O LEVEL MATHS SPECIMEN SYLLABUS A PAPER 1

Añadido: 2026-05-09

[00:00:00]Today we are going to discuss on this question paper that is specimen mathematics syllabus A, okay? So this is for all levels and today we are going to put more of our focus on this paper one.

[00:00:09]So we have I have done a video of paper two syllabus A as well as syllabus B.

[00:00:14]Paper one for syllabus will be as well.

[00:00:16]So just subscribe to our channel so that you'll be notified every time we post a new video. And I do believe guys that you've already done this paper before so that you can simply make use of this same to improve yourself assess yourself. And also doing Zimsec it is very very important to read these instructions uh to candidate. So in this question paper we are required to answer all questions. We're going for all questions and it has a total of 100 marks, okay? And uh but if you're not doing Zimsec just pay attention to the data. Just pay attention to what we are going to discuss in this question paper because we are going to dissect explain to uh expand our explanation on each and every question, okay? So let us dive into our question so I want number one.

[00:00:51]So there's this very very important instruction here. Uh so the instruction says neither mathematical tables nor slide rules nor calculators may be used in this paper, okay? So we are not going to use calculators. We are not going to use mathematical tables. We are not going to use slide rules, okay? So let us start with in question number one. So the question says list the prime numbers which are between 10 and 15. So we are required to list the prime numbers which are between 10 and 15.

[00:01:18]Okay? So the first thing is to define what are prime numbers. So prime numbers these are numbers that are only divisible by one and itself, okay? So we only divide these numbers by one and itself. Uh these are what we call prime numbers, okay? We are required to give prime numbers that are between 10 and 15. So the first step is to write uh the the numbers that are between 10 and 15.

[00:01:39]So we have 10, we have 11, we have 12, we have 13, we have 14, and we also have 15. So these are numbers that are between 10 and 15. So we want to pick uh the prime numbers. So the prime numbers are 11 and 13. These are two prime numbers.

[00:01:56]So for us to define uh 11 such that we can get a whole number, we should divide it by one and itself. And for us to divide divide 18 such that we can get a whole number, we should divide this 18 by one and itself. So, our prime numbers is um 11 and 18. These are two prime numbers that are like that lie between 10 and 15. Uh 10 is not a prime number, 12 is not a prime number, 14 is not a prime number, as well as 15 is not a prime number. And let us move on to the next part of the question, that is part B.

[00:02:26]So, the question says, "Identify an integer from the following numbers: 1, square root of 5, uh 2, 3/4, and 3.25." So, here as well, we are required to define the term integer. So, integer is just a whole number. Integer is just is just a whole number. So, from the following numbers, we're required to identify an integer.

[00:02:45]So, an integer is just one. Uh square root of 5 is not a whole number. 2 and 3/4 is not a whole number. 3.25 is not a whole number. So, the only integer that we have in this uh in these numbers it is one. That is the only integer that we have.

[00:02:59]And then, uh let us move on to question number two. So, the question number two says, "Round off 3 um 3/4 to the nearest whole number." So, we're required to round this to the nearest whole number. So, what you can simply do it is that you can uh just express this as a decimal. You can express this as a decimal, and then after that, we can now round off to the nearest whole number. All right. So, what we're going to do it is that we're going to express this as an improper fraction first. So, to express uh 3 3/4 as an improper fraction, it will be multiply and plus. So, we're going to multiply 3 by 4. It gives us what? It gives us 12 + 3. It gives us 15. And then, we divide everything by 4. So, it is now 15 / 4. So, what is 15 / 4? 15 / 4 it gives us 3.75.

[00:03:40]Okay? It gives us 3.75. So, this is um 3 3/4 as a decimal. Okay? But, the question requires us to round this as a whole number to the nearest whole number. So, when uh we are rounding a decimal to the nearest whole number, we look at this number just after the comma. Uh and then we see is it greater than five or it is less than five? And if it is less than five, uh we just leave it as three, okay? But if it is greater than five, we round three to four, okay? So, seven is greater than five, so to round 3.75 as a whole number, we're going to get four. That is our final answer. And then let us move on to part B of question number two.

[00:04:17]So, the question [music] says, "Round off 292 to the nearest 10." So, to the nearest 10, we are more concerned with this here, this part here.

[00:04:25]Uh so, is it greater than five [clears throat] or it is less than five?

[00:04:29]Two is less than five, so to round 292 to the nearest 10, we're going to have 290, okay? So, this it is 290. Why?

[00:04:37]Because two is less than five. And then uh for the nearest hundreds, we're going to focus on 92, okay? We're going to focus on 92. 92 is it greater than 50 or it is less than 50? If it is less than 50, you're going to have 200. But if it is greater than uh 50, we're going to have 300. [music] So, to the nearest hundred, it gives us 300 because 92 is greater than 50, okay?

[00:04:59]So, to the nearest 10 is 290 and to the nearest hundred, it gives us 300, okay?

[00:05:05]And then let us move on to question number three. So, we we are on question number three. So, the question says, "Simplify." We're given hours, minutes, and seconds.

[00:05:13]>> [music] >> Uh so, we're having three hours, 45 minutes, and 30 seconds. We also having two hours, 20 minutes, and 42 seconds.

[00:05:19]So, what you should know [clears throat] whenever you are adding or subtracting time is that uh you should relate the addition and subtraction of time to number bases. So, if you still [music] remember in number bases, we having base two, base three, base four, or whatever base that we we were having that time.

[00:05:33]So, uh let's say we are I adding, let's say I adding uh 23 in base five to 23 in base five, okay? So, 3 + 3 gives us six, okay? 3 + 3 gives us six. And then after we have six is above five. So you can't have six in base five. So we're going to say six divided by five. So if we divide six by five we're going to have one remainder one. And then we put remainder and add one here. And two plus two plus one it gives us five. And five it is equal to five. So you can't have five inside the base five. So we're going to divide five by five it gives us one remainder zero. So we're going to put zero and then we add one here. So it's one here. So this is our 23 in base five plus 23 in base five it gives us 101 in base five. So this is the same concept that we're going to apply whenever we are adding or subtracting time. So time it is in base 60. Time is in base 60, okay? So we're going to have 30 plus 42. So what is 30 plus 42? 30 Let me just write 30 plus 42 here. So if we add 30 to 42 we're going to get two.

[00:06:38]Three plus four it gives us 72. And then we divide 72 by 60. So if we divide 72 by 60 we're going to get one remainder one remainder what?

[00:06:50]It is one remainder. So it is 72 minus 60.

[00:06:55]So this gives us two one. So it is remainder 12. So we're going to add one here. And then we are going to put 12 here. So it is 12 seconds and then we add one minute there. And then for my minutes we're going to say it is 40 46 plus 20. So 46 plus 20 this gives us what? This gives us six 66.

[00:07:16]>> [music] >> And then we divide 66 because the base for minutes it is 60 minutes.

[00:07:21]So we're going to divide this by 60 and then we'll get one remainder six. Get one remainder six. And we're going to put six here and then we add one here, okay? And then we say three plus two plus one it gives us what? It gives us four.

[00:07:35]So our final answer it is four hours six minutes 12 seconds. [music] Four hours six minutes 12 seconds. It's as simple as that. And then let us move on to part B of the question. So, part B of the question says, "2 and 1/2 days in hours." So, we're going to calculate 2 and 1/2 days in hours. So, 2 and 1/2 days in hours, first thing it is [music] to say, "1 day it has a total of 24 hours." Okay? 1 day it has a total of 24 hours. So, what is 1/2 of 24 hours?

[00:08:01]1/2 of 24 hours it gives us what? It gives us 12 hours. So, this is 1/2 day.

[00:08:05]1/2 day it has a total of 12 hours. And what about [music] 2 days? So, 2 days it is 2 by 24 and this gives us 48 hours.

[00:08:13]So, this is uh the total time for 2 days. So, we're going to say [music] it is now 48 + 12 to find the total for 2 and 1/2 uh days in hours. So, this gives us what? This gives us 0 here and uh we're going to get 4 + 1 + 1 that is 60.

[00:08:29]It gives us 60 hours. So, 2 and 1/2 days in hours it gives us a total of 60 hours. Or you can simply use proportion.

[00:08:36]If 1 day uh let me use proportion here. Use proportion.

