IIT JEE Hairy Trig and Algebra (Part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:01.370
Welcome back.

00:00:01.370 --> 00:00:03.600
Where we left the last
problem, we essentially

00:00:03.600 --> 00:00:06.750
found ourselves having to solve
this equation right over here.

00:00:06.750 --> 00:00:09.260
And where we left off, we
said, well, maybe, just maybe,

00:00:09.260 --> 00:00:11.610
we might be able to
simplify this a little bit

00:00:11.610 --> 00:00:14.180
if we can find some common
factors between the numerator

00:00:14.180 --> 00:00:15.200
and the denominator.

00:00:15.200 --> 00:00:17.879
And I want to do it here because
if I multiplied this out,

00:00:17.879 --> 00:00:19.670
and multiply it times
the square root of 3,

00:00:19.670 --> 00:00:22.620
and then subtract
it or add it to that

00:00:22.620 --> 00:00:25.367
so I got just a straight-up
polynomial, that polynomial

00:00:25.367 --> 00:00:27.200
is going to have the
square root of 3 in it.

00:00:27.200 --> 00:00:28.960
It's going to be a really,
really messy thing.

00:00:28.960 --> 00:00:30.760
And it's still going
to be fourth degree.

00:00:30.760 --> 00:00:34.080
And at least this way,
I have some chance

00:00:34.080 --> 00:00:35.220
of being able to solve it.

00:00:35.220 --> 00:00:37.910
At least I just have integer
coefficients over here.

00:00:37.910 --> 00:00:40.000
And the only way that
this is going to simplify

00:00:40.000 --> 00:00:43.370
is if it has any common factors
with the stuff down here.

00:00:43.370 --> 00:00:46.110
And when you look
over here, this right

00:00:46.110 --> 00:00:48.680
here is actually going
to have complex roots

00:00:48.680 --> 00:00:52.550
if you did the negative b plus
or minus the square root of b

00:00:52.550 --> 00:00:54.350
squared minus 4ac over 2a.

00:00:54.350 --> 00:00:56.675
The b squared
minus 4ac over here

00:00:56.675 --> 00:00:58.050
is actually going
to be negative.

00:00:58.050 --> 00:01:00.130
So it's not going to
have any real roots.

00:01:00.130 --> 00:01:04.569
So that is not going to have any
easy, real roots, but this is.

00:01:04.569 --> 00:01:10.390
The x squared plus 1, that
has x plus 1 and x minus 1.

00:01:10.390 --> 00:01:13.670
Or another way to think
about it is roots to this

00:01:13.670 --> 00:01:15.800
are negative 1 and positive 1.

00:01:15.800 --> 00:01:17.722
So if either of those
are roots to this,

00:01:17.722 --> 00:01:18.680
then we're in business.

00:01:18.680 --> 00:01:21.050
And actually, just
looking at this,

00:01:21.050 --> 00:01:23.630
there's a decent chance,
because in general,

00:01:23.630 --> 00:01:28.640
if you want to think about the
roots of really any n-th degree

00:01:28.640 --> 00:01:30.819
polynomial, the roots,
if you think about them--

00:01:30.819 --> 00:01:32.110
let me just write it over here.

00:01:32.110 --> 00:01:36.453
If I have a bunch of roots--
x minus r1, x minus r2,

00:01:36.453 --> 00:01:40.220
x minus r3, and
then let's say we

00:01:40.220 --> 00:01:41.867
have a fourth
degree, x minus r4.

00:01:41.867 --> 00:01:43.200
So let's say we have four roots.

00:01:43.200 --> 00:01:44.360
When you multiply
this whole thing,

00:01:44.360 --> 00:01:47.110
you're going to have x to the
fourth plus blah, blah, blah,

00:01:47.110 --> 00:01:48.760
blah, blah, all the way.

00:01:48.760 --> 00:01:56.000
And then the last term over here
is going to be r1, r2, r3, r4.

00:01:56.000 --> 00:01:58.680
It's going to be the
product of all of these.

00:01:58.680 --> 00:02:00.420
And so if you wanted
to put this equation

00:02:00.420 --> 00:02:02.130
right or this
expression in this form,

00:02:02.130 --> 00:02:06.490
you divide the whole
expression by 2.

00:02:06.490 --> 00:02:09.009
Or you can kind of say, well,
let's just look at this thing

00:02:09.009 --> 00:02:09.987
and divide it by 2.

00:02:09.987 --> 00:02:11.820
And then you would have
x to the fourth plus

00:02:11.820 --> 00:02:14.070
2x, all of that plus 1/2.

00:02:14.070 --> 00:02:17.900
And so this thing would
have to, in this context,

00:02:17.900 --> 00:02:19.910
be equal to 1 over 2.

00:02:19.910 --> 00:02:22.200
And away, this is
kind of a-- it's

00:02:22.200 --> 00:02:24.410
not a fail-proof
technique, but it

00:02:24.410 --> 00:02:26.480
is one way to kind of
discover roots, is you,

00:02:26.480 --> 00:02:30.300
say, OK, the roots could
be the factors of this

00:02:30.300 --> 00:02:32.630
divided by the factors of that.

00:02:32.630 --> 00:02:35.012
And 1 is definitely a
factor of both of those.

00:02:35.012 --> 00:02:36.970
And actually, if you
think about it, negative 1

00:02:36.970 --> 00:02:37.678
could be as well.

00:02:37.678 --> 00:02:38.820
So we feel good.

00:02:38.820 --> 00:02:40.070
But let's actually try it out.

00:02:40.070 --> 00:02:41.986
Instead of going through
all of this argument,

00:02:41.986 --> 00:02:43.890
let's actually see if
either 1 or negative 1

00:02:43.890 --> 00:02:46.130
are factors of
this thing up here.

00:02:46.130 --> 00:02:47.880
If they're not,
then we just have

00:02:47.880 --> 00:02:52.942
to move ahead and multiply
this thing out, multiply it

00:02:52.942 --> 00:02:54.650
by square root of 3,
and then just solve.

00:02:54.650 --> 00:02:55.860
But we can hope.

00:02:55.860 --> 00:02:59.200
Usually the test writers
aren't that big sadists.

00:02:59.200 --> 00:03:02.100
Otherwise, it would be
impossible to solve.

00:03:02.100 --> 00:03:04.550
So I'm feeling good about this.

00:03:04.550 --> 00:03:06.170
So let's just try it with 1.

00:03:06.170 --> 00:03:08.280
So if we try 1 here,
so let's put 1 here

00:03:08.280 --> 00:03:09.770
and see if we get 0 over here.

00:03:09.770 --> 00:03:13.709
So if we get 2 times 1 to
the fourth plus-- so let

00:03:13.709 --> 00:03:14.500
me just write that.

00:03:14.500 --> 00:03:17.810
So 2 times 1 to the fourth
is just 2, plus 2 times

00:03:17.810 --> 00:03:22.130
1 to the third is just 2, minus
3 times 1 squared is just 3,

00:03:22.130 --> 00:03:24.270
minus 2 plus 1.

00:03:24.270 --> 00:03:27.100
So this is 4 minus 5 plus 1.

00:03:27.100 --> 00:03:30.050
That indeed equals 0.

00:03:30.050 --> 00:03:32.500
And let's try negative 1.

00:03:32.500 --> 00:03:34.390
So this is 2 times
negative 1 to the fourth.

00:03:34.390 --> 00:03:35.800
That's still 2.

00:03:35.800 --> 00:03:39.410
Plus 2 times negative 1 to the
third, so that's negative 2.

00:03:39.410 --> 00:03:43.480
Minus 3 times negative 1
squared, that's negative 3.

00:03:43.480 --> 00:03:47.340
And then minus 2 times negative
1, so that's plus 2 plus 1.

00:03:47.340 --> 00:03:51.860
So this is 0 minus
3 plus 2 plus 1.

00:03:51.860 --> 00:03:52.860
This also cancels out.

00:03:52.860 --> 00:03:55.100
This also gets us 0.

00:03:55.100 --> 00:03:58.900
So we now know that both
x plus 1 and x minus 1

00:03:58.900 --> 00:04:00.790
are factors of
this thing up here.

00:04:00.790 --> 00:04:02.720
Or another way to
think about it is

00:04:02.720 --> 00:04:05.150
their product is a factor
of this thing up here. x

00:04:05.150 --> 00:04:07.330
squared minus 1 is
a factor of this.

00:04:07.330 --> 00:04:09.824
So if we were to divide x
squared minus 1 into this,

00:04:09.824 --> 00:04:11.240
we should get a
reasonable answer.

00:04:11.240 --> 00:04:13.365
We shouldn't get a remainder,
I guess I should say.

00:04:13.365 --> 00:04:14.280
So let's do that.

00:04:14.280 --> 00:04:16.155
So we're going to get
a little practice doing

00:04:16.155 --> 00:04:17.630
polynomial long division.

00:04:17.630 --> 00:04:23.890
So if we divide x squared minus
1 into this thing up here,

00:04:23.890 --> 00:04:28.470
2x to the fourth plus
2x to the third minus

00:04:28.470 --> 00:04:33.970
3x squared minus 2x plus 1.

00:04:33.970 --> 00:04:36.330
So this is good practice on
polynomial long division--

00:04:36.330 --> 00:04:38.750
highest degree term going
into highest degree term.

00:04:38.750 --> 00:04:43.720
x squared goes into 2x to
the fourth 2x squared times.

00:04:43.720 --> 00:04:47.070
So let's put that in
the 2x squared spot.

00:04:47.070 --> 00:04:51.370
2x squared times x squared
is 2x to the fourth.

00:04:51.370 --> 00:04:55.650
2x squared times negative
1 is negative 2x squared.

00:04:55.650 --> 00:04:57.400
And now let's subtract
this from up there.

00:04:57.400 --> 00:05:00.510
So we'll multiply by it
negative 1 and then add it.

00:05:00.510 --> 00:05:07.550
And so this gives
us 2x to the third.

00:05:07.550 --> 00:05:13.770
Negative 3 plus 2, so it's
minus x squared minus 2x plus 1.

00:05:13.770 --> 00:05:20.360
And then x squared goes into
2x to the third 2x times.

00:05:20.360 --> 00:05:23.930
So this is plus 2x.

00:05:23.930 --> 00:05:26.730
2x times x squared
is 2x to the third.

00:05:26.730 --> 00:05:32.400
2x times negative
1 is negative 2x.

00:05:32.400 --> 00:05:34.370
And then we can subtract
this from there,

00:05:34.370 --> 00:05:35.840
or you could say
add the negative.

00:05:35.840 --> 00:05:37.131
So let's just add the negative.

00:05:37.131 --> 00:05:38.760
It's easier to keep
track of things.

00:05:38.760 --> 00:05:39.900
So those cancel out.

00:05:39.900 --> 00:05:44.550
You get negative x squared--
these cancel out-- plus 1.

00:05:44.550 --> 00:05:47.790
x squared goes into negative
x squared minus-- or negative

00:05:47.790 --> 00:05:49.020
1 times.

00:05:49.020 --> 00:05:52.310
Negative 1 times x squared
is negative x squared.

00:05:52.310 --> 00:05:54.830
Negative 1 times
negative 1 is positive 1.

00:05:54.830 --> 00:05:58.620
And indeed, we get no remainder.

00:05:58.620 --> 00:06:00.500
So we can write
this thing up here.

00:06:00.500 --> 00:06:03.610
We can write it as
the product of this.

00:06:03.610 --> 00:06:06.150
So it is equal
to-- so this thing

00:06:06.150 --> 00:06:12.590
we can rewrite as 2x squared
plus 2x minus 1 times x

00:06:12.590 --> 00:06:13.520
squared minus 1.

00:06:13.520 --> 00:06:16.890
So this times this is
equal to this, so times

00:06:16.890 --> 00:06:18.690
x squared minus 1.

00:06:18.690 --> 00:06:20.660
So we just factored it.

00:06:20.660 --> 00:06:22.934
And so that simplifies
things a good bit.

00:06:22.934 --> 00:06:24.350
It doesn't make
it trivial, but it

00:06:24.350 --> 00:06:25.880
simplifies things a good bit.

00:06:25.880 --> 00:06:29.200
So now we can cancel
that with that.

00:06:29.200 --> 00:06:31.940
And now at least we have
things in second degree.

00:06:31.940 --> 00:06:35.190
We got rid of these third
and fourth degree terms.

00:06:35.190 --> 00:06:37.990
So let's now rewrite things.

00:06:37.990 --> 00:06:40.480
So now let's multiply both
sides of this equation

00:06:40.480 --> 00:06:42.300
by x squared plus x plus 1.

00:06:42.300 --> 00:06:44.980
So we're going to get
the square root of 3-- so

00:06:44.980 --> 00:06:46.420
let me just write it out.

00:06:46.420 --> 00:06:49.420
So I'm going to get the
square root of 3 times

00:06:49.420 --> 00:06:54.820
x squared plus the
square root of 3 times x

00:06:54.820 --> 00:06:57.780
plus the square
root of 3 times 1--

00:06:57.780 --> 00:07:01.170
so I just multiplied the
left-hand side by what we have

00:07:01.170 --> 00:07:05.584
left in the denominator-- is
equal to what we have left

00:07:05.584 --> 00:07:07.250
in the numerator,
because if we multiply

00:07:07.250 --> 00:07:09.320
the right side by this,
it obviously cancels out.

00:07:09.320 --> 00:07:18.440
So this is going to be equal
to 2x squared plus 2x minus 1.

00:07:18.440 --> 00:07:24.900
And now if we want to-- now
let's see what we can do here.

00:07:24.900 --> 00:07:30.130
Well, we can subtract the
stuff on the right side

00:07:30.130 --> 00:07:32.230
from the left-hand side.

