IIT MADRAS MATHS QUALIFIER EXAM PYQ SOLUTION | PART -1 | NOVEMBER | 2022️ IITM

[00:00:00]hello Learners hope you all of it welcome to iitm student Community Channel today we'll be solving the maths question of iitm qualifier paper the latest one so let's start so our first question is consideration are defined on the state a for eign and the relation are defined as X comma y so that y equals to X plus 2 which among the following is or are correct okay our first we have to find the relation r so R is defined like this y equal stroke X plus two so when x equal to 4 y equals to 6 when x equals to 6 y equals to 8 when x equals to 8 y equals to team when x equals to 10 y equals to 12 when x equals to 12 y equals to 14 but this 14 is not a element of the set a so this is not an element of the set R so R is looks like this 4 comma 6 6 comma 8 comma 10 and 10 comma 12 as 14 is not a element of the set a this cannot be H element of the relation R right so what is our first option first option is R is neither reflect R is neither replaced in nor symmetric and not transitive this is transitive okay so first we have to know what is the reflexive relation reflexive relation reflexes so a relation is a reflexive relation if and only if every element of the set a maps to itself what does it mean for every element a belongs to a a comma a belongs to r that means 4C say 10 12 these are elements of a so in this set are 4 comma 4 6 comma 6 and like 12 comma 12 these elements should be in R to be a reflexive relation right but we can see in the state R there is no 4 comma 4 6 comma 6 like this element so R is not a reflexible relation next is symmetric thank you what is symmetric relation if relation is said to be a symmetric relation if a comma B belongs to R and also B comma e belongs to R for all a comma B belongs to a so if there is the a comma B there is also a b command so in the set R we can see 4 comma 6 belongs to R right so to be a symmetric relation there should be a 6 comma 4 belongs to R but here 6 comma 4 not belongs to R right so R is not a symmetrical lesson next one is transitive relation now what is transitive relation foreign if a comma B belongs to R and B comma C also belongs to R then there should be a comma C belongs to R that means here in the state are 4 comma 6 belongs to R right next 6 comma 8 also belongs to R so to be a transitive relation 4 comma C 8 have 2 4 comma 6 and 4 comma 8 also belongs to R but in the state R there is no element like 4 comma eight so four command not belongs to R so it's not not a transitive relation so our first option R is neither reflective nor symmetric and not transitive this option is correct next of R is an anti-symmetric relation now we have to know what is anti-symmetric relation first clear it anti-symmetric right [Music] and this is a metric what is the anti-symmetriculation so if a comma B belongs to R then if a not equals to B then B comma a should not belongs to R okay in other words we can say if a belongs to R and B comma a also belongs to r that means a equals to B now in our set our relation are we can say 4 comma 6 belongs to R right also we can see 4 not equals to 6 so 6 comma 4 6 comma 4 should not belongs to R and yes 6 comma 4 not belongs to R so we can say R is an anti-symmetric relation as it holds the property of scientific relation right next one are even it's our correct option next one R is an identification now what is identity relation identity what is identity relation every element of a state is related to itself only then we can say the relation is an identity relation what is that if a com a belongs to a a then a comma e belongs to R so 4 is a 4 is an element of a so 4 comma 4 should be in R right in our set there is no 4 comma 4 6 comma 6 like these elements so R is not an identity relation wrong option R is an equivalence relation what is equivalence reduction is relation can be said can be say say say the equivalence relation if R holds the property of R holds the property of R is the reflexive symmetric and also in transitive relation then only we can say R is an equivalent relation okay but here we can see or is neither reflective nor symmetry not transitive so R is not an equivalence relation so these also an incorrect option next question here we are given a polynomial configured a polynomial function pH equals to these choose the correct options from the following so first we have to draw the graph right from the function we can see it is a it is an odd degree function degree equals to 9. now the graph looks like these the two end points are in opposite direction these are the turning points foreign looks like this one right also we can see these are the turning points right okay so what is our first option PX is one one one function when X belongs to minus infinity to minus 2 even 4 comma Infinity so what is the reason this region minus infinity to minus 2 is region and 4 to Infinity this region right now first we have to know what is 1 1000 what is one one function IF function means one one function if no two elements in the domain corresponds to the same element in the rate what does it mean if there are two points X1 and X2 right if X1 not equals to A x 2 then f of x 1 should not equals to f of x 2.

