The distance between the foci of the ellipse
\[ \frac{(x+2)^2}{9} + \frac{(y-1)^2}{4} = 1 \]
is ______.
If \( \vec{a} = (1,1,-1), \vec{b} = (-1,2,1) \) and \( \vec{c} = (-1,2,-1) \), then
\[ \left| (\vec{a} + \vec{b}) \times (\vec{b} + \vec{c}) \right| \]
The value of
\[ \lim_{x \to \infty} \frac{x^3 \sin\left(\frac{1}{x}\right) - 2x^2}{1 + 3x^2} \]
In an A.P., if the 5th term is \( \frac{1}{7} \) and the 7th term is \( \frac{1}{5} \), then the sum of the first 35 terms is ______.
Let
\[ f(x) + 2f\left(\frac{1}{x}\right) = \frac{1}{x} - 5 \]
Then
\[ \int_{1}^{2} 3f(x)\,dx \]
equals ______.
Let \( f(x) \) be differentiable and
\[ \int_{0}^{t^2} x f(x)\,dx = \frac{1}{2} t^4 \quad \text{for all } t. \]
Then the value of \( f(17) \) is ______.
Let \( f(x) = \sin x - \cos x \). Then the value of
\[ \lim_{x \to \frac{\pi}{2}} \frac{f(x) - f\left(\frac{\pi}{2}\right)}{x - \frac{\pi}{2}} \]
The degree of the differential equation
\[ \left[1 + \left(\frac{dy}{dx}\right)^2 \right]^{3/2} = \frac{d^2 y}{dx^2} \]
