 Hello, good afternoon getting from sender academy. Welcome to the J main problem solving session So I would request the students who have joined in please type in your names in the chat box So that I know who all are attending the session Can all of you see my screen? Can all of you hear me clearly? Okay, great. All right, so we are starting from the first question and I think most of you have started answering it Great, so which of the following is not true Which of the following is not true For any of the two statements P and Q of course Statement one is going to be correct Statement two is not correct because it is Contradicting de Morgan's law. It is contradicting the de Morgan's law. Okay So we all know de Morgan's law negation of P or Q is Equivalent to saying negation of P and negation of Q. So this symbol should not be the symbol of This should be the symbol of and not for the symbol of all. Okay, so option number two becomes correct in this case All right, so long. Let's move on to the next one Let R be on the real line consider the following subsets of the plane of R cross R so S is a Set of ordered pairs X comma Y Such that Y is GIF of X X belonging to zero to four and T is all ordered pairs X comma Y So is that X minus Y is an integer Which of the following is true? Please give me a response after stating the question number so that I know which question number You're trying to answer. Good afternoon Rohan Aditya Sanjana Okay, again, I'll wait for three people to respond at least Okay, so all of you are saying option number one. Okay, let's check it out So the options are saying T is equivalent solution on R, but S is not of course, you can see if you take an element like 1.2 comma 1 it clearly belongs to S But the moment you flip the position that is 1 comma 1.2. This does not belong to us Okay, the reason being 1 is the GIF of 1.2, but we cannot say 1.2 is the GIF of 1 so this relation is not correct So the symmetric relation is violated by the set S and therefore S cannot be an equivalent solution So the statement which is true is going to be having the fact that S is not an equivalent solution Now X minus Y is an integer will always be an equivalent solution because it will be reflexive symmetric Transitive and hence equivalent so 1 is correct because it says T is an equivalent solution, but on R but not S So the first one to answer this was Sondarya. Well done. Good. Let's move on to the next one The complete solution set of mod X minus sine X less than 1 in the interval 0 to pi Okay Alright, so let's discuss this So we know that in the interval 0 to pi your mod of sine X is going to behave as sine X itself right and From pi onwards you can say pi to 2 pi it will behave as negative of sine X right So this is the definition of mod of sine X. So if I take this interval Then in the interval 0 to pi you would get something like this 0 Less than 1 0 less than 1 is obviously true That means all the values in the interval 0 to pi would be your answer. So this is an accepted Region so basically in 0 to pi all values of X are going to satisfy this given inequality Now the moment I am between pi to 2 pi so in the interval where X belongs to pi to 2 pi This inequality will convert itself to minus of 2 sine X less than 1. That means Sine X is less than negative half Okay, sorry sine X is greater than negative half. Remember the inequality sign is going to flip So when you say sine X is greater than negative half and you're looking from the interval pi to 2 pi So this is your negative half line Right and you want your sine X graph to be above that So it has to be in the zone where This part of the graph is line. So it has to be above the graph of y is equal to negative half Okay, so this point here is 7 pi by 6 undoubtedly this point here is going to be 11 pi by 6 Okay, so I can say in this interval your solution zone will be from pi to 7 pi by 6 Okay union 11 pi by 6 To 2 pi Right now finally what I have to do I have to take the union of 0 to pi with this interval as well Right because there were two cases this was your first case Where you had taken your x between 0 to pi and this was your second case where you had taken your x between pi to 2 pi So if you combine it your answer will turn out to be from 0 to 7 pi by 6 union 11 pi by 6 to 2 pi Which is clearly your option number 3 option number 3 is going to be correct in this case and the first one to answer this again was Why should we this Monday very good? Keep it up by should be Let's move on to the next question, which is your fourth question. So we are going at a good rate It's almost like three minutes Three minutes per question. We are taking almost right Ayantika. Correct Yeah, now again a easy question if a express B y equal to 1 touches the circle x square plus y square is equal to p square Then the locus of a comma B is Okay, I need one more response before I start solving this alright, so basically it's a case where there's a circle of Radius p units having a center at the origin and There is a tangent and the tangent a x plus B y equal to 1 which touches the circle So it's very obvious that the distance of the center from this tangent should be equal to p units Okay, so the distance of the center from the tangent that is going to be mod of 1 by under root of a square plus B square and That should be equal to p. That means a square plus B square is 1 by p square and if you generalize this if you generalize this by replacing your a with x and b with y Because you're finding the locus of a comma B. You are going to end up getting a Circle of radius 1 by p So option number 4 becomes the right option in this case absolutely correct again an easy question So without much waste of time will go on to the next one question number 5 if Integral of x times 1 minus x square to the power of 1 third into x to the power minus 4 dx is a Times x to the power b minus 1 whole to the power c plus k Then which of the following option is correct So this is how you may write it 1 minus x square whole a to the power of 1 third By x to the power 4 dx. This is given to us as a times x to the power b minus 1 whole to the power c Plus a constant. Okay, Shantanu says the third option Let's wait for fewer people to respond Okay, so Roger and thinks differently. He thinks option number 4 is correct Okay, so mostly people are saying option number 4. Let's check this out First of all, I can write this expression as x minus x cube. There's to be power of 1 3rd let me divide this x to the power 4 into x and x cube if I introduce this x within the Cube root symbol it will become x minus x cube by x cube all to the power of 1 3rd times 1 by x cube dx right Now this is clearly Integral of 1 by x square minus 1 Whole to the power one-third into 1 by x cube with respect to dx, right? Now let us do one thing. Let me take 1 by x square minus 1 as t So if we do that negative 2 by x cube dx will become dt That's nothing but dx by x cube will start acting as negative half dt Right, so this entire integral will convert to negative half t to the power one-third dt So that's going to be negative half t to the power of 4 by 3 Okay, divide it by 4 by 3 so it will become a 3 by 4 on top Okay plus c and If you replace our t back, this is what we are going to get Okay, so from this expression. It is very clear that I Can write 1 by x square as x to the power negative 2 as well So it is very clear that my a is negative 3 by 8 My b is negative 2 and my c is 4 by 3 Right, so ideally speaking neither 1 is correct nor 2 is correct nor 3 is correct And let's check a ta plus b plus 3 c plus 1 a ta is going to be minus 3 B is going to be negative 2 3 c is going to be 4 and of course 1 is going to be 1 So when you add a 1 you will definitely get a 0 So option number 4 is the right option in this case