00:07:32.230 --> 00:07:37.170
And so we would get the
square root of 3 minus 2 times

00:07:37.170 --> 00:07:50.530
x squared plus the square
root of 3 minus 2 times x--

00:07:50.530 --> 00:07:52.780
just subtracting this
stuff from that side--

00:07:52.780 --> 00:07:58.750
plus the square root
of 3 plus 1, right?

00:07:58.750 --> 00:07:59.860
Did I do that right?

00:07:59.860 --> 00:08:02.270
Square root of 3, I subtracted
the x squared from that,

00:08:02.270 --> 00:08:05.064
so square root of 3 minus
2, square root of 3 minus 2.

00:08:05.064 --> 00:08:06.480
And you have the
square root of 3.

00:08:06.480 --> 00:08:09.310
Subtracting a negative 1,
I get a plus 1 over here.

00:08:09.310 --> 00:08:11.310
And since I subtracted
all of this stuff

00:08:11.310 --> 00:08:12.856
from both sides
of the equation--

00:08:12.856 --> 00:08:14.230
this is just the
left-hand side--

00:08:14.230 --> 00:08:16.530
the right-hand side is now 0.

00:08:16.530 --> 00:08:19.680
I subtracted this from the
right-hand side to get 0.

00:08:19.680 --> 00:08:22.464
So now we have-- this almost
looks ridiculously simple.

00:08:22.464 --> 00:08:24.630
Even though I have these
ridiculous radicals and all

00:08:24.630 --> 00:08:29.280
that, I have a straight-up
quadratic equation right here,

00:08:29.280 --> 00:08:31.750
so I can use the
quadratic formula.

00:08:31.750 --> 00:08:35.400
So the roots here, so x is
going to be equal to negative b.

00:08:35.400 --> 00:08:36.740
This is b right here.

00:08:36.740 --> 00:08:38.640
So negative b, you
can swap those.

00:08:38.640 --> 00:08:47.092
So it's 2 minus the square root
of 3 plus or minus b squared.

00:08:47.092 --> 00:08:48.300
So let me just write it here.

00:08:48.300 --> 00:08:54.580
So the square root of 3 minus
2 squared minus 4 times a.

00:08:54.580 --> 00:08:59.350
So that's the square root
of 3 minus 2 times c.

00:08:59.350 --> 00:09:01.820
Well, c is the entire constant
term right over there.

00:09:01.820 --> 00:09:10.920
So it's times square root of
3 plus 1, all of that over 2a,

00:09:10.920 --> 00:09:13.650
so all of that over
this business over here.

00:09:13.650 --> 00:09:17.081
So that's 2 root 3 minus 4.

00:09:17.081 --> 00:09:18.830
So we thought we were
in the home stretch,

00:09:18.830 --> 00:09:22.140
but now we have to do some
major radical simplification.

00:09:22.140 --> 00:09:23.990
So once again,
these test writers

00:09:23.990 --> 00:09:25.740
are not pleasant people.

00:09:25.740 --> 00:09:30.520
So let's try our
best to tackle this.

00:09:30.520 --> 00:09:35.180
So let's go into-- let's
go inside the radical right

00:09:35.180 --> 00:09:36.300
over here.

00:09:36.300 --> 00:09:38.840
Let's go inside of the
radical right over here.

00:09:38.840 --> 00:09:41.806
And you have the square
root of 3 minus 2.

00:09:41.806 --> 00:09:43.430
Actually, I'm going
to leave you there.

00:09:43.430 --> 00:09:44.950
I'm approaching 10
minutes, and frankly, I

00:09:44.950 --> 00:09:45.870
need a drink of water.

00:09:45.870 --> 00:09:49.225
This problem is maybe the most
tiring problem I've ever done.

00:09:49.225 --> 00:09:50.100
I'll leave you there.

00:09:50.100 --> 00:09:53.690
We just have to simplify
this in the next video.


IIT JEE Hairy Trig and Algebra (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.680
"Let ABC be a triangle,
such that angle ABC

00:00:04.680 --> 00:00:05.890
is equal to pi over 6."

00:00:05.890 --> 00:00:07.070
Let me draw it.

00:00:07.070 --> 00:00:12.950
So that's A, B, and
C. And they tell us

00:00:12.950 --> 00:00:18.490
that angle ACB, this angle
right over here, is pi over 6.

00:00:18.490 --> 00:00:20.800
I'm assuming it's
pi over 6 radians.

00:00:20.800 --> 00:00:23.790
Or we could view
that as 30 degrees.

00:00:23.790 --> 00:00:25.710
And then they tell
us, "Let a, b, and c

00:00:25.710 --> 00:00:28.947
denote the lengths of the sides
opposite to capital A, B, C,

00:00:28.947 --> 00:00:29.530
respectively."

00:00:29.530 --> 00:00:35.130
So this is a, this is b, and
this is c, right over there.

00:00:35.130 --> 00:00:38.730
The values of x, for which
a is equal to this thing, b

00:00:38.730 --> 00:00:41.791
is equal to this thing, and
c is equal to this thing is?

00:00:41.791 --> 00:00:42.790
So let's think about it.

00:00:42.790 --> 00:00:45.540
One, we need to find a
relationship between A, B,

00:00:45.540 --> 00:00:47.144
and C in this angle.

00:00:47.144 --> 00:00:48.810
And then once we have
that relationship,

00:00:48.810 --> 00:00:51.360
we can substitute them
with these expressions.

00:00:51.360 --> 00:00:53.995
And then essentially
try to solve for x.

00:00:53.995 --> 00:00:56.120
And the first thing you
want to think about is, OK,

00:00:56.120 --> 00:00:57.970
how can I relate
the length of b,

00:00:57.970 --> 00:01:02.150
this lowercase c, this lowercase
a with this angle here.?

00:01:02.150 --> 00:01:04.129
And what should pop
out into your brain

00:01:04.129 --> 00:01:05.870
is the law of cosines.

00:01:05.870 --> 00:01:09.550
Law of cosines.

00:01:09.550 --> 00:01:12.300
Law of cosines that
tells us that c squared

00:01:12.300 --> 00:01:15.592
is equal to a squared
plus b squared.

00:01:15.592 --> 00:01:18.050
It's essentially the Pythagorean
theorem with an adjustment

00:01:18.050 --> 00:01:20.170
right here, because this
isn't a right triangle.

00:01:20.170 --> 00:01:23.830
So a squared plus b
squared minus 2ab times

00:01:23.830 --> 00:01:26.970
the cosine of C.

00:01:26.970 --> 00:01:29.930
Now in this case,
angle C right here--

00:01:29.930 --> 00:01:33.610
or I should say, in this case,
it's angle ACB-- this over here

00:01:33.610 --> 00:01:37.950
is going to be 30 degrees.

00:01:37.950 --> 00:01:40.860
So what's the cosine
of 30 degrees?

00:01:40.860 --> 00:01:43.180
And just in case you
forgot it, although I

00:01:43.180 --> 00:01:45.960
assume if you're going to
take the IIT exam, the JEE,

00:01:45.960 --> 00:01:50.200
you're going to need all the
help you need in terms of time.

00:01:50.200 --> 00:01:51.620
So this is 30 degrees.

00:01:51.620 --> 00:01:53.070
That is 1.

00:01:53.070 --> 00:01:56.560
Then this is 1/2, the side
opposite the 30-degree side.

00:01:56.560 --> 00:01:57.720
This will be 60 degrees.

00:01:57.720 --> 00:01:59.370
It's going to be a 30-60-90.

00:01:59.370 --> 00:02:01.350
And so this is square
root of 3 over 2.

00:02:01.350 --> 00:02:03.980
Cosine is adjacent
over hypotenuse,

00:02:03.980 --> 00:02:06.720
so square root of 3/2 over 1.

00:02:06.720 --> 00:02:08.720
So this, in this
example, it's going

00:02:08.720 --> 00:02:16.100
to be c squared is equal
to a squared plus b squared

00:02:16.100 --> 00:02:17.965
minus 2ab.

00:02:17.965 --> 00:02:21.280
And then this whole thing
is square root of 3/2.

00:02:21.280 --> 00:02:25.140
So times square root of 3/2.

00:02:25.140 --> 00:02:27.200
And these 2's cancel out.

00:02:27.200 --> 00:02:31.660
And so we get c squared is equal
to a squared plus b squared

00:02:31.660 --> 00:02:36.130
minus the square root of 3ab.

00:02:36.130 --> 00:02:40.715
Now the next thing I'd want
to do is substitute for a,

00:02:40.715 --> 00:02:42.900
I'd substitute with this
expression over here.

00:02:42.900 --> 00:02:46.160
For b, I want to substitute
with this expression over here.

00:02:46.160 --> 00:02:49.160
And then for c, I want to
substitute this expression

00:02:49.160 --> 00:02:50.106
right over here.

00:02:50.106 --> 00:02:51.980
But I want to be careful,
I want to make sure

00:02:51.980 --> 00:02:54.230
that it's in a form that I
can, in some way, simplify.

00:02:54.230 --> 00:02:56.900
Because if I just take this
a over here-- this x squared

00:02:56.900 --> 00:02:58.505
plus x plus 1-- if
I square it, I'm

00:02:58.505 --> 00:03:00.630
going to end up with
something to the fourth power.

00:03:00.630 --> 00:03:03.407
This b, I'm going to end up with
something to the fourth power.

00:03:03.407 --> 00:03:05.740
And then I have the square
root of 3 business over here.

00:03:05.740 --> 00:03:09.150
So let's see if we can at least
rearrange it, or at least maybe

00:03:09.150 --> 00:03:11.250
quarantine the square
root of 3, little bit,

00:03:11.250 --> 00:03:13.880
to make it at least
something that we might have

00:03:13.880 --> 00:03:16.390
some hopes of factoring,
or solving in some way.

00:03:16.390 --> 00:03:19.260
And I want to make a little
clarifying point here.

00:03:19.260 --> 00:03:22.020
In most Western exams,
if you're taking the SAT

00:03:22.020 --> 00:03:24.560
or you're taking any type of--
even at a math competition--

00:03:24.560 --> 00:03:27.000
if you find yourself
grinding through math,

00:03:27.000 --> 00:03:30.730
multiplying huge polynomials
times each other,

00:03:30.730 --> 00:03:32.540
you're probably doing
something wrong.

00:03:32.540 --> 00:03:35.724
Most Western exams--
like the SAT--

00:03:35.724 --> 00:03:37.140
you kind of have
to see the trick,

00:03:37.140 --> 00:03:39.110
you kind of have to see the
elegance in the problem.

00:03:39.110 --> 00:03:40.840
But once you see the
elegance, the problem

00:03:40.840 --> 00:03:42.000
is usually pretty
straightforward.

00:03:42.000 --> 00:03:43.760
In fact, if you're
grinding through math,

00:03:43.760 --> 00:03:46.480
it's usually a giveaway that
you're doing something wrong.

00:03:46.480 --> 00:03:48.867
On these JEE math
problems-- and I've

00:03:48.867 --> 00:03:50.450
only recently gotten
exposure to them.

00:03:50.450 --> 00:03:53.640
I obviously never took them,
not having grown up in India.

00:03:53.640 --> 00:03:55.904
But on these, you have
to kind of see the trick.

00:03:55.904 --> 00:03:57.570
And then you have to
grind through math,

00:03:57.570 --> 00:04:00.360
and you have to use every
trick at your disposal

00:04:00.360 --> 00:04:04.510
to make the grinding through
math a little bit easier.

00:04:04.510 --> 00:04:08.280
So the test writers for the
JEE are definitely a little,

00:04:08.280 --> 00:04:10.050
actually, a lot more
cruel than the test

00:04:10.050 --> 00:04:14.302
writers for most Western
exams that I've seen.

00:04:14.302 --> 00:04:15.760
So what I want to
do is, let's just

00:04:15.760 --> 00:04:17.800
isolate the square root of 3ab.

00:04:17.800 --> 00:04:19.649
So let's add it to both sides.

00:04:19.649 --> 00:04:23.510
So the square root
of 3ab plus-- so

00:04:23.510 --> 00:04:26.680
let's just add that--
square root of 3ab.

00:04:26.680 --> 00:04:29.930
And let's get the c squared
onto this side of the equation.

00:04:29.930 --> 00:04:32.970
So let me subtract c
squared from both sides

00:04:32.970 --> 00:04:34.210
of the equation.

00:04:34.210 --> 00:04:36.430
So minus c squared.

00:04:36.430 --> 00:04:39.040
And then minus c squared.

00:04:39.040 --> 00:04:42.540
And so I will have, on this
side, those cancel out.

00:04:42.540 --> 00:04:45.410
And I just have the
square root of 3ab

00:04:45.410 --> 00:04:51.960
is equal to a squared plus b
squared-- these cancel out--

00:04:51.960 --> 00:04:53.990
minus c squared--
I'm just rearranging

00:04:53.990 --> 00:04:56.290
the whole thing--
minus c squared.

00:04:56.290 --> 00:04:59.370
All of that, well right
now, that's all it is.

00:04:59.370 --> 00:05:02.390
And now I can divide
both sides of this by ab.

00:05:02.390 --> 00:05:04.200
So if I divide
that by ab and then

00:05:04.200 --> 00:05:08.015
I divide the right-hand side
by ab, that cancels out.

00:05:08.015 --> 00:05:10.140
And I've kind of quarantined
that square root of 3.

00:05:10.140 --> 00:05:12.310
So I'll hopefully only
have integer stuff

00:05:12.310 --> 00:05:14.050
on this side of the equation.

00:05:14.050 --> 00:05:19.270
So I have the
square root of 3 is

00:05:19.270 --> 00:05:22.237
equal to a squared plus b
squared minus c squared.

00:05:22.237 --> 00:05:24.320
So let's figure out what
each of these things are.

00:05:24.320 --> 00:05:27.720
Let's expand them out,
actually, before we rewrite it.

00:05:27.720 --> 00:05:29.980
So let's figure out
what a squared is.

00:05:29.980 --> 00:05:32.890
So let's think about a squared.

00:05:32.890 --> 00:05:37.100
a squared is going
to be equal to--

00:05:37.100 --> 00:05:39.000
and this is the part
where they clearly

00:05:39.000 --> 00:05:42.280
want to see whether you can
multiply multiple polynomials.

00:05:42.280 --> 00:05:46.510
That's going to be x squared
plus x plus 1 squared,

00:05:46.510 --> 00:05:52.040
or x squared plus x plus 1
times x squared plus x plus 1.

00:05:52.040 --> 00:05:55.130
And so this is going to be
1 times this whole thing,

00:05:55.130 --> 00:05:58.280
is x squared plus x plus 1.

00:05:58.280 --> 00:06:02.610
x times this whole
thing, is x plus x

00:06:02.610 --> 00:06:05.270
squared plus x to the third.

00:06:08.017 --> 00:06:09.600
And you may or might
not have noticed,

00:06:09.600 --> 00:06:11.880
I'm doing these a little faster
than I traditionally do them,

00:06:11.880 --> 00:06:13.680
just in the hope of
being able to finish

00:06:13.680 --> 00:06:17.430
this problem in a
reasonable amount of time.

00:06:17.430 --> 00:06:19.840
And then finally, x squared
times this whole thing,

00:06:19.840 --> 00:06:23.270
x squared times 1 is x squared.

00:06:23.270 --> 00:06:26.250
x squared times x
is x to the third.

00:06:26.250 --> 00:06:28.785
And then x squared times x
squared is x to the fourth.

00:06:33.700 --> 00:06:38.280
So a squared is going to
be x to the fourth plus

00:06:38.280 --> 00:06:46.250
2x to the third plus 3x
squared plus 2x plus 1.

00:06:46.250 --> 00:06:48.020
So that's a squared.

00:06:48.020 --> 00:06:48.870
What's b squared?

00:06:48.870 --> 00:06:50.960
That looks a little bit
more straightforward.

00:06:50.960 --> 00:06:52.760
b squared, so that's b.

00:06:52.760 --> 00:06:57.250
So b squared is going to be--
well, that squared is just

00:06:57.250 --> 00:07:01.830
x to the fourth minus
2x squared plus 1.

00:07:01.830 --> 00:07:03.894
And then c squared.

00:07:03.894 --> 00:07:05.060
That's more straightforward.

00:07:05.060 --> 00:07:06.940
So at least they
weren't as painful

00:07:06.940 --> 00:07:16.520
as this. c squared is going to
4x squared plus 4x-- 2 times

00:07:16.520 --> 00:07:21.040
the product of these
two, so plus 4x-- plus 1.

00:07:21.040 --> 00:07:23.170
So the numerator of
our expression here.

00:07:23.170 --> 00:07:25.830
So let's rewrite this thing
that we were simplifying.

00:07:25.830 --> 00:07:30.290
It now becomes the square
root of 3 is equal to-- oh,

00:07:30.290 --> 00:07:32.180
and we don't want
to-- well, let's just

00:07:32.180 --> 00:07:33.290
do one step at a time.

00:07:33.290 --> 00:07:36.920
So it's equal to a squared,
which is this thing over here.

00:07:36.920 --> 00:07:39.340
So it is x to the
fourth-- actually,

00:07:39.340 --> 00:07:41.830
let me scroll down a little bit.

00:07:41.830 --> 00:07:46.366
It's this thing plus this thing,
minus this thing over here.

00:07:46.366 --> 00:07:47.240
So let me just do it.

00:07:47.240 --> 00:07:49.830
It's this thing
plus the blue thing,

00:07:49.830 --> 00:07:55.440
plus x to the fourth
minus 2x squared--

00:07:55.440 --> 00:07:58.530
it'll be easier to add
them up this way-- plus 1.

00:07:58.530 --> 00:08:02.120
And then it's minus this
thing, minus c squared.

00:08:02.120 --> 00:08:02.980
So minus.

00:08:02.980 --> 00:08:08.870
So this is going to be a
minus 4x squared minus 4x.

00:08:11.690 --> 00:08:13.810
And then you have a minus 1.

00:08:13.810 --> 00:08:16.394
And then when you
add them all up-- so

00:08:16.394 --> 00:08:18.310
this will be the numerator
of this expression.

00:08:18.310 --> 00:08:20.870
We have the square root
of 3 is equal to-- so let

00:08:20.870 --> 00:08:22.440
me just do it down here.

00:08:22.440 --> 00:08:25.210
So we have 2x to the
fourth-- let's scroll

00:08:25.210 --> 00:08:26.770
to the left a little bit.

00:08:26.770 --> 00:08:32.150
So we have 2x to the fourth
plus 2x to the third.

00:08:32.150 --> 00:08:38.039
And then 3x squared minus 6x
squared is minus 3x squared.

00:08:38.039 --> 00:08:43.130
2 minus 4 is minus 2x, and
then you have 2 minus 1

00:08:43.130 --> 00:08:47.310
is-- then you have just plus 1.

00:08:47.310 --> 00:08:50.650
So the numerator, this thing
over here, simplifies to this.

00:08:50.650 --> 00:08:56.510
It's equal to 2x to the fourth
plus 2x to the third minus

00:08:56.510 --> 00:09:01.010
3x squared minus 2x plus 1.

00:09:01.010 --> 00:09:05.232
And the denominator is ab.

00:09:05.232 --> 00:09:07.550
a is this thing over here.

00:09:07.550 --> 00:09:12.640
a is x squared plus x plus 1.

00:09:12.640 --> 00:09:16.490
And b, we saw it up
here, b is-- actually,

00:09:16.490 --> 00:09:21.020
we have it written up over
here. b is x squared minus 1.

00:09:21.020 --> 00:09:27.280
So b is x squared minus 1.

00:09:27.280 --> 00:09:29.040
Now at this point, we
need to solve for x.

00:09:29.040 --> 00:09:31.248
If we just multiply this
stuff times square root of 3

00:09:31.248 --> 00:09:33.690
and went backwards, kind of
unsolved it back to the form

00:09:33.690 --> 00:09:38.039
that we had way up here, we'd
have a fourth degree polynomial

00:09:38.039 --> 00:09:39.330
with square roots of 3's in it.

00:09:39.330 --> 00:09:41.780
We still have a fourth
degree polynomial.

00:09:41.780 --> 00:09:44.220
But at least this one has
all integer coefficients,

00:09:44.220 --> 00:09:46.094
which at least
gives me some hope

00:09:46.094 --> 00:09:47.510
that I might be
able to factor it.

00:09:47.510 --> 00:09:49.340
I might be able to
find some roots.

00:09:49.340 --> 00:09:51.220
And if I can find
some roots that

00:09:51.220 --> 00:09:53.460
are common to the numerator
and the denominator,

00:09:53.460 --> 00:09:56.967
maybe I can reduce this
into a lower degree problem.

00:09:56.967 --> 00:09:58.300
So I'm going to leave you there.

00:09:58.300 --> 00:09:59.258
I'm pushing 10 minutes.

00:09:59.258 --> 00:10:02.690
In the next video, we're going
to try to factor this beast up

00:10:02.690 --> 00:10:04.930
here and see if it has any
common factors with this

00:10:04.930 --> 00:10:05.510
over here.

00:10:05.510 --> 00:10:09.340
And then try to solve for x.


IIT JEE Diameter Slope

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.130
Let A and B be two distinct
points on the parabola

00:00:04.130 --> 00:00:06.960
y squared is equal to 4x.

00:00:06.960 --> 00:00:10.770
If the axis of the parabola
touches a circle of radius r,

00:00:10.770 --> 00:00:13.600
having AB as its
diameter, then the slope

00:00:13.600 --> 00:00:16.309
of the line joining
A and B can be.

00:00:16.309 --> 00:00:17.900
So let's draw this.

00:00:17.900 --> 00:00:20.000
So let me draw my axes here.

00:00:20.000 --> 00:00:22.360
That's my y-axis.

00:00:22.360 --> 00:00:27.520
And here is my x-axis.

00:00:27.520 --> 00:00:31.250
And then the parabola y
squared is equal to 4x

00:00:31.250 --> 00:00:32.450
would look like this.

00:00:32.450 --> 00:00:33.755
Something like that.

00:00:33.755 --> 00:00:35.869
That would be in
the first quadrant,

00:00:35.869 --> 00:00:37.660
and then it would be
symmetric around that.

00:00:37.660 --> 00:00:39.010
It would look like that.

00:00:39.010 --> 00:00:43.160
So this would be y
squared is equal to 4x.

00:00:43.160 --> 00:00:46.080
Or if I just focused on
this top part over here,

00:00:46.080 --> 00:00:48.300
I could take the principal
root of both sides.

00:00:48.300 --> 00:00:52.600
And I get y is equal to 2
times the square root of x.

00:00:52.600 --> 00:00:54.540
Just taking the square
root of both sides,

00:00:54.540 --> 00:00:58.090
the square root of 4x is 2
times the square root of x.

00:00:58.090 --> 00:01:01.260
This down here would be y
is equal to negative 2 times

00:01:01.260 --> 00:01:02.150
the square root of x.

00:01:02.150 --> 00:01:04.500
So the positive square root,
the negative square root

00:01:04.500 --> 00:01:05.560
of both sides.

00:01:05.560 --> 00:01:08.920
But this one defines this
whole parabola implicitly.

00:01:08.920 --> 00:01:12.870
And then they say, they
tell us that if the axis

00:01:12.870 --> 00:01:14.820
of the parabola--
that's essentially

00:01:14.820 --> 00:01:18.210
the x-axis in this drawing--
if the axis of the parabola

00:01:18.210 --> 00:01:22.810
touches a circle of radius r
having AB as its diameter--

00:01:22.810 --> 00:01:25.670
So let's take two
points on this parabola.

00:01:25.670 --> 00:01:28.890
So let's say that
that is point A,

00:01:28.890 --> 00:01:33.440
and let's say that
this is point B.

00:01:33.440 --> 00:01:36.460
It is the diameter
for some circle.

00:01:36.460 --> 00:01:38.930
So that's the diameter
for some circle

00:01:38.930 --> 00:01:40.860
that just touches this axis.

00:01:40.860 --> 00:01:43.410
It'll just touch on here.

00:01:43.410 --> 00:01:46.380
So let me see how well
I can draw that circle.

00:01:46.380 --> 00:01:49.560
Obviously, this will
not be an exact drawing.

00:01:49.560 --> 00:01:57.000
But let me try my best to draw
the circle under question.

00:01:57.000 --> 00:01:59.150
So the circle will
look-- I can do a better

00:01:59.150 --> 00:02:06.570
job than that-- the circle
will look like that.

00:02:06.570 --> 00:02:08.430
That's my best shot at it.

00:02:08.430 --> 00:02:10.389
And then it has a radius r.

00:02:10.389 --> 00:02:12.330
So if we say the
center of the circle,

00:02:12.330 --> 00:02:13.930
and obviously the
diameter will go

00:02:13.930 --> 00:02:15.680
through the center
of the circle,

00:02:15.680 --> 00:02:17.910
this distance right here is r.

00:02:17.910 --> 00:02:23.280
So another way to think
about it, this y value

00:02:23.280 --> 00:02:27.800
right over here,
that y value is r.

00:02:27.800 --> 00:02:31.600
Now, let's think a little
bit about the center

00:02:31.600 --> 00:02:32.440
of this circle.

00:02:32.440 --> 00:02:34.273
Because maybe with the
center of the circle,

00:02:34.273 --> 00:02:36.670
we can relate the A's
and the B's to the r's.

00:02:36.670 --> 00:02:41.340
So let me define A and B. So
let me define the point A. Let's

00:02:41.340 --> 00:02:45.200
say the point A,
just for convenience,

00:02:45.200 --> 00:02:49.780
let's make x is equal
to lowercase a squared.

00:02:49.780 --> 00:02:52.730
If x is lowercase a squared,
then the square root of that

00:02:52.730 --> 00:02:54.730
is a times 2.

00:02:54.730 --> 00:02:56.660
Then y will be 2a.

00:02:56.660 --> 00:03:01.480
So this right here is
the point a squared 2a.

00:03:01.480 --> 00:03:05.730
And let's make B the point
using the same logic.

00:03:05.730 --> 00:03:07.849
Let's just say that's
lowercase b squared.

00:03:07.849 --> 00:03:10.140
And I'm doing that so that
when I take the square root,

00:03:10.140 --> 00:03:10.973
it'll become simple.

00:03:10.973 --> 00:03:16.110
So the square root of b
squared is b times 2 is 2b.

00:03:16.110 --> 00:03:19.200
Now, let's get the
coordinates for the center

00:03:19.200 --> 00:03:20.424
of that diameter.

00:03:20.424 --> 00:03:22.090
Now, the center of
that diameter is just

00:03:22.090 --> 00:03:24.960
the midpoint of this
point and that point.

00:03:24.960 --> 00:03:28.070
So let me write it down here.

00:03:28.070 --> 00:03:33.810
The midpoint of
the points A and B

00:03:33.810 --> 00:03:37.360
is going to be equal to the
average of the x values,

00:03:37.360 --> 00:03:41.860
a squared plus b squared over
2, and the average of the y

00:03:41.860 --> 00:03:42.480
values.

00:03:42.480 --> 00:03:48.690
So this will be 2a plus
2b, all of that over 2.

00:03:48.690 --> 00:03:50.740
And of course the 2s cancel out.

00:03:50.740 --> 00:03:54.620
And so this is going
to be a plus b.

00:03:54.620 --> 00:03:56.530
That is the y value
of the midpoint.

00:03:56.530 --> 00:03:58.910
So this point right
over here is going

00:03:58.910 --> 00:04:08.200
to be a squared plus b
squared over 2 comma a plus b.

00:04:08.200 --> 00:04:12.360
Now, we just saw that this
height right here is r.

00:04:12.360 --> 00:04:16.680
So a plus b, this y value right
over here is also equal to r.

00:04:16.680 --> 00:04:19.450
This is equal to a plus b.

00:04:19.450 --> 00:04:23.050
Write the y value in
between 2a and the y value

00:04:23.050 --> 00:04:25.440
in between 2b right over here.

00:04:25.440 --> 00:04:25.940
2b.

00:04:25.940 --> 00:04:28.060
This is a plus b.

00:04:28.060 --> 00:04:30.390
So just like that,
we were able to get

00:04:30.390 --> 00:04:35.010
a relationship
between a, b, and r.

00:04:35.010 --> 00:04:39.210
So we know a plus
b is equal to r.

00:04:39.210 --> 00:04:43.150
Now, let's try to find the
actual slope of this diameter.

00:04:43.150 --> 00:04:45.540
That's what they want to ask us.

00:04:45.540 --> 00:04:48.400
The slope of the line
joining a and b can be.

00:04:48.400 --> 00:04:51.450
So let's figure out the
slope of this actual line,

00:04:51.450 --> 00:04:52.920
and let's see if
we can relate it

00:04:52.920 --> 00:04:55.869
to r in some way maybe
using this information.

00:04:55.869 --> 00:04:58.160
Because all the answers they
give us are in terms of r.

00:04:58.160 --> 00:05:00.430
So we need to have an
answer in terms of r.

00:05:00.430 --> 00:05:06.310
So the slope is going to be
change in y over change in x.

00:05:06.310 --> 00:05:15.470
So 2a minus 2b over a
squared minus b squared.

00:05:15.470 --> 00:05:20.690
This is equal to 2 times
a minus b over a plus

00:05:20.690 --> 00:05:23.650
b-- this is just a difference
of squares-- a plus b times

00:05:23.650 --> 00:05:24.846
a minus b.

00:05:24.846 --> 00:05:27.190
The a minus b's cancel out.

00:05:27.190 --> 00:05:29.710
The slope of the
diameter, the slope

00:05:29.710 --> 00:05:37.360
of the line that we care about
is going to be 2 over a plus b.

00:05:37.360 --> 00:05:40.650
It's 2 over a plus b.

00:05:40.650 --> 00:05:42.800
Now, that's clearly
not one of the answers.

00:05:42.800 --> 00:05:44.610
We need our answer
in terms of r.

00:05:44.610 --> 00:05:46.870
But we just figured
out the y-coordinate

00:05:46.870 --> 00:05:49.230
of the midpoint,
which was a plus b.

00:05:49.230 --> 00:05:50.850
That is equal to r.

00:05:50.850 --> 00:05:54.760
So this is the
same thing as 2/r.

00:05:54.760 --> 00:06:00.070
So the slope of this line
right over here is 2/r.

00:06:02.850 --> 00:06:07.370
Now, you might be tempted
to say OK, the answer is

00:06:07.370 --> 00:06:09.979
C. And I actually probably
should have told you

00:06:09.979 --> 00:06:11.770
this at the beginning
of this video in case

00:06:11.770 --> 00:06:14.240
you wanted to pause it
and try it yourself,

00:06:14.240 --> 00:06:17.860
is that this is actually
a multiple correct answer

00:06:17.860 --> 00:06:18.360
problem.

00:06:18.360 --> 00:06:20.401
So you could actually have
multiple choices here.

00:06:20.401 --> 00:06:23.012
So we want to make sure that
this isn't the only answer.

00:06:23.012 --> 00:06:24.220
And it's not the only answer.

00:06:24.220 --> 00:06:29.730
Because we could have put
a here and b here and drawn

00:06:29.730 --> 00:06:30.885
the circle like this.

00:06:33.880 --> 00:06:35.380
Essentially the
exact same circle,

00:06:35.380 --> 00:06:36.920
but in the fourth quadrant.

00:06:36.920 --> 00:06:38.934
So I could have drawn
the circle like this.

00:06:38.934 --> 00:06:40.350
And then I would
have been dealing

00:06:40.350 --> 00:06:42.924
with the slope of this
diameter would be in question.

00:06:42.924 --> 00:06:44.590
And you can see
everything is symmetric.

00:06:44.590 --> 00:06:47.030
It's just flipped around
the axis of symmetry.

00:06:47.030 --> 00:06:48.990
It's flipped around the x-axis.

00:06:48.990 --> 00:06:51.402
Whatever is the slope up
here, the negative of it

00:06:51.402 --> 00:06:52.860
is going to be the
slope over here.

00:06:52.860 --> 00:06:55.600
So this is also a
potential slope.

00:06:55.600 --> 00:06:58.550
So if this is 2/r, then
this slope over here

00:06:58.550 --> 00:07:00.790
is negative 2/r.

00:07:00.790 --> 00:07:04.990
So this is also an option.

00:07:04.990 --> 00:07:08.500
And just based on the
reasoning we just did,

00:07:08.500 --> 00:07:10.090
you could have ruled
out two of these.

00:07:10.090 --> 00:07:12.425
If you knew that
this slope is a,

00:07:12.425 --> 00:07:14.800
or if you know that this
slope up here is going to be,

00:07:14.800 --> 00:07:18.015
I don't know, if this slope
up here was going to be r,

00:07:18.015 --> 00:07:20.140
then the slope over here
is going to be negative r.

00:07:20.140 --> 00:07:22.570
If the slope up here is
going to be a plus b,

00:07:22.570 --> 00:07:24.510
this slope would be
negative a plus b.

00:07:24.510 --> 00:07:26.420
So you know that you would
have to have two answers that

00:07:26.420 --> 00:07:27.650
are the negative of each other.

00:07:27.650 --> 00:07:30.066
So when you look at just the
choices, you could have said,

00:07:30.066 --> 00:07:33.030
oh, it's either going to
be A and B because they're

00:07:33.030 --> 00:07:37.036
the negatives of each other,
or it's going to be C and D.

00:07:37.036 --> 00:07:38.410
So you could have
ruled out-- you

00:07:38.410 --> 00:07:40.690
could have either just picked
these two or these two.

00:07:40.690 --> 00:07:42.275
So you had a 50/50
chance of guessing.

00:07:42.275 --> 00:07:43.400
But we don't want to guess.

00:07:43.400 --> 00:07:44.520
We want to solve it.

00:07:44.520 --> 00:07:45.480
And we just did.

00:07:45.480 --> 00:07:47.990
Anyway, hopefully
you enjoyed that.


IIT JEE Function Maxima

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.690
"Let f, g, and h be
real-valued functions

00:00:03.690 --> 00:00:06.760
defined on the
interval, 0 to 1."

00:00:06.760 --> 00:00:08.885
And it includes the
boundaries 0 and 1.

00:00:08.885 --> 00:00:10.260
That's what the
brackets tell us.

00:00:10.260 --> 00:00:11.676
It wouldn't include
the boundaries

00:00:11.676 --> 00:00:12.840
if we have parentheses.

00:00:12.840 --> 00:00:14.400
And then they define
the functions.

00:00:14.400 --> 00:00:16.880
f of x is equal to e
to the x squared plus e

00:00:16.880 --> 00:00:18.470
to the negative x squared.

00:00:18.470 --> 00:00:20.670
g of x looks very
similar to f of x,

00:00:20.670 --> 00:00:24.690
except we have this x out here
in front of the first term.

00:00:24.690 --> 00:00:27.360
And then we have h of x,
which looks very similar to g

00:00:27.360 --> 00:00:31.520
of x and f of x, except
it's got an x squared term

00:00:31.520 --> 00:00:32.790
right over there.

00:00:32.790 --> 00:00:35.420
And then they tell us,
"If a, b, and c denote,

00:00:35.420 --> 00:00:38.520
respectively, the
absolute maximum of f, g,

00:00:38.520 --> 00:00:40.460
and h on the interval, then?"

00:00:40.460 --> 00:00:45.710
So a is the maximum of
f, b is the maximum of g,

00:00:45.710 --> 00:00:47.319
and then c is the maximum of h.

00:00:47.319 --> 00:00:48.860
And then they have
a bunch of choices

00:00:48.860 --> 00:00:51.900
that compare the maximum values.

00:00:51.900 --> 00:00:54.360
Now, the first thing that may
or may not jump out at you,

00:00:54.360 --> 00:00:56.705
just looking at these
equations over here-- now,

00:00:56.705 --> 00:00:59.030
we're over the interval, 0 to 1.

00:00:59.030 --> 00:01:02.587
And let's just look at
the functions themselves.

00:01:02.587 --> 00:01:04.420
Let's just look at the
functions themselves.

00:01:04.420 --> 00:01:06.590
The difference is this
one, you could view,

00:01:06.590 --> 00:01:08.460
has a 1 coefficient out front.

00:01:08.460 --> 00:01:12.460
This has an x, and
this has an x squared.

00:01:12.460 --> 00:01:15.130
So over that interval,
how are these functions

00:01:15.130 --> 00:01:16.720
going to relate?

00:01:16.720 --> 00:01:18.760
Over that interval.

00:01:18.760 --> 00:01:25.030
We know that over the
interval, from 0 to 1,

00:01:25.030 --> 00:01:31.460
we know that x squared is going
to be less than or equal to x,

00:01:31.460 --> 00:01:34.650
which is going to be
less than or equal to 1.

00:01:34.650 --> 00:01:36.720
They all equal each other
at the right interval,

00:01:36.720 --> 00:01:39.090
at actually being equal to 1.

00:01:39.090 --> 00:01:41.930
So this is going to
be less than this,

00:01:41.930 --> 00:01:43.680
which is going to
be less than that.

00:01:43.680 --> 00:01:45.480
And these are all
positive values.

00:01:45.480 --> 00:01:48.610
And the rest of the
functions are the same.

00:01:48.610 --> 00:01:50.970
This is the same as that,
that's the same as that.

00:01:50.970 --> 00:01:53.210
And that is the same as that.

00:01:53.210 --> 00:01:59.210
So over the interval,
we know that h of x

00:01:59.210 --> 00:02:04.010
is going to be less than or
equal to g of x, which is going

00:02:04.010 --> 00:02:07.820
to be less than or
equal to f of x.

00:02:10.520 --> 00:02:12.891
That just comes straight
out of, this coefficient

00:02:12.891 --> 00:02:15.140
is going to be less than or
equal to this coefficient,

00:02:15.140 --> 00:02:17.060
which is going to be
less than or equal to 1,

00:02:17.060 --> 00:02:18.910
over the interval.

00:02:18.910 --> 00:02:20.390
Now with that said,
let's actually

00:02:20.390 --> 00:02:22.705
try to figure out
the maximum values.

00:02:22.705 --> 00:02:23.580
So I'll start with f.

00:02:26.960 --> 00:02:29.430
So f of x-- let me just
tried it the boundaries.

00:02:29.430 --> 00:02:36.430
f of 0 is equal to e
to the 0 squared plus e

00:02:36.430 --> 00:02:37.540
to the negative 0 squared.

00:02:37.540 --> 00:02:39.830
That's just e to the
0 plus e to the 0,

00:02:39.830 --> 00:02:42.780
which is 1 plus 1, which is 2.

00:02:42.780 --> 00:02:50.070
And then f of 1 is e to the
1, plus e to the negative 1.

00:02:50.070 --> 00:02:53.560
This is e to the 1, or
just e, I should say,

00:02:53.560 --> 00:02:56.130
plus e to the negative 1.

00:02:56.130 --> 00:02:59.736
Now this is clearly larger
than that. e is 2.71

00:02:59.736 --> 00:03:00.876
and keeps going.

00:03:00.876 --> 00:03:02.250
e to the negative
1, that's still

00:03:02.250 --> 00:03:04.980
going to be a positive
number, obviously.

00:03:04.980 --> 00:03:07.640
So this, between
the two endpoints,

00:03:07.640 --> 00:03:10.391
this seems to be a candidate
for the maximum point.

00:03:10.391 --> 00:03:12.890
We don't know that's the maximum
point, because the function

00:03:12.890 --> 00:03:15.190
might have done something
funky in between.

00:03:15.190 --> 00:03:17.690
So let me draw an axes.

00:03:17.690 --> 00:03:20.240
I can draw a straighter
line than that.

00:03:20.240 --> 00:03:24.250
So we know at 0, we
took on the value 2,

00:03:24.250 --> 00:03:32.135
and at 1, we took on the
value e plus 1 over e,

00:03:32.135 --> 00:03:33.420
or e to the negative 1.

00:03:33.420 --> 00:03:35.440
So that's that value
right over there.

00:03:35.440 --> 00:03:37.750
Now, we don't know what the
function did in between.

00:03:37.750 --> 00:03:39.550
This would be the
maximum point--

00:03:39.550 --> 00:03:43.930
this is the maximum point,
if the function did something

00:03:43.930 --> 00:03:44.890
like this.

00:03:44.890 --> 00:03:47.360
If it's constantly
increasing the entire time,

00:03:47.360 --> 00:03:49.090
this would be the maximum point.

00:03:49.090 --> 00:03:50.650
It would not be
the maximum point

00:03:50.650 --> 00:03:54.670
if the function looks
something like that.

00:03:54.670 --> 00:03:57.040
And the way we can tell
what the function is doing

00:03:57.040 --> 00:04:00.150
is we can take the derivative
of f, which tells us

00:04:00.150 --> 00:04:02.990
the slope at any point
over this interval.

00:04:02.990 --> 00:04:05.310
And if the derivative
is constantly positive,

00:04:05.310 --> 00:04:07.520
that means we always
have an upward slope.

00:04:07.520 --> 00:04:09.442
If the derivative is
not always positive,

00:04:09.442 --> 00:04:11.150
then we have something
like this and then

00:04:11.150 --> 00:04:15.730
we could actually look for
that actual maximum point.

00:04:15.730 --> 00:04:17.810
So let's calculate
the derivative.

00:04:17.810 --> 00:04:26.330
So f prime of x is equal to--
let's see the derivative of e

00:04:26.330 --> 00:04:29.420
to the x squared is
2x e to the x squared,

00:04:29.420 --> 00:04:30.870
just using the chain rule.

00:04:30.870 --> 00:04:32.800
Derivative of e to
the negative x squared

00:04:32.800 --> 00:04:37.770
is negative 2x e to
the negative x squared.

00:04:37.770 --> 00:04:41.260
And let's see, we
could factor out--

00:04:41.260 --> 00:04:51.210
this is equal to 2x e to the
negative x squared times--

00:04:51.210 --> 00:04:55.360
so if you divide this by e
to the negative x squared,

00:04:55.360 --> 00:04:59.380
that's like multiplying it
times e to the x squared.

00:04:59.380 --> 00:05:07.160
So this becomes e to
the 2x squared minus 1.

00:05:07.160 --> 00:05:10.400
Now, our question is-- this is
the slope over the interval--

00:05:10.400 --> 00:05:12.080
what is this doing at any point?

00:05:12.080 --> 00:05:14.260
Well, this value
right here, this right

00:05:14.260 --> 00:05:17.270
here is always going to be
greater than or equal to 0.

00:05:17.270 --> 00:05:19.115
It'll be 0 at the
left-hand interval.

00:05:19.115 --> 00:05:20.490
Just remember,
this is the slope.

00:05:20.490 --> 00:05:23.170
The slope is 0 at the left-hand
interval, which tells us

00:05:23.170 --> 00:05:25.490
it's either a maximum
or minimum point.

00:05:25.490 --> 00:05:27.880
But then, it's going to
be positive after that.

00:05:27.880 --> 00:05:30.460
Because this is going to be
a positive value, and that

00:05:30.460 --> 00:05:32.002
this is going to be
a positive value.

00:05:32.002 --> 00:05:34.460
Remember negative exponent does
not mean a negative number.

00:05:34.460 --> 00:05:35.860
It means a fractional value.

00:05:35.860 --> 00:05:37.680
It's going be 1 over
e to the x squared.

00:05:37.680 --> 00:05:40.150
So this is always going to be
greater than or equal to 0.

00:05:40.150 --> 00:05:41.670
And so what we
need to figure out,

00:05:41.670 --> 00:05:46.530
is this always greater
than or equal to 0?

00:05:46.530 --> 00:05:52.110
So we want to figure out, is
e to the 2 x squared minus 1

00:05:52.110 --> 00:05:55.324
greater than or equal
to 0 over the interval?

00:05:55.324 --> 00:05:56.740
Which is the same
thing as asking,

00:05:56.740 --> 00:05:59.940
is e to the 2x
squared greater than

00:05:59.940 --> 00:06:01.930
or equal to 1 over the interval?

00:06:01.930 --> 00:06:05.030
I just added 1 to both
sides of this inequality.

00:06:05.030 --> 00:06:06.530
And that's the same
thing as asking,

00:06:06.530 --> 00:06:08.800
if you take the natural
log of both sides,

00:06:08.800 --> 00:06:12.060
the natural log of
this is to 2x squared.

00:06:12.060 --> 00:06:15.100
The natural log of 1 is 0.

00:06:15.100 --> 00:06:19.550
So is 2x squared greater than
or equal to 0 over the interval?

00:06:19.550 --> 00:06:20.350
Well, yeah, sure.

00:06:20.350 --> 00:06:23.300
It's going to be equal
to 0 at x is equal to 0.

00:06:23.300 --> 00:06:25.580
But then after every other
value, as it approaches 1,

00:06:25.580 --> 00:06:26.920
it'll be positive.

00:06:26.920 --> 00:06:28.670
So this is true.

00:06:28.670 --> 00:06:29.760
This is true.

00:06:29.760 --> 00:06:32.600
So this thing is always
going to be greater than 0

00:06:32.600 --> 00:06:34.505
over the interval,
which tells us

00:06:34.505 --> 00:06:36.097
that the two parts
of the derivative

00:06:36.097 --> 00:06:38.180
are always going to be
greater than or equal to 0.

00:06:38.180 --> 00:06:41.010
So the derivative itself,
the slope of the function,

00:06:41.010 --> 00:06:43.840
is always greater than 0.

00:06:43.840 --> 00:06:46.130
So always greater than
0-- over the interval

00:06:46.130 --> 00:06:49.940
I should say-- always
greater than or equal to 0

00:06:49.940 --> 00:06:55.220
over the interval from 0 to 1.

00:06:55.220 --> 00:06:58.390
Which tells us that the
slope is never negative.

00:06:58.390 --> 00:07:01.342
And actually, we know
that the slope is 0 at 0.

00:07:01.342 --> 00:07:03.300
So that tells us that
the graph looks something

00:07:03.300 --> 00:07:09.060
like this, which tells us
that the right endpoint--

00:07:09.060 --> 00:07:14.220
that f of 1, when x equals 1--
f of 1 is the maximum point.

00:07:14.220 --> 00:07:21.930
So this, indeed, is equal to a.

00:07:21.930 --> 00:07:24.910
So we can actually write this
here. f of x is less than

00:07:24.910 --> 00:07:29.700
or equal to a.

00:07:29.700 --> 00:07:33.910
Now, let's think about
h of x and g of x.

00:07:33.910 --> 00:07:36.890
So f of 1 was its maximum point.

00:07:36.890 --> 00:07:41.240
But g of 1-- if you
take g of 1 or h of 1,

00:07:41.240 --> 00:07:43.860
it'll evaluate to f of
1, because if you put a 1

00:07:43.860 --> 00:07:45.132
here-- let's do that.

00:07:45.132 --> 00:07:46.590
Let me just do it
to make it clear.

00:07:49.580 --> 00:07:52.170
So we already evaluated
f of 1 right over there.

00:07:52.170 --> 00:07:57.570
g of 1 is equal to--
let me make sure we

00:07:57.570 --> 00:08:01.430
can see the-- so the
g of 1 is equal to 1.

00:08:01.430 --> 00:08:08.090
1 times e to the 1 plus
e to the negative 1.

00:08:08.090 --> 00:08:10.520
That's g of 1, which is
the same thing as f of 1.

00:08:10.520 --> 00:08:12.990
So this is also equal to a.

00:08:12.990 --> 00:08:16.820
And h of 1 is the same thing.

00:08:16.820 --> 00:08:21.170
So 1 times-- it's 1 squared,
which is 1, times e to the 1

00:08:21.170 --> 00:08:26.369
plus e to the negative 1,
which is also equal to a.

00:08:26.369 --> 00:08:27.910
So what do we have,
a situation here?

00:08:27.910 --> 00:08:31.935
We know that h of
x over the interval

00:08:31.935 --> 00:08:33.809
is less than or equal
to g of x, is less than

00:08:33.809 --> 00:08:36.115
or equal to f of x, which
is less than or equal to a.

00:08:36.115 --> 00:08:40.169
And actually, the only point
where they all equal each other

00:08:40.169 --> 00:08:41.770
is at x is equal to 1.

00:08:41.770 --> 00:08:46.274
If this right here
is f of x, g of x

00:08:46.274 --> 00:08:47.815
is going to look
something like this.

00:08:50.330 --> 00:08:53.110
And h of x is going to
look something like this.

00:08:53.110 --> 00:08:56.670
h of x might look
something like that.

00:08:56.670 --> 00:08:58.821
And so they all finally
equal each other.

00:08:58.821 --> 00:09:00.320
The only place they
equal each other

00:09:00.320 --> 00:09:02.680
is at a, which is
the maximum of f.

00:09:02.680 --> 00:09:05.570
So that also has to be
the maximum of g and h,

00:09:05.570 --> 00:09:08.649
because they can both
get to that value.

00:09:08.649 --> 00:09:10.690
And they can't get to any
value higher than that.

00:09:10.690 --> 00:09:12.430
That's an upper bound on them.

00:09:12.430 --> 00:09:14.870
So this is also
equal to b-- remember

00:09:14.870 --> 00:09:17.430
b is just the maximum of g.

00:09:17.430 --> 00:09:20.800
And it's also equal to c,
which is the maximum of h.

00:09:20.800 --> 00:09:23.590
So a, b, and c are all
equal to each other,

00:09:23.590 --> 00:09:27.598
or we can say choice D.


IIT JEE Algebraic Manipulation

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.260
What I want to do
in this video is

00:00:02.260 --> 00:00:03.740
something a little
unconventional.

00:00:03.740 --> 00:00:06.950
Because I tried this problem
before I started this video,

00:00:06.950 --> 00:00:09.070
and I kept getting
an answer that

00:00:09.070 --> 00:00:11.100
isn't one of these
answers over here.

00:00:11.100 --> 00:00:14.680
So assuming that the exam
writers did this properly--

00:00:14.680 --> 00:00:16.020
and this is a serious exam.

00:00:16.020 --> 00:00:19.529
It's to get into a very
prestigious set of universities

00:00:19.529 --> 00:00:21.070
in India, so I'm
assuming they didn't

00:00:21.070 --> 00:00:22.910
make any major mistakes here.

00:00:22.910 --> 00:00:26.000
I want you to tell me where
I'm making the mistake.

00:00:26.000 --> 00:00:27.870
So let's just try
to do the problem.

00:00:27.870 --> 00:00:32.049
Let p and q be real numbers
such that p does not equal 0,

00:00:32.049 --> 00:00:35.250
p to the third does not
equal q, and p to the third

00:00:35.250 --> 00:00:37.180
also does not equal negative q.

00:00:37.180 --> 00:00:41.120
If alpha and beta are
nonzero complex numbers

00:00:41.120 --> 00:00:44.740
satisfying alpha plus beta
is equal to negative p

00:00:44.740 --> 00:00:48.510
and alpha cubed plus
beta cubed is equal to q,

00:00:48.510 --> 00:00:54.510
then a quadratic equation
having alpha/beta and beta/alpha

00:00:54.510 --> 00:00:55.538
as its roots is.

00:00:55.538 --> 00:00:56.746
Did I just call it b up here?

00:00:56.746 --> 00:00:57.850
It's beta.

00:00:57.850 --> 00:00:59.834
So what could that quadratic be?

00:00:59.834 --> 00:01:01.500
Well, these are the
two roots, so that's

00:01:01.500 --> 00:01:02.530
pretty straightforward.

00:01:02.530 --> 00:01:05.670
We could say that the quadratic,
a possible quadratic here,

00:01:05.670 --> 00:01:15.460
would be x minus alpha/beta
times x minus beta/alpha

00:01:15.460 --> 00:01:16.450
is equal to 0.

00:01:16.450 --> 00:01:18.730
Clearly, the roots of
this equation right

00:01:18.730 --> 00:01:21.730
here are alpha/beta
and beta/alpha.

00:01:21.730 --> 00:01:27.110
Either one of those would
satisfy this equation.

00:01:27.110 --> 00:01:32.310
Now let's multiply this
out. x times x is x squared.

00:01:32.310 --> 00:01:40.960
x times negative beta/alpha
is negative beta/alpha x.

00:01:40.960 --> 00:01:43.230
Then you have
negative alpha/beta

00:01:43.230 --> 00:01:48.110
x, so negative alpha/beta x.

00:01:48.110 --> 00:01:50.130
And then you have
negative alpha/beta

00:01:50.130 --> 00:01:51.880
times negative beta/alpha.

00:01:51.880 --> 00:01:53.090
The alpha's cancel out.

00:01:53.090 --> 00:01:54.010
The beta's cancel out.

00:01:54.010 --> 00:01:55.390
The negatives cancel out.

00:01:55.390 --> 00:02:01.801
And you get plus
1 is equal to 0.

00:02:01.801 --> 00:02:03.800
And then let's see, we
might be able to simplify

00:02:03.800 --> 00:02:04.670
this a little bit.

00:02:04.670 --> 00:02:09.139
If we multiplied everything
times alpha beta,

00:02:09.139 --> 00:02:10.310
what will happen?

00:02:10.310 --> 00:02:12.000
So I'm just going to
multiply both sides

00:02:12.000 --> 00:02:16.150
of this equation
times alpha beta.

00:02:19.120 --> 00:02:21.344
I don't like having
things in denominators.

00:02:21.344 --> 00:02:23.510
Clearly, the right-hand
side is still going to be 0.

00:02:23.510 --> 00:02:30.380
The left-hand side now
becomes alpha beta x squared.

00:02:30.380 --> 00:02:34.590
Alpha beta times beta/alpha,
the alphas cancel out.

00:02:34.590 --> 00:02:36.080
You have a beta times a beta.

00:02:36.080 --> 00:02:45.020
So it's negative beta squared
x minus-- the betas cancel out.

00:02:45.020 --> 00:02:51.330
And now you have an alpha times
an alpha minus alpha squared x.

00:02:51.330 --> 00:02:57.540
And now you have plus
alpha beta is equal to 0.

00:02:57.540 --> 00:02:59.620
And then we can combine
these terms over here.

