KAKAMEGA JOINT EXAMINATION, MATHS PAPER 1 SECTION 1

Added: 2026-05-05

[00:00:03]Thank you for joining today's lesson. We shall be looking at the the Kakamega mathematics joint examination 2026.

[00:00:12]This is a paper that has been done by a group of very many schools. It has been set by serious examiners and it's my hope that through the interaction with this paper you are going to get something. So we start with the section one number one simplify 3 n^ 2 - 12 out of 3 - 1 + n. So as usual I've been advising you people to always start with simplifying the numerator first later you simplify the denominator then you combine the two simplified forms.

[00:00:53]Meaning that from the numerator we shall talk of whatever is common now in the two terms of the numerator that is three we remain with n^ 2 and -4 I hope we are together in that then when you look at whatever is inside the brackets now ladies and gentlemen this is now a difference of two squares because there is a square root for n squ there's a square root for 4 and in between them is are negative meaning that this is a difference of two squares. Now according to the trigonometric identities ladies and gentlemen we usually talk of x^ 2 - y^ 2 and to simplify this difference of two squares you have to pick the square root of the first minus the square root of the second into square root again but now with a positive.

[00:01:46]To prove this you can expand the right hand side you shall get the difference of two squares. Therefore, you don't have to simplify now using any long method. You just follow the what you call quadratic identities about a difference of two squares. So this now shall be 3 into n with a minus 2 then n with a positive2. So now we have fully factorized the numerator. Let us get to the denominator whereby at the denominator we can open the bracket there. This is three then minus multiplies 1 and minus again multiplies n meaning that when we group these terms it shall be now positive2 3 - 1 and n like that. Now we can combine the numerator and also the denominator. In this case we talking about 3 into nus 2 uhhuh n + 2 then out of this is out of uhhu n - 2. The idea of now combining numerator and denominator is to examine whether there is a common factors that can simplify out. So when you look at the factors they are the same because they have two there is n that means now when we play along with the negatives.

[00:03:24]Now for example if we multiply both the numerator and the denominator with a1 actually look at that step because we can see this factor is not the same as this one and it's not the same as this one. What about when we multiply both sides by1 by1 and by1?

[00:03:51]Let's check the next step and you shall realize that uh this is becoming -3 then n - 2 into n + 2 then at the denominator ladies and gentlemen at the denominator now when you multiply by -1 you know it's like this by1 so the two becomes negative and the n becomes positive meaning that we can now rearrange them and have n -2. So it will be n - 2 like that. So at that step now you can check there is a n minus 2 here and another n minus 2 here. So now this is -3 into n + 2 over 1. So this is now the simplified format of the expression.

[00:04:54]Remember whenever you can notice that some factors are bearing the same terms but they can't be simplified please remember to multiply both sides by1.

[00:05:08]That way one of the terms shall get rearranged and the factors will simplify out and you get a simplified expression.

[00:05:18]If you getting something, type I am learning. As you continue learning, please be reminded of our holiday programs here. We invite you. It is never late. Starting date 13th, 24th of April, 2026. Use the number which is already on the board here 07041536.

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[00:06:27]Mhm. We can now check number two in this paper.

[00:06:34]The question says trivia answer basic salary of 24,000 Kenya shillings per month in addition is entitled to a commission of 10% on goods sold up to 100,000.

[00:06:52]He also receives a commission of 25% on the goods above 100,000.

[00:07:00]In a certain month, he received a total of 45,450 Kenya shillings. Calculate the total sales that month. So, you can see the earnings have been broken down for this gentleman. That's a tribion. We told that he's earning a basic salary of 24,450.

[00:07:26]Then the others are commissions. Meaning that if the total earning in this certain month was 45,450, then we can get the amount that came as commission. So how much did did he earn as commission?

[00:07:40]So the commission was you subtract the total earning from the basic salary because anything else now is commission.

[00:07:56]Uh-huh. This is 21,000. So this gentleman is getting 21,000 as commission. Then in this commission now we are saying that there is 25% uhhuh 10% up to 100,000. So the first 100,000 the first 100,000 attracted a commission that is equal to 10% which is now 10,000.

[00:08:31]meaning that the commission for the amount of sales that was beyond 100 now is going to be the other part of the commission. I want us to agree on that. Total commission 21,000.

