 Good evening everyone is the screen clear to you is the screen clear to you, okay fine, so Today's class I'll I'll do two things the plan of the classes like this first would be to solve questions based on Cubes and there is no particular sheet for this the second one is Because all the type of questions I'll be doing here in the class itself you can refer to the class There are only two to three type of questions here That's why I have not planned any sheet because you can make only two to three questions And then then you can keep on changing the same question here and there and second would be direction best questions so in direction best question there are Questions on direct I mean fine directions and then you have questions on the basis of Distance calculation and then you have questions on the basis of Angles so angle based questions Now I'm starting with the first type of questions which is Question based on cubes now Now I'm giving directly a question and the question is there is Three into three into three type of cube this cube is cut into smaller cubes of size One into one into one now all six pieces of The Cubes are painted then find out how many cubes are one side painted Two side painted three sides painted and no sides painted. This is the question Now how many of you want me to do this question? Before telling you the answer and and how many of you want to attempt this question and then want me to explain the question Okay, fine. You attempted. I give you time to attempt it. Yeah, I'm giving you time to attempt it. Don't worry Attempt it. Okay. So should I solve it? Okay, so I'm solving now what I have done is I've drawn a particular cube on on The screen now when you draw first, let me calculate how many cubes smaller cubes are possible so first calculation is number of smaller cubes So try to understand if Three into three into three cube is there so total volume would be seven Now Volume this is volume of bigger cube volume of smaller cubes would be one into one into one Which is one so number of cubes would be equal to volume of bigger cubes divided by volume of smaller cubes Which is 27 divided by one that is equal to 27 So there would be total 27 smaller cubes Now if there are total 27 smaller cubes I have to understand how many of them would be three side painted or more than three side painted or something like that Please try to understand look at this portion this corner of the cube. So corner of the cube. There are three sides You look at here this cube This particular cube would be three side painted. This will be one left side This is front side and this top side would be printed. So all corners this corner one two three four five six seven eight So eight corner cubes would be three sides painted Now how many of them would be two side painted? So try to understand two side painted means common to two surfaces Common to two surfaces means any particular cube Which is falling on the edge like this cube here the cube in between these two corner cubes This cubes these this cube would be two sided So one on the top and one on the front So all the cubes the rule is try to understand corner cubes three sided Cubes on edges like this this cube would be a Certain portion of this cube would be on the top and this is on the front so top and front This is painted and this is painted. So two sided painted. So cubes On Edges would be two sides painted third is cubes on Surfaces that would be one side painted and fourth would be Cubes inside the bigger cube that would be zero side painted Just write it down. I'm explaining everything Have you written it? Just let me know once you have written it down Is the screen not clear to you guys? Okay, fine. Is it clear now? Okay, so now look at here. How do I solve this question if it is a three is two three is two three type I'll take one surface. I know that This let me make So always the first step is to take one surface in a cube. How many surfaces would be there in a cube or cuboid 12 edges six surfaces Eight corners are there. So try to understand These are three side painted the corner ones The one on edges are two side painted so And these are one side painted now this is the story of one surface So try to understand three side painted would be eight Because there are eight corners two sides painted would be Count the number of edges. Remember, whatever I am saying for two sides painted count the number of sides or Edges whatever you call it In the cube or cuboid the second step is calculate or Count the smaller cubes on one edge except corner cubes So here if I take this particular as Except corner cubes. How many cubes are there only one cube there? So it means that for as One cube is two sides painted. So there are 12 edges So two sides painted would be 12 cubes So this comes out to be how much? Now one side painted Remove three sides and two sides only one cube is left out on one surface and there are six surfaces So one side painted would be one on one surface multiplied by six. This gives me six cubes Now total how many cubes three side painted are eight two sides painted are 12 One side painted are six. This comes out to be 26. So out of 27 smaller cubes 27 smaller cubes 26 I have counted so zero side painted would be 27 minus 26 that comes out to be one So this is your answer Just note it down if you have any doubt ask me Okay, I hope you have written this I'm giving you another question the question is I Have a five into five into five cube painted on all six Surfaces then cut into smaller pieces of One is to one is to one find The number of cubes Which have three sides painted two sides painted one side painted and zero side painted So I've got a