Question:

Let A = {1967+1686isinθ73icosθ:θR\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 ______

Updated On: Oct 9, 2024
Hide Solution
collegedunia
Verified By Collegedunia

Correct Answer: 281

Solution and Explanation

281.(7)+218.(6isin(x)73i.(cosx)\frac{281.(7)+218.(6i\,sin\,(x)}{7-3i.(cos\,x)}

281(7+6isinx73icosx)\Rightarrow 281(\frac{7+6i\,sin\,x}{7-3i\,cos\,x})

Multiplying both the numerator and denominator by 7+3isinx7+3i\,sin\,x,

281(49+21i(cosx+2sinx)+18sinxcosx49+9cos2x)\Rightarrow 281(\frac{49+21i(cos\,x+2\,sin\,x)+18\,sin\,x\,cos\,x}{49+9cos^2x})

(49+18sinxcosx49+9cos2x+i21(cosx+2sinx)49+9cos2x)\Rightarrow (\frac{49+18\,sin\,x\,cos\,x}{49+9\,cos^2x}+i\frac{21(cos\,x+2\,sin\,x)}{49+9cos^2x})

Hence we can say that,

21cosx+2sinx49+9cos2x=0\frac{21\,cos\,x+2\,sin\,x}{49+9\,cos^2\,x}=0

Hence, the value of n will be 281.

Was this answer helpful?
2
1

Top Questions on complex numbers

View More Questions

Questions Asked in JEE Advanced exam

View More Questions

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.