Maths-
General
Easy
Question
If
then
![a b less or equal than 0](data:image/png;base64,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)
![a b greater or equal than 0](data:image/png;base64,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)
![a plus b equals 0](data:image/png;base64,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)
0
Hint:
In the question we will simplify the LHS using trigonometric identities.
The correct answer is: ![a b less or equal than 0](data:image/png;base64,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)
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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,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)
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,iVBORw0KGgoAAAANSUhEUgAAAHcAAAAVCAYAAAB8BXHbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAAopJREFUeNrtWU8oREEcnvawOUg5OMhBSZKDlIMkuThJcnWUkiQHFwdJctkcHNycJElJTpKSJG1ykRwclKMkN0mSWr+pTz2z8+fN23m7817vq692377Zt998M9/8ZpaxDElHCfwmFontWZekDzniLPE264r04ivrgnTG3hDxMsv89MVeM/q/TXUD//GFhIov4PlxaPMh9nT6OojHMFiKbuJ1wjO/CB1xaXMeew70cUNPiA2mhj0JET+KpUBEL/HKQtsnEsdJ7DkAn4HLxDvF5zJ93NhO3Zf24IfHmvmOUE98UJjLIL43hDZeG9xbdLo29hxhlzit0SbTV5LwH9aJcxWOuCjiSxGetUWc0rSdF9ZWlbZx4n7IQWuMvRgqdBZSnxGnxEFFkbRKfEU1zOOi1aF4W3MHiOeGtv2Be3TaVohL0PeCWuFCMkCNsVdlc0V9RrwT85LrfDZuEhth9AFGvCvxNubmEaOthrZ56DFp41qeiIsBfRuS2WyMvSqbK+oz4kdybQIdEMQOccSheJt7C0K86tp+G7RxPBK7hGsNth1XA3NFfZHMvUAEBHEpieUoBx46qrZpRYsOMJmbQ6Usoq5Cc6Pqi9VcWXSJ2wT++qPKI/QPN6z85KuSWO7D4K14PUtCLJ+hWAniTVLM3NbIXJvZIBok08aXnG3Jcxawz/TZXOsBKNsuPCOm/sBFH9bIXJu2c9Cj08aLxElFwdbsubmiPiNkG/017ClzWGf3UBn7bm6YQ4wjFFTDeN+Ca5PMD9gcYoTCNSs/t1xCxbyMWfwq2Qr51AGq40dRG9cxBoN/sH8f90STbslR6TMizOF6E/Mbcf9x4Ku+UJhhbv4WqwX4OjSdUm1h9JXhF4t14ft4W9OtAAABJXRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZmVuY2VkIHNlcGFyYXRvcnM9InwiPjxtcm93Pjxtc3VwPjxtaT5hPC9taT48bW4+MjwvbW4+PC9tc3VwPjxtbz4rPC9tbz48bW4+NDwvbW4+PC9tcm93PjwvbWZlbmNlZD48bXN1cD48bWZlbmNlZCBzZXBhcmF0b3JzPSJ8Ij48bXJvdz48bXN1cD48bWk+YjwvbWk+PG1uPjI8L21uPjwvbXN1cD48bW8+JiN4MjIxMjs8L21vPjxtbj4xPC9tbj48L21yb3c+PC9tZmVuY2VkPjxtbj4yPC9tbj48L21zdXA+PC9tYXRoPuxgY6IAAAAASUVORK5CYII=)
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
![](data:image/png;base64,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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