Explanation

Explanation

Explanation

Explanation

The identity element e should satisfy ae = ea = a for all a in G.

Here, e = 6 works since 26=2, 46=4, 66=6, 86=8.

Explanation

Explanation

Explanation

When throwing a die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6).

When drawing a letter at random from the English alphabet, there are 26 possible outcomes (A, B, C, ..., Z).

Since the two events are independent, the total number of points in the sample space is the product of the two:

6 (die outcomes) × 26 (letter outcomes) = 156

Explanation

Since f(x) = x^2 is always non-negative, there is no real x such that f(x) = -9.

So, the inverse image of -9 is the empty set:

f^(-1)(-9) = Ø

Explanation

To evaluate the limit:

lim (x→0) (√(x^2+100) - 10)/x^2

Rationalize the numerator:

= lim (x→0) ((x^2+100) - 100) / (x^2(√(x^2+100) + 10))

= lim (x→0) x^2 / (x^2(√(x^2+100) + 10))

Cancel x^2:

= lim (x→0) 1 / (√(x^2+100) + 10)

As x→0, √(x^2+100) → √100 = 10:

= 1 / (10 + 10)

= 1 / 20

Explanation

1. • The height $h$ of the equilateral triangle is $\frac{\sqrt{3}}{2}x$.

   • Using the area formula for a triangle ($\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$):

     $$\mathrm{<b><font\; color="\#0000ff">A</font></b>}<b><font\; color="\#0000ff">=</font></b>\frac{\mathrm{<b><font\; color="\#0000ff">1</font></b>}}{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">\times </font></b>\mathrm{<b><font\; color="\#0000ff">x</font></b>}<b><font\; color="\#0000ff">\times </font></b>\frac{\sqrt{\mathrm{<b><font\; color="\#0000ff">3</font></b>}}}{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}\mathrm{<b><font\; color="\#0000ff">x</font></b>}<b><font\; color="\#0000ff">=</font></b>\frac{\sqrt{\mathrm{<b><font\; color="\#0000ff">3</font></b>}}}{\mathrm{<b><font\; color="\#0000ff">4</font></b>}}{\mathrm{<b><font\; color="\#0000ff">x</font></b>}}^{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}$$