[00:10:49]in other words we can say if X1 equals to x 2 then f of x 1 equals to FX2 now in our function if we take two points here X1 here and X2 here so what is f x 1 this point right it is if X1 and what is X2 this point if X2 right here x 1 not equals to x 2 and we can see that if x 1 not equals to f x 2 so in this region p x is in one on function it's clear now in this region here also you can take six one and it is X to point and so it is f x 1 right and it is FX2 pardon me for my bad handwriting and bad drawing so here also x 1 not equals to x two and we can see it takes one not equals to f x 2 so in this region also PX is one one function so this is our correct option okay next option is the total number of turning points of PX is so count the turning points these are the turning points one two three four five six seven eight so it's also our right option the total number of turning points of pH is eight next option p x greater than 0 when X belongs to minus 2 to minus 1 okay this region this region in this region right but here is you can see PX is not greater than zero this is less than zero right so we don't need we have not need to check the other region but if we check okay one comma 2 in this region in this region yes here p x greater than 0 2 comma 3 here also p x greater than 0 3 comma 4 greater than 0 but only in this region p x Less Than Zero so this is our wrong option as in this region p is not greater than 0. so it's not a correct option next option is there are exactly eight points on PX where the slope of the tangent is zero so when the slope of the tangent is 0 in this point that means in the turning points the tangent is zero so there are eight turning points so they are exactly eight points of on p x where that slope of the time it is 0. so Theta leads also characters okay next question in a survey of 500 people it was found that 49 percent like to watch comedy movies 53 person liked thriller movies 62 percent liked romantic movies also 27 percent likes was both comedian Thriller right 29 percent like to watch both Thriller and romantic and 28 percent like watching both comedy and romantic and five person liked none of the other sinners right based on these information and the given subsequences for questions okay so you can see 49 percent liked to watch comedy movies so here we represents comedy movies T thriller movies are romantic movies first we have to draw the Venn diagram okay here will be three set comedy thriller and romantic right this is our comedy this is trigger and this is romantic so enough C we are given 49 percent comedy movies nft 3D movies 53 percent n of R is 62 percent and both comedy and Thriller the percentages percentage is 27 percent both Thriller and romantic percentage is 29 percent and comedy and romantic both this is 28 percent and also given that five percent people like none of the genus so enough C Union T Union are complement is five percent from these we can say enough free Union the union R equals to 95 percent as five percent people like none of the dinner so 95 percent people like at least one of the genus okay now if we take if we take equals to X person that means X person people what the three dinners all three dinners that is comedy thriller romantic this ex-person people like to watch comedy thriller romantic these three genres okay this is percent okay so we know a formula like n of a union B Union C formula is n of a plus n of B plus n of C minus pin of a mid B [Music] minus n of we need three minus enough a plus enough a made to be made C so like this formula here we can see enough pH [Music] equals to mean of C plus n of t plus n of r minus n of C meet d sorry x minus B enough r minus n of see meat are Plus enough see me R right now we have to put the values it is the value of n of C Union t n r this is 95 okay n of C is 49 plus n of t it is 53 53 n of R 62 minus n of P met t 27 29 you now see me there it's 28 28 and we have taken enough primitive with r as X right now we have to find the value of x as we have questioned how many people like all the three movies so this is the people percentage of people who likes all the three movies right so we have to find the value of x so if we calculate we get x equal to 15 so this is the percentage so we have to find the number of people total we have 500 people is 15 percent in my hundred and eight fifteen into five that means 75 so this is the number of people who like all three movies so what answer is 75 next question find the number of people who like only one of the three geners okay let's draw the green makeup this is comedy this trailer and this is romantic and in the previous question we found this word which is 15 percent and we are given and we have to find the number of people we like only one of the three winners okay so we have to find this portion this person likes only the comedy Jenners and this person who likes only the Thriller and also this person who likes only the Romantic channel right okay so we are given that both comedy and Thriller the people who like both comedians winner percentages percentage is 27 percent okay comedy and Thriller this person total this person is 27 percent to get only this person what is the percentage so we have to substant fifteen percent from 27 percent so this video is 27 minus 15 that is 12 okay again we have to find this person only what is given who like to both Thriller and romantic that is 29 percent so we have substrate 15 that is 14 and who liked comedy and romantic also percentages 28 percent so this person is 28 minus 15.

