Straight Lines - Class 11

Let $\vec{r} = x\hat{i} + y\hat{j}$ be the position vector of a point $P$ moving on a closed curve $C$ in the $xy$-plane. Let $\vec{v}_1 = x\hat{i} + y\hat{j} + \hat{k}$ and $\vec{v}_2 = y\hat{i} - x\hat{j} + \hat{k}$. The curve $C$ is defined by the locus of $P$ such that $|\vec{v}_1 \times \vec{v}_2 + (x^2+y^2)\hat{k}| = 2\sqrt{2}$. Let $f(P) = [\alpha] + [\beta] + [\gamma]$, where $\alpha, \beta, \gamma$ are the scalar components of $\vec{v}_1 \times \vec{v}_2$ along $\hat{i}, \hat{j}, \hat{k}$ respectively, and $[ \cdot ]$ denotes the greatest integer function. Find the number of points on the curve $C$ where $f(P)$ is discontinuous.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_0^x \left( t^3 \sin\left(\frac{1}{t}\right) + t^2 \right) dt \quad \text{for } x \neq 0, \text{ and } f(0) = 0.$ For each $x \in \left(0, \frac{1}{3}\right)$, let $c_x \in (0, x)$ be the unique point guaranteed by the Lagrange Mean Value Theorem such that the tangent to the curve $y = f(x)$ at $x = c_x$ is parallel to the chord joining the points $(0, 0)$ and $(x, f(x))$. Find the value of $\lim_{x \to 0^+} \frac{c_x}{x}$

A point $P$ with position vector $\vec{p}$ moves in space such that $\vec{p} \times (\hat{i} + \hat{j} + \hat{k}) = \hat{i} - 2\hat{j} + \hat{k}$. Let $M$ and $N$ be two points with position vectors $2\hat{i} + 2\hat{j} - \hat{k}$ and $10\hat{i} + 10\hat{j} - 5\hat{k}$ respectively. Let $Q$ be the orthogonal projection of $P$ onto the line passing through $M$ and $N$. If the locus of $P$ is restricted such that $Q$ lies strictly on the line segment $MN$, find the maximum value of $|\vec{p}|^2$.

Let $y = (x^2 - 6x + 8)^{e^x}$ be a real-valued differentiable function defined on its maximal domain $D \subset \mathbb{R}$. The logarithmic derivative $\frac{1}{y}\frac{dy}{dx}$ can be expressed in the form $e^x \ln(u) + e^x \frac{v}{u}$, where $u$ and $v$ are coprime polynomials in $x$ and the leading coefficient of $u$ is $1$. If a moving point $P(u, v)$ traces a curve in the $uv$-plane for all $x \in D$, which of the following represents the complete locus of $P$?

Let a point $P(x, y)$ move such that the following system of linear equations in variables $U, V, W$: $(t^2+1)U + e^t V + \ln(t) W = 0$ $(t^3+t)U + (t e^t + 5x + 12y - 39)V + (t \ln(t) + \sin(t))W = 0$ $\left(t+\frac{1}{t}\right)U + \frac{e^t}{t} V + \left(\frac{\ln(t)}{t} + \sqrt{t-1}\right)W = 0$ has a non-trivial solution for all $t \in (1, 2)$. If the locus of $P$ is a curve $L$, then the shortest distance from the origin to $L$ is:

Let $y = y(x)$ be the solution to the differential equation $\frac{dy}{dx} + y = |x - 1|$ satisfying the initial condition $y(0) = 2$. The minimum value of $y(x)$ for $x \in \mathbb{R}$ is:

Let $f: [-59, 13] \to \mathbb{R}$ be defined as $f(x) = \sqrt[3]{(x-5)^2}$. Let $S$ be the set of all points $c \in (-59, 13)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the points $(-59, f(-59))$ and $(13, f(13))$. Then the number of elements in $S$ is:

Let $L$ be a directed line in the $xy$-plane passing through the origin, having a slope of $-\frac{5}{12}$ and directed such that its direction vector has a positive $x$-component. Let $\vec{v} = 2\hat{i} + 10\hat{j} + 11\hat{k}$. Let $p$ be the algebraic projection of $\vec{v}$ onto the directed line $L$. If $\vec{w}$ is the reflection of the vector $\vec{v}$ in the plane passing through the origin and perpendicular to $L$, find the value of $\vec{w} \cdot \vec{v} + 5p^3$.

