Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A. x = nrac{π}{12} + (-1)^{n}rac{π}{12} B. x = nrac{π}{2} + (-1)^{n}rac{π}{12} C. x = nrac{π}{2} + (-1)^{n+1}rac{π}{12} D. x = nrac{π}{2} + (-1)^{n}rac{π}{12} A particle is moving in simple harmonic motion. The displacement of the particle is x and its velocity, v, is given by the equation v^{2} = n^{2}(2kt - x^{2}), where n and k are constants. The particle is initially at x = k. Which function, in terms of time t, could represent the motion of the particle? A. x = k ext{cos}(n t) B. x = k ext{sin}(n t) C. x = 2k ext{cos}(n t) - k D. x = 2k ext{sin}(n t) + k

SSCE HSC Mathematics Extension 1 Integration of sin^2x and cos^2x: Which of the following is a gene

Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Question 11

Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A. x = nrac{π}{12} + (-1)^{n}rac{π}{12} B. x = nrac{π}{2} + (-1)^{n}rac{π... show full transcript

Worked Solution & Example Answer:Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Step 1

Which of the following is a general solution of the equation sin 2x = -rac{1}{2}?

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Answer

To find the general solution for the equation $\sin(2x) = -\frac{1}{2}$ :

Identify the angles: The sine function equals $-\frac{1}{2}$ at specific angles. These angles are typically $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$ in the range from $0$ to $2\pi$ .
General form: The general solutions can be expressed as:
- For $\frac{7\pi}{6}$ : $x = \frac{7\pi}{12} + n\pi$ (since $\sin$ has a period of $\pi$ )
- For $\frac{11\pi}{6}$ : $x = \frac{11\pi}{12} + n\pi$
Combine results: When combining these with the factor $(-1)^{n}$ due to periodicity, we arrive at the most comprehensive general solution format.

Thus, the answer is C: $x = n\frac{\pi}{2} + (-1)^{n} \frac{\pi}{12}$ where $n$ is any integer.

Step 2

Which function, in terms of time t, could represent the motion of the particle?

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Answer

In simple harmonic motion, the general form of displacement can be expressed as:

Understanding displacement and velocity: Given the relation $v^{2} = n^{2}(2kt - x^{2})$ , we recognize that a typical solution is either a sine or cosine function.
Position function: The displacement function of a particle in harmonic motion can be expressed as a sine function:
- Given the form resembles oscillatory motion, we can assume $x(t) = A\sin(nt + \phi)$ where $A$ is the amplitude and $\phi$ is the phase shift.
Evaluate options: Considering the given options:
- A. $x = k\cos(nt)$ is valid but depends on the phase.
- B. $x = k\sin(nt)$ fits well.
- C. $x = 2k\cos(nt) - k$ includes a constant adjustment.
- D. $x = 2k\sin(nt) + k$ does as well, but might need further analysis.

Thus, the most versatile choice in harmonic models, considering amplitude is B: $x = k \sin(nt)$ .

Which of the following is a general solution of the equation sin 2x = - rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Worked Solution & Example Answer:Which of the following is a general solution of the equation sin 2x = - rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Which of the following is a general solution of the equation sin 2x = - rac{1}{2}?

Which function, in terms of time t, could represent the motion of the particle?

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Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Worked Solution & Example Answer:Which of the following is a general solution of the equation sin 2x = -rac{1}{2}? A - HSC - SSCE Mathematics Extension 1 - Question 11 - 2018 - Paper 1

Which of the following is a general solution of the equation sin 2x = -rac{1}{2}?