Modulus+Logs+Circles+Vectors!

Calculus Level 5

Let $k_1, k_2$ be any two integers given in the relation $|(n-a)!-t| + |t-(n-b)!| + | a+b-k_1n-k_2| \leq 0$ $\forall n$ such that $a < b \leq n$ and $a,b,n,t \in N$ .

Let $P , Q$ be any two points on the curve

$y = \log_{\frac{1}{2}} \left({x + \dfrac{k_2}{2}}\right) + \log_2 \sqrt{4x^2 + 4k_2x +(k_1 + k_2)}$ .

Also $P$ lies on the circle $x^2 + y^2 = k_1^3 - 2k_2$ and $Q$ lies inside the given circle such that its abscissa is a non-zero integer.

O is the centre of the circle.

1) Find the minimum possible value of $\vec{OP} \cdot \vec{OQ}$ .

2) The maximum length of PQ

2 ; 3

4 ; 2

0 ; 3

2 ; 4

3 ; 1

1 solution

Nishant Rai
Apr 26, 2015

@Nishant Rai The solution has made so many assumptions!! If $Q$ has it's abscissa as integer, then why not on the negative $X-axis$ , namely $(-2,1)$ and $(-1,1)$ , as they are integers as well. That will give the value of $OP.OQ$ as negative. Please check this.

Secondly, in your question, what do you mean by length of $OP.OQ$ . That is the scalar product and how do you define the length of a scalar. It would be good if you could change that.

Thirdly, due to the anomaly mentioned in the first point, the value of length of $PQ$ also changes. It comes out to be 5 as maximum, i.e. between $P = (3,1)$ and $Q = (-2,1)$ .

Thanks! :)

A Former Brilliant Member - 5 years, 3 months ago

No, the $x$ coordinate of $P$ and $Q$ must be $x > 1/2$ , so that the $log$ functions are well-defined. Notice that the question simply uses $log$ and not $|log|$ .

However I agree on the 'length' aspect. He should change that.

Adwait Godbole - 5 years, 2 months ago

Can you please try the problem named 'spicy 1'?It is a tough question on definite integrals. If you are able to solve it please give me an outline of the procedures.Please do reply here.

Manish Maharaj - 5 years, 2 months ago

Oh ok! Thanks! I kinda solved it by taking Mod in my copy!

A Former Brilliant Member - 5 years, 2 months ago

Which is this book @Nishant Rai

Arpit Kharbanda - 3 years, 9 months ago

Modulus+Logs+Circles+Vectors!

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