Let $f: \left[0, \frac{3\pi}{2}\right] \to \mathbb{R}$ be defined by $f(x) = \frac{x}{2} - \sqrt{1 - \cos^2 x}$. If $M$ and $m$ are the absolute maximum and absolute minimum values of $f(x)$ respectively, then the value of $M - m$ is:

Consider the curve $C: |y| = x^2 - x$. Let $\Gamma_1$ be the parabola $y = x^2 - x$ and $\Gamma_2$ be the parabola $y = x - x^2$. For a variable line $L$, let $P_1$ and $P_2$ be the poles of $L$ with respect to $\Gamma_1$ and $\Gamma_2$ respectively. Let $S$ be the set of all lines $L$ such that $P_1$ lies on $\Gamma_2$ and $P_2$ lies on $\Gamma_1$. If the lines in $S$ enclose a bounded region with the $y$-axis, find the area of this region.

A line $L$ passing through the point $P(2, 3)$ intersects the curve $C: |x - 2| + |y - 3| = 4$ at two distinct points $A$ and $B$. If the area of the triangle $OAB$ (where $O$ is the origin) is $10$ sq. units, then the sum of all possible values of the slope of the line $L$ is:

Let $L$ be the line given by the symmetric equations $X = Y = Z = x$, where $x$ is a real parameter. Let $P(x)$ be a point on $L$ and $d(x)$ be the perpendicular distance from $P(x)$ to the plane $X + Y - Z = 0$. Let $f(x) = \sqrt{3} d(x)$. A continuous function $y(x)$ satisfies the differential equation $x \frac{dy}{dx} + f(x) y = 3x^3$ for all $x \in \mathbb{R} \setminus \{0\}$. If $y(-1) = -3$, then the value of $y(1)$ is:

Consider the curve defined parametrically by $x(t) = t^3 + t$ and $y(t) = |t|$ for $t \in [-1, 2]$. Since $x(t)$ is strictly increasing, the curve can be represented by an explicit function $y = f(x)$ on the interval $[-2, 10]$. Let $S$ be the set of all $x \in (-2, 10)$ where the tangent to the curve is parallel to the secant line joining its endpoints. Let $P$ be the logical proposition "Lagrange's Mean Value Theorem is applicable to $f(x)$ on $[-2, 10]$". Which of the following options is correct?

Let $L_1$ and $L_2$ be two rays originating from the origin in the first quadrant, making angles $\theta_1 = \tan^{-1}(2)$ and $\theta_2 = \tan^{-1}(5)$ with the positive $x$-axis, respectively. A vertical line $x = r$ ($r > 0$) intersects $L_1$ and $L_2$ at points $P$ and $Q$, respectively. Let $y_1(r)$ and $y_2(r)$ be the ordinates of $P$ and $Q$. A function $f: (0, \infty) \to \mathbb{R}$ is defined as $f(r) = \int_{y_1(r)}^{y_2(r)} \left( \arcsin(\sin t) + \arccos(\cos t) \right) dt$ Find the value of $f'\left(\frac{3\pi}{5}\right)$.

Let $\Pi$ be the family of planes passing through the line of intersection of the planes $x = 9$ and $y = 6$. Let $P_\lambda$ be a plane in $\Pi$ whose equation is given by $x + \lambda y - (9+6\lambda) = 0$ for $\lambda \ge 0$. Let $d(\lambda)$ be the perpendicular distance from the origin to the plane $P_\lambda$. Consider the function $f(\lambda) = [d(\lambda)] - 2\lambda$, where $[ \cdot ]$ denotes the greatest integer function. The global maximum value of $f(\lambda)$ on the interval $[0, 2]$ is:

Let $S$ be the set of all complex numbers $z$ that satisfy the system of equations: $(1-i)z + (1+i)\bar{z} = 4$ $z^3 - 6z^2 - 2z\bar{z} + 16z + 6\bar{z} - 16 = 0$ Find the sum of all elements in $S$.

Let $D$ be the set of all real numbers $m$ satisfying the inequality $\log_{0.5}(m-1) \ge -1$. Let $R$ be the region in the first quadrant consisting of all points $(x, y)$ such that $x \ge 1$ and $0 \le y \le \sup_{m \in D} \left( \frac{2m}{x^2} - \frac{m^2}{x^3} \right)$ If $A$ is the area of the region $R$, find the value of $A$.

