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
64 views

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ ...
2
votes
4answers
88 views

Probability that team $A$ has more points than team $B$

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the ...
2
votes
4answers
131 views

How do I solve $2^x + x = n$ equation for $x$?

I need to solve the equation $$2^x + x = n$$ for $x$ through a programming-based method. Is this possible? If not, then what would be the most efficient way to approximate it?
2
votes
2answers
230 views

Total number of divisors of factorial of a number

I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where ...
2
votes
1answer
63 views

Geometry in Vectors

Let ${A} = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix}$ and ${B} = \begin{pmatrix} 1 & 0 \\ 0 & -1 ...
2
votes
1answer
53 views

Need help solving linear equations with elimination and substiution method

I need help solving system of equations. I never worked with these equations before, so I'm not sure how this works. If anyone could show me the steps so I could go from there, that would be great. ...
2
votes
1answer
79 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
2
votes
3answers
140 views

How many times between $2$ pm and $4$ pm does the minute hand coincides with second hand.?

How many times between $2$ pm and $4$ pm does the minute hand coincides with second hand.? options $a.)\quad 118 \\ b.)\quad 119\\ c.)\quad 120\\ d.)\quad 121\\$ Number of rounds of full circle ...
2
votes
3answers
64 views

Can we subtract a trigonometric term from a polynomial?

Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations? thanks.
2
votes
2answers
91 views

Why isn't the domain of $y=x^{1/3}$ just $x \in \Bbb R$?

Why isn't the domain of $y=x^{1/3}$ just $x \in \Bbb R$? Could someone explain it to me? I got the answer, and went to check if i was right on a graph plotter, and found out that I got the domain ...
2
votes
2answers
91 views

Show that $4^n + n^4$ is always composite $\forall n > 1$ [duplicate]

I have to show that: $4^n + n^4$ is always composite $\forall n > 1$. I know that composite numbers are integers greater than one but not prime, but I am finding difficult to solve this ...
2
votes
4answers
93 views

$x^2+y^2+9=3(x+y)+xy$ Find all pairs of real $x,y$ that meet this equation

$\frac{(x-y)^2}{(y-3)(3-x)} = 1$ That was my attempt, I can't think of anything else here. I'd prefer a hint
2
votes
0answers
60 views

Zeilbergers algorithm in Maple

I try to prove several hard combinatorial identities. One of them is following \begin{align*} \sum_{s=0}^{\min\{k,n-1\}} \sum_{i=0}^{k-s} (-1)^{i} {2n+k-i-1 \choose k-s-i} {i-n \choose s} {n+i-1 ...
2
votes
1answer
109 views

Prove $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ - (A+B)$. ...
2
votes
3answers
114 views

If $x,y,z \geq 1/2, xyz=1$, showing that $2(1/x+1/y+1/z) \geq 3+x+y+z$

If $x,y,z \geq 1/2, xyz=1$, showing that $2(1/x+1/y+1/z) \geq 3+x+y+z$ I tried Schturm's method for quite some time, and Cauchy Schwarz for numerators because of the given product condition.
2
votes
1answer
56 views

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I'm trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I'm not really sure how to approach finding the solution. Can anyone help me out? I've spent well over an ...
2
votes
5answers
223 views

Is $0.\overline{0}1$ possible? [duplicate]

Is $0.\overline{0}1$ possible in mathematical terms? I know that if you have a repeating decimal, the number is infinite and doesn't end. Thus, the 1 at the end here would stop the repeating decimal, ...
2
votes
1answer
57 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$? [duplicate]

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
2
votes
2answers
73 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
2
votes
6answers
123 views

Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$ [duplicate]

Show $${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$$ A hint is given to consider the expansion $(x-y)^n$ However, when I plug in a number for $n$, I don't get an ...
2
votes
4answers
257 views

Are these proofs logically equivalent?

Here are two proofs, firstly: x = 0.999... 10x = 9.999... = 9 + 0.999... = 9 + x 9x = 9 x = 1 And secondly: ...
2
votes
2answers
147 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
2
votes
3answers
121 views

Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
2
votes
0answers
86 views

Help calculating Combinations

A boy has n objects to paint, ordered in a row and numbered form left to right starting from 1. There are totally c colors, numbered from 0 to c-1. At the beginning all objects are colored in color ...
2
votes
1answer
37 views

How do you prove a function is defined for a a certain set?

Spin-off from here. Context: Highschool textbooks often ask students to find the domain of functions. Let's say $f(x) = x+2$. The domain is $\mathbb{R}$...suppose a student (highschool or o/w) asks ...
2
votes
2answers
124 views

How do you prove the domain of a function?

