IIT JEE circle hyperbola intersection

WEBVTT
Kind: captions
Language: en

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The circle x squared plus
y squared minus 8x is

00:00:04.010 --> 00:00:06.770
equal to 0, and the
hyperbola x squared

00:00:06.770 --> 00:00:10.090
over 0 minus y squared
over 4 is equal to 1,

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intersect at the points
A and B. In problem 46,

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they want us to find equation
of the circle with AB

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as its diameter.

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So let's visualize the
circle and the hyperbola.

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The equation of the
circle x squared

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plus y squared minus 8x.

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Let me write it this way.

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This can be rewritten as x
squared minus 8x plus y squared

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is equal to 0.

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You can add 16 to both sides.

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And I'm doing that to complete
the square for the x term.

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So plus 16.

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This over here becomes
x minus 4 squared,

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plus y squared is equal to 16.

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And so the circle's going
to look something like this.

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That's my x-axis.

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That's my y-axis.

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Its center is at 4, 0.

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Zero.

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So 1, 2, 3, 4-- 4, 0.

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And its radius is 4.

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Its radius squared is 16.

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So its radius is 4.

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So let me go 1, 2,
3, 4-- 1, 2, 3, 4.

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Then we go up 4-- 1, 2, 3, 4.

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So all of these points, that
point, that point, that point,

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and that point are all
going to be on the circle.

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Let me draw that a
little bit neater.

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That point, that
point, and that point

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are all going to
be on our circle.

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So my circle's going to
look something like this.

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It's not drawn as
neatly as it could be.

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But I think you get
the general idea.

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This is our circle.

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Then they have a hyperbola.

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x squared over 9 minus y squared
over 4 is going to equal 1.

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This hyperbola is going to
open to the left and the right

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since our x term is positive.

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It's going to look
something like this.

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You can actually figure
out the asymptotes.

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Actually, let me just do that.

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Let me solve for y.

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So you would get, let
me write it this way,

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negative y squared over 4 is
equal to negative x squared

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over 9 plus 1.

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I just subtracted x squared
over 9 from both sides.

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And then we would get,
let's multiply both sides

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by negative 4.

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You get y squared
is equal to 4/9.

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X squared minus 4.

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Or y is equal to--
and now this will just

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be the positive part over here.

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But I'm doing that so that we
can understand its asymptotes.

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y is equal to the square root
of 4 over 9x squared minus 4.

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So as x gets larger
and larger and larger,

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this term right here is going
to stop mattering so much.

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So as x approaches
infinity, this

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is going to approach the
square root of 4/9 x squared.

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This constant's not
going to matter much.

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And so it's going to approach,
this thing right over here

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is going to be 2/3 x.

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So it's going to, if you
imagine a slope, let's see, 2/3.

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If you run 3, you rise 2.

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So the asymptote is
going to look like that.

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It's going to approach
that line over there.

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And it's going to be symmetric.

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So it's also going to
approach-- so 3 and 2,

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it's also going to approach
that line over there.

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And if we want to see where
it intersects the x-axis,

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you just set y is equal to 0.

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You get x squared
over 9 is equal to 1.

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So x is going to be
equal to plus or minus 3.

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So the positive intersection
is going to be over there.

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So our hyperbola
on the right side's

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going to look something like,
something, something like that.

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It's also going to show
up on the left side.

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But that's less interesting
because it's not

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doing anything with the circle.

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Now they told us that the
hyperbola and the circle

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intersect at the points A
and B. So this right here

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is the point A. This
right here is the point

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B. In this question, we want the
equation of the circle with AB

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as its diameter.

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So AB as the diameter.

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I could have drawn that
a little bit straighter.

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So the equation of that
circle right over there.

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So essentially, we
just need to figure out

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where does this hyperbola
intersect this circle.

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Now, the easiest way to
do this is to-- well,

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we have two constraints here.

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Let's solve for y
squared for the circle.

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And we could substitute that
in for y squared right here

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and see what x values,
the intersect that.

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And then we want the
x value out here.

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So something that
looks like this value

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is the one we want to use.

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Then we could figure
out the y value.

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The y value is going to be
the radius of our circle.

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The x value comma 0
will be the center.

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And then we'll
have our equation.

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So let's do that.

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So this up here, we have--
let me do it in yellow.

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So if we subtract x squared and
negative 8x from both sides,

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the equation of
the circle can be

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rewritten as y squared
is equal to-- I'm

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going to add 8x to both sides.

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So it's 8x.

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And then subtract x
squared from both sides.

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8x minus x squared.

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I really just moved
to that and that

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over to the right hand
side of the equation.

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Now I can take
this and substitute

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for y squared in the
equation of the hyperbola.

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So the hyperbola's
equation is x squared

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over 9 minus y squared
over 4 is equal to 1.

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Instead of running a
y squared over there,

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we know that it has to
satisfy this equation as well.

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So for y squared, I'm going
to put an 8x minus x squared.

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And let's see if we can
solve this right here.

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This is just a straight
up quadratic equation,

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although it might not
look like it just yet.

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Let's simplify this.

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So this tells us that x
squared over 9 minus 8x/4.

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So minus 2x plus x squared
over 4 plus x squared over 4.

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I just distributed
essentially the negative 1/4

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over both these terms.

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Is equal to 1.

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Let's see, we can multiply
the whole thing times 36

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to get rid of these fractions.

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So that's 4 times 9.

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So 36 divided by 9 is 4.

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So this is 4x squared
minus 36 times 2 is 72.

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72x plus 36 divided by 4 is 9,
plus 9x squared is equal to 36.

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These two terms right
here, we can add them.

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We get 13x squared, minus 72x.

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And then we could subtract
36 from both sides.

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So minus 36 is equal to 0.

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So now we just have a straight
up quadratic equation.

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We just have to find
the x's, find the roots.

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So let's use the
quadratic formula here.

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So x is going to be
equal to negative B

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So that's negative negative 72.

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So it's 72 plus or minus the
square root of 72 squared.

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I'll just write that 72
times 72 minus 4 times

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A times C. C is negative 36.

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So that negative, you
could put it out here.

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And just put the 36 out back.

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And then all of
that over 2 times a.

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So all of that over 26.

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So the hard part is
really to simplify this.

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But it looks like we have some
interesting stuff going on.

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So let me rewrite
this part over here.

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Let me write it over here.

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72 times 72 is the same thing
as 2 times 36 times 2 times

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36, right?

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Each of those are 72.

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That's the same
thing as 72 times 72.

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And then we're adding to
that 4 times 36 times 13.

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And we're taking the square
root of this whole thing.

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I'm just doing the work
out here so we don't.

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waste this real
estate over here.

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Now we can factor
out a 4 times 36.

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This is 4 and this is a 36.

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So this is equal
to-- I know it's

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getting-- let me write
it a little bit neater.

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This is equal to the square root
of, we can take a 4 times 36.

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And actually 4 times 36 is 144.

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I'll just write 4
times 36 right here.

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4 times 36.

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Now over here we use
the both 2s in the 36.

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So we have a 36 left.

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We use this 4 times 36.

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So we have plus 13.

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So this becomes the square
root of, 4 times 36 is 144.

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36 plus 13 is 49.

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So we're lucky that
it actually worked out

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to 2 perfect squares.

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So this is the square root
of 144 times the square root

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of 49, which is 12, times
7 or this is equal to 84.

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So this business over here
simplifies to 72 plus or minus

00:09:09.340 --> 00:09:10.450
84.

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This whole thing over
here is simplified to 84,

00:09:13.330 --> 00:09:15.800
all of that over 26.

00:09:15.800 --> 00:09:18.634
Now if we were to
subtract 84, we

00:09:18.634 --> 00:09:20.050
would get something
over here that

00:09:20.050 --> 00:09:22.920
doesn't make sense
in this context.

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We're looking for something
a positive x value.

00:09:25.550 --> 00:09:34.520
So let's only consider the
plus 84 and see what we get.

00:09:34.520 --> 00:09:36.600
So the first thing to
do, actually let me just

00:09:36.600 --> 00:09:38.710
divide the numerator and
the denominator by 2.

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So this is the same thing
as 36 plus 42 over 13.

00:09:45.390 --> 00:09:50.190
And this is the same
thing as 78 over 13.

00:09:50.190 --> 00:09:53.437
And it looks like 78
is divisible by 13.

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78 divided by 13 is 6.

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This is equal to 6.

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So the x-coordinate here is 6.

00:10:03.570 --> 00:10:04.570
It's right there.

00:10:04.570 --> 00:10:05.760
So it's 6.

00:10:05.760 --> 00:10:07.600
We don't know what
the y-coordinate is.

00:10:07.600 --> 00:10:08.860
So let's do that.

00:10:08.860 --> 00:10:09.960
It's pretty easy to solve.

00:10:09.960 --> 00:10:13.330
We could substitute x equals
6 into any of these equations.

00:10:13.330 --> 00:10:15.860
This one is probably
easier right over here.

00:10:15.860 --> 00:10:18.520
So we get y squared
is equal to 8 times

00:10:18.520 --> 00:10:28.070
6 is 48, minus 48 minus
36, which is equal to 12.

00:10:28.070 --> 00:10:31.610
And so y is equal to
the square root of 12.

00:10:31.610 --> 00:10:33.852
We could simplify that
radical if we want.

00:10:33.852 --> 00:10:34.935
But we know the point now.

00:10:34.935 --> 00:10:38.240
It's 6 comma the square root--
Sorry, this point over here

00:10:38.240 --> 00:10:42.230
is 6 comma 0, but the
actual intersect, the center

00:10:42.230 --> 00:10:45.460
is 6 comma 0, but the
actual intersection point

00:10:45.460 --> 00:10:51.120
is going to be 6 comma
positive square root of 12?

00:10:51.120 --> 00:10:53.190
And I should say this is
a positive or negative

00:10:53.190 --> 00:10:54.560
square root of 12.

00:10:54.560 --> 00:10:56.150
And this row over
here is going to be

00:10:56.150 --> 00:10:59.240
6 negative square root of 12.

00:10:59.240 --> 00:11:01.390
This right here is 6 comma 0.

00:11:01.390 --> 00:11:03.610
That's the center of our circle.

00:11:03.610 --> 00:11:06.330
So what's the equation of
this new circle going to be?

00:11:06.330 --> 00:11:09.440
What's the equation of this
new circle going to be?

00:11:09.440 --> 00:11:10.440
Well we know its center.

00:11:10.440 --> 00:11:14.420
Its center is at 6 0.

00:11:14.420 --> 00:11:15.750
Let me write it down here.

00:11:15.750 --> 00:11:18.000
Its center is at 6 0.

00:11:18.000 --> 00:11:23.520
So it's going to be x minus 6
squared plus y minus 0 squared

00:11:23.520 --> 00:11:25.020
is equal to the radius squared.