Let $A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. A point $P(x, y)$ in the Cartesian plane is represented by the column vector $P = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$. A line $L$ with equation $ax + by + c = 0$ is represented by the column vector $V = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ such that $V^T P = 0$. Under the transformation $P' = M P$, where $M = (AB)^T$, the line $L: 3x - 4y + 5 = 0$ is transformed to a new line $L': a'x + b'y + c' = 0$. If $\gcd(|a'|, |b'|, |c'|) = 1$ and $a' > 0$, find the value of $a' + b' + c'$.

Let $P_1$ be the plane containing the lines $L_1: \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$ and $L_2: \frac{x-2}{1} = \frac{y-3}{2} = \frac{z-5}{3}$. Let $P_2$ be a plane parallel to $P_1$ such that the perpendicular distance between $P_1$ and $P_2$ is $2\sqrt{3}$. A line $L_3: \frac{x - |a-2|}{1} = \frac{y - |a-5|}{2} = \frac{z - 1}{3}$ lies entirely in $P_2$. Let $S$ be the set of all possible real values of $a$. For each $a \in S$, a point $Q_a$ is defined as $(a, a, a)$. If $S = \{a_1, a_2\}$, let $I_1$ and $I_2$ be the images of $Q_{a_1}$ and $Q_{a_2}$ respectively in the plane $P_1$. The distance between $I_1$ and $I_2$ is:

Let $L_1$ and $L_2$ be two families of straight lines defined by: $L_1: (\sin \theta) x + (\cos \theta) y = \sin 2\theta$ $L_2: (\tan \theta) x + y = 2\sin \theta + |\cos 2\theta|$ where $\theta \in \left(0, \frac{\pi}{2}\right)$. Let $d(\theta)$ be the perpendicular distance between $L_1$ and $L_2$. Let $S$ be the number of points in $\left(0, \frac{\pi}{2}\right)$ where $d(\theta)$ is non-differentiable, $M$ be the number of local extrema of $d(\theta)$ in $\left(0, \frac{\pi}{2}\right)$, and $L = \lim_{\theta \to \frac{\pi}{2}^-} d(\theta)$. Find the value of $S + M + L$.

Let $E_p(N)$ denote the exponent of the highest power of prime $p$ dividing $N$. Consider two parallel lines $L_1: Ax + By + C_1 = 0$ and $L_2: Ax + By + C_2 = 0$, where: $A = E_2\left( \sum_{r=0}^{32767} \binom{32767}{r}^2 \right), \quad B = E_3\left( \sum_{r=0}^{6560} \binom{6560}{r}^2 \right)$ $C_1 = E_5\left( \sum_{r=0}^{125} (-1)^{125-r} \binom{125}{r} r^{125} \right), \quad C_2 = E_5\left( \sum_{r=0}^{265} (-1)^{265-r} \binom{265}{r} r^{265} \right)$ Find the distance between the lines $L_1$ and $L_2$.

Consider two lines $L_1$ and $L_2$ in $\mathbb{R}^3$. Line $L_1$ passes through the point with position vector $\vec{a}_1 = \hat{i} + 2\hat{j} - \hat{k}$ and is parallel to the vector $\vec{d}_1 = \vec{u} \times (\vec{v} \times \vec{w})$, where $\vec{u} = \hat{i} + 2\hat{j} - \hat{k}$, $\vec{v} = 2\hat{i} - \hat{j} + 3\hat{k}$, and $\vec{w} = \hat{i} - \hat{j} + \hat{k}$. Line $L_2$ passes through the point with position vector $\vec{a}_2 = 2\hat{i} - \hat{j} + \hat{k}$ and is parallel to the vector $\vec{d}_2 = \vec{x} \times (\vec{y} \times \vec{z})$, where $\vec{x} = \hat{i} - \hat{j}$, $\vec{y} = \hat{i} + \hat{j} + 3\hat{k}$, and $\vec{z} = \hat{i} + \hat{k}$. The shortest distance between $L_1$ and $L_2$ is:

Let $R$ be the region in the $xy$-plane defined by the inequalities $|x - 2y| \le 4$ and $|2x + y| \le 5$. If a point $P(x, y)$ is chosen at random from the region $R$, the probability that $P$ satisfies the inequality $|x + y| \ge 2$ is

Let $C$ be the curve defined by the equation $y = -x^3 + 5x^2 - 7x - 4$ for $x \in [0, 3]$. Let $L$ be the line given by $x - y = 0$. What is the minimum distance between a point on the curve $C$ and the line $L$?

