Post mathematic related questions here

[Thetrueblueviking]

Lol - I committed a blunder so 1 is not the ans.

I could tell you how to go about it but its a tedious task.

First take the x-sinx out of the integral.

Then integrate the function in t by some method - I used substitution.

Then Substitute the limits and you have function in x.

Now solve for final integral/x-sin x (x tends to 0) = 1.

Apply LH a couple of times and use the condition that if limit is to exist and since denom is 0 - then num should as well be 0 and find 'a' OR apply LH till the 0 in the denom vanishes and equate to 1 to get a.

 

[rst]

Lol - I committed a blunder so 1 is not the ans.

I could tell you how to go about it but its a tedious task.

First take the x-sinx out of the integral.

Then integrate the function in t by some method - I used substitution.

Then Substitute the limits and you have function in x.

Now solve for final integral/x-sin x (x tends to 0) = 1.

Apply LH a couple of times and use the condition that if limit is to exist and since denom is 0 - then num should as well be 0 and find 'a' OR apply LH till the 0 in the denom vanishes and equate to 1 to get a.

You put t = ??
i.e you put what value to t (in sustitution method)

 

[Thetrueblueviking]

Take a new variable say P=sqrt(a+t)

So P^2 = a + t
So 2PdP = dt
and also t^2 = (P^2-a)^2

so integral t^2 dt /sqrt (a+t) = [(P^2-a)^2.2P.dP]/P = 2(P^2-a)^2.dP

Note after integrating in P - the limits will be changed.

 

[rst]

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NEW OBJECTIVE QUESTION

*img713.imageshack.us/img713/5994/limitint.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
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you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

We can also solve this question by putting P = a + t
then it will give ∫(x+a,a) [(P-a)² / √P ] dP {here x+a is upper limit and a is lower limit }

∫(x+a,a) [P^(3/2) +a² P^ (-1/2) + 2a √P] dP

Solving we ,get
2/5 (x+a)^ (5/2) +2a² √(x+a) +(4/3) a * (x+a)^(3/2) - [2/5 (a)^ (5/2)] -[2 a² √a] -[(4/3) a * (a)^(3/2) ]

This is easier to do in LH rule

By solving with LH rule ,we get

1/ √a =1

which is possible for a=1

 

[Thetrueblueviking]

We can also solve this question by putting P = a + t

By solving with LH rule ,we get

1/ √a =1

which is possible for a=1

You get 1 even by doing what I said
A bit tedious though

 

[rst]

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NEW OBJECTIVE QUESTION

*img545.imageshack.us/img545/2017/intexp.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

 

[rst]

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NEW OBJECTIVE QUESTION

*img545.imageshack.us/img545/2017/intexp.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

Put x= t²

Then integral will become
2 ∫(1,2) t e^t dt {here 1 is lower limit}

Then by using integration by parts, we get

ans 2e² (option 4)

 

[rst]

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NEW OBJECTIVE QUESTION

*img14.imageshack.us/img14/8696/kvalue.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

 

[rst]

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NEW OBJECTIVE QUESTION

*img14.imageshack.us/img14/8696/kvalue.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

In ∫(0 to 1) (tan⁻¹ x /x )dx

Put x =tanθ

Then integral will become
∫(0 to π/4 ) [ (θ sec² θ) /tanθ ]dθ
∫(0 to π/4 ) [ (2θ /sin2θ ]dθ

Put 2θ =t ,we get
∫(0 to π/2 ) [ (t/2sint )dt

Using in given question ,we get
k=1/2

 

[rst]

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NEW OBJECTIVE QUESTION

*img811.imageshack.us/img811/6514/domaint.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

 

[mastercool8695]

Solve these (all are ISI interview Questions) :


1. Suppose n lines are drawn on a plane. Some of them can be concurrent (pass through same point). How many different regions created in this process? (find the least and the greatest number of regions that can be created).

2. Suppose there are 'n' circles no three of which pass through the same point and all of which intersect every other circle at two points. How many regions are created?

3. Suppose there are 100 points on a plane no three of which are on the same straight line. Can you draw a line on the plane such that 50 points are on one side of it?

4. A game is played between two players. There is a round table and unlimited supply of stones (dimension of the stone is unimportant). In each turn of the game a player can put one stone on the table. Whoever fails to find space one the table looses the game. Find a winning strategy for the first player.

