Let A = {\(\frac{1967+1686 i\,sin\theta}{7-3i\,cos\theta}:\theta\in R\) }. If A contains exactly one positive integer n, then the value of n is ______
Solution and Explanation
Let's write the numbers \( (1967, 1686) \) as factors of 7 and 3:
\( 1967 = 7 \times 281 \) and \( 1686 = 6 \times 562 \)
So, the expression will be as follows:
\[ \frac{281(7) + 281(6i\sin(x))}{7 - 3i(\cos(x))} \]
Thus, we have:
\[ 281\left(\frac{7 + 6i\sin(x)}{7 - 3i\cos(x)}\right) \quad \text{(i)} \]
Now, by multiplying both the numerator and denominator by \( 7 + 3i\sin(x) \), we get:
\[ = 281\left(\frac{49 + 21i(\cos(x) + 2\sin(x)) + 18\sin(x)\cos(x)}{49 + 9\cos^2(x)}\right) \]
Which simplifies to:
\[ \Rightarrow 281\left(\frac{49 + 18\sin(x)\cos(x)}{49 + 9\cos^2(x)} + i \frac{21(\cos(x) + 2\sin(x))}{49 + 9\cos^2(x)}\right) \]
Since we know the expression will be an integer after simplification, we can conclude that the imaginary part of the expression is zero.
Therefore, we can say that:
\[ \left(\frac{21(\cos(x) + 2\sin(x))}{49 + 9\cos^2(x)}\right) = 0 \]
After further simplification, it turns out to be:
\[ \cos(x) = -2\sin(x) \]
Now, substitute the value of \( \cos(x) \) back into the equation, and the equation becomes:
\[ \Rightarrow 281\left(\frac{7 + 6i\sin(x)}{7 + 6i\sin(x)}\right) = 281 \]
Final Answer:
The value of \( n \) is \( \boxed{281} \).
Top JEE Advanced Mathematics Questions
- In a $\triangle$ ABC w ith fixed base BC, the vertex A moves such that cos B + cos C = 4 $ sin^2 \frac{A}{2}$. If a, b and c denote th e lengths of th e sides of th e triangle opposite to the angles A, B and C respectively, then
- Let $P=[a_{ij}]$ be a $3\times3$ matrix and let $Q=[b_{ij}]$ where $b_{ij}=2^{i+j} a_{ij}$ for $1 \le i, j \le $.If the determinant of $P$ is $2$, then the determinant of the matrix $Q$ is
- For the equation $3x^2+px+3=0,p>0,$ if one of the root is square of the other, then p is equal to
- The equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4x-1}$ has
Let \(f(x)=x+log_{e}x−xlog_{e}x,\text{ }x∈(0,∞)\).
- Column 1 contains information about zeros of \(f(x)\), \(f'(x)\) and \(f''(x)\).
- Column 2 contains information about the limiting behavior of \(f(x)\), \(f'(x)\) and \(f''(x)\) at infinity.
- Column 3 contains information about increasing/decreasing nature of \(f(x)\) and \(f'(x)\).
Top JEE Advanced Complex numbers Questions
- For $x \in(0, \pi)$, the equation $\sin x+2 \sin 2 x-\sin 3 x=3$ has
- Let S be the set of all complex numbers z satisfying $\left|z-2+i\right|\ge\sqrt{5}.$ If the complex number $z_0$ is such that $\frac{ 1}{\left|z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{ 1}{\left|z_{0}-1\right|}: z\,\in\,S\right\},$ then the principal argument of $\frac{4-z_{0}-z^{-}_{0}}{z_{0}-z^{-}_{0}+2i}$ is
- Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$ In the set of complex numbers, the number of distinct roots of the equation $\bar{z}-z^2=i\left(\bar{z}+z^2\right)$ is ___
- Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{ z ^2}$ are integers, then which of the following is/are possible value(s) of $|z|$?
- Let $z$ be a complex number with non-zero imaginary part If $\frac{2+3 z+4 z^2}{2-3 z+4 z^2}$ is a real number, then the value of $|z|^2$ is ___ .
Top JEE Advanced Questions
- Yellow light is used in a single slit diffraction experiment with slit width of 0.6 mm. If yellow light is replaced by X-rays, then the observed pattern will reveal
- The size of the image of an object, which is at infinity, as formed by a convex lens of focal length 30 cm is 2 cm. If a concave lens of focal length 20 cm is placed between the convex lens and the image at a distance of 26 cm from the convex lens, calculate the new size of the image.
- A concave lens of glass, refractive index 1.5 has both surfaces of same radius of curvature R. On immersion in a medium of refractive index 1.75, it will behave as a
- The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation $\vec{r}(t)=\alpha t^{3} \hat{i}+\beta t^{2} \hat{j}$ where $\alpha=\frac{10}{3} \,ms ^{-3}, \beta=5\, ms ^{-2}$ and $m =0.1 \,kg$. At $t =1\, s$, which of the following statement (s) is (are) true about the particle?
- In an n-p-n transistor circuit, the collector current is 10 mA. If 90% of the electrons emitted reach the collector
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
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.