Let $C$ be a curve passing through the point $P$, where $P$ is the point of contact of the tangent to the parabola $y^2 = 4x$ that is perpendicular to the line $x - y = 0$. If the curve $C$ satisfies the differential equation $\frac{dy}{dx} = y - x + 1$, and intersects the line $x = 1 + \ln 2$ at the point $(1 + \ln 2, y_0)$, then the value of $y_0$ is

Let $R$ be the feasible region in the $xy$-plane defined by the system of inequalities: $y \le 3x$ $2y \ge x$ $2x + y \ge 8$ If the set of all possible values of $\alpha$ for which the point $(\alpha, \alpha^2)$ lies in the region $R$ is the interval $[m, n]$, then the value of $m+n$ is:

Let $y = f(x)$ be the solution to the differential equation $\frac{dy}{dx} = \frac{y^2 - 2xy}{x^2}$ satisfying $f(1) = 1$. The general solution can be expressed in the form $1 - \frac{3x}{y} = C x^3$, where $C$ is the constant of integration.

Let the vectors $\vec{u} = C\hat{i} + \hat{j} + \hat{k}$, $\vec{v} = \hat{i} + C\hat{j} + \hat{k}$, and $\vec{w} = \hat{i} + \hat{j} + k\hat{k}$ be coplanar.

If the three lines $L_1: kx + 2y + 1 = 0$, $L_2: x + ky + 1 = 0$, and $L_3: 2x + 2y + \lambda = 0$ are concurrent, find the value of $\lambda$.

Let $L_i: a_i x + b_i y + c_i = 0$ ($i=1,2,3$) be three distinct, non-parallel lines in the $xy$-plane that are concurrent at the point $P(\alpha, \beta)$, where $\alpha \beta \neq 0$. A new system of lines is defined as $L_i': (a_i + k c_i)x + (b_i + m c_i)y + c_i = 0$ for $i=1,2,3$, where $k, m \in \mathbb{R}$. If the system of lines $L_i'$ is inconsistent, which of the following conditions must hold true?

Consider three lines in the $X-Y$ plane parameterized by a real number $x$: $L_1: X - Y = x - \{x\}(1-\{x\})$ $L_2: X + Y = x + \{x\}(1-\{x\})$ $L_3: X - 2Y = [x]$ where $\{x\}$ and $[x]$ denote the fractional part and greatest integer function, respectively. Let $C$ be the locus of the intersection point of $L_1$ and $L_2$ as $x$ varies over $\mathbb{R}$. Let $S$ be the set of all $x \in [0, 20]$ such that $L_1, L_2$, and $L_3$ are concurrent at a point $P$, and the curve $C$ (expressed as $Y = f(X)$) is differentiable at $P$. Find the sum of all elements in $S$.

Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$. Let $N = BA$. The rows of $N$ represent the coefficients $(a_i, b_i, c_i)$ of three lines $L_i: a_i x + b_i y + c_i = 0$ for $i \in \{1, 2, 3\}$. It is given that the lines $L_1, L_2, L_3$ are concurrent at a point $Q$. Let $S$ be the sum of the eigenvalues of the matrix $M = AB$. If a line $L^*$ passes through $Q$ and has a slope equal to $S$, then the equation of $L^*$ is:

A new coordinate system $(X, Y, Z)$ is obtained by shifting the origin of the original system $(x, y, z)$ to a point $S(\alpha, \beta, \gamma)$ without rotating the axes. In this new system, three lines $L_1, L_2, L_3$ are given by the equations: $L_1: \frac{X-2a}{a} = \frac{Y-2b}{b} = \frac{Z-2c}{c}$ $L_2: \frac{X - (a+2b)}{b} = \frac{Y - (b+2c)}{c} = \frac{Z - (c+2a)}{a}$ $L_3: \frac{X - a + c}{c} = \frac{Y - b + a}{a} = \frac{Z - c + b}{b}$ where $a, b, c$ are distinct non-zero real numbers. It is given that the lines $L_1, L_2, L_3$ are coplanar. Let $\Pi$ be the plane containing these three lines. If $\Pi$ intersects the original coordinate axes at points $A, B,$ and $C$, and the volume of the tetrahedron $OABC$ (where $O$ is the original origin) is $36$ cubic units, then the value of $|\alpha + \beta + \gamma|$ is:

Let $A(x_A, y_A)$, $B(x_B, y_B)$, and $C(x_C, y_C)$ be the vertices of a triangle given in counter-clockwise order, with an area of $18$ sq. units. Point $P$ lies on the line segment $BC$ such that $BP:PC = 1:2$. Point $Q$ lies on the line $CA$ outside the segment $CA$ such that $CQ:QA = 2:1$. Point $R$ lies on the line segment $AB$ such that $AR:RB = 1:2$. Find the value of the determinant $\Delta = \begin{vmatrix} x_P & x_Q & x_R \\ y_P & y_Q & y_R \\ 1 & 1 & 1 \end{vmatrix}$.