[00:25:23]that is 13 right so but we have to find this portion this person this person and this person so to get the percentage only this person who liked only the common designers that is we are given who liked comedy that is 49 percent right so the people like who only the People Like Only The Comedy winners percentage is 49 minus I have to substitute this portion that is 12 this portion that is 15 and this portion also to get only this person okay that is minus 30. so it's equal to 9.

[00:26:28]so what is the percentage who like only the Thriller dinner on the percentage of people who liked three levels of Thriller it is 53 percent right 53 percent so we have to subtract the Sportsman this person and this person that is minus 12 minus 15 and minus 14.

[00:27:01]that's equal to 12.

[00:27:07]okay and what is the percentage of this portion we are given who like romantic dinner 62 percent people 62 subtract these portion this person this person minus 13 minus 50 minus 14 okay it's equal to 20 right so this percentage of people this percentage of people and this percentage of people like only one of the three genasis that is only comedy only female Thriller and only romantic so if we add these three percentage we get 20 plus 12 plus 9 that is 41 percent these 41 percent people like only one of the three genus now we have to find the number of people so total number of people is 500 and it's 41 percent like this if we can then by 5 this 2 0 5 so this is the number of people who like only one of the three generous right okay next question find the number of people who like at least two of the given genasis what does it mean at least one of the two genasis that means among the three genus find the first we have to find the percentage of people who like any of the two generous or the all three genus as we are given we have to find at least two of the given generous so any of the two or we have an R all three generous so from the diagram we can see this percentage of people liked comedy and Thriller these two generals this percentage of people light the Thriller and romantic Zen are both and this percentage of people like comedy and romantic both dinner and these 15 percent people liked all the three Jenners so if we add this percentages like 12 plus 13 plus 14 plus 15.

[00:30:08]equal to 54 percent this percentage of people liked at least two of the given genres so we have to find the number of people so to find the number of people 500 it's 54 percent by five I will get that is 5 into 4 22 270 okay this is our right answer 270 people like at least two of the given answer here I can see this is the letter okay now next question click next question okay this is our next question the polynomial the polynomial PX equals to A into x minus 4X minus 6 x minus 8 x minus 10 plus it passes through the vertex of the quadratic function QX equals to minus x minus 7 by whole Square minus 9 this is a equation of parabola based on this information answer the given for questions okay what is our first question enter the x coordinate of the vertex of key waves so this is the equation of keyword when it is the equation of parabola so now we know that the vertex form of the parabola is looks like this into x minus 8 whole square plus k and here H comma each comma k is the vertex of that Parabola now we can see these four and this form I looked alike so what is the vertex of this Parabola 7 comma minus 9 this is the vertex and what is your question find enter the x coordinate of the vertex of Q x what is the coordinate of butter that is given okay right yes enter the y coordinate of the vertices or next question so minus 9 is the y coordinate of vertex now next question enter the value of a so how can we find the value of a from these two equations p x and q F here in the question it is given that this function passes through the vertex of that quadratic function q w x so to find the value of a if we equate these two functions and we know that it passes through the vertex that means passes through 7 comma minus 9 so we create these two function at 7 comma minus 9 so if we put the value of x as 7 and equate these two function we can find the value of a Okay so a into we have to put the value of x might be where it passes through then minus 4 7 minus 6 7 minus eight greater than minus 10 equal to N is a minus 7 minus 7 it will be 0 minus 9 and if we calculate this this is 3 is 1 this is minus 1 this is minus 3 and this is minus 9 right Sports become line a equals to 9 that means b equals to A is a minus equals to minus 1 so the value of a is minus 1.

[00:34:53]let's see into the value of waves minus 1 yeah yeah my answer is minus 1 okay next question next question Ramis and crumbly recite at four comma minus 18.9 and 4 comma 18.9 respectively okay they planned a rotary Ramesh decided to go by road which is defined by a function r1a is constant and come let's choose the root defined by a function up to X okay they decided to meet where these rules intersect so to know where these two functions are will they intercept we have to draw the graph first so I have drawn this graph these blue line is the graph of function r 1 x and this red line is the graph of function r2x I hope you know the process of drawing a graph of a function so what is our question choose the state of correct option okay the first option is the number of if intercepts of R1 if and R2 it's are same so from the these are the x-intercept from the graph and the function we can see the x intercept of area r 1 x and R 2 x are same so this is our correct option the number of intersection points next option is the number of interception points of R1 if and r2h can be found using r1s equals to R2 here we know the process how to find the number of intersection points of two function we have to equate to function it is also write option number of intersection points of R1 if and r2x can be found using R1 is equal to minus accuracy no it's not it's R1 is equal to R 2 x Etc it's also a correct option what is our next question in total how many places can be made so in the question it is said that oh what is it that they decided to meet where these roads are intersex so from the graph we can see the intersect points this is the intersect Point these also a interesting point this is the interesting point it's interesting point it also it also so if we count The Intercept points are one two three four five six six into sixth point so in total in six basis we can meet the indices Point yes in six places they can meet okay next question very parallelogram whose side are represent by the line for equation of layer four equation of lines based on these information apps at the given sub questions okay we call a parallelogram is defined as a quadrilateral increase both pairs of opposite sides are parallel okay so we are given for equation of lines using these four lines we have to draw the parallelogram right so let's start with that let's draw the graph I have already drawn the graph so you can see these two equations that means these two lines are pass through origin so this line what is the equation of the line this is minus 2X plus y equals to zero this is the equation of this line and the other one which passes through origin X plus 2 y equals to 0 this line now we have to draw this line X Plus 2y minus 5 so how to draw this line we can see you can see that X Plus to y equals to 5. so if we convert it by you click on the date in intercept form then you can see do this so it intersect the the line intersect the exactly at 5 comma 0 and the Y is at 5 by 2 comma 0 into y f is at 5 by 2 comma 0. so this is the line why actually intercepted 5 by 2 that means 0.5 comma 0 2.5 comma 0 and here is the x-axis interest is that 5 comma zero right and the other one minus 2X plus y we have to convert it at intercept form that is 5 by 2 minus plus y by 5 equals to 1 that means it intersects the x axis at minus 2.5 minus 2.5 comma 0 and y axis at 5 comma 0.

[00:41:34]so this is the line which intersect the effect with at minus 2.5 comma 0 and y axis got the y axis at five comma zero so we can see this is our parallelogram right now what is our question choose the correct option from the following first option is the line minus 2x plus y equals to 0 and x plus to y equals to 0 passes through origin yeah from the equation we can see that and also graph we can see that minus 2x plus y equals to 0 and x plus y equals to 0 passing through origin this is our right option next option the lines minus 2x plus y equals to 0 and equals to y equals to 0 are parallel to each other from the graph we can see these are not parallel to each other they are intersecting so it's a incorrect option next option the line minus 2x plus y equals to 0 and 8 plus 2 y equals to 0 or perpendicular to each other yeah from the graph you can see these two lines these lines and these lines are perpendicular to each other at origin and then it's also our correct option next option the line minus 2x plus y equals to 0 and minus 2x plus y minus 5 are parallel to each other let's check in the graph the equation minus 2x plus y equals 2 0 and minus 2x plus y minus 5.

[00:43:36]so from the graph you can see minus 2x plus y equal to 0 and minus 2x plus y equals to 5 these two lines are parallel to each other so it's also a correct option next question the equation of the diagonal that does not pass through the origin so with the uh which diagonal which does not pass through the auditing buttons this this line This is the diagonal and we have to find the equation of these diagonal so to find the equation first we have to find this point these intersecting points and by these two intersecting points we can find the equation of a line As We Know infer two points are given we can find the equation of line so first our test is to find the point of this interesting point so this point so this point is intersecting point of these two lines those are X Plus 2y equals to 5 and and these plan minus 2X plus y equals to 0 so from here we can see y equals to 2x if we put this value of y in this equation we will get X Plus 4X equals to 5 x equals to 1 so y equals to 2 so these intersecting point is 1 comma 2 and what is the point of intersection this point is intersecting points of the lines minus 2X plus y ma equals to 5 and X Plus 2y X Plus 2y equals to 0 so this is the interesting points of these two lines from here we will get x equals to minus 2y and putting this value in this equation we will get 4X it's 4y plus y equals to 5 so y equals to 1 and x equals to minus 2 so the point is minus 2 comma 1 right so we get two points so we now we have to find the equation of the diagonal which passes through the points minus 2 comma 1 and 1 comma 2 so what is the equation of diagonal y minus 1 x minus 2 equals to equals to which is 1 minus 2 and minus 2 comma minus 1 so this is x minus X Y minus y1 by x minus x 1 equals to Y2 minus y 1 by X to minus x 1 this is our formula of the line if we calculate minus 1 plus minus 3 this is 3y minus 3 x minus 2 implies x minus 3y x minus 3y plus 1 equals to 0.

[00:48:14]so it pops in so check the options OMG no option like this hope I made a mistake let's check clear this we have to find the equation of the diagonal passing through the points minus 2 and 1 and 1 comma 2.

[00:48:56]so the equation like y minus y1 1 x minus X1 Y2 minus y1 x 2 minus X1 y minus 1 X plus 2 one by three let's calculate 3y minus 3 equals 2 X plus 2.

[00:49:39]x minus 3y Plus 5.

[00:49:45]now check the option x minus 3y plus 5 equals to 0.

[00:49:51]yeah this is our option so we get our right answer and also we can see if we multiply this equation by minus 2 we will get this equation minus 2 s plus 6 by minus 10 equal to 0 so it's also our correct Ops right now next question here is our question consider the function f x equal to 4 mod is minus 10.

[00:50:27]so our function is like f x equals to 4X minus 10 when X greater than 0 minus 4X my next team when it's less than 0 and if it's equals to minus D I mean x equals to 0.

[00:50:54]right now check the options slope of the function it is that every point is minus 4 X less than 0 for a x less than 0 the slope of the function is minus 4. so this is our right option next is the minimum value of f x is -10 we will get the minimum value of Ethics when we put the value x equals to 0.

[00:51:24]so it is our minimum value the y-intercept of f x is minus 10 yeah we can see the y-intercept of f x so that means if we put x equals to 0 now Y intercept is minus 10.

[00:51:43]so is also our correct option and the minimum value of f x is 0 no the minimum value of f x is minus 10 and it is the graph of the equation looks like this next question so here is the question of SSC this is our next question a function if it's feed for the data given in the table one table one recorded by student is y equal to FX equal to x minus 2 x minus 4 into x minus 6 Plus K we have to find the value of K so that SSE will be minimum so first we find the value of f x putting the value of x let's create a table it's equal to one three five six y equals to minus 11 7 1 4 and so if we put the value of x equals to 1 in if it's we get minus 15.

[00:53:02]plus k next put the value of x as 3 here we get if x as 3 plus a National value of x is 5 put the value of x 5 here we get minus 3 plus k and the last one is 6 if we put the value of 6 value of x equals to 6 this term becomes 0 and we'll get if x as K now what is the formula to calculate SSP equals to summation of FX I minus y i per whole Square R equals to 1 2 in so if we calculate here we have four points first one minus v 15 plus K minus 11 that is K minus 4.

[00:54:21]4 whole square plus 3 plus K minus 7 that is also K minus 4 next one minus 3 plus K minus 1 K minus 4 and last one is K minus 4 if x minus 5 that whole square right like this 4 into K minus 4 the whole Square or into K Square minus 8 k plus 16 equals to 4K Square minus 12 32k plus 64. now our question is find the value of K when ACC will be mean by so when will win the SSC minimum when the value of K is minimum so it's the equation of a parabola and the value of the equation will be minimum when when k equals to the vertex at vertex this is the minimum this equation becomes the minimum so P equals to x coordinate of the vertex math is minus B by 2A this is the formula that means minus 32 by 2A that is eight so it becomes 4 so this is the value of K this is our answer right okay next go to next question Sonia and heena want to solve a quadratic equation Sonia made a mistake in writing down the constant term and end it up in Roots 3 and 4 okay and heena made a mistake in writing down the coefficient of x and I got the roots two and three okay consider the leading coefficient to be 1 in all cases the correct roots of the equation are so we have to find the correct roots of the equation right so Sonia wrote the equation where he from that equation he gets the roots as 3 and 4. okay so first Sonia erotic equation whose roots are three and four this saying the question that leading coefficient to V1 in all cases okay so the equation is minus a One X plus 12 it is said that Sonia made a mistake in writing down the constant term so this is our constant term and we need to make a mistake writing down this constant term so this is wrong this constant term is wrong and he knows then Franklin whose roots are 2 and 3. right so it's the equation 5X plus 6 and it is also said that I we made a mistake in writing down the coefficient of x so these term is wrong for Hina so for Sonia this time is wrong and foreign and we have to find the correct equation and the correct roots of the equation so in the equation for Hina these term is wrong and Sonia wrote the equation who in that equation this term is correct it so it is 7x and Sonia wrote the equation in that equation these constant term is wrong but in enough equation the constant term is okay so our actual equation is it and this is this is our actual equation so from this equation we can easily find the roots this is minus 6x minus X plus six x minus six x minus 1 so the correct roots are 6 and 1. this is our sorry eight then one this is now next question so it is our last question thank you so in this question it is that consider to quadratic polynomials PX and Q wave whose roots are shown in figure one we have given a graph like this the leading coefficients of PX and QX are plus 1 and minus 1 okay the y coordinate of the vertex of p x is C and the y coordinate of the vertex of p x is minus the find the value of d so we are given this figure we have to find the value of d so from the graph we can see PX and Q wave these are two parabola and 1 comma 0 and D comma 0 are the roots of the equation of p x and D comma beta and the 19 comma 0 are the roots of the function QX so DX equals to x minus one into x minus d we are also proved that the leading coefficient of PX is plus 1 and QX is minus 1.

[01:01:13]so it is like x squared minus D plus 1 DX plus d and the equation of Q X root size d and 19.

[01:01:28]x minus E and if minus 19 and it's the reading coefficient is -1 okay so it is minus x squared Plus d it is not plus it's minus D Plus 90 into X Plus 90 d if we multiply the minus 9 it will become minus x square plus d Plus 90 X minus 19 d now in the question it is given the y coordinate of the vertices of p x is C now what is the formula of vertex the y coordinate of the vertex we know the x coordinate of the vertex is minus B by 2 a and the y coordinate y coordinate of vertex formula is 4 AC minus b square by 40.

[01:03:02]don't confuse with this C and the given C those are different so from using this formula we can find the y coordinate of the vertex of from these two equation right so the y coordinate of vertex of t f is C so c equal to bhatics of digitation so 4 AC minus P Square right 4 is B that is 4D minus b square that is D plus 1 whole Square by 4A that is 4. and the y coordinate of vertex period is minus C all right so minus C Matrix of this equation minus equal to 4 into minus 19 t four into minus 19 d minus B plus 90.

[01:04:26]whole Square by by 4 into minus 1 here also there is a minus one right so this is 3 and this is minus C so to equate these two if we multiply a minus 3 here we can equate these two and we can find the value of d right so this is minus 4D minus D plus 1 whole Square by 4 and here it is 4 into 19 the 90 minus D plus 1 whole Square and it is minus 4.

[01:05:37]two side minus canceled you get and also 4 canceled minus B plus 1 whole Square equals to 4 into 19 d minus now we have to calculate the value of t is only the calculation path 2019 D minus t if we simplify all these we will get the value of b equal to 10 this is our Chi sensor right so that's all thank you