(a) An aqueous solution containing 10 g of optically pure fructose was diluted to 500 mL with water and placed in a polarimeter tube 20 cm long. The measured rotation was $-5.20^{\circ}$. Calculate the specific rotation of fructose. (b) If this solution were mixed with 500 mL of a solution containing 5 g of racemic fructose, what would be the specific rotation of the resulting fructose mixture? What would be its optical purity?

A certain natural product having $[\alpha]_{{D}}+40.3^{\circ}$ was isolated. Two structures have been independently proposed for this compound. Which one do you think is more likely to be correct? Why?

Diltiazem is prescribed to treat hypertension, and simvastatin is a cholesterol-lowering drug. Locate the chirality centers in each.

Which of the following compounds has a stereoisomer that is a meso compound? A 2,4-dibromohexane, B 2,4-dibromopentane, C 2,4-dimethylpentane, D 1,3-dichlorocyclohexane, 1,4-dichlorocyclohexane, F 1,2-dichlorocyclobutane

Identify the chirality centers, if any, in (a) 2-Cyclopentenol and 3-Cyclopentenol (b) 1,1,2-Trimethylcyclobutane and 1,1,3-Trimethylcyclobutane

Muscarine is a poisonous substance present in the mushroom Amanita muscaria. Its structure is represented by the constitution. (a) Including muscarine, how many stereoisomers have this constitution? (b) One of the substituents on the ring of muscarine is trans to the other two. How many of the stereoisomers satisfy this requirement? (c) Muscarine has the configuration 2S, 3R, 5S. Write a structural formula of muscarine showing its correct stereochemistry.

Specify the configuration of the chirality center as R or S in each of the following. (a) (-)-2-Octanol (b) Monosodium L-glutamate (only this stereoisomer is a flavor-enhancing agent)

Define $T: P_{2} \rightarrow R$ by $T(p)=\int_{0}^{1} p(x) d x$. What is the kernel of T?

Find $y^{\prime}.$ $y=\frac{1}{\left(x^{2}-3 x\right)^{2}}$

An example showing that the probability of winning the Maine lottery is 1/5,245,786. Find the probability of winning the indicated lottery. New York Take Five: Select the winning five numbers from $1,2, \ldots, 39.$

Determine whether each of the following statements is true or false, and explain why. Integration by parts should be used to evaluate $\int_{0}^{1} x e^{10 x} d x$.

What are some mechanisms for controlling a genetically modified organism other than making it dependent on synthetic amino acids?

For each of the compounds described by the following names, 1. draw a three-dimensional representation. 2. star (*) each chirality center. 3. draw any planes of symmetry. 4. draw any enantiomer. 5. draw any diastereomers. 6. label each structure you have drawn as chiral or achiral. (a) (S)-2-chlorobutane. (b) (R)-1,1,2-trimethylcyclohexane. (c) (2R,3S)-2,3-dibromohexane. (d) (1R,2R)-1,2-dibromocyclohexane. (e) meso-hexane-3,4-diol, $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}(\mathrm{OH}) \mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{2} \mathrm{CH}_{3}$. (f) $(\pm)$-hexane-3,4-diol

Which of the following bones in the neck is U-shaped?

A. hyoid bone

B. atlas

C. axis

D. coccyx

E. none of these