few answers not Only seven eight answers out of 19 people So I'll wait for two more answers. So I have made a five into five One surface I have made you also make one surface Identify what is happening around So one surface try to understand in any kind of cube three side painted would be corner cubes So three side painted would be corner cubes So this would be three side painted. I'm sorry. Let me use another color. This would be three side painted This would be three side painted and this would be three side painted So and and and four cubes on the other corner. So total it will be eight cubes Now two side painted. I told count number of cubes on one is I'm taking this particular is and I'm counting total number of cubes here one two Three four five, but I also told that don't count the corner cube. So this cube. I'll not count this cube I'll not count One two three cubes are Two sides painted would be three cubes on one edge So that would be equal to three into twelve 36 cubes then I have One side painted so try to understand. I am cutting these cubes because they are two side painted So all the cubes on edges would be cut down Because they are two sides painted. So how many cubes are left out for one side painted for one side painted For one side painted on one surface There are one two three four five six seven eight nine cubes on one surface Nine cubes are there So on six surfaces, it would be nine into six 54 cubes One side painted or zero side painted would be so let me count Total smaller cubes would be total small cubes would be equal to five into five into five divided by one into one into one which is equal to one twenty five Now if it is equal to one twenty five, let's count So three sides painted are eight Two side painted are here. Look at here 36 One side painted are 54 So which comes out to be how much 98 So zero side painted would be 125 minus 98 That is equal to 27 So this is the answer most of you are getting this right So I will solve one more question On this basis one or two more questions So let's take a bigger cube seven into seven into seven cube find the same thing Okay, I've got a few answers. I'll wait for a few more answers Okay, most of you have given me the answers now try to understand So it is something like this So I have bifurcations like this and Something like this If you make one surface you get clarity and then I give you a generalistic formula for this Why didn't you one two three four five six seven? So now we have So this is three side painted So three side painted remains eight now number of smaller cubes are equal to seven into seven into seven divided by One into one into one which is 343 Two sides painted would be one two three four five On one as they are five so five Into 12 That comes out to be 60 now you remove this this this and this So this comes out to be one two three four five And there are five lines like this one two three four five So there are 25 on one surface So one sides would be 25 into 12 300 So total Sorry 25 into six. I'm sorry 25 into six it'll be 125 so Three sides painted are eight Two sides are equal to 60 One sides are equal to 125 So how much it comes I'm doing calculation mistake again and again So 25 into six is 150. So This is nothing but 150 So how much it comes it comes out to be 150 plus 60 is 210 Plus eight is 218. So this is 150 plus 60. This is 218 So zero sides painted would be 343 minus 218. This gives me a value of 125 Now I'm giving you a generalistic formula write it down three side painted are eight which is On corners two sides painted are Whatever are you try to understand when it was three into three into three What I did I made this and I Separated like this. So I removed corner ones. How much I got I got one when it was five into five into five I made five Like the one two three four five So I removed this these two and I take one two three. So what I what what is happening is whatever Kind of cube it is if it is three into three into three side I'm removing the corner cube. So I'm reducing two from here. Similarly, if it is five into five into five I'm removing two from one edge. So how much I'm getting three on one edge If it was seven into seven into seven if you look at this If it was seven into seven into seven, how many Edges I got on two. So seven minus two I got five so What is happening is from each side? So try to understand it would be 12 into n minus two where n is size of size of One edge of cube Now one side painted would be You look at here again. I'm going back to the previous one if this is Seven into seven from one side. I am removing this from this side. I am removing one So it means from length if this is length from length I am removing two sides. So this becomes length becomes l is equal to n minus two And from width if this is my width b So from b also, I am removing two the upper one and the lower one. So b becomes how much n minus two So on one surface, how much how many cubes I'm getting I'm getting n minus two into n minus two cubes So try to understand It would be one side painted would be Six multiplied by n minus two into n minus two Why six multiplied because n minus two into n minus two you are getting on one cube and you multiply it with six Now zero side painted would be n minus two cube So you see here in Three side painted zero side painted came out to be one. So it became three minus two So one cube in five side painted it came out to be 27. You check your answer. So it was five minus two Cube, which is three cube. That's why it came out to be 27. Then it came out to be seven minus two Cube, which is five cube that is equal to 125 So it will be n minus two cube So that that would be your answer for this kind of question Now if somebody asks you What for cuboid? So I'm giving you one question Or let me give you a generalistic formula for cuboid so So cuboid what happens? The volume is not n into n into n it becomes l into b into h So what will happen on different sides? This is l. This is b. This is this is l. This is b. This is h So here different number of cubes are there here different number of cubes are there here different number of cubes are there But what happens? Three side painted would be eight Two sides painted would be what I was doing in cube if you have written down the formula in cube. I did 12 into n minus two So it would not be 12 into n minus two It would be try to understand Four sides would be for length Four sides would be for width width and four sides would be for height Let me make it for you so that you can understand it So this is l one This is length two This is length three This is length four This is width b one This is b two This is No, this is h one. Sorry. This is h two This is h three and this is length four And width is what? So width is this is b one This is b two This is b three And this is b four I'm not saying this so I'm saying this l one is equal to l two is equal to l three is equal to l four I'm just showing you that there are four sides which are for length Then b one is equal to b two is equal to b three is equal to b four which is b And then h one is equal to h two is equal to h three is equal to h four, which is equal to h So what is happening here or for two sides painted from length also? I'll reduce two cubes which are on corner from width I'll reduce two cubes which are on corner and from height I'll remove two cubes which are on corner So it becomes four into l minus two Plus four into b minus two Plus four into h minus two So it comes out to be four into l plus b plus h minus six This is your answer for two sides painted Now let me do it for one side painted So one side painted I have already told you That it would be when it will be see there are there would be three kind of combinations And three kind of combinations would be first combination would be l into b Second combination would be b into h third combination would be h into l And for each combination I'll have two surfaces To total six surfaces I'll have So you look at here What I'm trying to say see this combination where where the pointer is there this combination Is l into h similarly the back one would be l into h Now this upper one is l into b lower one is l into b so two like that and this Right and left one is h into b kind of Surface now what happens on l into b kind of surface how many Cubes I I'll have a smaller cube so I reduce Two from length and two from width so l minus two Into b minus two And two surfaces are there so I multiply it with two Similarly b minus two H minus two two surfaces are there I multiply it with two And then two into h minus two l minus two so one side painted would be equal to two l minus two Into b minus two plus b minus two into h minus two Plus h minus two into l minus two That's the value That is what you have to get and zero side painted would be I'm writing here in the corner l minus two Multiplied by b minus two multiplied by h minus two. This would be zero side painted So that's your generalistic value. Okay, so now let me Have you written it all? Just let me know Did you all understand it? Okay, great fine. So let's do one thing Let's move to another topic and which is directions So directions direction sense, perhaps we all know this is north. This is south This is east. This is west Now one thing one proper One thing properly we should look at is That you have to focus on Left and right properly So if I am saying that For an example that a person is going towards east So he is going in this direction And then he takes right now. I am standing here like this One person standing like this. So what would be right? This would be my right hand and this would be my left hand So which side I should move this side. I should move. This is the right So put your face on this side. Just assume that you are facing this side Your right would be this your left would be this most of the people who do the Who get these kind of questions wrong are in forced Sorry are I mean they are the people who do mistake it while taking left and right from east west north south and And all these things. So try to understand that Whenever you are taking or whenever you are taking Directions make sure that you doubly check it and after checking it make sure that You draw the diagram properly. So this kind of question the second instruction that I want to give you that don't solve it in mind Do draw the direction Diagram on your copy and then solve it. Don't solve it in your brain People tend to make mistake in in in while you're doing that So now what I do is I give you few questions to solve so that you can understand Just give me a moment. Okay before I move with the question. Please please understand this The one thing which I didn't discuss was Apart from north south direction north south east west direction There are other angles which are talked about so the angle best question would be on the basis of northeast Now when somebody is saying you northeast you please assume that this angle is 45 degrees Now then until and unless some angle has been mentioned like 30 degree east of North so if I am saying you 30 degree east Of north So this is my north and from here I make 30 degrees towards east east is which direction this direction So 30 degrees towards east This particular angle would be 30 degree so until and unless somebody is giving me some kind of angle There is no necessity to assume any random angle if northeast has been given I will take 45 degree similarly for southeast. I'll take this 45 degrees. This becomes my northwest And this becomes my southwest So this is the one thing that you have to take Okay. Now now write down a question. I'm I'm orating this question. Please write it down and the question is two ncc groups a and b A is facing east and b is facing west sorry Yes, east and west Now what are the directions given to them? The directions given to them are Please understand left on the first direction is left on then second is about turn And about turn is nothing but u turn third direction is right turn and fourth is again left turn now you have to tell now finally in which direction a and b are facing respectively Okay, I have got few answers okay Please try to understand here that if I draw this north south direction like this So what is happening? a is facing east like this And b is facing west like this If I do it for one Automatically the other one would be opposite because they are taking the same kind of turn the first turn is left So if first turn is left So left means From east they'll start facing north Now second turn is Let me check. I just forgot it second turn is about turn. Okay about turn means if they are facing North They will start facing south. So this is my first turn. This is my second turn The third turn is right turn. So right turn from here would be if they are facing south somebody is facing south This is right So right turn is this So which is this direction? This direction is west And then they are taking Left turn So from here from west left turn would again be what? This side which is south So this particular guy a would be a would be facing south So a faces in which direction this is my third turn. This is my fourth turn in fourth turn a is facing south So if a is facing south no need to do it for b b will face north Because they were in opposite direction and same turns are taken. So this is your answer Now let me move to another question So second question is you go north then first turn right Then second turn again right And third turn Is left then tell me in which direction You are going now So the answer over here is If you go if I'm going north The right would be this direction. So this is my first turn Again, I take right. It means that I go in this direction And then I take left So if I'm going south here And from south I have to take left. So south this is right and this is left So again, I come back to this direction now try to understand This second right and left are cancelling them themselves if you go right and left Then whichever direction you have been going in the first move the same direction you will be going So if the person was going north Now it will go east. So answer is east here So I have a few questions. Just solve those those questions and then perhaps we will Wrap up the session So these are the questions. There are 11 questions. Just solve first first two questions Whenever you are answering answer with the question number so that I can come to know which question you are answering Okay, a lot of you are saying First answer is a So let me just check the answer So Rohit walk 25 meters towards south So suppose this is he's starting from origin. This is a So 25 meters in south Then he turned left So This would be left And he's walking 20 meters So this is b He then turned to his left. So this would be left and then he walked 25 meters. So he came down 25 meters He is going up by 25 meters And then he turned right And walked for 15 meters. So this distance is how much 20 meters and then he's turning right And going 15 meters. So first he is traveling in each direction and from origin. He is 35 20 plus 15 35 meters east. So this is my answer. So option a is my answer Now solve question number two So second answer a lot of you are saying it's b So village q is To the north of village p. So suppose this is village p So village q would be here to the north Village r is to the east of village q. So village r would be somewhere here The village s is to the left of village p. So village s Would be here in which direction village s with respect to village r So which direction this is This direction is southwest direction. That's why the answer is b Now let's do another questions question number three and four Question number three. What is the answer you are giving most of you are giving answer three a So rada moves towards southeast east at a distance of seven kilometers So southeast So let me make this This is origin. This is a seven kilometers Then she moves towards west and travels at a distance Travels at distance of 14 kilometers. So west would be at this direction So then she is going west And she is reaching this point to be and this distance is 14 kilometers From here She moves towards northwest Northwest wins this direction, which is again seven kilometers from here Finally she moves four kilometers towards the east So this comes out to be four kilometers How far she is now from the starting point the answer is a why because if you look at this The directions the southeast directions And northwest directions here would be parallel lines And they are of equal length And this line, which is west and east would also be parallel So if I match C and o from here if c o is matched then I'll get a parallelogram in parallelogram Opposite sides are suppose. This is b point opposite sides are parallel and equal So if this side is completely of 14 kilometers and I have traveled four kilometer already I'm left out with 10 kilometers. So answer would be 10 kilometers Question number four Question number four also most of you are saying me for b So a river flows from west west to east. So this is west to east And on the west turns And on the west turns left and goes into semicircle round a hillock So a river flows from west to east And on the west turns left And on the west turns left Goes in a semicircle. This is the semicircle around a hillock So semicircle would be somewhere here And then turns left at a right angle Which answer you guys are giving me you are giving me answer of Have I done something wrong? Let me just check a river flows West to east And on the west turns left and goes in the semicircle So it's hillock And on the west this side Tons left and goes in the semicircle like this And then take left Okay semicircle would be Like this and then turns left. Okay understood. So something like this And like this so it goes in the east direction. Okay, so that's why you're giving me the answer So this is going in the east direction now solve question number five No, four c is not the right answer four b is the right answer I made three fourth quarter of the circle. That's why it came wrong Those who are not understanding The semicircle would be like this And from here left is east So that is why this question answer was east Question number five. What's the answer? Question number five. You guys are giving me answer five a lot of you So Raju walks a distance of three three kilometers towards north So Raju is walking three kilometers One second three kilometers towards north oa Then he turns to his left Walks two kilometers. He reaches b Then he turns to his left So this is three and then again three kilometers. So he's coming at the same level of o So c and or at same level. So this is three At this point he turns to his left And walks for three kilometers So left would be this direction and walks for three kilometer in here. So this is two kilometer He is one one kilometer from o so answer would be five a which is one kilometer Now let's Solve question six and seven Okay, what are what are the two answers you have been giving me? Six a and seven a both of them you are giving me a only so Six a and seven a is right and most of you have done it right. So I'm not doing it. Let's go to question number eight and nine What is the answer you have been giving me? Question number eight. I'm getting a lot of different answers Okay, I'll solve question number eight. Give me the answer of question number nine Eight b is a wrong answer I think Okay, let me solve question number eight for you If southeast is called east Try to understand This is east in north south West So southeast is called east. Now this becomes east. This is southeast This is called east now Northwest is called west. So obviously Northwest would be called west exactly opposite to it And southwest is called south So what is happening in clockwise direction? I am shifting my Directions by 45 degree. So east becomes southwest southeast West becomes northwest becomes west so 45 direction in 45 In uh this thing in uh clockwise direction So what would be north? called north will also go in uh 45 degrees So this will be This is northeast Now north will become This will become north So answer is b. So those who are giving answer be they are right. I'm sorry. I thought I read the question wrong. So that's why I thought to be uh, this is not Eight b is a right answer question number nine Nine all of you are giving me answer a so I'm not solving it a is a Right answer solve question number 10 and 11 and then let us wrap up the session Okay, let me check your answers Question number 10 a lot of you are saying b one of you have said a 10 b is Let me check Rina walked from a to b in the east 10 fields So a to b is In the east 10 fields See then turned right. So this is right And walk three fields So three sorry three fields This is c See again turned right And walks 14 feet far from a So this is 14 fields So this is 10 feet. This is how much? Four feeds from here. So this is d And this is e And this is how much three feet. So how far from here this a to e is a right angle triangle here This length is three and this This is four. So under root three square plus four square that comes out to be how much Under root 25, which is five. So she's five feet away So that's the answer one morning after sunrise sures was standing facing a pole The shadow of the pole fell exactly to the right To which direction he was facing So try to understand if there is a pole standing here And if it is a sunrise so light would be coming from east From this direction and the shadow would be falling on west direction So shadow would be somewhere here in the west direction and shadow if shadow is in west direction And sures is seeing it to the right It means that sures is facing this direction south because right of south is west So sures is facing what south so both of these answers are b That's how you can solve it. So i'm done with the sheet Now what I do is I give you a homework Write down your homework. There is only one question for homework and that is a cuboid Or let's say two questions A cuboid of side seven into six into five Painted all sides All six surfaces then one count And then cut into one into one into one So then count three side painted Two side painted One side painted Zero side painted So this is your first homework second homework is suppose you have a cube of six into six into six Again all sides painted All surfaces painted and then you have one into one into one cutting Same you have to find here also So two questions I have written on the board. Please solve it And let's meet another time So thank you so much for joining the session Okay, thank you so much for joining the session. See you next time. Thank you