Suppose we have a function, say, $f(x) = x+2$. Its domain is $\mathbb{R}$. How do you prove this? Or is this something not needed to be proven since it is "defined" $\forall$ x $\in \mathbb{R}$? If ...
2
votes
3answers
271 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
2
votes
2answers
63 views

Prove that $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$. [duplicate]

Prove that the binomial coefficient $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$ if $n$ is odd and maximum at $k=n/2$ ...
2
votes
3answers
180 views

Proof that $\sqrt{x}=-\sqrt{x}$ [duplicate]

$\sqrt{x}=\sqrt{1\cdot x}=\sqrt{(-1)^2\cdot x} = \sqrt{(-1)^2} \cdot \sqrt{x} = (-1) \cdot \sqrt{x}=-\sqrt{x}$ The idea popped into my head while I was evaluating an integral. I have a feeling that I ...
2
votes
3answers
2k views

Finding surface area of part of a plane that lies inside a cylinder???

I have a question:: Let $S$ be the part of plane $x+2y+3z=1$ that lies inside cylinder $x^2 + y^2 = 3$ They want me to find the surface area of S?? This is a way harder question than all my ...
2
votes
2answers
32 views

no. of Digit in $x^y\;,$ where $x,y\in \mathbb{N}$

$(1)$:: Calculation of no. of Digits in $2^{100}$ .$(2)$:: Calculation of no. of Digits in $3^{100}$. If it is given that $\log_{10}(2)=0.3010$ and $\log_{10}(3) = 0.4771$ $\bf{My\; Try::}$ I have ...
2
votes
4answers
117 views

Notation: is it correct to state $3a=a3$?

If $a$ is a real constant, do you regard $3a$ and $a3$ as equal or different?
2
votes
2answers
752 views

Derivation of slope of line formula

The formula for slope of a line as we know: $y_2 - y_1/x_2 - x_1$ or just rise / run What is the derivation for this formula? E.g. Why is it not rise times run for example?
2
votes
2answers
11k views

How to calculate the percentage of increase/decrease with negative numbers?

I feel like an idiot for asking this but i can't get my formula to work with negative numbers assume you want to know the percentage of an increase/decrease between numbers ...
2
votes
5answers
271 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
2
votes
6answers
140 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
2
votes
2answers
295 views

Finding the “triangular root” of a number.

A triangular number is a number that is the sum of the natural numbers up to some $n$. The closed form is $x = \frac{n(n+1)}{2}$. How do I get $n$ on one side? I've been looking at it from every ...
2
votes
2answers
869 views

Exponent rule and square roots?

For some $x$, $\sqrt{x^2} = |x|$ However, for $x= -1$. $\sqrt{(-1)^2} = (-1^2)^{1/2} = (-1)^{2/2} = (-1)^1 = -1$ Isn't this paradoxical?
2
votes
3answers
112 views

Combinatorial proofs of the identity $(a+b)^2 = a^2 +b^2 +2ab$

The question I have is to give a combinatorial proof of the identity $(a+b)^2 = a^2 +b^2 +2ab$. I understand the concept of combinatorial proofs but am having some trouble getting started with this ...
2
votes
1answer
48 views

Simplify the following indices

$3^{x+4} * 5^{x+7} * 15^{2x-1}$ I tried it in this way: $3^{x+4}*5^{x+1}*(3*5)^{2x-1}$ Then: $3^{x+4}*5^{x+1}*3^{2x-1}*5^{2x-1}$ And what about next?
2
votes
4answers
349 views

Simplify the surd expression.

Simplify the surd. $(2\sqrt 3 + 3\sqrt 2)^2$ I know I should us this formula: $(a^2+2ab+b^2)$ But this gets complicated later. Please explain. :(
2
votes
2answers
76 views

Determining the domain of a function

What is the domain of: $$\left(\frac{5x+4}{x^2+9x+8}\right)^{1/3}$$ I got $(-\infty, -8) \cup (-8,-1) \cup (-1, \infty).$ But according to Wolfram Alpha it is $(-8, -1) \cap [-4/5, \infty)$. Could ...
2
votes
3answers
234 views

Quadratic formula - math error

I'm attempting a past paper and I have been asked to compute the derivative for $(x^2-2x+2)$ and from this I calculated $2x-2$. Once I completed this, I was then asked to find and classify the ...
2
votes
2answers
32 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
2
votes
1answer
78 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
2
votes
1answer
2k views

Determine the cube roots of -8 in polar form

Exam time tomorrow and I am not entirely sure if I am doing this right. I first write -8 as a complex number $z^3 = -8 = -8 \times 0i$ Calculate the modulus of z $|z| = \sqrt{-8^2} = 8$ Get the arg ...
2
votes
2answers
74 views

How to prove: $-\frac{1}{\sec2x}=\frac{\cos^3x-\sin^3x}{\cos x +\sin x}+\frac{\cos2x}{(\cos x +\sin x)^2}$

How do you do it? I'm really stuck on this proof. Can someone please explain? Thanks
2
votes
3answers
114 views

Prove that for real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \le 1$ or $x \ge 4$.

Its another homework question that I'm having trouble understanding. The full question is write a detailed structured proof that uses a proof by cases to prove that for real numbers $x$, if $x^2 - 5x ...
2
votes
1answer
115 views

Help with understanding a proof that $f$ is bounded on $[a,b]$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ ...
2
votes
3answers
152 views

Inequalities - Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?