[00:08:42]So, the proportion says, "If 1 day is equal to 24 hours, what about uh so 2 and 1/2 in uh as an improper fraction this gives us 5/2. So, what about 5/2? So, 5/2 it gives us more." Okay? Gives us more so we're going to say it is now 5/2 / 1 * [music] 24. Okay? So, we're going to say 5/2 * 24. So, 2 into 24 it gives us 12 and then 12 by 5 it gives us 60 hours.

[00:09:11]So, our final answer it is 60 hours. So, you can simply use proportion to calculate the total hours.

[00:09:17]Uh and let us move on to the next part of the question which says, "Express the time uh 19:28 in 12-hour notation." So, we're going to express 19:28 as um in 12-hour notation. So, this 19:28 it is 28 past 7 in the evening. Okay? So, in um 24 in 12-hour notation it will be 7 uh 28 p.m. That is our 12-hour notation. And then uh let us move on to question number four. So, the question says A is a set of whole numbers from 2 to 8, and B is a set of prime numbers from 2 to 8, okay? So, the numbers that are between 2 and 8, it is 2 uh 2 3 4 5 6 7 and 8, okay? So, the whole numbers uh that are between 2 and 8, we're having 2 3 4 5 6 7 8. And then [music] uh the prime numbers that are found between 2 and 8, we have 3. 3 is a whole number.

[00:10:13]So, we also have 5 as a whole as a prime number, and then 7 as a prime number.

[00:10:18]So, set B, it is equal to 3 5 and 7. And then set A, it has 2 3 4 5 6 7 8, okay? So, that is what we have.

[00:10:29]So, we are required to show on the Venn diagram uh when I label it Venn diagram.

[00:10:34]So, when I label it Venn diagram, I'm going to have something like this.

[00:10:39]Something like this, okay? So, I'm going to have set B inside set A.

[00:10:45]So, here is our set B, and here is our set A. So, in set B, we're having 3 5 and 7.

[00:10:52]And then set A, we're going to have 2 We're going to have uh 4 We're going to have 6. We're going to have [music] 8. So, the relationship for the second part of the question, it says use the notation show the relationship between uh set A and set B. The relationship between set A and set B, it is that set B is a subset of set A. Set B is a subset of set A. So, it is B is a subset of set A, okay?

[00:11:22]So, let us move on to question number five. So, the question number five says, "Simplify the ratio 4 is to 18." So, you're required to simplify the ratio of 4 is to 18. So, to simplify, we're going to reduce this in lowest terms. So, we're going to divide everything by two.

[00:11:35]So, if we divide four by two, we're going to get two. And uh if we divide 18 by two, we're going to get nine. So, this is the simplest form of this ratio, two is to nine. And then the question says um $18 is shared in the ratio of 4:18, find the smaller share. So, you can uh use this ratio that is two is to nine. You can simply use this ratio, two is to nine. And then after that, you calculate the total parts or the total ratio. So, the total ratio, it'll be two plus nine, that is equal to what? That it gives us 11. And then And then after that, we are going to calculate the smaller share. So, the smaller share, it'll be equal to two over 11 multiplied by 88. [music] And then we divide 88 by 11. This gives us what? This gives us eight. And then we multiply uh two by eight. This gives us 16, okay? So, this gives us 16 here.

[00:12:18][music] So, the smaller share, it will be $16. And then let us move on to the next part of the question paper, that is number six. So, question number six says, evaluate 1/3 of 1,200 cubic centimeters. So, we are having 1/3 multiplied by 1.20 uh in cubic [music] centimeters. So, we're going to divide this 12 by three.

[00:12:40]It gives us four zero zero zero zero.

[00:12:43]So, our final answer, it is 400 cubic centimeters. And then uh the next part of the question, that is part B, says, simplify 2m³ - 0.5m³ >> [music] >> + 7.2m³.

[00:12:57]So, I think this m³ it is um cubic meter, okay? I think it's cubic meter.

[00:13:01]So, they want to simplify here. So, to simplify here, we're going to use the BODMAS, where B is for bracket, O is for of, and then D is for division, M is for multiplication, A is for addition, and S is for subtraction. So, we're going to collect the like terms. We're going to add these two to these seven by 7.2. So, 7.2 + 2, this gives us what? This gives us 9.2. And then after that, we subtract 0.5 from 9.2. So, it is 9.2 - 0.5 and then we add 10 here, we add 1 here. So, it is 12 - 5. So, 12 - 5 it gives us what? It gives us 7. 9 - 1 it gives us 8. So, this gives us 8.7. So, our final answer it is 8.7. And let us move on to question number seven. So, we are on question number seven. So, the question says, "5.9 * 10 ^ -2 as a in in ordinary form as a number in ordinary form." So, we are going to express this as a number in ordinary form. So, to express this as a number in ordinary form, we are going to say 10 ^ -2 can be expressed as 1 uh over 10 ^ 2 and 10 ^ 2 it is equal to what? It is equal to 100. [music] So, this is equal to 1 over 100. So, we are going to have 5.9 [music] * 1 over 100. So, this is equal to 5.9 / 100. So, 5.9 / 100 we are going to get 0. 0 uh 5 9, okay?

[00:14:29]So, it is 0.0 59, okay? And then let us move on to the next part which says, "80,000 as a number in standard form." So, we are going to express this 80,000 as a number in standard form. So, we are going to say it's 8 >> [music] >>.0 * 10 ^ So, we want to generate 80,000 again from 8. So, we are going to multiply 8 by 10 ^ what to get 80,000 again? So, how many zeros do we have? 1 2 3 4. So, we then just erase 10 ^ 4.

[00:14:58]So, 80,000 as a number in standard form it is equal to 8 * 10 ^ 4. That is our final answer. And then let us move on to part B of the question. So, part B of the question says, "Find the square root of 49." So, the square root of 49 it is equal to 7. This is a very very simple question. One mark uh this is a free mark.

[00:15:17]The square root of 49 it is equal to 7.

[00:15:19]And then let us move on to question number eight. So, the question number eight says, calculate the simple interest of $180 invested for two years at a rate of 5% per annum. So, interest is equal to principal times rate times time divided by 100.

[00:15:38]Okay? So, that is our interest. It is equal to our principal. It is 180 multiplied by our rate. That is five multiplied by our time. That is two divided by 100. Divided by 100. So, we're going to say uh two by five it gives us 10.

[00:15:54]>> [music] >> And then we multiply the 10 by uh with 180. So, 10 with 180 it gives us what?

[00:15:59]1,800.

[00:16:00]>> [music] >> And then we divide 1,800 with 100. So, these two zeros cancels and we're going to be left with $18.

[00:16:08]So, $18 it is our interest. So, our interest is $18.

[00:16:12]And then let us move on to question number nine. So, the question number nine says, expand 4 uh 1 4 2 3 in base five in powers of the given bases. Okay?

[00:16:21][music] Okay, so we're going to to say here we're having zero, one, two, three.

[00:16:26]Okay? [music] So, we're going to say it's five to the power three by one plus five to the power two by four plus five uh to the power one times two plus uh five.

[00:16:41]Uh plus five to the power zero times three. Okay? So, five to the power three it gives us what? What is five to the uh to the power three? It gives us 125. So, the question says in powers of the given [music] base. So, we're going to leave the answer like this. So, we're going to leave our answer like this.

[00:16:58]Okay? Like this.

[00:17:00]And then let us move on to the next part of the question which says, convert 43 um in base 10 to a number in base five.

[00:17:06]So, 43 in base 10 to a number in base five we're going to say 43 here. We have five here. And then we divide 43 by five. 43 divided by five it gives us eight remainder three. And then uh you divide this eight by five. It gives us one, remainder three. And then, we divide one by five. It gives us zero, remainder one. And then, we move in this direction, okay? After we move in this direction, we're going to have 133 in base five. So, this is our 43 in base 10 as a number in base five. [music] It is 133 in base five, okay? And the last part of the question says, "Simplify" Uh the question says, "Simplify 110 in base two plus 11 in base two, giving the answer in base two." So, we are having 110 in base two, 11 in base two. So, we're going to add this. So, 1 + 0 it gives us one in base two. And then, 1 + 1 it gives us two. And then, we divide this two by two. We're going to get one, remainder zero. So, we're going to put zero here. And then, we add one. 1 + 1 it gives us two again. We're going to divide two by two. We get one, remainder zero. And then, we put zero. And then, we add one. 1 + 0 gives us one. So, our final answer it is 1001 in base two. That is our final answer.

