Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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2
votes
2answers
63 views

Why do we need to rationalize fractions? [duplicate]

Teachers often take off points from students who write 1/sqrt(2) instead of sqrt(2)/2. Why do we need to write it as sqrt(2) / 2 ? Where did that convention come from? Do we need to even do it? Why do ...
2
votes
2answers
56 views

Is $ (-1)^n(x-a)^n = (a-x)^n?$ If not, why?

I came across this during an attempt at a Taylor series expansion (which I'm not very good at yet), and assumed this would be true because $(ab)^n = a^nb^n$. Plugged it into Wolfram Alpha, though, and ...
0
votes
4answers
53 views

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$. I do know how to solve this problem using trigonometry, however I need to solve it by using vectors. ...
3
votes
2answers
71 views

Problem in deducing the number of onto functions

Let $A, B$ have $m, n$ elements ($m > n$). Therefore, the number of onto functions from $A$ to $B$ is: $$\sum_{k = 0}^n (-1)^k \binom{n}{k} (n - k)^m.$$ How can one use the IE (Inclusion/Exclusion) ...
0
votes
0answers
23 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
0
votes
3answers
54 views

Looking for value of equation

I have been trying to solve this MONBUKAGAKUSHO past test papers, and I am completely stuck. I have no single idea how to solve it, and I´ve tried many different things, but without any results. Here ...
0
votes
1answer
14 views

What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$?

What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$? Obviously, the polynomial $f(x)=x-2-i$ satisfies the constraint but does not have ...
1
vote
3answers
28 views

Divisibility of integers by integers

We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. Factorizing the second bracket ...
-1
votes
0answers
16 views

distance between normal and axis? [on hold]

The job is finding the distance between the normal to the plane $x-y+4z-10=0$ and the $z$ axis...I cannot really imagine how can one do this?
-2
votes
0answers
17 views

angle between normal and axis? [on hold]

I was tasked with finding the angle between the normal to the plane $x-y+4z-10=0$ and the $z$ axis...I cannot really imagine how can one do this?
0
votes
3answers
61 views

What is $n+1$ factorial or $(n+1)!$? [on hold]

I have to prove by induction but first I want to know what $(n+1)!$ is? I know that $n!=n \cdot (n-1) \cdot (n-2)...$
1
vote
0answers
13 views

Find the equation of intersection of a torus and a circle on a plane without using iterative methods.

I have the equation of a circle on the plane (where $p_0$ is the centre, $\theta$ is the angle of the circle, and $w$ and $v$ are pair of orthogonal vectors from $p_0$ to the circle (having equal ...
5
votes
2answers
201 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
-5
votes
0answers
17 views
1
vote
3answers
29 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
1
vote
1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
4
votes
4answers
77 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
0
votes
1answer
22 views

Find the largest segment

I have seven lines with different measures. The length of each line it's a positive integer and the shortest length is equal to 1 cm. It is known that's impossible to choose three of them that makes a ...
-3
votes
1answer
60 views

Can someone help me why this equation equals zero?

I played around with some numbers and stuff and made this weird equation: $$\huge x^{- \frac{n}{n^{-x}}}$$ So the thing is, with every number I tried typing this into a calculator, I got 0. Can ...
-1
votes
1answer
28 views

Multiplying logarithms of different bases [on hold]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
0
votes
2answers
24 views

How to prove that eventually $(x^p/e^{x^q}) < 1/(x^2) $ for $p,q>0$

How to prove that eventually $x^p/\exp(x^q) < 1/(x^2) $ for $p,q>0$. I tried showing that $x^{p+2} > \exp(x^q)$ by using the Taylor expansion of e but this didn't really work.
3
votes
2answers
189 views

Simple 2 equations and 2 unknowns

I am reading the second partial derivative test example, but I am suck on the following step: $$f(x,y) = -x^3 + 4xy - 2y^2 + 1$$ And we have the partial derivatives as follow... $$f_x(x,y) = -3x^2 ...
4
votes
2answers
34 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
4
votes
1answer
188 views

Partial fractions - different results when done in steps than not

We have: $\frac 1 {(1-x)(1+x)(1-2x)}$ If I do the partial fractions straight: $\frac 1 {(1-x)(1+x)(1-2x)}= \frac a {1-x} + \frac b {1+x} + \frac c {1-2x}$ I get: $a=-\frac 12, b = \frac 1 6, c=\frac ...
3
votes
1answer
30 views

Is there any solution to this quadratic Diophantine 3 variables equation?

