Application of Derivatives - Class 12

A family of curves $C_1$ satisfies the differential equation $x dy - \lambda y dx = 0$, where $\lambda = \frac{(\alpha-2)(\alpha-5)^2 (\alpha-8)^4}{\alpha+3}$ and $\alpha \in \mathbb{R} \setminus \{-3\}$. Let $C_2$ be the family of orthogonal trajectories to $C_1$. If every real, non-degenerate curve in $C_2$ is unbounded, find the sum of all positive integers $\alpha$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_{x}^{2x} t \sin^{-1}(\sin t) \, dt$ Find the value of $f''\left(\frac{5\pi}{6}\right)$.

Let $\vec{a}(t)$ and $\vec{b}(t)$ be twice differentiable vector-valued functions such that at $t=1$: $\vec{a}(1) = \vec{b}(1) = 2\hat{i} + \hat{j} + \hat{k}$ $\vec{a}'(1) = 2\hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{b}'(1) = \hat{j} + 2\hat{k}$ $\vec{a}''(1) = 2\hat{j} + 6\hat{k}, \quad \vec{b}''(1) = \vec{0}$ Let $\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$. Evaluate the limit: $\lim_{t \to 1} \frac{|\vec{a}(t) \times \vec{b}(t)|}{|[\vec{a}(t), \vec{b}(t), \vec{c}]|}$ (where $[\vec{x}, \vec{y}, \vec{z}]$ denotes the scalar triple product).

Let $S = \{z \in \mathbb{C} : \text{Im}(z^2) = 32 \text{ and } |z + 4 + 4i| \le 4\sqrt{2}\}$. A particle $P$ representing the complex number $z = x+iy$ moves along the locus $S$ such that its imaginary part increases at a constant rate of $4$ units/sec. What is the rate of change of $|z|^2$ at the instant when $|\text{Re}(z)| = 2$?

Let the curve $C$ be defined by $y = f(x)$, where $f(x) = \begin{vmatrix} \sin(\pi x) & \cos(\pi x) & 2e^{x-1} \\ -\cos(\pi x) & \sin(\pi x) & \ln(x) \\ -2e^{x-1} & -\ln(x) & \tan(\pi x) \end{vmatrix}$ Let $N(x)$ be the function representing the normal line to the curve $C$ at $x = 1$. Find the value of $N(1 + 3\pi)$.

The sum of all values of $t \in (0, 20\pi]$ for which the normal to the curve given parametrically by $x = \cos t + t \sin t$ and $y = \sin t - t \cos t$ passes through the point $(1, -1)$ is:

A plane is tangent to the surface $S: z = 4 - x^2 - y^2$ at a point $P(x_0, y_0, z_0)$ in the first octant ($x_0 > 0, y_0 > 0, z_0 > 0$). The tangent plane intersects the coordinate axes at points $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$. If $x_0 = y_0$ and the vector $\vec{v} = \frac{1}{a}\hat{i} + \frac{1}{b}\hat{j} + \frac{1}{c}\hat{k}$ has a magnitude of $\frac{1}{2}$, what is the volume of the tetrahedron $OABC$ (where $O$ is the origin)?

Let $f: [1, \infty) \to \mathbb{R}$ be a differentiable function satisfying the differential equation $x \frac{dy}{dx} - y = \sqrt{x^2 - y^2}$ with the initial condition $f(1) = 0$. The value of $f(e^\pi)$ is

Let $f(x) = \int_{0}^{x} (4t + 5\text{sgn}(t-1)) \, dt$, where $\text{sgn}(\cdot)$ denotes the signum function. Let $m_1$ and $m_2$ be the left-hand and right-hand derivatives of $f(x)$ at $x=1$ respectively. Let $L_1$ and $L_2$ be the lines passing through the point $(1, f(1))$ with slopes $-\frac{1}{m_1}$ and $-\frac{1}{m_2}$ respectively. If $x_1$ and $x_2$ are the x-intercepts of $L_1$ and $L_2$, find the value of $x_1 x_2$.