[00:18:15]So, let us move on to question number 10. So, the question says, "A salesman's monthly salary consists of a basic salary that is $150 and a commission of 2 and 1/2% of the total sales of the month, okay?" So, we are going to calculate his total salary in a month with a total of sales of $3,000. $3,000.

[00:18:36]So, the first step it is to calculate the commission. So, we're going to say, uh 2 and 1/2% it is 5/2%. So, we're going to divide this 5/2 by 100. So, what is 5/2 by 100? So, you are going to divide this by 100, okay?

[00:18:49]Divide this by 100. So, if you are dividing this, you are going to replace this division by a multiplication, and then we invert the second fraction. So, it will be 5/2 multiplied by 1/100, okay? So, 5/2 multiplied by 1/100 it gives us 5/200.

[00:19:06]And then, we multiply 5/200 multiplied by 3,000, okay? So, this gives us uh we're going to cancel these zeros, these zeros, >> [music] >> then we're now left with 5 over 2 multiplied by 30. So, 30 divided by 2, it gives us what? It gives us 15. And then 5 multiplied by 15, so we're going to say 5 multiplied by 15, this gives us 25 here, and then you carry the 2. 5 by 1 it gives us 5 by plus 2 it gives us 75. So, 15 by 5 it gives us 75. So, the commission it is equal to $75. That is the commission. [music] And then you add $75 to 150. So, it is now 150 plus 75. So, if we add here, we're going to get um Let me rub this. Okay. So, we are having we're having 150 plus 75 again.

[00:19:57]So, this gives us 5, uh and then 12 here. Uh 1 plus 1 it gives us what? 2.

[00:20:03]So, the total salary it is equal to $225.

[00:20:07]That is the total salary of the salesman. And uh let us move on to the next part of the question, that is number 11. So, we are on number 11. So, the question says, "Make R the subject of the formula X is equal to P a factor of Q plus R." So, as I said in my previous videos, to make something the subject of the formula, number one, you should uh make the power of that that uh letter positive one, as well as the coefficient of the letter it must be positive one.

[00:20:33]So, we are having X being equal to P a factor of Q plus R. So, the first step is to divide both sides by P. So, it will be X divided by P being equal to [music] Q plus R. And then we shift this Q to that side of the equation, so it will be X over P minus Q is equal to R.

[00:20:50]Okay? So, as you can see here, the power of X is positive one, and the coefficient the power of R is positive one, and the coefficient of R is positive one. So, R is equal to X over P minus Q.

[00:21:02]And we are done. And then let us move on to number 12. So, the question number 12 says, "Solve uh the following inequalities. We are going to solve this inequality. So we are having 2x + 5 < 1.

[00:21:14]And then we shift this if this crosses this inequality sign, it becomes negative. So this gives us 2x < 1 - 5 that is -4. And then we divide both sides by [music] 2. Therefore, x is < -4 / 2 it gives us -2. So x is < -2. x is < -2. Okay.

[00:21:36]All right. So we can just pick any number that is less than -2. So any number that is less than -2 it is -3. -2 is less than -3 is less than -2.

[00:21:44]>> [music] >> So if we substitute -3 here, we're going to get 2 the factor of -3 + 5. So 2 by -3 it gives us 6 + 5. This gives us -1.

[00:21:54]So -1 is less than 1 and then we might have managed to satisfy this inequality.

[00:21:58]And let us move on to the next part of the question that is part B. So the question says, illustrate the answer in one a number line. So we are going to illustrate the answer on a number line.

[00:22:08]So the number line it will be like this.

[00:22:10]So we are having we are having what? We are having x being less than -2. So you can just have -2 here. So x is less than what? -2. So we are taking the values of x in this direction.

[00:22:24]Okay. Anything that is in this direction that is less than 2 [music] is correct for x, okay? And then we move on to number 13. So the question number 13 says, the universal set is subset m and n such that the universal set is 1 2 3 4 5 6 7 8 9. And then the set m it is 2 4 6 8. The set n it is a set of multiples of 3. So we are going to list all the elements of the set n. So the elements of the set n are multiples of 3. [music] So it is 3 6 9. Okay? And then the next part of the question requires us to calculate the union of these two sets m union N, okay?

