 Okay guys, we'll solve a few questions out here. So what I'll do is that I have downloaded a few sample question papers. Okay, so I'm uploading a paper out here and you will have to solve the questions one by one. So as soon as you have solved with the questions, you let me know. Okay, so here are the few instructions that you need to read and once you are done with reading the instructions, so I am reading it for you. There are 30 questions which are divided into four sections A, B, C, D. Section A contains of six questions of one marks each. Similarly, section B, six questions of two marks each. Section C is the largest section which has questions of three marks and section B has eight questions of four marks each. So internal choices are there, so there you can select. Calculators are not allowed. Now start doing the question. So I'm starting with the first six questions. I'll revise first. I'll do the first six questions after suppose how much time you will take for five, six questions. Suppose eight minutes or ten minutes. So solve these questions, six questions. After five minutes, I'll scroll down so that you can write down the first so that you can see the sixth question. You have to solve first six questions. You can see five questions on your screen after two, three minutes. I'll scroll down so that you can see sixth question. Okay, okay, next first six questions. I'm going to solve. So what's the answer of first six questions? Can you tell me the answer of first question? Russel, I'll go to the next set, but I need to solve the first six questions. At least I need to give you answers. Param, your answer is correct, non-terminating, but it is non-terminating, repeating or non-terminating, non-repeating. So seven by 75, no other option. Again, it does not go. So zero. Then how many times it will go? It will go nine times. This will be 675. This will be 25. So it will go three times. And this will be 225. And this will again be 25. This will again go three times. So this will be 225. This will again be 25. So we see that the answer would be 0.093 bar. So what would be the answer 0.093 bar? So it is non-terminating, but it is repeating. Now question number two, question number two tells me that find the values of k for which quadratic equation has equal roots. So if quadratic equation has equal roots, it means that b square is equal to 4ac. And b square, because d is equal to zero, so b square minus 4ac is equal to zero. So b over here is minus k root 3 whole square, and that is equal to 4. A is 3 here, and c is 4 here. So if I take square of it, it becomes k square into 3, and that becomes 4 into 3 into 4. This 331, k square is equal to 16. k can be equal to plus minus 4. Question number three tells me that find the 11th term of the series. So I can see that a is equal to 23, b is equal to, a is equal to 27, I'm sorry, b is equal to 23. So I can see that b is equal to minus 4 here. So t11 would be equal to a n minus 1 dt. So it becomes 27, n is 11 minus 1 is 10 into minus 4. So that comes out to be 27 minus 4t, and that is equal to minus 13. Question number four, find the coordinates of the point on y-axis, which is nearest to the point. So this is a point minus 25, and I have to find out a point on y-axis, which is nearest to the point. So nearest to the point means from this point on y-axis, I draw a perpendicular. So this is the point on y-axis, which is nearest to this point. So what is the y value here? 5. What is the x value on y-axis? That is 0, 5. So that's the answer. Now question five I need to do. Question five tells me that, so st by rq, one second, st by rq, st and rq are parallel. Okay, there is no relationship has been given between st and rq. So st and rq are parallel. It means that triangle, I can prove that triangle pts and triangle pqr are similar. And if they are similar, then what I have to find out? How much PS has been given? PS has been given as 3, and SR has been given as 4. So relationship would be PS divided by PR. And PR is nothing but PS divided by PS plus SR. PR is PS plus SR. So this is 3 and this is 3 plus 4. So how much it comes? This comes out to be 3 divided by 7. So if corresponding sides are in ratio 3 by 7, area would be in ratio of square of the ratio of corresponding side. So the ratio of PST divided by area pqr would be 3 by 7 square. So which is equal to 9 by 49. Now question number 6 tells me that cos A is equal to 2 by 5. So this is base and this is hypotenuse. For tan A, I need to find out perpendicular divided by base. So perpendicular would be hypotenuse square minus base square. So hypotenuse