If $f(1) = 3,\; f'(1) = 2$, then $\dfrac{d}{dx}\left\{\log\left[f\left(e^x + 2x\right)\right]\right\}$ at $x = 0$ is ______.
The line $L$ is passing through $(1, 2, 3)$. The distance of any point on the line $L$ from the line $\vec{r} = (3\lambda - 1)\hat{i} + (-2\lambda + 3)\hat{j} + (4 + \lambda)\hat{k}$ is constant. Then the line $L$ does not pass through the point ______.
If the area of a parallelogram, whose diagonals are $\hat{i} - \hat{j} + 2\hat{k}$ and $2\hat{i} + 3\hat{j} + \alpha \hat{k}$ is $\dfrac{\sqrt{93}}{2}$ sq. units, then find $\alpha$.
The number of integral values of $p$ for which the vectors $(p + 1)\hat{i} - 3\hat{j} + p\hat{k},\; p\hat{i} + (p + 1)\hat{j} - 3\hat{k}$ and $-3\hat{i} + p\hat{j} + (p + 1)\hat{k}$ are linearly dependent, is ______.
The distance of the plane $\vec{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$ from the origin is ______.
\((\int \frac{\sin 7x}{\cos 9x \cos 2x} dx)\) is equal to
The solution of the equation \((\frac{dy}{dx} = \frac{1}{x+y+1})\) is
The value of
\[ \int_{0}^{\pi} |\sin^3 x| \, dx \]
is ______.
\(( \int_0^1 \log(x + 1) , dx = )\)
If
\[ A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \]
where
\[ a = 7^x,\quad b = 7^{7^x},\quad c = 7^{7^{7^x}} \]
then
\[ \int |A| \, dx \]
is equal to ______.
\[ \int \frac{dx}{\cos x \, (1 + \cos x)} = \; ? \]
The approximate value of
\[ \frac{1}{(2.002)^2} \]
The derivative of
\[ y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots \infty}}} \]
\[ {}^{47}C_{4} + \sum_{j=1}^{5} {}^{(52-j)}C_{3} \]
If \( X \sim B(35, p) \) such that \( 7P(X = 0) = P(X = 1) \), then the value of
\[ \frac{P(X = 15)}{P(X = 20)} \]