$(20$ points$)$ For the following multiple choice questions, choose the correct answer and write

a brief reason for your choice. There $isno$ penalty for incorrect choices. A fully correct

question must contain both the correct choice and correct reasoning.

$(1)$ Let $(t)=(x(t),y(t))be$ a continuous parametrized curve $in{R}^2$ defined $on$ its maximal

domain. $Itis$ impossible for $(t)to$ contain a circle $or$ any loop $if$:

$(A)x(t)=\frac{dy}{dt}(t)$

$(B)y(t)>0$ for all $t$

$(C)x(t)=t^3$

$(D)x(t{)}^2+y(t{)}^2=k$ for a constant $k.$

$(E)$ None $of(A)-(D)$ guarantee $(t)$ won't contain a circle $or$ loop.

Reason:

$(2)$ Let $(t)=(\sqrt[\phantom{2}]{16-t^2},y(t),t)be$ a continuous parametrized curve $in{R}^3$ defined $on$ its

maximal bounded domain. The following choice ensures $(t)$ will never intersect the

$\{z=3\}$ plane:

$(A)y(t)=-3$

$(B)y(t)=e^{-t}$

$(C)y(t)=ln(2-t)$

$(D)$ All $(A)-(C)$

$(E)$ None $of(A)-(C)$

Reason:

$(3)L_1(t)=p_1+t{v}_1$ and $R^3{v}_1*v_2=||v_1|{|}^2{L}_1{L}_{2}P=\{2x+3y+5z=1\}($:$2,3,5$:$)*(p+kv)=1k($:$2,3,5$:$)*v0($:$2,3,5$:$)v0($:$2,3,5$:$)*(p+k^{2}v)=kkL(t)=p+tv,-.$

Then $it$ must $be$ true that:

$(A)($:$2,3,5$:$)*(p+kv)=1$ for some $k$

$(B)($:$2,3,5$:$)*v0$

$(D)($:$2,3,5$:$)v0$

$(D)($:$2,3,5$:$)*(p+k^{2}v)=k$ for some $k$

$(E)$ None $of(A)-(E)L_2(t)=p_2+t{v}_{?}$