and The first one to get this correct was I guess Ramcharan was the first one to get this correct. Okay Well done. Very good. Let's now move on to the sixth question Six questions says the equation of the circle which cuts the circle given to you on your screen and the lines Lines means pair of straight lines Orthogonally is which of the following? So there is a circle which cuts the given circle as well as these pairs of lines Orthogonally, Orthogonally means at 90 degrees So which could be that possible circle? Okay Got the first response So I also backs him up anyone else I Need one more person to answer Now that doesn't mean you'll type anything so that I start solving it so please try it out All right, guys So first of all just a question in general to all of you that if a line is cutting a circle Orthogonally, so let's say this is a line which cuts the circle orthogonally What are the meaning of that? The meaning is very clear and simple that the line must pass through the Center of the circle then only it can cut the circle orthogonally in that case We actually do not use the word orthogonally. We use the word normal Orthogonally is generally used when we are talking about two curves when you're talking about a line in a curve We use the word normal and when we talk about two lines being perpendicular, we use the word perpendicular Anyhow, so they are pair of lines That means there are two lines and both of them are basically cutting the circle orthogonally means both are normal to the circle and That can only happen when the meeting point of these two lines happens to be the center of the circle So whatever circle I'm going to find out that's the circle of the center of that circle must lie at the point of Intersection of these two lines Which are collectively expressed by this expression Correct. I can easily find that out because if you take the first and the third term you get something like this and You take minus 2 common you get x minus 1 equal to 0 That means y minus 2 times x minus 1 equal to 0 is the two lines which comprise or which constitute this pair of straight lines Now this clearly implies that the meeting point has to be 1 comma 2 right Now unfortunately all the options given to us have centers at 1 comma 2 So this is not going to help us to that extent So we need to now move on to check What should be the radius of that circle? now The radius of the circle that we are looking for Should be such that It cuts this circle Okay, so basically The tangent drawn at the point of intersection Must pass through the center right So basically the tan this line or any one of the lines Sorry from the center of the circle. We are going to draw a tangent to this given circle Right because if they're cutting orthogonally Let's say these two circles are cutting orthogonally then the tangent drawn at the point of intersection Right must pass through the center of the other Okay, so this line must pass through the center of the required circle. So this circle is required circle This circle is given circle Okay, okay So this point is 1 comma 2 we have already figured out over here So in order to find the radius I'm just going to find the length of the tangent drawn from 1 comma 2 To the given circle that is x square plus y square plus 2x plus 4 y minus 4 That length would be under root of s1 So your radius r will be under root of s1 which we call as the power of the point with respect to the circle Right so under root of s1 means under root of x1 square y1 square 2x1 plus 4 y1 minus 4 and your x1 y1 is nothing but 1 comma 2 so that would be 1 plus 4 and This will be 2 plus 8 minus 4. That's going to be root of 11 Okay, so whichever of these circles has a root of 11 That would be your desired circle. So if you see The second option that is 1 square plus 2 square plus 6 This actually gives you root of 11 So it is satisfied by the second equation and none of the other equation satisfied So option number two becomes the right option and the first one to answer this correctly was Kushal Okay Awesome great. So hope you have recalled through this problem The concept of length of a tangent and when does two circles meet each other orthogonally So when two circles meet each other orthogonally the tangent drawn at the point of intersection Will pass through their respective centers. So it will pass through C1 and C2 Respectively, okay. So this was the concept which we used All right, let's move on now to the seventh question for the day Seventh question says there are four points in a plane of which five are collinear. So five of them are collinear Barring these fives no three are collinear Then the number of quadrilaterals that can be formed using these points in an easy question should be you should be able to do This within one minute. Yes. Yes any response All right Okay option D So I say the option three. Okay, let's discuss this out now if I have to make a quadrilateral I have to choose all the four points in such a way That not more than two of them should be collinear, right? So I have seven of them which are definitely non collinear Okay, and five of them are collinear I can make a quadrilateral in following ways I can choose all the four points from the non collinear ones, which will be seven C4 in number Correct. So all the seven points Which are non collinear from there I can choose for such points and make a quadrilateral so that will be seven C4 ways or It can happen that I can choose three points from these non collinear ones and One point from these collinear ones that will also lead to a quadrilateral Okay, third case could be you can choose two points from the non collinear ones Correct and again two points from the collinear one. So that's going to be five C2 Okay, so these are the only cases that you can form Seven C4 would be Five into six into a seven by six that's going to be 35 in number Okay, and Seven C3 will be again 35 into five That's going to be 175 in number And seven C2 is going to be 21 into 10 that's going to be 210 in number. So when you add it you get 420 quadrilaterals possible which is clearly your option number four that is correct, right? I thought it was a very simple question and you would answer this within well within one minute Okay So great so first one to answer this was Ramcharan again awesome great Let's not move on to question number eight the first sum of an infinite geometric series is two and Its sum is denoted by s Says that mod of s minus two is less than one by ten Then the two set of the range of the common ratio of the series is Okay Psi son area they have already given the response Great. So let's discuss this we all know that the sum of infinite series for this case will be two by one minus r R being the common ratio. So this is less than one by ten. So two Minus one bar minus two should lie between minus one by ten to one by ten That means two by one minus r should lie between two minus one by ten to two plus one by ten That's going to be nineteen by ten to twenty one by ten Right now. We all know that if this has to be an infinite series Mod of r has to be less than one Correct that mean r has to be a fraction that can be between minus one to one So one minus r will always be a positive term. So let me solve this as two separate inequalities one is This now in this case, I can always write this as one minus r is less than 20 by 19 the reason I could do that is because I know one minus r is a positive term Right that means r should be greater than minus one by 19 Okay, in a similar way, I can stick a two by one minus r less than 21 by 10 Which again means one minus r is greater than 20 by 21 So r is less than r is going to be less than one by 21 and of course r cannot be zero Okay So in this case in a geometric progression, we can never have the common ratio as zero So your r should belong to the interval minus one by 19 to one by 21 Excluding zero r cannot be zero. So obviously option number four is going to be the right option Simple question just was testing your concept of inequalities. That's it So without much ado, we'll move on to the ninth question two equal sides of an isosceles triangle with fixed base six centimeter Are decreasing at a rate of three centimeter per second? How fast is the area decreasing when? Two equal sides are equal to the base Okay, I'm getting a mix of spawns kushals is for Vashnavi says option three Alright, so this is because this so let's say these two equal sides are of length xx each, okay? and Let's say this is my a b and c sides and I drop a perpendicular When I drop a perpendicular, okay, this is going to be three the perpendicular let's say ad is Is going to be under root of x square minus nine So area of this triangle is half base into height That's going to be three times in the root x square minus nine This is area and if you want the area to be How fast is the area decreasing when two equal sides are equal to the base that means I want to find dA by dt The dA by dt will be linked To this expression by two under root of x square minus nine times two x into dx by dt so that's going to be a Three x by under root of x square minus nine into dx by dt So dx by dt is given to us as Negative three because it is saying that it is decreasing at a rate of three centimeters per second So when we put over here, I get dA by dt as three x Now x has to be equal to six So 36 minus nine is going to be five into negative three, okay? Sorry, it will be equal to 36 minus nine is going to be 27 small mistake here So that's going to be a root 27 root 27 is three root three Okay, so root three and this will go off giving you the answer as negative six root three centimeters square per second So it's always saying decrease. So the negative sign is already accounted for so option number four is going to match The required result So the first one to get this correct is I Think same here No, sorry kushal kushal kushal got this correct great Let's move on to the next one question number 10 in a complex way in the point a and b are at z1 and z2 and There's a point P which moves in such a way that mod of z minus z1 is twice mod of z minus z2 Then the maximum area of Pab anyone and response Nobody So these kind of questions are going to you know take a bit of your time if you are not aware of the concept here so guys all of you would have you know come across You know expressions of this nature right more of this is equal to k correct Remember here that if k is equal to one it actually is the locus of z is actually going to be a Line which is exactly bisecting the joint of z1 and z2 so basically if you join this Then z would be lying on the bisector only if k is equal to one right so z can lie anywhere on this line Right because when case one it is just like saying z minus z1 is equal to mod z minus z2 That means the distance of z from z1 has to be equal to the distance of z from z2 That can only be possible if your z is located on the on the Perpendicular bisector of the line segment joining z1 and z2 However, if k is not equal to one then what happens it actually results in a circle which we call as the aplonious circle Okay, what is a circle Aplonious circle is basically a circle In this case which you'll find that Has diameters Let me draw a circle just to you know make it more clear to you. So it would be a circle Which would pass in such a way that the extremities of the circle The extremities of the circle Let's say point a and point b would be in such that a divides a divides z1 and z2 in the ratio of k is to 1 write the same k internally internally and B divides this point B as you can see over here this B would divide the join of z1 and z2 in the ratio of k is to 1 again But this time externally in the ratio of k is to 1 But this time externally this is very important and it saves a lot of time So right now in my given problem I can see that z1 z minus z1 mod by z minus z2 mod is equal to 2 Right. So k divides sorry a divides z1 and z2 internally in the ratio of 2 is to 1. So this is 2 is to 1 So if I know the coordinates of z1, which is going to be 5 comma negative 2 and Z2 coordinates, which is 1 comma 1 Can I find out the coordinates of a? Of course, we can we can use a section formula to get the coordinates of a so that's going to be 2 into 1 1 into 5 that's 7 upon 3 comma 2 into 1 and 1 into negative 2 that's going to be 0 so a is going to be 7 by 3 comma 0 a is going to be 7 by 3 comma 0 Similarly, we can easily find out the coordinates of point B by using our external division formula You just have to take your 2 as negative 2. So it's going to be a negative 2 Okay, so negative 2 into 1 again a minus 2 by negative 2 plus 1 comma and again negative 2 negative 2 I'm sorry negative 2 Plus 5 there would be a 5 over here. My mistake. I wrote a 2. I'm sorry. That's a 5 and you'll have a negative 2 Into minus negative 2 by minus 2 plus 1 Right So if I'm not wrong the B coordinates is going to be negative 3 comma Negative 3 comma 4 is that fine? So if you found A and B what is going to be the What is going to be the? Radius of this circle AB is the diameter. So what is going to be the radius? Let me find some place to write so radius would be nothing but half of the distance between A and B, right? So the distance between A and B would be 7 by 3 plus 3 square and 4 square. That's going to be 9 9 to 7 16 16 by 3 that's 16 square by 3 square plus 16 under root Let me pull out a 16 out as 4 So inside I will have a 16 by 9 plus 1. That's going to be 5 by 3. So 4 into 5 by 3 That's going to be 20 by 3 and half of that. I'm sorry. I forgot the half everywhere So half of that would be 10 by 3 Correct So now you are going to make a triangle with P AB. So P is basically moving So when do you when do you think your triangle will have the maximum area when its height is maximum because base is fixed Correct. So if you are able to draw a triangle like this, that triangle will have a maximum area So its height is going to be 10 by 3 And its base is going to be the distance between these two points So I think the distance between 5 comma negative 2 and 1 comma 1 would be under root of 4 square plus under root of 3 square that's going to be 5 So the answer is going to be half base into height Half base into height that gives you 25 by 3 which is clearly option number 2, which is correct Well, we learned a lot of things over here in case you have not practiced a lot of problems of this type Let me tell you this is a commonly seen expression and it will always result into an a plonious circle Whose diametrical opposite ends would be lying at points A and B where A divides the join of Z1 and Z2 in the ratio of case to 1 internally and B divides the join of Z1 and Z2 in the same ratio case to 1 but externally So the moment I know this concept I could actually find out the points A and B and Then after I could get the you know You can you can get the radius by halving the distance between A and B and For the area of this triangle to be maximum since the base is fixed Its height has to be maximum and height can at the max be the radius of this a plonious circle and therefore the result Right. So good learning in case You were not aware of this So none of you could answer this question. So let's move on Let's move on to the next one, which is question number 11 for the day For the system of equations as you can see on your screen, which of the following is false Which of the following is false Okay, so mostly I don't know why all if you are saying One because if you see if your P is equal to Q is equal to R this thing will become 0 0 0, right? That means it will result into a system of homogeneous equation, right and Homogeneous equations are always known to have a solution Either the trivial solution that we all know as x equal to 0 y equal to 0 or z equal to 0 Or they have infinitely many solutions So the fact that they have solution makes them consistent. So this statement can never be false This is correct statement because If it is consistent, that means it has a solution, right? So even P is equal to Q is equal to R. It will have a solution. So this is a true statement So my question is when when is the statement false? Please do not, you know, get confused here. It's very important all homogeneous system of linear equations are consistent So one definitely cannot be your answer. Okay? So Andrea says option number four. Okay, again a equal to b equal to c is not the decider because if these numbers are all zero Okay, you can have infinitely many Solutions for that. So it can be consistent even in case of fourth But if you look at option number two, if you make your ABC equal and your PQR are distinct Okay, then what will happen? It'll actually result in let's say if you take PQR as distinct So it'll actually let's say I take ABC all equal to one Okay, so at one case, I will get something like this at another case I will get something like this at another case. I'll get something like this Okay Now unless until P minus Q is equal to R minus P is equal to P minus R Only in that case both will represent the same equation as they would represent parallel planes Right? If they represent parallel planes, then they will know there'll be no solution Then in that case it can become Inconsistent so it would be wrong to say that it is consistent for all such cases Right, so I could figure out a case where in this case it would be non consistent or inconsistent whatever you call it So this Option number two. I do not agree with so this is a false statement So two becomes your right answer because they're asking us which of the statement is false So absolutely, I think Ramcharan gave the answer first Okay, so guys be very very clear with your idea of Solution of system of linear equations and what conditions are responsible for consistency and inconsistency Right, so without much I do we'll move on to the next question, which is the 12th question a simple one. I think You would have come across this question before as well Okay, so Ganesh has responded Okay, so mostly people are going for option number two. So let's look into this First of all we can write our dy by dx as xy upon x square plus y square It's a clear-cut case of a homogeneous differential equation So let me take y as vx implies dy by dx will be v plus x dv by dx so when you make these substitutions we get v plus x dv by dx is equal to v square by 1 plus v square so Sorry v by 1 plus v square So let me take the v on the other side. So this we let me bring it to the other side Okay, so this will become negative of v cube by 1 plus v square. This is x dv by dx Right So you'll get 1 plus v square by v cube dv is negative dx by x. Let's integrate both sides Okay, so this one will give me integration of 1 by v cube Integration of 1 by v cube is going to be negative 1 by 2 v square plus integration of 1 by v is going to be ln v minus ln x plus c Okay, that's like saying minus x square by 2 y square Plus ln y by x is equal to Negative ln x plus c now we can use this initial value condition that when x is 1 y is 1 So when x is 1 y is 1 gives me 0 is equal to negative ln x Sorry, this is also going to be 1 So your c is going to be negative half Okay, so putting back over here. I Get minus x square by Nick by 2 y square Plus ln y by x is equal to negative ln x minus half Now they're saying that when x is x naught your y is e So if I substitute my x with x naught and y with e By the way, I can write this as ln y minus ln x that will get cancelled over there Okay, in this case, this is going to be e and Lne is going to be 1 so this I can replace with 1 okay one goes on the other side it becomes Negative 3 by 2 negative negative on 2 to go on so x naught square is equal to 3 square So x naught is going to be a root 3 e in fact Plus minus root 3 so one of the values They should write x naught can be x naught can be plus root 3 E which is option number 2 which is correct, okay? Absolutely correct. The first one to answer this was Rithvik Next is your question number 13 This says if f of x is this function, then which of the following is correct Okay. All right, so let's do this question. So derivative of this I Can simply follow product rule Now if this is 0 first of all we can take e to the power 2x minus 1 common Okay, so that would leave us with 2x minus 2 plus x square Minus 2x plus 1 so e to the power 2 x minus 1 and This will give me x square minus 1 equal to 0 so definitely f dash 1 is going to be 0 So if I put 0 I get 2 points 1 is plus minus 1. I hope I've done this calculation correctly. So let me just quickly Check this up. So x square minus 2x plus 1 so Those off it becomes x square minus 1. Yeah, that's absolutely correct. Oh There's a into 2 as well into 2 as well. So there's a slight error over here So if I have to differentiate e to the power 2x minus 1 a factor of 2 will definitely come inside So that would make it 2x minus 2 plus 2x square minus 4x plus 2 Right, so that would give you actually 2x square minus 2x Yeah, but anyhow When I put this to 0 I get two critical points One is x equal to 1 other is x equal to 0 Okay. Now, let me make the analysis of the nature of these critical points through the wavy curve sine scheme Now this is always positive, right? This term will always be positive So if I draw the wavy curve for 2x square minus x That means I draw the wavy curve for x into x minus 1. This will be positive negative positive right So at 1 you see that the curve is coming from negative slope to positive slope Negative slope to positive slope that means there's a formation of a minima there Okay So of course f dash 1 cannot be less than 0. So this is a wrong option now Zero is a point from where it's coming from positive slope to negative slope correct So it is a point of maxima in a point of maxima the double derivative is going to be negative so 4 is also not correct and Of course minus 1 is not a point of extrema. So 1 is also not correct and Of course 2 is going to be correct because we figured this out Now why I'm doing all these things is because if they ask this in a multiple option correct question You should be aware of all the possibilities But since you know here is one only one option correct. That's why you can save your time the moment you realize that minus 1 sorry x minus 1 is appearing as a factor for this Okay Great, so let's move on to the next question now The mean and variance of n observations x1 x0 till xn are 5 and 0 so x bar is 5 and Variance is 0 Summation of x i square is 400 find the value of n You may expect one statistics question to be asked in your j main However, it's a very easy chapter, but many people ignore this because they don't find it interesting enough But it's very scoring So mathematical reasoning and statistics, please do revisit the formulas at least the basic ones which you have in your ncrd textbook So this is your question number 14. Okay. All right. So in this case, it is a very easy question We all know that the variance formula is 1 by n summation x i square Okay, minus the mean square and this is given to us as 0 Right. So 1 by small n summation x i square is going to be 400 minus 5 square is going to be 0 So n is going to be 400 by 25, which is 16. It's a hardly 10 second question Okay, so please be aware of your formulas or type of formulas There are actually four formulas for finding the variance or the standard deviation one of them is the one