00:11:25.020 --> 00:11:26.280
Now what's the radius?

00:11:26.280 --> 00:11:28.700
The radius is this
height right over here.

00:11:28.700 --> 00:11:31.380
Or it's just equal to the y
value at the intersection.

00:11:31.380 --> 00:11:33.110
It's the square root of 12.

00:11:33.110 --> 00:11:35.050
Now this is going to
the radius squared.

00:11:35.050 --> 00:11:36.400
The radius is squared of 12.

00:11:36.400 --> 00:11:39.300
Squared of 12 squared is 12.

00:11:39.300 --> 00:11:41.490
So the equation is
going to be x minus 6,

00:11:41.490 --> 00:11:43.770
and see what form
they have up here.

00:11:43.770 --> 00:11:45.534
So they actually
multiply everything out.

00:11:45.534 --> 00:11:46.450
So let's just do that.

00:11:46.450 --> 00:11:52.280
So this is going to be x
squared minus 12x plus 36 plus y

00:11:52.280 --> 00:11:55.310
squared is equal to 12.

00:11:55.310 --> 00:11:58.010
And then we could subtract
12 from both sides.

00:11:58.010 --> 00:12:00.240
Subtract 12 from both sides.

00:12:00.240 --> 00:12:07.460
And we get x squared minus
12x plus 24 plus y squared

00:12:07.460 --> 00:12:09.390
is equal to 0.

00:12:09.390 --> 00:12:14.400
Let's see which of the
choices match that.

00:12:14.400 --> 00:12:16.370
Let me copy it.

00:12:16.370 --> 00:12:19.820
And then let me
paste it up here.

00:12:19.820 --> 00:12:21.550
Let me paste it up here.

00:12:21.550 --> 00:12:24.150
And it looks like we have
an x squared positive y

00:12:24.150 --> 00:12:29.750
squared, a negative 12x
x and then a positive 24.

00:12:29.750 --> 00:12:32.340
Negative 12 x positive 24.

00:12:32.340 --> 00:12:35.860
So it looks like
our answer is a.

00:12:35.860 --> 00:12:37.630
Did I do that right?
x squared plus y

00:12:37.630 --> 00:12:40.540
squared minus 12 x plus 24.

00:12:40.540 --> 00:12:43.710
Yep, our answer is a.


IIT JEE Divisible Determinants

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.440
So in problem 44, once again
we have the exact same setup

00:00:03.440 --> 00:00:05.830
that we had in
problems 42 and 43.

00:00:05.830 --> 00:00:07.330
But now they're
asking us the number

00:00:07.330 --> 00:00:10.440
of A in T sub p such
that the determinant of A

00:00:10.440 --> 00:00:15.040
is not divisible by p.

00:00:15.040 --> 00:00:19.120
Well, we've already done
some work in problem 43

00:00:19.120 --> 00:00:21.750
in figuring out some
set of the members that

00:00:21.750 --> 00:00:22.882
are divisible by p.

00:00:22.882 --> 00:00:25.340
So why don't we just figure
out the total number of members

00:00:25.340 --> 00:00:29.840
of T sub p, and then subtract
the total members whose

00:00:29.840 --> 00:00:32.000
determinants are divisible
by p, and then we'll

00:00:32.000 --> 00:00:33.890
get this answer right over here.

00:00:33.890 --> 00:00:44.050
So what's the total
members in T sub p?

00:00:44.050 --> 00:00:46.880
Well, for a, I have p
possible choices, right?

00:00:46.880 --> 00:00:50.590
If I'm picking between-- up to
p minus 1, but I'm including 0,

00:00:50.590 --> 00:00:52.200
there's actually p
possible choices.

00:00:52.200 --> 00:00:54.362
So I have p possible
choices for a,

00:00:54.362 --> 00:00:56.860
p possible choices for
b by the same logic,

00:00:56.860 --> 00:00:59.270
and p possible choices for c.

00:00:59.270 --> 00:01:01.790
So I have p to the
third possibilities.

00:01:01.790 --> 00:01:06.640
So there are a total of p to
the third members in T sub p.

00:01:06.640 --> 00:01:10.790
Now, what did the
last problem tell us?

00:01:10.790 --> 00:01:14.580
This was number 43.

00:01:14.580 --> 00:01:17.840
It told us the members-- maybe
I should write it this way.

00:01:17.840 --> 00:01:26.560
The number of members, numbers
of A's, such that the trace

00:01:26.560 --> 00:01:28.270
of A-- this was
the language I used

00:01:28.270 --> 00:01:31.300
in the last video or the last
problem-- such that the trace

00:01:31.300 --> 00:01:41.220
of A not divisible by p,
but the determinant of A is.

00:01:41.220 --> 00:01:47.270
And we got the answer
as p minus 1 squared.

00:01:47.270 --> 00:01:49.270
Now, if you remember that
problem, and even just

00:01:49.270 --> 00:01:51.500
to do that problem,
when we thought

00:01:51.500 --> 00:01:54.390
about this statement,
such that the trace of A

00:01:54.390 --> 00:01:57.630
is not divisible by
p, we figured out

00:01:57.630 --> 00:02:05.670
that this was exactly equal to
saying that A does not equal 0.

00:02:05.670 --> 00:02:08.800
So one way to view the
result of problem 43

00:02:08.800 --> 00:02:12.920
is if you assume that
A is not equal to 0,

00:02:12.920 --> 00:02:17.480
there are p minus 1 squared
members of T sub p where

00:02:17.480 --> 00:02:19.840
the determinant
is divisible by p.

00:02:19.840 --> 00:02:25.570
So we're starting to count some
of the A's whose determinant

00:02:25.570 --> 00:02:26.340
is divisible by p.

00:02:26.340 --> 00:02:29.860
But these are the only--
the matrix A's whose

00:02:29.860 --> 00:02:31.900
a entry is not equal to 0.

00:02:31.900 --> 00:02:33.530
So if we want to
count all of them,

00:02:33.530 --> 00:02:37.380
we have to also count the
ones where a does equal 0,

00:02:37.380 --> 00:02:40.630
so where a does equal 0.

00:02:40.630 --> 00:02:43.060
So let's think about that case.

00:02:43.060 --> 00:02:46.070
We can figure out how
many matrices there

00:02:46.070 --> 00:02:49.710
are where a does equal 0,
and the determinant of A

00:02:49.710 --> 00:02:51.780
is divisible by p.

00:02:51.780 --> 00:02:52.790
Add it to that.

00:02:52.790 --> 00:02:55.992
Then we have the total
number of A's whose

00:02:55.992 --> 00:02:57.325
determinants are divisible by p.

00:02:57.325 --> 00:02:59.660
And we could subtract
that from the total number

00:02:59.660 --> 00:03:02.430
of possibilities, and hopefully,
we'll get one of the answers

00:03:02.430 --> 00:03:03.157
up here.

00:03:03.157 --> 00:03:04.740
So let's think a
little bit about what

00:03:04.740 --> 00:03:06.450
will the determinant
look like over here?

00:03:06.450 --> 00:03:09.830
So in this situation, the
determinant of our matrix

00:03:09.830 --> 00:03:12.070
will be-- it's a squared.

00:03:12.070 --> 00:03:17.180
It's a times a minus b
times c, a squared minus bc.

00:03:17.180 --> 00:03:19.840
Now, we're assuming
that a is equal to 0.

00:03:19.840 --> 00:03:24.600
If a is equal to 0, it's
equal to negative bc.

00:03:24.600 --> 00:03:27.530
And we need to think
about how is this

00:03:27.530 --> 00:03:29.615
going to be a
multiple, or how is

00:03:29.615 --> 00:03:33.350
this going to be divisible by p?

00:03:33.350 --> 00:03:38.700
So this has got to be equal
to some multiple of p.

00:03:38.700 --> 00:03:42.370
So the first thing, just
think about b's and the c's.

00:03:42.370 --> 00:03:44.110
They are both non-negative.

00:03:44.110 --> 00:03:49.740
So this value right
here cannot be negative.

00:03:49.740 --> 00:03:52.950
So if you put a negative sign
there, it cannot be positive.

00:03:52.950 --> 00:03:55.970
So it can't be any
positive multiples of p.

00:03:55.970 --> 00:03:56.880
It could be 0.

00:03:56.880 --> 00:03:59.190
So we could have negative bc.

00:03:59.190 --> 00:04:02.250
We could have negative
bc is equal to 0, which

00:04:02.250 --> 00:04:04.690
is equivalent to saying
that bc is equal to 0.

00:04:04.690 --> 00:04:07.450
That's one possibility,
where it equals 0 times p.

00:04:07.450 --> 00:04:08.790
That's still a multiple.

00:04:08.790 --> 00:04:10.560
Or maybe it could be negative.

00:04:10.560 --> 00:04:16.420
Maybe we could have negative bc
is equal to negative 1 times p.

00:04:16.420 --> 00:04:18.990
But when you think about
it here, if this was true,

00:04:18.990 --> 00:04:23.460
that would mean that bc
is equal to p, so b or c

00:04:23.460 --> 00:04:25.250
would be factors of p.

00:04:25.250 --> 00:04:30.120
Now, we know that p
is a prime number.

00:04:30.120 --> 00:04:33.182
Its factors are only 1 and p.

00:04:33.182 --> 00:04:36.780
b and c-- I mean, one of
these guys could be 1,

00:04:36.780 --> 00:04:38.530
but then the other guy
would have to be p.

00:04:38.530 --> 00:04:39.360
They can't be p.

00:04:39.360 --> 00:04:41.620
They only get as
high as p minus 1.

00:04:41.620 --> 00:04:43.340
This is not an option.

00:04:43.340 --> 00:04:46.420
So the only possibility, if a is
equal to 0, for the determinant

00:04:46.420 --> 00:04:49.580
of capital A to still
be divisible by p,

00:04:49.580 --> 00:04:51.460
is for bc to be equal to 0.

00:04:51.460 --> 00:04:54.010
So let's think about how
many possibilities there

00:04:54.010 --> 00:04:54.976
are over here.

00:04:54.976 --> 00:04:56.850
So let's think about
all of the combinations.

00:04:56.850 --> 00:05:01.030
So there's the one situation,
where b and c are equal to 0.

00:05:01.030 --> 00:05:03.299
b and c is equal to 0.

00:05:03.299 --> 00:05:04.840
Now let's think
about the other ones.

00:05:04.840 --> 00:05:10.560
There's one where b is
equal to 0 and c isn't.

00:05:10.560 --> 00:05:13.680
bc will still equal
0 in this situation.

00:05:13.680 --> 00:05:15.580
So how many
possibilities are there?

00:05:15.580 --> 00:05:17.860
How many possibilities
are there for this?

00:05:17.860 --> 00:05:21.190
Well, b is equal to 0.

00:05:21.190 --> 00:05:24.420
C over here can be any
value, although we've already

00:05:24.420 --> 00:05:26.630
considered the case
where c is equal to 0.

