GRADE 9 MATHEMATICS, END TERM 2, KJSEA PREPARATIONS

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[00:00:08]This is KGA Kenya Junior School Education Assessment Preparations and we're going to look at a mathematics paper which has been set for term two and term. Remember to subscribe, remember to share, remember to invite a friend to this great presentation.

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[00:03:08]So we can now check the section B of this paper. uh the structured questions.

[00:03:15]Number 21.

[00:03:17]A number P is formed by listing all prime numbers between zero and 10 in ascending order. Another number Q is formed by listing all odd numbers between 0 and 10 in ascending order.

[00:03:36]Express Q minus P in words.

[00:03:41]Uh-huh. Let's check what the examiner intends uh you to do in this particular case. First, we can check the number P.

[00:03:51]This one is formed by listing all the prime numbers. The first prime number is two, then three, then five and 7 between 0 and 10. We together in that? Uhhuh.

[00:04:06]Then we can check Q which is formed by what? Again in ascending order we're talking about the odd numbers. Odd numbers start with 1 3 5 7 and 9. So we have a P and Q. What we are finding is the difference between P and Q. Then we express it in words. So 13 57 9 - 23 57.

[00:04:35]Now when you carry out the difference you get uh 1 one triple 2 that is now the difference between you and P in words means we talk about 11,000 11,000 Uh-huh. 222.

[00:05:04]2 and 22 202.

[00:05:16]Yeah. 2 and 22.

[00:05:20]That part A has been squared properly.

[00:05:23]Done. Look at part B of the same question. How many times is the digit in thousand's position greater than the digit in the on's position in the number formed by Q minus P. So in this number this one we are asked how many times is the digit in the thousand's position. Thousand's position. What is there? Ones, tens, hundreds, thousands. There is a one and its total value.

[00:05:59]How many times is the digit in thousand's position greater than the digit in the on's position. All right.

[00:06:10]Not the total value but just the value.

[00:06:17]How many times is the digit? So it's just about the digits. So in the thousand's position, the digit is one greater than the digit in the on's position. So once position, which digit is there? It's a two. So the examiner is claiming that a one is greater than two by a number of times. And for such a case, you are supposed to divide.

[00:06:46]Therefore 1 is greater you divide by 2. So 0.5 times yeah 0.5 times when you multiply 0.5 times by two you should get one that means 1 is greater than 2 0.5 times actually I thought they are talking about the total values but now the total values of the digits is not mentioned they just asking how many times is the digit in the thousand's position greater than the digit in the on's position that way.

[00:07:29]So me that is how it's supposed to be done.

[00:07:36]Look at number 22 ladies and gentlemen in a meeting three out of five of the attendants were women. A quarter of the remainder were men and the rest were children.

[00:07:49]There were 80 men in that meeting. How many children were there? Now this is a question that should be done systematically and with a lot of order. First we can talk about women.

[00:08:04]Women were three out of five.

[00:08:09]A quarter were men and it's a quarter of the remainder. A quarter of the remainder were men. So the remainder is 2 over 5.

[00:08:20]Meaning that men shall be a quarter of 2 over 5. Uh this is 1 over 5. Is that okay? 1 5 1 / 10. Sorry.

[00:08:36]Yeah. 1 / 10.

[00:08:42]This is 1 / 10. That is the ratio for men. The rest were chinton. So chinton shall be the total proportion minus the fraction that has gone to the women and the one that is getting to the children.

[00:08:58]Uhhuh. The fraction for women and the fraction for men when added then we subtract now 1 minus the combination of men and women we shall get the proportion for chint. So chint are going to pick uh this particular ratio. I'm adding I'm adding up and I'm getting three out of 10. So three out of 10 proportions this one represent this one represent children. Now we go to the question how many children were there? You know the point is the point is that there were 80 men.

[00:09:44]Just check men. We are saying one proportion equals to 80. What about 10 proportions?

[00:09:52]Because men are 1 / 10. So one proportion represent the number of men.

[00:09:57]What about 10 which is the total proportions to be 10 by 80 giving us 800. So we had we had 800 human beings in this particular meeting which is actually not a main thing. We would still have gone to one proportion for 80. What about the proportions for children are only three out of 10. So three proportions shall be what? 80 by 3 giving us 240 children. If one proportion equals to 80, what about three proportions? One proportion for children. What about one proportion for men? According to this ratio, what about three proportions for children according to what we have gotten here. You get like that. And later you can be telling me why is it that in every meetings in a meeting and I think they should have been specific in a party in a wedding in a graduation ceremony you'll find that the proportions for men and children are usually greater than those of men. I've never known why. So you can also do research on the same and update us. You can update us on the comments section. Why is it that men are fewer the mature men the adults ceas why are we the minority in terms of numbers?