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation : $$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$ Here $a, b, c$ are fixed positive integers.
0
votes
1answer
27 views

Precalculus: Velocity addition

A boat is rowed 6.4 km up a river and back again and this takes in total 2 hours. The stream velocity is 2.4 km/h. What velocity would the boat have been moving in if the water was standstill? I ...
4
votes
2answers
42 views

If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$

If $a,b,c\in \mathbb{R}$ and If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$ is, Where $-1 \leq x\leq 1$ $\bf{My\; Try::}$ Put $x=1$ in ...
12
votes
7answers
1k views

How do we prove this logarithm?

Given: $$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$ We have to show that : $$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$ I made three equations using cross multiplication : ...
1
vote
1answer
16 views

Finding orthonormal basis using orthogonal basis

I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for ...
4
votes
3answers
52 views

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x)$

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x).$ My Solution:: Let $$\displaystyle y = \sin^4 x+\cos^2 x \leq \sin^2 x+\cos^2 x=1$$ ...
0
votes
0answers
10 views

Generalization of minimisation problem

First I would like indtroduce my problem ! There is an easy way to solve this one : Find the value of $$ \inf_{(a,b)\in \mathbb{R}^2} \int_0^1 (t^2-at-b)^2 dt $$ and precise for which values $a$ ...
-4
votes
0answers
47 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
0
votes
3answers
30 views

When is $\theta$ obtuse or acute in sin, cosine, tan when they are positive, negative or both?

My textbook gives a non intuitive answer and tells us to memorize when the ratios are positive or negative or both based on some arbitrary rule that I don't understand. I know how to do both of ...
0
votes
1answer
24 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
2
votes
1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
-3
votes
1answer
52 views

What grade does Bob need on his final to pass his math class?

Bob's teacher has a syllabus where the grading breakdown is as follows: Homework: 10% Tests (2): 60% Final Exam: 30% Bob receives all 10% from the homework Bob receives 78/100 on Test 1 (78%) ...
0
votes
2answers
28 views

Vector Magnitude problem

I do not understand how to set up the following problem: "Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force." An actually picture would ...
3
votes
1answer
23 views

How to find the numbers that the product of it's digits is equal to ten times the sum of them.

I formulated this so that the number be in range $[111,999]$, it was narrowed so that $a,b,c$ is not equal to zero. $$a\cdot b\cdot c = (a+b+c)\cdot 10$$ With this we can see that $\frac{a\cdot ...
-2
votes
1answer
43 views

An algebra word problem in one variable on percentages - assistance required [on hold]

I'm new here. I'm not sure how this works, but I was wondering if someone could help me with this problem. I'm having difficulty solving it, and I was hoping someone could help me. Thank you! Assign ...
-1
votes
2answers
72 views

Word problem on the perimeter and sides of a triangle [on hold]

Having difficulty answering this question. I'm not sure how to do it, but if anyone could show me the steps; so I could answer it myself, that would be great. Thanks Assign x, make an equation and ...
16
votes
3answers
130 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
-2
votes
0answers
13 views

Central orbits ellipse change of centre of force [on hold]

a body is decribing an ellipse of eccenticity e under the action of central force directed towards the focus, and when at the nearer apse, the centre of force is transferred to the other focus, Find ...
2
votes
2answers
49 views

Advanced (for me) algebra and mean

I have been struggling over questions like these which my maths teacher has been throwing into our weekly papers for about a week now, and it has stumped all of us. Can you help? Question: There are n ...
-6
votes
1answer
60 views

A trick and interesting math SUM [duplicate]

If $x=(a/b)^{2ab/(a^2-b^2)}$ I want to prove that $$( (ab)/(a^2+b^2) )(x^{a/b} + x ^ {b/a})=(a/b)^{(a^2+b^2)/(a^2 - b^2)}.$$
3
votes
2answers
33 views

Stuck in a problem in permutation and combination.

I am solving problems in permutation & combination and stuck in this problem. Two players $P_1$ and $P_2$ play a series of $2n$ games. Each game can result in either a win or a loss for $P_1$. ...
-4
votes
0answers
62 views

Prove this enticing equation [on hold]

If $$x=\left(\frac{a}{b}\right)^{\frac{2ab}{a^2-b^2}}$$ I want to prove that $$\left(\frac{ab}{a^2+b^2}\right)\left(x^{\frac{a}{b}} + x ...
1
vote
1answer
26 views

Total no. of Transitive Relation on $A = \{a,b,c\}$

Calculation of total no. of Transitive Relation on $\displaystyle A = \left\{a,b,c\right\}.$ $\bf{My\; Try::}$ First We will calculate Total no. of Relation on $A$, Which is $\displaystyle = 2^{3^2} ...
7
votes
2answers
382 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
1
vote
3answers
50 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
1
vote
3answers
51 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...