is 5 square 25, base square is 2 square 4. So perpendicular comes out to be under root 21. So tan A is under root 21 divided by base is 2. So what I have to find out? I have to find out 4 plus 4 tan square A. So this is nothing but 4 plus 4 into tan square would be 21 divided by 2 square 4. This 4 and 4 gone. 4 plus 21 comes out to be 25. So that's your answer. So I have solved these six questions. Now let me move to another section. So this is, so question 7 to 12 is, oh Russel, I forgot it says 11 term from the last. So okay, you guys can solve that question. I didn't see that it is asking 11 term from the last and I'm looking at your message now. So I will solve that question. Let's solve second part. I'll solve question number 3 again. Yes, answer is minus 25. There is no doubt the answer is minus 25. I didn't see it that it is asking 11 term from the last. I'll solve that question again. Yes, I know from the last it has been told. I will do that question. I didn't see it. The answer is minus 25. Okay, should I solve these questions or should I give you only answers? Okay, fine. So I'll give you only answers. Okay, I'll solve question number 8. Question 7 answer is you don't need to have that was to prove question 8. I'll solve anyway those who have done the answer is a 16 is equal to 65 and question 9 k is equal to 1 question 10 question 11 question 10. I'll just see guys question 11. First answer is P of question 9. I'll do guys just wait. I'll do question 8 question 9. I'm just giving answers. So P is 113 and for question second part multiple of 7 it will be 6 divided by 113 and question 12 x is equal to 3. So these are the answers. Now let me solve question 8 and 9 for you guys. So question 8 tells that s n is equal to 2 n square plus 3 n. So s 1 is equal to 2 plus 3 5 and what do you mean by s 1 sum of the first term sum of the first term means what it is only the first term so a 1 is equal to 5 and s 2 is equal to 2 into 2 square plus 3 into 2. So how much it comes 4 into 2 8 8 plus 6 is 14. So D comes out to be s 2 minus s 1 that is equal to 9. So 16 term would be a plus 15 D. So how much it would be just give me a moment. Have I done something wrong? So okay yes D would not be 14 if s 2 is 14 it means that a 1 plus a 2 is equal to 14. So a 1 is equal to 5 so a 2 would be 9 not D would be 9. So a 2 is 14 so a 2 is equal to 9. So D comes out to be how much D comes out to be 4. So how much it would be this would be 5 plus 15 into 4. So that comes out to be 5 plus 60 is equal to 65. So this is how you have to do question 8. Now what is there in question 9? The question 9 is telling me that let me read the question first of all. Yes question 9 is find the value of k for which you have infinitely many solutions for infinitely many solutions if I have equations like this a 1 by a 2 is equal to for this linear equation a 1 x plus b 1 y is equal to c 1 and a 2 x plus b 2 y is equal to c 2. So a 1 by a 2 is equal to b 1 by b 2 equal to c 1 by c 2. So I have k x plus y is equal to k square and I have x plus k y is equal to 1. So k by 1 is equal to 1 by k and that is equal to k square divided by 1. So you take anywhere if you take these two equations also k square comes out to be 1. So if k square is 1 k would be how much k can be anything k can be how much guys k can be either plus or minus 1. So I will check if I keep minus here. So you cannot take negative value here because if you take k negative so this becomes minus 1 this becomes minus 1 but this remains positive because this is k squares. So I cannot take this negative value. So I will have to take positive value so k is equal to 1. So this is what question 9 is. Any other doubt guys from question number 7 to 12 just tell me. Otherwise I will move to question number 13 to 20. Sorry 13 to 22 I think 13 to 22. Okay nobody is asking me anything. So I am moving to question number 13 now. Earth question I have already explained. Who is asking earth question? I have already explained earth question. Earth question was like this. Where were you when I was solving earth question? Earth question I once again solve s n is equal to 2 n square plus 3 n. s n means sum of first n terms. So s 1 is equal to 2 into 1 square plus 3 into 1 that is equal to 5. So s 1 in sum of first term means it is only the first term. s 2 is equal to 2 into 2 square plus 3 into 2. So that is 8 plus 2 that is equal to 4 8 plus 6 sorry that is equal to 14. So s 2 is what sum of first 2 terms a 1 plus a 2 is equal to 14. So this is 5 plus a 2 is equal to 14 a 2 comes out to be 14 minus 5 that is equal to 9. So a 1 is 5 and a 2 is 9. So d would be equal to a 2 minus a 1 which is 9 minus 5 that is equal to 4. So a 16 would be equal to a 1 plus so 5 plus 16 minus 1 is 15 5 plus 15 into 4. So how much it comes? 