which involves the Summation of the square of the data itself, which is this formula in front of your screen next question number 15 for the day Okay anybody else So far only Purvig has responded. Okay. So in this case first of all we'll go for f of 1 by x When you say f of 1 by x your upper limit is just going to be replaced with 1 by x Now if I want if I see the options are all in the limit 1 to x 1 to x So what I'll do is I'll put an obvious substitution Let 1 by x be Let's say you Okay, that means x is 1 by you which means dx is going to be negative 1 by u square du So in that situation your entire expression becomes in of 1 by you 1 plus 1 by u into negative 1 by u squared u Okay, sorry. In fact, I've put t as 1 by you not x t as 1. Okay So if you simplify this expression you get Okay, by the way limits will be 1 by 1 to 1 by 1 by x which is going to be x again So it's 1 by x negative ln u This will become 1 plus u into Minus 1 by u square into you Right, so negative negative will take care of itself. So it's going to be 1 by x ln u by u 1 plus u du now there's nothing in the name you can replace your u back with t So it's going to be ln t by t 1 plus t dt and that's going to match with your option number 2 Okay, however, if you treat this as your multiple options correct question Okay, then you should always check one of the three and four because this cannot be the possibility. Okay So always see the different types of questions linked in the multiple option correct So here you can see they're also inquiring about f of x plus f of 1 by x Okay, so let's try that out. So having figured out that 2 is correct I'm just going to spend some time verifying whether any one of 3 and 4 is correct or not. Okay So I'm just going to clear this up So f of x plus f of 1 by x Is going to be 1 to x ln t by 1 plus t plus ln t by t times 1 plus t dt Okay, if you take ln t by 1 plus t common It becomes 1 plus 1 by t which is 1 plus t by t That cancels off 1 plus t 1 plus t now it becomes a simple integral to do Where we can take ln t as let's say z then 1 by t dt will become dz Okay, so the limit of integration will become 0 to ln of x Right and this becomes This becomes Z dz which is going to be z square by 2. So the answer is going to be ln of x square by 2 So neither option 3 matches neither option 4 matches only option 2 is going to be the right one and hence only I think poor week Who it was the first one to give the correct answer for that? So we'll now move on to question number 16 This is a binomial theorem chapter especially related to the properties of binomial coefficients Okay. Yeah, so we have done this question Many times so if you expand this term you get C0 plus C1 x plus C2 x square C3 x cube C4 x 4 on the way till Cn xn right and If you write the expansion of 1 plus x to the power n we get C0 x to the power n Cn x to the power n minus 1 C2 x to the power n minus 2 C3 x to the power n minus 3 C4 x to the power n minus 4 and so on Till we see it. Okay now we actually looking for the The sum of these series and the series is obtained when you see that these two terms actually multiply So C1 C taught C0 C2 is obtained when these two terms multiply Similarly C1 C3 is obtained when these two terms multiply and Similarly C2 C4 is obtained when these two terms multiply. You can see that in all these cases C1 C2 Sorry C0 C2 C1 C3 C2 C4 etc It forms the coefficient of x to the power of n minus 2 Right it comes along with x to the power n minus 2 So basically what I'm going to do. I'm going to find the coefficient of X to the power n minus 2 In the multiplication of these two terms, which is nothing but 1 plus x to the power 2n Right, which is obvious. It's 2n Cn minus 2. That's nothing but 2n factorial by n minus 2 factorial n plus 2 factorial and We also know that in general if basically this is what is happening CR plus 1 C2 CR plus 3 etc Then in that case my formula becomes 2n factorial by n minus r factorial n plus r factorial So it's very important that you know this so that you can save a lot of time If you know this you can do this within two seconds. So option number two is going to be the right option Okay, great We'll not talk about the next question, which is question number 17. Can you see the screen now? What Yeah, now I can now I can see the screen I guess Are you able to see the screen guys, and now I think some movement has started to happen Wait wait wait, yeah okay, so Sondarya says 17 Option 2, okay. All right, so let's discuss this. So first of all I can say that If x plus y plus z is equal to a that means mod of x plus y plus Z is going to be Mod a mod a is going to be 2 right So if you square this it's going to be 2 squared Correct and if I expand this I'm going to get mod x square plus mod y square mod z square plus twice of x dot y y dot z Z dot x that's going to be four Okay, and since x y z are unit vectors each of them are going to be one one each So I can conclude that x dot y y dot z Z dot x is going to be 3 by 2 Sorry 1 by 2 not 3 by 2. I'm sorry. So it's going to be 1 by 2. Yeah, is that fine? Okay, now if you look at the options the options are all hinting towards How are the dot products of these unit vectors related to one another that means? First one is basically trying to hint whether y dot z is 0 or not second option is just trying to hit whether y dot b is 0 or not Third option sees whether x dot y is positive or not Fourth option is like whether x dot z is negative or not So basically it's all about knowing the values of these x dot y y dot z and z dot x from this expression So we'll try to tackle it one by one. So first of all From this expression if I take the dot product with respect to a Because I have been given information about a dot x and a dot y I would get a dot x plus a dot y Plus a dot z is equal to 2 is equal to 2.2 2 into 2 which is 4 So let me write 4 over here. Okay now Does this give us any information about any one of these required options? So we know that first a dot x a dot x a I can write it as x plus y plus z So a dot x is given to us as 3 by 2. So this is not helping us out in any way So I'm going to just erase this off So this gives you x dot x which is mod of x square x dot y Plus x dot z is equal to 3 by 2 that means x dot y Plus x dot z is going to be 3 by 2 minus 1 which is going to be half So these two expressions. I know which clearly implies Y dot z is going to be 0. Okay That means y and z are perpendicular to each other. That's obvious So option number one cannot be correct because they're asking us Which statement is false? So statement one is true So it's not going to be not going to be the correct option Right in a similar way. I know that a dot y is equal to 7 by 4 So if I take the dot product with y I should get 7 by 4 that means x dot y plus Y dot z is equal to 3 by 4 So we already know x dot y plus y dot z plus z dot x which is equal to half so from here it is Evident that z dot x would be half minus 3 by 4 right half minus 3 by 4 is going to be negative 1 by 4 So the angle between z and x Has to be obtuse therefore option For which says angle between x comma z is obtuse is the right statement and hence this also cannot be my answer Now I haven't yet used this particular expression yet So we all know the vector triple product So when we say a cross b cross c Right, what is the formula for that? a dot c times b vector minus a dot b times c vector Right, so in this case I can write this as negative of z cross x cross y Okay, let me make some space for myself Let me use this space over here Yeah, so this would be if you expand this it's going to be a z dot y Times x Okay minus x dot z times y is equal to negative of b we can say here Now z dot y how do I get the value of z dot y? I think we figured out the value of z dot y earlier That's was