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Questions in mathematics

[Done] What is the exact value of [tex]$\cos \frac{7 \pi}{12}$[/tex]? A. [tex]$\frac{\sqrt{6}-\sqrt{2}}{4}$[/tex] B. [tex]$\frac{\sqrt{6}+\sqrt{2}}{4}$[/tex] C. [tex]$\frac{2 \sqrt{2}-\sqrt{6}}{4}$[/tex] D. [tex]$\frac{\sqrt{2}-\sqrt{6}}{4}$[/tex]

[Done] Perform the indicated operation: $\frac{-[-45 \div(-5)(3)]}{(7-10)}$

[Done] Which of the following is an example of an improper fraction? A) $3 / 10$ B) $4 / 5$ C) $6 / 7$ D) $10 / 3$

[Done] What is the value of \( \log_3 81 \)? Options: 1) 2 2) 3 3) 4 4) 5

[Done] Evaluate the following: 1. [tex]\int \frac{dx}{(x^2-16)^3}[/tex] 2. [tex]\int \frac{dx}{(1-x^2)^2}[/tex] 3. [tex]\int (x^2-1)^{\frac{5}{2}} dx[/tex] 4. [tex]\int (4-x^2)^{\frac{3}{2}}dx[/tex] 5. [tex]\int \frac{\sqrt{9-4x^2}}{x} dx[/tex] 6. [tex]\int \frac{x^2}{\sqrt{x^2-4}} dx[/tex] 7. [tex]\int \frac{x^2}{\sqrt{x^2-4}} dx[/tex] 8. [tex]\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6} dx[/tex] 9. [tex]\int \frac{dx}{x\sqrt{9+4x^2}}[/tex] 10. [tex]\int \frac{dx}{x^2\sqrt{9-4x^2}}[/tex]

[Done] Determine [tex]D_x\left[\frac{\sqrt[3]{x^2}-x^{-\frac{3}{2}}}{\sqrt{x}}\right][/tex]

[Done] I live on Yonge Street where there are 6 houses on my side of the block. The house numbers are consecutive even numbers. The sum of all 6 house numbers is 8790. You don't know which block I live on, and it's a long street, but I will tell you that I live in the lowest number on my side of the block. What's my address?

[Done] Rewrite the product as an exponent: $3^3 \cdot 3^3$

[Done] The function $a(b)$ relates the area of a trapezoid with a given height of 14 and one base length of 5 with the length of its other base. It takes as input the other base value, and returns as output the area of the trapezoid. $a(b)=14 \bullet \frac{b+5}{2}$ Which equation below represents the inverse function $b(a)$, which takes the trapezoid's area as input and returns as output the length of the other base? A. $b(a)=\frac{a}{7}-5$ B. $b(a)=\frac{a}{7}+5$ C. $b(a)=\frac{a}{5}+7$ D. $b(a)=\frac{a}{5}-7$

[Done] 4. (a) Find the first 10 terms of the sequence, defined as follows; the first two terms are 1 and 3 respectively, each later term is formed by multiplying its predecessor by 3 and subtracting the next previous term. (b) Evaluate the following (i) [tex]$\sum_{r=1}^5 r^3$[/tex] (ii) [tex]$\sum_{r=3}^7 3^r$[/tex]