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Mohamed J.

Experienced Tutor. Math Enrichment at Rochester Math and Science Academy.

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Linear Algebra

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Question:

Solve the following system of equations $$ x+y+z=27$$ $$x+y-2z=9$$ $$2x+3y+z=59$$

Mohamed J.

Answer:

This is equivalent to the matrix system: $$ \begin{bmatrix} 1 &1 &1 \\ 1&1 &-2 \\ 2 &3 &1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 27\\ 9\\59 \end{bmatrix} $$ Using row operations, we get the reduced row echelon matrix $$ \begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0 & 0 &1 \end{bmatrix} \begin{bmatrix} 10\\ 11\\ 6 \end{bmatrix} $$ So the solution to the system is: $$ x=10, y=11 and z=12$

Number Theory

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Question:

Prove that the product of two consecutive integers is even?

Mohamed J.

Answer:

Let the two integers be represented by a,a+1 Since the integers are consecutive, one of them must be even. Suppose that the first integer is even (and the second will be odd). We can re-write the integers as $$ 2k, 2k+1$$ $$ a*(a+1) =2k*(2k+1)$$ $$2k*(2k+1)=2[k*(k+1)]$$ Since, $$k*(k+1)$$ is an integer, a*(a+1) is even. Therefore, the product of two consecutive integers is even.

Statistics

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Question:

Cumulative SAT scores are approximated well by a normal model, $$ N(\mu=1500, \sigma=300)$$ a) What is the probability that a random student will score greater than 1600 on their SAT? b) Given that a student scores in the 90th percentile, what is their SAT score?

Mohamed J.

Answer:

a) First calculate the z-score that corresponds to a 1500 score. Recall that $$ Z=\frac{x-\mu}{ \sigma}$$ $$ Z=\frac{1600-1500}{300} $$ =0.33 Using the standard normal table, the probability of getting a Z-score of 0.33 is 0.63 or 63%. b) Since the student scored in the 90th percentile, their Z-score is 0.90 (90% of students score more than them). Using the standard normal table, Z=1.29 So, $$ Z=\frac{x-1500}{300} $$ =1.29 Solving the above equation for x, we get: $$ x=1887$$ Therefore, the students scored 1887 on their SAT.

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