65 that's the answer. Which one is seventh question? Seventh question seventh question is like this p is equal to a cube a p is equal to a square b cube q is equal to a cube b. So LCM would be highest power of a in both of them that is a cube and HCF would be how much sorry LCM and highest power of b in both of them so that is b cube HCF would be lowest power of a in both of them that is a square and lowest power of b in both of them that is b. So LCM into HCF would be equal to a cube b cube into a square b that is equal to a to the power 5 into b to the power 4 and when I multiply p into q so I have a square b cube into a cube b so that is also equal to 2 plus 3 5 and 3 plus 1 4 so both are equal. Should I scroll down? Just let me know when I have to scroll down. Should I scroll down? Where are you guys? Should I scroll down? I solved may be the seventh question already. Look at here those who have not done seventh question time and again where were you when when I was solving? Here is a seventh question on the screen question number 17 is either this question this question. Guys don't discuss among yourself I'll solve the questions yeah I'll explain 17th question okay so guys try to understand and I'm going to solve questions now and then we will solve rest of the questions so this is how you need to solve question number 13 all of you look at the screen now don't ask me later sir how did you do question number 13 n n plus 2 n plus 3 there are three kind of multiples of three or there can be three types of number only either the number would be 3k divisible by 3 or it would be one more than divisible by 3 or it it can be two more than divisible by 3 so first I assume n is equal to 3k if n is equal to 3k n would be divisible by 3 so one number is divisible by 3 no need to check n plus 2 and n plus 4 if n is 3k plus 1 n is not divisible but n plus 2 would be 3k plus 1 plus 2 is equal to 3k plus 3 3k plus 3 is divisible by 3 now n is equal to 3k n plus 2 is equal to 3k plus 2 n plus 4 is equal to 3k no sorry n is 3k plus 2 so n plus 2 is 3k plus 4 and n plus 4 is 3k plus 6 now this is not divisible by 3 this is not divisible by 3 this is divisible by 3 so in all three cases I have proved that one number is divisible by 3 one second let me solve this these questions I am coming to the next set now question 14 it's so question 14 let me do it so x minus root 5 by 3 and x plus root 5 by 3 are my factors I will multiply these factors I will get x square minus 5 by 3 which is nothing but 1 by 3 3x square minus 5 it means that 3x square minus 5 is factor of 3x to the power 4 plus 6x cube minus 2x square minus 10x minus 5 I will divide this by 3x square minus 5 so this will go x square times so this becomes 3x to the power 4 minus 5x square so I get minus and plus here so I have 6x cube plus 3x square so this will give me plus 2x here so I will get 6x cube minus sorry here I write minus 10x and minus 10x will come from above also so you see here if I subtract this this is gone and this minus minus becomes positive so this is gone and this is gone I have 3x square minus 5 here which will go one time so if I take it one time this will become 3x square minus 5 so this is gone I have x square plus 2x plus 1 as other two factors so x square plus 2x plus 1 is 0 it means that x plus 1 square is 0 so x is equal to minus 1 this is it now let me go to question 15 7 times a two digit number is equal to either I don't think how many of you have not done question 15 if you have not done then only I'll solve otherwise I'll not solve these are easy questions if anyone wants me to solve question 15 just post it in the nobody wants me to solve question 15 okay no issues I'm going to question 16 now then so question 16 what is it telling first part is telling me that in what ratio does x axis divide the line segment joining the points minus 4 6 so I am taking the two points minus 4 6 points are minus 4 minus 6 and this has minus 1 minus 7 and suppose x axis this divides it in 1 is to k ratio so if it divides it into 1 is to k ratio how much it becomes it it is so this is x 1 this is x 2 so this is x 1 into k so minus 4 k x 2 into minus 1 so this is minus 1 divided by 1 plus k and this is y 1 and this is y 2 so y 1 into k minus 6 k plus y 2 into 1 so plus 7 divided by k plus 1 now this has to be x axis coordinate so here here y is equal to 0 so which is the y axis this is my y axis this has to