[Music] Hello everyone. Today we'll be going through IGCSC Cambridge mathematics paper 4 extended 058042 October number 2024. This is calculator paper extended and question number one says envy buy a new car. The price of the car is this much. she is given a 7.5% discount. Calculate the amount uh she pays. So we know uh for discount we are taking 100 minus whatever discount which is 7.5. So 100 - 17 7.5 divided by 100 and multiply by the total amount which is 28 uh 282 4 and this will be calculated as uh 100us 100 minus this will be 7.5 over this will be 100 and this whole thing will be multiplied divide by 282 4 and this will be giving us 2612 2612 this will be dollars and this will be our answer for two marks part second. The fuel tank in the new car has a capacity of 45 lit. This is 72% of the capacity of the fuel tank in her old car. calculate the capacity of the fuel tank in her old car. So, reverse percentage question u 45 72% of the old car. So, 72% of the old car which we don't know. So we can suppose that was x is equal to 45 lit and x will be equal to 45 over this will be 72 over 100 and this one we can calculate 45 over uh 72 over 100 and this will be equal to 62.5 62.5 l and 62.5 lit over here this is two mark question. Hopefully you understood this. Now B part B part says Eddie buys a new car costing this much. He pays for the car using a finance plan. Finance plan is a deposit 47 equal monthly installments of 330 each. So we will be calculating these installments. We can calculate 330 * 47 and this will be 330 * 47 is equal to 1 1510. So 1 510 and then a final payment final payment of this much. So we'll be adding this. So and they are saying using this finance plan add pays a total of this much. So if we add uh this uh 15 1 510 which is installments plus the final payment which is 1 1490 and these two will be adding up to 11490. These two are adding up to 270. So $270 and the final payment which he has paid is uh is this much. So 313 1 900 finally he has uh total he has paid uh and minus this will be 270 and the rest will be uh deposit. So we can calculate this will be 3 1 900 minus answer is giving 49 uh 490. So they are saying calculate the deposit. This is the deposit which we have uh calculated. They're saying calculate the deposit paid as a percentage of this. So after calculation of the deposit which we have done over here we need to calculate the percentage. So 490 percentage of 28 triple0 and this will be multiply by 100 and this we can calculate. So this will be answer over this will be 280 and multiply by 100 and this will give us 17.5%. 17.5% will be our final answer for four marks. Hopefully you understood this. Now next C part. C part says a car travels 64 kilometer and use 2.5 L of uh 2.5 L of fuel. It then travels 128 kilometers and use this much. So total distance we can calculate total distance will be equal to 64 + 128 and this will be 64 + 128 is equal to 64. Okay. So this will be 190 192 kilm. 192 kilm is the total distance and uh fuel will be total fuel will be equal to 2.5 L + 6 L. So this will be [Music] 8.5 L use this uh fuel. So they are saying calculate the rate at which the car uses fuel during the whole journey. Give your answer in liters per 100 kilometers. So 100 kilometer we can check for this uh basically um distance we know speed is equal to distance over time or uh basically distance is not required in this case. So what we'll be doing uh travel this much distance. Okay. So total distance covered is 192 and divided by 8.5 lit and we can calculate how many kilometers per liter 192 over this will be 8.5 and this will be equal to 22.588. So 22.588 kilome per liter but we need 100. So how many uh liters in 100 kilometers? So 100 kilometer divided by 22.588 and this will give us liter per 100 kilometer. So we will take 100 over this will be answer is giving us 4.427. So 4.427 will be lit per 100 kilometer. 4.4 427 and this is uh answer for two marks. Hopefully you understand this depart. At the start of 2021, the car value, the value of the car was 46 uh 500. At the end of 2021, the value of the car was 21 20% lesser. So at at the end of 2022, the value of the car was 15% less than its value at the end of 2021. Calculate the value of the car at the end of 2022. So what we will do? We will check end of uh we can write end of 20 21. What will be the situation? Uh car value is lesser. So 20% lesser. So we'll calculate 100 minus 20 over 100. This is percentage decrease we are calculating. And the amount was this much 4 6 500. And this will be calculated as uh 100 minus this will be 20 divided by basically we can divide here. So this will be -20 over this will be 100 and this will be multiplied by 4 65 0 and this is giving 37 2 3 7200 this will be dollars at the end of 2021. Now we'll have to proceed at the end of uh end of 2022. 2022 what will be the situation? Further reduction of 15%. So 15% reduction they have mentioned over here. So this will be for this for this amount we will be calculating the reduction. So this will be uh 37 uh 2 0 and if we calculate this this will be 100 minus this will be 15 over this will be 100 and multiplied by 37 2 and this will be equal to 316 2 3 1 6 2 0 this will be in dollars. So 31 620 will be answer at the proper given location and this is equal to mass. Hopefully you understood this as well. Now this is question number two and table of values is given but some values are missing. This one is missing and this one is missing. Two values are missing. So we need to fill this. So very easy method is just type the equation alpha x and cube. So this will be cube and this will be + 4 and x² and next will be min -4. After typing this we'll be using the calculate button to check the value is uh we have typed the equation properly or not. We can check for calculate for calculate for x = -4.5. Our answer should be -14.1. So -4.125 correct. Now we can again calculate for calculate for uh this will be calculate for minus4. So -4 will be equal to it will be giving minus4. So here value will be -4 and the next value for x= calculate for x = -1 and this is giving us minus1. So this is also minus1. So these are the two values which we were unknown and we have filled and this is for two marks. Hopefully you understood this. Then complete the table which we have done basically it was a part b part on the grid draw this curve. So these values whatever in the table these values we need to draw in uh what do you call uh this uh grid and first value is -4.5 and -4.1 so -4.1 will be 14.1 can be over here and Next is what? Uh - 4.5 not 4 - 4.5. So 4.5 and 14.1 will be over here 1 2 3 4 5 and 14.1 we can write over here. And the next value