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
3answers
23 views

Find the equation of the circle.

Find the equation of the circle whose radius is $5$ which touches externally the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
0
votes
1answer
12 views

equation solving approach for the given equation

$y^4-10y^3+5y^2+100y+100=0$ how to solve this equation ? what will be the approach to solve this question. Breaking $100$ into $84$ and $16$ and then solving could be one .
1
vote
2answers
52 views

This expression is always a perfect square [on hold]

How to show that for $x,y\in \Bbb R$, the expression $xy+\left(\frac{x-y}{2} \right)^2$ is always a perfect square? For example $x=7, y=3$, $7\times 3+\left(\frac{7-3}{2} \right)^2=25=5^2$
2
votes
2answers
24 views

Parallelogram ABCD

There's a parallelogram $ABCD$. I'm given point $A(3,12)$ and point $B(-1,5)$. Given the equations of the lines $BC$ and $AC$ are $y=8x+13$ and $y=3x+3$ respectively. How to find the coordinates of ...
1
vote
1answer
54 views

How to find x,y,z such that $\frac{1800}{x}=a$, $\frac{1800}{y}=b$, $\frac{1800}{z}=c$, $\frac{1800}{a+b+c}=d$

I'm really fascinated by how questions and problems are designed in mathematics. So, I was designing a simple word problem, and in the course I fell into this situation: a,b,c,d are natural numbers. ...
1
vote
0answers
29 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
1
vote
2answers
43 views

Intuition: inverse function

Disclaimer: I'm a beginner with inverse functions. Can anyone explain what I'm doing wrong here? I'd like to avoid using "y" -- that is, I want to show everything in terms of x and f(x). Thanks!
0
votes
3answers
46 views

If a given # is $70$% of $X$. How do you determine what $X$ is? [duplicate]

Given I have the number $50,000$ which is $70$% of $X$. How do I calculate what $X$ is without guessing. Thanks
-1
votes
1answer
12 views

How to find the bearing and velocity of a boat on a flowing river

Point A is on the west bank of a river and point C is directly across from it on the east bank. The river is 648.6 meters wide and flows south at 2.45 km/hr. A boat wants to cross the river from point ...
1
vote
1answer
35 views

How to solve a equation task

I have found a equation task that is: y=0,5x+2,5 y=3x-1 The answer and its process is: ...
-2
votes
1answer
48 views

Algebraic Problem regarding Cubes. [on hold]

If $a^3 + b^3 + c^3 = 3abc$ where $a \ne b \ne c$, find the value of: $(a + b + c)$
-9
votes
1answer
39 views

Algebra 2 help! [on hold]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
1
vote
2answers
32 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
4
votes
4answers
107 views

Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer?

The problem is the following: Prove that this number $$x = \left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$$ is an integer. Show which integer it is. I thought that it ...
0
votes
0answers
37 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
0
votes
1answer
20 views

How to prove that $(a-b) \mod N = a \mod N + ((-b) \mod N)$?

I've gone through the similar post Modulo of a negative number . But that post is not about proof and I'm asking for the proof in general. This question is another follow up question of my previous ...
0
votes
1answer
26 views

Question about the polynomial remainder theorem

Given that $f(x) = x^3 - x^2 - ax - b$ has a factor $x - 3$ but leaves a remainder of $13x - 11$ when divided by $x + 4$, find $a$ and $b$. I get that in order for $x-3$ to be a factor then according ...
1
vote
1answer
48 views

AMT - Three whole numbers add up to 149 and multiply to give 987. What is the largest of the three number

So about this question I'm not too sure... Can't find out what I should start off with. If anyone can help me I'll be very greatly appreciated. The question is: Three whole numbers add up to 149 ...
-3
votes
1answer
53 views

Help me understand probability..I don't get it!! Exam tomorrow

Postcodes can be made from 4 digits --> 1234 How many different postcodes beginning with 2 are possible? How do I truly understand probability questions? Nothing beyond the level of ordered / ...
1
vote
3answers
61 views

how can i prove this trigonometry equation

I need help on proving the following: $$\frac{\cos {7x} - \cos {x} + \sin {3x}}{ \sin {7x} + \sin {x} - \cos {3x} }= -\tan {3x}$$ So far I've only gotten to this step: $$\frac{-2 \sin {4x} \sin {3x} ...
3
votes
4answers
74 views

Solve for $x$ - Logarithm Equation $\ln x+\ln(x+1)=\ln 2$

My attempt: $\ln x(x+1)=\ln 2$ $e^{\ln x(x+1)}=e^{\ln 2}$ $x(x+1)=2$ $x^2+x-2=0$ $(x-1)(x+2)=0$ therefore $x=1, -2$
1
vote
1answer
32 views

Find all ordered triplets $(p,q,r)$

Let $A,G^2,H$ be the roots of the cubic equation $x^3+px^2+qx+r=0$ which are in G.P., where $p,q$ are integers and $A,G,H$ are respectively AM,GM,HM of two positive numbers. If $p,q \in (-100,100)$, ...
0
votes
0answers
37 views

Why in order to have the greatest term in the expansion of$ (1 + x)^n$, $x$ can't be greater than unity?