Let $A(\vec{a})$ and $B(\vec{b})$ be two distinct points such that $\vec{a}$ and $\vec{b}$ are non-collinear vectors. A point $P(\vec{p})$ divides the line segment $AB$ strictly internally. If $\vec{p} = \lambda \vec{a} + \mu \vec{b}$, where $\lambda = \log_2(4\sin x)$ and $\mu = \log_2(2\cos x)$, find the sum of all possible values of $x$ in the interval $[0, 2\pi]$.

Let $E$ be the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ and $S(3,0)$ be its focus. A variable point $P$ lies on $E$. The point $Q$ divides the line segment $SP$ internally in the ratio $2:1$ (i.e., $SQ:QP = 2:1$). Let $C$ be the locus of $Q$. If a line $L$ passing through the origin is tangent to $C$, then the square of the slope of $L$ is:

A sequence of points $P_1, P_2, P_3, \dots$ on the X-axis is constructed such that $P_1 \equiv (0,0)$ and $P_2 \equiv (10,0)$. For every integer $n \ge 3$, the point $P_n$ divides the directed line segment $P_{n-2}P_{n-1}$ externally in the ratio $1:3$. If $x_n$ denotes the abscissa of $P_n$, find $\lim_{n \to \infty} x_n$.

Let $L_1: \frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $L_2: \frac{x}{-1} = \frac{y}{-2} = \frac{z}{-2}$ be two lines. Let $B$ be the line which is the acute angle bisector of $L_1$ and $L_2$. A plane $\Pi$ contains the line $B$ and the point $P(1, -1, 1)$. Let $L_3$ be the line passing through the point $Q(4, 5, 2)$ and perpendicular to the plane $\Pi$. What is the shortest distance between the line $B$ and the line $L_3$?

Let $C_1$ be the curve given by $y = \frac{x^3 - 4x^2 + x + 6}{x - 2}$ and $C_2$ be the line $y = -3$. Let $P$ be the valid point of intersection of $C_1$ and $C_2$. If $L$ is the angle bisector of the tangents to $C_1$ and $C_2$ at $P$ such that $L$ has a positive slope, then the $x$-intercept of $L$ is:

Let $\alpha$ and $\beta$ ($\alpha < \beta$) be the critical points of the function $f(x) = \begin{vmatrix} x & 1 & 0 \\ 0 & x & 1 \\ 72x & 21x & 2x \end{vmatrix}$. Consider two lines $L_1: \beta x - \alpha y + 10 = 0$ and $L_2: (\alpha \beta)x + (\beta^2 - \alpha^2 - 2)y + 26 = 0$. Let $m$ be the slope of the acute angle bisector of $L_1$ and $L_2$. Let $M = \begin{pmatrix} 8m & 2 \\ -2 & 8m \end{pmatrix}$. If the characteristic equation of $M$ is $\lambda^2 - p\lambda + q = 0$, find the value of $p+q$.

Let $L_1: x - y = 0$ and $L_2: x + 7y - 8 = 0$ be two lines in the $xy$-plane. Let $B_1(x,y) = 0$ be the acute angle bisector of $L_1$ and $L_2$, normalized such that the coefficient of $x$ is $1$. Let $B_2(x,y) = 0$ be the obtuse angle bisector of $L_1$ and $L_2$, normalized such that the coefficient of $x$ is $3$. Consider the feasible region $R$ defined by the system of inequalities: $B_1(x,y) \ge 0$ $B_2(x,y) \le 0$ $x \ge 0$ Find the maximum value of the objective function $Z = 4x - y$ on the region $R$.

Consider the points $A(0, 5)$ and $B(27, 8)$ in the $xy$-plane. Let $P$ be a point on the $x$-axis whose abscissa is given by $x = 2\sec^2 t - 4\sec t + 14$ for some $t \in \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$. Find the minimum possible value of $PA + PB$.

Let $P(2, 3, 4)$ be a point in space. Let $\Pi$ be the plane $x + y + z = 3$ and $L$ be the line $\frac{x-1}{1} = \frac{y-1}{-1} = \frac{z-1}{1}$. If $P_1$ is the reflection of $P$ across the plane $\Pi$, and $P_2$ is the reflection of $P_1$ across the line $L$, then the coordinates of $P_2$ are: