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Questions in mathematics

[Done] Choose the improper fraction equivalent to the mixed number below. [tex]$5 \frac{5}{9}$[/tex] A. [tex]$\frac{48}{9}$[/tex] B. [tex]$\frac{49}{9}$[/tex] C. [tex]$\frac{51}{9}$[/tex] D. [tex]$\frac{50}{9}$[/tex]

[Done] Find the HCF of [tex]$x^7-\frac{27 x^4}{x^3}$[/tex] and [tex]$x^2+\frac{9}{x^2}-6$[/tex]

[Done] Suppose \(T \in \mathcal{L}(V)\) is such that \(\|Tv\| \leq \|v\|\) for every \(v \in V\). Prove that \(T - \sqrt{2}I\) is injective.

[Done] Consider the following rectangle with sides 3x and 5x. How can we express the area of the rectangle in terms of the variable x?

[Done] Determine the equation of a circle with a center at $(-4,0)$ that passes through the point $(-2,1)$ by following the steps below. 1. Use the distance formula to determine the radius: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$. 2. Substitute the known values into the standard form: $(x-h)^2+(y-k)^2=r^2$. What is the equation of a circle with a center at $(-4,0)$ that passes through the point $(-2,1)$? A. $x^2+(y+4)^2=\sqrt{5}$ B. $(x-1)^2+(y+2)^2=5$ C. $(x+4)^2+y^2=5$ D. $(x+2)^2+(y-1)^2=\sqrt{5}$

[Done] 14. There are 480 boys and 378 girls in a school. How many students are there in all? 15. There are 750 oranges, 858 apples, and 456 bananas in a store. How many fruits in all are there in that store? 16. Fill in the blanks: (a) 276 + 1 = 276 (b) 483 + 0 = (c) 374 + (d) 450 + 1 = 450 (e) 500 + 0 = (f) 600 +

[Done] $6 \frac{m+n}{m-n}+\frac{m-n}{m+n}$

[Done] The product of two numbers is 21. If one of the numbers is -3, what is the other number?

[Done] If $P \times Q =\{(4,5),(4,6),(4,7),(5,5),(5,6),(5,7)\}$, find P and Q . Also show that $n ( P \times Q )= n ( P ) \times n ( Q )$.

[Done] A cone has a volume given by the formula: [tex]$V=\frac{1}{3} \pi r^2 h$[/tex] where [tex]$V$[/tex] represents the volume, [tex]$r$[/tex] represents the radius, and [tex]$h$[/tex] represents the height. If the cone has a volume of [tex]$100 cm^3$[/tex] and a height of 5 cm, find the length of the radius.