I was reading Binomial theorem of any index of Higher Algebra by Hall & Knight; there a section was attributed to find the greatest term in the expansion of $(1+x)^n$ for any rational value of ...
0
votes
2answers
55 views

For which $x, y\in\mathbb{R ^+}$ do we have $|xy-\frac{1}{xy}|\le|x-\frac{1}{x}|+|y-\frac{1}{y}|$?

I need to find all $x, y\in\mathbb{R^+}$ such that the following inequality holds. $$\Big| xy-\dfrac{1}{xy}\Big|\le\Big|x-\dfrac{1}{x}\Big|+\Big|y-\dfrac{1}{y}\Big|$$ If I substitute $x=2$ and $y=3$ ...
2
votes
6answers
55 views

Why does $|x_1| = |x_2| \implies x_1 = \pm x_2$

I was doing a 'prove this is not surjective' practice problem and the step leading from my hypothesis, as listed, to the conclusion was not defined. I don't recall being exposed to a situation where ...
0
votes
0answers
41 views

Monotonicity of a discrete fucntion

I am trying to show that the following function is increasing in $m$: $$ f(m)=\sum_{i=0}^{m-1}(1-x)x^{i}\frac{1-x_{1}^{m-i}}{1-x^{m+1}}, $$ where $m$ is a positive integer, and $x$ and $x_{1}$ are ...
-2
votes
1answer
38 views

How would you divide a polynomial by another polynomial whose power is greater than its nominator? [on hold]

I have a polynomial which is: $$\frac{(x^3-4x)}{(4x^2-4x+1)} = -10$$ Is there a way to do this? I have thought about doing long division which was not helpful...
3
votes
3answers
157 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$. If $J=\dfrac{pI}{q}$, then find the value of $p+q$ where $p$ and $q$ are ...
0
votes
2answers
31 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
-2
votes
3answers
114 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
1
vote
2answers
39 views

How do I interpret this question: Do I multiply, divide, subtract first?

Which of the following expresses $6p+2py-4p$ in its simplest form? (A) $2p+2py$ (B) $4py$ (C) $4p^3y$ (D) $10p+2py$ Im not really sure how to go about it...
6
votes
5answers
109 views

Calculate the limit $\lim_{x\to 0} \left(\dfrac 1{x^2}-\cot^2x\right)$ [on hold]

The answer of the given limit is $2/3$, but I cannot reach it. I have tried to use the L'Hospital rule, but I couldn't drive it to the end. Please give a detailed solution! $$\lim_{x\to 0} ...
0
votes
4answers
43 views

Lawn mowing problem solving

Kate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them? I know the answer ...
-3
votes
1answer
32 views

Express as a single logarithm [on hold]

Hi I need to express the following and have no clue how to do so. $$\ln(x+3)-3\ln(x-7)-\ln(x+8)$$ Can someone please help
-4
votes
2answers
59 views

How to simplify $(x+1) / (x^3-x)$ [on hold]

For all $x$ in the domain of the function $\frac{x+1}{x^3-x}$, this function is equivalent to which of the following? (A) $\dfrac{1}{x^2}-\dfrac{1}{x^3}$ (B) ...
1
vote
6answers
169 views

Sum of cosines of complementary/suplementary angles

Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero? Could you prove it by some identity?
14
votes
7answers
1k views

Why do remainders show cyclic pattern?

Let us find the remainders of $\dfrac{6^n}{7}$, Remainder of $6^0/7 = 1$ Remainder of $6/7 = 6$ Remainder of $36/7 = 1$ Remainder of $216/7 = 6$ Remainder of $1296/7 = 1$ This pattern of ...
9
votes
7answers
134 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
-1
votes
0answers
35 views

Find the solution to the following LPP by solving its dual. [on hold]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
0
votes
1answer
25 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.
0
votes
5answers
137 views

Why is reminder of $8^{30} / 7$ same as that of $1^{30} / 7$

I am not able to figure out why the reminder of $8^{30} / 7$ is same as that of $1^{30} / 7$. I know Euclid division $a=bq+r$ but I don't know modular arithmetic, so please explain without referring ...
1
vote
1answer
28 views

How to solve problems on alligation and mixture when three types are given?

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...
4
votes
7answers
293 views

Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
2
votes
2answers
58 views

Proper way to solve function notations?

I'm just starting to use function notation and I'm wondering if I'm solving correctly. If $f(x) = 4x - 11$, determine a. $f (1/4)$ $f(x) = 4x - 11$ $f(1/4) = 4 (1/4) - 11$ $f(1/4) = 1 - 11 $ ...
0
votes
4answers
39 views

How to find perpendicular vectors in 3D

Find all values of a such that the vector $q = \langle 2, a, –2\rangle$ is perpendicular to the vector $p = \langle –3, a, 5 \rangle$.
4
votes
2answers
63 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
1
vote
1answer
20 views

Property of an almost additive sequence of functions

We say that a sequene of functions $\Phi=(\phi_n)_n$ is almost additive if there exists a constant $C > 0$ such that for every $n,m \in \mathbb{N}$ and $x\in \Lambda$ we have \begin{equation*} -C + ...
0
votes
2answers
50 views

A question about Idempotent functions [on hold]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...
0
votes
2answers
14 views

Finding function inverse

Hi, This question is actually from KhanAcademy - Algebra 2. I managed to solve it using the rote method by swapping the x and y. But I would like to find out the reason for swapping x and y for ...
2
votes
2answers
78 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...