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Varun S.

Undergraduate at IIT Roorkee

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Pre-Calculus

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Question:

Find $\sin 2\theta$ if $\sin\theta=3/5$ and $\theta$ is in Quadrant II.

Varun S.

Answer:

We have that $\cos^2 \theta=1-\sin^2 \theta=1-9/25=16/25$, so $\cos\theta=-4/5$ since $\cos\theta<0$ in Quadrant II. Therefore $\sin 2\theta=2\sin\theta \cos\theta=2(3/5)(-4/5)=-24/25$.

Finance

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Question:

You plan to retire 33 years from now. You expect that you will live 27 years after retiring. You want to have enough money upon reaching retirement age to withdraw $180,000 from the account at the beginning of each year you expect to live, and yet still have $2,500,000 left in the account at the time of your expected death (60 years from now). You plan to accumulate the retirement fund by making equal annual deposits at the end of each year for the next 33 years. You expect that you will be able to earn 12% per year on your deposits. However, you only expect to earn 6% per year on your investment after you retire since you will choose to place the money in less risky investments. What equal annual deposits must you make each year to reach your retirement goal?

Varun S.

Answer:

You must solve this problem in two steps. First, calculate the PV at the time of retirement of the amount needed to give you the annuity and remaining sum wanted. Second, calculate the payment necessary each year over the period from now until retirement to generate the goal. n = 27 i = 6 FV = 2500000 PMT = 180000 solve for PV (answer: = $3,038,989.79) (make sure you are in begin mode) n = 33 i = 12 FV = 3038989.79 solve for PMT (answer: = $8,874.79) (make sure you are in end mode)

Algebra

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Question:

If x <2, simplify |x - 2| - 4|-6|

Varun S.

Answer:

This is a question of the modulus function Given the expression |x - 2| - 4|-6| If x < ;2 then x - 2 < 2 and if x - 2 < 2 the |x - 2| = -(x - 2). Substitute |x - 2| by -(x - 2) and |-6| by 6 . |x - 2| - 4|-6| = -(x - 2) -4(6) = -x -22

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