Let $f(x) = \begin{vmatrix} x^2-4x & |x-2| & 1 \\ -3 & 1 & 1 \\ 0 & 2 & 1 \end{vmatrix}$. Consider the application of Rolle's Theorem for the function $f(x)$ on the interval $[0, 4]$. If Rolle's Theorem is applicable, let $S$ be the sum of all valid values of $c \in (0, 4)$ such that $f'(c) = 0$. If Rolle's Theorem is not applicable, let $S = -1$. Find the value of $S$.

Let $f(x) = \frac{\{x\} (x^2 - 5x + 8)}{x}$ for $x \in (0, 5)$, where $\{x\}$ denotes the fractional part of $x$. Find the number of solutions to the equation $x f'(x) + f(x) = 0$ in the interval $(0, 5)$.

Let $A$ be a $3 \times 3$ matrix with real entries such that all its eigenvalues are real. Let $S = \{x \in \mathbb{R} : \text{tr}(\text{adj}(xI - A)) = 0\}$. Which of the following statements is \textbf{INCORRECT}?

Let $S = \min_{t>0} \left( t + \frac{25}{t} \right)$ and $P = 3 \cdot \min_{x,y,z>0} \frac{(x+y)(y+z)(z+x)}{xyz}$. Let $\alpha, \beta$ (with $\alpha < \beta$) be the roots of the equation $x^2 - Sx + P = 0$. Consider the function $f(x) = \frac{x^2 - Sx + P}{x - c}$, where $c \in (\alpha, \beta)$. Let $A$ be the set of all values of $c \in (\alpha, \beta)$ for which the equation $f'(x) = 0$ has at least one real root in the interval $(\alpha, \beta)$. Which of the following is true?

Let $A(x) = \begin{pmatrix} x^2+x & x & 1 \\ 2x & x^2 & 1 \\ 1 & 1 & 0 \end{pmatrix}$ and $f(x) = \det(A(x))$. By Lagrange's Mean Value Theorem applied to $f(x)$ on the interval $[0, 2]$, there exists a point $c \in (0, 2)$ such that the tangent to the curve $y = f(x)$ at $x=c$ is parallel to the chord joining the endpoints of the interval. If $M = \text{adj}(A(c))$, which of the following matrix equations is satisfied by $M$?

Let $f(x) = x \sin\left(\frac{\pi x}{2}\right)$ for $x \in [0, 1]$. By Lagrange's Mean Value Theorem, there exists $c \in (0, 1)$ such that the tangent $T$ to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the endpoints of the curve.

A curve $C$ in the complex plane is defined by the locus of $z = x + iy$ satisfying $|z - 5| - |z + 5| = 8$.

If a normal to the curve $C$ is parallel to $T$, then the equation of this normal is:

Let $f(x) = \sum_{k=1}^{8} \frac{x^2}{(k+[x])(k+1+[x])}$, where $[x]$ denotes the greatest integer less than or equal to $x$. Find the number of points $c \in (1, 3)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the chord joining the points $(1, f(1))$ and $(3, f(3))$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function satisfying the differential equation $x f'(x) - 2f(x) = 2x^4$ for all $x \in \mathbb{R}$. It is given that the application of Lagrange's Mean Value Theorem to $f(x)$ on the interval $[-2, 2]$ yields exactly three distinct points in the open interval $(-2, 2)$ where the tangent to the curve is parallel to the secant line joining the endpoints. If $S$ is the set of all possible integer values of $f''(0)$, find the sum of all elements in $S$.

Let $\alpha, \beta \in \mathbb{C}$ be the roots of the quadratic equation $z^2 - (4-2i)z + (4-3i) = 0$. Consider the complex-valued function $f(t) = t^3 - (\alpha+\beta)t^2 + \alpha\beta t$ defined for a real parameter $t \in [0, 2]$. The function $f(t)$ traces a curve in the complex plane. If the tangent vector to this curve at $t = c$ (where $0 < c < 2$) is parallel to the chord joining the endpoints $f(0)$ and $f(2)$, find the value of $15c$.

