sonu
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How to prove that $\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$ without squaring both sides
9 votes

First of all we're not trying to find a solution of the equation here, what you are suggesting is to prove that $\mathrm{lhs} =\sqrt2 $ To do so we square the lhs (first read it fully) and we get $2$...

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Alternate interior angles with two crossing transversals
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3 votes

does it make sense now? just consider the angles more arbitrary than the diagram you made

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Find the locus of perpendicular drawn from focus upon variable tangent to the parabola $(2x-y+1)^2=\frac{8}{\sqrt{5}}(x+2y+3)$
1 votes

There is a general identity that the locus of the foot of perpendiculars to all tangents of the parabola is the tangent at the vertex. Here is a link to the proof. So it is easier to find the line ...

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Plotting points on a halfcircle, given diameter and facing direction.
1 votes

Find the equation of the line $(L_1)$ joining the two given points, and find the point of intersection of the line with the circle. To calculate the intersection point, find the line $L_d$ ...

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Solving a sequence limit with floor
1 votes

Generally, whenever we encounter greatest integer functions, we create a bound for it and try to apply sandwich theorem for the limit. Here, $na-1\le[na]\le na$

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Unsure how to use initial conditions in characteristic value problem
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1 votes

$u_{xx} - 7u_{tx} + 12u_{tt} = 0$ ...$(i)$ $u(t,x) = sinx$ on $t+3x=0$ ...$(ii)$ $u(t,x)=x$ on $t+4x=0$ ...$(iii)$ First of all, we observe from equations $(ii)$ and $(iii)$ that $u(...

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Upper bound on the value of a variable.
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1 votes

It is impossible to find an upper bound because it does not exist. We can conclude this by observing the behaviour of the polynomial as $x→∞$, which again goes to $∞$ which is greater than zero. If ...

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for poisson distribution, show that $P(0<X<2(\lambda +1)) \ge \frac \lambda{\lambda +1}$
0 votes

So it turned out I was doing some calculation mistakes before. As suggested in the question, I got the answer this way: $$P(0<X<2(\lambda +1))=1-P(X=0)+P(X\ge 2\lambda +2)$$ $$\implies P(X=0)+P(...

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Number of garlands which could be formed by using 3 flowers of same type and 12 flowers of other type.
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0 votes

Your approach is correct if the three parts are linear or their order mattered like giving flowers to $3$ people. But here because it is a circle, the total order is not important here. out of the $12$...

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Find the diametre of the well
0 votes

I think the relations (i) and (ii) should be sufficient to solve for $\theta$ and $\phi$ and hence $AC$. (i) would give : $2\cos\theta = 3\cos\phi$ and (ii) : $2\sin\theta + 3\sin\phi = 6\sin\theta \...

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Determine whether the list in k^4 is linearly independent.
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0 votes

They are linearly dependent as you see, we have here three equations in four variables so we can always find solutions where all four are not zero. Hence the system is linearly dependent.

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Solving a square sequence limit
0 votes

$\lim_{n\to\infty}(3^n+7^n)^\frac1n=\lim_{n\to\infty}7(1+(\frac 37)^n)^\frac1n=7\cdot 1^0=7$

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Find the ratio of AM:MO
0 votes

Doing it by the method of vectors would be easiest I think. Consider a triangle $ABC$. Take point $B$ to be origin, $A$ to be pointed by a and $C$ by c, now $O$ is centroid(point of intersection of ...

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Compute $\int_3^6 \frac{\sqrt{3}}{x^2 - 6x +12} \,dx$
0 votes

If you want to use trigonometric substitution, it is bettar to do $$x-3=\sqrt3\tan\theta $$ so that the denominator reduces to a single term which helps make the solution easier which was not possible ...

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Solving $y’=(y^2-c^2)^{\frac{1}{2}}$
0 votes

Hint: substitute $y=c\sec \theta$

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What's the difference between these two approaches to find the equation of a parabola?
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0 votes

Yes, you are correct that they would result in the same equation. In the case of parabolas that have directrices that are parallel to one of the axes, you would think that the second method is better ...

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Reflection of a plane in a plane.
0 votes

The most basic, easy to understand and straightforward method according to me is this:- Consider any point on the plane that has to be reflected, here $P_1:2x+3y+4z−3=0$. Let $(0,0,\frac 34)$ (or any ...

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For what values of $x$ is the series convergent $\sum_{n=1}^∞ \frac{(-1)^n(x+3)^n}{n \cdot 5^n}$
0 votes

For power series,$$\sum_{n=1}^{\infty}a_nX^n$$ $a_n=\frac {(-1)^n}{n.5^n}$ , and $X=-(x+3)$ By ratio test, $$\lim_{n\to∞}|{\frac{a_{n+1}}{a_n}}|=\frac 15$$ Radius of convergence $=5$ So the series ...

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A Biased Coin Flip Problem
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In case of equal biasing in all coins. Let, for the biased coin, the probability of landing heads is $p$ and tails is $1-p$. Then if you understood the formula given in question, The change we need ...

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Exponential definite integral
0 votes

after taking out $e^{-2b^2a^2}$, substitute $\sqrt 2bx=t$. this leaves us with $\int_{-\infty}^{\infty}e^{-t^2}\text dt$ with some factor multiplied. Now we substitute $t = \sqrt { y}$ leaves us with $...

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notation clarifications
0 votes

I believe that both have the same meaning and the first and second both are short forms for what should have been $$ \sum_{i=1}^n(\sum_{j=1}^nX_{kj}) X_{ik}$$ If the first was like $[i=1,j=1],[i=2,...

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How to find length difference between two tangential points of a circle inscribed in a triangle?
0 votes

Let $r$ be the inradius of triangle ABC. The sides mentioned are - $a$ is the side opposite to angle $A$ and so on. Now we want the value of $ CM-AN=AC-AM-AN =AC-2AN = b - 2r$ (because of the square,...

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Find the general solution of: $y^"+3y^{'}+2y=\sin(x)$ (I'm using the Annihilator Method)
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-1 votes

sorry for bad quality but the answer is correct.

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my question is about inverse matrix
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-1 votes

$$A^2=I_n$$ pre multiplying $A^{-1}$ on both sides, gives $$A=A^{-1}$$

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