be 0 so for this to be 0 minus 6 k plus 7 divided by k plus 1 is equal to 0 so minus 6 k plus 7 is equal to 0 or k comes out to be 7 divided by 6 so so it would be 1 is to k ratio so 1 divided by 1 is to 7 by 6 which is nothing but 6 is to 7 ratio so that is how you have to solve this question now question 16 i am going to the next part so question 16 the next part is let me read the question for you so question tells that the points a b c d form a parallelogram find the length of the altitude of the parallelogram on base a b so how do you solve this question now take any three points take point a b c and find out the area of the triangle how do you find out area of the triangle with point a b c so to do this question what i am telling you is that these are a b c d listen to just look at this question this question is very important so this is a b c d so i am just taking points a points b and point c so i know the length of a b length of a b would be equal to so let me find out length of a b length of a b would be equal to 7 minus 4 and 2 plus 2 so how much it would be 7 minus 4 square plus 2 minus minus 2 is 2 plus 2 square so this is 3 square plus 4 square which is equal to 5 so this a b is 5 now i need to find out area of triangle a b c so area of triangle a b c is equal to 1 by 2 now suppose i am writing it here let me write a is 4 minus 2 b is 7 2 and c is how much c is 0 9 great so it would be x 1 y 2 minus y 3 plus x 2 y 3 minus y 1 plus x 3 y 1 minus y 2 and this will be wrong like this so 1 by 2 x 1 i am assuming to be 4 so this comes out to be 4 so y 2 would be 2 and y 3 would be 9 so this becomes 4 into 2 minus 9 x 2 i am assuming to be 7 so y 3 would be 9 y 2 y 1 would be minus 2 so plus 2 here minus minus plus and x 3 i am assuming to be 0 so i i don't need to take anything this this this is 0 so i'll close it here so how much it comes it comes out to be 1 by 2 4 into 4 into minus 7 so how much that is minus 28 plus 9 into 7 is 11 into 7 is 77 so how much it would be that would be 49 divided by 2 so area of triangle is 1 by 2 into altitude into a b is equal to 49 divided by 2 now all this 2 and 2 gone altitude would be equal to 49 divided by a b 49 divided by a b a b i have calculated as 5 so a altitude will come out to be 9.8 so this is how you have to do it now let me go to question 17 so question 17 tells me that question 17 first part i am solving so p q r and then i have a per angle 1 is equal to angle 2 and then i have s t m r n m so i have something like this this is p this is q this is r and on the same line this was m perhaps yes m and this is n and there was a line here and this was connected to this and this was connected to this this is this is s t so this is angle 1 this is angle 2 and it has been given that angle 1 is equal to angle 2 now what i have to prove i have to prove that pts so where is pts this is pts is similar to so this is this is sign of congruency so this is sign of congruency and this is sign of similarity congruency means angles are same the ratio of corresponding angle is 1 everything over here here i have to prove p r q okay great so pts pr q so i have to prove that pts is similar to pr q and i have been given that q n s q n s q is congruent to n s q is congruent to m tr okay so how do i prove it to prove that they are similar i have to prove that if i have to prove that this line is parallel to this line if i am able to prove that this this angle is 1 and this angle is 2 because p is common angle i will be able to prove that they two are similar so to prove that this angle are true i have to prove that these two lines are parallel to each other now how do i do it so now try to understand that first you see that this angle q q is equal to this angle r how so look at here here it has been told that s is congruent with 3 so it means that angle s q r is equal to is equal to angle m tr so these sorry not this angle this angle and this angle is equal now try to understand now angle p plus angle 1 plus angle 2 is equal to 180 degree and angle p plus angle q plus angle r is also equal to 180 degree so what i can do is that i can say that just just look at here guys i can say that angle p plus angle 1 plus angle 2 is equal to angle p plus angle q plus angle r why because they both are equal to 180 degree so angle p angle p is gone angle 1 plus angle 2 is equal to angle q plus angle r and try to understand you guys as angle 1 is equal to angle 2 so i can write it that you look at here once again so i can write it that two times angle 1 now you see here what's what's the criteria i am getting here so this angle