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

To find the slope of the tangent to the parabola y = 2x^2 - 13x + 5, we need to find the derivative of y with respect to x.

dy/dx = 4x - 13

We want to find if there is a point where the slope of the tangent is -1.

4x - 13 = -1

4x = 12

x = 3

Now, we need to find the corresponding y-coordinate.

y = 2(3)^2 - 13(3) + 5

y = 18 - 39 + 5

y = -16

So, the point of tangency is (3, -16).

The equation of the tangent line is:

y - (-16) = -1(x - 3)

y + 16 = -x + 3

y = -x - 13

Explanation

Explanation

Explanation

Calculation:

• Current total age = 80 years

• Three years ago, each person was 3 years younger.

• Since there are three people, the total reduction in age is $3\times 3=9$ years.

So, the total of their ages three years ago is:

$$\mathrm{<b><font\; color="\#0000ff">80</font></b>}<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">9</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">71</font></b>}\text{<b><font color="\#0000ff"> years</font></b>}$$

Explanation

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

Explanation

Explanation

When a pair of dice are tossed, there are 36 possible outcomes.

The combinations that result in a total of 7 or 11 are:

Total 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

Total 11: (5,6), (6,5) = 2 outcomes

Total favorable outcomes = 6 + 2 = 8

Probability = (Number of favorable outcomes) / (Total possible outcomes)

Probability = 8/36 = 2/9