# All Questions

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### Which method do i use ? Interpolation

I have table of 5 values i.e abscissa and ordinates are given. I have been asked to find derivative at particular point and also second derivative at that value. That value is between my given equi ...
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### Eigenvalues of matrix with all $1$'s.

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
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### Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection.

Two straight lines one being a tangent to $y^2=4ax$ and the other to $x^2=4by$ are at right angles.Find the locus of their point of intersection. I tried but could not reach final answer.The tangent ...
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### Product of measures on rectangular cylinders

Let $(\Omega_i,\mathcal A_i,\mu_i)$ be a $\sigma$-finite measure space. Why is \begin{split} ...
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### Looking for set of combinatorics problems

I'm preparing to Mathematics for Computer Science exam. What I learned from past edition of exams is fact of very often occurence of old problems. I mean more or less known problems, but possible to ...
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### Drunk Passenger Probability question

I don't know how to solve this question. A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. For convenience, lets say that ...
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### constant coefficient difference equations LTI, why do I need the initial conditions?

Consider the following difference equation $$y_{n}=-\sum_{k=1}^{q}a_{k}y_{n-k}+\sum_{m=0}^{p}b_{m}x_{n-m}$$ I know that this is supposed to be LTI iff $y_{-q}=y_{-q+1}=...=y_{-1}=0$. How does one go ...
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### Need help proving this geometry problem.

My friend asked me one question yesterday.It is as follows. Let there be two triangles ABD and ACD.D is a point on base BC such that BD=CD(given).Also,clearly side AD is common.Now we know median ...
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### How to prove that $\{0\} \cup \{1, \frac{1}{2}, \frac{1}{3}, …\}$ is not isolated

(As a subset of $\mathbb{R}$.) I am having trouble proving that $0$ is not an isolated point. If there exists an open ball with radius $\epsilon$ about $0$, I have to find a point of the form ...
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### Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
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### Proof for the fact that the method of characteristic equations is a valid method to solve recurrence relations

Could someone kindly point me to a proof of the fact that the method is characteristic equations is a valid way of solving recurrence relations? It seems fairly arbitrary to me. I would be grateful ...
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### null empty set has 2 subsets?

The question in the book was: How many subsets does $\{\emptyset\}$ have? a) 0, b) 1, c) 2, d) 3. The answer was c. How can an empty set have 2 subsets?
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### The dice is rolled 10 times and the results are added with given conditions.

Q: A dice has one of the first 6 prime number on each its six sides ,with no two sides having the same number .the dice is rolled 10 times and the results added.the addition is most likely to be ...
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### How would one work this out? (Fractions)

One fifth of all value lamps are already defected at the time of purchase. How many do you have to buy to ensure that you have 16 functioning lamps.? Anyone have any advice on how to layout this ...
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### Project point on plane - Unique identfier?

I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$. If I project a point $\vec{Q}$ (which does not ...
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### Integral $\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$

I need a hint how to start solving this integral: $$\int{ \frac{1}{\sqrt {1 - e^{2x}} } dx}$$
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### Hooke's Law - Modelling a 'Bungee Jump'

A question in my textbook asked me to find a general equation for depth fallen by an object of mass $75kg$ thrown from a bridge whilst tied to an elastic rope. Below the bridge there is a stream of ...
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### Monoid filtration

I lately been introduced to monoid filtrations and I have a couple of questions. Let $(\mathfrak{M},\star,1_\mathfrak{M})$ be a monoid with total order, $(A,+)$ the additive subgroup and ...
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### If $N$ is nilpotent then there exists $A$ such that $A^2=I+N$

Suppose $N\in M_{3\times 3}^{\mathbb{C}}$ is a nilpotent matrix. Prove that there exists $A\in M_{3\times 3}^{\mathbb{C}}$ such that $A^2=I+N$. Hint: find $A$ in the form $A=P(N)$ where $P$ is a ...
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### Endomorphisms of $\mathbb C^{\times}$

What are all continuous multiplicative endomorphisms of $\mathbb C$? As $\mathbb C^{\times} \simeq \mathbb R_{>0}^{\times}\times S^1$ this question can be reduced to the description of continuous ...
I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...