00:05:26.630 --> 00:05:28.190
So it could be any
value so that we

00:05:28.190 --> 00:05:31.050
don't double count, any value
where c does not equal 0.

00:05:31.050 --> 00:05:33.405
So it could be 1
through p minus 1.

00:05:33.405 --> 00:05:34.780
Or another way to
think about it,

00:05:34.780 --> 00:05:37.540
there's p minus 1 possibilities.

00:05:37.540 --> 00:05:43.400
Now let's think about c
is equal to 0 and b isn't.

00:05:43.400 --> 00:05:44.827
Well, exact same logic.

00:05:44.827 --> 00:05:46.535
Once again, we don't
want to double count

00:05:46.535 --> 00:05:48.559
that they're both
being equal to 0.

00:05:48.559 --> 00:05:49.350
So let me write it.

00:05:49.350 --> 00:05:51.920
And c isn't-- well,
we're saying it isn't 0,

00:05:51.920 --> 00:05:54.295
so we are actually already
putting that constraint there.

00:05:54.295 --> 00:05:57.960
So it has p minus
1 possibilities.

00:05:57.960 --> 00:06:02.200
So how many total
possibilities do we

00:06:02.200 --> 00:06:04.140
have where the
determinant of A is

00:06:04.140 --> 00:06:06.830
equal to 0 and a is equal to 0?

00:06:06.830 --> 00:06:09.060
So we have one here.

00:06:09.060 --> 00:06:12.030
We have one possibility here,
then p minus 1 and p minus 1.

00:06:12.030 --> 00:06:17.380
So our total possibilities
are p minus 1 plus p minus 1

00:06:17.380 --> 00:06:25.670
plus 1, which is 2p minus 2
plus 1, which is 2p minus 1.

00:06:25.670 --> 00:06:27.860
2p minus 1 incremental
possibilities

00:06:27.860 --> 00:06:30.970
when a is equal to 0
for the determinant of A

00:06:30.970 --> 00:06:32.760
to be divisible by p.

00:06:32.760 --> 00:06:34.550
So that's what we just
got in this video.

00:06:34.550 --> 00:06:38.040
In the last video, we had p
minus 1 squared possibility

00:06:38.040 --> 00:06:40.070
when a does not equal 0.

00:06:40.070 --> 00:06:42.066
So if we want the total
number of A's, where

00:06:42.066 --> 00:06:43.690
the determinant of
A is divisible by 0,

00:06:43.690 --> 00:06:46.770
we can add the number that
we got in the last video

00:06:46.770 --> 00:06:48.200
to the number we just got.

00:06:48.200 --> 00:06:49.590
So let's do that.

00:06:49.590 --> 00:06:53.060
So if I were to
rewrite this over here,

00:06:53.060 --> 00:06:58.370
this is equal to p
squared minus 2p plus 1.

00:06:58.370 --> 00:07:01.380
And then I want to
add this 2p minus 1

00:07:01.380 --> 00:07:05.560
here, so plus 2p minus 1.

00:07:08.680 --> 00:07:10.180
And what do I get?

00:07:10.180 --> 00:07:11.780
These cancel out.

00:07:11.780 --> 00:07:13.410
These cancel with that.

00:07:13.410 --> 00:07:15.220
And these guys cancel out.

00:07:15.220 --> 00:07:24.940
So there's a total of p
squared possible A's where

00:07:24.940 --> 00:07:30.370
the determinant of
A is divisible by p.

00:07:30.370 --> 00:07:33.340
Now, we're very close, but
this isn't what they're asking.

00:07:33.340 --> 00:07:36.270
They want the total members
where the determinant of A

00:07:36.270 --> 00:07:37.890
is not divisible by p.

00:07:37.890 --> 00:07:41.770
So all we have to do here is
take our total membership,

00:07:41.770 --> 00:07:47.570
which is p to the third,
and subtract from that

00:07:47.570 --> 00:07:50.272
the members whose determinants
are divisible by p.

00:07:50.272 --> 00:07:51.480
And we just figured that out.

00:07:51.480 --> 00:07:53.380
We subtract p squared.

00:07:53.380 --> 00:08:03.350
And now we'll get the number
of A's where determinant of A

00:08:03.350 --> 00:08:06.260
not divisible by p.

00:08:06.260 --> 00:08:09.440
Now let's see if that's one
of the choices, and it is.

00:08:09.440 --> 00:08:13.400
D, p cubed minus p squared.


IIT JEE Trace and Determinant

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.340
So we have the same
setup here as we

00:00:02.340 --> 00:00:05.450
had in the previous problem,
in problem number 42,

00:00:05.450 --> 00:00:07.710
where p is an odd prime number.

00:00:07.710 --> 00:00:11.020
And they say, "Let T sub p
be the following set of 2

00:00:11.020 --> 00:00:13.090
by 2 matrices," where
each of those matrices

00:00:13.090 --> 00:00:16.230
have the form a's along the
diagonals and then a b there

00:00:16.230 --> 00:00:17.140
and a c there.

00:00:17.140 --> 00:00:19.470
And each of those
numbers are between 0

00:00:19.470 --> 00:00:21.260
and p minus 1 inclusive.

00:00:21.260 --> 00:00:25.000
So they can be 0, 1, 2, 3,
all the way up to p minus 1.

00:00:25.000 --> 00:00:26.850
And then in number
43, they ask us,

00:00:26.850 --> 00:00:31.940
"The number of A in the set
such that the trace of A

00:00:31.940 --> 00:00:35.410
is not divisible by p,
but the determinant of A

00:00:35.410 --> 00:00:41.414
is divisible by p is?"

00:00:41.414 --> 00:00:43.330
And then, they tell us,
"The trace of a matrix

00:00:43.330 --> 00:00:45.299
is the sum of its
diagonal entries."

00:00:45.299 --> 00:00:47.090
It's actually the sum
of its main diagonal.

00:00:47.090 --> 00:00:48.130
It's not going to
be that diagonal.

00:00:48.130 --> 00:00:49.350
It's only the sum of this.

00:00:49.350 --> 00:00:53.310
So if you sum this, you're going
to get the trace of the matrix.

00:00:53.310 --> 00:00:56.120
So let's use this first
piece of information.

00:00:56.120 --> 00:00:59.940
The trace of A is
not divisible by p.

00:00:59.940 --> 00:01:03.760
So the trace of A is going
to be a plus a, or 2a.

00:01:03.760 --> 00:01:14.180
So 2a not divisible by p.

00:01:14.180 --> 00:01:17.860
So let's think of a situation
where 2a is divisible by p.

00:01:17.860 --> 00:01:22.540
So 2a could be equal
to 0, because 0

00:01:22.540 --> 00:01:27.140
is a multiple of p, in which
case a would be equal to 0.

00:01:27.140 --> 00:01:28.790
So this is a possibility.

00:01:28.790 --> 00:01:33.810
So if a is equal to 0,
then 2a is divisible by p.

00:01:33.810 --> 00:01:36.250
So one of the constraints,
from this sentence right here,

00:01:36.250 --> 00:01:38.070
is that a cannot be equal to 0.

00:01:38.070 --> 00:01:40.790
Because if a is 0, then the
trace is divisible by p.

00:01:40.790 --> 00:01:42.389
So a cannot be equal to 0.

00:01:42.389 --> 00:01:43.430
And let's think about it.

00:01:43.430 --> 00:01:45.730
Can 2a become any
other multiple of p?

00:01:45.730 --> 00:01:49.190
Can 2a equal p?

00:01:49.190 --> 00:01:52.760
Well, if 2a equaled p, then 2
and a would be factors of p,

00:01:52.760 --> 00:01:54.950
and then p would not be prime.

00:01:54.950 --> 00:01:56.810
And so we can cross that out.

00:01:56.810 --> 00:01:59.730
And then 2a can't
be equal to 2p.

00:01:59.730 --> 00:02:02.380
That would make a equal
to p, which is impossible,

00:02:02.380 --> 00:02:04.070
because a can only be p minus 1.

00:02:04.070 --> 00:02:05.990
So that's not possible.

00:02:05.990 --> 00:02:08.624
And then you can keep going
higher, higher multiples of p,

00:02:08.624 --> 00:02:12.610
not possible, because a is
going to be smaller than p.

00:02:12.610 --> 00:02:15.220
So the only constraint that we
get from this first sentence,

00:02:15.220 --> 00:02:16.920
the trace of a is
not divisible by p,

00:02:16.920 --> 00:02:19.780
tells us that a
does not equal 0.

00:02:19.780 --> 00:02:21.600
So now let's look at
the second constraint.

00:02:21.600 --> 00:02:27.340
The determinant of
A is divisible by p.

00:02:27.340 --> 00:02:29.640
So the determinant of
A-- let me write this.

00:02:29.640 --> 00:02:33.770
The determinant of A
is equal to a times

00:02:33.770 --> 00:02:37.990
a-- it's equal to a squared
minus b times c, or c times

00:02:37.990 --> 00:02:39.280
b, so minus bc.

00:02:39.280 --> 00:02:44.850
So this is going to
be divisible by p.

00:02:44.850 --> 00:02:49.900
This is going to be equal
to some integer times p.

00:02:49.900 --> 00:02:52.010
Now one thing just
looking at this,

00:02:52.010 --> 00:02:53.922
there's no obvious
simplification here.

00:02:53.922 --> 00:02:55.630
But one thing I'm
starting to appreciate,

00:02:55.630 --> 00:02:57.950
not having taken the
IIT Joint Entrance

00:02:57.950 --> 00:03:00.760
Exam when I was in high school,
not growing up in India,

00:03:00.760 --> 00:03:03.350
is that when they have
multiple choices, sometimes

00:03:03.350 --> 00:03:05.747
it might be faster to just
do the brute force method.

00:03:05.747 --> 00:03:07.330
And when I say the
brute force method,

00:03:07.330 --> 00:03:10.000
not just proving mathematically
which of these expressions

00:03:10.000 --> 00:03:13.190
are true, but just
let's try a simple p.

00:03:13.190 --> 00:03:15.760
Let's try a simple
p, and then let's

00:03:15.760 --> 00:03:20.850
find the members of the set
that would satisfy it for that p

00:03:20.850 --> 00:03:23.450
and then see which of these
expressions equal that p.

00:03:23.450 --> 00:03:24.660
Let's see what we get.

00:03:24.660 --> 00:03:26.010
So let's just simplify this.

00:03:26.010 --> 00:03:28.160
Let's just say p is equal to 3.

00:03:28.160 --> 00:03:31.710
And I pick p is equal to 3
because that's the smallest odd

00:03:31.710 --> 00:03:32.330
prime number.

00:03:32.330 --> 00:03:35.220
It's the smallest number that
satisfies this constraint

00:03:35.220 --> 00:03:36.130
up here.

