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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0
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1answer
30 views

Mid level algebra headscratcher.

I have 4 numbers. Base = 100 , Start = 0.4 , Count = 4 , Multiplier = 0.64448 I do 100 × 0.4 = 40 Then 4 times: 40 × 0.64448 = 25.7788368 25.7788368 × 0.64448 = 16.61371067 16.61371067 × ...
0
votes
1answer
28 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
1
vote
2answers
32 views

Finding (or rather expanding) the product $(5-xy)(3+xy)$

Given the product $(5-xy)(3+xy)$ I tried the following, As we know, $(x+a)(x+b)=x^2+(a+b)x+ab$ Here $x$ is $xy$. But $xy$ has two signs$-$ and $+$. How do I solve this.
0
votes
3answers
37 views

Finding the tangent line to the graph of $f(x)=(x+2)^{3/5}$ at $x=-2$

Does the graph of the function $f$ have tangent line at the given points? If yes, what is the tangent line? $f(x)=(x+2)^{3/5}$ at $x=-2$ solution: yes, $x=-2$ The derivative I found: ...
-1
votes
1answer
44 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
-3
votes
1answer
21 views

Fourth roots of a certain complex number [on hold]

Find the fourth roots of $81(\cos 320^\circ + i\sin 320^\circ )$. Write the answer in trigonometric form. \begin{array} \text{a.} & 3(\cos 160^\circ + i \sin 160^\circ ); & &3(\cos ...
0
votes
2answers
17 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
votes
1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
0
votes
1answer
8 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
4
votes
4answers
55 views

How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$?

I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state: $$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} ...
0
votes
3answers
53 views

Can anyone help me understand the simplification of $\frac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}\;$?

Can anyone help me understand the following simplification of the fraction? $$\dfrac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} = 5 + 2\sqrt 6$$ I cant understand how to simplify the left-hand side to get ...
2
votes
2answers
117 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
0
votes
1answer
18 views

Find equation of the straight line tangent to the curve at the point indicated

Find equation of the straight line tangent to the curve at the point indicated: $y=2x^2 -5$ at $(2,3)$ I think I have to use $y=m(x-x_o)+y_0$ etc but I'm not sure how to find the $m$! Thanks for ...
2
votes
3answers
35 views

Finding the perimeter of the room

If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by ...
0
votes
3answers
19 views

Finding the angles of a parallelogram.

In a parallelogram, one angle is $2/5th$ of the adjacent angles. Determine the angles of the parallelogram. I tried the following, Let the adjacent angles be $2x$ Let the other angle be $y$ ...
0
votes
2answers
23 views

The perimeter of a rectangle is 48 meters and its area is 135 m^2. Determine the sides of the rectangle.

The perimeter of a rectangle is 48 $m$ and its area is $135$ $m^2$. Determine the sides of the rectangle. I tried the following, Perimeter$=$$48$ $m$ Let the length be $x$ m and the breadth be $y$ m ...
3
votes
2answers
58 views

Mr. and Mrs. Ahuja weigh x and y kg. Find their present weights.

Mr. and Mrs. Ahuja weigh $x$ kg and $y$ kg respectively. They both take a dieting course at the end of which Mr. Ahuja loses $5$ kg and weighs as much as the wife weighed before the course. Mrs. Ahuja ...
1
vote
3answers
63 views

Solve the equation: $\frac{z}{z-5}+\frac{1}{3}=-\frac{5}{5-z}$

Solve the equation: $\frac{z}{z-5}+\frac{1}{3}=-\frac{5}{5-z}$ First $z$ cannot be equal to $5$. First, I multiplied $z$ with $3$, $1$ with $z-5$ and $-5$ with both. Eliminating the denominators ...
0
votes
0answers
34 views

Proof of Descartes' theorem

I came across the use of Descartes' theorem while solving a question.I searched it but I could only find the theorem but not any ...
0
votes
1answer
33 views

The denominator of a fraction is 4 more than twice the numerator. Determine the fraction.

The denominator of a fraction is $4$ more than twice the numerator. When both the numerator and denominator are decreased by $6$, the denominator becomes $12$ times the numerator. Determine the ...
3
votes
0answers
45 views

How to find $f$ and $g$ if $f\circ g$ and $g\circ f$ are given?

The question is: Let $f:\mathbb R\rightarrow \mathbb R$ and $g:\mathbb R\rightarrow \mathbb R$ be two functions such that $(f\circ g)(x)=4x^2+4x+1$ and $(g\circ f)(X)=x^2+2x+2$. Find $f(x)$ and ...
3
votes
2answers
52 views

Prove that every non-prime natural number $ > 1$ can be written in the form of $n+(n+2)+(n+4)+…+(n+2m) = p$

I'm trying to prove that every non-prime natural number greater than $1$ can is equal to a sum of consecutive even or odd numbers. This can be resumed in : « $p,m,n \in ℕ$» , «$p > 1$» , «$n > ...
2
votes
0answers
50 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
1
vote
3answers
48 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
2
votes
3answers
27 views

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$?

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$? I am learning trigonometric identities one identity I have to proof is the next: $ (1- \sin \alpha + \cos \alpha)^2 = ...
0
votes
3answers
36 views

Derivation of sine and cosine case

I am struggling to see this. I know that we can factor out $ a$, but I don't see how we can end up with the right hand side. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
0
votes
3answers
51 views

Find the values of parameter $x$ for which two graphs have 0, 1, or 2 intersection points

I have been given this question: $$y=\frac{4}{10}x+c\quad;\quad y=\frac 4x,$$ Investigate the values of $c$ that may provide, $0,1,$ or $2$ points of intersection. I'm really stuck, any ideas?
2
votes
4answers
56 views

Show that there is an angle $\theta$ such that $a=\cos\theta$ and $b=\sin\theta$

My problem is from Israel Gelfand's Trigonometry textbook. Page 50. Exercise 3: Suppose that $\alpha$ is some angle. If $a=4\cos^3\alpha-3\cos\alpha$ and $b=3\sin\alpha-4\sin^3\alpha$, show that ...
-2
votes
1answer
66 views

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$ I understood the solution given in my book which said  $$\cos(x)+\sin(x)\leq\sqrt{2}<90$$ $$\cos(x)<90-\sin(x)$$ But if ...
1
vote
1answer
37 views

A farmer sold a calf and a cow for Rs 760. Find the cost of each.

A farmer sold a calf and a cow for Rs. 760 Thereby making a profit of 25% on the calf and 10% on the cow. By selling them for Rs. 767.5 he would have raised a profit of 10% on the calf and 25% on ...
1
vote
3answers
48 views

How many even 3 digit integers greater than 700

How many even 3 digit integers greater than 700 with distinct non zero digits are there ? My answer is: the only hundred digit that are possible are 7, 8 and 9 (3) the only ten digit that ...
0
votes
4answers
109 views

How to calculate $k^0+k^1+k^2 + k^3+…+ k^{n-1}$ [duplicate]

How to simplify below expression or convert it to something simpler like $k^{n-1}$? $$ k^0+k^1+k^2 + k^3+...+ k^{n-1} $$
1
vote
1answer
126 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
3
votes
1answer
20 views

Intersection of a circumference with a the curve: $y=ax^k$

Given the circunference centered in the origin of a cartesian reference frame, its equation is: $x^2+y^2=r^2$, Assuming $r=1$, we have: $x^2+y^2=1$. The intersections of this curve with the curve ...
0
votes
1answer
17 views

Knowing the total price of stamps of two denomination, find the number of stamps of each kind

A man buys postage stamps of denominations $3$ paise and $5$ paise, for Rs $1$. He buys $22$ stamps in all. Find the number of $3$ paise stamps bought by him. (100p= 1 Rs) I tried, Let the number of ...
3
votes
1answer
26 views

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$

Sketch The Region In The Plane Defined By $\lfloor x + y\rfloor^2 = 1$ I would like for you guys to have a look at my approach and give my advice regarding the solution and whether there's a ...
0
votes
3answers
17 views

Sifting through data to find number of people in a group

Of the 240 campers at a summer camp, 5/6 could swim. If 1/3 of the campers took climbing lessons, what was the least possible number of campers taking climbing lessons who could swim? My Attempt: ...
5
votes
7answers
684 views

When the numerator of a fraction is increased by 4, the fraction increases by 2/3…

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction? I tried, Let the numerator of the fraction be $x$ and the denominator ...
3
votes
3answers
544 views

If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 5/8…

If the numerator of a fraction is increased by $2$ and the denominator by $1$, it becomes $\displaystyle \frac{5}{8}$ and if the numerator and the denominator of the same fraction are each increased ...
0
votes
4answers
24 views

Ratios: Algebra Problem Help

The present ages of Ram and Shyam are in the ratio 5:6. Five years ago, the ratio was 4:5. Find their present ages. I tried, Let the age of Ram be x Let the age of Shyam be y Accordingly, Ram's ...
0
votes
1answer
14 views

Prove $x_1$ is at least a $k$-fold root of polynomial $p$ if and only if $p(x_1) = p^{'}(x_1) = \dots p^{(k-1)}(x_1) = 0$?

Suppose $p: \mathbb R \rightarrow \mathbb R$ is a polynomial given by $x \mapsto a_nx^n + \dots a_1 x_1 + a_0$. How do I prove $x_1$ is at least a $k$-fold root of $p$ if and only if $p(x_1) = ...
0
votes
1answer
18 views

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R ...
2
votes
1answer
29 views

Trajectory of a rock

During the eruption of Mount St. Helens in 1980, debris was ejected at a speed of over $440$ feet per second ($300$ miles per hour). The height in feet of a rock ejected at angle of $75^ \circ$ is ...
-1
votes
5answers
72 views

Intersection of line with elliptic curve

How to obtain the point of intersection of the line $y=x-1$ and the curve $y^2=x^3+17$. Any help from experts is deeply appreciated.
0
votes
1answer
67 views

$a+b=\dfrac{a^2-1}{a-b}-\dfrac{b^2-1}{a-b}$, proof? [on hold]

$a+b=\dfrac{a^2-1}{a-b}-\dfrac{b^2-1}{a-b}$, where $a$ and $b$ are any integers.
0
votes
4answers
34 views

Solving inequality with division [on hold]

Question 4: Solve the inequality: $$\frac{(x-1)^3(x+1)^8}{(x+2)^4} >0$$
0
votes
1answer
42 views

How to get the answer choice

This is a GRE quantitative question. Without hints, how to pick the answer?
1
vote
2answers
25 views

Accuracy of linear approximations.

it's another day of calculus and I'm having trouble with linear approximations, perhaps you guys can help. I am unsure of how to calculate the 'accuracy' of these approximations, let me give you an ...
1
vote
3answers
58 views

If one number is thrice the other and their sum is $16$, find the numbers

If one number is thrice the other and their sum is $16$, find the numbers. I tried, Let the first number be $x$ and the second number be $y$ Acc. to question $$ \begin{align} x&=3y &\iff ...
-3
votes
0answers
21 views

Gaussian Elimination and Matrix [on hold]

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. \begin{cases} 4x - y + 3z = 12 \\ x + 4y + 6z = -32 \\ 5x + 3y + 9z = 20 ...