00:02:59.620 --> 00:03:04.180
So we have alpha beta
x squared minus-- you

00:03:04.180 --> 00:03:11.740
have an alpha squared plus a
beta squared x-- all right,

00:03:11.740 --> 00:03:13.780
I just took the
negatives out of it--

00:03:13.780 --> 00:03:20.730
plus alpha beta is equal to 0.

00:03:20.730 --> 00:03:21.250
All right.

00:03:21.250 --> 00:03:23.711
So everything I did is
just kind of-- I just said,

00:03:23.711 --> 00:03:24.710
OK, these are the roots.

00:03:24.710 --> 00:03:26.710
Let me just multiply it
out, and I got something

00:03:26.710 --> 00:03:28.040
in terms of alpha and beta.

00:03:28.040 --> 00:03:29.664
Now when you look at
it, their choices,

00:03:29.664 --> 00:03:32.190
clearly they want answers
in terms of p's and q's.

00:03:32.190 --> 00:03:36.050
So let's see if we can
convert this into p's and q's.

00:03:36.050 --> 00:03:40.050
So a good place to start, they
tell us that alpha plus beta is

00:03:40.050 --> 00:03:46.240
equal to negative p, or negative
p is equal to alpha plus beta.

00:03:46.240 --> 00:03:48.920
And so if we squared that,
what I want to do is somehow

00:03:48.920 --> 00:03:50.816
relate it to q as well.

00:03:50.816 --> 00:03:53.190
And so maybe if we eventually
take it to the third power,

00:03:53.190 --> 00:03:56.340
we can get some of these terms
over here and relate it to q.

00:03:56.340 --> 00:03:57.932
And also, when we
square it, I think

00:03:57.932 --> 00:04:00.390
we'll be able to relate some
of these things to it as well.

00:04:00.390 --> 00:04:02.790
So let's square both
sides of this equation.

00:04:02.790 --> 00:04:07.060
So you have negative
p squared, which,

00:04:07.060 --> 00:04:09.000
because the negative
canceled out,

00:04:09.000 --> 00:04:11.520
this is the same thing
as negative 1 times p

00:04:11.520 --> 00:04:13.760
squared, which is negative
1 squared times p squared,

00:04:13.760 --> 00:04:16.779
which is just p squared,
is equal to this expression

00:04:16.779 --> 00:04:18.890
over here squared,
which is just alpha

00:04:18.890 --> 00:04:25.740
squared plus 2 alpha
beta plus beta squared.

00:04:25.740 --> 00:04:26.340
Fair enough.

00:04:26.340 --> 00:04:28.690
And already we have
something interesting

00:04:28.690 --> 00:04:31.030
showing up over here.

00:04:31.030 --> 00:04:33.710
We can express alpha
squared plus beta

00:04:33.710 --> 00:04:39.982
squared in terms of p
squared and alpha beta.

00:04:39.982 --> 00:04:41.690
And then we simplify
things a little bit.

00:04:41.690 --> 00:04:42.856
Well, we'll just keep going.

00:04:42.856 --> 00:04:44.890
Let's just take this
to the third power.

00:04:44.890 --> 00:04:48.470
So if we have negative p
to the third power, which

00:04:48.470 --> 00:04:52.340
is the same thing as the
negative of p to the third,

00:04:52.340 --> 00:04:54.830
if you multiply this
thing times negative p,

00:04:54.830 --> 00:04:57.240
you're going to get the
negative sign p to the third.

00:04:57.240 --> 00:05:00.150
And this is equal to-- we
could just multiply again,

00:05:00.150 --> 00:05:01.410
do the distributive property.

00:05:01.410 --> 00:05:03.710
Or if you kind of remember
Pascal's triangles,

00:05:03.710 --> 00:05:07.350
you remember the coefficients
on it, or the binomial theorem,

00:05:07.350 --> 00:05:10.520
this would be a
faster way to do it.

00:05:10.520 --> 00:05:13.070
I'll do a Pascal's
triangle really quick here.

00:05:13.070 --> 00:05:14.222
So you have 1.

00:05:14.222 --> 00:05:15.350
1, 1 for binomial.

00:05:15.350 --> 00:05:18.990
1, 2, 1, when you take the
square of a binomial, 1, 2, 1.

00:05:18.990 --> 00:05:21.180
And then when you take
that to the third, when

00:05:21.180 --> 00:05:24.762
you go to the third
power, it's 1, 3, 3, 1.

00:05:24.762 --> 00:05:26.220
Bunch of videos on
this, especially

00:05:26.220 --> 00:05:28.570
in the videos on the
binomial theorem.

00:05:28.570 --> 00:05:30.566
So this to the third
power, this thing

00:05:30.566 --> 00:05:31.940
right here to the
third power, is

00:05:31.940 --> 00:05:37.110
going to be alpha
to the third plus

00:05:37.110 --> 00:05:46.750
3 alpha squared beta
plus 3 alpha beta squared

00:05:46.750 --> 00:05:47.920
plus beta to the third.

00:05:47.920 --> 00:05:50.964
I just decreased the exponent
on the alpha each time

00:05:50.964 --> 00:05:53.380
and started increasing the
exponent on the beta each time.

00:05:53.380 --> 00:05:55.460
Or you could just
multiply this times this,

00:05:55.460 --> 00:05:57.130
and you should also get this.

00:05:57.130 --> 00:05:59.300
And now this is
interesting because this

00:05:59.300 --> 00:06:03.575
is expressing everything
in terms of-- let's see

00:06:03.575 --> 00:06:05.460
if we can simplify
this a little bit.

00:06:05.460 --> 00:06:07.620
This is equal to--
so let me write

00:06:07.620 --> 00:06:10.830
this-- we have negative
p to the third, which

00:06:10.830 --> 00:06:13.900
is equal to negative
p to the third,

00:06:13.900 --> 00:06:15.730
just a difference
of the parentheses,

00:06:15.730 --> 00:06:20.300
is equal to-- we have this
alpha cubed plus the beta cubed.

00:06:20.300 --> 00:06:22.470
That's those two terms
right over there.

00:06:22.470 --> 00:06:25.570
And I separate those out
because those are our q.

00:06:25.570 --> 00:06:26.910
Those are our q.

00:06:26.910 --> 00:06:28.730
And then I have--
let's see, over here I

00:06:28.730 --> 00:06:35.380
can factor out a 3 alpha
beta, so plus 3 alpha beta.

00:06:35.380 --> 00:06:36.160
And what do I get?

00:06:36.160 --> 00:06:39.150
If I factor out a
3 alpha beta here,

00:06:39.150 --> 00:06:41.080
I'm just left with an
alpha on this term.

00:06:41.080 --> 00:06:42.750
And if I factor out
a 3 alpha beta here,

00:06:42.750 --> 00:06:44.870
I'm just left with a beta.

00:06:44.870 --> 00:06:46.780
I am just left with a beta.

00:06:46.780 --> 00:06:48.910
So what does this simplify to?

00:06:48.910 --> 00:06:53.370
So this is going to simplify
to-- this right here

00:06:53.370 --> 00:06:55.550
is equal to our q.

00:06:55.550 --> 00:07:00.670
So this is equal to
q plus 3 alpha beta.

00:07:00.670 --> 00:07:02.330
And what's alpha plus beta?

00:07:02.330 --> 00:07:04.540
Well, they already told us
what alpha plus beta is.

00:07:04.540 --> 00:07:09.490
Alpha plus beta is negative p.

00:07:09.490 --> 00:07:14.500
It is negative p.

00:07:14.500 --> 00:07:18.220
So let's see what
we can do over here.

00:07:18.220 --> 00:07:19.960
And this right here
is the same thing

00:07:19.960 --> 00:07:22.540
as negative p to the third.

00:07:22.540 --> 00:07:25.340
So here we can solve
for alpha beta,

00:07:25.340 --> 00:07:28.349
and so we'll have alpha beta
in terms of p's and q's.

00:07:28.349 --> 00:07:29.140
So that'll be good.

00:07:29.140 --> 00:07:31.760
So we can substitute the alpha
beta's with whatever we get

00:07:31.760 --> 00:07:33.570
in terms of p's and q's.

00:07:33.570 --> 00:07:37.875
And then this thing we can
express in terms of a p

00:07:37.875 --> 00:07:40.250
and an alpha beta, which means
we can express it in terms

00:07:40.250 --> 00:07:41.020
of p's and q's.

00:07:41.020 --> 00:07:43.279
So let me do what
I'm talking about.

00:07:43.279 --> 00:07:44.570
Let me do it I'm talking about.

00:07:44.570 --> 00:07:48.230
So let's just solve, first
of all, for alpha beta.

00:07:48.230 --> 00:07:50.360
So if we solve for
alpha beta, first we

00:07:50.360 --> 00:07:53.360
want to subtract q from
both sides of this equation.

00:07:53.360 --> 00:07:56.680
So you have negative
p to the third minus

00:07:56.680 --> 00:08:01.470
q is equal to-- we have this
negative sign over here,

00:08:01.470 --> 00:08:06.690
so it's equal to
negative 3 alpha beta p.

00:08:06.690 --> 00:08:08.510
And I want to solve
for alpha beta, so let

00:08:08.510 --> 00:08:12.080
me divide both
sides by negative 3.

00:08:12.080 --> 00:08:14.800
And so we get, on
the right-hand side,

00:08:14.800 --> 00:08:18.734
we get alpha beta is equal
to, if you divide-- so

00:08:18.734 --> 00:08:20.400
let me just write it
out because I don't

00:08:20.400 --> 00:08:21.780
want to make a careless mistake.

00:08:21.780 --> 00:08:30.210
So you have negative p to the
third minus q over negative 3p.

00:08:30.210 --> 00:08:32.210
Now if we multiply the
numerator and denominator

00:08:32.210 --> 00:08:38.710
by a negative sign, we get p
to the third plus q over 3p

00:08:38.710 --> 00:08:45.520
is equal to alpha beta.

00:08:45.520 --> 00:08:48.980
So we can now express this
and this term in terms

00:08:48.980 --> 00:08:50.860
of p's and q's.

00:08:50.860 --> 00:08:52.820
Now what about this
one right here?

00:08:52.820 --> 00:08:55.800
So what is alpha squared
plus beta squared?

00:08:55.800 --> 00:08:57.630
And we see right up
here, alpha squared,

00:08:57.630 --> 00:09:00.570
that's alpha squared
plus beta squared.

00:09:00.570 --> 00:09:02.740
So if we subtract 2 alpha
beta from both sides

00:09:02.740 --> 00:09:06.650
of this equation,
we get alpha squared

00:09:06.650 --> 00:09:12.550
plus beta-- let me do this
in a different color-- we get

00:09:12.550 --> 00:09:16.360
alpha squared plus
beta squared is going

00:09:16.360 --> 00:09:24.710
to be equal to p squared
minus 2 alpha beta.

00:09:30.690 --> 00:09:31.710
All right?

00:09:31.710 --> 00:09:34.720
I just subtracted 2 alpha
beta from both sides.

00:09:34.720 --> 00:09:39.640
And so this is going to be
equal to p squared minus 2 times

00:09:39.640 --> 00:09:49.927
this thing, minus 2 times p
to the third plus q over 3p.

00:09:49.927 --> 00:09:51.010
And what is this equal to?

00:09:51.010 --> 00:09:53.620
This is equal to-- let's put
on a common denominator--

00:09:53.620 --> 00:09:59.480
p squared over-- this is the
same thing as 3-- my cell

00:09:59.480 --> 00:10:00.000
phone rang.

00:10:00.000 --> 00:10:01.041
I had to pause it, sorry.

00:10:01.041 --> 00:10:04.890
So this is equal to
3p to the squared--

00:10:04.890 --> 00:10:08.380
if I want to put it
over 3p-- is equal to 3p

00:10:08.380 --> 00:10:10.625
to the third over 3p.

00:10:10.625 --> 00:10:11.500
Let me make it clear.

00:10:11.500 --> 00:10:13.995
This term right here
is this term over here

00:10:13.995 --> 00:10:15.120
if you were to simplify it.

00:10:15.120 --> 00:10:15.985
The 3's cancel out.

00:10:15.985 --> 00:10:17.970
You just have p squared.

00:10:17.970 --> 00:10:20.370
And then you have
minus-- I'm just

00:10:20.370 --> 00:10:25.040
going to distribute the negative
2-- minus 2p to the third minus

00:10:25.040 --> 00:10:28.400
2q, all of that over 3p.

00:10:28.400 --> 00:10:32.540
So it is equal to 3p
cubed minus 2p cubed

00:10:32.540 --> 00:10:43.190
is equal to p cubed minus
2q, all of that over 3p.

00:10:43.190 --> 00:10:46.250
So now we can write this
entire expression in terms

00:10:46.250 --> 00:10:47.140
of p's and q's.

00:10:47.140 --> 00:10:50.490
This first term over
here, alpha beta, this

00:10:50.490 --> 00:10:56.810
is p cubed plus
q over 3p, and we

00:10:56.810 --> 00:10:58.995
want to multiply
that times x squared.

00:11:01.540 --> 00:11:05.680
And then you have minus alpha
squared plus beta squared.

00:11:05.680 --> 00:11:07.680
We just figured out that
alpha squared plus beta

00:11:07.680 --> 00:11:09.050
squared is this thing over here.

00:11:09.050 --> 00:11:12.200
So minus-- I'll do it
in a different color,

00:11:12.200 --> 00:11:15.190
I'll do it in green--
minus this thing,

00:11:15.190 --> 00:11:19.270
p to the third minus 2q over 3p.

00:11:21.910 --> 00:11:26.980
And that's going to be
multiplied by an x, and then

00:11:26.980 --> 00:11:28.820
plus alpha beta again.

00:11:28.820 --> 00:11:34.450
So alpha beta is p to
the third plus q over 3p.

