Test Papers – ICSE – Class – X
Std 11, Physics, Vectors, Ch-02 Representation of vector in unit vector
Find the angle between given vector with the help of given angle. Not visible. Ok I will redraw the figure for you. Three vectors, we need to find angle between PQ, QR and PR. If I have given you some example 45, 45 and this angle 90 deg.
Tell me what is the angle between P and Q? What is the angle between Q and R?
What is the angle between R and P? Vector.
We have a rule to find the angle between two vectors and one information given.
What is the information? And information is you can shift the vector parallel to itself; it is not going to change. Now shift vector parallel to parallel and join tail to tail.
If I want to work out the angle between two vectors then I have shift both vectors and any one vector in such a manner that is join tail to tail. After joining tail to tail, smaller angle will be termed as angle between two vectors. So what will be angle in this particular scenario? Let us work out this problem.
Say if I want to find angle between P and Q then shift any one vector. We will do one thing we will shift P vector. Can I shift P vector parallel to itself? Yes and if I shift P vector parallel to itself, it will come in this particular place. now P and Q are in a state of tail to tail. Then this angle will define angle between segments. Isn’t it? What is that angle, 90 degrees? Done.
So angle between P vector and Q vector is 90 degrees. First answer.
Let us talk angle between Q and R. So shift Q or R, whichever you like which is easier. I feel if I shift Q upwards then it meets tail of R. see I am shifting R like this. Now they have joined tail to tail. Now tell me what is the angle? This angle will be between Q and R vector .Angle between two vectors will be anything between 0 to 180 degrees. So 135 is the angle. Done
Now angle between P and R.
Std 11, Physics,Vectors, Ch-01 Introduction to vectors
We will start with the vector. Note down the definition of the Vector
A physical quantity is said to be vector quantity if it carries direction as well as magnitude and it follows Triangle law of vector addition.
See when it comes to the understanding of vector, the definition includes 3 important aspects.
First, direction, quantity should carry the directional sense. We should automatically feel that we need to apply concept of direction to explain this quantity. It is necessary for me to give direction.
We have applied 10 Newton force, one question automatically comes that sir, in which direction you have applied the force. Have you pushed or pulled or you are pulling in which angle?
So when it comes to application of force by default an idea comes to mind like what force is applied and in what direction force is applied then we realise we have to explain this quantity completely then I have to explain directional sense to this quantity.
Second one, the magnitude. The value the numerical quantity. Done?
And the third and the most important thing about defining or the definition of vector is the Triangle Law of Vector Addition, this is very important, many times books and other things do not use this particular idea in defining the vector.
So I believe, definition of vector is complete only when you introduce this third point. So I believe the definition of vector is complete only when you apply this third point
Let me give you one very good example about this.
We all know electric current. Done. I is equal to 10 ampere. This is the magnitude of the current, done and we know that current has direction. Current flows from high potential to low potential done.
That means electric current carry magnitude as well as direction that means electric current should be a vector quantity, note down no hold on, but actually electric current is scalar quantity. This is the difference. Only magnitude and direction does not explain the situation. There is a third quantity, third idea that is the triangle law of vector addition. In real life current is a scalar quantity even if it carries direction and magnitude because it does not follow the triangle law of vector addition
So if I want to define vector, it must include triangle law of vector addition Ok class.
Similarly there is one more physical l quantity called area, what do you think area is scalar quantity or vector quantity?
Area is a vector quantity. Again area is assumed to be vector quantity. We are taking area as vector quantity because area is following Triangle law of vector addition in certain situation. Done?
Please note down these two examples. Write First point. Electric current is a scalar quantity
Even if it carries direction and magnitude because it does not follow triangle law of vector addition
Area is a vector quantity and direction of area vector can be decided by using two different points. What is first point?
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.
Std 11, Physics, Differentiation,Ch-4 Differentiation by substitution
Ok so it actually goes with chain rule.
What is chain rule? See dy/dx which explains you rate of change of Y with respect to X can also be written as product of two different rates dy/dt and
dt/dx.
First term gives you rate of change of Y with respect to t and second rate of change of t with respect to x, see class, this can also be written as dy/dt
multiplied by dt/dp multiplied by dp/dq and multiplied by dq/dx. You can form a chain even an infinitely long chain so that is why it is known as Chain
Rule.
For given differential dy/dx can be considered as product of two or minimum two separate differentials.
This particular approach will allow you to solve some good problems of differentials done.
Say if Y = sin (x ^2). Find dy/dx.
Now see this Y = sin (x^2)
This is not our standard formula. What is our standard formula? sine X, sine θ, sine of anything power 1 so you cannot say the differential will be
directly because differential of sin X is cos X so this will be cos (x^2) that is not correct. That means the rules or the list of formulae that we have
prepared is would not be applicable here directly because function is not falling in to required format. So what is required format? If it was sin X then
our problem would have been easy it would have been cos x of sin x. Then what to do? What will be our planning? Now listen carefully.