00:03:36.130 --> 00:03:38.120
So let's set p is equal to 3.

00:03:38.120 --> 00:03:43.480
So if p is equal to 3,
our possibilities for a

00:03:43.480 --> 00:03:46.090
are that a could be equal
to-- a can't be equal to 0.

00:03:46.090 --> 00:03:47.350
We know that already.

00:03:47.350 --> 00:03:48.440
a cannot be equal to 0.

00:03:48.440 --> 00:03:55.730
So a could be equal to 1
or a could be equal to 2.

00:03:55.730 --> 00:04:00.500
And if a is equal to 1,
then the determinant of A

00:04:00.500 --> 00:04:05.890
is going to be equal
to 1 minus bc, which

00:04:05.890 --> 00:04:12.000
needs to be equal to some
integer multiple of 3,

00:04:12.000 --> 00:04:14.100
because we set our
p is equal to 3.

00:04:14.100 --> 00:04:16.670
It must be equal to 3k.

00:04:16.670 --> 00:04:18.709
If a is equal to 2,
then the determinant

00:04:18.709 --> 00:04:21.970
is going to be 4,
because it's a squared.

00:04:21.970 --> 00:04:24.690
So it's going to
be 4 minus bc, must

00:04:24.690 --> 00:04:28.270
be equal to some
integer multiple of 3.

00:04:28.270 --> 00:04:30.390
So let's see how many
combinations of b and c

00:04:30.390 --> 00:04:33.250
we can find in each
of these constraints.

00:04:33.250 --> 00:04:35.240
So 1 minus bc.

00:04:35.240 --> 00:04:38.430
So one possibility--
let's think about it.

00:04:38.430 --> 00:04:43.040
So one possibility is 1 minus
bc is equal to 3 times 0.

00:04:43.040 --> 00:04:44.420
That's one possibility.

00:04:44.420 --> 00:04:51.894
Maybe another possibility
is 1 minus bc is equal to 3.

00:04:51.894 --> 00:04:53.810
Actually, let's think
about this a little bit.

00:04:53.810 --> 00:04:56.630
Let me try this
possibility here, first.

00:04:56.630 --> 00:04:58.370
So if 1 minus bc is
equal to 0-- this

00:04:58.370 --> 00:05:01.500
is pretty straightforward-- bc
would have to be equal to 1.

00:05:01.500 --> 00:05:04.950
And so b would have
to be equal to 1

00:05:04.950 --> 00:05:06.820
and c would have
to be equal to 1.

00:05:06.820 --> 00:05:12.130
So we have one combination
here, a could be equal to 1,

00:05:12.130 --> 00:05:14.940
b is equal to 1,
and c is equal to 1.

00:05:14.940 --> 00:05:18.490
That is one possibility
right over there.

00:05:18.490 --> 00:05:20.910
And then another
possibility-- now,

00:05:20.910 --> 00:05:26.000
no matter what bc we pick
here, the maximum value

00:05:26.000 --> 00:05:27.210
here is going to be 1.

00:05:27.210 --> 00:05:30.350
If bc are both 0, then this
expression is going to be 1.

00:05:30.350 --> 00:05:32.910
If they're both 1, this
expression is going to be 0.

00:05:32.910 --> 00:05:35.520
If they are any other value--
and the only other values

00:05:35.520 --> 00:05:37.750
that they could be are
some combination of 1 and 2

00:05:37.750 --> 00:05:41.220
or 2 and 2-- then we're going
to get a negative number.

00:05:41.220 --> 00:05:44.720
So the question is, is
a negative multiple of 3

00:05:44.720 --> 00:05:47.280
considered to be
a multiple of 3?

00:05:47.280 --> 00:05:49.770
Now let's just figure it
out assuming that it is,

00:05:49.770 --> 00:05:52.150
and then we can see if maybe
one of these expressions

00:05:52.150 --> 00:05:52.650
makes sense.

00:05:52.650 --> 00:05:55.420
And I'm going to do that in
white just because it depends.

00:05:55.420 --> 00:05:57.790
If you considered
negative 3 divisible by 3,

00:05:57.790 --> 00:05:59.020
then we should go with that.

00:05:59.020 --> 00:06:01.600
So maybe we should
say 1 minus bc

00:06:01.600 --> 00:06:05.580
could be equal to
negative 3, which

00:06:05.580 --> 00:06:07.760
is the same thing
as bc minus 1 is

00:06:07.760 --> 00:06:10.530
equal to 3, which is the
same-- I just multiplied

00:06:10.530 --> 00:06:13.710
both sides by negative 1--
which is the same thing as bc

00:06:13.710 --> 00:06:15.979
is equal to 4.

00:06:15.979 --> 00:06:17.270
And we have to be careful here.

00:06:17.270 --> 00:06:20.310
You might be tempted to say b
is equal to 1, c is equal to 4,

00:06:20.310 --> 00:06:21.810
or b is equal to
4, c is equal to 1,

00:06:21.810 --> 00:06:23.590
but they can't be equal to 4.

00:06:23.590 --> 00:06:24.980
4 is bigger than our p.

00:06:24.980 --> 00:06:26.440
They can only be up to 2.

00:06:26.440 --> 00:06:29.860
So this constraint is
only, b is equal to 2,

00:06:29.860 --> 00:06:31.700
and c is equal to 2.

00:06:31.700 --> 00:06:33.880
And of course, a is equal to 1.

00:06:33.880 --> 00:06:37.190
So this is, if you do
consider negative 3

00:06:37.190 --> 00:06:40.380
to be divisible by 3,
which it actually is.

00:06:40.380 --> 00:06:41.600
It's negative 1 times 3.

00:06:41.600 --> 00:06:42.690
It is a multiple.

00:06:42.690 --> 00:06:44.679
So this would be
one of the cases.

00:06:44.679 --> 00:06:46.220
But I'll put that
there just in case.

00:06:46.220 --> 00:06:48.678
You never know what the test
writers are actually thinking.

00:06:48.678 --> 00:06:50.900
So I'll put that there as
kind of a contingent case.

00:06:50.900 --> 00:06:54.750
And then this last situation,
4 minus bc is equal to 3k.

00:06:54.750 --> 00:07:01.810
So 4 minus bc could
be equal to 0.

00:07:01.810 --> 00:07:04.200
Or 4 minus bc--
let's think about it.

00:07:04.200 --> 00:07:06.710
4 minus bc could be equal to 0.

00:07:06.710 --> 00:07:10.300
4 minus bc could be equal to 3.

00:07:10.300 --> 00:07:13.280
That's definitely a possibility,
because this value right here

00:07:13.280 --> 00:07:15.640
will always be less than 4.

00:07:15.640 --> 00:07:18.030
And you can't get
anything less than 0

00:07:18.030 --> 00:07:21.020
here, because the highest
value for b and c are both 2.

00:07:21.020 --> 00:07:23.820
2 times 2 is going to be 4,
so you can't get anything less

00:07:23.820 --> 00:07:24.610
than 0 here.

00:07:24.610 --> 00:07:26.350
So you won't have
a case like this.

00:07:26.350 --> 00:07:27.610
So-- and I just said it.

00:07:27.610 --> 00:07:30.980
So if you have this equation,
that means 4 is equal to bc.

00:07:30.980 --> 00:07:34.440
The only possibility with
b and c being less than 3,

00:07:34.440 --> 00:07:38.350
being 0, 1, or 2, is
that b is equal to 2,

00:07:38.350 --> 00:07:39.512
and c is equal to 2.

00:07:39.512 --> 00:07:41.220
Neither one of those
could be equal to 4,

00:07:41.220 --> 00:07:42.990
because that's
bigger than our p.

00:07:42.990 --> 00:07:48.810
And so here, we have a is
equal to 2, b is equal to 2,

00:07:48.810 --> 00:07:50.750
and c is equal to 2.

00:07:50.750 --> 00:07:53.190
And then if 4 minus
bc is equal to 3,

00:07:53.190 --> 00:07:57.340
that means that this, that
bc, has to be equal to 1,

00:07:57.340 --> 00:07:58.550
bc is equal to 1.

00:07:58.550 --> 00:08:00.810
And so the only
possibility is that-- let

00:08:00.810 --> 00:08:03.600
me scroll to the
right-- b is equal to 1,

00:08:03.600 --> 00:08:07.770
and c is equal to 1, and
of course, a is equal to 2.

00:08:10.920 --> 00:08:14.060
So we've been able to list four
distinct possibilities, four

00:08:14.060 --> 00:08:17.450
distinct matrices, for
our p is equal to 3.

00:08:17.450 --> 00:08:21.470
Now let's see if any
of these evaluate to 4

00:08:21.470 --> 00:08:23.560
when p is equal to 3.

00:08:23.560 --> 00:08:24.840
Let me put this down here.

00:08:24.840 --> 00:08:29.600
So if we have 3 minus 2--
sorry, 3 minus 1, that's 2,

00:08:29.600 --> 00:08:32.490
3 squared, 9, minus
3 is 6, plus 1 is 7.

00:08:32.490 --> 00:08:33.460
So this is 2 times 7.

00:08:33.460 --> 00:08:34.330
That's 14.

00:08:34.330 --> 00:08:37.500
That doesn't evaluate to 4.

00:08:37.500 --> 00:08:45.120
This is 27 minus 4, this is
23, also doesn't evaluate to 4.

00:08:45.120 --> 00:08:47.170
3 minus 1 squared,
that's 2 squared.

00:08:47.170 --> 00:08:48.550
This evaluates to 4.

00:08:48.550 --> 00:08:49.842
So this is looking pretty good.

00:08:49.842 --> 00:08:51.841
We just have to hope that
something else doesn't

00:08:51.841 --> 00:08:52.660
evaluate to 4.

00:08:52.660 --> 00:08:57.610
This is 2, 3 minus
1 is 2, times 7.

00:08:57.610 --> 00:08:59.650
Once again, this is also 14.

00:08:59.650 --> 00:09:00.730
Not there.

00:09:00.730 --> 00:09:01.860
So I would go with C.

00:09:01.860 --> 00:09:05.440
And so the exam writers
really did want negative 3

00:09:05.440 --> 00:09:09.190
to be a possible-- or they
really do view negative 3

00:09:09.190 --> 00:09:11.009
as being divisible
by 3, which it is.

00:09:11.009 --> 00:09:12.550
But, you know,
sometimes people think

00:09:12.550 --> 00:09:15.040
of it only as kind of
positive multiples.

00:09:15.040 --> 00:09:18.403
But it clearly is divisible
by 3, so our answer is C.


IIT JEE Symmetric and Skew-Symmetric Matrices

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.600
Let p be an odd
prime number and T

00:00:03.600 --> 00:00:07.740
sub p be the following
set of 2 by 2 matrices.

