166 - Mathematics Questions Answers

$$\begin{gathered} Let the lines in the pair are m_{1}x-y=0 and m_{2}x-y=0 \\ Find perpendiculars from (x^{,},y^{,}) on lines and calculate their sum \\ Use formulae m_1+m_2 = -\frac{2h}{b} and m_1{m}_2= \frac{a}{b} \end{gathered}$$Inform for any issue

length for one turn = 3

Draw diagonal of a rectangle then turn it as a cylinder, you will get the construction given in the question.

$$\begin{gathered} In triangle l = \sqrt{3^2+n^2} \\ So total length of wire having n turn = n\sqrt{3^2+n^2} \\ Compare with 20 and solve \end{gathered}$$

$$\begin{gathered} The general solution of this kind of \\ differential eq. is y = e^{ax} \\ Let a function g\left(x\right) = e^{ax}f\left(x\right) \left(1\right) \\ differentiate eq. \left(1\right) thrice for getting g^{,,,}\left(x\right) \\ now for getting the same function as given \\ find a. you get the same function for \\ a = 1/2. clearly, for min 5 zeros of g\left(x\right), \\ there are min 2 zeros for g^{,,,}\left(x\right). \\ solve and inform if face any problem. \end{gathered}$$

$$\begin{gathered} Probability to get following differences \\ difference probability \\ 0 6/36 \\ 1 10/36 \\ 2 8/36 \\ 3 6/36 \\ 4 4/36 \\ 5 2/36 \\ So required probability =1-2\left[\frac{6}{36}\left(\frac{4}{36}+\frac{2}{36}\right)+\frac{10}{36}\left(\frac{2}{36}\right)\right] \\ now solve \\ problem is based on integral values so area is not very useful \end{gathered}$$

$$\begin{gathered} \frac{x}{1-x^2}=\frac{1}{1-x}-\frac{1}{1-x^2} \\ break all terms similarly and solve \end{gathered}$$

1+x²+x⁴ = (1+x+x²)(1-x+x²)

now solve