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Here are some of the most trickling Maths and Science problems, If you are able to solve it. Write Detailed solutions below, with your email id. The best performer may be highly rewarded!- Q1.)If a,b,c are distinct zeroes of polynomial x^3 - x^2 + x - 2 then a^3 + b^3 + c^3 = ? Q2.)In a Triangle ABC, AB=10, BC=5, AC=6. BC is extended to D such that Trngle ABC is similar to Trngle DCA then lenght of DC = ? Q3.) If 1/x + 2/y + 3/z = 0 and 1/x - 6/y - 5/z = 0 then find x/y + y/z + z/x? Q4.)If a+b= 1 and a^2+b^2= 2 then find a^5+b^5? Q5.)ABCD is a square. AB is extended to P and DP intersects AC and BC at Q and R respectively such that DQ= 3cm and QR=2cm , then find PR? Q6.)In Trngle ABC, B=90 degree, AB=6 and AC = 10 then the length of angle bisector AD from Angle A is?

Note by Aßhĩмanyu Singh
3 years ago

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First of all , I want to tell that they are no tricky problems. Tooooooo easy problems.

The complete solutions are -

1.) Clearly, by Vieta's Formula - \(a+b+c=1\), \(ab+bc+ca = 1\), \(abc=2\)

We know that \(a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca) = (a+b+c)((a+b+c)^2 -3(ab+bc+ca))\)

Plugging the values, we get

\(a^3+b^3+c^3-6=(1)(1^2-3(1))=-2 \Rightarrow a^3+b^3+c^3 = 4\)

2.) Since ABC is similar to DCA, the ratio of their corresponding sides must be equal. So,

\(\frac{AB}{DC}=\frac{BC}{AC} \Rightarrow DC = 12\)

3.) The given equations can be simplified to \(yz + 2xz + 3xy = 0\) and \(yz - 6xz - 5xy = 0\)

Subtracting both the equations, we get -

\(z + y = 0 \Rightarrow z=-y\)

Substituting \(z=-y\) in first equation, we get - \(x^2-y^2 = 0\)

Now, put \(z=-y\) in \(x/y + y/z + z/x\). So, what you will get is \(x/y + y/z + z/x = -1\)

4.) \(a+b = 1\), \(a^2+b^2 = 2\)

We can easily get \(ab = \frac{-1}{2}\)

Now -

\(a^5 + b^5 = a^5 + b^5 + a^3b^2 + a^2b^3 - a^3b^2 - a^2b^3\)

\( = a^3(a^2+b^2)+b^3(a^2+b^2) - a^3b^2 - a^2b^3 = (a^2+b^2)(a^3+b^3) - a^2b^2(a+b)\)

So, \( a^5+b^5 = (a^2+b^2)(a^3+b^3) - (ab)^2(a+b) = (a^2+b^2)(a+b)(a^2+b^2-ab) - (ab)^2(a+b)\)

Now plugging the values, we get - \(a^5+b^5 = \frac{19}{4}\)

5.) First, let \(\angle DRC = \angle BRp = \theta\) and \(RP = x\)

Then \(AD = (5+x)cos \theta\) and \(RC = 5cos \theta\)

Now observe that - triangle QAD is similar to QCR. So, on solving for PR gives \(PR = \frac{5}{2}\)

6.) Let \(BD = x\), \(\angle BAD = \angle DAC = \theta\)

Then, from triangle \(ABC \rightarrow sin 2\theta = \frac{8}{10} = \frac{4}{5}\)

from triangle \(BAD \rightarrow sin \theta = \frac{x}{\sqrt{36+x^2}}\) and \(cos \theta = \frac{6}{\sqrt{36+x^2}}\)

We know that \(sin 2\theta = 2sin\theta cos\theta\)

Plugging their values, we get the following quadratic - \(s^2-15x+36 \Rightarrow x = 12, 3\)

Since, \(BC = 8\), so we can't take \(x=12\)

Hence, \(AD = \sqrt{3^2 + 6^2} = \sqrt{45}\)

Next time when you post a question in discussion, please makes sure that they are of proper difficulty level. Kishlaya Jaiswal · 3 years ago

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@Kishlaya Jaiswal All these are Class X FTRE Sample Paper Questions. Bhargav Das · 3 years ago

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I'm not trying to be rude at all but these questions were really easy and they aren't science problems. Though if you are having trouble with anyone of them just let me know because I will help you. I can give you some hints if you like too: 1. this involves complex numbers 2. I used the cosine rule 3. this is just algebra 4. this just involves a quadratic 5. just a bit of trigonometry 6. this just involves Pythagoras/tan. Hope this helps. Joel Jablonski · 3 years ago

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that was easy. Now, help me solve this... \(x^{2}+5x+5=32\) Don't use trial and error method and the \(x\) value should be a single one. Fahad Shihab · 3 years ago

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@Fahad Shihab Are you f'ing kidding us? This is as easy as the ones posted above. Sorry. Guilherme Dela Corte · 3 years ago

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@Guilherme Dela Corte Not kidding.Not f'ing. but making use of situations. Fahad Shihab · 3 years ago

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@Fahad Shihab That's the question which the brilliant community should solve Fahad Shihab · 3 years ago

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@Fahad Shihab \(x=\frac{\pm \sqrt{133}-5}{2}\) Joel Jablonski · 3 years ago

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@Joel Jablonski thank you. that's all i need. please write the steps also. Fahad Shihab · 3 years ago

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These are very easier problems and way below than the level of Brilliant problems. You must not post these types of problems to this discussion room. Ram Prakash Patel Patel · 3 years ago

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