00:00:07.740 --> 00:00:10.599
So it's a set of matrices
where each member,

00:00:10.599 --> 00:00:12.140
we're just calling
it A where A could

00:00:12.140 --> 00:00:14.920
be made of lowercase,
a, b, c and a.

00:00:14.920 --> 00:00:19.570
And then these could take any
values between 0 and p minus 1.

00:00:19.570 --> 00:00:22.060
And remember, p was
an odd prime number.

00:00:22.060 --> 00:00:25.250
And frankly, all the prime
numbers greater than 2 are odd.

00:00:25.250 --> 00:00:29.320
So the number of A in Tp,
so the number of matrices

00:00:29.320 --> 00:00:34.220
such that A is either symmetric
or skew-symmetric or both,

00:00:34.220 --> 00:00:37.820
and the determinant of A is
divisible by p is-- so let's

00:00:37.820 --> 00:00:39.400
just make sure we
understand what

00:00:39.400 --> 00:00:41.820
symmetric and
skew-symmetric mean.

00:00:41.820 --> 00:00:44.330
So let's think about
symmetric first.

00:00:44.330 --> 00:00:48.230
So a matrix is symmetric
if it equals its transpose.

00:00:48.230 --> 00:00:54.520
So in this case, a matrix
a b, c, a-- and that's

00:00:54.520 --> 00:00:57.800
what the members of
the set look like.

00:00:57.800 --> 00:01:01.299
This is symmetric if this
is equal to its transpose.

00:01:01.299 --> 00:01:02.840
And a transpose of
a matrix, you just

00:01:02.840 --> 00:01:04.260
swap the rows and the columns.

00:01:04.260 --> 00:01:06.270
So this first row
here, a, b, will

00:01:06.270 --> 00:01:08.780
become the first column, a, b.

00:01:08.780 --> 00:01:12.040
And the second row, c, a,
becomes the second column,

00:01:12.040 --> 00:01:13.070
c, a.

00:01:13.070 --> 00:01:15.970
So that's what symmetric means.

00:01:15.970 --> 00:01:17.560
Now if you just
look at this, that

00:01:17.560 --> 00:01:20.194
means that a is equal
to a. a is equal to a.

00:01:20.194 --> 00:01:21.610
This essentially
means that b will

00:01:21.610 --> 00:01:24.990
have to be equal to c in
any of these matrices that

00:01:24.990 --> 00:01:26.130
are symmetric.

00:01:26.130 --> 00:01:28.720
Because that would have to
be equal to that and that

00:01:28.720 --> 00:01:30.400
would have to be equal to that.

00:01:30.400 --> 00:01:31.380
Fair enough.

00:01:31.380 --> 00:01:33.970
Now let's think about
what skew-symmetric means.

00:01:38.920 --> 00:01:41.980
Skew-symmetric just
means that a matrix

00:01:41.980 --> 00:01:44.350
is equal to the negative
of its transpose.

00:01:44.350 --> 00:01:47.592
So skew-symmetric
would mean that a,

00:01:47.592 --> 00:01:52.800
b, c, a is equal to the
negative of its transpose.

00:01:52.800 --> 00:01:55.860
This is its transpose, so
the negative of its transpose

00:01:55.860 --> 00:02:02.430
is negative a, negative
c, negative b, negative a.

00:02:02.430 --> 00:02:04.590
And so the two
results from this mean

00:02:04.590 --> 00:02:10.350
that a would have to be equal
to negative a, which essentially

00:02:10.350 --> 00:02:12.145
means that a has
to be equal to 0.

00:02:14.900 --> 00:02:16.860
The only number,
especially in this set,

00:02:16.860 --> 00:02:19.068
where it equals the negative
of each other-- frankly,

00:02:19.068 --> 00:02:22.080
the only number where you equal
the negative of yourself is 0.

00:02:22.080 --> 00:02:23.650
So that means a equals 0.

00:02:23.650 --> 00:02:27.280
And it also means that b
must be equal to negative c,

00:02:27.280 --> 00:02:30.520
or that c must be
equal to negative b.

00:02:30.520 --> 00:02:34.240
So it also means that b
is equal to negative c.

00:02:34.240 --> 00:02:36.790
So that's what
either of these mean.

00:02:36.790 --> 00:02:39.380
Now, either of these
are going to be true.

00:02:39.380 --> 00:02:42.760
We want to find the number
of members of the set, such

00:02:42.760 --> 00:02:46.210
that either of these are
true and the determinant

00:02:46.210 --> 00:02:49.820
of the matrix is divisible by p.

00:02:49.820 --> 00:02:51.840
And remember, the
determinant-- let me just

00:02:51.840 --> 00:02:53.680
write this over
here as a reminder.

00:02:53.680 --> 00:03:00.560
The determinant of a matrix
A is equal to a, b, c, a.

00:03:00.560 --> 00:03:03.720
And it's just equal to
a times a, or a squared,

00:03:03.720 --> 00:03:08.060
minus b times c.

00:03:08.060 --> 00:03:11.470
So in this situation over
here, the symmetric matrix,

00:03:11.470 --> 00:03:18.100
the determinant here is going
to be equal to a squared minus

00:03:18.100 --> 00:03:19.660
b times c.

00:03:19.660 --> 00:03:22.490
But we know that b is equal
to c because it's symmetric.

00:03:22.490 --> 00:03:26.510
So it's going to be minus--
we could say minus b squared.

00:03:26.510 --> 00:03:28.500
And over here, what's
the determinant

00:03:28.500 --> 00:03:29.900
going to be equal to?

00:03:29.900 --> 00:03:34.780
The determinant of a is going
to be equal to a squared.

00:03:34.780 --> 00:03:37.130
Well, we know a is 0, so
that's just going to be 0.

00:03:37.130 --> 00:03:39.275
Minus b times c.

00:03:44.210 --> 00:03:50.030
If we substitute for c, we know
that c is equal to negative b.

00:03:50.030 --> 00:03:52.120
We could write negative
b is equal to c.

00:03:52.120 --> 00:03:55.560
So this is the same thing as
negative b times negative b,

00:03:55.560 --> 00:03:58.440
which is equal to b squared.

00:03:58.440 --> 00:03:59.984
So if the matrix
is symmetric, we

00:03:59.984 --> 00:04:01.400
know that its
determinant is going

00:04:01.400 --> 00:04:02.980
to be a squared minus b squared.

00:04:02.980 --> 00:04:06.090
If it's skew-symmetric,
its determinant

00:04:06.090 --> 00:04:09.260
is going to be equal to b
squared, which is frankly

00:04:09.260 --> 00:04:12.330
going to be the same
thing as c squared.

00:04:12.330 --> 00:04:15.030
Now let's think about how
many matrices satisfy this

00:04:15.030 --> 00:04:19.209
where this determinant is
going to be divisible by p.

00:04:19.209 --> 00:04:20.810
So this needs to
be divisible by p.

00:04:20.810 --> 00:04:22.240
So first let's just factor this.

00:04:22.240 --> 00:04:24.550
This is the same thing, just
because it looks tempting,

00:04:24.550 --> 00:04:27.870
as a plus b times a minus b.

00:04:27.870 --> 00:04:30.430
So this is the determinant
in the symmetric case.

00:04:30.430 --> 00:04:33.120
And it needs to be divisible
by p, which essentially means

00:04:33.120 --> 00:04:38.770
it needs to equal some
integer constant times p.

00:04:38.770 --> 00:04:41.840
And remember, p was
an odd prime number.

00:04:41.840 --> 00:04:46.180
And it's an odd prime number
larger than any of the possible

00:04:46.180 --> 00:04:46.850
a's or b's.

00:04:46.850 --> 00:04:49.200
The a's and b's can
only go up to p minus 1.

00:04:49.200 --> 00:04:51.510
They start at 0 and they
can only go up to p minus 1.

00:04:51.510 --> 00:04:55.590
So this is larger than
any of these numbers.

00:04:55.590 --> 00:04:57.380
Now let's think about
how this would work.

00:04:57.380 --> 00:05:00.730
In order for this to
be divisible by p,

00:05:00.730 --> 00:05:03.890
that means either the first
term is divisible by p

00:05:03.890 --> 00:05:07.142
or the second term
is divisible by p.

00:05:07.142 --> 00:05:08.100
Let's think about that.

00:05:08.100 --> 00:05:12.300
Either the first term or the
second term is divisible by p,

00:05:12.300 --> 00:05:14.840
or I guess in theory,
both terms divisible by p.

00:05:14.840 --> 00:05:18.792
So we could say either-- let
me just write it like this.

00:05:18.792 --> 00:05:20.750
So one of those are going
to be divisible by p.

00:05:20.750 --> 00:05:23.480
So let's think about how
these could be divisible by p.

00:05:23.480 --> 00:05:26.270
So the only way that
this term right over here

00:05:26.270 --> 00:05:32.362
could be divisible by p, so
a minus b divisible by p So

00:05:32.362 --> 00:05:33.320
let's think about that.

00:05:33.320 --> 00:05:34.980
How can this be divisible by p?

00:05:34.980 --> 00:05:38.930
a and b are both less than p.

00:05:38.930 --> 00:05:45.550
So their difference is always
going to be smaller than p.

00:05:45.550 --> 00:05:48.550
Their difference, by definition,
is going to be smaller than p.

00:05:48.550 --> 00:05:51.214
So how can it be divisible by p?

00:05:51.214 --> 00:05:53.880
Well, the only way it's going to
be divisible by p is if it's 0.

00:05:53.880 --> 00:05:57.640
0 is the only number smaller
then p that is divisible by p.

00:05:57.640 --> 00:06:00.320
So the only way that this is
going to be divisible by p,

00:06:00.320 --> 00:06:05.610
if a minus b is equal to 0.

00:06:05.610 --> 00:06:10.650
Or that's the same thing as
saying that a is equal to b.

00:06:10.650 --> 00:06:16.451
Now how many possibilities are
there for a being equal to b?

00:06:16.451 --> 00:06:18.450
Well, we can just look
at all the possibilities.

00:06:18.450 --> 00:06:22.430
There are, if we look over
here, there are p possibilities.

00:06:22.430 --> 00:06:25.050
You have one, two, three.

00:06:25.050 --> 00:06:27.300
Let me write in
a a darker color.

00:06:27.300 --> 00:06:28.300
a could be 0.

00:06:28.300 --> 00:06:29.350
That's one possibility.

00:06:29.350 --> 00:06:31.610
Two possibilities,
three possibilities.

00:06:31.610 --> 00:06:33.410
And if you were to just count
all the way up, it would be p.

00:06:33.410 --> 00:06:34.530
Notice when I'm
counting it, it's

00:06:34.530 --> 00:06:36.113
one more than the
number I'm counting.

00:06:36.113 --> 00:06:38.190
So this was p possibilities.

00:06:38.190 --> 00:06:39.804
And then b has to
be the same thing,

00:06:39.804 --> 00:06:41.470
so there's not much
freedom in selecting

00:06:41.470 --> 00:06:43.200
a b once you select the a.

00:06:43.200 --> 00:06:46.045
So there are p possibilities
where a is equal to b.

00:06:51.650 --> 00:06:55.200
There are p possibilities
of that happening.

00:06:55.200 --> 00:06:57.750
And let's just
keep in mind-- so a

00:06:57.750 --> 00:06:59.472
might be equal to 0,
which is equal to b.

00:06:59.472 --> 00:07:01.430
So this would include
both of them equalling 0,

00:07:01.430 --> 00:07:04.230
both of them equaling 1, both
of them equaling 2, all the way

00:07:04.230 --> 00:07:06.220
to both of them
equaling p minus 1.

00:07:06.220 --> 00:07:08.570
So there are p
possibilities of this.

00:07:08.570 --> 00:07:13.430
And let's think about how a
plus b could be divisible by kp.

00:07:13.430 --> 00:07:18.090
So remember, both a
and b are less than p.

00:07:18.090 --> 00:07:19.647
Both a and b are less than p.

00:07:19.647 --> 00:07:20.980
So this is a thought experiment.

00:07:25.540 --> 00:07:29.980
So a plus b could be
equal to 0 times p.

00:07:29.980 --> 00:07:31.930
That is a possibility.

00:07:31.930 --> 00:07:33.570
But frankly, when
you think about it,

00:07:33.570 --> 00:07:35.861
that means that a would have
to be equal to negative b.

00:07:35.861 --> 00:07:37.714
But both of these
are positive numbers,

00:07:37.714 --> 00:07:38.880
so that's not a possibility.

00:07:38.880 --> 00:07:40.965
So actually, a plus
b can't be 0 times p.

00:07:40.965 --> 00:07:43.640
I said 0 times p based on just
these are both smaller numbers.

00:07:43.640 --> 00:07:46.056
But then they would have to
be the negatives of each other

00:07:46.056 --> 00:07:48.450
and both of these
numbers are not negative.

00:07:48.450 --> 00:07:50.720
Well, I guess let
me take that back.

00:07:50.720 --> 00:07:55.450
This would work in the situation
where both a and b are 0.

00:07:55.450 --> 00:07:58.510
But we are ready included
that in this over here.

00:07:58.510 --> 00:08:00.850
We already included that
in these p possibilities

00:08:00.850 --> 00:08:03.170
where a and b are
both equal to 0.

00:08:03.170 --> 00:08:04.880
So let's not add to that.

00:08:04.880 --> 00:08:08.280
Now let's think about, because
this isn't a new possibility.

00:08:08.280 --> 00:08:11.750
a plus b is equal to p.

00:08:11.750 --> 00:08:13.230
Well, that's
completely possible.

00:08:13.230 --> 00:08:17.640
How many situations are there
where a plus b is equal to p?

00:08:17.640 --> 00:08:19.140
So if you pick your
a, it completely

00:08:19.140 --> 00:08:20.723
determines what your
b is going to be.

00:08:20.723 --> 00:08:22.900
Your b is going to be p minus a

00:08:22.900 --> 00:08:26.850
So can a be equal to 0?

00:08:26.850 --> 00:08:27.350
Well, no.

00:08:27.350 --> 00:08:29.790
If a was 0, b
would have to be p.

00:08:29.790 --> 00:08:33.909
And we know that b cannot be
p. b can at most be p minus 1.

00:08:33.909 --> 00:08:37.564
So a can be 1 and then b would
be p minus 1. a could be 2

00:08:37.564 --> 00:08:38.980
and then b could
be p minus 2, all

00:08:38.980 --> 00:08:42.630
the way to a being p minus
1 and then b would be 1.

00:08:42.630 --> 00:08:52.310
So over here, there are
p minus 1 possibilities.

00:08:55.730 --> 00:08:57.090
And notice they don't overlap.

00:08:57.090 --> 00:09:01.860
And none of these possibilities
is either a or b equal to 0

00:09:01.860 --> 00:09:06.420
or is either a or b
equal to each other.

00:09:06.420 --> 00:09:08.440
So they're never going
to be equal to each other

00:09:08.440 --> 00:09:12.690
because p over 2-- and you don't
have a situation where a plus

00:09:12.690 --> 00:09:15.484
b is equal to p, where
they're the same thing.

00:09:15.484 --> 00:09:17.900
Because if they were the same
thing, you'd go to this one.

00:09:17.900 --> 00:09:20.399
And you know this is never going
to be the case because this

00:09:20.399 --> 00:09:21.390
is a prime number.

00:09:21.390 --> 00:09:24.620
If it was possible that b could
equal a and this equation hold,

00:09:24.620 --> 00:09:26.830
you would have 2a
is equal to p, a

00:09:26.830 --> 00:09:28.660
is equal to p over
2, which would

00:09:28.660 --> 00:09:30.200
mean that p is divisible by 2.

00:09:30.200 --> 00:09:30.850
But it's prime.

00:09:30.850 --> 00:09:32.660
It can't be divisible by 2.

00:09:32.660 --> 00:09:35.610
So these are all p minus
1 distinct possibilities.

00:09:35.610 --> 00:09:37.420
They don't overlap
with any of these.

00:09:37.420 --> 00:09:40.340
a and b are never equal to 0.

00:09:40.340 --> 00:09:43.760
And well frankly, a and
b never equal each other

00:09:43.760 --> 00:09:45.270
in these possibilities.

00:09:45.270 --> 00:09:47.650
So so far, just from
the symmetric case,

00:09:47.650 --> 00:09:50.840
we've counted up minus
1 plus p possibilities.

00:09:50.840 --> 00:09:56.100
So we've counted up a total
of 2p minus 1 possibilities.

00:09:56.100 --> 00:09:58.480
I just added up all of
these possibilities.

00:09:58.480 --> 00:10:00.070
Now let's think
about the situation

00:10:00.070 --> 00:10:01.790
of being skew-symmetric.

00:10:01.790 --> 00:10:03.790
When you're skew-symmetric,
the determinant of a

00:10:03.790 --> 00:10:04.845
is equal to b squared.

00:10:06.932 --> 00:10:08.390
And we want the
possibilities where

00:10:08.390 --> 00:10:12.070
the determinant
is divisible by p.

00:10:12.070 --> 00:10:18.740
So b squared would have
to be equal to k times p.

00:10:18.740 --> 00:10:20.050
So let's think about this.

00:10:20.050 --> 00:10:30.330
If b times b is divisible by
p-- and remember, both of these

00:10:30.330 --> 00:10:33.130
are smaller than
p and p is prime.

00:10:33.130 --> 00:10:35.380
That means that either one
of these, or essentially it

00:10:35.380 --> 00:10:38.460
means that b would have
to be divisible by p.

00:10:38.460 --> 00:10:42.370
It means that b would
have to be divisible by p.

00:10:42.370 --> 00:10:46.420
And the only way that b is
going to be divisible by p,

00:10:46.420 --> 00:10:51.150
since b is smaller than
p, is if b is equal to 0.

00:10:51.150 --> 00:10:53.295
0 divided by any
number other than 0--

00:10:53.295 --> 00:10:55.170
because when you divide
by 0, it's undefined.

00:10:55.170 --> 00:10:57.850
But 0 divided by any other
number you have 0 remainder,

00:10:57.850 --> 00:10:59.960
so you can view it as divisible.

00:10:59.960 --> 00:11:02.260
So you could say, OK, here's
another possibility here.

00:11:02.260 --> 00:11:04.570
But we have to be very,
very, very careful.

00:11:04.570 --> 00:11:07.680
We already included that
possibility over here.

00:11:07.680 --> 00:11:12.000
We already included
the possibility

00:11:12.000 --> 00:11:17.580
of b equaling 0 in
this combination

00:11:17.580 --> 00:11:21.050
here where a is equal
to b is equal to 0.

00:11:21.050 --> 00:11:21.980
And let me be clear.

00:11:21.980 --> 00:11:24.206
We already said over here--
so you might say, wait.

00:11:24.206 --> 00:11:26.080
Over here, we're just
saying b is equal to 0.

00:11:26.080 --> 00:11:27.954
We're not saying that
a and b are equal to 0.

00:11:27.954 --> 00:11:29.810
But remember, in the
skew-symmetric case,

00:11:29.810 --> 00:11:32.559
a has to be equal to 0.

00:11:32.559 --> 00:11:34.100
So in the case where
b is equal to 0,

00:11:34.100 --> 00:11:37.419
you're saying that a
and b are equaling 0,

00:11:37.419 --> 00:11:38.960
but that was one of
the possibilities

00:11:38.960 --> 00:11:40.290
that we considered here.

00:11:40.290 --> 00:11:41.810
So that would work,
but we already

00:11:41.810 --> 00:11:42.900
counted that possibility.

00:11:42.900 --> 00:11:47.390
So there's a total of
2p minus 1 possibilities

00:11:47.390 --> 00:11:51.850
where a is either
symmetric or skew-symmetric

00:11:51.850 --> 00:11:53.900
and the determinant of
a is divisible by p.

00:11:53.900 --> 00:11:58.272
So 2p minus 1, the only
answer choice is D.


IIT JEE Integral with Binomial Expansion

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:02.310
The values of the
definite integral

00:00:02.310 --> 00:00:06.260
from 0 to 1 of x to the fourth
times 1 minus x to the fourth,

00:00:06.260 --> 00:00:09.890
all of that over 1 plus
x squared dx is or are--

00:00:09.890 --> 00:00:11.930
and they say are because
more than one of these

00:00:11.930 --> 00:00:13.096
might be the correct answer.

00:00:13.096 --> 00:00:16.221
This is one of those multiple
correct answer problems.

00:00:16.221 --> 00:00:18.220
So this is just a straight-up
definite integral.

00:00:18.220 --> 00:00:19.310
And it looks like
the best way to do

00:00:19.310 --> 00:00:21.210
it is really just expand
out the numerator,

00:00:21.210 --> 00:00:23.300
see if we can simplify
that with the denominator,

00:00:23.300 --> 00:00:25.750
and then just try to
take the antiderivative.

00:00:25.750 --> 00:00:26.470
So let's do that.

00:00:26.470 --> 00:00:31.390
So first, let's expand
1 minus x to the fourth.

00:00:31.390 --> 00:00:33.750
And when you're taking a
binomial to the fourth power,

00:00:33.750 --> 00:00:35.624
to remember the
coefficients, you, of course,

00:00:35.624 --> 00:00:37.692
can use the traditional
binomial coefficients.

00:00:37.692 --> 00:00:39.775
But I find it easier just
to use Pascal's triangle

00:00:39.775 --> 00:00:41.920
to remember what the
coefficients are.

00:00:41.920 --> 00:00:44.350
So if you have a binomial,
if you square it,

00:00:44.350 --> 00:00:49.710
the coefficients are 1, 1
plus 1 is 2, and then 1.