zero actually because they were perpendicular So zero and x dot z also was negative one-fourth Okay So this gives us minus b vector as one-fourth of y vector which implies Y vector is parallel to b vector So why cannot be perpendicular to b? So in this case option number two becomes our correct answer because it is false Okay, so we need not look any further so option number Yeah, by the way, we can also find out x dot y the moment. We know y dot z By dot z was zero That dot x was negative of one by four. I Can use this to find my x dot y Okay, so x dot y will now become three by four Which is a positive quantity and hence the angle between x and y should be acute so option number three is also a correct statement So three also cannot be my answer in case we are actually solving a multiple option correct question So option number two is the right option and yes, Sondarya Purvik. You are correct Let's now move on to the next question, which is question number 18 We need to find integral of Ln Under root of this whole thing is under the Ln. Okay, so Ln is operating on the whole thing Okay Wow, so most of you are going with option number three So let's discuss this first of all, let me call this as I and I'm going to apply The King's property on this integral so when I do that I get I is equal to integral from zero to four a lawn of replace x with four minus x so that's going to give you under root of Four minus x minus two. Okay, that's going to be two minus x Square plus four and this is going to be four minus x minus two again going to be two minus x Okay So since this is subjected to the Square you can actually rewrite this as x minus two whole square plus four But here I will have to have a negative sign right Now the obvious step that comes in our mind is let's try to rationalize this term on the in the inside of Ln term So you would get x minus two whole square plus four plus x minus two Okay, and here you'll have x minus two whole square plus four negative x minus two whole square So this and this will get cancelled off Okay So this will leave you with I is equal to zero to four Ln four DX minus zero to four Ln X minus two square plus x minus two DX This is I again So two I is equal to zero to four Ln four that's going to be four Ln four Correct drop the factor of two. So I will be two Ln four, which is again four Ln two So option number three becomes the right option in this case So well done guys all of you have answered correctly Let's move on to the 19th question. Okay, Vaishnavi All right, so let's say the roots are alpha and beta So you want the sum of the squares of the loose roots to be leased so this should be leased Okay, so we already know that alpha square plus beta square you can actually express it as alpha plus beta whole square minus two alpha beta so alpha plus beta whole square will be 4a minus 6 whole square minus two alpha beta is going to be plus Two times four a plus four Okay, you want this expression to be leased Right, so if you want this expression to be leased you can use the derivative of this with respect to a that would be a faster Or you could complete the perfect square with respect to a So when we do that we get two times four a minus six times four plus eight is equal to zero That's let me just check if something everything is fine So two times four a minus x is eight Okay, so in this case I would get four a minus six plus one is equal to zero So four a minus five is zero that means a has to be five by four Okay, and since this is a quadratic expression in a it has to be Opening upwards opening upwards will have a minimum only it cannot be a maxima point So no need to do a double derivative test So option number two becomes the right option in this case I think most of you got this Let's take up the next question, which is the 20th question. So there's a function defined like this Which of the following is false? Again focus on the question. They're asking you which of them is false Okay So these kind of problems are best solved by plotting graphs So for x greater than equal to zero you have two functions one is x and other is x square So I will draw them in different colors. Let's say x is this line okay And x square is like this And when x is less than one But we have to take the minimum of the two minimum of the two means the part of the graph which is the minimum most So we'll have this part And then this part okay Similarly for the right-hand side x square minus one graph would be like this And y is equal to 2x graph will be like this correct Again, we have to take the minimum. Sorry the maximum of the two so maximum of the two would follow this graph And then this graph right so ultimately your figure is going to be Obtained like this right now This point is clearly going to be one and there is a King getting formed over here. So it is not differentiable at x equal to one So The fact that it is differentiable at x equal to one is a false statement. So this is my correct question answer By the way, we'll just quickly run through other options. This cannot be my right option because it is two It's continuous at zero It is not differentiable at exactly three points. That is also correct Because you can see that it is not differentiable here It is not differentiable here And of course at one so three points are there. So this is also a correct statement. So cannot be my answer It is continuous everywhere. That's correct. So this also cannot be my answer. So option number two is only The wrong statement and hence the right option Well done So we'll now move on to the next question question number 21 find the area bounded by y is equal to under root of five minus x square and y equal to mod of x minus one Yes So Rohan has responded. So this is the A graph for the given scenario Note that when they say y is equal to under root of five minus x square We we cannot make the full circle. We only have to take that part of the circle which has got y as positive Okay Mod x minus one is the graph of mod x shifted one units to the right So we can just take a shortcut. We can see that there are two small triangles formed over here, right? So let's say this triangle Let us find out the area of this triangle. Let's say I call it t1 triangle and t2 triangle Okay, so for finding the area of the t1 triangle I would need to know the x and the y coordinates of this point So I can say under root of five minus x square This line is going to be y equal to one minus x. This line is going to be y equal to x minus one So let's try to compare this. So five minus x square Is one plus x square minus two x That's going to give you two x square Uh minus two x minus four equal to zero. That's x square minus x minus two equal to zero This could be factorized as x minus two into x plus one equal to zero And this has to be x equal to negative one point Okay So this is negative one Okay, and this if I solve it Five minus x square is x square minus two x plus one I will get two x square Minus two x minus four equal to zero. That's x square minus x Uh minus two equal to zero same equation we'll get in fact So the other root which is x equal to two is the one which is uh to be used over here So x equal to two over here Now this triangle t1 the area would be half base into height Now the moment you put x as minus of one the y value is going to be two And here if I put x as one the y value is going to be one So area of t1 is half two into two which is two Area of t2 is Half into one into one That's going to be half So these two areas have to be subtracted that means Uh, I need to subtract Two plus half that is five by two from the area within the circle So for area within the circle, I can directly use my integration. So I'll just erase this part integration from minus one to integration of minus one to two of