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Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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Manipulating Algebraic Equations

I'm sorry if this question (or possibly multiple depending on how long this I intend for this to be) is a little too elementary, but I've been seriously struggling with this for the past week. I ...
Lucas's user avatar
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25 views

See body paragraph. Couldn’t make it fit into question character limit.

Suppose there are 5 people sick on the first week of an epidemic. By the end of the next week, they each make 4 people sick. How many people will be affected by the tenth week of the epidemic?
Lucas Town 's user avatar
1 vote
1 answer
30 views

Proof that two matrices are row-equivalent iff they have the same nullspace

The matrices are both of size m x n over some field F, obviously. The first direction of this proposition is clear enough, however the opposite direction (same nullspace -> row-equivalence) is ...
Blabla's user avatar
  • 179
-2 votes
0 answers
16 views

Confusion in the application of the concept of alligation in mixture problems [closed]

Ankit buys 8 pens and 4 pencils for Rs. 2400. He sells pencils at a profit of 20 percent and pens at the loss of 10 percent. If his overall profit is Rs. 240, then what is the cost price of one pen ...
Try's user avatar
  • 1
1 vote
2 answers
55 views

Find $x$ in this concyclic quadrilateral

$ABCD$ is a concyclic quadrilateral, with $\angle A = 60^\circ$, and $ AB = 10, BC = x , CD = x+2 , DA = x+4 $. Find $x$. My attempt: Using the vector method, we can express the horizontal and ...
c'est pas normale's user avatar
0 votes
1 answer
58 views

Formula for Root Series

Given the series: $S_n=\sqrt{c}+\sqrt{c-1}+\sqrt{c-2}+\sqrt{c-3}+...+\sqrt{c-n}$ in which $c$ can be any value but independent from n. Is there a formula for these kinds of series? Excluding the ...
Anthony's user avatar
0 votes
1 answer
35 views

Measuring circle offset for each angle

I try to calculate offset change at outer rim of a circle for each angle with respect to "0" but can not figure out. its is easy on the paper with geometry but very difficult to formulate it....
ebbac44's user avatar
  • 11
-2 votes
0 answers
36 views

Proving a tough inequality [closed]

enter image description here Take no heed of parts i) and ii) and instead directly to part iii) how to prove that? I could start by breaking up the squares algebraically, but other than that, I haven'...
Enshu Liu Dongfang's user avatar
-5 votes
0 answers
43 views

Trouble in understanding the series $\sum_{n = 1}^\infty \left[\frac{4}{n(n + 1)} - \frac{1}{3^n} \right]$ [closed]

Find the sum of the series $$\sum_{n = 1}^\infty \left[\dfrac{4}{n(n + 1)} - \dfrac{1}{3^n} \right]. $$ I can't seem to end on a finite sum for this series when you split it into twos because the $-1/...
Kermitheweeb's user avatar
3 votes
3 answers
99 views

Find $x$ in this quadrilateral

A quadrialteral $ABCD$ has $AB = 10$, $\angle A = 50^\circ, \angle B = 120^\circ$, $ BC = x , CD = x + 2 , AD = x + 4 $. Find $x$. My attempt: Applying the law of cosines to $\triangle DAC$ and $\...
c'est pas normale's user avatar
3 votes
4 answers
112 views

Find the exact value of $\DeclareMathOperator{\cosec}{cosec} \cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$

Find the exact value of $\cosec(10^\circ) + \cosec(50^\circ) - \cosec(70^\circ)$. The equation can be written as $$ \cosec(x) + \cosec(60^\circ-x) - \cosec(60^\circ+x), $$ where $x = 10^\circ$. I ...
Hitesh's user avatar
  • 71
0 votes
1 answer
43 views

Relationship between roots and coefficients of a cubic

Let $f(x) =x^3-px+q,p>0,q>0$ and all the zeroes of $f(x) $ are real. Prove that if $\alpha$ be the root with least absolute value then $|\alpha|$ lies in the interval $(q/p, 3q/2p) $. I have ...
YBR's user avatar
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0 votes
1 answer
84 views

Floor function with factorials

$x$ is an integer satisfying: $$\left[\frac x{1!}\right]+\left[\frac x{2!}\right]+\left[\frac x{3!}\right] + \cdots +\left[\frac x{10!}\right]=1001.$$ Find largest prime divisor of $x$, where $[~]$ ...
Physics enthusiast's user avatar
-3 votes
0 answers
67 views

Solving GIF questions [closed]

The value of [1+1/√3+1/√5+.......+1/√121] is equal to? A)10 B)11 C)9 D)12 Where [.] denotes greatest integer function How to proceed with these questions? I am thinking with approximation something ...
Loci's user avatar
  • 7
1 vote
1 answer
22 views

Prove the gambler can expect to play forever in the gambler’s ruin scenario

In mcs.pdf, Problem 21.2 says: In a gambler’s ruin scenario, the gambler makes independent $1$ bets, where the probability of winning a bet is p and of losing is $q ::= 1-p$. The gambler keeps ...
An5Drama's user avatar
  • 390
-1 votes
0 answers
26 views

Trigonometric formulas derivation? [closed]

We know that how other formulas like quadratic formula, distance formula, section formula are derived but what about trigonometry formulas? We are just told to memorize their values. Pardon me for ...
Payal Payal's user avatar
0 votes
1 answer
76 views

Is it possible for a quadratic function to not have $y$-intercept?

I know quadratic functions may have up to two $x$-intercepts, but can they have no $y$-intercepts? Edit: I mean quadratic functions which output parabolas (i.e. $f(x)=ax^2+bx+c$ where $a\neq0$). So ...
Vee's user avatar
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-1 votes
0 answers
89 views

Decide whether the polynomial $ x^{154} - 2x - 3 $ is divisible without a remainder by the polynomial [duplicate]

I have a question regarding polynomial division. I had this problem at one of my test. Decide whether the polynomial $ x^{154} - 2x - 3 $ is divisible without a remainder by the polynomial: a) $ x + 1 ...
peterparker321's user avatar
2 votes
2 answers
163 views

Find out $\int_{0}^{1} \frac{\ln(-x^2+x+1)}{x(1-x)} \rm{d}x$ without Feynman .

Find out $$\int_{0}^{1} \frac{\ln(-x^2+x+1)}{x(1-x)} \rm{d}x$$. They say when it comes to logarithms in integrals first thing to check if $[x → \frac{1}{x}]$ works or not . My try : $$I = \int_{0}^{1} ...
Ash_Blanc's user avatar
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0 votes
3 answers
79 views

Difficult algebra problem related to finding a specific value in terms of "$c$" or "$b$"

If $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$ and $ax^2+bx+c=0$ then the value of $b^2x^4+b^2c^2x^2+2b^2cx^3$ is A) $c^4$ $\quad$B) $-c$ $\quad$C) $-c^2$ $\quad$D) $b^4$ Can anyone please help me with this ...
MIND FORGE NEXUS's user avatar
0 votes
1 answer
59 views

Suppose that $-1\leq ax^2 + bx + c\leq 1$ for $-1\leq x\leq 1$, where $a, b$ and $c$ are real numbers, prove that $-4\leq 2ax + b\leq 4$. [duplicate]

I see that for $x = 0, |c|\leq 1$, for $x = -1, |a - b + c|\leq 1$ and for $x = 1, |a + b + c|\leq 1$. Thus, $|2a + 2c|\leq 2\Rightarrow |2a|< 4$ and $|b|\leq 1$. This, gives me $|2a + b|\leq 5$, ...
Shiv's user avatar
  • 231
1 vote
1 answer
50 views

In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$.

I was trying to solve this problem: In an isosceles triangle with base $a$ and congruent side $b$ the vertex angle is equal to $20°$. Prove that $a^3 + b^3 = 3ab^2$. After a long time of thinking ...
pie's user avatar
  • 5,475
-6 votes
2 answers
77 views

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ smartly than rest?

Integrate $ \int_{} \frac{\sin(x-a)}{\sin(x-b)} {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and lets see the no. of other ways to solve it. My try : $$I = \int_{}...
Ash_Blanc's user avatar
  • 837
1 vote
1 answer
57 views

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ quickly .

Integrate $ \int_{} 1 + \tan x\tan(x+\theta) {\rm{d}}x$ I am expecting better new elegant approaches for this simple integral and let us see how shorter it can be. My try : $$I = \int_{} 1 + \tan x\...
Ash_Blanc's user avatar
  • 837
-2 votes
0 answers
14 views

Is there any solution to a system of inequalities $Ax>b$, where $x$ is a column vector and $A$ is a symmetric positive matrix? [closed]

So I need to solve the inequality $Ax>b$ where $x$ is a column and $A$ a symmetric positive matrix (in that case).
Amine Lahmouz's user avatar
0 votes
1 answer
55 views

Expanding $\log_2(8x^3y^4)$ into addition form [closed]

So, I have this logarithmic expression $$\log_2(8x^3y^4)$$ I know how to evaluate this expression, but how do you expand one like this, with three logs, into addition form? I've been able to expand ...
Kayla's user avatar
  • 9
0 votes
3 answers
84 views

When solving $\sin{x}+\cos{x}=\sin{2x}+\cos{2x}$, where does the extra solution come from?