[00:11:39]During a mathematics lesson on indices the nine learners discussed about the laws of indices. After the discussion, the teacher of mathematics wrote the following equation. Half power 2 t * 2^ 3 is equal to 128. Solve this equation or solve the equation. One, we can talk about uh what you call bases, equiting bases and also working with what you call loss of indices. There's a law uh that we call the negative law which says whenever you have a certain base given to a negative power you can take the reciprocal of the base then you have the power changing its sign. So that can be applied in the first term here. Instead of f we make it two. And now the power changes its sign tot 2 * 2^ 3. And this is equivalent to 2^ 7 instead of 128. 2^ 7 128. So with this we can say on the left hand side we can just apply the law that says whenever you have a base raised to a given power multiplied by a different base to a different power. Okay the same base but now to a different power you can simply take the base and you add the two powers. You add the two powers. So at this point you have 2 raised to 2 t + 3. This is equal to 2^ 7. At this point now we can equate the indices -2 t + 3 this is equivalent to 7. Meaning that -2 t is equivalent to 7 - 3 which is 4. To remain with the value of t, we divide through by -2 and it will give us -2 as the final answer in such a case. A good question. Very very good question on indices. When you follow up with the shifting grades online school, you shall discover that we've solved a lot of indices questions, equations, expressions, numerical problems. We have done our best on this particular topic. So kindly find time and see how we have exposed how we have exhausted this particular topic. Looking at number 24 now. All right.

[00:14:39]Yeah. In number 24 guys, in number 24 we told that a sequence is given as 8 x 12.5.

[00:14:49]It continues if the numbers in the sequence are in proportion.

[00:14:55]Find the value of x. If terms in a given sequence are in a proportion, then it means they are giving us the same ratio.

[00:15:06]Like when we take x we divide it by 8 we get the same as when we take 12.5 and we divide by x.

[00:15:18]You see that 12.5 uh divided by x like that. At this point we can do a cross multiplication.

[00:15:30]A cross multiplication whereby x now becomes squared and this is 8 by 12.5.

[00:15:40]I'm getting 100 exactly. If x squ gives us 100 then x shall be the square root of 100. We take square root on both sides. The value of x is equivalent to exactly 10. So that is how number 24 is supposed to be approached. A good question on continuous proportions.

[00:16:07]25 right. Yeah. Number 25. In number 25, these are the instructions. Two integers -3 and -5 were written on board during a grade 8 mathematics lesson. Use the number line to compare the two. Use the number line to compare uh the two.

[00:16:33]Okay.

[00:16:36]Now to compare the two means to compare the two means. Now we supposed to state which one is greater than which.

[00:16:46]Uh so this is going to be 0 1 then -1 -2 -3 -4 -5 -6. Looking at the positions, one is at -5, another one is atu3.

[00:17:12]Therefore, the one to the the one to the left hand side is usually a smaller one. Therefore, -5 is less than -3. If this is true, then it will also be true to say -3 is greater than five. Any of those two statements will earn a mark for you because they mean the same. When I say you are taller than James, then James is shorter than you.

[00:17:43]It's logical statements. They mean the same though they might imply they might imply different things but the meaning is just the same. Now perform the operation below on integers. You see a number line one. You're supposed to know that there are two signs here -3 which is followed by - - - - - - - - - - - - - - - - - - - - - 5 it will convert this to a positive. So when the sign changes to a positive that is after after two consecutive uh positives then now we can change it to a plus1 -2 -3 -4 then 1 2 3 remember here we are adding to 3 5. So -3 is here. Adding five means five steps. Because it is addition, we go to the right hand side. 1 2 3 4 5. So the answer is positive2 according to the number line pos2. So we have added like that. Ah you can check number 26 ladies and gentlemen. In a patent six from Fili distributors, three p and nine books are sold at 750 shillings.

[00:19:21]If six p and five books are bought from the same seller, they cost 260 shillings less. Calculate the cost of two books and a pen. Two books and a pen. Now we can form two linear equations and we solve them simultaneously. Remember MP can be for pens and B for books. So 3 P three * the cost of a pen plus 9 * the cost of a book. This will cost 750.

[00:19:56]Then if six pens and five books, six pens and five books were bought from the same seller, they cost 206 less. So 750 - 260 they would cost for 90. That's after subtraction. So we have now two equations here. We have equation one. We have equation two. I know there are several methods that can be used to solve simultaneous equations. But for this allow me use substitution method.

[00:20:36]From the first equition 3 P to remain here to remain with 3 P on the left then we will have 750 - 9 B. meaning that P is equivalent to you divide 750 by 3 you get to 50 - 3 by B because we dividing everything by 3 now that we are expressing the value of P from the first equation then substitution should be done in the second equation 6 P so instead of P we're writing 250 uh minus 3 B + 5 B equivalent to 49.

[00:21:27]Opening of brackets.

[00:21:30]Uh-huh. 15 0 0 - 18 B + 5 B = 490.

[00:21:42]This is equal to 490.

[00:21:47]490 like that. So now we can start another business of grouping terms together.8 8 we add 5 this becomes -3 the value of B this is equal to 490 minus 18 0 0 this becomes -10 10 how do we remain with the value of B we divide both sides by3 we divide both sides by Allow me check closely.