Let $f(x) = |x^2 - 4|^{\ln |x^2 - 4|}$ for $x \in \mathbb{R} \setminus \{-2, 2\}$. The set of all values of $x$ for which $f(x)$ is strictly increasing is:

Let $f(x) = \int_0^x (t^2 - 3t + 2) \,dt$ and $g(x) = x^2 - 6x + 8$. If $F(x) = f(g(x))$, then the set of all values of $x$ for which $F(x)$ is strictly increasing is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function. For any $x \in \mathbb{R}$, the signed volume of the parallelepiped formed by the vectors $\vec{u} = e^{x^2}\hat{i} - 2xe^{x^2}\hat{j}$, $\vec{v} = f(x)\hat{i} + f'(x)\hat{j}$, and $\vec{w} = \hat{k}$ is $4x^3$. If $f(x)$ is strictly decreasing for all $x \in (0, \infty)$, then the minimum possible value of $f(0)$ is:

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $g(x) = (x^2 - 2x + 3)e^{-x}$. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined by $f(x) = \int_0^x (g(t) + t g'(t)) dt$. Which of the following correctly describes the intervals of monotonicity for $f(x)$?

Let $f(x) = \left| \det \begin{pmatrix} x^2 & x & 1 \\ 1 & x & x^2 \\ 1 & 1 & 1 \end{pmatrix} \right|$. Let $S$ be the set of all points where $f(x)$ attains a local extremum, $D$ be the set of all points where $f(x)$ is non-differentiable, and $C$ be the set of all stationary points of $f(x)$. If $s = \sum_{x \in S} x$, $d = \sum_{x \in D} x$, and $c = \sum_{x \in C} x$, find the value of $2s - d + 4c$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = x^7 + 3x^5 + 8x^3 + 2$. Let $g(x)$ be the inverse function of $f(x)$. If $H(x) = g(x^3 - 3x^2 + 4)$, find the sum of the squares of all critical points of $H(x)$.

Let $f(x)$ be a differentiable function such that $f'(x) = (x-2)^2 (x-3)(x-4)$. Let $S$ be the set of all points of local extrema of $f(x)$. If $A$ is the area of the region bounded by the curve $y = f'(x)$, the x-axis, and the vertical lines $x = \min(S)$ and $x = \max(S)$, then find the value of $60A$.

Let a function $f(x)$ be defined as: $f(x) = \lim_{N \to \infty} \sum_{n=1}^{N} (2n-1) \left( \frac{x^2-3x}{x^2+3} \right)^n$ Let $S$ be the set of all critical points of $f(x)$. Find the sum of the elements of $S$.

Let $\vec{a} = t \hat{i} + t^2 \hat{j} + \hat{k}$, $\vec{b} = 2 \hat{i} + 4 \hat{j} + \hat{k}$, and $\vec{c} = 4 \hat{i} + 16 \hat{j} + \hat{k}$ be three vectors parameterized by $t \in \mathbb{R}$. Let $\vec{p} = \vec{a} \times \vec{b}$, $\vec{q} = \vec{b} \times \vec{c}$, and $\vec{r} = \vec{c} \times \vec{a}$. For $t \in (0, 6)$ such that $[\vec{p} \, \vec{q} \, \vec{r}] \neq 0$, let $\vec{p}', \vec{q}', \vec{r}'$ be the reciprocal system of vectors for $\vec{p}, \vec{q}, \vec{r}$. If $f(t)$ is the continuous extension of the function $g(t) = \frac{1}{2 \sqrt{[\vec{p}' \, \vec{q}' \, \vec{r}']}}$ over the interval $(0, 6)$, find the sum of all values of $t$ at which $f(t)$ attains a local extremum.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as $f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) - |x|$ Let $M$ be the global maximum value of $f(x)$. Then $M$ is equal to:

Let $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 2x & 4x^2 & 8x^3 \\ 3x & 9x^2 & 27x^3 \end{vmatrix}$. Define a function $g(x) = f'(x) + 90x^4 - 240x^3$. If $x = \alpha$ is the unique point of local minimum for $g(x)$, find the value of $\alpha$.

Let $f(x) = \int_0^x \left( \lim_{n \to \infty} \frac{t^{2n} - (x-2)^{2n}}{t^{2n} + (x-2)^{2n}} \right) dt$ for $x \ge 0$. Find the area bounded by the curve $y = f(x)$ and the x-axis for $x \in [0, 4]$.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \int_0^x (\{t\}^3 - \{t\}) dt$ where $\{t\}$ denotes the fractional part of $t$. Which of the following statements is CORRECT?

Let $f(x) = \begin{vmatrix} 2 & 2 & x+1 \\ 2 & x+1 & 2 \\ x+1 & 2 & 2 \end{vmatrix}$. If $x_1$ is the point of local maximum and $x_2$ is the point of local minimum of $f(x)$, then the value of $2x_1 - x_2$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = x^3 - 3x^2 + 6x + k \tan^{-1}(x)$ where $k \in \mathbb{Z}$. Find the minimum integral value of $k$ such that $f(x)$ is an injective function.

Find the global maximum value of the function $f(x) = \sin^{-1}(\sin x) + \ln|x - \sqrt{3}|$ on the interval $[0, \pi]$.

Let $F(x) = \int_{0}^{x} (t^2 - 5t + 4) \ln(3 - t) \, dt$ be defined on its maximal domain $(-\infty, 3)$. If $x_0$ is the point of local maximum of $F(x)$, then the value of $F(x_0)$ is:

Let $f: \left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right] \to \mathbb{R}$ be a continuous function satisfying the differential equation $\sin x \frac{dy}{dx} + y \cos x = x \cos x$ for all $x \in \left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right] \setminus \{0\}$. If $M$ and $m$ are the global maximum and global minimum values of $f(x)$ respectively, then $M - m$ is equal to:

Let $f(x) = \begin{vmatrix} 0 & x^2-4x+1 & x^2-4x+1 & x^2-4x+1 \\ -x^2+4x-1 & 0 & 2x^2-8x+2 & 4x^2-16x+4 \\ -x^2+4x-1 & -2x^2+8x-2 & 0 & x^2-4x+1 \\ -x^2+4x-1 & -4x^2+16x-4 & -x^2+4x-1 & 0 \end{vmatrix}$. Find the global maximum value of $f(x)$ on the interval $[0, 5]$.

Let $S$ be the set of all values of the parameter $c$ such that the solution $y = f(x)$ of the differential equation $\frac{dy}{dx} - y = \cos x - \sin x$ satisfying $y(0) = 2c + 1$ has exactly two points of inflection in the interval $(0, \pi)$. Then $S$ is equal to:

Let $A(t) = \begin{pmatrix} t-3 & 1 & 1 \\ 1 & t-3 & 1 \\ 1 & 1 & t-3 \end{pmatrix}$ and $f(x) = \int_0^x (\det(A(t)))^{1/3} dt$. If $S$ is the sum of the abscissae of all the points of inflection of the curve $y = f(x)$, find the value of $S$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \begin{cases} \frac{1 + (x-1)^2 + \frac{(x-1)^4}{2} - e^{(x-1)^2}}{(x-1)^2} & \text{if } x < 1 \\ x^5 - 6x^4 + 14x^3 - 16x^2 + 9x - 2 & \text{if } x \ge 1 \end{cases}$ Let $S$ be the set of all points of inflection of the curve $y = f(x)$. Find the sum of the elements of $S$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_{0}^{x} (t^2 - 4)^3 \left( \int_{-t}^{t} \left( u^4 \ln\left(\frac{3+\sin u}{3-\sin u}\right) e^{|u|} + 55u^4 - 330u^2 \right) du \right) dt$ Find the sum of the squares of the abscissae of all points of inflection of the curve $y = f(x)$.