is -4 and -4. So -4, -4 will be this point. And the next value is -3 and 5. -3 and 5 will be this location. Then - 2 and 4. -2 and 4 also this one. And then min -1 and -1 will be over here. And then 0 and -4 0 and -4 is here. 1 and 1 1 and 1 will be here. And 1.5 and 8.4. So 2 3 4 5 uh 8.45 here. So 6 7 8 8.4 will be roughly over here. So after uh this one we need to draw the graph and we can try to draw a smoother graph. So we can draw a smoother graph like this one. So from here this will be till here almost smoother we have uh done. Now they are saying next is what next is uh draw draw the tangent to the graph at point 1 one. So one one tangent at the graph we need to draw and we can check this. It should be here. So this will be the graph. I will change the this one. Okay. So okay. So this will be the tangent line which we have drawn and we can take basically I will have to redraw this a little. So that it should be roughly we can take for for example we can take from here as well. We can take from here and we can this is touching at this point and we can if you want we can rectify this curve and this curve will be smoothly passing through this point and up to this one. Now it's better. It's uh I'm drawing on the screen that's why it's a little difficult to draw. But uh if you check this one, if you take this triangle like for example up to this one, it's always the graph answers are always a range but your answer should be within the limit of accuracy. So this triangle we can consider for the gradient. Rise we can take from 10 to 10 will be this will be 20 and run we will be taking as two. So m will be equal to rise over run. They are finding they are asking us to find the gradient of estimate at this one. So this will be we can write m is equal to rise over run which is 20 /2 and this is equal to 10. And this will be the gradient for this. Hopefully you understood this. Now they are saying by drawing a suitable state line on the grid uh solve the equation this. So if we check this uh equation basically the the given equation y is what? Y is this one. If I write the y, y is = xq - 4x^2 - 4. So this is basically the y which we can take to the location. Okay. So and form the equation. So this is our y. And if you check this uh the given equation this one uh I'm talking about this given equation. So x cq is same this one - x4. It is uh basically uh - x4 or + x4. Let me check. It is + x4. So this one I can change to plus. So + 4 x² is also there which is and -4. So this is -6 over here. So we'll split it into -4 and -2 will be remaining and - x will be remaining is equal to zero. So what we have done we have changed the format of this equation and we for this portion we can write as y. So our equation will be y - x - 2 =0 or we can write y = x - 2. This is the equation which we need to draw which we need to draw on the uh coordinate is x is y = x - 2. Now y is = x -2. So it's passing through uh x it's x + 2 I think. So yes x + 2 I've done mistake over here when you are shifting it will be x + 2. So y = x + 2 is our equation y = x + 2. So y intercept is 2. So it means it is passing through this point which is y intercept and the gradient is 1. So we can take like uh if you're taking three units up but three we don't have on the x-axis. So we'll take one and two units from here up to this point and then we we can move to the x-axis side and we will reach over here. So our line should pass through uh this point like for example if I draw the line this line should be like I'm trying to be as accurate as possible. So this line should be like this. And if I extend this line, this will be this will be like this one. So now we need the solutions. These three solutions are required over here. So we will be checking for x values from here. So one value we can check this is uh from here this is 3.9 - 3.9. So we can write as -3.9 over here. So -3.9 is one value. The second value we can take is uh -1 from here this will be -1 uh 2 we can take min -1.2 so -1.2 2 or even if you are getting minus 1.3 that will also be fine and the third value we can take from here which is uh 1.1 so 1.1 this will be positive value so these are the three values which we solutions we have uh given and that is for four marks hopefully you understood this question number three a part simplify this is two marks very simple question algebra algebraic expression we need to add this is 3x like terms we will be adding so 3 m - 4 m so 3 m - 4 m I'm arranging first then 5 - 5 n + 8 n so these two we can combine and this will be - m and these two we can combine and this will be + 3 n so we can write 3 n minus m and this will be our answer for two marks. Now this uh second part we will be taking separate parts. So 3^ 4 and a² raised to ^ 4 and cq raised to ^ 4 and then 3^ 4 is 81. So we can write 81 a^ 2 into 4 will be 8 and c^ 3 into 4 will be 12 and our answer will be 8 a raised^ 8 81 raised^ 8 uh c^ 12 this will be our final answer for two marks. Now uh this uh uh expression this one uh fraction is given we need to write in solve this basically. So what we can do is we can change the denominator equalize the denominator we can change into 30. So 6 * by 4x over 30 6 into 5us this will be 3 * 3x over 30 and plus this will be uh 3 into 2x over this will be basically uh multiply by two so we'll write here 2. So 2 into 2x over this will be 30. And then we can write this as 24x over 30 minus this will be 9x over 30 and plus this will be 4x over 30 which we can rewrite as we can simplify 24 + 4 will be 28. So 28 - 9 will be 19. So 19x over 30 this will be our answer 19x over 30 will be final answer for two marks. Hopefully you understood this. Now B part B part says this is triangle has a parameter of 35.5 and diagram is not to scale. Find the value of A. This side we need to find. So we know that uh this side is also 3 a + 2 because these two sides are equal. So parimeter will be what? parameter will be adding all the sides. So we can write it as this will be uh 2 uh 3 a basically 3 a + 2 into we can add again 3 a + 2 and + a and this is like parameter is equal to 35.5 35.5 so this will be 3 a + 3 a will be 6 a + 7 a so we can write 7 a + 2 + 2 will be 4. 