[00:22:27]Allow me check closely where maybe a small mistake might have happened because I'm getting a very rational number. Very rational.

[00:22:43]Mhm. We took 750 and we subtracted 260.

[00:22:46]We got for 90. That is okay.

[00:22:49]And now 250 by 6. That's okay. 15 0 0 it can be. So allow me check. Let me check where maybe a mistake could be happening.

[00:23:08]Where a mistake could be happening.

[00:23:11]This one we divided well by three both sides.

[00:23:16]Then now we substituted that this point.

[00:23:20]Uhhuh.

[00:23:22]Like that.

[00:23:27]So we have to be very careful at that point because now I'm getting decimal places here. I'm getting 10 10 / 13 which is giving me 77.70 or 77 69 Kenya shillings. Okay, we've done everything right. Let us stick to the values like that only that in most cases uh the money is all numbers. But now we can see that that a book is going for 77. What about a pen? According to this instruction, a pen now shall be 250 - 3 * b.

[00:24:17]So the value of P shall shall be 250 - 3 * answer giving us 16.92 16.92.

[00:24:35]Then after that now we get to the question. Calculate the cost of two books. So 2 * a book and a pen 16.92.

[00:24:51]The cost now here shall exactly be 172.31.

[00:25:02]Yeah.

[00:25:04]Simple. A very good question. Very very good question.

[00:25:12]Mhm.

[00:25:13]We are checking number 27 right now.

[00:25:16]Five was added to unknown number x. The results were multiplied by five then divided by two. Write the correct expression for this statement.

[00:25:29]One five is being subtracted from then the results were multiplied by five.

[00:25:40]Everything by five then divided by two.

[00:25:47]Everything now would be divided by two.

[00:25:50]According to me we have expressed everything.

[00:25:57]Uh-huh. Five added to a number x five was added. So this supposed to be adding sorry five was added. After it has been added the result is being multiplied by five. So it's everything by five. After which we now dividing by two. Now dividing uh by two like that.

[00:26:21]Mhm. 28.

[00:26:24]What is the value of r in the equation?

[00:26:28]r - 3 out of 3 + 2 + r out of 2 = to 5. So allow me talk about five as a whole number to be automatically over one. Then we talk about the LCM of the denominators. The LCM of 3 and 2 is 6. So it's now LCM divide by each denominator. LCM divide by 3 we get two.

[00:26:56]So R minus 3 should multiply 2 plus 6 / 2 we get three. Three should now multiply 2 + R 2 + R like that. This is equal to R5 also multiplied by the LCM. Remember we are dividing 6 by 1 to get six. Then now we multiply the result by the numerator.

[00:27:24]After that you are done with the LCM.

[00:27:27]You now expand by opening the brackets.

[00:27:31]Then you group terms together. After which you simplify to get the value of r. So this is twice the value of r. Then -6.

[00:27:42]You add 6 and 3 * r. This is equal to exactly that when we start grouping terms in a case like this we have 2 R and 3 R on the same side. So that totals to 5 R.

[00:27:58]The -6 + 6 that will cancel out. On the right hand side we are having 30 already - 6 + 6 that gives us a zero. Then the other side there is already 30. The value of r shall be you divide both sides by the by the coefficient of r.

[00:28:22]This will give us six. If this is going to give us exactly six like that.

[00:28:30]So that is the answer to the linear equation. That is the answer to the linear equation. It's a good question. A good question on linear equations. So grade nine learners are supposed to be in a position to work out such a question because this is grade eight work linear equations and it can be assessed in uh kg here.

[00:28:59]Are we looking at number 29 ladies and gentlemen? Now the cash price of a TV is 20,000 shillings. Mr. Tulo bought it on higher purchase terms. The total amount he paid was 25% more than the cash price. He paid a deposit.

[00:29:21]He paid a deposit Mhm. of 5,800 shillings and the rest in 12 equal installments.

[00:29:33]How much was each deposit? What you need to do first here is to ask yourself if the cash price is 20,000 what about the higher purchase price which is 25% more. So we need 125% of 20,000 so that we may know the higher purchase price. So the higher purchase price shall exactly be 25,000.

[00:30:11]H that is the higher purchase price.

[00:30:14]Then what will be the balance after the deposit have been made? Once the deposit is made, there shall be a balance. Now it is the balance which will now be paid in installments.

[00:30:28]So the balance is 19,200 19 19 uh 200 Kenya shell lines. Now with this balance you're supposed to know that it's being settled in 12 installments.

[00:30:43]12 installments. What about each installment?

[00:30:48]If all the installments now shall total to each installments okay all the installments shall total to 19,200 what about each installment so we having 19200 out of the 12 installments when you divide this value ladies and gentlemen you get exactly 1,600 Kenya shillings Kenya shillings A very good question on higher purchase that is simply on money.