[00:08:45]Commission from the first 100,000 of sales 10,000. Meaning that now remaining commission the other part of commission now the other part of the commission which is 11,000.

[00:09:07]Is it 11,000? Yeah.

[00:09:11]11,000.

[00:09:14]This is now 25% of the sales that are above 100,000.

[00:09:22]That is above 100,000. So, we're saying uh-huh 11,000 equals to 25% of some unknown cells.

[00:09:37]So, how much is these cells?

[00:09:40]We multiply this by entered over 25.

[00:09:45]Now this is giving us exactly 44,000.

[00:09:50]Yeah, 44,000.

[00:09:52]So the sales that were above 100,000 were 44,000. So now the sales now for this particular month there was the first 100,000 That one attracted a commission of 10,000.

[00:10:14]Then the next 44,000 whereby 25% of 44,000 is 11,000.

[00:10:26]That is now what attracted to commission of 21,000. So that the total earning now for this particular guy became 45,450 that particular month.

[00:10:47]So now the total sales was 144,000.

[00:10:53]Yeah. 144,000 as you can see.

[00:10:57]Kenya shillings. A very good question in number two. Can we check number three together?

[00:11:05]Okay. And if you're getting something on commissions, kindly type I am learning.

[00:11:11]Remember also to subscribe to this channel. Remember also to share the link with your friends. Now a room measuring 5.4 m by 4.8 m is to be fit with square tiles. Calculate the least number of tiles that can fit in the room without a remainder. It's good for you to know something here. You know this is application of GCD but again you supposed to know that the questions can confuse a learner with the application of GCD and I want to okay GCD and LCM sorry. So I want to make it very clear when you are given the dimensions of a room for you to get the dimensions of a square first then because the room is bigger than the square then it means you are getting a divider that's why you go for the GCD but in case you are given measurements of rectangular tiles for you to find the square room that they can completely fill dimensions of a square room then that should be LCM because for tiles to get the dimensions of a room then the room measurements should be a multiple of the tiles measurement but when we have the room's measurement the tile should be a divider.

[00:12:38]Yeah, the tiles are dividers but the room is a multiple of the tiles.

[00:12:45]And that is why from measurements of the room to measurements of the tile we find GCD a divisor.

[00:12:55]But from the measurements of a rectangular tile to get measurements of a room that they can fit that room should be a multiple of these tiles. So LCM.

[00:13:06]Thank you. So now for us to get the dimensions of the square tiles then we supposed to talk about the GCD of four uh 4.8 and 5.4. So the first thing that we do ladies and gentlemen I know when we are dealing with factors it's more encouraged to work with whole numbers.

[00:13:29]Therefore, allow me convert this measurements by multiplying each by 100.

[00:13:34]So that we have 540 cm and 480 cm. So we're using now cm.

[00:13:42]The units are now in cm. We check the GCD by two. This is a is it 270 and this one becomes 240. Yeah. Then we can divide by two again ladies and gentlemen.

[00:13:58]Uh-huh. This becomes 135 and this becomes 120. You see uh we can't divide further by two. So we can check another factor like three. By three we shall get 45 and 120 by 3 we shall get exactly 40 because of 40. Three cannot divide uniformly again. So we can proceed now to five. by five we shall get 9 and by five we shall get eight. So there is no other common factor now between them. So the GCD shall be 2 * 2 which is 4 * 3 which is 12 * 5 which is 60 cm. So the LCM of 540 cm and 480 cm is 60 cm. So the size of the square tile is 60 by 60.

[00:14:53]H the tile now the tile measurements is 60 by 60 60 by 60 cm. So the number of tiles number of tiles shall be given by 540 area of the old room out of 60 by 60 that is area of one square tile. So 540x 480 out of 60 by 60. This gives us exactly 72 tiles.

[00:15:30]So we shall use 72 tiles.

[00:15:34]72 tiles without a remainder. Exactly 72 tiles.

[00:15:40]Mhm. We can check now number four. And if you're getting something from the application of GCD list type I am learning. Let us know where you're learning from. Let us know your school.

[00:15:53]Let us know your county. Let us know whether you're getting something. Number four. In the figure below, Z Y is equal to 50. Z Y is equal to 50 m. Let me label that as we continue. Then the angle of elevation of Z.

