Koundinya-Mathematics Teachers Hub

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Friday, 20 November 2020

$If \,\,z=r(cos\theta+i\,sin\theta)\,\,and \,\,2w=r\left [ cos\frac{\pi}{4}+\theta+i\,sin(\frac{\pi}{4}+\theta) \right ],\,\, prove \,\,that \\z+w=\frac{r\,cosec\phi}{\sqrt{2}}\left [ cos(\frac{\pi}{4}+\overline{\theta-\phi})+\theta+i\,sin(\frac{\pi}{4}+\overline{\theta-\phi}) \right ]\,\,where \\tan\phi=2-\sqrt{2}$

Posted by Koundinya-Mathematics teacher's hub at 05:18 No comments:

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$Find\,\, the\,\, values\,\ of\,\, x\,\, satisfying\,\, the\,\, equation\,\, 3^{x}+1-\left | 3^{x}-1 \right |=2log_5^{\left | 6-x \right |}$

Posted by Koundinya-Mathematics teacher's hub at 04:50 No comments:

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Friday, 13 November 2020

$A\,\,triangle\,\,has\,\,vertices\,\,at\,\,the\,\,points\,\,represented \,\,by\,\, complex\,\, numbers\\Z_{1},\,\,Z_{2},\,\,Z_{3}.\,\,\, Show \,\,that\,\, the\,\, complex\,\, number\,\, associated\,\, with\,\, the \,\,orthocentre\,\, is\\\\ \frac{Z_{1}Tan\,A+Z_{2}Tan\,B+Z_{3}Tan\,C}{Tan\,A+Tan\,B+Tan\,C}$

Posted by Koundinya-Mathematics teacher's hub at 10:01 No comments:

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$If \,\,a>0, b>0;\, a\neq 1, b\neq 1, \,\,find\,\, the\,\, value\,\, of\,\, p\,\, for\,\, which\,\, the\,\, linear\,\, equations\\ px+(\log_b^a)y+(\log_a^b)z=0\\ x+(\log_a^b)y+(\log_b^a)z=0\\ x+(\log_b^a)y+(\log_a^b)z=0\\ has\,\ a \,\,non-trivial\,\, solution.\,\,What\,\, happens\,\, for\,\, p=1?$

Posted by Koundinya-Mathematics teacher's hub at 09:54 No comments:

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$Solve \,\,\,\,\,\, \frac{3}{2}\,\log_\frac{1}{4}^{(x+2)^2}-\,3=\log_\frac{1}{4}^{(4-x)^3}-\,\log_{4}^{(x+6)^3}$

Posted by Koundinya-Mathematics teacher's hub at 09:44 No comments:

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$If \quad S_{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....\frac{1}{n},\\ \quad \quad S_{n}^{'}=\frac{n+1}{2}-{\frac{1}{n(n-1)}+\frac{2}{(n-1)(n-2)}+.....+\frac{(n-2)}{3.2}},\\ then \,\,prove\,\, that\,\, S_{n}=S_{n}^{'}$

Posted by Koundinya-Mathematics teacher's hub at 09:35 No comments:

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Monday, 9 November 2020

$For\,\, n,\,\, a\,\, positive\,\, integer \geq 1,\,\, Prove\,\, by\,\, the\,\, method\,\, of\,\, mathematical\,\, induction\,\, that \\ \sum_{r=1}^{n}\frac{1}{sin\,2^{r}\,x}=cot\, x-cot\,2^{n}\,x$

Posted by Koundinya-Mathematics teacher's hub at 04:11 No comments:

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$\quad \quad Prove\,\, by\,\, the\,\,method\,\,of\,\, mathematical\,\, induction \,\,\\sin \,\alpha +sin(\alpha +\beta )+sin(\alpha +2\beta )+....+sin(\alpha +(n-1)\beta )=\frac{sin(\alpha+\frac{(n-1)\beta}{2})\,sin\frac{n\beta}{2}}{sin\frac{\beta}{2}}$

Posted by Koundinya-Mathematics teacher's hub at 04:05 No comments:

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Thursday, 5 November 2020

$n \,\,positive\,\, integers\,\, are\,\ taken\,\, at\,\, random\,\, and\,\, multiplied\,\, together.\,\,Show\,\, that \,\,\\the\,\, chance \,\,that\,\, the \,\,last\,\, digit\,\, of\,\, the\,\, product\,\, is \,\,zero\,\, is \,\,\frac{(2^n-1)(5^n-4^n)}{10^n}$

Posted by Koundinya-Mathematics teacher's hub at 06:02 No comments:

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$\quad \,\,\,\,\,Out \,\,of\,\, (4n+1)\,\, tickets\,\, consecutively \,\,numbered\,\, as\,\, 1,2,3,...(4n+1),\\ five\,\, tickets\,\, are\,\, chosen\,\, at\,\, random\,\, without\,\, replacement.\,\,Find\,\, the \,\,probability\,\,\\ that\,\, these\,\, tickets\,\, are\,\, in \,\,A.P.$

Posted by Koundinya-Mathematics teacher's hub at 05:55 No comments:

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Tuesday, 3 November 2020

$Prove \,\,by\,\, induction\,\,\frac{1}{1+x}+\frac{2}{1+x^{2}} + \frac{2^2}{1+x^{2^2}}+......+\frac{2^n}{1+x^{2^n}}=\frac{1}{x-1}+\frac{2^{n+1}}{1+x^{2^{n+1}}}$

Posted by Koundinya-Mathematics teacher's hub at 05:12 No comments:

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$Solve\,\,\,\,\log_{\sqrt{2}}\left ( {5^{x}-1} \right )\log_{\sqrt{2}}\left ( \frac{2\sqrt{2}}{5^{x}-1} \right )> 2$

Posted by Koundinya-Mathematics teacher's hub at 04:59 No comments:

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