The given expression is:
\(\frac{281.(7) + 218.(6i \, \sin(x))}{7 - 3i \, (\cos(x))}\)
We simplify this as:
\(\Rightarrow 281\left(\frac{7 + 6i \, \sin(x)}{7 - 3i \, \cos(x)}\right)\)
Next, we multiply both the numerator and denominator by \(7 + 3i \, \sin(x)\):
\(\Rightarrow 281\left(\frac{49 + 21i (\cos(x) + 2 \sin(x)) + 18 \sin(x) \cos(x)}{49 + 9 \cos^2(x)}\right)\)
This simplifies to:
\(\Rightarrow \left(\frac{49 + 18 \sin(x) \cos(x)}{49 + 9 \cos^2(x)} + i \frac{21 (\cos(x) + 2 \sin(x))}{49 + 9 \cos^2(x)}\right)\)
Hence, we can conclude that:
\(\frac{21 \cos(x) + 2 \sin(x)}{49 + 9 \cos^2(x)} = 0\)
Therefore, the value of \( n \) is 281.
Thus, the value of \( n \) is 281.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
A Complex Number is written in the form
a + ib
where,
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.