If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at 440^th position in this arrangement is:

If $ \tan(\theta - \phi) = \frac{3}{4} $ and $ \theta + \phi = \frac{\pi}{2} $, then the value of \[ \tan \left( \frac{\pi}{4} - \theta \right) + \tan \left( \frac{\pi}{4} - \phi \right) \] is:

If the tangent to the curve $ y = ax^2 + bx + c $ at the point $ (1,1) $ is parallel to the x-axis, and the curve passes through the point $ (2,3) $, then find the value of $ a + b + c $:

Let \[ f(x) = x^4 - 4x^3 + 6x^2 + k \] \text{If $f(x) \geq 0$ for all real $x$, then the value of $k$ is:}

Let $ f(\theta) = \begin{vmatrix} 1 & \cos \theta & -1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1 \end{vmatrix} $. Suppose $ A $ and $ B $ are respectively the maximum and minimum values of $ f(\theta) $. Then $ (A, B) $ is equal to:

A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 88 square meters, what is the width of the garden?

What is the distance between the points $(3, -2)$ and $(-1, 4)$?

If $A$ is a square matrix of order 3 such that $|A| = 2$, then $|\operatorname{adj}(\operatorname{adj}(A))|$ is:

A pair of linear equations $ a_1 x + b_1 y + c_1 = 0 $ and $ a_2 x + b_2 y + c_2 = 0 $ is such that $ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} $, then they are:

Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to:

If the components of $ \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} $ along and perpendicular to $ \vec{b} = 3\hat{i} + \hat{j} - \hat{k} $ respectively are $ \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) $ and $ \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is equal to:

Evaluate the integral: \[ \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx \]