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00:00 - 00:59 | hello everyone letter start this question this questions is both roots of quadratic quadratic equation X square + sin theta + cos theta x x + 3 upon 8 = 0 a positive and distinct then complete set of value of theta in 2022 closed interval is first option pivot 12255 12 open interval second option is open interval 13 12 to 17 512 third option is 7512 21112 and fourth option is 19122 23235 12 again and open interval first let's look at the condition that |

01:00 - 01:59 | the roots are distinct which implies that discriminant as the roots are distinct discriminant both roots are positive that it that is both roots are real and an equal to different should be strictly greater than zero different is sin theta + cos theta whole square that is b square minus 4 x 8 that is one x see that is 3 upon 8 it should be greater than zero at this finally implies that sin theta + cos theta whole square that is one not one sorry sin square theta + cos square theta + 2 sin theta cos theta minus |

02:00 - 02:59 | this becomes 3 upon 2 is greater than zero sin square theta + cos square theta is 1 so just replacing that this becomes 2 sin theta cos theta is basically greater than half this implies that sin theta and Cos theta have same sign why is it that sin theta and Cos theta sem sem sem sem sign because if sin theta and Cos theta different sin theta sin theta is positive and Cos theta is negative in sin theta cos theta is negative similarly if sin theta is negative and Cos theta is positive then again sin theta cos theta is negative so that sin theta and Cos theta both are greater than zero or sin theta cos theta both less than zero |

03:00 - 03:59 | then only can this equation hold so let's just put this year because this is very important now from here we can say that sin 2 theta is basically greater than half now also the sum of roots se the roots are Alpha 1 and alpha 2 then the sum of roots Alpha 1 + Alpha to that is minus b y that is minus sin theta + cos theta as per our question now to the roots again are positive both roots are positive positive positive is again positive so this should be greater than zero so this would imply that sin theta + cos theta is less than zero just multiplying with -1 this inequality would change its sign because -1 is negative for sin theta + cos theta is negative and |

04:00 - 04:59 | sin theta and Cos theta have same sign we have just found it happened that sin theta and Cos theta both positive because if sin theta and Cos theta both positive then sin theta + cos ringtone positive because positive plus positive is positive so sin theta and Cos theta from our conclusion is that sin theta is also negative and Cos theta is also negative because both have same sign so we can I want we can have only two options either sin theta cos theta both are positive aur both of them and negative if both of them were positive then this inequality would be a problem so the second choice this and didn't positive and negative this implies as theta sin 0 to pi by sorry 0225 this implies that theatre belongs to the third quadrant that is pi to 3 pi by 2 hair does data belong |

05:00 - 05:59 | this would imply that 2 theta would belong to 25 to 35 and in that we have sin2theta greater than half because sign is an increasing function also in 25 to 35 because in 25 to 35 sign Switzerland 2.2 to 5 + 2 Sin is an increasing function then to pi + 522235 it is an it is in a decreasing function so what we will do is make 2 theta to belong to 25 + 5 by 6 because Sin of 5S sixes half 35 - 56 this is where to theatre muscular so this would imply that theatre belongs to 13 pi by 12 |

06:00 - 06:59 | to 17 pi by 12 that is the correct answer for theatre so this is the second option is the correct one this completes our question thank you |