Introduction to Three-dimensional Geometry - Class 11

Let $f(x)$ be the volume of the tetrahedron formed by the origin $O(0,0,0)$ and the points $A, B,$ and $C$ whose position vectors are given by: $\vec{a} = [x] \hat{i} + \cos(\pi [x]) \hat{j} + \hat{k}$ $\vec{b} = \hat{i} + ([x]-1) \hat{j} + \cos(\pi [x]) \hat{k}$ $\vec{c} = \cos(\pi [x]) \hat{i} + \hat{j} + ([x]-2) \hat{k}$ where $[x]$ denotes the greatest integer less than or equal to $x$. The value of the integral $\int_0^4 6 f(x) \, dx$ is:

Let $P$ be a plane satisfying the equation $|x + 2y - 2z + 2| = 7$. The normal vector to $P$ drawn from the origin to the plane points into the octant where $x > 0$, $y > 0$, and $z < 0$. Find the coordinates of the image of the point $A(1, -2, 2)$ in the plane $P$.

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$. A vector $\vec{c}$ is collinear with $\vec{d} = 3\hat{i} + 2\hat{j} + \hat{k}$. The volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b},$ and $\vec{c}$ is $192$ cubic units. If the terminal point of $\vec{c}$ (when its initial point is at the origin) lies in the octant where all three coordinates are strictly negative, find the value of $[\vec{a} + \vec{b} \quad \vec{b} + \vec{c} \quad \vec{c} + \vec{a}]$.

A point $P(x,y,z)$ moves in 3D space such that its $L_\infty$ distance from the origin, defined as $\max(|x|, |y|, |z|)$, is at most 3. If the coordinates of $P$ also satisfy the inequality $\frac{x y z}{(x-1)(y-2)(z-1)} \ge 0$ what is the volume of the region representing the locus of $P$?

Let $\vec{u}(\theta) = 2\cos\theta \hat{i} + 2\sin\theta \hat{j}$ for $\theta \in [0, 2\pi)$. A set $S$ consists of all distinct vectors $\vec{v}$ defined by $\vec{v} = [\vec{u} \cdot \hat{i}] \hat{i} + [\vec{u} \cdot \hat{j}] \hat{j}$, where $[t]$ denotes the greatest integer less than or equal to $t$. Find the value of $\sum_{\vec{v} \in S} |\vec{v}|^2$.

Let $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$ be three non-collinear points in $\mathbb{R}^3$ such that the area of $\triangle ABC$ is $3\sqrt{5}$. Let $M = \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}$. It is given that $\text{adj}(\text{adj}(M))$ is a singular matrix. Let $\Delta_x$ and $\Delta_y$ be the areas of the orthogonal projections of $\triangle ABC$ onto the $yz$-plane and $zx$-plane, respectively. If $\Delta_x = 2\Delta_y$, what is the value of $\Delta_x$?

Let $M$ be an orthogonal matrix of order 3 with column vectors $\vec{c}_1, \vec{c}_2, \vec{c}_3$. Let $\vec{u} = \vec{c}_1$, $\vec{v} = \vec{c}_1 + \vec{c}_2$, and $\vec{w} = \vec{c}_1 + \vec{c}_2 + \vec{c}_3$. Two planes $P_1$ and $P_2$ have normal vectors $\vec{n}_1$ and $\vec{n}_2$ respectively. The scalar projections of $\vec{n}_1$ onto $\vec{u}, \vec{v}, \vec{w}$ are $1, \sqrt{2}, \sqrt{3}$ respectively. The scalar projections of $\vec{n}_2$ onto $\vec{u}, \vec{v}, \vec{w}$ are $2, 0, -\sqrt{3}$ respectively. If $\theta$ is the acute angle between the planes $P_1$ and $P_2$, then $\cos \theta$ is equal to:

Let $P(1, -1, 0)$ and $Q(1, 1, 2)$ be two points in $\mathbb{R}^3$. A point $R$ divides the line segment $PQ$ internally in the ratio $k:1$. Let $A, B,$ and $C$ be the projections of $R$ onto the $xy$, $yz$, and $zx$ planes respectively. If the area of $\triangle ABC$ is $\frac{3}{8}$, then the distance of $R$ from the origin is:

Let $\vec{a}, \vec{b}, \vec{c}$ be three unit vectors such that the angle between any two of them is $\frac{\pi}{3}$. Let $\vec{r} = \vec{a} - \vec{b} + 2\vec{c}$. If $\vec{P}_1 = \vec{a} \times (\vec{r} \times \vec{a})$, $\vec{P}_2 = \vec{b} \times (\vec{r} \times \vec{b})$, and $\vec{P}_3 = \vec{c} \times (\vec{r} \times \vec{c})$, then the volume of the parallelepiped determined by the vectors $\vec{P}_1, \vec{P}_2, \vec{P}_3$ is:

Let $S$ be the set of all real numbers $x$ such that the perpendicular distance from the point $P([x], [x]+|x|, |x|-[x])$ to the plane $2X - Y + 2Z - 202 = 0$ is exactly $22$. Find the sum of the absolute values of all elements in $S$. (Here $[t]$ denotes the greatest integer less than or equal to $t$, and $|t|$ denotes the absolute value of $t$)

Let $M = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 5 \\ 2 & 3 & 5 \end{pmatrix}$. The position vectors of the vertices $A, B,$ and $C$ of a tetrahedron $ABCD$ are given by the first, second, and third column vectors of $M$, respectively. Let $M'$ be the matrix obtained by interchanging the first and second columns of $M$. If the position vector of vertex $D$ is given by $(\det(M), \text{Tr}(M), \det(M'))$, find the distance between the centroid of the tetrahedron $ABCD$ and the vertex $D$.

Let $L_t$ be a variable line in 3D space. The orthogonal projection of the origin $O(0,0,0)$ onto $L_t$ is the point $Q(t) = (x(t), y(t), z(t))$, where for $t > 0$: $x(t) = \int_0^t (2u \sin u + u^2 \cos u) du$ $y(t) = \int_0^t (2u \cos u - u^2 \sin u) du$ $z(t) = \int_0^t 3u^2 du$ Find the minimum distance between the point $P\left(0, 0, \frac{5}{3}\right)$ and the projection point $Q(t)$.

Let $A(6, 0, 0)$, $B(0, 6, 0)$, $C(0, 0, 6)$, and $D(0, 0, 0)$ be the vertices of a tetrahedron. Let $P(x, y, z)$ be a variable point. The position vectors of $A, B$, and $C$ relative to $P$ are $\vec{a}, \vec{b}$, and $\vec{c}$ respectively. Let $V$ be the volume of the tetrahedron formed by the vectors $\vec{a}, \vec{b}$, and $\vec{c}$. Let $\vec{g}$ be the position vector of the centroid of the tetrahedron $ABCD$ relative to $P$. If the locus of $P$ satisfying the equation $V + 8 \vec{g} \cdot (\hat{i} + \hat{j} + \hat{k}) + 24 = 0$ is a plane, then the perpendicular distance of this plane from the origin is:

Let $A, B, C, D$ be four points in $\mathbb{R}^3$ such that the volume of the tetrahedron $ABCD$ is $7$. Let $M$ be the $4 \times 4$ matrix defined by $M = \begin{pmatrix} \vec{a} & \vec{b} & \vec{c} & \vec{d} \\ 1 & 1 & 1 & 1 \end{pmatrix}$ where $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are the position vectors of $A, B, C, D$ respectively, and $\det(M) > 0$. Let $\vec{g}$ be the position vector of the centroid of the tetrahedron $ABCD$. Let $N$ be a $4 \times 4$ matrix whose columns are $C_1, C_2, C_3, C_4$, where $C_1 = \begin{pmatrix} 3\vec{a} - \vec{g} \\ 2 \end{pmatrix}, \quad C_2 = \begin{pmatrix} 3\vec{b} - \vec{g} \\ 2 \end{pmatrix}, \quad C_3 = \begin{pmatrix} 3\vec{c} - \vec{g} \\ 2 \end{pmatrix}, \quad C_4 = \begin{pmatrix} 3\vec{d} - \vec{g} \\ 2 \end{pmatrix}$ If $P = \frac{1}{3} N$, what is the value of $\det(P)$?