Background This question is from the 1907 Victorian Universities and Schools Examination Board Trigonometry (Senior) Examination. My solution is given below, but apparently there is an extra solution ...
Red Five's user avatar
  • 2,673
1 vote
1 answer
43 views

Omitting a parameter from an inequality

Could someone help in omitting the parameter $n$ (represents an odd integer) from this inequality? I need to find an equivalent inequality in which $n$ is absent. $$\frac{c_1}{c_2-n \pi }\neq \frac{...
mattTheMathLearner's user avatar
0 votes
0 answers
38 views

If $\sum_{j=1}^{\infty} \vert R_j \vert < \infty$, how to show that $\frac1n\sum_{j=1}^njR_j\to0$? [duplicate]

Suppose we have a series $\{ R_j,j=1,2,\dots\}$ and it's absolutely convergent, i.e., $\sum_{j=1}^{\infty} \vert R_j \vert < \infty$. I am wondering how to show that the partial sum $\sum_{j=1}^{n-...
Percy Wong's user avatar
2 votes
2 answers
146 views

Maximise $xyz$ such that $x+xy+xyz=1$, $y+yz+xyz=2$, $z+zx+xyz=4$

$x, y, z$ are real numbers which satisfy the following: $$x + xy + xyz = 1$$ $$y + yz + xyz = 2$$ $$z + zx + xyz = 4$$ Then find the maximum value of $xyz$. I tried adding and subtracting a few ...
Sujal Motagi's user avatar
0 votes
1 answer
54 views

Simplifying an inequality by omitting a parameter

The answer to my question might be a clear "No", is there an equivalent way to write the following inequality in which we do not need the parameter $n$? $$\frac{1}{2} \pi (2 n+1)<a<\...
mattTheMathLearner's user avatar
1 vote
1 answer
61 views

Proving the implicit equation of the parabolic boundary of a plane domain defined by two parameters

$p,q \in [0,1]$ I have the equation: $(x,y) = pq(2,1) + p(1-q)(-1,-1) + (1-p)q(-1,-1) + (1-p)(1-q)(1,2)$ I want to show that the parabolic boundary connecting $(2,1)$ and $(1,2)$ is given by $5(x-y+1)^...
John Smith's user avatar
0 votes
2 answers
183 views

summation of $\cos{x} + \cos{2x} + \cos{3x}$ and so on is $-\frac{1}{2}$

$$\cos{x} + \cos{2x} + \cos{3x} \ldots = y$$ $$ 2\cos{x} + 2\cos{2x} + 2\cos{3x} \ldots = 2y $$ By grouping every alternate term by $\cos{A} + \cos{B} = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$ $$ \...
dhruvk's user avatar
  • 11
3 votes
2 answers
63 views

Given $a,b,c,d$ in arithmetic progression, can we express the solution to $\frac1{x-a}+\frac1{x-b}+\frac1{x-c}+\frac1{x-d}=0$ in terms of $a,d$ only?

Context This is a question I wrote for myself recently, heavily based on an old (1930s) examination paper for university admissions. Given $a,b,c,d$ which are real numbers and consecutive terms in an ...
Red Five's user avatar
  • 2,673
3 votes
2 answers
90 views

How to show $\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{m}{2}\rfloor\lfloor\frac{m-1}{2}\rfloor$ is equal to this expression?

I am trying to show equality between the two expressions $$\frac {\lfloor{\frac n 2}\rfloor(\lfloor{\frac n 2}\rfloor - 1)\lfloor{\frac m 2}\rfloor(\lfloor{\frac m 2}\rfloor - 1)} 4 + \frac {\lfloor{\...
Princess Mia's user avatar
  • 2,499
0 votes
0 answers
20 views

Medians from rows of numbers

You have three rows of numbers, $(a_1,a_2,a_3), (b_1,b_2,b_3), (c_1,c_2,c_3)$, each in the range $[0,1]$ and summing up to $1$. If the medians of the three columns are $m_1,m_2,m_3$ with sum $m$, you ...
user355066's user avatar
1 vote
1 answer
72 views

Solving the system $a^x=(x+y+z)^y$, $a^y=(x+y+z)^z$, $a^z=(x+y+z)^x$, for $x$, $y$, $z$

Given system of equations: $$ \left\{ \begin{array}{c} a ^ x = (x + y + z)^y \\ a ^ y = (x + y + z) ^ z \\ a ^ z = (x + y + z) ^ x \end{array} \right. $$ find the values of $x$, $y$ and $z$ where $...
Samiksha's user avatar
3 votes
1 answer
59 views

