Messages posted by KAB

1)

There is a similar worked example in HC Verma in page no. 93 Q.no.12

Do have a look at it and tell me if you have any doubts

6)

Use Charles and volume V=AL

Here V₁=50A T₁=300K V₂=XA T₂=384K where A is area of cross section

Using V₁/T₁=V₂/T₂

We get X=64cm

So the mercury level falls by 14cm

I agree that the 2 combos are different......Also I'm not able to find out the flaw in my method.

It is given that 6 different colours should be used.So I thought repetition is NOT allowed!

1st face can be coloured in 6 ways.Next in 5 ways.So on last face in 1 way.

Also all faces are alike.

So total no. of ways of colouring is 6!/6!=1

The answer has to be 1

Because all faces of the cube are alike and 6 different colours are used.

So the required no. of ways=6!/6!=1

So the answer is 1

Let the required no. of minutes be x.

The no. of minutes between 8am and 12noon is 240min.

It is given that 240-(x+60)=3x (Because one hour ago it was three times as many minutes after 8 a.m)

So x=45min.

Let P be (0,y)

So the slope of AP=(5a-y)/a

Slope of BP=(a-y)/4a

So tan

=|({(5a-y)/a}-{(a-y)/4a})/(1+(5a-y)(a-y)4a²)|

or tan

=|a(19a-3y)/(y-3a)²|..............(1)

Clearly tan

is maximum when y=3a ie tan

is infinity and hence

=

/2

You can also check this by differentiating (1) and putting d

/dy=0 you will get

y=3a or y=29a/3

When y=29a/3 tan

=9/40

Hence when y=3a

is maximum ie

/2

So the answer is B)

/2

@titun,

You were right till

For x = 0 the above limit is in 0/0 form. So just apply L. Hospital

Then we will get lnP=_{[x ][0 ]}2*3*(a^xlna+b^xlnb+c^xlnc)/(a^x+b^x+c^x)

Hence P=(abc)^2/3

You can also solve this by making use of _{[x ][0 ]}(1+x)^1/x=e but using L'Hospitals rule as done by titun is the shortest method

Congrats Himanshu!!!!!!

Why dont u guys check this http://integrals.wolfram.com/index.jsp

Just type the function there and u will get it integrated.......Simply dont waste time on such questions.

Hey aman531,
We cannot evaluate it.Suppose x is in 3rd or 4th quadrant then root(sinx) will be a complex number.But a definite integral like limit from 0 to pi/2 etc can be evaluated.

Refer this http://integrals.wolfram.com/index.jsp

bujji has told the right approach.

Let the position vectors of vertices be OA(x,0,0) , OB(0,y,O) and OC(0,0,z)

The vector AB=OB-OA= -xi+yj

The vector BC=OC-OB= -yj+zk

So area of the triangle ABC=AB*AC/2 (Where * is cross product of vectors)

AB*AC=| i j k|

| -x y 0|

|0 -y z|

or AB*AC=(yzi+xzj+xyk)

So Area =|AB*AC|/2=1/2

x²y²+y²z²+z²x²

So a) is the answer

Its based on multinomial theorem which is as follows,

(a₁+a₂+.....+a_m)ⁿ= {n!(a₁ⁿ¹)(a₂ⁿ²).......(a_m^nm)}/(n₁!n₂!......n_m!)

Where n₁+n₂+n₃+....+n_m=n

In this problem,

a₁=1 a₂=x a₃=y a₄= -z n=9

So

(1+x+y-z)⁹=

{9!(1^ax^by^c(-z)^d)}/(a!b!c!d!)

Where a+b+c+d=9

Here since co-efficient of x³y⁴z is required a=1 b=3 c=4 d=1

So the co-efficient of x³y⁴z is -9!/1!3!4!= -2*9C2*7C3

Yaar joyfrancis,

The co-efficient method is very important.Also many problems which r difficult to solve by finding out the number of cases , can be easily solved by this method.

Plz tell me where u r finding it difficult.

Yaar its gr888

Joyfrancis,

It is an important method and it will be easy to solve using this method.

Look I will explain the co-efficient method.

In one ball he can score either 0,1,2,3 or 4 runs So the powers of x vary from 0 to 4 in one ball.

Since there are six balls the expression becomes (x⁰+x¹+x²+x³+x⁴)⁶

If there were seven balls then it would be (x⁰+x¹+x²+x³+x⁴)⁷

Also a total of 14 runs is to be scored.So we must find the co-efficient of x¹⁴ in

(x⁰+x¹+x²+x³+x⁴)⁶

If a total of 15 runs is to be scored then we must find the co-efficient of x¹⁵ in (x⁰+x¹+x²+x³+x⁴)⁶

Hope your doubt is cleared!

Mostly the answer is d)CCl3CHO as the chlorine obtained by the electrolysis will oxidise ethyl alcohol to acetaldehyde and then it undergoes halogenation and finally chloral will be obtained.

I think the question should be

"If (

3 -i )ⁿ = 2ⁿ , n belongs to integers , then n is a multiple of.....?"

Now (

3 -i )ⁿ = 2ⁿ

So ((

3 -i )/2)ⁿ = 1

So (cis( -pi/6))ⁿ=1

Implies cis( -npi/6)=1

This will be true when npi/6=2kpi

or n=12k where k is any integer.

So n is a multiple of 12

In one ball he can score either 0,1,2,3 or 4 runs and there are 6 balls.Total of 14 runs should be scored.

So required no. of ways is the co-efficient of x¹⁴ in (x⁰+x¹+x²+x³+x⁴)⁶

ie co-efficient of x¹⁴ in (x⁵-1)⁶/(x-1)⁶

Now u can solve furthur

A matrix A is said to be of rank r if there is at least one non zero minor of order r and every minor of order r+1 is zero.

Eg:

Consider a matrix A,

[2 -1 5]

|3 0 6|

[4 1 7]

|A|=0

So the minor of third order is zero

There exists a minor of second order

|3 0|

|1 4|

which is equal to 3

Thus a minor of second order is not zero.

So rank of A is 2

Hey its very useful....thanx for posting it!

_{^{Given integral is I=}[ 0]}^{[pi/2 ]} dx/(1+

tanx)

_^I=[0]

^{[pi/2 ]} dx/(1+

tanx)=_{[ 0]}^{[pi/2 ]}

cosxdx/(

sinx+

cosx).......(1)

Also I=_{[ 0]}^{[pi/2 ]}

cos(pi/2-x)dx/(

sin(pi/2-x)+

cos(pi/2-x))=_{[ 0]}^{[pi/2 ]}

sinxdx/(

sinx+

cosx)........(2) (Using _{[ 0]}^[a]f(x)dx=_{[ 0]}^[a]f(a-x)dx)

Add (1) and (2)

we get I=pi/4

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