7 a + 4 is equal to 35.5. So 7 a is equal to 35.5 - 4. So this will be 31.5. So a will be equal to 31.5 / 7. So which we can solve 31.5 / 7 is equal to 4.5. So 4.5 will be our final answer for three marks. Hopefully you understood this. Now using quadratic formula they are saying using quadratic formula we need to solve this uh quadratic equation. If I write the quadratic formula x = - b + - b² - 4 a c / 2 a. This will the quadratic formula and a value is what? A value is 5. B value is equal to -4 and c value is equal to minus3 which we are getting from this equation. Now we can replace the values and always replace the values in brackets. So - into - b and + - b² means -4² -4 into a value is 5 and c value is -3 and then we can divide this by 2 a which is this one. And now we can use calculator to solve this the equation. So this will be fraction - into -4 bracket close and we'll take plus value and later on we will be taking negative value as well. So this will be -4 and this will be squared and -4 into 5 into -3 and over will be 2 into 5 and this is giving us uh 1.27. So 1.27 will be our one answer. We can write here 1.27 27 as x1 and x2 will be second value. We'll have to go here and change plus into minus and then we will get the second value which is this one which is -0.47. So -0.47 will be the second value and this is for three marks. Hopefully you understood this. Now D part D part says solve the simultaneous equation and we know that these two sides are y. So these two sides we can equate and equating will be x² - 4x + 5 is equal to 2x - 3. And further we can shift these to the other side. And this will be uh x² - 4x + 5 - 2x + 3 is = 0 and x² - 4x - 2x will be - 6x and 5 + 3 will be 8 is = 0. Now we can uh basically for find the x values factoriize mode 53. I'm using calculator for that. So x1 cofficient of x² is 1. So cofficient of x is -6 and then 8 and this is giving us 4 and minus2. So we can write x -4 and x + 2 opposite values we are taking. Here we're getting two basically four and minus 2. So -4 and minus2 should be here. So here this one if you multiply x² - 2. Yes. So in the intermediate step we can write x into x -2 and -4 into x -2 is = 0. This is equal to 0. So x - 4 =0 and x - 2 is equal to 0. So x = 4 and x = 2. So these are two values 4 and 2 we have written here. Now we can replace for y values and we know y is = 2x - 3. So we'll replace four. So 2 into 4 - 3 will be 8 - 3. 8 - 3 will be 5. So 5 will be this value. And y = 2x - 3. We'll be replacing value two. So this will be two here and minus 3. So 4 minus 3 is one. So this is five mark question. It was a very easy straightforward question which we have done. Now this is question number four. And question number four says the angles of the quadilateral. Quadilateral is having four sides. So four angles uh the ratio of the angles is given. Find the value of w. So we can write this uh uh this is uh basically uh ratio is w. First we can find the sum of ratios right. Sum of ratios will be equal to 3 + 5 which is equal to 8. So angle in a quadilateral is uh 360. So 360 divided by 8 will be equal to 360 / 8 will be equal to 45. So 45 is one part and we will be taking this ratio x + y + z which is equal to 3 ratio 5. So we can multiply 3 into 45 and ratio 5 into 45. So they are asking for this portion W which is 3 into 45 and 3 into 45 will be what? This will be 135. So W value will be 135°. Okay. So this is for two marks. Hopefully you understand this. Now B part B part is circle theorem and circle theorem is saying basically it's not to scale first thing then A B C D are points on the circle. So A is here B is here C and D is here. So they're saying PQ is the tangent at A. So PQ this line is tangent to the circle. BMD is a state line. BMD is a state line which they are telling this is the state line. And then they are saying uh angle ACD is 49. This is 49. Angle A M B is 105 and angle P A B is 45. This one is 45. So this is the information given. The part one is find angle B A and M. So B A M is required. B A M is which angle? B A M is this angle. So in in this triangle we have one angle 105 and if we find this top angle B then two angles of this triangle will be with us. So we'll be able to find the third angle. So because this is 49, this is top angle. So this will also be 49. Hopefully you understood this. So I'm writing here 49°. So this will be 49. So remaining will be 180 minus uh that angle. So 180 - 49 + 105 uh 49 + 105. So we can calculate this will be 180 - 49 + 105 and this will be giving us 26. So 26° is our answer for this two marks part. Find the angle B part uh that's second part A. Find angle B A D B A D. So before finding B a D I'm just writing the angle which we found which is 26° over here and B A D is which angle I'm going to indicate B A D. So we require this complete angle this much. So by alternate segment theorem, by alternate segment theorem, this angle will be equal to this angle. These two angles are equal by alternate segment theorem. So this angle will be 45°. And in triangle A B D we can find B A D will be equal to 180 minus these two angles one this one and second this one. So these two angles we will subtract from 180 and we will get the answer. So 180 - 49 + 45 and I will erase and write properly. So 49 + 45 and this will give us answer. So 49 + 45 over here calculation this will be 86. So 86° will be our answer for two marks. Hopefully you understood this as well. Now B part give a geometrical reason that BD is not the diameter of circle and that one B D is uh this one. So for example if BD is is uh uh B D is uh BD is again this watch this one. See if BD is a is a diameter then A B D should be right angle triangle which is not in this case. So we can say that angle in a semicircle is 90° which is not true in this case. So we will use the property angle uh in a semicircle semicircle is 90°. So this will be the thing which we'll be using and saying uh it's not uh not a diameter. Now this C part again circle theorem and diagram is not to scale. saying a b c d are points on the circle center o. This is center o and t a is equal to t cc. So t a is equal to tc this one and we can indicate also these two are equal. And next they are saying oa is equal to o a is equal to 6.75. So this is 6.75. This is also 6.75. And next they are saying OT is equal to 11 uh 11.5. So this is 11.5 cm and they're saying show a one part is show that the angle A O C is 108. A angle A O C. So this complete angle they are talking about we need to prove that this angle is 108.12 up to two decimal places. So what we will do we will consider this triangle and in this is right angle triangle. So 190 one angle is 90°. Now we will be uh considering this triangle. We will be focusing on this angle first. we will find this angle. So we can write in triangle uh O A T in triangle O A T uh O A O A over O T basically uh is so O A / O T is equal to cos of theta and we can take this as theta. So O A is what? O A A is 6.5. So 6.75 over OT is 11.5 is equal to cos theta. And we can write theta is equal to cos inverse of 6.75 / 11.5. And this one will be your calculator should be in radian mode. And this is shift