Mach equation algebraic manipulation

I'm trying to go from: $$ M^2 = \frac{M_S^2 + \frac{2}{\gamma-1}}{\frac{2M_S^2 \gamma}{\gamma-1} - 1}$$ to: $$ M^2 =1 - \left[\frac{\gamma +1}{2 \gamma} \right] \left[ \frac{M_S^2-1}{(M_S^2-1) + \frac{...
MicrosoftBruh's user avatar
0 votes
0 answers
21 views

Confusion about replacing a variable in a expression of paprameters

I have a diophantine equation of the form: $$f(x,a,b)+c=0.........(**)$$ and by reducing it modulo $p^n$ I get a congruence of the form $h(x,a,b)+c\equiv 0\pmod {p^n}$ where $x$ is the variable and $a,...
Safwane's user avatar
  • 3,834
3 votes
1 answer
200 views

Showing an inequality using another inequality

Let $ a,b,c,d,e ,k$ be positive with $k\geq 1$. Define $s_1=k(a+b)$ and $s_2=k(c+d)$. Then is it true that $$[a(s_2-c)+db][(s_2+e)s_1-s_2a]+s_1a[a(s_2-c)+db]-b(s_2+e)s_1d-s_1s_2ab$$ is positive given ...
stochs's user avatar
  • 400
2 votes
0 answers
53 views

Elementary functions: Can an arithmetic operation be a composition?

The elementary functions are the functions of one complex variable that are generated by applying finite numbers of $\rm{exp}$, $\ln$ and/or unary or multiary $\mathbb{C}$-algebraic functions. Let $n\...
IV_'s user avatar
  • 7,104
0 votes
1 answer
24 views

Oscillations of Lagrange interpolation polynomials

Let $I = [a,b]$ be a real closed interval. Let $n$ be a positive integer and let $x_i = a+i\frac{(b-a)}{n}$ for $i=0,...,n$. Let $p_j(x)$ be the Lagrange interpolating polynomial of the $n+1$ points ...
Alberto's user avatar
  • 503
0 votes
1 answer
23 views

Show $f(X) = X^n + aX + b$ has $f'(\alpha_i)=\frac{-n(a\alpha_i+b)+a\alpha_i}{\alpha_i}$.

I have the polynomial $f(X) = X^n + aX + b$, which has roots $\alpha_i$. I want to show that $$ f'(\alpha_i)=\frac{-n(a\alpha_i+b)+a\alpha_i}{\alpha_i}$$ for all $i$, as given in this answer. How can ...
Robin's user avatar
  • 3,463
-3 votes
0 answers
52 views

Given two numbers, how to find the number of digits in their sum, product, difference, and quotient? [closed]

Let $\{k,p\}\subset \mathbb{N}$. An arithmetic operation is performed on a natural decimal $k$-digit number and a natural decimal $(k+p)$-digit number. Knowing each of these numbers explicitly, how to ...
Rusurano's user avatar
  • 828
-4 votes
0 answers
34 views

Let $a,b,c,d \in \mathbb R$. if $a^2+4b^2+c^2+4d^2=2(ab+bc+cd+da)$, find $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$ [closed]

I tried using identities but not able to arrive at the sum of fractions
Dhopu K's user avatar
0 votes
1 answer
35 views

Applying successive discounts, an un-intuitive formula

I was reading a book where the author calculated the final price using a formula I can't understand how it was obtained. I would like to know how it was derived and how it could be generalized. ...
NoChance's user avatar
  • 6,524
-1 votes
1 answer
36 views

Arc length of a polar curve r = -8cos(t)

If I am being asked to find the arc length of the polar curve $r = -8cos(t)$ when I use the integral formula it gives me 16 $\pi$. But since this polar curve represents a circle with radius 4, should ...
Mathematican's user avatar
-2 votes
1 answer
56 views

Solve each inequality. Write your answer using interval notation. a) −4 < 5 − 3x ≤ 17 [closed]

Solve each inequality. Write your answer using interval notation. a) −4 < 5 − 3x ≤ 17
Ngirababyeyi Arnaud's user avatar
-3 votes
0 answers
47 views

Evaluating $\lim_{x\to\pi/3} \frac{1-2\cos x}{\pi-3x}$ without L'hopital [closed]

Evaluate using trigonometric identities, no L'hopital: $$\lim_{x\to\pi/3} \frac{1-2\cos x}{\pi-3x}$$
Sksi Dndkke's user avatar
-2 votes
2 answers
90 views

Advanced Algebra Problem Maybe linked with Vectors? [duplicate]

$x^2 +y^2 + xy = 25$ $y^2 + z^2 + yz = 49$ $z^2 + x^2 + zx = 64$ Find $(x + y + z)^2 -100$ Here's My Approach : $x^2 + y^2 -2xycos120 = 25$. This Equation looked too similar to the subtraction of the ...
memeguy's user avatar
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