[00:31:24]Money is a serious topic which cannot fail to be evaluated in any national examination. So all the concepts on money that you've already covered up to this point including profits and losses, discounts, commission, higher purchase, appreciation, depreciation, and we have the taxes in grade nine. You're supposed to be well conversant with them so that when assessed you are in a position to work them out. Number 30.

[00:31:59]The figure below shows a water trough.

[00:32:03]You can see the measurements. The diameter on the circular part is 98 cm.

[00:32:09]Then the overall length is 2 m. That means whether they are asking surface area or volume, we can't work with units that are not the same. So we can now convert this to 200 cm so that you work with similar units.

[00:32:28]This is a water trough meaning that the top is open. The container cannot be closed because it is used to to put some water. And in case you close it, that will be a lid and not the container.

[00:32:45]Therefore, it is having an open top here. The top is open. The rectangular part that means how many surfaces are we having here? There's a curved surface.

[00:32:57]Mhm. There's a curved surface and we can see it.

[00:33:04]And then there's also there's also two semicircular parts which eventually will join to become a full circle or we can say semicircular parts which are two the semicircles.

[00:33:20]Uh-huh. The semicircles will be given by half pi r². That is the formula of the area of a semicircle. meaning that it's half * 22 / 7 according to the instruction multiplied by 49 as the radius. Remember 98 is a diameter. Then times two such we see 2 and two will cancel and it becomes a full circle. So 22 / 7 * 49 will give us exactly radius squared. I think we are leaving out something in our substitution. We wrote our formula very well. P<unk> R² R P<unk> R². So R then squaring we multiply once again then by two so that the two cancels now. So by 49 I'm getting exactly 75 46 square cm that is now on the two circular parts. Then the curved surface, the curved surface, this one, the whole of this curved surface, the whole of it up to the other side. We usually say this that a curved surface area in a cylinder is usually given by 2 pi r multiplied by length. But now that it is half a semicircle, then we multiply by half everything. So 2 pi r * the length or 2 pi r * the height will give us will give us the curved surface of a full cylinder. But now we are only interested in the half of it. So this shall be 2 *<unk> * a radius of 49* a length of 200. But now we multiply this by half. The two can cancel out. Then we work with 22 7 * 49 * 200 giving us exactly 30,800 uh-huh square cm. So with this we can talk about total total surface area which will be the curved surface.

[00:35:53]We add uh the two semicircles 7546 7546 So we add 7546. This is exactly 38,346 square cm.

[00:36:15]38,346 square cm square cm.

[00:36:28]Yeah, they didn't regulate on anything.

[00:36:31]But in case they did, we would have left this one as 2 m and we remain with nine.

[00:36:39]Actually, we'll make it a zero point.

[00:36:43]Uh-huh. when we make it m it will be divided by 100 0.98 m.

[00:36:52]So when you use 0 uh 98 when you use 0.98 and the meters here you leave them as 2 m the answer you will get shall be in square m square m otherwise you can know that this area can still be converted to square m by dividing by 10,000 making it 3.8 8 3 4 6 square meters like that. Yeah, square m.

[00:37:27]So the two answers can work since we didn't have any preservations on the units to leave our final answers at.

[00:37:39]We are now at number 31.

[00:37:42]Yeah, at number 31.

[00:37:45]Uh-huh. Number 31, we told that this school water tank in the shape of a cylinder has a diameter of 2.4 and it's 3,000 mm high. Determine the capacity in L of water the tongue can hold when full. It's also very good for you to know that unit conversions is really being applied in this particular paper.

[00:38:15]unit conversions. So you're supposed to know that 1 m is equivalent to 1,000 mm.

[00:38:27]So the 3,000 mm here shall be equivalent to you divide by 1,000 to get 3 m. So the height of this particular water tank is 3 m. Now we talking about volume first. volume which is given by<unk> r² * the height. Uh-huh. Pi r 1.2 because this is diameter supposed to be very keen.

[00:39:03]You square this. Then you multiply by the height of 3. Remember you get cubic m. Now 22 / 7 * 1.2 2^ 2 * 3. This is giving us 13.5 77 4 m^ 3. And now you say uh 1 m^ 3 is equivalent to 1,000 L. So when the meters power 3 are 13.57714 she should be converted by multiplying by a,000 and this will exactly give us 13,000 7 okay 57 74 lit. So we multiply by a th00and to convert to liters. A good question. Very very good question, ladies and gentlemen. And thank you for following.

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[00:41:07]So candidates are on their own from force separately, grade nine separately each with their own examiners per day such that if it's a day for mathematics we shall exhaust mathematics to the fullest and allow our clients to ask questions and to respond to questions and to get the best from the best examiners that we are bringing from neck. Remember, we're also going to have guidance and counseling sessions on academic excellence and peer pressure management.

[00:41:45]Reach out through our number 07041536.

[00:41:51]Book your chance. We are doing it purely online. But remember if you are within Nairobi or the environments of Nairobi you can engage me through the same number for a physical mentorship program and that is purely over the weekend on Saturdays. So you can reach out for a mentorship program otherwise thank you so much for supporting us. Remember, we already talked about the learning areas that you will feature in our neck program. Please remember to enroll. Do not be left out. Please do not be left out. Are we together? Number 32. Look at this. Amos bought a car for 400,000 Kenya shillings. Each year, the value of the car depreciates by 8%.

[00:42:50]What will be the value of the car at the end of the second year? Let us look first. By the end of the first year, how will it have depreciated? So, it's going to depreciate by 8% of its current value.

[00:43:11]8% of the uh current value.

[00:43:16]So it shall be costing 3 and then 20.

[00:43:21]Wait, it's going to it's going How many zeros are here? 1 2 3. It's going to depreciate by 32,000.

[00:43:30]Meaning that new value shall be equal to it was originally 400,000.

[00:43:41]We now subtracting 32,000.

[00:43:45]Uh-huh.

[00:43:49]And this gives us exactly 368,000.

[00:43:59]368,000.

[00:44:01]So by the end of the first year the cash value shall be 368,000.

[00:44:08]Then after this we ask ourselves in the next depreciation you know it will be starting with now 368. So 8% for the second year now with 368,000.

[00:44:24]How much will we end the year?

[00:44:27]How much are we going to lose by the end of this year? We shall lose 29,440 at the end of the second year. Meaning that value at this year shall be Mhm. We subtract the value lost.

[00:44:52]We subtract the value lost.

[00:44:55]Mhm.

[00:44:57]This is going to give us exactly 338 after 2 years 560 Kenya shillings.

[00:45:07]So after 2 years this will be the value of the car. You are supposed to know how to treat depreciation and also how to treat appreciation which is the reverse process of depreciation.

[00:45:24]So kindly equip yourself. Please equip yourself.

[00:45:30]Are we looking at number 33 together? A model was made in the shape of a triangular prism below. Look at the prism. There is a height of six. There is a base of 8 and a length of 20.

[00:45:45]Determine the area of the prism in square meters. Now, when you're told to determine area, you're supposed to know that this supposed to be surface area.

[00:45:54]That's a typing error according to me.

[00:45:56]Area is always for a plain figure. But when it becomes a a solid like a prism, we talk about surface area. When we talking about surface area, it means every face in the solid and we add them.

[00:46:10]So this is surface area not and not area. So we identify if there could be any similar shapes like the triangle here. If we call this one A B C D E F when we talk about triangles A B C and F E D they are similar. Therefore, for the triangle parts, for the triangles, we have F times a base of 8 and a height of six. But they are two such 24 by 2 we get 48 uh square cm.

[00:46:58]Uh-huh. The next shall be rectangles.

[00:47:02]You know rectangles are many. So rectangles we can talk about rectangle.

[00:47:07]Rectangle B A E F. This one here B A E F. That's a rectangle. What is it measuring? This is 20 on CB. So BF is also 20. So this is measuring 6 by 20. 6 by 20 should give us 6 by 20 should give us 12 and 0.

[00:47:37]12 and zero.

[00:47:41]Uh-huh. Another rectangle. Rectangles are three, remember? So in a prism like this, we have five faces. Another rectangle which can be called uh-huh e.

[00:47:57]This one D E A C that's a rectangle.

[00:48:03]Uhhuh. And A C is not known. A C is an hypotenuse for ABC. Meaning that 6 2 + 8 2 then we take the square root this becomes 10 cm. So the rectangle now the sloping rectangle is measuring 10 by 20 giving us exactly 200 square cm. There is another rectangle ladies and gentlemen another rectangle at the base which is a B F DC at the base now which is 8 by 20.

[00:48:43]This is 16 and a zero.

[00:48:47]So we have all the faces, two triangles and three rectangles independently. So can we talk about total now?

[00:49:00]Total surface area.

[00:49:03]Uh-huh. 48.

[00:49:07]Yeah. 48 plus the first rectangle 120.

[00:49:12]The second one 200. The third one 160.

[00:49:17]Mhm.

[00:49:18]Let me add up. I'm starting with 48. I go to 120. I go to 200. Then I add 160.

[00:49:26]This is exactly 528 uh square cm. But look at the instructions. Back to the instructions.

[00:49:36]Square m. So we divide by 10,000 we get 0.05 28 square m 0.0528 square m. It's good for you to remember that instead of converting the final answer, you can convert this even at the you can convert this you can convert this even at the length level whereby instead of working with six you work with 0.06 m 0.08 08 m. We are dividing cm by 100 0.2 m. Then this one you work with 0.1 m. When you work with the units when they are already in meters, you automatically arrive at area which is already in square meters. But in this case we worked with cm and we converted the area itself. Either way, you shall arrive at the same answer.

[00:50:52]Uh-huh.

[00:50:54]That is good. We checking number 34 together, guys. We are checking number 34 together. In number 34, we being told to construct triangle ABC in which AB = to 7 cm.

[00:51:11]Uh-huh.

