 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that state which of the following functions are linear give reason a y is equal to 0.25 plus 0.5 into x minus 2 b a is equal to i into r square and c y is equal to 2x we know that a function is said to be linear if we can write it in the slope intersect form that is y is equal to n into x plus c with this key idea let us proceed to the solution now the first path says y is equal to 0.25 plus 0.5 into x minus 2 let us simplify it and therefore we get y is equal to 0.25 plus 0.5 into x that is 0.5x minus us 0.5 into 2 that is 1 therefore y is equal to now combining like times together we write 0.25 minus of 1 plus 0.5 x and therefore we get y is equal to 0.25 minus 1 is minus of 0.75 plus 0.5x we can write y as 0.5x minus of 0.75 and from the key idea we know that a function is said to be linear if we can write it in the slope intercept form that is y is equal to mx plus c now here we have written it in the form of y is equal to m into x plus c where m is given by 0.5 and the value of c is given by minus of 0.75 so we say that it is a linear relationship in the second path we have the relation a is equal to pi r square which is the area of the circle with radius r here variables r and r and here we see that the power of r is 2 so it cannot be written a is equal to m into r plus c form so we say that it is non-linear that is we can see if we put the value of r as 0 we get the value of a as 0 if we put the value of r as 1 we get the value of a as pi and similarly if we put the value of r as 2 we get the value of a as pi into 2 square that is pi into 4 which is equal to 4 pi now if we see change in r we get 1 minus 0 which is equal to 1 and 2 minus 1 that is also 1 so we can say that it is constant but change in a that is pi minus 0 is equal to pi and 4 pi minus pi is equal to 3 pi and we see that it is not constant so it is a non-linear relationship and in the third path we are given y is equal to twice of x and we can see that the equation y is equal to twice of x is of the form y is equal to m into x plus c where the value of c is 0 it is the equation of the straight line pattern through origin so we say that it is a linear relationship which is the required answer this completes our session hope you enjoyed this session