00:00:49.710 --> 00:00:52.247
And I've done several videos
where I explain why this works.

00:00:52.247 --> 00:00:53.830
And if you take it
to the third power,

00:00:53.830 --> 00:00:58.560
the coefficients
are 1, 3, 3, and 1.

00:00:58.560 --> 00:01:02.550
I'm just adding the two
branches that point to that one

00:01:02.550 --> 00:01:03.060
node there.

00:01:03.060 --> 00:01:04.410
I guess you could
view it that way.

00:01:04.410 --> 00:01:06.243
And then if you take
it to the fourth power,

00:01:06.243 --> 00:01:13.380
the coefficients are
1, 4, 6, 4, and 1.

00:01:13.380 --> 00:01:16.230
So if we want to take 1 minus
x to the fourth power-- and I'm

00:01:16.230 --> 00:01:17.396
going to write it like this.

00:01:17.396 --> 00:01:19.787
I'm going to write it
as negative x plus 1

00:01:19.787 --> 00:01:20.620
to the fourth power.

00:01:20.620 --> 00:01:22.203
And I want to do it
that way because I

00:01:22.203 --> 00:01:24.560
want the high-degree x
terms to be listed first.

00:01:24.560 --> 00:01:28.127
Then this is going
to be-- so you're

00:01:28.127 --> 00:01:30.460
going to start with this term
at the highest power, then

00:01:30.460 --> 00:01:31.790
this term at the 0 power.

00:01:31.790 --> 00:01:34.610
And then each successive
term, this term

00:01:34.610 --> 00:01:36.450
is going to lower in
power, and this one's

00:01:36.450 --> 00:01:37.450
going to go up in power.

00:01:37.450 --> 00:01:43.000
So it's going to be negative x
to the 1/4 times 1 to the 0--

00:01:43.000 --> 00:01:45.510
I'll just write it
here-- times 1 to the 0

00:01:45.510 --> 00:01:48.340
plus-- the coefficient of
that first one was just

00:01:48.340 --> 00:01:55.270
that 1, so plus 4 times
negative x to the third times 1

00:01:55.270 --> 00:02:00.130
to the 1 power, plus-- now
we have our 6-- plus 6 times

00:02:00.130 --> 00:02:04.550
negative x squared
times 1 squared.

00:02:04.550 --> 00:02:07.380
Or another way to think about
is the exponents on the 1

00:02:07.380 --> 00:02:09.720
and the negative x should
always add up to 4.

00:02:09.720 --> 00:02:14.200
And then you have plus-- now
we're at this 4-- plus 4 times

00:02:14.200 --> 00:02:19.540
negative x times 1 to
the third, and then

00:02:19.540 --> 00:02:22.990
plus 1 to the fourth power,
or negative x to the 0 times 1

00:02:22.990 --> 00:02:27.010
to the fourth, which is really
just 1 to the fourth power.

00:02:27.010 --> 00:02:30.360
So this is going to
simplify to-- so negative x

00:02:30.360 --> 00:02:31.380
to the fourth.

00:02:31.380 --> 00:02:34.010
Negative x to the fourth is the
same thing as x to the fourth.

00:02:34.010 --> 00:02:36.290
Negative 1 to the
fourth is just 1 times

00:02:36.290 --> 00:02:39.530
1 plus-- well, actually,
I should say minus,

00:02:39.530 --> 00:02:43.160
because negative x to the third
is negative x to the third,

00:02:43.160 --> 00:02:46.500
so this is minus
4x to the third.

00:02:46.500 --> 00:02:47.829
So we've done that term.

00:02:47.829 --> 00:02:49.120
This term will become positive.

00:02:49.120 --> 00:02:51.203
Negative x squared is the
same thing as x squared.

00:02:51.203 --> 00:02:54.340
So it's plus 6x squared.

00:02:54.340 --> 00:03:01.600
And then this is
negative 4x plus 1.

00:03:01.600 --> 00:03:03.850
Now, we want to multiply
this whole thing that we just

00:03:03.850 --> 00:03:07.040
expanded times x to the fourth.

00:03:07.040 --> 00:03:10.550
So let's multiply that whole
thing times x to the fourth.

00:03:10.550 --> 00:03:12.840
So the expression x
to the fourth times 1

00:03:12.840 --> 00:03:16.870
minus x, all of that
to the fourth power

00:03:16.870 --> 00:03:18.370
is going to be equal
to-- we'll just

00:03:18.370 --> 00:03:19.620
multiply this times
x to the fourth.

00:03:19.620 --> 00:03:20.870
That's pretty straightforward.

00:03:20.870 --> 00:03:24.840
It's going to be x to
the eighth minus 4x

00:03:24.840 --> 00:03:30.390
to the seventh plus 6x
to the sixth minus 4x

00:03:30.390 --> 00:03:35.580
to the fifth plus
x to the fourth.

00:03:35.580 --> 00:03:38.230
Now we want to divide
this bottom expression

00:03:38.230 --> 00:03:39.460
into the top expression.

00:03:39.460 --> 00:03:42.690
So we can just use some
algebraic long division.

00:03:42.690 --> 00:03:47.160
So we want to divide-- and I'll
write it as x squared plus 1.

00:03:47.160 --> 00:03:50.980
We want to divide x squared
plus 1 into this big thing,

00:03:50.980 --> 00:03:52.180
into this thing right here.

00:03:52.180 --> 00:03:55.322
So let me copy and paste that.

00:03:55.322 --> 00:03:57.030
We want to divide it
into this, and we're

00:03:57.030 --> 00:03:59.680
going to break out some
algebraic long division

00:03:59.680 --> 00:04:00.930
to do it.

00:04:00.930 --> 00:04:05.560
Let me draw the division
symbol right over there.

00:04:05.560 --> 00:04:08.460
And so we just say we'd look
at the highest-degree terms

00:04:08.460 --> 00:04:11.300
on every step. x squared
goes into x to the eighth x

00:04:11.300 --> 00:04:12.570
to the sixth times.

00:04:12.570 --> 00:04:15.640
So we put in the x
to the sixth place.

00:04:15.640 --> 00:04:19.079
x to the sixth times x
squared is x to the eighth.

00:04:19.079 --> 00:04:23.000
x to the sixth times
1 is x to the sixth.

00:04:23.000 --> 00:04:25.740
And then we subtract this
from the number up here,

00:04:25.740 --> 00:04:26.979
the expression up here.

00:04:26.979 --> 00:04:29.020
x to the eighth minus x
to the eighth is nothing.

00:04:29.020 --> 00:04:31.120
Negative 4x to the
seventh minus nothing

00:04:31.120 --> 00:04:34.200
is negative 4x to the seventh.

00:04:34.200 --> 00:04:41.002
6x to the 6 minus x to the
sixth is plus 5x to the sixth.

00:04:41.002 --> 00:04:43.210
And then these, you're
subtracting nothing from them,

00:04:43.210 --> 00:04:48.070
so it's minus 4x to the
5th plus x to the fourth.

00:04:48.070 --> 00:04:51.350
Now, x squared goes
into x to the seventh,

00:04:51.350 --> 00:04:54.810
or it goes into negative 4x
to the seventh, negative 4x

00:04:54.810 --> 00:04:55.725
to the fifth times.

00:05:00.760 --> 00:05:02.470
Let's scroll down a little bit.

00:05:02.470 --> 00:05:04.920
Negative 4x to the
fifth times x squared

00:05:04.920 --> 00:05:08.750
is negative 4x to the seventh.

00:05:08.750 --> 00:05:14.560
Negative 4x to the fifth times
1 is negative 4x to the fifth.

00:05:14.560 --> 00:05:16.900
Now we subtract this from that.

00:05:16.900 --> 00:05:17.810
These cancel out.

00:05:17.810 --> 00:05:18.800
These cancel out.

00:05:18.800 --> 00:05:28.330
And we are left with 5x to the
sixth plus x to the fourth.

00:05:28.330 --> 00:05:33.060
x squared goes into 5x to the
sixth 5x to the fourth times,

00:05:33.060 --> 00:05:35.380
so plus 5x to the fourth.

00:05:35.380 --> 00:05:36.830
I just arbitrarily
switched colors

00:05:36.830 --> 00:05:39.050
since yellow was
becoming monotonous.

00:05:39.050 --> 00:05:41.711
5x to the fourth times
x squared-- scroll down

00:05:41.711 --> 00:05:42.210
some more.

00:05:42.210 --> 00:05:48.010
5x to the fourth times x
squared is 5x to the sixth.

00:05:48.010 --> 00:05:51.410
5x to the fourth times
1 is 5x to the fourth.

00:05:54.410 --> 00:05:56.670
And now we subtract.

00:05:56.670 --> 00:05:57.420
We just subtract.

00:05:57.420 --> 00:06:00.740
Those cancel out. x to the
fourth minus 5x to the fourth

00:06:00.740 --> 00:06:04.220
is negative 4x to the fourth.

00:06:04.220 --> 00:06:05.580
Nothing to bring down.

00:06:05.580 --> 00:06:08.390
x squared goes into
negative x to the fourth.

00:06:08.390 --> 00:06:12.725
It goes into it negative
4x squared times.

00:06:15.720 --> 00:06:18.870
Negative 4x squared
times x squared

00:06:18.870 --> 00:06:22.140
is negative 4x to the fourth.

00:06:22.140 --> 00:06:28.050
Negative 4x squared times
1 is negative 4x squared.

00:06:28.050 --> 00:06:31.100
And now we subtract
that from that.

00:06:31.100 --> 00:06:35.660
You subtract this from 0, where
0 minus negative 4x squared

00:06:35.660 --> 00:06:39.450
is going to be
positive 4x squared.

00:06:39.450 --> 00:06:41.326
All of these steps,
we're always subtracting.

00:06:41.326 --> 00:06:43.325
Maybe I should draw the
subtraction symbol right

00:06:43.325 --> 00:06:44.020
over there.

00:06:44.020 --> 00:06:45.797
And then x squared
goes into 4x squared.

00:06:45.797 --> 00:06:46.880
We're in the home stretch.

00:06:46.880 --> 00:06:50.200
It goes into it 4
times, so plus 4.

00:06:50.200 --> 00:06:53.880
4 times x squared is 4x squared.

00:06:53.880 --> 00:06:57.450
4 times 1 is 4.

00:06:57.450 --> 00:07:00.130
And then we subtract
one last time.

00:07:00.130 --> 00:07:01.650
We subtract this whole thing.

00:07:01.650 --> 00:07:02.770
These cancel out.

00:07:02.770 --> 00:07:07.290
And then we're just left with
a remainder of negative 4.

00:07:07.290 --> 00:07:11.110
So this whole thing, this
whole expression over here,

00:07:11.110 --> 00:07:15.440
simplifies to this plus negative
4 over x squared plus 1.