q are you guys listening to me is my voice clear i am not getting any comment nothing i am just solving the question so i thought i'll ask in between that do guys listen to me i was listening to me or not okay fine so after this what i need to do is that i need to find find once again and by cpcte can i prove that angle q is equal to angle r so you see here this angle is opposite to angle r and this side this side is opposite to angle r and this side is opposite to angle q and both are same sides so by cpcte i can prove that angle q is equal to r so i can write down that 2q is equal to 2 times 1 so angle q is equal to angle 1 and similarly angle r would be equal to angle 2 so i have proved that angle q is 1 and angle r is 2 p is common so i have angle angle angle similarity or so that's how you have to prove 17th question now let me go to oh just with the the this the method that you are talking is correct i am just i was just trying to prove angle angle angle similarity now let me go to b part of this so b part is telling me that in an equilateral triangle abc d is the point on bc such that bd is 1 by 3 bc prove that 9 ad square is equal to 7 ab square so how do you prove it so it has been given to me as abcd and bd is equal to 1 by 3 bc so i have to prove that 9 ad square is equal to 7 ab square so what i do is that i draw a perpendicular from here now this perpendicular would be perpendicular bisector because in in any equilateral triangle if from vertex i draw a perpendicular to the opposite side that perpendicular is perpendicular bisector so what happens over here is i can find out the length of of of this perpendicular bisector in terms of ab or whatever you are saying so whatever so if this is ap so what happens is ad square is equal to ap square plus pd square and this pd square can be so what is pd pd is equal to pb minus bd so pb is bc by 2 half of it and bd is 1 by 3 bc so bc divided by 3 so how much it comes it comes out to be bc by 6 so pd is equal to bc by 6 and and ap square would be equal to i am converted converting it ap square plus bp square is equal to ab square so ap square is equal to ab square minus bc square by 4 so that is how it is so you see here this will be equal to ad square ap square is ab square minus bc square by 4 and here i am getting pd square is plus bc square by 36 so how much i get i get ad square is equal to ab square and this becomes 9 minus how much oh and it is ab is equal to bc also because this is equilateral triangle so this is this this becomes ab square minus ab square by 4 plus ab square by 36 so how much it comes 36 plus 37 37 minus 37 minus 36 plus 37 this is ab square so somewhere i'm going calculation mistake i'm doing guys just give me a moment i'll do it once again i've done some calculation mistake so let me find it out for you once again ap square is equal to ap square plus a dp square is equal to ad square and ap dp can be written as dp is equal to dp minus bd and this is dp and ap square can be written as so let me ap square can be written as ap square plus dp square is equal to ab square so ap square dp is how much ab by 2 so ab square by 4 is equal to ab square so ap square is equal to 3 ab square by 4 so this is nothing but 3 ab square by 4 and dp square is equal to how much bp bp is how much bc by 2 which is also ab by 2 and bd is how much 1 by 3 bc so 1 by 3 bc i can also write 1 by 3 ab so ab by 2 so this is comes out to be ab by 6 so dp square is nothing but ab square divided by 36 so this will be how much this will be 36 and this was 9 times of 27 ab square plus ab square is equal to ad square so now i got 28 ab square divided by 36 is equal to ad square so this is 7 this is 9 so i can write here 7 ab square is equal to 9 ad square as simple as that so now we will solve other questions solve from question number 18 solve question number 18 then you have question number 19 okay question number 18 i have put this is question number 19 this is question number 20 so first i will show you question number 19 if you want to write write it down write it down then i go to question number 20 question number 20 is on two pages okay let me go to question number 20 now write down this part of question number 20 on your copy okay this is question number 20 then question number 21 is also on the screen now okay i will solve question number 20 any other request for any other question from question number 18 to 22 list down the questions that i need to solve okay question 19 part 2 i'm giving you one minute to tell you which question i need to solve then i'll move ahead start solving the question 