# Tagged Questions

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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### Prove the given condition from given two quadratic equation

Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$ Till yet, I had figured the common ...
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### how many questions did D answer correctly

Each of A, B, C, and D took a test. Each of them answered at least one question correctly, and altogether they answered 67 questions correctly. A had more correct answers than anyone else. B and C ...
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### Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$

Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$ My calculator gives a decimal for $\arctan(-4)$ so I don't know what answer is expected.
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### Bearings and distances

Two ships leave port at the same time. One travels at $5$ km/h on a bearing of $46$ degrees. The other travels at $9$ km/h on a bearing of $127$ degrees. How far apart are the ships after $2$ hours?
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### Proving $\binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n$ without induction [duplicate]

I have to prove that: $$\binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n$$ I don't want a complete solution, but only a hint.
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### Find all real x such that $\cos x$, $\cos2x$, $\cos 4x$, $\cos 8x$, etc. ($\cos 2^n$ for all non negative $n$) are all negative

I think I got that $|\cos (2^nx)|$ must be less than $|\sin (2^nx)|$ for all non negative $n$.
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### What is the meaning of $\log^2n$ and how should it be read in word form?

$\log^2n$ is what I need assistance with. How is this read in word form? What exactly does this mean? No matter how much I read about logarithms, they still seem new to me.
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### If the quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots, then $k^2 – 2k – 8 > 0$ [closed]

The quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots. Show that $k^2 – 2k – 8 > 0$. I'm not sure what you're meant to do here- it's a 2 mark question.
How would I go about solving an equation like this? $3^{4/3}b^{5/3} - b^3 = 1$ I thought about rearranging to get $3^{4/5}b = (1 + b^3)^{3/5}$, but that didn't seem to lead anywhere as I couldn't ...