[00:51:13]Then uh BC 6 cm and AC 5 cm. Now draw a circle that touches the vertices of the triangle.

[00:51:29]Find the radius of the circle. find the radius of the C or rather uh measure determine the radius of the circle.

[00:51:44]Mhm. We start by our straight line at line AB which is 7 cm.

[00:51:58]Let me get my ruler so that I may measure accurately.

[00:52:03]7 cm accurately. Seven uh cm.

[00:52:18]So that is point A.

[00:52:22]Then we measure we measure Mhm. Seven.

[00:52:28]So exactly 7 cm shall be at that point and it becomes our B 7 cm that way. Then from there BC from B to C. So I go to my ruler I measure 6 cm. Then now I do an arc. I do an ar 6 cm. Therefore, from B I use six cm because it's an arc, it touch a different point.

[00:53:04]Mhm.

[00:53:08]All right.

[00:53:13]So, let me let me do this arc 6 cm.

[00:53:17]So, I'm just doing an arc from point B.

[00:53:21]uh 6 cm long.

[00:53:25]That's an arc using 6 cm. Then I'm supposed to do the same but now using 5 cm I think uh 5 cm for AC. See I think this arc can be brought closer.

[00:53:42]The ark for six.

[00:53:45]The ark for 6 cm can be brought closer.

[00:53:52]It can be made longer because the other one is aha. We have our ar there using 6 cm.

[00:54:02]What about a to c? We use 5 cm. So an ark length of 5 cm. an arch length of 5 cm.

[00:54:13]Uhhuh.

[00:54:17]An arch length of five.

[00:54:23]Uh five five cm like that. So you can see.

[00:54:35]Mhm. So the two acts are meeting at a point that is what we call point C. Then now we use our our ruler to join the point of intersection of the two ars.

[00:54:50]the point of intersection of the two ars to be yeah like that we have joined the point of intersection which is our C and that means we are having a triangle and now we we draw a circle ladies and gentlemen we draw a circle that touches we draw a circle that touches es Mhm.

[00:55:23]the vertices of the triangle. Meaning that we're going to bict any two lines.

[00:55:29]We bisect any two lines. And it's a perpendicular bisection, not just any biscting.

[00:55:38]A perpendicular bisection. Yeah, perpendicular bisection. So the first line to bisect shall be AB.

[00:55:50]So we bisect A like that.

[00:55:54]Then we also biseect the line BC.

[00:55:59]Then we check the meeting point of the bis sectors.

[00:56:04]We examine the meeting point of the bis sectors. You see the bis sectors are meeting at a point here.

[00:56:15]That becomes our center. Oh, that becomes our center. Now with our center, ladies and gentlemen, with our center, we can now use a suitable radius.

[00:56:28]We can now use a suitable radius to come up with uh to come up with the circle. Now, uh-huh. We check whether it touches point C.

[00:56:44]Okay.

[00:56:47]Uh-huh. Then we check whether it touches point A.

[00:56:51]Check through point A. Good. We check whether it touches point whether it touch point B.

[00:57:05]Mhm. Good one. Very good. So now using that radius can first make axle here. I'm just going to do it where our circle uh shall be passing.

[00:57:24]Very clear. Very clear pathway.

[00:57:28]Very clear pathway.

[00:57:32]Mhm. Very clear pathways. Okay.

[00:57:51]That is what we'll be tracing.

[00:57:54]That is what we shall trace.

[00:57:59]Uhhuh.

[00:58:05]Yeah. I think this is now good.

[00:58:08]This is now good. We are We are good to go. We are good.

[00:58:16]We are good to start our our circle.

[00:58:23]We are good to start our We are good to start our sle.

[00:58:35]Uhhuh.

[00:58:37]So allow me use now.

[00:58:42]Let me use my blue to do the circle.

[00:58:49]Mhm.

[00:58:51]And remember we are just going to lift uh the compass just the way we are using it. We lift it directly.

[00:59:03]We lift it directly.

[00:59:07]We lift it directly and measure the radius. And this is exactly 3.6 cm.

[00:59:16]there is an error of plus or minus one cm. So the people with 3.5 the people with 3.7 they are allowed to score something but it's a good question. Very very good question.

[00:59:34]Uhhuh.

[00:59:36]Looking at number 35 now use the graph paper provided to solve the following simultaneous equations. x + y = 5 and 2x - y = 4. The first thing that you supposed to do in this case, essential that you come up with what you call a table of values. A table of values for the first equation or the first equation and allow me also give you uh something that will help you. You know, it's good for you to be aware that the intersects are the best when it comes to coming up with an equation. We call this one table one.

[01:00:26]Uh-huh.

[01:00:28]When x is zero and when y is zero. When x is zero, what is y? We substitute x with a zero, you know, it will just cancel like that and y becomes 5. Then we substitute again in the same equation. When y is a zero, x shall be five. So we are now having we are having the intercepts or the first line. Can we check this second line? Table two.