19 part 2 and question number 20 okay so i'm solving question number 19 now and i'm solving question number so question number 19 part 2 i'm solving sin theta plus cos theta is equal to root 2 so let's take square of both the sides sin square theta plus cos square theta plus 2 sin theta cos theta is equal to 2 and this is nothing but one so one plus 2 sin theta cos theta is equal to 2 so 2 sin theta cos theta is equal to 1 sin theta cos theta is equal to 1 by 2 now i know that sin square theta plus cos square theta is equal to 1 so this is equation 1 this is equation 2 i'll divide equation 2 by equation 1 so i get sin square theta plus cos square theta divided by sin theta cos theta is equal to 1 divided by 1 by 2 which is equal to 2 so sin square theta divided by sin theta cos theta is nothing but sin theta sin theta gets cancelled out that comes out to be sin theta by cos theta which is tan theta and here cos square theta cancelled out cos theta so you have cos theta divided by sin theta that is equal to cot theta and that is equal to 2 so that's how you have to solve question number 19 part b okay question number 20 i'm solving now and question number 20 tells me that what is the ab pc is a quadrant of the circle of radius 14 centimeter ab pc okay is a quadrant of a circle of radius and semicircle is drawn with a bc as diameter okay find the area of shaded region so this is a this is b this is c and this is how this quadrant of the circle and this is where another semicircle is so to find out area of this region what i need to do is to find out the area of this semicircle to find out area of the semicircle this is p so and suppose this is and q so area of bqc is equal to so not bqc this is bqc so this is suppose this is d so that would be area of semicircle bdcq minus area of area from here to here how do you find out area from here to here to find out area so that is minus area of quadrant of a circle circle is what ab pc minus area of triangle because this area would be fine found out only if area of triangle abc so area of triangle abc so first i am i am i am looking at the values given to me the values which have been given to me is radius is 14 centimeters so if radius is 14 centimeter both ac and ab would be 14 centimeters so what do i get i get or let me find out suppose this is radius r this is radius r so this is root 2 r so area of semicircle would be 1 by 2 pi root 2 r square minus area of quadrant so what would be 1 by 4 pi r square minus area of this triangle so 1 by 2 into r square so how much i am getting i am getting pi by 2 into 2 r square minus pi by 4 r square plus 1 by 2 plus r square by 2 so this 2 and 2 gone so i have 3 pi by 4 r square pi s and plus r square by 2 now let me put the values so how much i am getting i will get the values as oh guys i have taken root 2 here this will be one second this will be root 2 divided by 2 because that's diameter so this is nothing but root 2 divided by 2 so this is root 2 only so this becomes let me just solve you let me just solve it once again so this becomes 1 by 2 pi r by root 2 square is r square by 2 minus pi r square by 4 plus r square by 2 and this and this is gone so you get only r square by 2 so r is how much r how much r has been given 14 so 196 by 2 that is equal to 98 that's the answer so now let me move to the other second section do you want to continue guys or you want to do it some other time i have uploaded the other section start solving okay fine so if you want to continue continue with the questions we are left out with only eight questions do it quickly yes ask first you solve these questions then if i get time i'll explain that okay fine so let others solve it i'll explain sometimes later there is nothing in this those who have solved it how to prove it i'll i'll tell you how how do you prove it so psi the similarity ratio is side by side area is side square into side square now each side i mean each side i mean if numerator is a and denominator is b so a1 by b1 is is is is similarity so a1 by b1 is 3 by a5 now area of the triangle would be 1 by 2 a1 into a2 divided by 1 by 2 b1 into b2 now it becomes a1 square a1 and a2 a1 a1 by b1 and a2 by b2 are same so a1 into a1 into a2 divided by b1 into b2 can be taken as a1 square divided by b1 square that is how it is proved okay fine okay now let me know if you have any doubt otherwise okay great yes rushill i have uploaded question 30 okay guys if any health is needed you just let me know otherwise i'll bind up the session because i have to start a class again at seven okay finally as i'm wrapping up the session now so