Harvard Admission Exam circa 1869

Original doc: link.

Algebra

Problem 1. Reduce the following expression to its simplest form:

$(9a^2{b}^2-4b^4)(a^2-b^2)-(3ab-2b^2)\cdot (3a[a^2+b^2]-2b[b^2+3ab-a^2])b$

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Problem 2. Divide $36a^2+1-64a^4-12a$ by $6a-1-8a^{2.}$

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Problem 3. Explain why when different powers of the same quantity are multiplied together their exponents are added.

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Problem 4. Reduce to one fraction with the lowest possible denominator:

$\frac{3a+2b}{a+b}-\frac{25a^2-b^2}{a^2-b^2}=\frac{a}{2b}$

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Problem 5. Divide $x+y$ by $\frac{x^2+xy}{x-y}$ and reduce the answer to its lowest terms.

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Problem 6. Solve for $x$ in terms of $a$, $b$, and $c$ from

$\frac{a-2x}{b}=\frac{cx-bc}{a}$ ,

and then find $x$ when $a=2$, $b=-1$, $c=3$

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Problem 7. A man bought a watch, a chain, and a locket for $216. The watch and locket together cost three times as much as the chain, and the chain and locket together cost half as much as the watch. Find the price of each item.

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Problem 8. Solve the equation $\frac{5x}{x+12}-\frac{8-3x}{3x-1}=1.$ .

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Problem 9. Find ${(a-b)}^6$ and ${(xy-\frac{x^2}{2y})}^6$ by the Binomial Theorem.

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