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Assume coordinates of the vertices of triangle as

(5, 10), (3, 8) and (1, 2)

now use distance formula to obtain the results.

please try to send the problem rather than reference.

Dear Drishti

I think the formula u are trying to highlight is

R = R_o A^1/3

where R_o = 1.07 x 10^-15 m

and A = mass number of the nucleus = neutrons + protons

But this formula is applicable for RADIUS OF THE NUCLEUS BUT NOT FOR

ATOMIC RADIUS.

y = (x¹⁰)/11 + 6 / x

x dⁿy / dxⁿ + n d^n-1y / dx^n-1 = 0 .... (1)

To find the value of n

Solution) Since, y = (x¹⁰)/11 + 6 / x

Therefore,

dⁿy / dxⁿ = (10) (9)...[10 - (n - 1)] x^10-n / 11 + (-1)ⁿ n! 6 / xⁿ⁺¹

or dⁿy / dxⁿ = (10) (9)...(11 - n) x^10-n / 11 + (-1)ⁿ n! 6 / xⁿ⁺¹

Therefore,

x dⁿy / dxⁿ = (10) (9)...(11 - n) x^11-n / 11 + (-1)ⁿ n! 6 / xⁿ ...(2)

also,

n d^n-1y / dx^n-1 = n (10) (9)...(12 - n) x^11-n / 11 + n (-1)^n-1 (n-1)! 6 / xⁿ

or n d^n-1y / dx^n-1 = n (10) (9)...(12 - n) x^11-n / 11 + (-1)^n-1 n! 6 / xⁿ ...(3)

Substituting (2) and (3) in equation (1) we obtain

(10) (9)...(11 - n) x^11-n /11+ (-1)ⁿ n! 6 / xⁿ + n (10) (9)...(12 - n) x^11-n / 11

+ (-1)^n-1 n! 6 / xⁿ = 0

Here, (-1)ⁿ n! 6 / xⁿ + (-1)^n-1 n! 6 / xⁿ = 0

So,

(10) (9)...(11 - n) x^11-n /11 + n (10) (9)...(12 - n) x^11-n / 11 = 0

or, (10) (9)...(11 - n) + n (10) (9)...(12 - n) = 0

or, (10) (9)...(12 - n) [ 11 - n + n ] = 0

or, (10) (9)...(12 - n) 11 = 0

Which is possible if (12 - n ) = 0,

Hence, n = 12

Let ABC be a triangle inscribed in the circle with center O and radius r.

If the area of this triangle is maximum, then vertex C should be at a maximum distance from the base AB i.e. CD must be perpendicular to AB.

Hence, ABC is an isosceles triangle.
If

BCD =

, where D is the mid-point of BC, then

BOD = 2

so, AB = 2 BD

= 2r sin 2

CD = CO + OD = r + r cos 2

If S be the area of the triangle ABC, then

S = (1/2) AB x CD

= (1/2) x 2r sin 2

(r + r cos 2

)

ds/d

= r²[sin2

(-2 sin2

) + (1 + cos2

)(2 cos2

)]

= 2r²[cos²2

- sin²2

+ cos2

] = 2r²(cos4

+ cos2

)

For maximum and mimimum

ds/d

= 0

or, cos4

+ cos2

= 0

or, 2 cos3

cos

= 0

so, Either cos3

= 0 or, cos

=0

If cos

= 0 or 3

=

/2 or

=

/6

(d²s/d

²) = -ve, for

=

/6

ACB = 2

= 2(

/6) =

/3 =

ABC =

BAC

so

ABC is an equilateral triangle.

Reflected ray will be rotated by the same angle only.

Well according to lenz's law the current is induced in the circuit in such a manner that it opposes the change caused due to external agency.

Keeping this in mind we proceed as below.

Direction of current and magnetic field is given by RIGHT HAND THUMB RULE.

So if we curl the fingers of right hand then if the direction of curled fingers show direction of current then that of thumb shows the magnetic field. And obviously the magnetic field is perpendicular to the plane containing current of the coil or conductor.

Now in the given picture x sign shows that magnetic field is perpendicular and going inside the plane of paper

So direction of current is such that it opposes/resists the change causing it.

Here as the circular wire moves in Right direction the emf is induced because of change in magnetic field.
But remember here change in magnetic field is such that the magnetic flux is decreasing. So current is induced to resist the change in magnetic flux, which means that current is in CLOCKWISE DIRECTION so that it opposes the decrease in magnetic flux and thus due to this induced current the magnetic field due to induced current is also perpendicular to the plane of paper.

can u post the figure somehow ?

Maximum value of acceleration of the table is at amplitudes which is given by

a_max= -w²A (at amplitudes and directed towards mean position as shown by -ve sign)

So force on the block = ma_max

Where m is mass of the block.