00:11:34.450 --> 00:11:36.770
Now, when we look at any
of the choices up here,

00:11:36.770 --> 00:11:37.790
it looks simplified.

00:11:37.790 --> 00:11:39.930
We don't see any 3p's
in the denominator.

00:11:39.930 --> 00:11:43.680
So let's see if we
can simplify that.

00:11:43.680 --> 00:11:46.345
Oh, and we don't want to
lose-- we had an equals 0 here

00:11:46.345 --> 00:11:46.970
the whole time.

00:11:46.970 --> 00:11:49.260
So this is equals 0.

00:11:49.260 --> 00:11:52.030
So when you multiply both
sides of this equation by 3p,

00:11:52.030 --> 00:11:53.190
what do you get?

00:11:53.190 --> 00:11:59.070
This first term becomes p to the
third plus q times x squared.

00:11:59.070 --> 00:12:05.310
The second term becomes minus
p to the third minus 2q x.

00:12:05.310 --> 00:12:12.540
And then this last term becomes
plus p to the third plus q,

00:12:12.540 --> 00:12:15.370
and this is going
to be equal to 0.

00:12:15.370 --> 00:12:19.620
Now, let's see how this
compares to the choices.

00:12:19.620 --> 00:12:23.090
And I think I actually did
not make my careless mistake

00:12:23.090 --> 00:12:23.900
in this video.

00:12:23.900 --> 00:12:26.220
I actually got the right answer.

00:12:26.220 --> 00:12:27.700
Let me see.

00:12:27.700 --> 00:12:30.700
It shows you, when
you do it carefully,

00:12:30.700 --> 00:12:32.360
you're less likely
to make errors.

00:12:32.360 --> 00:12:32.860
OK.

00:12:32.860 --> 00:12:33.580
So let's see.

00:12:33.580 --> 00:12:37.390
How does this match up
to what I have over here?

00:12:37.390 --> 00:12:38.880
Well, this looks exactly.

00:12:38.880 --> 00:12:41.670
I'm very excited because
I tried this twice

00:12:41.670 --> 00:12:43.400
before actually
making the video,

00:12:43.400 --> 00:12:44.990
and I kept making
a careless mistake.

00:12:44.990 --> 00:12:47.390
But this time I didn't
make the careless mistake,

00:12:47.390 --> 00:12:51.350
and I got B, p to the third
plus q x squared minus p

00:12:51.350 --> 00:12:56.206
to the third minus 2q x
plus p to the third plus q.

00:12:56.206 --> 00:13:01.312
So that right there, that
right there is our answer.

00:13:01.312 --> 00:13:03.520
So I take back what I said
at the beginning of video.

00:13:03.520 --> 00:13:04.760
I didn't need your help.

00:13:04.760 --> 00:13:06.510
But I guess just the
process of explaining

00:13:06.510 --> 00:13:08.070
it was the help that I needed.

00:13:08.070 --> 00:13:11.060
Anyway, hopefully you
found that useful.


IIT JEE Integral Limit

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.870
The value of the limit
as x approaches 0 of 1

00:00:03.870 --> 00:00:07.490
over x to the third times
the definite integral from 0

00:00:07.490 --> 00:00:10.890
to x of t natural
log of 1 plus t, all

00:00:10.890 --> 00:00:14.530
of that over t to the
fourth plus 4dt is.

00:00:14.530 --> 00:00:16.830
Now, when you see an
integral like this,

00:00:16.830 --> 00:00:20.170
you're first temptation is
to try to solve the integral,

00:00:20.170 --> 00:00:23.770
try to find the antiderivative
of this expression here.

00:00:23.770 --> 00:00:27.320
And you might try to experiment
maybe with inverse trig

00:00:27.320 --> 00:00:30.120
functions, maybe arctangent
looks tempting over here.

00:00:30.120 --> 00:00:31.700
But you have this natural log.

00:00:31.700 --> 00:00:33.970
Maybe you can do it with
integration by parts.

00:00:33.970 --> 00:00:36.460
But very quickly, or
maybe not so very quickly,

00:00:36.460 --> 00:00:39.490
you'll realize that this is
a nearly impossible integral

00:00:39.490 --> 00:00:43.170
to simply solve using
analytical methods.

00:00:43.170 --> 00:00:45.890
But, lucky for us,
this problem isn't just

00:00:45.890 --> 00:00:47.130
about solving this integral.

00:00:47.130 --> 00:00:50.010
It's actually about
evaluating this a limit.

00:00:50.010 --> 00:00:52.410
And if we rearrange this
expression a little bit,

00:00:52.410 --> 00:00:55.120
it might become a
little obvious to you

00:00:55.120 --> 00:00:56.910
how we can approach
this problem.

00:00:56.910 --> 00:01:02.992
So this is the limit
as x approaches 0 of.

00:01:02.992 --> 00:01:04.450
Let me write this
in the numerator,

00:01:04.450 --> 00:01:07.269
because this is could be
multiplying by 1 over x

00:01:07.269 --> 00:01:09.810
to the third is the same thing
as dividing by x to the third.

00:01:09.810 --> 00:01:15.050
So this is the integral from
0 to x of t natural log of 1

00:01:15.050 --> 00:01:20.900
plus t, all of that over
t to the fourth plus 4dt.

00:01:20.900 --> 00:01:25.810
And all of that
over x to the third.

00:01:25.810 --> 00:01:27.660
And now, this is
probably ringing bells

00:01:27.660 --> 00:01:30.220
in some of your brains.

00:01:30.220 --> 00:01:32.290
And what we see here
is we don't know

00:01:32.290 --> 00:01:34.620
how to evaluate this
integral over here.

00:01:34.620 --> 00:01:38.390
But what does this integral
approach as x approaches 0?

00:01:38.390 --> 00:01:41.270
What is the limit of
this numerator expression

00:01:41.270 --> 00:01:44.340
as it approaches 0?

00:01:44.340 --> 00:01:45.340
So let me just write it.

00:01:45.340 --> 00:01:49.910
The limit as it
approaches 0, so the limit

00:01:49.910 --> 00:01:57.490
as x approaches 0 of 0 to x,
all of that stuff inside dt

00:01:57.490 --> 00:01:59.892
is going to equal
the integral from 0.

00:01:59.892 --> 00:02:01.350
It's going to
approach the integral

00:02:01.350 --> 00:02:04.950
from 0 to 0 of that stuff dt.

00:02:04.950 --> 00:02:08.310
And a definite
integral from 0 to 0,

00:02:08.310 --> 00:02:11.810
we're not moving at all along
the horizontal or the t axis.

00:02:11.810 --> 00:02:14.552
And so, this is
just going to be 0.

00:02:14.552 --> 00:02:16.510
And if we do the same
thing in the denominator,

00:02:16.510 --> 00:02:17.343
that's much simpler.

00:02:19.760 --> 00:02:24.850
The limit as x approaches
0 of x to the third, that's

00:02:24.850 --> 00:02:27.600
clearly also going to be 0.

00:02:27.600 --> 00:02:29.860
So we took the limits of the
numerator and denominator

00:02:29.860 --> 00:02:31.710
and we got 0/0.

00:02:31.710 --> 00:02:33.460
That's the indeterminate form.

00:02:33.460 --> 00:02:37.740
And so now the bells should
be ringing very loudly

00:02:37.740 --> 00:02:40.170
in your brain and
screaming L'Hopital's rule.

00:02:45.100 --> 00:02:47.840
And I've done several videos on
this in the calculus playlist.

00:02:47.840 --> 00:02:49.880
But this just tells
us if we evaluate

00:02:49.880 --> 00:02:51.860
the limit of the
numerator as it approaches

00:02:51.860 --> 00:02:54.420
0 and the limit of the
denominator approaches 0,

00:02:54.420 --> 00:02:56.490
and we get an
indeterminate form,

00:02:56.490 --> 00:02:59.150
then this limit is going
to be the exact same thing,

00:02:59.150 --> 00:03:02.400
assuming the limit
exist, of the limit

00:03:02.400 --> 00:03:06.190
as x approaches 0 of the
derivative of this expression

00:03:06.190 --> 00:03:08.867
over the derivative
of this expression.

00:03:08.867 --> 00:03:11.200
Now let's do the denominator
first, since that's easier.

00:03:11.200 --> 00:03:13.350
What's the derivative
of x to the third?

00:03:13.350 --> 00:03:15.930
Well, that's just 3x squared.

00:03:15.930 --> 00:03:18.680
And what's the derivative
of the numerator?

00:03:18.680 --> 00:03:20.180
And just as a
reminder, and I'm just

00:03:20.180 --> 00:03:21.950
going to break out the
fundamental theorem

00:03:21.950 --> 00:03:22.530
of calculus.

00:03:22.530 --> 00:03:26.780
The derivative with respect
to x from some constant to x

00:03:26.780 --> 00:03:34.400
of f of t dt is just
equal to f of x.

00:03:34.400 --> 00:03:38.170
You're undoing the
integration here.

00:03:38.170 --> 00:03:41.230
You just get f of x, it's just
the function with respect to x.

00:03:41.230 --> 00:03:44.440
So the derivative of
this with respect to x

00:03:44.440 --> 00:03:47.900
is just going to be this
function with respect

00:03:47.900 --> 00:03:49.030
to x instead of t.

00:03:49.030 --> 00:03:52.780
So it's going to be
x natural log-- that

00:03:52.780 --> 00:03:57.310
was an ugly natural
log-- of 1 plus x,

00:03:57.310 --> 00:04:02.230
all of that over x
to the fourth plus 4.

00:04:02.230 --> 00:04:04.750
Now, when we try to evaluate
the limit here, what happens?

00:04:04.750 --> 00:04:07.510
If we do the numerator, if we do
the same thing we did up here,

00:04:07.510 --> 00:04:09.300
this numerator still
evaluates to 0.

00:04:09.300 --> 00:04:11.010
You get 0 times
natural log of 1.

00:04:11.010 --> 00:04:11.940
That's 0 as well.

00:04:11.940 --> 00:04:13.260
So that's all 0.

00:04:13.260 --> 00:04:15.950
The denominator
still evaluates to 0.

00:04:15.950 --> 00:04:18.060
So we have indeterminate
form again.

00:04:18.060 --> 00:04:21.110
So we might be tempted to just
go straight, do L'Hopital's

00:04:21.110 --> 00:04:23.700
rule again, take the
derivative of the numerator

00:04:23.700 --> 00:04:25.700
and then take the derivative
of the denominator.

00:04:25.700 --> 00:04:26.570
And that would work.

00:04:26.570 --> 00:04:29.320
Or maybe you'll see that
this is a pretty hairy thing

00:04:29.320 --> 00:04:30.530
to take a derivative of.

00:04:30.530 --> 00:04:33.113
And maybe we might have to take
a derivative again after that.

00:04:33.113 --> 00:04:35.800
So this could get
very, very, very hairy.

00:04:35.800 --> 00:04:37.510
So, instead of just
straight up just

00:04:37.510 --> 00:04:39.210
taking a derivative of the
numerator and derivative

00:04:39.210 --> 00:04:40.668
of the denominator
again, let's see

00:04:40.668 --> 00:04:42.560
if we can simplify
this a little bit.

00:04:42.560 --> 00:04:43.720
And we can.

00:04:43.720 --> 00:04:44.960
So this is the same thing.

00:04:44.960 --> 00:04:48.062
This is equal to the
limit as x approaches 0.

00:04:48.062 --> 00:04:49.770
I'm just going to put
the x to the fourth

00:04:49.770 --> 00:04:51.970
plus 4 in the denominator.

00:04:51.970 --> 00:04:57.640
So it's x natural log of 1 plus
x, all of that over this 3x

00:04:57.640 --> 00:05:00.400
squared.

00:05:00.400 --> 00:05:03.134
And if I'm dividing by something
and dividing by something else,

00:05:03.134 --> 00:05:05.050
that's just like dividing
by that first thing.

00:05:05.050 --> 00:05:08.190
So this is x to
the fourth plus 4.

00:05:08.190 --> 00:05:10.870
I just put this in
the denominator,

00:05:10.870 --> 00:05:12.340
and we can simplify this.

00:05:12.340 --> 00:05:15.760
We can divide the numerator
and denominator by x.

00:05:15.760 --> 00:05:17.600
And now, let's try to
evaluate this limit.

00:05:17.600 --> 00:05:19.120
As x approaches 0
in the numerator,

00:05:19.120 --> 00:05:20.410
we get natural log of 1.

00:05:20.410 --> 00:05:21.820
So that's 0.

00:05:21.820 --> 00:05:23.702
So this is approaching 0 over.

00:05:23.702 --> 00:05:25.160
And if we do it in
the denominator,

00:05:25.160 --> 00:05:27.280
we have 3 times 0 times 4.

00:05:27.280 --> 00:05:28.700
So that's going to be 0.

00:05:28.700 --> 00:05:30.660
Once again, indeterminate form.

00:05:30.660 --> 00:05:32.830
So now let's apply
L'Hopital's rule again.

00:05:32.830 --> 00:05:35.070
But we can apply it on
a simpler expression,

00:05:35.070 --> 00:05:39.460
on natural log of 1
plus x over 3x times x

00:05:39.460 --> 00:05:41.150
to the fourth plus 4.

00:05:41.150 --> 00:05:42.050
So let's do that.

00:05:42.050 --> 00:05:44.700
So this is going to be
equal to, assuming the limit

00:05:44.700 --> 00:05:47.770
exists, this is
equal to the limit

00:05:47.770 --> 00:05:52.120
as x approaches 0 of the
derivative of the numerator.

00:05:52.120 --> 00:05:55.530
The derivative of natural log of
1 plus x, that's just 1 over 1

00:05:55.530 --> 00:05:58.810
plus x over the derivative
of the denominator.

