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Nauman S.
Engineering graduate from the University of Nottingham
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Calculus
TutorMe
Question:

Find the definite integral $$\int_{0}^{\infty}cos(x)e^{-x}dx$$

Nauman S.

The successful answer to this question requires knowledge of the technique of integration by parts: $$\int u \frac {dv}{dx} dx = uv - \int v \frac {du}{dx} dx$$ Thereby, the solution is a simple matter of performing integration by parts twice. The working is thus: 1. Let the original integral be denoted as $$I$$, i.e. $$I=\int_{0}^{\infty}cos(x)e^{-x}dx$$. Then taking integration by parts once, we have: 2. $$I = [-cos(x)e^{-x}]_{0}^{\infty} - \int_{0}^{\infty}sin(x)e^{-x}dx$$ $$= [0-(-1)]- \int_{0}^{\infty}sin(x)e^{-x}dx$$ 3. Taking integration by parts a second time, we have: $$I = 1 - ([-sin(x)e^{-x}]_{0}^{\infty} + \int_{0}^{\infty}cos(x)e^{-x}dx)$$ $$= 1 - [0-0] - \int_{0}^{\infty}cos(x)e^{-x}dx$$ $$= 1 - I$$ (since the 3rd term in the previous expression is equal to the original integral $$I$$) 4. Finally we have: $$2I = 1$$ $$\therefore I = \frac 12$$

SAT
TutorMe
Question:

Solve the following SAT: Writing question. Decide whether there is an error in the sentence at the underlined word/s and, if so, choose what the word/s should be replaced with. Otherwise, choose 'NO CHANGE': $$\textbf{The United Nations is an international organization that} \space \underline{take} \space \textbf {action to}$$ $$\textbf{resolve numerous serious issues confronting humanity in the 21st century.}$$ $$\textbf {(A) NO CHANGE}$$ $$\textbf {(B) have taken}$$ $$\textbf {(C) takes}$$ $$\textbf {(D) are taking}$$

Nauman S.

The answer is (C) Explanation: The first step to answer this question is to look at the underlined verb: $$\underline{take}$$. In this context, the verb is in plural form, and in the present tense. Is this the correct form of the verb for this sentence, or should we choose the verb in a different form from the last three choices? Consider this: 'Who is taking the action? Is it a singular entity, or many entities?' At first glance, one might think that the subject is plural, because it is called the $$'\textbf{United Nation}s'$$, with the $$s$$ in the end. However, the word that immediately follows is $$' \textbf{is}'$$, rather than $$' \textbf{are}'$$, which shows that the United Nations is, in fact, a singular entity. This means that the current underlined verb is definitely wrong. The verb needs to be used in its singular form. Therefore, choice (A) is not the right answer. From the other three choices, choices (B) and (D) are, again, plural forms of the verb 'take', with (B) being in the present perfect tense, while (D) is in the present continuous tense. Therefore, neither of these is the right choice. The only choice that is left is (C), which is the singular form of the verb in the present tense. Therefore, (C) is the correct answer.

SAT II Chemistry
TutorMe
Question:

Solve the following SAT II: Chemistry question. Choose the correct option from the choices provided. What is the approximate percentage composition by mass of the element carbon in the compound C_6 H_12 O_6? (A) 25% (B) 40% (C) 60% (D) 50%

Nauman S.

The answer is (B). Explanation: The question is essentially asking, if you are given 1 mole of the compound, what percentage of the total mass of 1 mole of the compound is made up of Carbon atoms? To answer this question, we must first find the mass of 1 mole of the compound. As can be seen, one molecule of the compound has 6 atoms of Carbon, 12 atoms of Hydrogen, and 6 atoms of Oxygen. The mass of 1 mole of each of these atoms is 12 g, 1 g and 16 g, respectively. Therefore, the total mass of 1 mole of the compound is given by: 6x12 + 12x1 + 6x16 = 180 g Then, we must find what percentage of this total mass is made up of Carbon atoms. The mass of the 6 moles of Carbon atoms is given by: 6x12 = 72 g Finally, the percentage composition of Carbon is given by: 72/180 x 100% = 40% Therefore, the correct answer is (B).

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