GR0568 GRE Math Test Solutions

I have decided to post some solutions to the GRE Mathematics Test Sample Practice Past Year Papers that are available online. Hope it helps!

The GRE (Graduate Record Examinations) Mathematics Test is a standardized test by the company ETS (Educational Testing Service), and meant for students considering PhD in Math in the United States.

Question 1) In the xy-plane, the curve with parametric equations x=\cos t and y=\sin t, 0\leq t\leq\pi, has length

(A) 3
(B) \pi
(C) 3\pi
(D) \frac{3}{2}
(E) \frac{\pi}{2}

Solution:

Using the integral formula for arc length, \displaystyle s=\int_0^\pi \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\ dt, we obtain

\displaystyle \begin{aligned}\int_0^\pi \sqrt{(-\sin t)^2+(\cos t)^2}\ dt &=\int_0^\pi 1\ dt\\    &= \pi\end{aligned}

Thus, the answer is option (B).
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