
This ODE has characteristic equation


which has roots at
. Then the characteristic solution to the ODE is


Given:
The equation of a line is:

A line passes through the point (-5,-3) and perpendicular to the given line.
To find:
The equation of the line.
Solution:
Slope intercept form of a line is:
...(i)
Where, m is the slope and b is the y-intercept.
We have,
...(ii)
On comparing (i) and (ii), we get

We know that the product of slopes of two perpendicular lines is always -1.



Slope of the required line is
and it passes through the point (-5,-3). So, the equation of the line is:



Using distributive property, we get




Therefore, the equation of the line is
. Hence, option A is correct.
Gaming a video-game designer is using the expression 6n3 in a program to determine points earned, where n is the game level.
Given expression is
. the given expression is for the nth level.
To simplify the expression for the
level, we plug in
in the place of 'n' in the given expression
(multiply the exponents)
<h3>
Answer: Choice C) 40 </h3>
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Work Shown:
Plug in x = 0

This indicates that (0,1) is on the curve. This is the y intercept.
Do the same for x = 2

So we know that (2,81) is another point on this curve.
We need to find the slope of the line through (0,1) and (2,81) to get the slope of the secant line we want.

The slope of the line through (0,1) and (2,81) is m = 40. This value of m is exactly the slope of the secant line your teacher is asking for. This is why the answer is choice C.
By the binomial theorem,

where

Then the coefficients of the
terms in the expansion are, in order from
to
,