cos uh 6.75 over this will be 11 not radian mode sorry it should be your calculator should be in degree mode this one so 54.0 0 uh 5 will be or you can write 54.1 54.1 your calculator should be again I'm saying should be in degree mode radian mode is used mostly in SC level so but you will have to check it should be in degree mode for these calculations calculate the length of minor arc ABC so this is basically 54.1 we calculated which is half angle which is up to this one only. Right? So we need double and complete angle which is this one. So double of this angle whatever whatever we found we need to double this angle and this is 54.1 * 2 will be 108 one or two whatever 108.12 will be our final answer for 3 months. Hopefully you understand this. Then they are saying next calculate the length of the minor arc ABC. Minor arc is which one? I'll have to indicate. Minor ark is this arc which is of lesser angle. Angle we already have. So we can apply the length formula. So length formula is what? L is equal to theta / 360 into 2 pi r. And length uh this will be theta is 108.12 divided by 360 into 2i and r value we already have which is 6.75. So 6.75 and this answer we can calculate this will be what 108.12 over this is not minus so divided by 360 so over this will be 360 and we have to multiply also by 2 pi and radius which is uh 6.75 five and this is giving us answer 12.737. So I'll write 12.74. So 12.74 uh this is in cm. So 12.74 this is the answer for two marks. Hopefully you understand this. Next they are saying calculate the area of the major arc. So major arc now we need major arc area. So this angle we will be requiring which this angle will be what this angle will be 180 minus uh 10 uh what do you call 108.12 and this major arc angle will be 180 - 108.12 180 this one and this is equal to what this is equal to [Music] 71.88. So 70 no basically we have to subtract from 360 not 180 because circle angle is 360. So this will be equal to 251.88 251.88 and this should be 360 here. So this is the angle which we'll be using for calculation of major arc. So theta / 360 theta again this same formula this is basically that was length this is p r² for area. So area this angle we have taken as 251.88 88 and divided by 360 into pi and r² will be 6.75 squared and calculation will be what calculation we can write this uh 251.88 88 over this will be 360 and ultip by 2 pi and then multiply by 6.75 squared and this is giving us 200 um to 200 2 it's not 2 pi it's basically pi r² so we need to rectify this this will be equal to 100.14 4 9 so 100.15 or 100.2 so we will be writing 100.2 here so 100.2 2 will be our area cm square. Hopefully you understand this. Question number 5 a part. The commumulative frequency diagram shows information about the distance traveled by each of 80 motorists in a month. This side is commumulative frequency and this side is kilometer. Distance in kilometers. Use the commumulative frequency diagram to find an estimate for the median. Median you know is the middle value. Half of 80 will be 40. So we will be taking the middle value of 80 will be reference. which will be taken from here and from here on this side we can check the reference and this will be this will be uh [Music] 1,480. So we can write here 1,480 will be reference and this is uh one more question. Next we can find the intercartile range. Intercartile range is upper cartile minus lower curtile or Q3 minus Q1. So that we will be checking again the reference from here. So this side is 60 and this side will be 1,600 upper cartile and lower cartile up to 20. So this one and from here we can take up to this one which is uh uh 1,600US this will be 1,160. So we can write 1,600US 1,160 and this will be 440 basically. So we can write 440 as our answer which is two marks. Now the next part they're saying one of the motorist is picked at random. Find the probability that this motorist travels more than 1,800 kilometer. So more than 1,800 kilome will be where? So in the middle of this one, 1600 and 1,00 uh 2,000 will be 1 2 3 4 5. So up to middle will be this one. And we can check the reference from here and on the other side on y-axis also we can check the reference and this reference will be up to this one. And this is uh if you check this side is uh from here till here is 8. So probability will be 8 out of uh 80. So we can write 8 out of 80. This will be answer or we can write 1 / 10. Uh so this will be two marks question. Hopefully you understood this. Now B part the distance around the racing track is uh this 5.104 km. The time taken by the car to complete one lap of the track is 1 minute and 18 second. Calculate the average speed of the car. Speed you know is equal to distance over time. Speed is equal to distance over time and distance is uh 5.104 kilm and time taken is 1 minute and 18 seconds. So 1 minute 18 second we can convert 1 minute into fraction we need to convert and this will be how we will be easily converting is by the help of calculator. So 0 hours 1 minute and 18 second time button we are using and press equal to and then this uh fraction so s and d button this one 13 over 600. So we can write 13 over 600 and further we will be just dividing 5.104 over this will be 13 / 600 and this will give us answer. So we can uh write 5.104 104 over this will be answer is equal to uh 235.57. So 235.57 kilome per hour. So 235.57 will be answered over here for three marks. Hopefully you understood this question. Now this is the C part and C part says the top speed v kilometer per hour of uh each of 160 cars is recorded. The histogram uh shows this information. This side is frequency density and this side is top speed. And we need to find show that okay part I show that uh there are eight cars with the top speed interval 120 to 160 to 160. 120 is this one and 160 is this one. So this histogram basically we will be calculating. How we'll be calculating this will be we know frequency is equal to frequency density into class width and frequency density for this one is is what this is like uh 0.2 we can take this as 0.2 so 0.2 2 and multiply by class width will be 120 to 1 uh 60 will be 40. So multiply by 40 will be 8. If you check this so 0.2 into 40 will be equal to 8. So for one mark question uh this is basically show that this is 8 which we have already shown. So we can write here uh 0.2 working I have already provided but I'm just writing for your understanding. And second part is this one calculate the estimate of the mean top speed you show you must show all your working. So now for mean uh speed we need to calculate each bar. This is six mark. So we can calculate this is uh like for example if we if we check this this class width which class width I'm talking about I'm talking about this class width is 20. So we can write 20 ult* by height which is up to this one which is um 1.1. We can write 20 into 1.1. This is multiplication. So this will be equal to uh 20 into 1.1 will be equal to 22. So this number will frequency will be 22 for this bar. And for the next bar, for next bar mean uh this much only this will be up to this will be 200 and so width is 20 again. So 20 and multiply by this uh height will be 1.8. So 1.8 8 is equal to 20 * 1.8 is equal to 36. So this bar is representing 36. And next bar which is width is this much is this much width is 200 to 240. So width is 40 up to this width is 40. So 40 ult* by height will be 1.6. So this is representing 40 * by 1.6 will be equal to 64. So this bar is representing 64. And the last one this bar is from 240. This is 240 to 300. So this will be 60 ultiplied by the height of the bar is up to 0.5. So 0.5 and this will be 60 * 0.5 is equal to 30. So this is 30. Now we got all the frequencies and frequency multiply by uh frequency multiply by this uh mid-range. we'll be taking mid-range also. So uh formula which we will be applying will be what will be sigma f_sub_x over sigma f. This will be for mean and we will be calculating the mid range for each bar. We'll be taking the mid value of the range. So mean is equal to first bar is uh um 8 basically we calculated this 8. So 8 * by 140 will be the middle of uh 120 and 160 plus next bar is 22 multiply by middle will be 170 and plus next will be 30 m uh 36 not 30 36 multiply by middle will be 190. So 190 plus next will be 60 * 220 * 220 and last one is 30 * by middle of 240 and 300 will be 3 uh 270. So multiply by 270 and 1 2 3 4 5 1 2 3 4 5 All bars are done and divided by the total which is uh total we know this is given 160. So we'll be dividing by 160. Now before calculating on the calculator uh we need to rectify the mistake which is over here 40 * 1.6 640 ult* by 1.6 for this bar we are getting 64 which we have written over here as 60. So this one is the rectification I'm making 64. And here also this small mistake I'm just changing to 64. And now we can calculate this will be uh this will be 8 * by 140 plus this will be 22 * 170 and plus this will be 36 * 190 and plus this will be 64 * 220 and + 30 * uh 27 70 and divided by 160 and this is giving us the value 211.75 211.75 and this is uh the answer which is for six marks. It was a good question. Now this is question number six and question number six is column vectors. there's a work out. So just we need to calculate this. So we can multiply this as uh 2 into 3 will be 6 and - 10 and -2 and - 2 and - 7 and this will be equal to 6 - 2 and - 10 and + 7 - will be plus and this will be equal to 4 and -3. So this will be our answer for two marks. Hopefully you understand this. Now for B part MN is given M is the point this one find the value of N. So we know that uh we don't have N value. So N value we can suppose as A and B and we can write M n value will be first. So it will be a b and minus will be m value which is 2 n and -5 and this is given that this is equal to -6 and 4. So we can form the equation and that will be a -2 is equal to -6. So this implies a will be equal to 4 and uh a will be equal to -4. So -4 will be a value and b will be equal to what? B will be equal to - - + so + 5 is equal uh wait erase this so b + 5 b + 5 is equal to 4. So it means this implies b is equal to uh b is = minus1. So minus1 will be this will be answer hopefully you understand this. Now mod is required find mn mod. So that will be under root of x value which is uh uh -6 squared we are using from here and - 6 squared and + 4 squared. This will be m n mod and this will be this can calculate under root of uh -6 n² and + 4² is equal to 2 under3 and decimal value will be what decimal value will be 7.21 7.21 will be our final answer for two marks. Hopefully you understand this. Now C part C part says diagram is not to scale. O A CB is a trapezium with O B. O B is uh this is the relation given basically O is equal to 2 A C. So A C is half of O. So we can say a c is uh c because uh o b is uh 2 uh 2 c. So and further they are giving o a is equal to a o b is equal to 2 c and a p ratio pb a ratio pb is 4 ratio 4 ratio 1. So total is divided into five parts. So 4x 5 and 1x 5 this one done. Then further they are saying aq is equal to 4x 5 a c. So aq is equal to 4x 5 a c. So this will be 4x 5 of c and this will be 1 by five of c. This one and this one. Now this is the information which we got and question is write each of the following terms in terms of a c a and c give your answer in the simplified form. So a b we know a b is which one? A b is this. So we can write always you have to write the root better to write the root even for one mark question to avoid mistake always write the root. So like for example a B is equal to what? A B is equal to A O + O B. This is what I meant that you will have to write this step so that there should be no mistake. So this will minus A and plus this will be 2 C and answer will be minus A + 2 C or you can write 2 C minus A and this is done. Next they're saying CB and CB is which one? C B is this one. So we can again write the root. This is C B. We have to start from C and reach to B. And this is equal to C A + A O + A B and these are the vectors and we'll be taking C A is minus C. So we can take minus C plus A. O is minus A. Basically we will have to take it as minus A. So this will be minus A and plus O uh A basically C A O then O B. It should be O B here. So O B here. Yes. And now uh O B will be 2 C + 2 C. So answer will be C min - A. After simplification there also one more question I have explained properly so that you can understand better. This will be OP. OP is required. So OP we can draw first. OP O2 P will be this. OP will be this direction. Now OP we can take uh O B plus BP. So we will be writing this O P vector is equal to O B vector plus BP vector. So we can write this is OB is 2 C and this will be 1x 5 of uh BA 1x 5 of BA so we can write here 1x 5 of AB we have so we will change BA and change sign A minus 2 C will be B A uh this is uh basically It's not B A, it's BP. So 1x 5 of BA. Yes, correct. Now we will be simplifying this. So we can open this will be 2 C + 1 by 5 A minus this will be 2x 5 C. So further we can simplify uh 2 - 2 over 5 will be equal to 8 by 5. So we can write 1x 5 a + 8 by 5 c that will be our final answer. Uh basically 1 by 5 even we can take common also. So a + 8 c we will be writing outside this will be final answer for two marks. Hopefully you got this. Qp is also required. So QP is from where? QP. Okay. So QP is uh if we draw QP QP is this one from Q to P. So we can take the root QP is equal to Q A plus A P. So we'll be writing here. I will solve here q qp is equal to q a + a p. So this one uh q a will be 4x 5 of 4x 5 - 4x 5 of c because opposite direction and plus a we already have which is uh 4x 5 of a b which is uh 4x 5 of a b is - a + 2 c this one and further we need to just uh simplify this I will shift this to the proper proper location so that I have solved here closer to the drawing so that you can understand the root. Now is just some simplification so I will be solving here and this one we can solve here. So we will open - 4x 5 C and - 4x 5 A and plus this will be 8 by 5 C. Now we can calculate -4 over 5 and + will be 8 over 5 and this will give us 4x 5. So 4x 5 C - 4x 5 A 4 by 5 A and 4x 5 we can take common and answer I will write in different color so that you can identify 4x five inside will be C minus A. So this will be our answer for two marks. Hopefully you understood this. Then they are saying use your answer to make two statements about the relationship between lines QP and CB. QP and CB. So QP is where QP we just found which is uh this one. I will just use another color. Okay, this one QP is this one and the other one is uh CB and CB is this one. So if you check this, if you multiply CB by 4x 5, we will get uh QP. So one thing which we can write is QP is equal to 4x5 of CB. And this is what we need to mention. This is our first point. And second point we can say because they are multiple of each other. So we can say that QP QP and CB are parallel to each other. parallel to each other. And this is two mark question which we have answered. Hopefully you understood this. Now this is question number seven. A part diagram is not to scale. The diagram shows a field ABC D in both uh in B with B north of A. So okay B is here and north of A. So it means that uh this is the north. This complete one is towards the north. So BD is a path across the field. Okay. So AB is equal to 85 m. A D is equal to 72 m. BD is equal to 129 m. All three sides are given for this triangle. And the angle BDC is 39 and angle BCD is 60°. This angle is also given. And we have to show that CBD is uh [Music] 81°. So how we will be showing CBD is what? CBD we need to show that this is uh uh 81° we need to show. So what we will be doing? We will be taking CBD is this angle and this is uh uh 60 is there 39 is there. We are considering this triangle and in this triangle we can write in triangle BC D in triangle B C D uh angle uh angle C B D is equal to 180 minus 39° + 60° and this will be equal to uh 180 180 minus this will be 39 plus other angle is 60 and this will be this will be 81° this is what we have shown now they are saying calculate CD C D is uh okay so this 81 I will mention first this one is 81° and we need to find this side which is uh uh which is CD. Okay. So we can use uh uh we can use sign rule basically in in this triangle. We can use uh sine rule and sine rule will be what? Sine rule will be uh sine rule will be a uh a b a basically this one cd. So CD over CD over sin of 81 is equal to the other side is 129 / sin 60 129 / sin of 60° and this implies CD is equal to 129 sin of 81 / sin of 60° and This value we can calculate this value will be 81 129 sin of 81 and over this will be sin of 60 and this will be equal to [Music] 147.122 147.122 or approximately we can write 147 and this will be in meters And this is uh 147 m. We found this side. And this is basically our answer for uh three marks. Now they are saying show that angle AB D is 31.6 correct to one decimal place. AB D is which one? A B D A B D is this angle. So we have all the sides. So we can apply cosine rule in this triangle. And that will be uh I will be solving here then I will shift. So it will be a d a d² is equal to a b² + uh b d² b d² - 2 into a b into b d and cos of uh angle b. So a d is what? 80 is uh 72 squared is equal to 85 squar + 129 squar - 2 into 85 into 129 and cos of b and this is what we need to shift to the proper location and we will solve I have written this closer to the diagram so that you can check the angles and step side by side. Now we can write 72 squared - 85 squar - 129 squared and divided by -2 into 85 into 129 and this will be equal to b and we can shift cos inverse on this side and this will be our answer and we can write here shift cause of uh this one which is 72 squared and - 85 squared and - 129 squared 1 129 squared and over this will be -2 into 85 into 129 and bracket close and this is giving + 31.58 31.58° this will be answer 31.8° 8° I'm just mentioning over here 31.58 basically 31.58 yes this is done now for part four find the shortest distance from uh A to B D shortest distance from A to B D we will have to draw the straight line so like for example this is a straight line and this is making an angle of 90°. So we can consider now the triangle this triangle and in this triangle uh we can take this point as E. So we can take in triangle in triangle A B E A B E uh so we need A. So a e over uh a b will be equal to sin of 31 58 and this one ae e is what ae we need to find and a is equal to 85 sin of uh 31 31.58 and this is the thing which we need to calculate and we and change this to the proper location for our answer over here. So this will be our answer and we can calculate this will be 85 sin of uh 31.58 and this is giving us 4 uh 4.189 or we can write um 85 it's 85 basically diagnistic so 85 so it will be 44.51 so a e is equal to 44.51 and 44.51 m this will be our answer for three marks hopefully you understand this [Music] so ae also we found I will I just want to copy here so 44 4 51 is a E. Now we can move to the next part. Five. This is saying find the bearing of B and B from C. Bearing of B from C is where. So we can draw north over here. This will be our north. Bearing of B from C. So we need this whole angle up to this one. So we can find this angle which is uh 180 minus this angle will be what? 180 minus 31.58 plus this will be 81 and this will be equal to uh 180 minus in brackets 31.58 and + 81 and bracket close is equal to 67.42 42 67.42 is this angle I will write here 67.42 now uh remaining angle which is starting um so this angle will be this angle will be what this angle will be 180 - 180 - 67.42 42 which will be 180 minus answer is equal to 112.58 112.58 and this angle which angle I'm talking about this angle till here will be 180 minus uh 180 - 112 12.58 and this will be equal to uh we can write 180 minus answer is equal to 67 uh it's not 180 it should be 360 minus 360 not 180 because circle angle is 360 so we can check for this this will be 360 and this will be this will be 292. Let me calculate again because we changed the number. 360 - 112 - 58 is equal to uh 247.42 247.42 42 and this calculation I want to move to the proper location and this one will be our answer for this part this one. Okay. So this is done 247 247.42 this will be bearing so it's degrees. Now the next part they're saying trees are planted in the field. The number of trees planted is is 1,100 per hectare and calculate the total number of trees planted in the field. So hector is 10,000 m²ared. So we need to find the area of two triangles. One triangle is uh one triangle is this whole triangle. this one and the second triangle will be this triangle. So area for triangle AB D. So we can write area of triangle uh ABD A B D will be equal to half into base into height. Pace will be BD which we can take 129 into height we can take 44.51 which we calculated and this area will be uh 0.5 into in brackets 129 into 44.51 and this is 28 2870 2870 0 something 895.895 8 95 this one and area of the other triangle which is uh uh BCD you can write triangle B C D will be what this will be we have two sides so half into A B sin C we will be using so half into one side which is 129 B and C we can take 147 which we have calculated and sine of angle 39° and this one will be equal to I will shift this at proper location at the end. So 0.5 into 129 into 147 sin of uh 39° and this will be resulting to 596.9 596 596.9 and if we write two digits more so 01 so these are the two areas which we found. So now I will move this and rest of the calculation we will be doing at the proper location and we can complete this question. Hopefully you are understanding each and everything. Don't forget to subscribe the channel and recommend this channel to your friends and class fellows so that they can get benefit also like you. So total area total area will be equal to 2870.895 and plus this will be what? Plus this will be 5966.901 and this will be total area this already there. So + 2870.895 895 and this is giving us837796. So837.796 and now this is the total area but they're saying in hectares in one hectare it's 10,000. So area in hectares area in hectares will be what? 8837.796 divided by 10,000 and this will be if we divide by 10,000 will be this one will be equal to 0.88 0.88 88 and 37 uh we can write 3 37 96. Okay. So 378 we can write 378 and then u they are saying this many plants in one hectare. So number of plants number of plants will be equal to 0.88 88 378 * 1 1 0 0 and this will be multiplied by 11 1 and this is equal to 972 972.1 uh 157 so I will be leaving that so 972 will be our answer for four marks this is basically uh this was a little uh there are a lot of calculations. So many students are asking for some small questions uh but uh I'm unable to answer. I will be requesting to the teachers who are watching uh this YouTube channel. I know from the data of YouTube that uh more than 40% around 45 46% teachers are watching this uh these videos and uh this is a positive sign for me and uh I I will be expecting that uh they should come forward and answer the small queries which students are putting in their in comment section. So I request specifically to the teachers from the age group and uh uh from the YouTube data I know that uh you uh teachers are watching. So they they should help us and answer small queries of the students in comment section so that uh everybody can get better uh benefit from this channel. Okay. So thank you in advance for the teachers. And now that this B part B part we can go through uh rectangle has an area of 9,400 correct to so this is basically rectangle which we can draw. Okay I will take the rectangle and this one we can redraw this. So this one and this is basically the rectangle which I'm looking for and they are saying this is uh uh 80 cm correct to this rectangle length is 80. So 80 and correct to the nearest 10. So upper bound and lower bound upper bound will be plus - 5 right. So this will be 85 here and 75 here. upper bound lower bound and then they are giving an area. So area is 940 correct to the nearest 100. So upper bound and lower bound we can write this will be adding half of 100 will be 50. So 9450 will be upper bound and 9350 will be lower bound. Now we know that area is equal to what? area is equal to length into width. So width will be equal to uh area over length. And if we need area they are saying calculate upper bound of the width. So if we need upper bound of width so we will use uh we will use what we will use upper bound of area and lower bound of length. This is what we will be using and our calculation will be what? Upper bound of area is uh 9450 and divided by lower bound of this will be 75 and if we calculate this will be uh 9450 over this will be 75 is equal to 126. So 126 will be our final answer for this question. Hopefully you understood this. Now question number eight. And question number eight says, a bag contains 24 colored beads. Some of the some are red, some are blue and 10 are yellow. One bead is picked at random from the bag. Find the probability that the bead is yellow. So there are 10 out of 10 by 24. 10 out of 24 will be our answer. Or you can simplify basically that will be 5 by 12. So 5x 12 you can provide answer for this one. Next part the bead is not yellow. Not yellow will be 1 - 10 by 24 which we can write or uh 1 - 10x4 will be what? 7 by 12. You can use calculator for that. And this will be answer for one mark. The B part. Another bag contains five green marbles, six white marbles, and four black marbles. Mera pick picks two marbles at random from the bag without replacement. Find the probability that the first marble is black and second marble is white. So first one should be black. So for this one should be back. For for black it is uh there are four blacks out of 15. So probability will be four out of 15. Next we need white. So for white it will be uh six out of one less because one is gone because uh so it will be 6 out of 14 and we can calculate the probability that will be 4 out of 15 into 6 out of 14 and this will be equal to uh 4 out of 15 into basically over here multiply by 6 out of 14 and this will be equal to 4 by 35 4 by 35 35 4x 35 so 4 over 35 will be answer for this two mark question hopefully you understood this then they are saying both marbles have different color. So same one minus same color 1 minus same color will be giving us the answer. So same color can be what? It can be green green it can be white white or it can be black right. So there are three types only. Yes. So this one so one minus green probability will be what? 5 by 15. 5x 15 and the second one will be one less 4x4 plus white are six. So 6 by 15 6 by 15 and 5 by 15 and plus uh not 15 14 and next will be there are how many four uh blacks so four out 15 into 3 out of 14 and this is what we need to calculate and that will be our answer. So 1 - this will be 5 over 15 into this will be 4 over 14 and plus this will be 6 over 15 and this should be plus here I written plus here and here it should be multiplication 5 over 14 And then it will be + uh 4 over 15 and multiplication 3 over 14 and bracket will be close and answer is 74 or 105. So 74 over 105 74 74 74 over 105 this is answer for four mark question hopefully you understood this question number nine functions question f ofx is equal to 2x - 5 and g ofx is also given find what part a f of seven so we'll be replacing seven in f of x. So this will be 2 into 7 - 5. 14 - 5 will be 9. So answer will be 9. One more question. And then uh g of f of 7 f of 7 we just found 9. So g of 9 we can find which is 9 squared. Uh we'll be replacing in this 9 square - 2 into 9. So this will be 81 - 18 and 81 - 81 - 18 will be equal to 63. So answer will be 63. This is also one mark question. Then they are saying f inverse f inverse is we will be taking f ofx and f ofx we can write like this one and we'll have to make x as subject. So we can write 2x = y + 5 and x will be equal to y + 5 / 2 and further we can write swap x and y. So y is = x + y by 2 and x + y by 2 will be our final answer for this inverse. And this is uh two mark question. Next they're saying find g of f ofx minus 3 g of x and answer then require in quadratic format. So g of f of x is equal to what? g of f ofx is f ofx is 2x - 5. So g of 2x - 5 in g of x we'll be replacing and g of x is x² - 2x. So 2 fx - 5² - 2x - 5. We have replaced x by 2x - 5. Now we will be opening this and this will be 4x² + 25 and minus this will be 20x. If you multiply uh 2x - 5 into 2x - 5, you will get the same thing which I have written over here hopefully or you can directly apply the formula as well. And this will be 2x uh 2x - 2x - 5 2x - 5 like this. Now we can open the brackets. So 4x² and - 20x and - 2x and + 25 and - - plus so + 5 and this will be what uh uh there is two we missed basically 2x it's 2x over here so we have to write here two and two here so this will be 4x uh which one will be this will be 4x and this will be 10. So we can write here 4x and this will be 10. So further we can write the step 4x² - 20x and - 4x. So -4x and this will be + 35 and this is equal to uh 3 g of x. So we can write 4x² - 24x not equal to - 3. So x + 35 and - 3 into g of x and g of x is what? x² - 2x. So we can write uh - 3 into x² - 2x and we will be opening the bracket. So we'll get 4x^ 2 - 24x and - 35 and - 3x² and plus this will be 6x and further we need to basically add the like terms. So 4x² - 3x² so only x² and 24 and 6 - 24 and 6 will be - 18x and further plus it should be + 35 here. Yes s + 35 and this is the answer which we were looking for. X² it's in the same format which was required. Hopefully you understand this. This was uh this is four mark question which we have completed. Now question number 10. Question number 10 is differentiation. They're saying a curve has this equation. Differentiate this. So differentiate mean we can write dy by dx is equal to what this is equal to 3x² you know we are pulling power first and reducing power by 1. So - 9 into 2 will be 18 x and - 48. This will be the answer which we can write over here. It's only two mark question which we have done. And next is B part. They're saying find the coordinates of the turning point of this curve. You must show all your working. So dy by dx is equal to zero which we already found. It's the same equation if you check this equation and this equation is same. So derivative is this one which we'll be using and we will write 3x^2 - 18x - 48 is = 0. And we will be finding the x value. So we can take common and inside will be x² - 6x and - uh 16 is = 0. And we know that 3 is not equal to 0. This implies x² - 6x - 16 is equal to 0. And we can solve this by factorization. So mode 53 coefficient of x is 1. Coefficient of uh coefficient of x² is 1 cofficient of x is - 6. and the constant is -6 and the value will be 8 and -2. So x - 8 and x + 2 is equal to 0. And intermediate step also we can give like this one. Uh this will be x + 2 =0. Hopefully you are understanding. So x - 8 is =0. This implies x = 8 and x + 2 is =0. So x= -2. So 8 and -2. Two values we found. Now the second value we can find by replacing this in the equation which we already have which is y is equal to x cq - 9x^ 2 - 48x. So this will be we'll be replacing 8 first. So this will be 8 cub - 9 into 8² - 48 into 8 and this will be um 8 q and - 9 into 8² and - 48 into 8 we can write like this and this is uh there is uh some mistake I think so no it's fine basically -4 48 -4 48 so when we replace 8 value we are getting -448 and if we replace -2 if we replace -2 y = -2 cq - 9 into -2² and - 48 2 - 2 - 2 here. So this will be what this will be in brackets -2 q and next will be -9 into -2 -2 and this is square and - 48 into -2 is here -2 is equal to -30 36 we had done again some mistake but we'll have to double check - 2 cq - 9 into -2² and - 48 basically we have written 4 and this is giving us 52 and this is correct this is uh 52 yes so this is answer for four marks hopefully You understood this. Then C part determine whether each of the turning points is maximum or minimum. Give your answer correct to the nearest. Okay. So uh give reason for your answer. So d second order derivative test we can apply d2y / dx² and derivative is this one which we have already taken 3x² - 18x - 48. So second derivative will be 6x - 18. So this will be what? Not 6x - 18. Yeah. So we can even take the derivative of uh this one. Okay. So 6x - 18 we can use 6x - 18 and this will be if we will be replacing 8 this one - 18 and this will be what? This will be 6 into 8 - 18 will be equal to 30 which is uh so this will be this will be this point a which point 8 and -4 48 this will be our minimum because this is greater than zero so this will be minimum minimum and if we replace uh uh 6 x - 18 in this one. If we replace -2, so this will be 6 into -2 -8 is equal to -8. I'll write here properly. So this will be giving us what? This will be giving us 6 into -2 and -8. And this is giving us -30. So which is less than zero. So this point the other point which is uh -2 and 52 is maximum. Hopefully you understand each and everything. With this one we are done with the paper and don't forget to subscribe the channel and recommend this channel to your friends and class followers and introduce this channel to your respective teachers. Thank you very much for watching. Take care. Bye. [Music]