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
15 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
33 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
22 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 ...
2
votes
2answers
46 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
57 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
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0answers
31 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
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1answer
32 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
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0answers
41 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
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2answers
48 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
49 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
47 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
26 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. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
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3answers
50 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 ...
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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 ...
0
votes
1answer
36 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
47 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
108 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
123 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
14 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
675 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
541 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 ...
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votes
5answers
71 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
66 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
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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
57 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 ...
1
vote
3answers
72 views

How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $?

I've the answer for a question in my textbook to be: $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $ However my textbook states $-\sqrt ...
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votes
2answers
60 views

How can I simplify the expression $\frac{\sqrt[5]{x^2}}{x^2}$?

$$\dfrac{\sqrt[5]{x^2}}{x^2}$$ I'm doing a summer math packet for calculus. I need to simplify the above. I think I may know the answer, but I'm not sure. Can someone help me, please?
0
votes
2answers
56 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
0
votes
3answers
40 views

Help needed verifying a trigonometric identity

I have the following identity: $$ \frac{\tan (t + h) - \tan(t)}{h} = \left( \frac{\tan (h)}{h} \right)\left( \frac{\sec^2(t)}{1 - \tan (t)\tan (h)} \right)$$ Having tried various approaches, ...
0
votes
4answers
58 views

Point of intersection of tangent line with curve [on hold]

How to determine the point of intersection of the tangent line at $(0, 0)$ on the curve $$y^2 + y = x^3 + x^2$$
4
votes
1answer
54 views

Are there more examples of functional equations which are also valid for the identity map?

I find the co-incidence of the identity: $$\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B$$ very pleasing. So, I was wondering if there are more of these type of identities. To make my question precise: ...
3
votes
3answers
306 views

Cannot follow the algebra

The following equality is stated in my text book and I cannot follow the algebra that makes it true. Please help me step through this to show how $$\frac{4^x}{3^{x-1}} = 4 ...
-4
votes
2answers
52 views

SAT question stuck? [on hold]

I am preparing for sat and this question, I have no idea how to solve it. Please provide step wise solution also. If $2|x+3|=4$ and $\frac{|y+1|}{3}=2$, then $|x+y|$ could equal of the following ...
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votes
1answer
34 views

Find the total area of the path? [on hold]

A grassy plot is 80 meters into 60meters two cross paths each 4meters which are constructed at right angles through the center of the field such that each path is parallel to one of side of the ...
1
vote
1answer
29 views

Finding the equation of more than one tangent line

I ran into a problem I have no idea how to begin, maybe you guys can help me out. I think maybe it has something to do with parametric equations? But this is just a guess. Find equations of both the ...
1
vote
3answers
48 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
0
votes
4answers
50 views

Correct this problem in complex numbers

Prove for all $|z| = 3$, $$\frac{8}{11} \leq \left | \frac{z^2 + 1}{z^2 + 2}\right | \leq \frac{10}{7}.$$ Here is what I did, $$\frac{8}{11}\leq \frac{8}{z^2 + 2}=\frac{|z^2| - 1}{z^2 ...