Let $F(x) = \int_{0}^{x^2} \frac{e^{t-4} - \cos(t-4)}{t^2 - 16} \, dt$. Using differentials, the approximate value of $F(2.04) - F(2)$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that for $x \neq 1$, $f(x) = \frac{20x^{21} - 21x^{20} + 1}{(1-x)^2}$ and $f(1) = \lim_{x \to 1} f(x)$. Using the method of differentials, the approximate value of $f(1.005)$ is:

Let $y = f(x)$ be a differentiable function satisfying the differential equation $x \frac{dy}{dx} + y = x^2 y^2$ for $x > 0$, with the initial condition $f(1) = -\frac{1}{3}$. Using differentials, the approximate value of $f(2.032)$ is:

Let $\vec{u}, \vec{v}, \vec{w}$ be three unit vectors such that the angle between any two of them is $\frac{\pi}{3}$. Define three new vectors using the vector triple product: $\vec{a} = \vec{u} \times (\vec{v} \times \vec{w}), \quad \vec{b} = \vec{v} \times (\vec{w} \times \vec{u}), \quad \vec{c} = \vec{w} \times (\vec{u} \times \vec{v})$ A variable parallelepiped is formed by the coterminous edges $\vec{A} = \vec{u} + x\vec{a}$, $\vec{B} = \vec{v} + x\vec{b}$, and $\vec{C} = \vec{w} + x\vec{c}$, where $x$ is a scalar parameter. Let $\Delta V$ be the exact change in the volume of this parallelepiped as $x$ changes from $0$ to a small value $\epsilon > 0$. If $dV$ is the estimated change in volume using the first-order differential approximation at $x=0$, find the value of $\Delta V - dV$.

Let $z_1, z_2, z_3$ be the roots of the equation $z^3 - 1 = 0$ such that $\text{Im}(z_1) = 0$ and $\text{Im}(z_2) > \text{Im}(z_3)$. Consider the complex-valued function $S(\theta) = 2z_1 + z_2 e^{i\theta} + z_3 e^{i2\theta}$ Using differentials, the approximate value of $|S(0.02)|$ is:

Let $L_1$ be the line given by $\frac{x-1}{1} = \frac{y-1}{2} = \frac{z-1}{2}$ and $L_2(t)$ be the family of lines given by $\frac{x - (3t - t^2/2)}{2} = \frac{y - 2|t-2|}{1} = \frac{z + 8}{-2}$, where $t \in \mathbb{R}$. If $d(t)$ represents the shortest distance between the lines $L_1$ and $L_2(t)$, then the minimum value of $d(t)$ is:

Consider a right circular cylinder with base radius $r(x) = x^2 - 9x + 18$ and height $h(x) = e^{-x}$. The volume $V(x)$ is defined only for those $x \in \mathbb{R}$ where the cylinder has strictly positive dimensions. Let $S$ be the set of all $x$ where $V(x)$ attains a local maximum. Find the sum of the elements in $S$.

Let $S$ be the set of all points $(a, b)$ lying on the curve $a^2 + b^2 - 6a - 8b = 0$ such that the function $f(x) = \frac{x^3}{3} - ax^2 + (a^2 - b^2 + 17b - 72)x + 2024$ is strictly increasing on $\mathbb{R}$. Find the maximum value of $a+b$ for $(a, b) \in S$.

Let $y = f(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} = \frac{y}{x} + 3 - \left[ \frac{y}{x} \right], \quad x \ge 1$ with the initial condition $f(1) = 0$, where $[ \cdot ]$ denotes the greatest integer function. Let $S$ be the set of all points $x > 1$ where $f(x)$ is not differentiable. Which of the following represents the product of all elements in $S$?