A certain tennis player makes a successful first serve \( 71 \% \) of the time. Assume that each serve is independent of the others. Suppose she serves 76 times in a match. Complete parts a through d below. a) What are the mean and standard deviation of the number of good first serves expected? The mean is \( \square \). The standard deviation is (Type integers or decimals. Round to two decimal places as needed.)

If \( \frac{1}{\sqrt{m}+\sqrt{m+1}}=\sqrt{m+1}-\sqrt{m} \), determine, without the use of a calculator, the exact value of: \( \frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\ldots+\frac{1}{\sqrt{1680}+\sqrt{1681}} \)

Elercicio \( 1[15 \) puntos] Calcula: \( \begin{array}{lll}\text { a) } \lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x-2} & \text { b) } \lim _{x \rightarrow+\infty}\left(1-\frac{2}{3}\right)^{x} & \text { c) } \lim _{x \rightarrow 3}\left(\frac{\sqrt{x+1}-2}{x-3}\right)\end{array} \)

8. \( 3 \times 4 \times=120 \)

Use the References to access important values if needed for this question. The equilibrium constant, \( K_{\mathrm{c}} \), for the following reaction is 55.6 at 698 K . \[ \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{HI}(\mathrm{g}) \] Calculate the equilibrium concentrations of reactants and product when 0.263 moles of \( \mathrm{H}_{2} \) and 0.263 moles of \( \mathrm{I}_{2} \) are introduced into a 1.00 L vessel at 698 K . \( \left[\mathrm{H}_{2}\right]=\square \mathrm{M} \) \( \left[\mathrm{I}_{2}\right]=\square \mathrm{M} \) \( [\mathrm{HI}]=\square \)

Solve the equation below. \( \log _{2} \sqrt{(x+2)}-\log _{4} x=1 \) If \( \sin A=\frac{5}{13} \) and \( \cos B=\frac{4}{5} \) where A and B are acute angles, determine the values of \( \cos (A-B) \) 5 marks (a)Use binomial theorem to expand \( \sqrt{\frac{3+4 x}{3-4 x}} \) up to the third term, by setting \( x=\frac{1}{64} \) in the expanded form determine the value of \( \sqrt{x} \) (b) The cost of 3 maps, 2 tape measure and 5 plumb bob is ksh. 821 . The cost of 5 maps, 2 tape measure and 7 plumb bob is ksh. 1143 . The cost of 10 maps, 4 tape measure and 5 plumb bob is ksh 1674 . Use Cramer's rule to determine the cost of each type of instrument. (10 marks)