The four complex numbers $z_1, z_2, z_3, z_4$ are the roots of the polynomial equation $z^4 - 4(w - 3 + 4i)z^3 + \alpha z^2 + \beta z + \gamma = 0$ where $w, \alpha, \beta, \gamma \in \mathbb{C}$. These points represent the projections of the vertices of a tetrahedron onto the complex plane. Given that the sum of the squares of their distances from the origin is $64$, find the maximum possible value of $|w - 11 + 19i|$.

Let $A(4, -2, -2)$, $B(-2, 4, -2)$, and $C(-2, -2, 4)$ be three vertices of a tetrahedron $ABCD$. The centroid $G$ of the tetrahedron lies on the plane $x + y + z = 3$. The distance of $G$ from the point $(1, 1, 1)$ is strictly greater than $\frac{1}{\sqrt{2}}$ and strictly less than $\sqrt{\frac{3}{2}}$. If the fourth vertex $D(x, y, z)$ has integer coordinates, find the number of possible positions for $D$.

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that $|\vec{a}| = |\vec{b}| = |\vec{c}| = 2$ and the angle between any two of them is $\frac{\pi}{3}$. Let $\vec{a}^*, \vec{b}^*, \vec{c}^*$ be the corresponding reciprocal system of vectors. A tetrahedron has vertices at the origin $O$ and at points $P, Q, R$ with position vectors $\vec{p} = \vec{a}^* + 2\vec{b}^* + 3\vec{c}^*$, $\vec{q} = 2\vec{a}^* + 3\vec{b}^* + \vec{c}^*$, and $\vec{r} = 3\vec{a}^* + \vec{b}^* + 2\vec{c}^*$. If $G$ is the centroid of the tetrahedron $OPQR$, find the value of $32 |\vec{OG}|^2$.

Let $M = \begin{pmatrix} 0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}$. The characteristic equation of $M$ is given by $\det(\lambda I - M) = \lambda^3 + c_1\lambda^2 + c_2\lambda + c_3 = 0$. Consider the vectors $\vec{u} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ and $\vec{v} = -2\hat{j} - \hat{k}$. Let $A(3, 0, 0)$, $B(0, 3, 0)$, and $C(0, 0, 3)$ be three fixed points in space. A point $P(x,y,z)$ moves such that the vector $\vec{w} = PA^2\hat{i} + PB^2\hat{j} + PC^2\hat{k}$ is linearly dependent on $\vec{u}$ and $\vec{v}$. Which of the following best describes the locus of $P$?

Let $P(x, y, z)$ be a point in $\mathbb{R}^3$. Define two complex numbers associated with $P$ as: $w_1 = (2x + y) + i(y + z)$ $w_2 = (2x - y) + i(y - z)$ Let $S$ be the locus of points $P$ such that $|w_1|^2 + |w_2|^2 = 16$ and $w_1 \bar{w_2} + \bar{w_1} w_2 = 0$. Find the maximum distance between any two points on the locus $S$.

A point $Q(x,y,z)$ moves on the sphere $x^2+y^2+z^2=4$ such that the ratio of its distance from the $xy$-plane to its distance from the $z$-axis is equal to $x$. Let $S$ be the set of all points on the locus of $Q$ where the distance of $Q$ from the $xy$-plane is maximized. Find the value of $\sum_{P \in S} (x_P^2 + y_P^2 + z_P^2)$.

Let $\pi$ be a permutation of the set $\{1, 2, 3, 4\}$ chosen uniformly at random. Define indicator variables $I_k$ for $k \in \{1, 2, 3\}$ as follows: $I_k = \begin{cases} 1, & \text{if } \pi(k) > \pi(k+1) \\ 0, & \text{otherwise} \end{cases}$ Consider the region $R = \{(x,y) \in \mathbb{R}^2 : |x| \le 4, \text{ and } |y| \le 4\}$. Let $A$ be the area of the locus of points $(x,y) \in R$ that satisfy the inequality: $I_1 x^2 + I_2 y^2 \le 4 I_3$ Find the expected value of $A$, denoted as $E[A]$.

Let $S$ be the locus of a point $P(x,y,z)$ in 3D space such that the square of its distance from the origin is equal to $4(d_1 + d_2 + d_3) - 9$, where $d_1, d_2,$ and $d_3$ are the perpendicular distances of $P$ from the $yz$, $zx$, and $xy$ planes, respectively. If $M$ is the maximum possible distance between any two points lying on $S$, find the value of $M^2$.

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$, and $\vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k}$ be three coplanar vectors. A point $P$ with position vector $\vec{p} = x\hat{i} + y\hat{j} + z\hat{k}$ divides the line segment joining $A(0, 4, 2)$ and $B(2, 0, 4)$ internally in the ratio $k:1$. If $\vec{p}$ satisfies the system of equations $\vec{p} \cdot \vec{a} = 14$ $\vec{p} \cdot \vec{b} = 20$ $\vec{p} \cdot \vec{c} = \mu$ and the system is consistent, then the value of $\mu$ and the set of all possible values for $k$ are:

Let $\vec{a}$ and $\vec{b}$ be the position vectors of two distinct points $A$ and $B$. A point $P$ with position vector $\vec{p}$ is uniquely determined by the relation: $8(\lambda^2 - 4)\vec{p} = (\lambda^3 + 3\lambda^2 - 4\lambda - 12)\vec{a} + (5\lambda^2 - \lambda^3 + 4\lambda - 20)\vec{b}$ where $\lambda \in \mathbb{R}$. If $P$ divides the line segment $AB$ strictly internally, then the set of all possible values of $\lambda$ is:

The plane $\Pi: x - y + z - 9 = 0$ intersects the line passing through the points $P(1, 0, 2)$ and $Q(4, 3, 5)$ at the point $R$. If $A(1, 2, 3)$ and $B(2, 4, 1)$ are two points in space, and $\Delta$ is the area of $\triangle RAB$, find the value of $4\Delta^2$.

Let $A(0,0)$ and $B(12\pi, 0)$ be two points. A point $P$ divides the line segment $AB$ internally in the ratio $k:1$. Let $C$ be the curve defined by $y = 4 + \sin x + \sqrt{3}\cos x$. If $k$ is chosen such that the vertical distance from $P$ to the curve $C$ is minimized, and the horizontal distance from $P$ to the point $M(6\pi, 0)$ is also minimized among all such possible positions of $P$, then the value of $k$ is:

Let $A(-1, 0, 2)$ and $B(1, 0, 1)$ be two points in $\mathbb{R}^3$. A transformation $T: \mathbb{R}^3 \to \mathbb{C}$ maps a point $(x,y,z)$ to the complex plane via $T(x,y,z) = z + (x+iy)e^{i\pi/2}$. Let $L$ be the line passing through $A$ and $B$. There exist exactly three distinct points $P_1, P_2, P_3$ on $L$ such that for each point $P_j$, its image $w_j = T(P_j)$ satisfies the condition that $w_j^3$ is a purely real number. If $P_j$ divides the directed line segment $\vec{AB}$ in the ratio $k_j:1$ (i.e., $\vec{AP_j} = \frac{k_j}{k_j+1} \vec{AB}$ where $k_j \in \mathbb{R}$), find the value of $\left(\sum_{j=1}^3 k_j\right)^2 + 2 \prod_{j=1}^3 k_j$.

Let $A(2, 0, 0)$, $B(0, 3, 0)$, and $C(0, 0, 6)$ be three points in $\mathbb{R}^3$. Let $\vec{n}_1 = \vec{AC} \times \vec{AB}$ and $\hat{n}_1$ be the unit vector along $\vec{n}_1$. Let $P_2$ be the plane $2x - y + 3z = 4$ with normal vector $\vec{n}_2 = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{n}_2$ be the unit vector along $\vec{n}_2$. Let $\Pi$ be the plane bisecting the acute angle between the plane $ABC$ and $P_2$. If $\vec{n}_3$ is the normal vector to $\Pi$ expressed in the form $\vec{n}_3 = \alpha \hat{n}_1 + \beta \hat{n}_2$ where $\alpha > 0$, find the scalar projection of the vector $\vec{v} = 5\hat{i} - \hat{j} + 4\hat{k}$ onto $\vec{n}_3$.

Consider a space curve $C$ parameterized by $\vec{r}(t) = 3\sin t \hat{i} + 2\sin 2t \hat{j} + \cos(e^t) \hat{k}$ for $t \in [0, 2\pi]$. Let $C'$ be the orthogonal projection of $C$ onto the $xy$-plane. What is the total geometric area of the region(s) enclosed by $C'$?

Let $S$ be the surface defined by $z = x|x| + y^2$. Let $E$ be the set of all points $T(x_0, y_0, z_0)$ on $S$ (where $x_0 \neq 0, y_0 \neq 0$) such that the tangent plane to $S$ at $T$ intersects the $x$-axis at $A(-2, 0, 0)$ and the $y$-axis at $B(0, 1, 0)$. Find the number of elements in $E$.

Let $C: y = f(x)$ be a curve in the first quadrant with $f'(x) > 0$ for all $x > 0$. For any point $P(x,y)$ on $C$, let the tangent at $P$ meet the $y$-axis at $A$ and the normal at $P$ meet the $x$-axis at $B$. Let $T_x$ and $T_y$ be the lengths of the projections of the line segment $PA$ onto the $x$-axis and $y$-axis, respectively. Let $L_y$ be the length of the projection of the line segment $PB$ onto the $y$-axis. If the curve satisfies the relation $T_x T_y = T_x^2 + T_x L_y - L_y^2$ and passes through the point $(1, 1)$, then the value of $f(2024)$ is

Let $A = (1, -1, 0)$, $B = (0, 1, -1)$, and $C = (-1, 0, 1)$ be three points in $\mathbb{R}^3$. A fourth point $D$ is given by the coordinates $D = \left( \text{sgn}(x^2 - 2x - 3), \text{sgn}(-2x^2 + x + 1), \frac{x}{\text{sgn}(x)} \right)$ where $\text{sgn}(\cdot)$ denotes the signum function. Let $S$ be the set of all real values of $x$ for which the points $A, B, C$, and $D$ are coplanar. Find the product of all elements in $S$.

Let $\vec{v}_1 = 2\hat{i} + 2\hat{j} + \hat{k}$, $\vec{v}_2 = 4\hat{i} + 7\hat{j} - 4\hat{k}$, and $\vec{v}_3 = 8\hat{i} - \hat{j} + 4\hat{k}$. Let $\vec{b}_1$ be a vector along the acute angle bisector of $\vec{v}_1$ and $\vec{v}_2$, and $\vec{b}_2$ be a vector along the acute angle bisector of $\vec{v}_2$ and $\vec{v}_3$. If the vector $\vec{u} = 14\hat{i} + \lambda\hat{j} + 2\hat{k}$ is coplanar with $\vec{b}_1$ and $\vec{b}_2$, then the value of $\lambda$ is:

