Maths-
General
Easy
Question
The sum of the series, upto n terms, is
Hint:
In the question we will simplify each term using trigonometric identities separately and then try to find any pattern in the series, then we will add the terms to get the sum of the series.
The correct answer is: ![1 half open square brackets tan space 3 to the power of n theta minus tan space theta close square brackets](data:image/png;base64,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)
Sn=
upto n terms.
![N o w comma space sin theta. s e c 3 theta equals fraction numerator sin theta over denominator cos 3 theta end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 sin theta. cos theta over denominator 2 cos 3 theta. cos theta end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half. fraction numerator sin 2 theta over denominator cos theta cos 3 theta end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin open parentheses 3 theta minus theta close parentheses over denominator cos theta. cos 3 theta end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin 3 theta cos theta minus cos 3 theta sin theta over denominator cos theta. cos 3 theta end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin 3 theta over denominator cos theta end fraction minus fraction numerator sin theta over denominator cos theta end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses tan 3 theta minus tan theta close parentheses
S i m i l a r l y comma space sin 3 theta. s e c 9 theta equals 1 half open parentheses tan 9 theta minus tan theta close parentheses comma space sin 9 theta. s e c 27 theta equals 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses comma space sin 3 to the power of open parentheses n minus 1 close parentheses end exponent theta. s e c 3 to the power of n theta equals 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses
S o comma space S n equals 1 half open parentheses tan 3 theta minus tan theta close parentheses plus 1 half open parentheses tan 9 theta minus tan theta close parentheses plus 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses plus..... plus 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses
space space space space equals 1 half open parentheses tan 3 theta minus tan theta plus tan 9 theta minus tan theta plus tan 27 theta minus tan 9 theta plus...... plus tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses
space space space space equals 1 half open parentheses tan 3 to the power of n theta minus tan theta close parentheses
space space space space space space space space space space space 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)
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The value of
is
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is
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If the equation
=0 has at least one solution, then the sum of all possible integral values of 'a' is equal to
If the equation
=0 has at least one solution, then the sum of all possible integral values of 'a' is equal to
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=
=
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If
d then the minimum value of
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If
d then the minimum value of
is equal to (where is variable)
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and
then
=
and
then
=
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=
If
then
=
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The maximum value of the expression ![open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar](data:image/png;base64,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)
where and x are real numbers is
The maximum value of the expression ![open vertical bar square root of open parentheses sin squared space x plus 2 a squared close parentheses end root minus square root of open parentheses 2 a squared minus 1 minus cos squared space x close parentheses end root close vertical bar](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASUAAAAbCAYAAAA6YJkeAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAABSZJREFUeNrtXG9kVlEYPyaTJKbNyqRMZmZmJLVWkpLJzMQ+TGJZZpL0NTXKPjRFkb6kksxEJqlJJGtmkpL0/0sf+pDKPkRlzdvy9jz2vLrezr333POeP/duz4+fvfee98+5z/k9z3nOvwnxD3nBYDAY7pFPXMBgMBgclBgMBgclzS9cSp9lpp8uwbpgqurReFBqAz7lYM9gXTDSkimdB55muzJYF4y0BKV3wM1sVwbrgpGGoLQO+A24JAUPhcwBp4DruZ29Ii26YGTDh4wGpcPAGykybBnwEPA5C8grfOqCO6js+ZDRoHQX2BNRjg835MGws4tYQEN0zyfSoIvF0kGl3YdU9GgsKGFq/gO4KqS8CfjYgzG3AycWuYCmyP4+kDZdzAqGbx+K06OxoBS35IsVaTaVxiliNf1ubcoauRU4CvxOw4oXwH0WBbQBOOnpWZPqwrVtTKIOOEB1Xiiw4UNxejQWlM4CB0PKmunBjI0tFQUyRkZNG9AxuoHL6bqB7NNtUUCTJAbXSKoLH7YxhWFgn1g4pyFs+lCUHo0FJVzy3RpSdkbMT3a6jO73gCscN2IpYlwLfGlRQEeEn/k8E7qwbZs06SBNGZJNH4rSo5GgtE5EL/neB24ruoc91jhwRoQfe8iHXHcBP9D8APasa4reh8asT2ik3hAjDVKZCzHK5jtwgvYU8GtgOLNWQ0AtwIeOha2jCx+2yVpQaiXdo00+AvcXlbcD35JN8G+H5Dvi/M+2D0XpMXFQqpTc6xfRS744P1BedO9ZiLHiglIX9bAF8R2QzBPonvPClZnqwDV+90VHYmwJGeJiD38BWEFOeBPYqSGgcmoHWzClCx+2yVJQwiHvND1nGWWBI4HyTdRhN9J1I123JPQ/2z4UpUfloIT7O67R/Y1FZXFLvnOSezMkpqRBaYekt8wZanCclD1Hr3dpZBalHGJ+Qj1gEN3kaEFcB+7RFFDOgpOY1oUv22QlKI3S0CeqvF2SOd3W8D/bPpQrNSidIPF9obQ7KBpM0VclFF+niJ/AzCtW1qQIHoj5VRocCqxUEF+pJ/ErSDC7JWXjkh5uQjJEUUXOwvOY1oUv25hqT1U96v7W95ihqCz7lGUlKv5n24dKDkoFHAX+ItEVouGrmEqGpem4wnKZxr11KQlKR8lYTZqiToJacrqwXcYzlAkGs8Kfms9le/hmUheubZOlTCmvWZ7T8D+bPmRk+BbEb+BBeo1LvicVImdrRPkxId9p6zooFfbInNPsQZLUo54EsSziPdOS+unuSHYx0W1KF65tk6WgNG0oU1LxP5s+ZHSiG/EI+IZeq5z+jlv6DZsXchmUsGe+Q46APcgzheGbbj2qaT4k7oDqp0DmgRigBtfBYWoHmzChCx+2yVJQuqwwp9QhGard0vA/mz4UpUetoLQT+Ae4BfhZQUBxmyd7aY7AV1CqEvOrNEED4uTgsCUxjgm1FSFcSr1EosG5khGqpw5cbJ40oQsftslSUELHx20Ae+nZa4BXA+W42ICrbQ103UTXrRr+Z9OHrGyexEnM90L99PfjojFmYTJvjh5mtaegVE69SI2kDJ+tzZJoVSc3j1PmMECZwVfx/7J3HFweMzGlC1e2sd2uNoDB/An5zgtJZtRObYDZz+sQm6j4ny0fsnbM5Aq9p0ex4r4O5C40VGl8xuWBXJ+6qGJ5ZALWDuRW03sqE1QGN9MNcZs4BY7b+xz+HuuCUaoeSzpm0s82ZrAuGBaGwdrgf2/KYF0wrOIv+LUBdj6q5s4AAAISdEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mZW5jZWQgY2xvc2U9InwiIG9wZW49InwiIHNlcGFyYXRvcnM9InwiPjxtcm93Pjxtc3FydD48bWZlbmNlZCBzZXBhcmF0b3JzPSJ8Ij48bXJvdz48bXN1cD48bWk+c2luPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48bW8+KzwvbW8+PG1uPjI8L21uPjxtc3VwPjxtaT5hPC9taT48bW4+MjwvbW4+PC9tc3VwPjwvbXJvdz48L21mZW5jZWQ+PC9tc3FydD48bW8+JiN4MjIxMjs8L21vPjxtc3FydD48bWZlbmNlZCBzZXBhcmF0b3JzPSJ8Ij48bXJvdz48bW4+MjwvbW4+PG1zdXA+PG1pPmE8L21pPjxtbj4yPC9tbj48L21zdXA+PG1vPiYjeDIyMTI7PC9tbz48bW4+MTwvbW4+PG1vPiYjeDIyMTI7PC9tbz48bXN1cD48bWk+Y29zPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4mI3hBMDs8L21vPjxtaT54PC9taT48L21yb3c+PC9tZmVuY2VkPjwvbXNxcnQ+PC9tcm93PjwvbWZlbmNlZD48L21hdGg+PC5y3QAAAABJRU5ErkJggg==)
where and x are real numbers is
Maths-General
Maths-
If
and
then ![open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared](data:image/png;base64,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)
If
and
then ![open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared](data:image/png;base64,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)
Maths-General
Maths-
The maximum value of
under the restrictions
and
is
The maximum value of
under the restrictions
and
is
Maths-General
Maths-
Let
be two sets. Then
Let
be two sets. Then
Maths-General
Maths-
If
then
is
If
then
is
Maths-General
physics-
A long cylindrical tube carries a highly polished piston and has a side opening. A tuning fork of frequency n is sounded at the sound heard by the listener charges if the piston is moves in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of n is
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)
A long cylindrical tube carries a highly polished piston and has a side opening. A tuning fork of frequency n is sounded at the sound heard by the listener charges if the piston is moves in or out. At a particular position of the piston is moved through a distance of 9 cm, the intensity of sound becomes minimum, if the speed of sound is 360 m/s, the value of n is
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)
physics-General
Maths-
The value of
The value of
Maths-General