P_2$ respectively, such that their $\hat{k}$ components are strictly positive and they both have a magnitude of $\sqrt{2}$. If $\theta$ is the angle between the vectors $\vec{n}_1$ and $\vec{n}_2$, what is the value of $\cos \theta$?

Let $\omega = e^{i \frac{2\pi}{5}}$. Let $\text{Arg}(z)$ denote the principal argument of a complex number $z$, such that $-\pi < \text{Arg}(z) \le \pi$. Consider two planes $P_1$ and $P_2$ in 3D space defined by the equations: $P_1: x \text{Arg}(\omega^2) + y \text{Arg}(\omega^3) + z \text{Arg}(\omega) = 1$ $P_2: x \text{Arg}(\omega^2) + y \text{Arg}(\omega^4) + z \text{Arg}(\omega^3) = 2$ If $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then the value of $\cos \theta$ is: