Test Papers – ICSE – Class – X
Std 11, Physics, Graph, Ch-03 Area under The curve
Look, class. Whenever a parabolic graph is made, so there are two types of scenario. If power of x is 2 and power of y is 1. But x can also have power of 1, if x power is 2, it is necessary to be 2. This is a parabolic graph. What it is showing, how the parabolic graph between x and y will be? It will be Parabolic. What is the interesting in this, many times what happens is that c is zero and the function will look like that. Many times c is also zero as well as b is also zero. So the graph looks like this. In all these cases, what will be the nature of the graph? It will be Parabolic only. Or only b will be zero, in this case also what will the nature of the graph, it will be parabolic. Suppose x is square then the graph between x and y will be this type of parabolic graph, upward opening parabola or downward opening parabola. How we will come to know, that it is upward or downward opening? This one value, the sign of this one value will tell you if it is upward opening or downward opening. If the co-efficient of x squared is positive then it will be upward opening parabola and if the co-efficient of x squared is negative then it will be downward opening parabola. If power of y is 1 and power of x is 2 then either it will be upward opening parabola or downward opening parabola, okay, children. And suppose this relation will be like this that power of y is 2 and power of x is 1, it can be like this also. The face of the function can be anyhow but the power of y is 2 and power of x is 1 then also the graph will be parabola. But what will happen this time, children. The opening will be rightward or leftward. If the co-efficient of y squared is positive then it will be rightward opening parabola or the y squared co-efficient will be negative then it will become leftward opening parabola. Get my point? So if it is y squared then right or left opening parabola graph and if x squared then upward or downward opening parabola graph. These kinds of functions are called Parabolic graphs.
We came to know the trend but we will come to know the exact graph only through the parameters of the questions.
See, graph between v and x or velocity and position or velocity and displacement for constant acceleration. For constant and acceleration motion, how will the graph be between v and x. Let’s see the graph between velocity and position for constant acceleration.
Std 11, Physics, Graph,Ch-02 Application of Slope
Std 11, Physics, Graph 02
Application of Slope
(No. 16)
The position time graph, this is position time graph A, B and C. And it is given that this is a parabola. If it is a parabola, acceleration will be constant. I will explain you why it is. Acceleration will be constant. And if it is upward opening parabola, acceleration will be positive and constant. So, for upward opening parabola, if position time graph is upward opening parabola, acceleration will be positive and constant.
So for entire motion from A to B and from B to C, for the complete motion acceleration is positive and constant. So for this section I will say A is positive and for this section I will say that A is positive. So for upward opening parabola, acceleration will be positive and constant. Any doubt?
Now listen carefully. A to B what about the velocity, will it be negative or positive, negative or positive, how is that defined? Slope of the tangent, greater than 90, tan theta negative. So during this reason v is negative and what about B to C, velocity is positive.
Now listen class, very carefully, B to C both velocity and accelerations are having the same signs that means particle is accelerating and velocity and acceleration are having the opposite sign that means particle is retarding. A to B particle is retarding and B to C particle is accelerating. Any doubt? Listen carefully, let’s discuss one important point.
Std 11, Physics, Graph,Ch-01 Graphs
Application of differentiation and integration in graph analysis.
See, let’s say some quantity y is the function of x, that means y varies with x. There are two ways of representing the variation of y with x, one is we write the function, and the other way is to give the graph between y and x. If function is given then how do we discuss that, we have learnt that with the help of integral and differentiation. So now we will discuss that if graph is given between y and x, then how can differentiation and integral used, okay. If y is function of x, then we had defined two things, average rate of change of y with respect to x, and that was defined by delta y by delta x. And another thing we had defined was instantaneous rate of change of y with respect to x, and that was defined by dy by dx. See if I know the function of yx then for average rate I will take change in y, meaning delta y divide dx, and for instantaneous change dy by dx, that means differentiation of y with respect to x. if I have to get these two information through the graph then how do we do that using a graph, average rate of change and how will we get instantaneous rate of change.
See what the graph is explaining. What does this point tell us? That if I make a vertical line then what is the value of x coordinate. And if I make a horizontal line then what is the value of y coordinate. When x is equal to x1 the y is equal to y1. Similarly when x is sum x2 to y will be sum y2. So this point represents x2y2 and this point represents x1y1, is this clear. Now listen carefully, then what is that distance. It will be x2 minus x1 and it will be delta x. And what is that distance? Delta y or y2 minus y1. That means if I will connect these two points, let’s say I connect these two points with a line. So what will this line connecting the curve be called? Secant or chord. What is chord or secant, a straight line with intersects the curve at two different points. Can you remember we defined two different things, secant of a curve and tangent of a curve. What is tangent? Line which touches the curve at one unique point. And when a line intersects the curve at two different points that is called secant. Now secant is a straight line, yes or no? Now class, let me ask you something? Do you know what is the meaning of slope? What is this slope? Slope is defined as tan theta. What is it? If I make a straight line between y and x, then the angle made by straight line with positive x axis is known as the slope. And then if we make two points here only then you will remember that this y2 minus y1 upon x2 minus x1, and this can also be written as delta y by delta x. So for a straight line delta y by delta x represent tan theta which is equal to the slope of a straight line. So, if we consider two points A and B and if we make a straight line passing through those two points. So then will there be a slope here of this straight line? Yes or no? So slope of secant which is tan theta is equal to delta y by delta x. So, average rate of change of y with respect to x is basically the slope of secant which passes through two different points of the curve. If I give you a graph and I ask you delta y by delta x, please hold and write it later. Let’s say, listen carefully….
Std 11, Physics,Integration,Ch-03 Application of Calculus
One example. Let’s write, position of a particle, position of a particle in its linear motion varies with acceleration, position of a particle in its linear motion varies with acceleration according to given function. Find the velocity at x = 2 meter if v = 0 for x = 0 meter per second. Acceleration of particle varies with x and we need to find the velocity when the x coordinate or the position is 2 meters if the initial velocity is zero when the position is x = 0 meter.
I think we have the acceleration function and we need to calculate the velocity. Listen class, stop writing. Step one is identification. Sir, we have an acceleration function and we need to find the velocity, so that it’s for sure that we need to integrate. And when we go for the process of integration, the next step would be to use the acceleration differential definition. Now we have two formulas or two definitions for acceleration differential definition. dv by dt or vdv by dx. Now let me ask you which one is useful here. vdv by dx. Because the variable of function is x, so we are supposed to use vdv by dx. Done. So.
Std 11, Physics, Integration, Ch-02 Application of Integration
Let’s discuss the definite integral. Let us discuss the definite integral. Now listen carefully. This is the indefinite integral of a function, okay. Now what we do is we integrate this function from one initial value to one final value. This is called definite integral of f(x), definite integral of f(x) with respect to x from, this is known as the lower limit and this is known as the upper limit. So this kind of terminology and this kind of expression is basically indicate the definite integral of a function from lower limit x = a to upper limit x = b. Done? And what it will be? f(x), listen class carefully, why we first go for the indefinite integral because the solution of definite integral can be found out only by indefinite integral, listen carefully.
What is the first step in solving an indefinite integral? That is forget; about the lower limit and upper limit and that is a standard indefinite integral, f(x) dx. And with the help of standard formulas or with the help of standard rules, integrate that function and you will get the new function, that is if you integrate the sine x, you will get what, -cos x. So you will integrate f(x) in an very ordinary manner, in the same manner as you do with indefinite. Done.
Then we will draw a line over here and we will write x = a, the lower limit and x = b, the upper limit, and then finally the final answer of your definite integral will be the value of new function at upper limit, value of your new function at upper limit minus value of new function at lower limit. This is called definite integral of a function f(x), from lower limit to upper limit. Done class? Now note one thing, the answer which we will get here would be a definite value or a definite function. That won’t be a familial function.
Observe in this answer there is no “+c” at the end of the value. Now we will just understand why did we name indefinite integral as indefinite integral, because we do not get any particular or any definite function from those operations. All we used to get is familial function that is there can be infinite number of possible answers. Instead of indefinite integrals, if we go for definite integral process, then we get as answer a definite value or a definite function; it’s not an indefinite value, it’s either a fixed value or a fixed function. There is no “+c” that is why it is named as definite integral.
So for this particular expression, relax take it easy, listen attentively, now you have to think in just one manner, we don’t have to bother about how or from where did this came about, if it is written in this manner, so how do we read it? We will first understand how to read it. How will you read out the expression stated in front of you? Sir it is the definite integral of function f(x) from lower limit x = a to upper limit x = b with respect to x. how to solve this? Step number one, forget about the limits and solve the general indefinite integral. You will get a new function, done.
Take one example. Don’t write now. We have to integrate x2 dx from a to b. you forgot? What is the integral of x2 dx? We will apply the standard formula or standard rule.
Std 11, Physics, Integration, Ch-01 Introduction
If x is replaced by ax + b in all standard formulas, then their differential follow this route. Suppose my dear there is some function f(x), and after differentiating you get g(x), for example if you differentiate sin x, you get cos x. so if the function if ax + b, and I am differentiating it, so notice you will get the same answer.
The answer would be same and this value would be same but only the entire answer would get multiplied by a. only the entire answer would be multiplied by a. So if in a general function, in general standard formulas, if x is replaced by ax + b, then their differential will follow this particular route, for example, if I want to write differentiation of sin (ax+b), then what will be the answer? I will differentiate cos in ordinary manner, sine multiplied by a. In all formulas. One more example. Y = (2x+5)99.
So I will consider 2x+5, so dy by dx, will become nxn-1 multiplied by 2. The function will undergo ordinary differential but it will be multiplied by the exponent of the x. You will multiply x by the coefficient of the x. Done. For an example, one more example, if I am saying y = ln (ax+b), then my dear y dot will be what is the differential of ln(x), 1 upon x. So what will happen to this? 1 upon ax+b and then it will be multiplied by a.
Let’s start another important application of the calculus. There are two mathematical operations that are very, very important in physics. One is differentiation and another one is the integration. We have gone through the differential. Now let us work out for the integration.