Condition required so that the block does not slip is

mg = ma_maxwhere

is coefficient of static friction

or

mg = ma_max

or

g = -w²A

or A = -

g / w²HERE -VE SIGN CAN BE DROPPED AS IT SIMPLY SHOWS THE DIRECTION OF FORCE AND THUS ACCELERATION TOWARDS THE MEAN POSITION

SO A =

g / w²OR A = 0.72 g / w²

The induced emf is given by

e = -d(BA)/dt = -B dA/dt

Since angular velocity is w, so we can find rate of change of area i.e dA/dt

So here e = (1/2)B w a²

Here resistance of the wire = R

so induced current = i = e/R = (B w a² / 2R)

Thus thermal power generated in the loop = i²R,

substitute the value of i to obtain the desired result.

Well done Asmita.

Rounding Decimals to the Nearest Tenth

Rounding decimals is very similar to rounding other numbers. If the hundredths and thousandths places of a decimal is forty-nine or less, they are dropped and the tenths place does not change.

For example, rounding 0.843 to the nearest tenth would give 0.8.

If the hundredths and thousandths places are fifty or more, the tenths place is increased by one. The decimal 0.866 rounded to the nearest tenth is 0.9

Rules for Working with Significant Figures:

Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see.
Addition or Subtraction:
The last digit retained is set by the first doubtful digit.
Multiplication or Division:
The answer contains no more significant figures than the least accurately known number.

EXAMPLES:

Example	Number of Significant Figures	Scientific Notation
0.00682	3	6.82 x 10^-3	Leading zeros are not significant.
1.072	4	1.072 (x 10⁰)	Imbedded zeros are always significant.
300	1	3 x 10²	Trailing zeros are significant only if the decimal point is specified.
300.	3	3.00 x 10²
300.0	4	3.000 x 10²

EXAMPLES

Addition		Even though your calculator gives you the answer 8.0372, you must round off to 8.04. Your answer must only contain 1 doubtful number. Note that the doubtful digits are underlined.
Subtraction		Subtraction is interesting when concerned with significant figures. Even though both numbers involved in the subtraction have 5 significant figures, the answer only has 3 significant figures when rounded correctly. Remember, the answer must only have 1 doubtful digit.
Multiplication		The answer must be rounded off to 2 significant figures, since 1.6 only has 2 significant figures.
Division		The answer must be rounded off to 3 significant figures, since 45.2 has only 3 significant figures.

Notes on Rounding

When rounding off numbers to a certain number of significant figures, do so to the nearest value.
- example: Round to 3 significant figures: 2.3467 x 10⁴ (Answer: 2.35 x 10⁴)
- example: Round to 2 significant figures: 1.612 x 10³ (Answer: 1.6 x 10³)
What happens if there is a 5? There is an arbitrary rule:
- If the number before the 5 is odd, round up.
- If the number before the 5 is even, let it be.
  The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out.
- example: Round to 2 significant figures: 2.35 x 10² (Answer: 2.4 x 10²)
- example: Round to 2 significant figures: 2.45 x 10² (Answer: 2.4 x 10²)
- Of course, if we round to 2 significant figures: 2.451 x 10², the answer is definitely 2.5 x 10² since 2.451 x 10² is closer to 2.5 x 10² than 2.4 x 10².

SCIENTIFIC NOTATIONS:

Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10^-9. So, how does this work?

We can think of 5.6 x 10^-9 as the product of two numbers: 5.6 (the digit term) and 10^-9 (the exponential term).

Here are some examples of scientific notation.

10000 = 1 x 10⁴	24327 = 2.4327 x 10⁴
1000 = 1 x 10³	7354 = 7.354 x 10³
100 = 1 x 10²	482 = 4.82 x 10²
10 = 1 x 10¹	89 = 8.9 x 10¹ (not usually done)
1 = 10⁰
1/10 = 0.1 = 1 x 10^-1	0.32 = 3.2 x 10^-1 (not usually done)
1/100 = 0.01 = 1 x 10^-2	0.053 = 5.3 x 10^-2
1/1000 = 0.001 = 1 x 10^-3	0.0078 = 7.8 x 10^-3
1/10000 = 0.0001 = 1 x 10^-4	0.00044 = 4.4 x 10^-4

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,

46600000 = 4.66 x 10⁷ This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example, 0.00053 = 5.3 x 10^-4 This number has 2 significant figures. The zeros are only place holders.

Well you are talking about the phenomenon of TOTAL INTERNAL REFLECTION.

You are also asking about the range of angle. I think here by angle you mean incident angle. You can cange incident angle but to a certain extent only.
the maximum angle of incidence can be = i = 90.
And if you try to increase beyond this then the incident ray is going to change its medium, that is for angles i < 90 if incident ray travels from denser to rarer medium and if you try to increase i beyond 90 this will simply not be possible as incident ray will change the medium without actually travelling and there is violation of the condition required for TOTAL INTERNAL REFLECTION to take place, I suppose you are able to visualise the situation. Thus there is a limit to the range of angle of incidence too.