00:05:58.810 --> 00:06:00.730
We just have to do a
little product rule here.

00:06:00.730 --> 00:06:03.970
Derivative of the first
function is derivative of 3x

00:06:03.970 --> 00:06:07.290
is just 3 times the
second function, which

00:06:07.290 --> 00:06:13.260
is x to the fourth plus 4 plus
the first function, 3x, it's

00:06:13.260 --> 00:06:15.070
just the product rule,
times the derivative

00:06:15.070 --> 00:06:16.180
of the second function.

00:06:16.180 --> 00:06:20.690
The derivative of this
is 4x to the third.

00:06:20.690 --> 00:06:25.180
Now, when we approach 0,
what happens to this value?

00:06:25.180 --> 00:06:28.120
What is the numerator and
the denominator approach?

00:06:28.120 --> 00:06:32.850
The numerator approaches 1
over 1 plus 0, or just 1.

00:06:32.850 --> 00:06:36.440
The denominator
approaches 3 times 0

00:06:36.440 --> 00:06:40.700
plus 4 plus this is
going to be 0, plus 0.

00:06:40.700 --> 00:06:44.630
So this is equal to 1/12.

00:06:44.630 --> 00:06:46.240
And we're done.

00:06:46.240 --> 00:06:49.550
This crazy looking
limit with this integral

00:06:49.550 --> 00:06:55.030
that we couldn't solve
is just equal to 1/12.

00:06:55.030 --> 00:06:57.480
So thank goodness we
had L'Hopital's rule

00:06:57.480 --> 00:07:00.170
so we didn't have
to actually take

00:07:00.170 --> 00:07:04.660
the antiderivative
of this expression.


IIT JEE Position Vectors

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:05.240
Let P, Q, R, and S be the points
on the plane with position

00:00:05.240 --> 00:00:12.480
vectors negative 2i minus j,
4i, 3i plus 3j, and negative 3i

00:00:12.480 --> 00:00:14.480
plus 2j, respectively.

00:00:14.480 --> 00:00:17.810
The quadrilateral
PQRS must be a--

00:00:17.810 --> 00:00:21.010
So let's just graph each
of these positions vectors,

00:00:21.010 --> 00:00:23.690
or graph the points
that they're specifying.

00:00:23.690 --> 00:00:28.170
So let me draw my coordinate
axes right over here.

00:00:28.170 --> 00:00:32.840
So this is my vertical.

00:00:32.840 --> 00:00:34.570
I can draw a
straighter line that.

00:00:34.570 --> 00:00:35.916
There you go.

00:00:35.916 --> 00:00:37.290
That's about as
good as I can do.

00:00:37.290 --> 00:00:40.450
Let me draw the horizontal.

00:00:40.450 --> 00:00:43.600
That's a better job, I think.

00:00:43.600 --> 00:00:44.900
And then let me mark it off.

00:00:44.900 --> 00:00:45.400
Let's see.

00:00:45.400 --> 00:00:47.060
We go as high as 4.

00:00:47.060 --> 00:00:49.012
We go up to 4i.

00:00:49.012 --> 00:00:52.650
So 1, 2, 3, 4.

00:00:52.650 --> 00:00:54.650
We goes as low was negative 2i.

00:00:54.650 --> 00:00:57.430
So 1, 2.

00:00:57.430 --> 00:00:59.350
Just making sure I
make it a little bit

00:00:59.350 --> 00:01:01.308
even, because we're going
to have to figure out

00:01:01.308 --> 00:01:02.560
the shape of whatever we draw.

00:01:02.560 --> 00:01:06.940
Then we're going to go up 3.

00:01:06.940 --> 00:01:10.220
1, 2, 3.

00:01:10.220 --> 00:01:11.755
And we actually go
down 1, as well.

00:01:11.755 --> 00:01:14.380
So I think that'll give us-- and
we could keep marking them off

00:01:14.380 --> 00:01:16.600
if we wanted to, but
we don't need to,

00:01:16.600 --> 00:01:19.640
based on the positions
that have been specified.

00:01:19.640 --> 00:01:20.870
So let's draw these vectors.

00:01:20.870 --> 00:01:23.100
So let's draw vector P first.

00:01:23.100 --> 00:01:26.200
Negative 2i minus j.

00:01:26.200 --> 00:01:29.465
So negative 2i, i is
along the horizontal,

00:01:29.465 --> 00:01:31.340
or as you can imagine,
it's the x-coordinate.

00:01:31.340 --> 00:01:34.130
So negative 2i minus j.

00:01:34.130 --> 00:01:37.170
So it is this position
vector right over here.

00:01:37.170 --> 00:01:39.230
If we put it in
standard form, starting

00:01:39.230 --> 00:01:41.330
with its base at the origin.

00:01:41.330 --> 00:01:42.460
It's that position vector.

00:01:42.460 --> 00:01:44.600
But it's specifying
that point over there,

00:01:44.600 --> 00:01:49.417
which is the coordinate
negative 2, negative 1.

00:01:49.417 --> 00:01:50.500
Let me just write it down.

00:01:50.500 --> 00:01:52.880
Negative 2, negative 1.

00:01:52.880 --> 00:01:55.830
Now let's do 4i.

00:01:55.830 --> 00:02:01.480
So 4i is literally-- that's
just 4 in the x direction.

00:02:01.480 --> 00:02:03.420
1, 2, 3, 4.

00:02:03.420 --> 00:02:07.890
There's no j component,
no vertical component.

00:02:07.890 --> 00:02:12.520
So it specifies that
position right over there,

00:02:12.520 --> 00:02:16.540
or it specifies the point 4, 0.

00:02:16.540 --> 00:02:20.510
Then we have 3i plus 3j.

00:02:20.510 --> 00:02:22.400
So 1, 2, 3.

00:02:22.400 --> 00:02:23.860
1, 2, 3.

00:02:23.860 --> 00:02:26.470
That is that point right there.

00:02:26.470 --> 00:02:29.610
The position vector
specifies that point,

00:02:29.610 --> 00:02:32.700
or specifies the point 3, 3.

00:02:32.700 --> 00:02:35.567
And then finally,
we have negative 3i.

00:02:35.567 --> 00:02:36.900
Let me do this in another color.

00:02:36.900 --> 00:02:37.800
Negative 3i.

00:02:37.800 --> 00:02:41.340
So 1, 2, 3, plus 2j, 1, 2.

00:02:41.340 --> 00:02:44.120
So right over there.

00:02:44.120 --> 00:02:46.460
So that is that last
position vector,

00:02:46.460 --> 00:02:50.640
and it's specifying the
point negative 3, 2.

00:02:50.640 --> 00:02:53.030
And they want to know,
what's the quadrilateral made

00:02:53.030 --> 00:02:54.770
by these four positions?

00:02:54.770 --> 00:02:55.980
So let's draw it.

00:02:55.980 --> 00:02:57.800
Let's connect the dots.

00:02:57.800 --> 00:03:03.280
So this is P, go to Q. It gets
us that line right over there.

00:03:03.280 --> 00:03:09.300
Then from Q to R, we're
going to go like that.

00:03:09.300 --> 00:03:15.874
And then from R to S, we're
going to go like that.

00:03:15.874 --> 00:03:17.790
Actually, it's going to
go a little bit more--

00:03:17.790 --> 00:03:22.050
it's going to go like that.

00:03:22.050 --> 00:03:29.320
Sorry, that's from R
to S. And then S to P

00:03:29.320 --> 00:03:31.250
is going to be like that.

00:03:31.250 --> 00:03:33.695
So just eyeballing it,
it's clearly not a square.

00:03:36.830 --> 00:03:38.230
It's clearly not a rectangle.

00:03:38.230 --> 00:03:40.830
It's coming off at
different angles.

00:03:40.830 --> 00:03:43.840
It's clearly not a rhombus.

00:03:43.840 --> 00:03:45.850
A rhombus would have
all of the same sides.

00:03:45.850 --> 00:03:48.510
So a square would
be all 90 degrees,

00:03:48.510 --> 00:03:51.522
would be all the same sides and
all the angles are 90 degrees.

00:03:51.522 --> 00:03:52.480
That would be a square.

00:03:52.480 --> 00:03:53.330
It's clearly not a square.

00:03:53.330 --> 00:03:55.610
A rectangle would be all
the angles are 90 degrees,

00:03:55.610 --> 00:03:57.950
but two sides are
going to be the same,

00:03:57.950 --> 00:04:00.170
and the other two sides
are going to be the same.

00:04:00.170 --> 00:04:04.510
These do not look
like 90 degree angles.

00:04:04.510 --> 00:04:06.050
And you could try if you want.

00:04:06.050 --> 00:04:07.675
I mean, just eyeballing
it, it's pretty

00:04:07.675 --> 00:04:09.470
clear these aren't
90 degree angles.

00:04:09.470 --> 00:04:12.730
But if you wanted to, you
could find the slope of this

00:04:12.730 --> 00:04:15.100
and find the slope of that
line, and if they're not

00:04:15.100 --> 00:04:16.579
the negative inverse
of each other,

00:04:16.579 --> 00:04:18.537
then they're not
perpendicular, and they're not

00:04:18.537 --> 00:04:19.890
coming at 90 degree angles.

00:04:19.890 --> 00:04:21.970
And so the last one
is are they a rhombus.

00:04:21.970 --> 00:04:25.210
A rhombus is a parallelogram
while the sides are the same.

00:04:25.210 --> 00:04:29.510
So a rhombus would
look like that.

00:04:29.510 --> 00:04:30.800
Clearly not a rhombus.

00:04:30.800 --> 00:04:32.320
These sides are
clearly much longer

00:04:32.320 --> 00:04:33.880
than those sides over there.

00:04:33.880 --> 00:04:35.940
Now just from
deductive reasoning,

00:04:35.940 --> 00:04:38.580
it looks like the answer
is A. it's a parallelogram.

00:04:38.580 --> 00:04:39.570
But we can verify it.

00:04:42.410 --> 00:04:45.000
In order for this to
be a parallelogram,

00:04:45.000 --> 00:04:47.110
these two sides
have to be parallel,

00:04:47.110 --> 00:04:50.802
and these two sides
have to be parallel.

00:04:50.802 --> 00:04:52.260
Or another way to
think of it, they

00:04:52.260 --> 00:04:53.610
would have to have
the same slope.

00:04:53.610 --> 00:04:54.420
So we could verify it.

00:04:54.420 --> 00:04:56.090
If we were taking this on
exam under time pressure,

00:04:56.090 --> 00:04:58.100
we would just cross these
out, and it looks pretty good.

00:04:58.100 --> 00:05:00.420
We would just go with A.
But let's just verify it.

00:05:00.420 --> 00:05:05.220
So what's the slope down here?

00:05:05.220 --> 00:05:07.000
So it's change in
y over change in x.

00:05:07.000 --> 00:05:15.940
So 0 minus negative 1,
over 4 minus negative 2,

00:05:15.940 --> 00:05:20.460
or this is equal to 1
over 4 minus negative 2,

00:05:20.460 --> 00:05:21.880
is positive 6.

00:05:21.880 --> 00:05:23.330
This is 1/6.

00:05:23.330 --> 00:05:25.660
Or another way to think
about it, we ran 6

00:05:25.660 --> 00:05:27.040
and we only rised 1.

00:05:27.040 --> 00:05:28.210
This is 1/6.

00:05:28.210 --> 00:05:29.450
Now what happens over here?

00:05:29.450 --> 00:05:30.630
Well, it's the same thing.

00:05:30.630 --> 00:05:33.840
We're going from
negative 3 to three.

00:05:33.840 --> 00:05:35.420
So we're running 6.

00:05:35.420 --> 00:05:40.060
So we say 3 minus
negative 3 is equal to 6.

00:05:40.060 --> 00:05:41.680
And how much did we rise?

00:05:41.680 --> 00:05:45.920
We have 3 minus 2,
which is equal to 1.

00:05:45.920 --> 00:05:48.440
So this is also a slope of 1/6.

00:05:48.440 --> 00:05:51.270
So these two lines
are clearly parallel.

00:05:51.270 --> 00:05:53.220
And now let's do it
for these other two.

00:05:53.220 --> 00:05:56.160
What's the slope over here?

00:05:56.160 --> 00:05:58.110
And actually, you could
just do rise over run.

00:06:02.020 --> 00:06:03.960
So that's exactly what
we were doing before,

00:06:03.960 --> 00:06:05.876
but just to give you the
taste for everything.

00:06:05.876 --> 00:06:07.420
So we ran, to go
from here to here,

00:06:07.420 --> 00:06:09.070
we have to move to the right 1.

00:06:09.070 --> 00:06:10.540
So we run 1.

00:06:10.540 --> 00:06:12.280
And how much do we rise?

00:06:12.280 --> 00:06:15.140
Well, we had to go down 3.

00:06:15.140 --> 00:06:16.220
So it's negative 3.

00:06:16.220 --> 00:06:18.520
So the slope here is negative 3.

00:06:18.520 --> 00:06:19.710
And what about here?

00:06:19.710 --> 00:06:23.340
To go from this point to this
point, we went to the right 1.

00:06:23.340 --> 00:06:24.890
So we ran 1.

00:06:24.890 --> 00:06:27.090
And we had to go
down-- we started at 2,

00:06:27.090 --> 00:06:28.720
and we ended up at negative 1.

00:06:28.720 --> 00:06:30.060
We had to go down 3.

00:06:30.060 --> 00:06:32.690
So the slope here
is also negative 3.

00:06:32.690 --> 00:06:35.240
So these lines
are also parallel.

00:06:35.240 --> 00:06:38.440
So we're definitely dealing
with a parallelogram, which

00:06:38.440 --> 00:06:41.924
is neither a rhombus
nor a rectangle.