00:07:15.440 --> 00:07:16.810
Let me rewrite it.

00:07:16.810 --> 00:07:18.780
Let me copy and paste it.

00:07:18.780 --> 00:07:20.730
So it turns into this.

00:07:20.730 --> 00:07:24.720
So copy and then let
me paste it down here.

00:07:24.720 --> 00:07:26.630
I wasted a lot of real estate.

00:07:26.630 --> 00:07:27.460
Paste.

00:07:27.460 --> 00:07:35.440
So it turns into this, this,
and then we have the minus 4

00:07:35.440 --> 00:07:38.000
over the x squared plus 1.

00:07:38.000 --> 00:07:45.844
So minus 4 over x squared
minus 4 over x squared plus 1.

00:07:45.844 --> 00:07:47.510
So this is the entire
expression that we

00:07:47.510 --> 00:07:50.300
have inside of the integral.

00:07:50.300 --> 00:07:52.170
So this is the same
thing as the integral

00:07:52.170 --> 00:07:59.729
of this from 0 to 1 times dx.

00:07:59.729 --> 00:08:02.020
So now we just have to take
the antiderivative of this.

00:08:02.020 --> 00:08:03.740
And this looks pretty
straightforward.

00:08:03.740 --> 00:08:06.720
The only complicated
part is this right here.

00:08:06.720 --> 00:08:10.910
But you might recognize that
the derivative with respect

00:08:10.910 --> 00:08:17.290
to x of arctangent of x is equal
to 1 over x squared plus 1.

00:08:17.290 --> 00:08:20.050
So that kind of simplifies
this part for us.

00:08:20.050 --> 00:08:21.427
So let's tackle the problem.

00:08:21.427 --> 00:08:23.010
So let's just take
the antiderivative.

00:08:23.010 --> 00:08:25.270
This is pretty straightforward.

00:08:25.270 --> 00:08:26.820
Antiderivative, we
just have to make

00:08:26.820 --> 00:08:28.403
sure we don't make
a careless mistake.

00:08:28.403 --> 00:08:30.280
Antiderivative of
x to the sixth is

00:08:30.280 --> 00:08:33.435
going to be x to
the seventh over 7.

00:08:33.435 --> 00:08:35.870
Or maybe I should say
1/7 x to the seventh.

00:08:35.870 --> 00:08:41.080
Antiderivative of
this is negative 4/6,

00:08:41.080 --> 00:08:45.600
which is the same thing as
negative 2/3 x to the sixth.

00:08:45.600 --> 00:08:50.451
Antiderivative of that is x to
the 5th, plus x to the fifth,

00:08:50.451 --> 00:08:50.950
right?

00:08:50.950 --> 00:08:52.410
I'm just raising the exponent.

00:08:52.410 --> 00:08:53.830
So that's where I get
the x to the fifth.

00:08:53.830 --> 00:08:55.288
And then I divide
by that exponent.

00:08:55.288 --> 00:08:57.190
So 5 divided by 5 is 1.

00:08:57.190 --> 00:09:01.880
And then I have x to the
third, or it's negative 4/3 x

00:09:01.880 --> 00:09:12.650
to the third plus 4x, and
then minus 4 arctangent of x.

00:09:18.085 --> 00:09:19.710
So let's think about
this a little bit.

00:09:19.710 --> 00:09:22.950
So we're going to evaluate it
at 1 and then subtract from that

00:09:22.950 --> 00:09:23.910
it, and evaluate at 0.

00:09:23.910 --> 00:09:26.610
So we need to evaluate this
at 1, and then from that,

00:09:26.610 --> 00:09:27.600
subtract it at 0.

00:09:27.600 --> 00:09:34.050
So if you evaluate it at
1, you have 1/7 minus 2/3

00:09:34.050 --> 00:09:42.460
plus 1 minus 4/3 plus 4.

00:09:42.460 --> 00:09:45.120
That's plus 4 right
over there, and then

00:09:45.120 --> 00:09:47.545
minus 4 arctangent of 1.

00:09:52.860 --> 00:09:56.550
So let's think about what
the arctangent of 1 is.

00:09:56.550 --> 00:09:58.460
So the arctangent
of 1 is essentially

00:09:58.460 --> 00:10:02.360
saying if I have
a right triangle,

00:10:02.360 --> 00:10:05.420
if I take the opposite over
the adjacent-- so the opposite

00:10:05.420 --> 00:10:10.710
is telling us give us the angle.

00:10:10.710 --> 00:10:13.110
So we're looking for this angle.

00:10:13.110 --> 00:10:17.840
We're looking for the angle
where the tangent of this angle

00:10:17.840 --> 00:10:20.050
is equal to 1.

00:10:20.050 --> 00:10:22.980
Or another way to say it is that
the opposite over the adjacent

00:10:22.980 --> 00:10:24.410
is equal to 1.

00:10:24.410 --> 00:10:26.110
Or another way to say
it is the opposite

00:10:26.110 --> 00:10:27.590
is equal to the adjacent.

00:10:30.280 --> 00:10:32.690
So this side has to
be equal to this side.

00:10:32.690 --> 00:10:36.070
This is going to be a
slope of 1 right over here.

00:10:36.070 --> 00:10:40.140
Now, we could try to look at
different types of triangles.

00:10:40.140 --> 00:10:42.170
And it might pop out at
you what kind this is,

00:10:42.170 --> 00:10:45.190
but you could just go to basic
geometry to think about this.

00:10:45.190 --> 00:10:46.817
This is a right
angle right here.

00:10:46.817 --> 00:10:47.900
These two sides are equal.

00:10:47.900 --> 00:10:49.024
These two angles are equal.

00:10:49.024 --> 00:10:51.780
So these two angles have to
evenly split the remaining 90

00:10:51.780 --> 00:10:54.570
degrees in the triangle
because it adds up to 180.

00:10:54.570 --> 00:10:58.760
So this is a 45-degree angle.

00:10:58.760 --> 00:11:01.270
But in general, when
they don't tell you

00:11:01.270 --> 00:11:03.660
whether you're in
radians or degrees,

00:11:03.660 --> 00:11:05.110
at least on most
exams I've seen,

00:11:05.110 --> 00:11:08.130
you should just assume
that you are in radians.

00:11:08.130 --> 00:11:11.530
So 45 degrees is the
same thing-- let's see.

00:11:11.530 --> 00:11:13.410
Pi is 180 degrees.

00:11:13.410 --> 00:11:16.630
Pi/2 is 90 degrees.

00:11:16.630 --> 00:11:19.630
Pi/4 is 45 degrees.

00:11:19.630 --> 00:11:24.280
So this is equal to-- that right
over there is pi/4 radians.

00:11:24.280 --> 00:11:28.290
So this over here
is pi/4 radians,

00:11:28.290 --> 00:11:30.080
or I could just say pi/4.

00:11:30.080 --> 00:11:32.200
Or another way to think
about it, tangent of pi/4

00:11:32.200 --> 00:11:32.870
is equal to 1.

00:11:32.870 --> 00:11:34.660
So this is just the
inverse function.

00:11:34.660 --> 00:11:38.200
So these two cancel out.

00:11:38.200 --> 00:11:40.490
So I just evaluate it at 1.

00:11:40.490 --> 00:11:42.310
Before I move forward,
let me subtract

00:11:42.310 --> 00:11:45.020
from that this whole
thing evaluated at 0.

00:11:45.020 --> 00:11:47.040
And this is a little bit
more straightforward.

00:11:47.040 --> 00:11:49.980
All of these things are going to
be 0 when you evaluate it at 0.

00:11:49.980 --> 00:11:53.060
And what's the arctangent of 0?

00:11:53.060 --> 00:11:54.740
Well, tangent of 0 is 0.

00:11:54.740 --> 00:11:58.160
If this angle is 0, if
you just had a flat line,

00:11:58.160 --> 00:11:59.550
then your slope is flat.

00:11:59.550 --> 00:12:00.900
So this is just going to be 0.

00:12:00.900 --> 00:12:03.770
So this whole thing evaluated
at 0 is going to be 0.

00:12:03.770 --> 00:12:06.010
So the whole value of
our definite integral

00:12:06.010 --> 00:12:08.090
is going to be this
thing over here.

00:12:08.090 --> 00:12:09.740
So it's going to be
all of this stuff,

00:12:09.740 --> 00:12:14.870
and then this last
term right over here

00:12:14.870 --> 00:12:19.065
is negative 4 times pi/4, so
it's going to be negative pi.

00:12:19.065 --> 00:12:21.190
And all of these are going
to add up to some number

00:12:21.190 --> 00:12:22.340
without a pi in it.

00:12:22.340 --> 00:12:24.010
So if you are under
time pressure,

00:12:24.010 --> 00:12:26.252
it's actually a good idea
to just look at the answers

00:12:26.252 --> 00:12:28.085
before we even add up
all of those fractions

00:12:28.085 --> 00:12:30.382
and say, which of these
have a negative pi in them?

00:12:30.382 --> 00:12:32.340
And only one of them has
a negative pi in them,

00:12:32.340 --> 00:12:35.897
so our answer is most
probably going to be this.

00:12:35.897 --> 00:12:38.230
If you're under time pressure,
I would just circle that.

00:12:38.230 --> 00:12:39.479
It's clearly not any of these.

00:12:39.479 --> 00:12:41.347
None of these have a
negative pi in them.

00:12:41.347 --> 00:12:43.180
And there's no way that
that negative pi can

00:12:43.180 --> 00:12:44.740
get altered by this
stuff over here.

00:12:44.740 --> 00:12:46.500
So you wouldn't have to
worry about adding it.

00:12:46.500 --> 00:12:48.833
But since we want to make
sure we have the right answer,

00:12:48.833 --> 00:12:50.980
let's add up all
of these fractions.

00:12:50.980 --> 00:12:53.370
So let's think about
what we can do here.

00:12:59.420 --> 00:13:00.350
So let's see.

00:13:00.350 --> 00:13:03.320
Negative 2/3 minus 4/3, let
me take those two terms.

00:13:03.320 --> 00:13:09.090
Negative 2/3 minus 4/3 is
negative 6/3, or negative 2.

00:13:09.090 --> 00:13:16.240
So those become negative 2 plus
1 plus 4, and then plus 1/7.

00:13:19.010 --> 00:13:23.607
So this is 3 plus 1/7.

00:13:23.607 --> 00:13:25.190
And of course, we
have the negative pi

00:13:25.190 --> 00:13:26.880
out there, minus pi.

00:13:26.880 --> 00:13:29.550
And this is the
same thing as 21/7

00:13:29.550 --> 00:13:36.230
plus 1/7 minus pi, which
is equal to 22/7 minus pi.

00:13:36.230 --> 00:13:41.730
So that is indeed the only
answer-- 22/7 minus pi.

00:13:41.730 --> 00:13:43.590
And we're done.