[01:01:00]Table two.

[01:01:07]In table in table two x y uh-huh 0 0 when x is a zero what shall be y when x is a zero this part everything becomes a zero. So 2x = 4 and you divide through by 2 and this value becomes also 2. So y intercept is a two.

[01:01:33]What about when x is a zero? When x is a zero, it's the all of this part that will become a zero. So y = 4 meaning that meaning that we can conclude now and say x = to 4 like that. This is our table two.

[01:01:54]Table two like that. Uhhuh. Now that we have the intercepts, we can plot our graphs. Yeah, we can plot our graphs to solve simultaneously.

[01:02:10]Supposed to plot the two equations on the same cartesian plane. Same cartesian plane. Largest values are going up to five on x and five on y.

[01:02:26]So can have our yaxis at this point.

[01:02:34]Yeah.

[01:02:38]Mhm. So this is zero. This is 1 2 3 4 5 6. This is one 2 3 4 5.

[01:02:54]Uh-huh.

[01:02:57]Then we are talking about -1 -2 3 -4 -5 -1 on the wire -2 3 4 -5.

[01:03:13]Then now we start the plotting. When plotting, let me use a black here. When plotting, we should start with the intercepts. The first y intercept is a five this one and x intercept five. So we can come up with this particular line. Now joining the two intercepts 5.

[01:03:43]Okay. We can also write the equation.

[01:03:47]This is x + y.

[01:03:54]x + 1 = to 5. You see x + y = 5. Uh-huh. After that we can get to table two about the other equation. The x intercept is at two. That point the y intercept is auh 4. which is here. So using a straight line we come up with the second line. So we join the two intercepts the two intercepts.

[01:04:34]And wait what have we said? X Y intercept is a pos2 of the first equation pos2 which is at this point Y intercept is a pos2 but X intercept is a -4 there X intercept is -4 not Y intercept XIS -4 so now we can do straight line we do a very straight smooth line.

[01:05:12]Uh-huh.

[01:05:14]This line is line 2x - y = to 4. Now that we have the two equations plotted on the same cartian plane, we can now extract the point of intersection. We can now examine the point of intersection of the two lines.

[01:05:34]This is the point of intersection. So the point intersection on the x part we can read a two there meaning that the value of x in the simultaneous equation equals to two.

[01:05:46]Then on the y-axis what are we able to read along the y ais the value we can read here is a three. So now the simultaneous solution is x = 2, y = 3 according to the point of intersection coordinate.

[01:06:05]The point of intersection coordinate 2 3 2 3. A very good one. 2 3 like that. 2 3 then number 36.

[01:06:20]The exterior angle of a regular polygon is 2/3 times the interior angle.

[01:06:29]Determine the number of sides. Hence find the number of triangles in the polygon. What you should know ladies and gentlemen is that interior plus exterior gives 180. So if interior is let to be x in this case it means that the exterior shall be 2/3 x and so according to the first statement here x + 2/3 of x should give us 180. every interior ankle plus the corresponding exterior should give us 180. So this is 1 and 2/3 of x giving us 180.

[01:07:23]This is the same as 5 / 3x being equal to 180. And therefore it means finding x means you multiply by 3 over 5 both sides. So 180 by 3 uh 180 by 3 over 5 giving us 108.

[01:07:49]So an interior angle equals to 108 108 meaning that exterior shall be exterior shall be 72. 2/3 of this 2/3 of this is 72. If it is 72. Now we say that 360 over 72 = to 360 over 72 equals to exactly five sides.

[01:08:22]You're supposed to be remembering that when you pick 360, we divide by an exterior angle. We should get the number of sides.

[01:08:31]Simple. You get the number of sides. Now that you have the number of sides, how about the number of triangles? Triangles is usually given by number of sides minus two. Therefore, it means a pentagon which is five sided can only be divided into three triangles. So the first the final answer is a exactly three three triangles. So number of triangles equals to sides minus two. A good one. A very very uh good one. Number 37. Ladies and gentlemen, number 30. Uh seven. Now the figure below shows a regular polygon.

[01:09:24]Regular one. Okay.

[01:09:30]uh calculate the value of x. If it's a polygon and it's regular, let's count the number of sides. 1 2 3 4 5sided.

[01:09:50]Now in a five sided figure you're supposed to know you know we've said that 360 out of exterior gives us n then it means when we take 360 and we divide by number of sides which is five we shall get exterior in simpler terms being a regular polygon it means number of times or number of sides times each exterior should give us 360.

[01:10:23]Number of sides times each exterior should give us 360. So to remain with exterior angle size of an exterior you need to divide by n through. Therefore 360 over 5 give us exterior which is exactly 72.

[01:10:43]Exactly 72. We are dealing with the pentagon that we are dealing with in number 36. In other words, find also the value of the angle on the other side. If it's if it's regular, it means theta is also the same as the interior on any other point. All of them are das because they are equal in a regular polygon.