under root five minus x square dx That's going to be x under root x by two five minus x square Plus five by two sine inverse Sine inverse x by root five Is that fine? So if you substitute two Yeah, if you substitute two you will get uh one into one And this will become five by two sine inverse two by root five Correct and if you substitute a minus one you get a minus half Uh, this will again be two Okay, and it'll give you uh minus five by two sine inverse one by root five Okay, so when you Open the brackets you already get one plus One plus and this side will give you again a one Okay And you'll have five by two sine inverse two by root five Plus five by two sine inverse one by root five Is that fine? So let me make some space for myself Yeah So two plus five by two you can use the formula of sine inverse x plus sine inverse y for this That's going to be sine inverse x into Under root of one minus One by five that's going to be four by five So it'll be two by a root five again Okay, plus one by root five Into one by root five Okay, so that's going to be a four by five plus one by five that's going to be one So that's two plus five by two into five by two and you subtract a five by two from here So it's five five by four minus half That's five five Minus two by four that's option number three is going to be correct So a bit of formula that is used over here from inverse signometry That's sine inverse x under root one minus y square plus y under root one minus x square Okay, great. So most of you got this correct. So we'll move on to the next one That's a question from probability question number 22 Please read the question very carefully Okay See, uh, if you read this question a coin is tossed 10 times and the outcomes are listed listed H i be the event that ith outcome is a head And am be the event that the list contains exactly m heads So when you toss a coin 10 times you get exactly m heads in that That is your am Okay H i is basically the probability of getting the ith Outcome as a head Which is very simple. That's going to be half right any outcome head is half because it's a it's a fair coin But what is p a m? So if you want exactly m heads to come that means 10 minus m tail should be there, right? So, of course, you will say 10 cm half to the power of m and half to the power of 10 minus m correct So in both in all the possibilities, this is going to be 10 cm half to the power of 10 So i've done almost 60 of your solution here After that you proceed I think Rohan and Ramchiran have already responded to this All of you please remember that if An event is independent Then the product Let's say a and b are independent Then the product of a into p a into p b is product of is the The probability of a intersection b is the product of probability of a and probability of b So this is when your a and b are independent events Okay, so if I say what is the probability of h i intersection a m That means you want your ith outcome to be head And you want exactly m heads to appear that means If ith outcome is known to be head that means the remaining Nine throws should have given me m minus one heads correct So if the ith outcome is the head That means remaining nine minus remaining nine throws would have given me m minus one heads correct So probability for that to happen would be choosing any m minus one throws from the nine which will give you a head That's going to be this And the ith throw which is going to be a head is going to be half That's nothing but nine c m minus one into half to the power of 10 If you want this to be independent that means this expression that is nine c m minus one into half to the power 10 it should be the product of This and this separately so 10 c m half to the power of 10 into half So I think we can cancel off uh this term from both the sides so nine c Let me make some space So nine c m minus one is half times 10 c m Right, so I can use this formula n cr is n by r into n minus one cr minus one So I can use here 10 by m into nine c m minus one is equal to nine c m minus one. So this and this gets cancelled So I finally get five by m is equal to one. So m has to be five so a in independent of i a five A five will be independent with anything correct, so a five is independent Of any event so this option three will be correct Okay, it is independent of i i can be anything so only dictating factor is the five Which is only present in option three So unfortunately nobody could give the right answer to this Which part purvic you did not understand See i'll repeat this the entire process once again first of all P a m is clear that means if you want to get Exactly m heads out of 10 throws it will be a case of binomial distribution And getting a head on the ith throw would be just half That's also understood So p a m and p h i are clear Okay, now what about p h i intersection a m that means you want the ith throw to be head as well as you want m heads to appear in total out of 10 times that you throw so if you have already account for one head in the ith throw that means The remaining nine throws should have resulted into m minus one heads So this is what I have done the remaining nine throws should have resulted in m minus one heads Not exactly this it is without the half expression Okay This expression and this half is because you want the ith throw to give you A head so again a multiplication of half is required So the first part deals with getting uh m minus one heads in nine throws And the last half denotes getting a head on the ith some ith I can be anywhere in between Okay So this expression multi this expression together should be the product of these two separately So I saw them got the value of m So they would be independent only under the situation when m is five great great So let's now move on to the next question. That's question number 23 Okay, so most of you are going with one. So let's discuss this So if I have to find the inverse of this Okay, uh, we all know that we can use the substitution x equal to cos of let's say theta Okay, but remember the moment you say your x lies in the interval minus one two minus half Right Your theta will actually belong to the interval Two pi by three to pi Right So three theta will belong to the interval two pi to three pi So when I make this substitution over here, I get four cos cube theta minus three cos theta Which clearly is the formula for cos of three theta Which is nothing but saying Now be careful about the intervals of x Okay, let's say I want to take cos inverse on both the sides. I take on both the sides Now be very very careful here because Your three theta is not in the principal value branch. It is in the branch where your x is between two pi to three pi And you would remember the graph of cos inverse cos theta Cos inverse cos theta graph versus theta. It's like this zero two pi three pi So you are actually dealing with this branch You're actually dealing with this branch correct And this branch equation is going to be x minus two pi Or theta minus two pi whatever you can call it. So this is going to give you three theta minus two pi as your answer right cos inverse pi And cos inverse y is three theta three theta means three cos inverse x minus two pi correct, so let me take the Two pi on the other side. I have to make x the subject of the formula. So it becomes two pi plus cos inverse y So cos inverse x will be two pi by three plus one third cos inverse y So you can take cos on both the sides freely So your answer will be this and this is your f inverse y So f inverse x is going to be cos two pi by three plus one third cos inverse x And to the surprise of all of you option number four is going to be correct Okay, so I think after ramchar and people started realizing That option four is correct and not three. So these values should not be ignored when you are dealing with inverse signometric functions They're very very sensitive to the interval of x that you're dealing with Okay, so this question is an eye-opener to most of you. Please do not take inverse signometric functions very lightly They are very sensitive to the interval of x in which you are working with next Question number 24 Okay Undefined, this is by the way minus two Now please change your answer. Oh now everybody started taking domain very seriously In the last question what had happened fine. So let us first make uh dy by dx the subject of the formula over here So it's a x cube minus two x square minus x plus two Is equal to uh f of x times x square minus four Okay, so I can say that f of x is actually x cube minus two x square minus x plus two by x square minus four Okay, now the moment you put a two you would realize that it's actually giving you Zero by zero right so eight minus eight minus two plus two okay, but Two is a part of the domain because I'm just excluding negative two So there has to be a value for two and that value has to be nothing but the limit of this function as x-institute correct And for that I can factorize this Okay Cancel out a factor of x minus two from both the places and the moment we put two we get three by four That's option number two. That is correct. Again, none of you got it correct Surprisingly, is that fine? Did you find your mistakes? Great. So let's move on to the next one That's question number 25 So there is a function from r to r which satisfies this functional equation That is f of x into f of y minus f of x y is equal to x plus y for all x belonging all x and y belonging to real numbers f of one is given to be positive H of x is the product of f of x and f inverse x Then the length of the longest interval in which The function h of sine x plus cos x is strictly decreasing Yes, guys any idea anyone First the full focus should be on getting your function itself find the function And when when you know the function you can always find its inverse And then you would know your h of x Okay, so he has come up with one answer anyone else Okay, so see here first of all in order to figure out what is my f of x Uh, I would like to first find out what is the value of f of one. So let's put x and y both as one in this So f of one square minus f of one is equal to two Right, okay So this gives me a quadratic in f of f f of one. So let's say k is f of one So it gives you quadratic k square minus k minus two equal to zero which is factorizable And this is factorizable as x minus two times x k plus one that's zero So k value is two or k value is negative one But since they have given that f of one is positive this value is rejected. So f of one is going to be two Now, let us do one thing. Uh, let's put x as one sorry x as Sorry, let's put y as one Let's put y as one and let x be as it is so f of one minus f of x Is equal to x plus one So f of one is already two. So two f of x minus f of x is equal to x plus one That means f of x is going to be x plus one Right, this is a very big achievement for us because now we know f of x So f inverse x is also very simple. That's going to be x minus one, right So the product of f of x into f inverse x That's your h of x is going to be x square minus one. This is something which is important Now in this function, if you put your x itself as sine x plus cos x That means this will give you sine x plus cos x whole square minus one And I can actually write that as sine of two x Now they're asking you which is the longest interval in which sine two x is decreasing Okay, so sine two x if I just draw the graph of sine two x Any interval you can draw, okay You can see that sine two x. Let's say I take this as pi by four Okay, this will be pi by two. This will be three pi by four So the longest interval in which you will find it decreasing Will be from let's say I take an example three pi by four to pi by four interval That's going to be of length pi by two itself So option number two is going to be correct in this case. Is that fine? So, yeah, now I think sigh me here Got the right answer Okay Great. So we'll all move on to the next one. That's question number 26 This is a single integer type question If x plus y plus one equal to zero is a common tangent to the parabola y square is equal to four x and x square is equal to lambda y these two curves This is a common tangent Then find the value of lambda So please type in your answer in your chat box Okay size says four Okay, so mostly people are going for option four So let's check this out. So if this is a tangent to this of course, uh, let's see Let's take the fact that We are trying to solve these two simultaneously. So when we solve them simultaneously ideally I should be getting a quadratic in, uh, let's say Let's try to make a quadratic here in x So why I'm going to replace with, uh negative one negative x So I'm getting x square plus lambda x plus lambda equal to zero So this is a common tangent This must always give me the discriminant as zero discriminant means b square minus four ac should be equal to zero So lambda can either be zero or lambda can be four Now if lambda is zero this actually becomes the y-axis Now this cannot be a tangent to the y-axis because it is definitely crossing y-axis. So lambda has to be four So very clear cut option. Uh, the answer is going to be four Okay, let's move on to the next question. That's uh, sorry Question number 27 Yes done It's pretty simple if the same vector is If the same vector x is perpendicular to three vectors simultaneously means abc should be coplanar okay that means the Scala triple product should be zero. So absolutely that was a simple question Should have not taken you more than two seconds to answer this Next is question number 28 a circular sector is formed by a thread of length l. What is the ratio of the maximum area of the sector? To maximum area of the rectangle formed by the same thread Two plus Yes, yes, it's an integer type Okay, I think you're taking a lot of time in this. So first of all the area of the sector Let's say I consider that this sector of radius r Okay, we know that half r squared theta is the area of the sector So we also know the fact that the total length of the wire will be equal to two r plus r theta So I can say it's half r square in place of theta I can write it as This which is half r times l minus two r Is that fine? So, uh, if I differentiate this With respect to this is my area of the sector if I differentiate this with respect to r. I should get zero So that would give me half l minus four r equal to zero that means r has to be l by four right If r has to be l by four That means theta has to be uh l minus l by two divided by l by four right That's going to be two radians Okay, so this will become area will become half r square That is l square by 16 into two. That's going to be l square by 16 itself. Okay, and if you want the Rectangle to the maximum area We all know that it has to be a square a square is basically a rectangle with maximum area So in this case it will be again l square by 16 So they're asking you what is the ratio ratio is going to be one in this case Is that fine? So guys two more questions are there. Uh, I would just Leave them for your homework Uh, not very difficult ones. One is the limit question other is a Shorter distance between skew lines ocean. So I would leave them as your homework so We'll call this session end Thank you so much everybody for coming online Uh, I think on second of april most of you would be having your board exams So there would not be any class on those days Okay And in case you have any doubt Or you want to solve more mock papers do connect with us Thank you so much over and out from my side Have a good day and best of luck