= Solution
The <helium mass fraction> is $Y=1-X-Z$. For the two compositions, $Y_1=0.28$ and $Y_2=0.18$. The <mean molecular weight> is therefore
$$
\mu_1^{-1}=1.62,\quad \mu_1=\frac{50}{81}\simeq0.6173,\qquad
\mu_2^{-1}=1.745,\quad \mu_2=\frac{200}{349}\simeq0.5731.
$$
At fixed mass and with the same prescribed opacity coefficient, part (a) gives $T_e\propto\mu^{1/2}R^{1/8}$ and $L\propto\mu^2R^{5/2}$. Eliminating radius gives
$$
\boxed{L\propto\mu^{-8}T_e^{20},\qquad
T_e\big|_L\propto\mu^{2/5}.}
$$
This is the <composition shift of a convective pre-main-sequence track>. The higher-<helium mass fraction> case $X_1=0.7$ is hotter at a given <luminosity>, lying to the left of the $X_2=0.8$ case in a <Hertzsprung-Russell diagram>. Its fixed-luminosity temperature ratio is $(349/324)^{2/5}\simeq1.030$.
Both stars contract, so both $L$ and $T_e$ decrease in this particular opacity model. Their nearly vertical <Hayashi tracks> therefore run \b[downward and slightly to the right], with arrows in that direction:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-63-premain-sequence.png]
{title=Composition-dependent pre-main-sequence tracks with contraction arrows}
The plotted <luminosity> and <effective temperature> are normalized to arbitrary common references; no opacity normalization or absolute age was specified. The comparison holds the given opacity coefficient fixed.
These paths cease to apply when their assumptions fail. <Deuterium-burning energy-generation rate> can temporarily slow contraction; sustained <hydrogen burning> ends the purely gravitational phase. Formation of a <radiative core> breaks the fully convective adiabat and can lead to a <Henyey track> toward hotter <effective temperature>. Partial <ionization>, a changing atmospheric <opacity>, and <electron degeneracy pressure> alter the <equation of state> or boundary relation. <Accretion luminosity>, changing mass, and magnetic inhibition of <convection> can also modify a young star's path.
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