[01:11:05]According to this statement, everything is equal. So, we need 180 minus 72 to give us 108. Each interior angle each interior angle is equal to 108.

[01:11:21]Yeah. 108. A very good question. Very very good. Uhhuh.

[01:11:30]Find the sum of the interior angles of the polygon. You see for us to find the sum of interior angles we can say the interior angles are five and each is measuring 108 according to Roman 2 meaning that we have 540.

[01:11:47]Alternatively we can say that the sum of interior angles in an nsided figure is given by 180 into nus 2. So 180 into 5 - 2. This will still give us 540°.

[01:12:08]540°.

[01:12:10]So that is the answer. 540.

[01:12:15]Uh-huh.

[01:12:17]Number 38. This is a proportional division of a line. Construct line mn which is equal to 5 cm.

[01:12:29]Are we together there? 5 cm. Then from there we are also told to divide line mn into four equal parts.

[01:12:39]Proportional division of a line grade eight work. The first thing that we supposed to do is to ensure that we draw the line.

[01:12:48]Mhm. 5 cm. The line is m and 5 cm. Let me check. 5 cm.

[01:13:01]Uh-huh.

[01:13:03]At this point, at this point, we having a m that is 5 cm.

[01:13:12]Now, with the five cm that way for us to do what we call proportional division of a line, we supposed to produce a line at a suitable angle here.

[01:13:26]just a straight line. You produce it through a suitable very suitable angle like that one. Then because you intend to divide to four proportions. Now you divide the line to equal parts of course reasonable radius like this one.

[01:13:46]Mhm. This one.

[01:13:49]This one. This one. How many proportions are those? 1 2 3 4. Yeah. Five. Pour proportions like that. Pour proportions like that. Then you join the last proportion to the last point. You join the last proportion to the last point.

[01:14:10]This way.

[01:14:12]Therefore now you are supposed to draw other lines which are parallel to this one. Assuming that this is a b c d. Now we draw lines.

[01:14:30]Draw lines uh parallel to DN at the point at point C, B and A. At those points, you draw other lines which are very parallel to the one that you've already established. So you can use a set square to slide the ruler and by such doing you'll find that eventually mn has been subdivided into four parts. So that is proportional division of a line. Internal proportion.

[01:15:16]Internal division of a line.

[01:15:19]A good question. Anyway, can we look number 39?

[01:15:24]A number is selected at random from the list of all numbers from 10 to 20.

[01:15:33]All the numbers from not between. Look at that instruction.

[01:15:39]10 to 20. not between but from 10 to 20.

[01:15:44]What is the probability that the number selected is a square number? The number selected is a prime number. Now what we need to do is to list the numbers first.

[01:15:55]10 itself. 11 12 13 14 15 16 17 18 19 20 10 11 12 13 14 15 16 17 18 19 20. Now we have all the numbers here. Then we pick a number at random and we check whether the probability is a square number. So how many square numbers are here? A number which is having a perfect square root like 16. The square root is four. We don't have another one. The next square number is a uh 4 squ 16. So 5 square is 25. So it's beyond here.

[01:16:39]Before 16 there should be 3 squar which is a 9. It's not within. So, it's only one number out of 1 2 3 4 5 6 7 8 9 10 11. One out of 11 possibilities. This is our answer. One out of 11. Then a prime number. Out of 11, how many prime numbers do we have in this case? 1 2 3 4. We have four prime numbers out of 11 possibilities.

[01:17:18]So that is now those are the probabilities. Remember probability equals to the number of favorable outcomes out of the number of total possible outcomes. Total possible outcomes. So that is how the question is supposed to be done. Is it not a good one?

[01:17:38]Definitely is looking at number 40 now. Looking at number 40. In number 40, we told that in a school the total average marks for nine subjects is 540.

[01:17:58]Now this course in eight subjects are Mhm. Eight subjects are 60, 5, 55, 72, 50, 49, 56, 60, and 60. Part one of this question.

[01:18:21]In part one, we being told to calculate the score in the ninth subject. The total average marks, the total average marks for nine sub.

[01:18:34]So the ninth subject which can be let to have scored X shall be given by when we take the marks of the nine subjects minus the available eight 60 55 72 50 45 56 60 and 60.

[01:19:03]This is going to be 540 minus 458.

[01:19:11]So this ninth subject will be equal to 82. Exactly 82. Uhhuh. And from there we determine in Roman 2 the mean mark of this code.

[01:19:29]The mean mark we need to get the total marks.

[01:19:33]out of the subjects involved and when you divide this you get exactly 16. This means that you can score Roman 2 even when you don't have an idea of Roman one as long as you know what mean is the average of the total marks. Now state the model mark. When you talk about the mode or model mark, it means the mark that occurs most.

[01:20:01]For example, 60 1 2 3 60 is occurring three times. There is no other mark which has occurred such a number of times. meaning that 60 now becomes our answer as the mod which is also the mean. So these are now the answers to number 40. A very good question. So thank you guys for following. Please remember to enroll to our neck program.

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