[00:16:13]The angle of elevation of Z from W.

[00:16:20]Uhhuh. Zed from W.

[00:16:24]Let me just read that one carefully.

[00:16:26]Kindly let's repeat that. The angle of elevation of Zed M from W. I can see they on the same level ground. So I don't understand why angle of elevation from Z of Z from W are 24 and 35 respectively. Okay, let me correct the statement. The angle of elevation of X this is from X of X from Uhhuh.

[00:17:01]from Z and W R. Yes. So this is the angles sorry. Yeah. The two angles of elevation. One angle measured at Z another angle measured at W are 25 and 35 respectively. So the angle of elevation when you are at Z it's 24. The angle of elevation to X the angle of elevation of X from Z and from W respectively.

[00:17:34]So here 35 like that.

[00:17:40]Mhm. Like that. So we can check now the question.

[00:17:44]Uhhuh. Calculate the length WY. The length WY. So let's first find the height. the height xy this one we can call it h applying trigonometry because we given z y the all of this length and we are having an angle here and now we need the opposite length we can see that the tan of 24 is equal to opposite h out of the adjacent length 50. So to remain with h to remain with h we need to multiply 50 by the tan of 24 because we multiply by 50 both sides.

[00:18:27]Yeah. By the turn of 24 60 tan 24. This is exactly 22 26 22.26 uh is it m? Yes. Now at that point again allow me also say this. Now that we've gotten the vertical height, then we can use it together with an angle of 35. Now to get the length wy this way, length wy can even be let to be x. And we can even make it simpler by having this angle also calculated because we have 35 considering the triangle which is right angled at this point. WXY this is going to be 55 because this already 90 and this is 35.

[00:19:20]So the other angle there should be 55.

[00:19:22]So can we say that the turn of 55 equals to uh the opposite length out of the adjacent length which is now the height that you've gotten the height that you've gotten which is 22.26.

[00:19:41]So how do you remain with x? You multiply 22.26 by the turn of 55. You just multiply Mhm. by the turn of 85.

[00:19:58]This becomes 31.79.

[00:20:01]31.79.

[00:20:04]Let me write the answer. 31.7926.

[00:20:10]We've been told this should be estimated to 3dP. So this is 31.793 m. So that is now the length wy but remember we could still have applied tangent and with the angle of 35 and we still get the same answer. So it's all the same application of trigonometric ratios on angles of elevation and depression getting something type.

[00:20:52]So number five given that 9^ 2x * 2 power y + 10 = to 72 raised to the power of x + 10. Uhhuh. Find the values of x and y. So when you look at this particular question you realize that the left hand side is having two parts.

[00:21:20]There is a base of nine. There's a base of two. Then on the other side there is a we having 72.

[00:21:29]Mhm. 72. Can 72 be broken into 9 and two?

[00:21:35]Uh-huh.

[00:21:37]Can that one happen into 9 and two?

[00:21:41]Let's check.

[00:21:44]Let's check. So we are now saying that 72ide by 9 we get 8 then 8. So this is like 9 * instead of 8 2^ 3 that is what now yeah that is what I was trying now to come up with so that we also have on the left hand side the same same bases.

[00:22:12]Now we have 9^ 2 x * 2^ y + 10. This is equal to we pick 9 we raise it to the power times 2^ 3 we also raise it to the same power like that. So at this point now now now that we are having base 9 and base 2 we can equate their respective powers. So that you see the powers of 9 which is 2x should be equal to the powers of 9 x + y and this becomes equation one.

[00:22:52]Though we can simplify it further. This x coming to the left becomes a x.

[00:22:59]So equation one x = to y. Then equation two, we equate the powers of y the powers of two. So y + 10 = 2. Uh-huh. On the other side, the powers of two shall be 3x + 3 y.

[00:23:19]This is our equation two. But can we have some simplification, some grouping of terms, you know.

[00:23:27]So this is going to be 10 = 2. Uhhuh.

[00:23:32]This is a 3x. This is 2 y. This is our equation two. So at this point now from equation one we can substitute 10 = 3 and instead of x we have y then + 2 y.

[00:23:49]You see this is now 5 y being equal to 10. Meaning that the value of y shall be you divide through by five giving you two. And because the value of x is also equal to the value of y then x is also two. So the value of y is two. The value of x is also two. The value of x is also two. I'm concluding from equation one.

[00:24:10]The value of x is given by just the value of y. So x and y are equal and they equal to two. That's a very good question on number five. Checking number six now. And if you are learning something on indices, type yes, I learn number six. Given that 2, the sin of 2x + 30 = to 3x or 3 cosine of 2x + 30.

[00:24:41]Solve for the s of 90 - 2x to four decimal places. The first thing that we supposed to do in number six is to ensure thath we find the value of x first before we get to what we are solving here. Let's find x first. Uh-huh. From this point, allow me divide by 2 so that I have the s of 2x + 30 = 3 / 2.

[00:25:13]the cosine of 2x + 30. There's something interesting that you're supposed to notice at this point that the angle inside the brackets is the same. So we can divide through by the cosine and say this is sin 2x + 30 over the cosine of 2x + 30. This is equal to 3 / 2. We're remaining with 3 / two alone like that.

[00:25:43]Then again from trigonometric identities the ratio of tan okay tan beta equals to the ratio of sin over nko and cosine of nko. So we can now see on the left hand side it's like we having now tan of 2x + 30 because it's already s of cosine of the same value then it can become it's tangent according to this trigonometric identity according to the trigonometric identity.

[00:26:20]Therefore at this point now we can say 2x + 30 then = to the tan inverse of 3 / 2. So can we find the value of 2x + 30s?

[00:26:35]Uhhuh. I type a shift tan 3 out of 2.

[00:26:41]This gives me 56.31.

[00:26:47]Meaning that 2x = to we subtract 30 on that side. Uhhuh. We get 26.31.

[00:26:58]So the value of 2x is 26.31.

[00:27:03]26.31. Then what are we finding the s of? Now I'm thinking so quickly because I had seen it sign.

[00:27:14]We are finding the sin of 90 - 2x. We already having 2x. So it is 90 - 26.

[00:27:24]Instead of dividing by two to multiply by two again we can think ahead of time. So this is the sign of 63.69.

[00:27:39]Uhhuh. So the sign of this from my calculator is exactly 0. We've been told to four decimal places. So 8 96 4 is an accuracy of four decimal places. A very good question on trigonometric identities. If you have understood something on trigonometry, please remember to type yes island.

[00:28:08]Yes, I learn.

[00:28:11]Mhm. Number seven. The shortest sides of a of the right angle triangle are given as x - 3 12 while the longest is x + 5.

[00:28:25]Calculate the area of the right angle triangle. In other words, we have the hypotenuse and the other two sides. So, let's first give a sketch.

[00:28:38]The busation that information this is right angle this is 12 yeah it's among the short sides is x - 3 and here it's a x + 5 that we so now we know pythogore that says the two short sides squared should be equal to the longer side squared so x - 3 2 + 12 2 should give us x + 5 2. Then we expand x^2 - 6 x + 9 + 144 = x^2 + 10 x uh that's 10 x then + 25. Now when we start grouping of terms x^2 and x2 shall cancel. Then 10 x and 6x become 16 on the right hand side.

[00:29:42]Then we shall have 9 + 144 + no minus 25 we get 128.

[00:29:53]Yeah. Yeah. 25 becomes minus. So the value of x shall be exactly 8. Now if x is 8 then it means we can get the vertical height which is x - 3 x - 3 which is five. Now that we have this we can find the area of the triangle. Area is given by half base * 8. So half * a base of 10 * height of five. This shall give us exactly 30 square 30 square cm.

[00:30:34]That is now the area. Yeah. 30 square cm.

[00:30:38]30 square cm. That's a very good question. Application of algebra. You're given unknown sides for you to apply Pythagoras theorem. Then after applying Pythagoras theorem now you can find the value of X which will lead you to find the vertical height then the area. Now if you're getting something type I am learning then we check number eight together. Now the figure below shows a sector of a circle cut from the center.

[00:31:13]Uh-huh. The distance AB = to 8.66 and the radius 5.0.

[00:31:23]Calculate the area of the shaded region.

[00:31:25]The one thing that you're supposed to understand, you can't find the area of a certain region unless you first understand what this region is made of.

[00:31:36]For instance, this is a minor segment.

[00:31:41]And the area of a minor segment, let's write it.

[00:31:46]Area of a minor segment is usually given by area of a minor sector minus area of a triangle minus area of a triangle like that.

[00:32:09]Yeah. Minus area of a triangle. So at this point ladies and gentlemen at this point now we can now start manipulating this particular diagram to see how the area of each can be found.

[00:32:27]Remember we need the angle at the center for us to work out area of a sector.

[00:32:34]Uhhuh. This angle here I'm calling it angle beta. Angle alpha.

[00:32:40]Okay. So angle alpha can be given by you see when you subdivide like that this becomes 4 33. Yeah 4.33.

[00:32:49]So now we talking about s of alpha being equal to the opposite length out of the hypotenuse. So that alpha becomes the sin inverse of 4.33 out of five.

[00:33:12]So the value of alpha equals to uh this is exactly 59.99 97.

[00:33:28]Yeah. 59.997.

[00:33:31]So if I approximate this to 4 DP ladies and gentlemen it shall become exactly 60°.

[00:33:43]Then now with the 60° like that, 60° like that, we can talk about now the bigger angle A O B which is going now to be 2 * 60 which is 120.

[00:34:05]At this point now with 180 like that with 180 like that now we can find we can find now the areas whereby we shall say for the sector which uses the over 360 r 2 are we restricted to any pi not really so we can have 120 over 360 beta 3.142 radius squared. So what is the area of the sector? We pick 120 over 360 * 3.142 * 25. This is exactly 26.18 26.18.

[00:34:57]What about the triangle whereby we having a five five and the angle at the center? So half one side the other side sign of the angle between them. So we talk about half * 25 the s of 120. This gives us exactly 10.83 10.83.

[00:35:26]Sorry this is a cm squared because it's area. Now with this we can talk about the shaded part. The shaded part is a minor segment. So 26.18 - 10 yeah subtract a triangle from a sector you remain with a minor segment.

[00:35:49]A minor segment you remain with a minor segment. So 15.35 15 35 square cm. So that is now the area Mhm. That is now the area of the shaded part which is a minor segment. A very good question ladies and gentlemen. A very very good question at number nine now. And if you're getting something on area of a sector, area of a triangle, area of a minor segment, please type I am learning number nine together. Illustrate the following inequalities on the grid provided below.

[00:36:34]Mhm. 2 y greater than x 3 y + 4x less than or equal to 12 and x greater than zero. Hence mark the region R that satisfies the inequalities plotting of inequalities. I will start by just giving you an idea something that will just keep you going as long as there's a region to be bounded. Don't struggle to check the side of the lines that is supposed to be shaded.

[00:37:10]So just plot your three your four lines.

[00:37:14]Then they shall automatically outline a common region and through that you'll be able to know that the other sides the other sides of the line are the ones to be shaded. So the first thing that I will do I am going to translate every inequality to a mirror line. So 2 y greater than x which is an inequality is represented by what we call a border line equals to x. So if we have values of x and values of y when the value of y is one the values of x are twice. When the value of x is two, the value of y shall be twice. So these four coordinates are enough. 21 and 42 21 2.

[00:38:11]Mhm. And I hope you can observe the scale very well because the examiner is already having a scale for us. On the vertical axis, it is intervals of two but on the horizontal it is intervals of one. So from one for for coordinate 21 for coordinate 21 on the x should be two but one is here 21. What about 42? 4 and two.

[00:38:41]42 like that. So we've shaded.

[00:38:46]Uhhuh. We have now plotted so we can connect. So that we come up with what with the border line. Then before I plot sorry look at the nature of the inequality strictly greater than. So it's supposed to be a dotted line strictly greater than. If it was greater than or equal to the line would have been complete.

[00:39:12]So here we use dotted lines.

[00:39:16]We use dotted lines.

[00:39:20]Yeah. So that is line 2 y = x the first line. The second line is this one. 3 y + 4x = This one will be represented by 3 y + 4x = 12 x y. Allow me the intercepts for this one. y and x. When y is zero, you substitute y with 0. What shall be x? So this becomes 0. 4x = to 12. So we get 3. Uhhuh. When x is zero, what shall be y? So when x is zero, we substitute zero here. 4x becomes a 0. 3 y = 12. So y = 4. So the x intercept is 3. The y intercept is four. 3 4. Then now we plot.

[00:40:18]Yeah, we now plot. And from the nature of the inequality symbol, it's a complete line. So we join using complete lines.

[00:40:29]We join using complete lines like that.

[00:40:32]So line 3 x no 3 y + 4x 3 y + 4x = 12. That is the line. After that line, we can have the last line x greater than zero. It's also going to be dotted greater than zero. So we use line border line x = to0 which is now our y ais. Line x= to0 is the y ais. So now we use the y ais.

[00:41:06]is the yaxis and it's supposed to be dotted.

[00:41:13]The yaxis doted.

[00:41:17]We supposed to be very careful with whether lines are complete or dotted.

[00:41:22]Uh-huh. At this point now we can see there is the common region now that has been formed. This is the common region R which has been bounded by the three border lines. After binding them like that, we can choose to say that the region R can be shaded or the outer sides of the R. So we have the region R now shaded like that. So the the area that has been shaded wire is the common region bounded by the three inequalities. So you plot the three inequalities then you examine the common area that they shall be shielding that becomes our r a very good question type I am learning then we check number 10 togetherh in a certain month a family's expenditures were recorded as follows rent a fifth of the total loan repayment 0.375 of the salary. Then Kenya shillings 26,000 on domestic expenditure and saved the rest. Saved the rest. Uh-huh. If they saved 8,000 Kenya shillings, calculate the monthly salary, the total salary. So we saying that let the full salary let salary the monthly let it be x then we say a fifth of the total means 1 over five of the total this one went to uh rent rent payment then loan repayment 0 375 of X. This one is loan repayment.

[00:43:25]Then we are talking of now the others. Now Mhm. the others 26 that one is domestic expenditures and eight.

[00:43:41]So savings and expenditures which is the other noun the last noun. So 26 and 8 gives us 34 right 34,000 now 34,000 this now becomes the other the other things.

[00:43:59]So all of them when we add we should get the total salary. So 1 5 x + 0.375x when we add 34,000 we supposed to get the total salary x.

[00:44:18]So x which is 1 - 0.375 and minus again 1 / 5 we shall get 0.4 4 25x then this is 34,000.

[00:44:40]So how would you remain with X which is our total salary?

[00:44:44]You divide 34,000 by 0 and you get exactly 80,000.

[00:44:53]So this family was earning 80,000 per month. 80,000 per month.

[00:45:04]80,000 per month.

[00:45:08]80,000 per month.

[00:45:13]Uh-huh. If you're learning something on fractions, you can type I am learning. Then we check 11 together.

[00:45:26]Uh-huh. Number 11. Now 15 men working for eight hours a day can complete a certain job in exactly 24 days. For how many hours a day must 16 men work in order to complete the same job in 20 days. So we can write our we can write our variables here. We have men. We have the number of days that they working.

[00:46:01]Okay. The number of hours. Let hours come first.

[00:46:05]Let hours come first. Then the days they take. So we're talking about 15 men working 8 hours a day, taking 24 days to complete a piece of work. For how many hours a day? This is now where the question is.

[00:46:23]Mhm.

[00:46:25]must 16 men in order to complete in 20 days like that. So now after analyzing the question you're now going to know that 8 is what shall be varied here. So we start with the ratio of men then we go to the ratio of days. Then we ask ourselves 15 men must work 8 hours. What about 16? When the men increase to finish the same work, this men because they have increased they can finish the work by working lesser hours because 16 is greater than 15. Then the number of men increasing means the hours taken should decrease. So we decrease these hours by the ratio of men. Then after that we proceed to the days. Now we say uhhuh for the work to end in 24 days we need 8 hours a day. What about for the work to end in 20 days? For it to work in 20 days then we need to work more hours per day so that it can end in lesser days at the end of the day. So more hours. So we are increasing eight in the ratio of days.

[00:47:49]We are increasing like that. So I'm multiplying 8 * 15 / 16 * 24 over 20. This is exactly 9.

[00:48:04]9 hours a day or 9 hours per day. When they work 9 hours per day, now they will be able to finish the same work in 20 days.

[00:48:19]So that is how the question was supposed to be.