5. On a 20 by 20 board a special knight is moving. In each turn the knight moves 1 step in a direction and 5 steps in a direction perpendicular to it. The knight is allowed to take as many turns as required. Can it come back to any of it's four adjacent squares of the square from which it started moving?

6. We often say that log (1+x) = x - x^2/2 + x^3/3 etc. Is the statement true for x = 500 (or may be larger values?) If not, why?

7. Find a point on the plane of a triangle such that the sum of it's distances from three vertices is minimum.

 

[rst]

6. We often say that log (1+x) = x - x^2/2 + x^3/3 etc. Is the statement true for x = 500 (or may be larger values?) If not, why?

log (1+x) = x - x^2/2 + x^3/3 -......

It is valid for |x|<1
As 500 >1
So it is not true for x=500

7. Find a point on the plane of a triangle such that the sum of it's distances from three vertices is minimum.

Such point is known as fermat point.
For solution here is the link; *en.wikipedia.org/wiki/Fermat_point

 

[mastercool8695]

thanks..
forwarding it to him..
great knowledge for me too.

thanks..

 

[rst]

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NEW OBJECTIVE QUESTION

*img811.imageshack.us/img811/6514/domaint.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

Function is defined for |[ |x|-1]|-5 >0 {As denominator can't be 0}
|[ |x|-1]| >5

=> [ |x|-1] >5 or [ |x|-1] < -5
=> [ |x|]-1>5 or [ |x|]-1 < -5 {As [x-1]=[x]-1 }
=> [ |x|]>6 or [ |x|]< -4
=> |x|≥ 7 or |x|< -4
=> |x|≥7 {here |x|< -4 rejected as modulus function can't be negative}
=> x≥ 7 or x≤ -7

So ans is option (1)

 

[Vignesh B]

Prove 0! = 1.
Quite easy one though, but I found the proof quite interesting.

 

[rst]

Prove 0! = 1.
Quite easy one though, but I found the proof quite interesting.

n! =n (n-1)!

n! /n =(n-1)!

put n=1 ,then we get
1! /1 =0!
1= 0!

 

[Vignesh B]

n! =n (n-1)!

n! /n =(n-1)!

put n=1 ,then we get
1! /1 =0!
1= 0!

Correct!
I had seen this on numberphile. They themselves gave the solution that you have give, but later stated that by this formula, -1! = 0!/0 = 1/0 which negates mathematic's rule. They have give another solution too.

 

[rst]

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NEW OBJECTIVE QUESTION

*img19.imageshack.us/img19/7158/inverseq.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

 

[rst]

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NEW OBJECTIVE QUESTION

*img19.imageshack.us/img19/7158/inverseq.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋

We can rewrite above question as
S= Σ r (1 to ∞) tan⁻¹ [(2r )/(2+r² +r⁴)]

S= lim (n→ ∞) Σ r (1 to n) tan⁻¹ [(2r )/(2+r² +r⁴)]

S= lim (n→ ∞) Σ r (1 to n) tan⁻¹ {[(1+r² +r )-(1+r² -r )] / [1+(1+r² +r )(1+r² -r )]}

S= lim (n→ ∞) Σ r (1 to n) tan⁻¹ (1+r² +r ) - tan⁻¹ (1+r² -r )

S= lim (n→ ∞) tan⁻¹ (1+n² +n ) - tan⁻¹ (1)

S= lim (n→ ∞) tan⁻¹ [(n² +n )/(n² +n+2)]

S=tan⁻¹ (1)

S= π /4

 

[rst]

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NEW OBJECTIVE QUESTION

*img827.imageshack.us/img827/8467/rangeb.png

Plz explain your answer

(Also you can ask your mathematics related questions or queries
we will try to solve them)
----------------------------------------------------------------------------------------------------------------------
you can use following mathematic symbols(just do copy and paste)
£ ω ∴ ∂ Φ γ δ μ σ Є Ø Ω ∩ ≈ ≡ ≈ ≅ ≠ ≤ ≥ • ~ ± ∓ ∤ ◅∈ ∉ ⊆ ⊂ ∪ ⊥ ô ∫ Σ → ∞ Π Δ Ψ Γ ∮ ∇∂ √ °α β γ δ ε ζ η θ ι κ λ ν ξ ο π ρ σ τ υ φ χ ψ
x⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ⁺ ⁻ ⁿ
x₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₊ ₋