[00:23:03]So, the union is um the elements that are in set N as well as the elements that are in set M.

[00:23:09]>> [music] >> So, we're going to have two uh three four uh six eight and nine, okay? This uh this is M union N. And uh for the numbers of elements in M union N, I'm going to count 1 2 3 4 5 6.

[00:23:26]>> [music] >> So, here it gives us six, that is our answer. And then let us move on to the next part, that is number 14. So, the question says simplify -15 + 3. So, we're going to simplify -15 + 3. So, -15 [music] + 3 this gives us -12, okay? And then uh the next equation says evaluate 3 + -15 / 3. So, the first step it is to use our BODMAS and uh we're going to start with the division.

[00:23:51]So, I'm dividing this -15 / 3. It gives us what? It gives us -5.

[00:23:57]So, -15 / 3 it gives us -5. So, here it will be 3 + -5. So, 3 uh -5 it gives us what? -2. So, our final answer it is -2.

[00:24:08]And then let us move on to the next part of that equation, that is part B. So, the question says during the day the temperature was 15° C.

[00:24:17]Uh at night the temperature fell by 10° C.

[00:24:20]So, we're required to calculate the temperature at night, okay?

[00:24:23]>> [music] >> So, during the day the temperature was 15° C and then it falls by 10° C.

[00:24:29]So, we're going to subtract 10° C >> [music] >> here. So, our final answer it will be 5° C.

[00:24:34]So, this is the temperature at the night. The temperature at the night it is 5° C, okay? And um we are on question number 15. So, the question number 15 says solve the following equation 3x = 15. We're going to divide both sides by 3 and then if we divide both sides by 3 we're going to have x being equal to 5.

[00:24:53]x being equal to 5. And then uh 6 a factor of x + 2 being equal to 19, we're also going to solve this equation. The first step is to expand the bracket on the left hand side. So this gives us 6X [music] plus 12 being equal to 19. And then we shift this 12 to that side, becomes negative. So we're going to have 6X being equal to 19 minus 12. So what is 19 minus 12? It gives us seven. And then we divide both sides by six.

[00:25:19]Therefore X it is equal to seven over six. That is our final answer. All right, so let us move on to question number 16. So we are on question number 16.

[00:25:28]>> [music] >> So the question says express 2X minus 3 over 5 plus X plus 4 over 7 as a single fraction. So we are going [music] to express these two fractions as a single fraction. So for you to understand how to express these two fractions as a single fraction, let us start with a simple one, okay? Let us start with a simple one. So let's say we are having 1 over 4 plus 1 over 3.

[00:25:51]First step is to look for a common denominator.

[00:25:54]So our common denominator just is the product of these denominators, that is 4 by 3. It gives us 12.

[00:25:59]And then we say 4 into 12 it gives us three. And then we multiply that three with the numerator here, that is one. So it gives us three. And then we say 3 into 12 it gives us four. And then we multiply this by one, it gives us four.

[00:26:10]>> [music] >> So our final answer it will be equal to seven over 12, okay? So we are going to use the same idea to solve for this 2X minus 3 [music] over 5 plus X plus 4 over 7. So our common denominator it is 5 by 7, that is 35.

[00:26:29]>> [music] >> And then we say 5 into 35 it gives us seven. So it is seven multiplied by the numerator, that is 2X minus 3. 5 into 35 it gives us five. And then we say plus [music] five a factor of X plus four, okay? And then we simplify on the numerator by just expanding the brackets. So we're going to get >> [music] >> 14X minus 21 plus 5X plus 20, okay?

[00:26:51][music] So 14X + 21 it gives us what?

[00:26:55]14x + 5x it gives us what? It gives us 19x. And then -21 + 20 it gives us -1.

[00:27:02]And then we divide everything by 35. So, this is our final answer as a single fraction, okay?

[00:27:08]And then let us move on to question number 17. So, we are on question number 17. So, the question says point C and D have coordinates 4, -1, 2, and 3 respectively. So, I require to find the gradient of the line passing C and D.

[00:27:22]So, the gradient Our gradient is equal to the change in Y over the change in X, okay? So, the change in Y it is equal to the change in Y it is equal to 3 -1. That is the change [music] in Y. And then the change in X it is equal to 2 - 4. That is the change in X. [music] So, for the change in Y you are having 3 - - it gives us a plus. So, it's 3 + 1 that is 4. And then for the change in X you are having 2 - 4 that is -2.

[00:27:49]Therefore, the gradient [music] it is equal to 4 / -2.

[00:27:54]>> [music] >> So, that gives us -2. So, -2 is the gradient of the line that is passing through C and D. And let us move on to part B of the question. So, the question says hence or otherwise find the equation of the line passing through C and D. So, the equation of a straight line is of this form Y is equal to mx + c where this m is the gradient and then c it is the Y intercept, okay? So, the Y intercept occurs when the line crosses the Y axis and it also occurs when the X coordinate at that point is equal to zero, okay? So, at Y intercept the X coordinate it is equal to zero.

[00:28:31]So, the question requires us to calculate the gradient. So, for you to calculate the gradient of the straight line you need two things. At least one point on that line and number two the gradient of that line, okay? So, we have this general formula to write the equation of a straight line which says Y - Y1 is equal to m a vector of x minus x one, okay? So, we're going to say uh let us just take one [music] point. So, we can take this point that is two three. But we can also take this one. So, you choose between these two points the one that you want to use, okay? So, if we use two three, we're going to say it is y minus three being equal to our gradient that is minus two. So, use minus two a vector of x minus two, okay? So, this gives us y minus three. And then we expand this bracket to get [music] minus two x plus four. And then after that, we shift this to that side. So, we're going to have y being equal to minus two x >> [music] >> and uh minus three it will when it crosses the equal sign, it gives us a positive. So, it is four plus three that is [music] seven, okay? So, this is our equation of the line, okay? The equation of the line it is equal to y is equal to minus two x plus seven.

[00:29:41]That is our final answer. And let us move on to number 18. So, we are on number 18. So, the question says solve simultaneous equations. So, we're given this. So, this is our equation number one and this is our equation number two.

[00:29:54]So, we are going to solve this.

[00:29:55]So, to solve this, it's either we're going to use the elimination method or we use the substitution method. But we can simply use the elimination method.

[00:30:02]Why? Because here we have this y as well as this y. So, we're going to say um if you shift this y here, it becomes negative. So, this gives us three x minus y um being equal to 16. And then we have minus x plus y being equal to minus 12.

[00:30:20]minus two being equal to minus two.

[00:30:24]Okay? And then after that, we uh we add. Why are we adding? Because we want to eliminate y. So, if we add here, we're going to get two x. And then minus y plus y it gives us zero. And then here we're going to get 16 minus two, that is 14. And then uh we are having two x being equal to 14. We divide both sides by two. Therefore, x it is equal to seven. X is equal to seven.

[00:30:48]Uh so, after that after we have found that x is equal to seven, we're going to substitute x in either equation one or equation two. So, let's just substitute seven into equation number two. So, we're going to get y minus seven being equal to minus two.

[00:31:02]>> [music] >> And then we shift minus seven to the right-hand side. This gives us y being equal to seven minus two.

[00:31:08]>> [music] >> So, this gives us y being equal to five.

[00:31:10]So, x is equal to seven and y is equal to five. And let us just test these two values back into the first equation. If they satisfy the equation, they are correct, okay? So, we are having 3x minus y being equal to 16. That is our first equation. 3x minus y is equal to 16. So, let us substitute the value of x that is seven.

[00:31:31]So, it is three by seven minus five is equal to 16. Three by seven it gives us 21 minus five it is equal to So, six 21 minus 15 it gives us 16.

[00:31:44]So, this is equal to 16 as per question.

[00:31:47]So, we have managed to satisfy the first equation using the value of x that is seven as well as the value of y that is five. So, y is equal to five and then x is equal to seven. And let us move on to number 19. So, we are on question number 19. So, the question says uh matrices x is equal to 3 2 1 -1 0 7 and matrix y it is equal to 9 4 -6 8 -5 11. So, we are required to find x + y.

[00:32:17]We are required to find x + y. So, we are having uh 2 1 0 [music] 7 plus 9 4 -6 8 -5 11. So, we're going to we're going to add three uh plus nine it gives us 12. Two plus four it gives us six.

[00:32:35]One plus minus six it gives us minus five. Minus one plus 8, it gives us 7.

[00:32:41]And then 0 + - 5, it gives us - 5. 7 + 11, it gives us 18. So, this is our X + Y. Okay? And let us move on to the next part which says simplify - 4 6 2 5 - 1 3 2 7. [music] So, you are going to simplify this. So, it is - 4 6 2 5 1 3 2 7. So, to simplify here, we are going to We are going to have - 4 - 1, that is - 5. Uh 6 - 3, it gives us what? 6 - 3, it gives us 3. And then 2 - 2, it [music] gives us 0. 5 - 2, it gives us - uh 5 - 7, it gives us - 2. Okay?

[00:33:25]So, here it's - 2.

[00:33:30]It is - 2. So, it gives us - 2 - 5 3 0 - 2. That is our final answer. And uh let us move on to question number 20. So, the question says express 48 as a product of its prime factors. So, we are going to express >> [music] >> 48 as a product of its prime factors.

[00:33:48]So, we are having 48 2. So, 2 into 48, it gives us 2 4, that is 24. 2 into 24, it gives us 12. 2 into 12, [music] it gives us 6. 2 into 6, it gives us 3. 3 into 3 into 3, it gives us 1. So, 48 as a product of its prime factors, it is 2 to the power 4 multiplied by 3.

[00:34:08]2 to the power 4 multiplied by 3. And then let us move on to the next part of the question which says find the highest common factor of 14 and 63. So, the first step it is to express 14 as a product of its prime factors as well as 63 as a product of its prime factors.

[00:34:23]So, we are going to have um 14 here. We divide by 2, we get 7. We divide by 7, we get 1.

[00:34:30]>> [music] >> And then for 63 we are going to divide by three. So, if we divide by three, we're going to get 21. We divide by three, we're going to get seven. Divide by seven, we're going to get one. So, 14 is equal to two by seven. And then 63 is equal to three squared multiplied by seven. That is our 63. So, the highest common factor, it is the number that is common both in 14 and 63. So, don't have three. Three is not common. Two is not common. So, the only that is common in both 14 and 63, it is seven. So, our highest common factor, it is seven. And the question number 21 says, "Find The diagram shows The diagram below shows AOBOC meeting at point O." So, we are required to calculate the value of X. So, uh seeing that this is a complete revolution, we're going to say 5x + 3x + 4x is equal to 360. So, 5x + 3x gives us 8x + 4x gives us 12x being equal to 360, okay?

[00:35:36]>> [music] >> And then we divide both sides by 12.

[00:35:41]So, X is equal to 360 [music] divided by 12.

[00:35:44]So, 12 into 36 it gives us three, and then into zero gives us zero. Therefore, X is equal to 30 degrees. X is equal to 30 degrees.

[00:35:55]Okay. And the next part of the question says, "Find AOB." So, we are required to find uh AOB. So, AOB it is 3x. So, three uh by three uh by 30 it gives us 90 degrees. So, AOB it is equal to 90 degrees.

[00:36:12]Okay. And let us move on to question number 22. So, the question number 22 says, "2/3 + 1/5." We are required to simplify. So, the common denominator, it is the product of the denominators, that is 15. Three into 15 we get 5. We multiply 5 by the numerator, we get 10.

[00:36:31]Uh 5 into 15 it is 3. We multiply 3 by the numerator, we get 3. So, our final answer it is 18 over 15. That is our final answer. And then now the second question says express 3/5 as a decimal.

[00:36:44]So, you're having 3/5 and we want to express this as a decimal. So, we say 5 into 3 gets 0 and then we carry 0 here. 5 into 30 it gives us 6.

[00:36:54]Okay? So, our final answer it is 0.6.

[00:36:57]And uh the next question that is part B says find the 45% of 200. So, we're required to calculate the 45% of 200.

[00:37:05]So, percent it means over 100. You're going to say 45/100 multiplied by 200. Okay? This uh these guys cancels, so we're going to be left with 45 by 2. So, 45 by 2 it gives us 90. So, 45% of 200 it is 90.