Let $g(t) = 4t^3 - 12t^2 + 8t$ and let $\alpha$ be the point of local maximum of the function $h(x) = \int_{0}^{x} g(t) dt$. A plane $P$ makes intercepts $a, b,$ and $c$ on the $x, y,$ and $z$ axes respectively, where $a = f(-1)$, $b = f(2)$, and $c = f(3)$, with $f(x) = \int_{\alpha}^{x} g(t) dt + 3$. If the point $D(2k, k, -2k)$ is coplanar with the points where the plane $P$ meets the coordinate axes, then the value of $k$ is

Let $\vec{u}_k = \binom{10}{k} \hat{i} + \binom{10}{k+1} \hat{j} + \binom{10}{k+2} \hat{k}$ and $\vec{v}_k = \binom{11}{k+1} \hat{i} + \binom{11}{k+2} \hat{j} + \binom{11}{k+3} \hat{k}$ for any integer $k$, where $\binom{n}{r} = 0$ if $r < 0$ or $r > n$. Let $S$ be the set of all integers $k \in [-3, 11]$ such that the points with position vectors $\vec{u}_k$ and $\vec{v}_k$ are collinear with the origin $O$. Find the number of elements in $S$.

Let $O$ be the origin. The coordinates of three points $A, B,$ and $C$ are determined by the roots of the following equations: For $A$: $t^3 - 7t + 6 = 0$ For $B$: $t^3 + 2t^2 - 5t - 6 = 0$ For $C$: $t^3 + 3t^2 - 4t - 12 = 0$ The coordinates $(x, y, z)$ of each point are a permutation of the roots of their respective equations, satisfying the following conditions: (i) $A$ lies in Octant II, and its $y$-coordinate is strictly less than its $z$-coordinate. (ii) $B$ lies in Octant VIII, and its $y$-coordinate is strictly greater than its $z$-coordinate. (iii) $C$ lies in Octant VI, and the absolute value of its $x$-coordinate is strictly greater than the absolute value of its $z$-coordinate. Let $V$ be the volume of the parallelepiped with coterminous edges $\vec{OA}, \vec{OB},$ and $\vec{OC}$. If $S = [\vec{OA}\ \vec{OB}\ \vec{OC}]$ is their scalar triple product, find the value of $2V - S$.

Let $S$ be the set of points $P(x,y,z)$ in $\mathbb{R}^3$ satisfying the equations $x+y+z=1$ and $|x|+|y|+|z|=1$. Let $A$ be the point $(1, 1, 1)$ and $B$ be the point $(-1, 2, 2)$. The minimum value of $PA + PB$ as $P$ varies over the region $S$ is:

Let $V$ be the region in $\mathbb{R}^3$ defined by $|x| + |y| + |z| \le 2$. Evaluate the integral: $I = \iiint_V \left( x^2 \{x\} + y^2 \{y\} + z^2 \{z\} \right) \,dx\,dy\,dz$ where $\{t\}$ denotes the fractional part of $t$.

In $\triangle ABC$, the angles satisfy the relation $\cos A \cos B + \sin A \sin B \sin C = 1$. Let $\lambda = \frac{\sin(A+B)}{\cos(A-B)} - 4$. A point $R$ divides the line segment joining $P(5, -1, 2)$ and $Q(1, 3, -2)$ in the ratio $\lambda : 1$. If $R$ lies in the $n^{\text{th}}$ octant, find the value of $n^2 + |\vec{OR}|^2$, where $O$ is the origin.

Let $L$ be a variable line passing through the origin such that its direction cosines $(l, m, n)$ satisfy $l \ge m \ge 0$ and $n = \frac{1}{\sqrt{2}}$. Let $Q$ be the foot of the perpendicular drawn from the point $P(3, 4, 5)$ to the line $L$. The maximum value of the distance $OQ$, where $O$ is the origin, is: