 Well, hello everyone. Let us look at the solution of J-Advanced 2023, question number one, paper number one. So it is from the chapter continuity and differentiability. Let us read the question first. The question is, let S be the set interval 0 open to 1 open, union 1 open to 2 open, union 3 open to 4 open. And T be a finite and discrete set which contains 4 elements only 0, 1, 2 and 3. Then which of the following statements are true? So let us go to each of the options one by one. So your option number A is, there are infinitely many functions from set S to set T. This is very much true because you know the formula for number of functions, right? If you know the number of elements in domain and code domain. So set S is acting as domain and set T is acting as code domain for this function. All right. Now what is the number of elements in domain? If I talk about number of elements in S, because it is an interval, of course it is infinity. And number of elements in the set T is only 4. Now what is the formula for number of functions? It is let us say if domain has m number of elements and code domain has n number of elements, then the number of functions is n raised to the power m. And because n is 4 and m is infinity, the answer will be 4 raised to the power infinity, which itself should be infinity. All right. So 4 raised to the power infinity itself is infinity. So the option number A is very much true. Let us look at the next option. Option number B is saying that there are infinitely many strictly increasing functions from S to T. Now let us try to draw the graph of the function, okay? And let us see if we are able to draw any strictly increasing function for the given domain and the given code domain, okay? So the domain contains the elements. Let's mark the numbers. 1, 2, 3, 4, 0, 1, 2, 3. And we don't need to go beyond that, okay? 1, 2, 3. Now let us suppose in the left most side the interval starts from 0 to 1. So you started with output y equals to 0. Then what is the definition for strictly increasing function? The definition is f of a minus h needs to be strictly less than f a, needs to be strictly less than f of a plus h. And because it has to be strictly increasing, if there are infinite elements in the domain, there needs to be infinitely many elements in the code domain. Otherwise, it is not possible to have a strictly increasing function. Because if you look at the first point, let's say the value is 0. At the next point, let's say the value is 1. At the next point, the value is 2. At the next point, the value is 3. And now we are out of points. Now we are very much out of the points, but there are still infinitely many points left in the domain. So statement b cannot be true. Then next option number c is the number of continuous functions from s to t is at most 120. So this is true because if you want to make a continuous function, it cannot jump from 0 to 1 in a particular interval. Let's say if it has to be continuous in 0 to 1, it needs to be constant. Because if it jumps from 0 to 1, it will need all the numbers between 0 to 1. But we don't have that. So if it has to be continuous, let's say the value of y between 0 to 1 is simply 1. Then between 1 to 2, it is let us say 2. And between let's say 3 to 4, the value is 3. So any such continuous function will be piecewise constant. That means in each of these intervals, it will be only a constant function. And how many choices are there for that output in this particular interval? There are four choices for the first interval, four choices for the second, and four choices for the third interval. So 64. And the option number c saying it is at most 120. So you have to interpret it like that 64 is less than equals to 120. So there is no problem with option number c, it is correct. And because y equals to c, constant function is differentiable, then you can say that all the continuous functions from s to t will be differentiable also. So option number d is also correct. All right, everyone, stay tuned. We'll come up with more solutions of previous year papers as well as more practice questions. All the best.