Chapter 4 Motion in a Plane Handwritten Notes Class 11 Physics

Chapter 4 Motion in a Plane Class 11 Notes Physics

Chapter 3 Motion in a Straight Line Handwritten Notes Class 11th Physics

Chapter 3 Motion in a Straight Line Class 11 Notes Physics

Motion In A Straight Line

In this chapter, we shall learn how to describe motion. For this, we develop the concepts of velocity, acceleration and relative velocity. We also develop a set of simple equations called Kinematic equations.

• Motion: Motion is change in position of an object with time.
• Rectilinear motion: The motion along a straight line is called rectilinear motion.
• Point object: If the distance travelled by the body is very large compared with its size, the size of the body may be neglected. The body under such a condition may be taken as a point object. The point object can be represented by a point.

Example:

• The length of bus may be neglected compared with the length of the road it is running.
• The size of planet is ignored compared with the size of the orbit in which it is moving.

Position, Path Length And Displacement

1. Reference point, Frame of reference:
In order to specify position of object, we take reference point and a set of axes. Consider a rectangular coordinate system consisting of three mutually perpendicular axes, labelled x, y, and z axes. The point of intersection of these three axes is called origin (O) and serves as the reference point.

The coordinates (x, y, z) of an object describe the position of the object. To measure time, we place a clock in this coordinate system. This coordinate system along with a clock is called a frame of reference.

To describe the motion along a straight line we can choose x-axis. The position of a carat different time are given in figure 3.1. The position to the right of 0 is taken as positive and to the left of 0 as negative. The position coordinates of point P and Q are +360m +240m. The position coordinate of R is-120m.

2. Path Length (Distance):
The total length of the path travelled by an object is called path length.
Explanation:
Consider a car moving along straight line. The positions of car at different time are given in the x-axis. (See figure 3.1)
Case-1:
The car moves from 0 to P. In this case the distance moved by car is OP = +360.
Case-2:
The car moves from 0 to P and then moves back from P to Q.
In this case, the distance travelled is OP + PQ = +360 + (+120) = +480m.

3. Displacement:
The distance between initial point and final point is called displacement.
OR
The change of position of the particle in a particular direction is called displacement.
Explanation:
Consider a car moving along a straight line. The positions of car at different time is given in the x-axis.
See figure (3.1)
Let us take two cases
Case-1:
The car moves from 0 to P, in this case displacement = (360 – 0) = 360
Case-2:
The car moves from 0 to P and moves back from P to Q.
In this case,
Displacement = 240m
Let x₁ and x₂ be the positions of an object at time t₁ and t₂. Then displacement in time Dt = (t₂ – t₁) can be written as Dx = x₂ – x₁
If x₁ < x₂, Dx is positive and if x₂ < x₁, Dx is negative.
Note: The magnitude of displacement may or may not be equal to the path length traversed by an object.

4. Position Time Graph:
Motion of an object can be represented by a position-time graph.
Position time graph for a stationary object:
For a stationary object, the position does not change with time. Hence the position time graph will be a straight line parallel to time axis.

Question 1.
The position-time of a car is given below. Analyze the graph and explain the motion of car.

Answer:
The car starts from rest a time t=0s from the origin 0 and picks up speed till t=10s. After 10 sec, the car moves with uniform speed till t=18 sec. Then the brakes are applied and the car stops at t = 20s and x = 296m.

Question 2.
Draw the position-time for an object

1. moving with positive velocity
2. moving with negative velocity.

Answer:

Average Velocity And Average Speed

1. Average Velocity:
The average velocity of a particle is the ratio of the total displacement to the time interval.

Explanation:
To explain average velocity, consider a position-time graph of a body given below.

Let x₁ be the position of body at a time t₁ and x₂ be the position at t₂.
The average velocity during the time interval Dt = (t₂ – t₁)

where Dx = x₂ – x₁, and Dt = t₂ – t₁,
¯¯¯vv¯ is the average velocity.

Question 3.
Find the slope of position-time graph given below of uniform motion and explain the result.

Answer:

Slope of displacement time graph gives average velocity.

Question 4.
Displacement time graph of a car is given below.

1. Find the average velocity during the time interval 5 to 7 sec.
2. Find the average velocity by taking slope in the interval 5 to 7 sec.

Answer:
1.

2. Slope, tan q

In this case, slope and average velocity are equal in the same interval.

2. Average Speed:
Average speed of a particle is the ratio of the total distance to total time taken.

Question 5.
A car is moving along a straight line. Say OP in figure. It moves from 0 to P in 18s and returns from P to Q in 6s. What are the average velocity and average speed of the car in going?

1. From 0 to P? and
2. from 0 to P and back to Q. (See Figure 3.1)

Answer:
1. Average velocity

Average speed

In this case the average speed is equal to the magnitude of the average velocity.

2. In this case
Average velocity

Average speed

In this case the average speed is not equal to the magnitude of the average velocity. This happens because the motion here involves change in direction. So that the distance is greater than displacement.
Note: In general, the velocity is always less than or equal to speed.

Instantaneous Velocity And Speed

Nonuniform Motion:
A body is said to be nonuniform motion, if it undergoes unequal displacements in equal intervals of time.

OR

A body moving with varying velocity is called nonuniform motion.

1. Instantaneous Velocity:
Question 6.
Why the concept of instantaneous velocity is introduced?
Answer:
In nonuniform motion the average velocity tells us how fast the object has been moving over a given interval. But it does not tell us how it moves at different instants during that interval. For this we define instantaneous velocity. The velocity at an instant is called instantaneous velocity.
Explanation:
Position-time of a body moving along a straight line is given below.

Let us find average velocity in the interval 2 sec (3s to 5s), centered at t = 4 sec. In this case, the slope of line P₁P₂ give the value of average velocity, ie. Slope of P₁P₂,

Decrease the value of Dt from 2.to 1 sec. (ie. 3.5 to 4.5 sec). Then line P₁P₂ becomes Q₁Q₂. Then the slope of gives average velocity overthe interval 3.5 sec to 4.5sec.
ie. slope of Q₁Q₂

In the limit Dt ® 0, gives the instantaneous velocity at t = 4sec and its value is nearly 3.84m/s.

Question 7.
When average velocity of a body becomes instantaneous velocity?
Answer:
In the limit, Dt goes to zero, the average velocity becomes instantaneous velocity.

But lim lim

\Instantaneous velocity,

Here dx/dt is the differential coefficient of x with respect to time. It is the rate of change of position with respect to time at an instant.

Question 9.
The position of an object moving along x-axis is given by x = a + bt² where a = 8.5m, b = 2.5 m/s² and t is measured in seconds

1. What is the velocity at t = 0s and t = 2s.
2. What is the average velocity between t = 2s and t = 4s?

Answer:
1.

when t = 0
we get v = 2 × 2.5 × 0
v = 0
when t = 2sec
v = 2 × 2.5 × 2 v = 10m/s.

2. The average velocity

Note: If a body is moving with constant velocity, the average velocity is the same as instantaneous velocity at all instants.

2. Instantaneous Speed:
The speed at an instant is called instantaneous speed.
Note:

• The average speed over a finite interval of time is greater or equal to the magnitude of the average velocity.
• Instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.

Acceleration

1. Average Acceleration:
Average acceleration of a particle is ratio of the change in velocity to the time interval.

Explanation
Consider a body moving along a straight line. Let v₁ and v₂ be the instantaneous velocities at time t₁ and t₂ respectively.

where Dv = change in velocity, Dt = Time interval

2. Instantaneous Acceleration:
Acceleration at any instant is called instantaneous acceleration.
Explanation
In the limit Dt ® 0, (Dt goes to zero) the average acceleration becomes instantaneous acceleration.
ie. Instantaneous acceleration

Instantaneous acceleration is the rate of change of velocity with respect to time.

3. Uniform Acceleration:
A body is said to be in uniform acceleration if velocity changes equally in equal intervals of time.

Question 10.
The velocities of two bodies A and B are given in the tables. From this table, find which body is moving with uniform acceleration. Explain.

Answer:
Body A is moving with uniform acceleration be-cause the velocity of body increases at the rate of 2 m/s².
The body B is moving with constant velocity. Hence this motion is called uniform motion.

4. Velocity-Time Graph For Uniformly Accelerated Motion:

An example for velocity-time of a uniformly accelerated motion is given in the above figure.
Let v_t1 and v_t2 be the velocities at instants t₁ and t₂respectively.
The slope of graph in the interval (t₂ – t₁) can be written as,

∴ tan q = acceleration
Thus the slope of the velocity-time gives the acceleration of the particle.

Question 11.
The Velocity-time of a body is given below. From this graph draw the corresponding acceleration time graph.

Answer:
The slope of velocity-time graph increases in the interval (0 – 10) sec which means that acceleration of the body increases in this interval.

Velocity is constant in the interval (10 – 18) sec. Hence ’ the slope is zero which means that acceleration is zero in this range.

The slope in the interval (18 – 20) sec is constant and negative. Hence acceleration in this is a negative value. The acceleration – time graph for the above motion is given below.

Question 12.
The position-time graph of a car is given below.

1. Draw corresponding velocity-time graph. Explain the reason for your answer.
2. From velocity-time graph draw acceleration-time graph and identify the regions of

• positive acceleration
• Negative acceleration
• zero acceleration.

Answer:
1. In the time interval (0 – t₁) sec, the slope of x – t graph increases which means that velocity is increasing in this time interval.

In the time interval (t₁ – t₂) sec, slope is constant. Hence velocity remains constant in this time interval.

In the time interval (t₂ – t₃) sec, the slope is decreasing and finally becomes zero. Which means that velocity decreases to zero.

2. Slope is constant throughout the interval (0 – t₁) sec which means that acceleration constant.

In the interval (t₁ – t₂) sec, slope is zero. Which means that acceleration is zero in this region.

Slope is constant (but negative) in the interval (t₂ – t₃)sec. Hence acceleration is constant and negative in this time interval.

Question 13.
Find the region of

1. positive acceleration
2. zero acceleration
3. negative acceleration from the above x-t graph

Answer:

1. Region OA – Positive acceleration
2. Region AB – zero acceleration
3. Region BC – Negative acceleration

Question 14.
Match the following.

Answer:
1) – d, 2) – c, 3) – b, 4) – a.

5. Area Under Velocity-Time Graph:
Area under velocity-time graph represents the displacement over a given time interval.
Explanation
Consider a body moving with constant velocity v. Its velocity-time graph is given below.

Kinematic Equations For Uniformly Accelerated Motion

For uniformly accelerated motion, we can derive some simple equations.

1. Velocity-time relation
2. Position-time relation
3. Position-velocity relation

These equations are called kinematic equations for uniformly accelerated motion.

Consider a body moving along a straight line with uniform acceleration ‘a’. Let ‘u’ be initial velocity and ‘v ‘ be the final velocity at time t.
We know acceleration a =  Change in velocity  Time interval  Change in velocity  Time interval 
a = v−utv−ut
at = v – u

Consider a body moving along a straight line with uniform acceleration a. Let ‘u’ be initial velocity and ‘v’ be the final velocity. ‘S’ is the displacement travelled by the body during the time interval ‘t‘.
Displacement of the body during the time interval t,
S = average velocity × time
S=(v+u2)tS=(v+u2)t _____(1)
But v = u + at ____(2)
Substitute eq.(2) in eq.(1), we get

3. Position-Velocity Relation:
S=(v+u2)tS=(v+u2)t _____(1)
But v = u + at
v−uav−ua = t _____(2)
Substitute eq.(2) in eq.(1)

Free-fall:
An object released (near the surface of earth) is accelerated towards the earth. If air resistance is neglected, the object is said to be in free fall. The acceleration due to gravity near the surface of earth is 9.8 m/s².
Note: Free-fall is a case of motion with uniform acceleration.

Question 15.
A body is allowed to fall freely. Draw the following graph.

1. Acceleration-time
2. Velocity-time
3. Position-time

Answer:
1.

2.

3.

Stopping distance of vehicles:
When brakes are applied to a moving vehicle, the distance it travels before stopping is called stopping distance.

Question 16.
Derive an expression for stopping distance of a vehicle in terms of initial velocity (u) and retardation (a).
Answer:
Let the distance travelled by the vehicle before it stops be ‘s’.
Then we can find ‘s’ using the formula
v² = u² + 2as
0 = u² + -2as

3.7 Relative Velocity
Suppose the distance between two bodies changes with time in magnitude, or in direction or in both. Then each body is said to have a velocity relative to the other.

For example, consider two cars A and B moving in the same direction with equal velocities. To a person in A, the car B would appear to be rest.

Hence the velocity of B relative to A is zero.
ie. V_BA = 0
Similarly, the velocity of A with respect to B is zero.
or V_AB = 0
Let A be moving with a velocity V_A and B be moving with a greater velocity V_B in the same direction. Then the person in A feels that the car B is moving away from him with a velocity V_BA. The velocity of B relative to A

For an observer in B, car A is going back with a velocity. The velocity of A relative to B
V_AB = -(V_B – V_B).

Question 17.
The position-time graph of two bodies A and B (at different situations) are given in the following graphs. Find the relative velocities of the following graph.

Answer:
a) The slope of Aand B are equal. Hence velocity of A and B are equal. So velocity of A with respect to B, V_AB = 0

b) The body A and B meet at t = 3sec

Velocity of A w.r. to B, V_AB = V_A – V_B
= 20-10 = 10 m/s Velocity of B w.r. to A, V_BA = V_B – V_A
= 10 – 20 = -10 m/s

c) The body A and B meet at t = 1 sec.
The velocity of body in the interval t = 1 sec,

Velocity of A w. r. to B,
V_AB = V_A – V_B
= 20 – 10 = 30 m/s
Similarly velocity of B w.r. to A,
V_BA = V_B – V_A
= 10 – +20 = -30 m/s
The magnitude of V_BA or V_AB (=30 m/s) is greater than the magnitude of velocity A or that of B.

Chapter 2 Units and Measurement Handwritten Notes Class 11th Physics

These Notes are Useful for CBSE, MPSB & Other state boards. Textbook with Our notes is sufficient for your XI Examination.

Chapter 2 Units and Measurement Class 11 Notes Physics

Introduction

a. Fundamental or base quantities:
Physics is based on measurement of physical quantities. Certain physical quantities are chosen as fundamental or base quantities. Length, mass, time, electric current thermodynamic temperature, amount of substance and luminous intensity are such base quantities.

b. Units: Fundamental Units and Derived Units Unit:
Measurement of any physical quantity is made by comparing it with a standard. Such standard of measurement are known as unit. If length of rod is 5 m, it means that the length of rod is 5 times the standard unit ‘metre’.

Fundamental Unit:
The unit of fundamental or base quantities are called fundamental or base units. The base units are listed in table.

Base quantity	Base unit
Length	Metre
Mass	kilogram
Time	Second
Electric current	Ampere
Thermodynamic Temperature	Kelvin
Amount of Substance	mole
Luminous Intensity	Candela

Derived Unit

The units of other physical quantities can be expressed as combination of base units. Such units are called derived units.
Example: Unit of force is kgms^-2 (or Newton). Unit of velocity is ms^-1.

The International System Of UnitsDerived Unit
System of Units: A complete set of fundamental and derived units is called a system of unit.

a. Different system of units:
The different systems of units are CGS system FPS (or British) system, MKS system and SI system. A comparison of these systems of unit is given in the table below, (for length, mass and time)

Note: The first three systems of units were used in earlier time. Presently we use SI system.

b. International System Of Unit (Si Unit):
The internationally accepted system of unit for measurement is system international d’ unites (French for International System of Units). It is abbreviated as SI.

The SI system is based on seven fundamental units and these units have well defined and internationally accepted symbols, (given in table – 2.1)

c. Solid Angle and Plane Angle:
Other than the seven base units, two more units are defined.
1. Plane angle (dq): It is defined as ratio of length of arc (ds) to the radius, r.

The unit of plane angle is radian. Its symbol is rad.

2. Solid Angle (dW): It is defined as the ratio of the intercepted area (dA) of spherical surface, to square of its radius.

The unit of solid angle is steradian. The symbol is Sr.

Measurement Of Length

Two methods are used to measure length

• direct method
• indirect method.

The metre scale, Vernier caliper, screwgauge, spherometer are used in direct method for measurement of length. The indirect method is used if range of length is beyond the above ranges.

1. Measurement Of Large Distances:
Parallax Method:
Parallax method is used to find distance of planet or star from earth. The distance between two points of observation (observatories) is called base. The angle between two directions of observation at the two points is called parallax angle or parallactic angle (q).

Parallax Method
The planet ‘s’ is at a distance ‘D’ from the surface of earth. To measure D, the planet is observed from two observatories A and B (on earth). The distance between A and B is b and q be the parallax angle between direction of observation from A and B.

AB can be considered as an arch A h B of length ‘b’ of a circle of radius D with its center at S. (Because q is very small, bDbD<<1], Thus from arch-radius relation.

Thus by measuring b and q distance to planet can be determined. The size of planet or angular diameter of planet can be measured using the value of D. If the angle a (angle between two directions of observation of two diametrically opposite points on planet) is measured using a

Where d is diameter of planet.

2. Estimation Of Very Small Distances:
Size Of Molecule
Electron microscope can measure distance of the order of 0.6A0 (wavelength of electron).

3. Range Of Lengths:
The size of the objects in the universe vary over a very wide range. The table (given below) gives the range and order of lengths and sizes of some objects in the universe.

Units for short and large lengths
1 fermi = 1f = 10^-15m
1 Angstrom = 1A° = 10^-10m
1 astronomical unit = 1AU = 1.496 × 10¹¹m
1 light year = 1/y = 9.46 × 10¹⁵m
(Distance that light travels with velocity of 3 × 10⁸ m/s in 1 year)
1 par sec = 3.08 × 10¹⁶m = 3.3 light year
(par sec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second).

Measurement Of Mass

Mass is basic property of matter. The S.l. unit of mass is kg. While dealing with atoms and molecules, the kilogram •is an inconvenient unit. In this case there is an important standard unit called the unified atomic mass unit( u).
1 unified atomic mass unit = lu
= (1/12)^th of the mass of carbon-12

1. Range Of Masses:
The masses of the objects in the universe vary over a very wide range which is given in the table.

Measurement Of Time

To measure any time interval we need a clock. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium clock sometimes called atomic clock.

Definition of second:
One second was defined as the duration of 9, 192, 631, 770 internal oscillations between two hyperfine levels of Cesium-133 atom in the ground state.
Range and Order of time intervals

Accuracy, Precision Of Instruments And Errors In Measurement

Error:
The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.

Systematic errors:
Systematic errors are those errors that tend to be in one direction, either positive or negative.

Sources of systematic errors

1. Instrumental errors
2. Imperfection in experimental technique or procedure
3. personal errors

1. Instrumental errors:
Instrumental error arise from the errors due to imperfect design or calibration of the measuring instrument.
eg: In Vernier Callipers, the zero mark of vernier scale may not coincide with the zero mark of the main scale.

2. Imperfection in experimental technique or procedure:
To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, velocity……..etc) during the experiment may affect the measurement.

3. Personal Errors:
Personal error arise due to an individual’s bias, lack of proper setting of the apparatus or individual carelessness etc.

Random errors
The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (eg. unpredictable fluctuations in temperature, voltage supply, etc.)

Least Count Error
The smallest value that can be measured by the measuring instrument is called its least count. The least count error is the error associated with the resolution of the instrument. By using instruments of higher precision, improving experimental technique etc, we can reduce least count error.

1. Absolute Error, Relative Error And Percentage Error:
The magnitude of the difference between the true value of the quantity and the measured value is called absolute error in the measurement. Since the true value of the quantity is not known, the arithmetic mean of the measured values may be taken as the true value.

Explanation:
Suppose the values obtained in several measurements are a₁, a₂, a₃,………,a_n. Then arithmetic mean can be written as

The absolute error,
∆a₁ = a_mean – a₁
∆a₂ = a_mean – a₂
∆a_n = a_mean – a_n

a. Mean absolute error:
The arithmetic mean of all the absolute errors is known as mean absolute error. The mean absolute error in the above case,

b. Relative error:
The relative error is the ratio of the mean absolute error (Da_mean) to the mean value (a_mean).

c. Percentage error:
The relative error expressed in percent is called the percentage error (da).

Example:
Question 1.
When the diameter of a wire is measured using a screw gauge, the successive readings are found to be 1.11 mm, 1.14mm, 1.09mm, 1.15 mm, and 1.16 mm. Calculate the absolute error and relative error in the measurement.
Answer:
The arithmetic mean value of the measurement is

The absolute errors in the measurements are
1.13 – 1.14 = 0.02mm
1.13 – 1.14 = -0.01mm
1.13 – 1.09 = 0.04mm
1.13 – 1.15 =-0.02 mm
1.13 – 1.16 = 0.03mm
The arithmetic mean of the absolute errors

Percentage of relative error

2. Combination Of Errors:
When a quantity is determined by combining several measurements, the errors in the different measurements will combine in some way or other.

a. Error of a sum or a difference:
Rule: when two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.
Explanation:
Let two quantities A and B have measured values A ± DA and B ± DB respectively. DA and DB are the absolute errors in their measurements. To find the error Dz that may occur in the sum z = A + B,
Consider
z + ∆z = (A ± ∆A) + B ± ∆B = (A + B) ± ∆A ± ∆B
The maximum possible error in the value of z is given by,

Similarly, it can be shown that, the maximum error in the difference.
Z = A – B is also given by

b. Error of product ora quotient:
Rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
Explanation:
Suppose Z=AB and the measured values of A and B are A + DA and B + DB. They
Z + DZ = (A + DA) (B + DB)
= AB ± BDA ± ADB ± DADB
Dividing LHS by Z and RHS by AB, we get

c. Errors in case of a measured quantity raised to a power:
Suppose Z = A²

Hence, the relative error in A² is two time the error in A.
In general, if Z=APBqCTZ=APBqCT
Then

Hence the rule: The relative error in a physical quantity raised to the power K is the K times the relative error in the individual quantity.

Significant Figures

Every measurement involves errors. Hence the result of measurement should be reported in a way that indicates the precision of measurement.

Normally, the reported result of measurement is a number that includes all digits in the number that are known reliable plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
Example:

• The length of a rod measured is 3.52cm. Here there are 3 significant figures. The digits 3 and 5 are reliable and the last digit 2 is uncertain.
• The mass of a body measured as 3.407g. Here there are four significant figures. The figure 7 is uncertain.

When the measurement becomes more accurate, the number of significant figure is increased.
Rules to find significant figures:
1. All the non zero digits are significant.
Example:
Question 1.
Find significant figure of

• 2500
• 263.25

Answer:

• In this case, there are two nonzero numbers. Hence significant figure is 2.
• In this, there are 5 nonzero numbers. Hence significant figure is 5.

2. All the zeros between two nonzero digits are significant, no matter where the decimal point is,
Example:
Question 2.
Find the significant figure

• 2.05
• 302.005
• 2000145

Answer:

• Significant figure is 3
• Significant figure is 6
• Significant figure is 7

3. If the number is less than 1, the zeros on the right of decimal point but to the left to the first nonzero digits are not significant.
Example:
Question 1.
Find the significant figure of

• 0.002308
• 0.000135

Answer:

• 4 significant figures
• 3 significant figures

4. The terminal zeros in a number without a deci¬mal point are not significant.
Example:
Question 1.
Find the significant figure of

• 12300
• 60700

Answer:

• 3
• 3

Note: But if the number obtained is on the basis of actual measurement, all zeros to the right of last non zero digit are significant.
Example: If distance is measured by a scale as 2010m. This contain 4 significant figures.

5. The terminal zeros in a number with a decimal point are significant.
Example:
Question 1.
Find the significant figure of

• 3.500
• 0.06900
• 4.7000

Answer:

• 4
• 4
• 5

Method to find significant figures through scientific notation:
In this notation, every number is expressed as a × 10^b, where a is a number between 1 and 10 and b is any positive or negative power. In this method, we write the decimal after the first digit.
Example:
4700m =4.700 × 10³m
The power of 10 is irrelevant to the determination of significant figures. But all zeros appearing in the base number in the scientific notation are significant. Hence each number in this case has 4 significant figures.
Significant figures in numbers:-

Numbers	Significant figures
1374	4
13.74	4
0.1374	4
0.01374	4
013740	5
1374.0	5
5100	2
51.00	4
5.100	4
3.51 × 103	3
2.1 × 10-2	2
0.4 × 10-4	1

a. Rules for Arithmetic operations with significant figures:
1. Rules for multiplication or division:
In multiplication or division, the computed result should not contain greater number of significant digits than in the observation which has the fewest significant digits.
Examples:
(i) 53 × 2.021 =107.113
The answer is 1.1 × 10² since the number 53 has only 2 significant digits.

(ii) 3700 10.5 = 352.38
The answer is 3.5 × 10² since the minimum number of significant figure is 2 (in the number 3700)

2. Rules for Addition and Subtraction:
In addition or substraction of given numbers, the same number of decimal places is retained in the result as are present in the number with minimum number of decimal places.
Examples:
(i) 76.436 +
12.5
88.936
The answer is 88.9, since only one decimal place is found in the number 12.5.

(ii) 43.6495 +
4.31
47.9595
The answer is 47.96 since only two decimal places are to be retained.

(iii) 8.624 –
3.1726
5.4514
The answer is 5.451

(iv) 6.5 × 10^-5 – 2.3 × 10^-6 = 6.5 × 10^-5 – 0.23 × 10^-5
= 6.27 × 10^-5
The answer is = 6.3 × 10^-5

Dimensions And Dimensional Analysis

All physical quantities can be expressed in terms of seven fundamental quantities. (Mass, length, time, temperature, electric current, luminous intensity and amount of substance). These seven quantities are called the seven dimensions of the physical world.

The dimensions of the three mechanical quantities mass, length and time are denoted by M, L and T. Other dimensions are denoted by K (for temperature), I (for electric current), cd (for luminous intensity) and mol (for the amount of substance).

The letters [L], [M], [T] etc. specify only the nature of the unit and not its magnitude. Since area may be regarded as the product of two lengths, the dimensions of area are represented as [L] × [L] = [L]².

Similarly, volume being the product of three lengths, its dimensions are represented by [L]³. Density being mass per unit volume, its dimensions are M/L³ or M¹L³.

Thus, the dimensions of a physical quantity are the powers to which the fundamental units of length, mass, time must be raised to represent it.
Note: The dimensions of a physical quantity and the dimensions of its unit are the same.

Dimensional Formula And Dimensional Equations

An equation obtained by equating a quantity with its dimensional formula is called dimensional equations of the physical quantities.
Examples:
Consider for example, the dimensions of the following physical quantities.
1. Velocity: Velocity = distance/ time = L/T = L¹T^-1 \The dimension of velocity are, zero in mass, 1 in length and-1 in time.

2. Acceleration:
Acceleration =  Change in velocity  time =L1T−1T=L1T−2 Change in velocity  time =L1T−1T=L1T−2

3. Force: Force = mass × acceleration
Dimensions of force = M × L¹T^-2 = M¹L¹T ^-2
That is, the dimensions of force are 1 in mass, 1 in length and -2 in time.

4. Momentum: Momentum = mass × velocity
Dimensions of momentum = M × L¹T^-1 = M¹L¹T ^-1

5. Moment of a force: Moment = force × distance
Dimensions of moment = M¹L¹T^-2 × L = M¹L²T ^-2

6. Impulse: Impulse = force × time
Dimensions of impulse = M¹L¹T^-2 × T = M¹L¹T ^-1

7. Work: Work = force × distance
Dimensions of work = M¹L¹T^-2 × L = M¹L²T ^-2

8. Energy: Energy = Work done
Dimensions of energy = dimensions of work = M¹L²T^-2.

9. Power: Power = work/time
Dimensions of power =M2L2T−2Tp=M2L2T−2Tp = M¹L²T^-3

Dimensional Analysis And Its Applications

The important uses of dimensional equations are:

1. To check the correctness of an equation.
2. To derive a correct relationship between different physical quantities.
3. To convert one system of units into another.

1. Checking the correctness of an equation:
For the correctness of an equation, the dimensions on either side must be the same. This ‘ is known as the principle of homogeneity of dimensions.

If an equation contains more than two terms, the dimensions of each term must be the same. Thus, if x = y + z, Dimensions of x = dimensions of y = dimensions of z
Example :
Question 1.
Check the correctness of the equation s = ut + 1/2at² by the method of dimensions.
Dimensions of, s = L¹
Dimensions of, u = L¹T^-1
Dimensions of, ut = L¹T^-1 × T¹ = L¹
Dimensions of, a = L¹T^-2
Dimensions of, at² = L¹T^-2 × T² = L¹
The constant 1/2 has no dimensions. Each term has dimension L¹.
Therefore, dimensions of, ut + 1/2 at² = ¹
Thus, either side of the equation has the same dimen¬sion L1 and hence the equation is dimensionally correct.
Note: Even though the equation is dimensionally correct, it does not mean that the equation is necessarily correct. For instance the equation s = ut + at² is also dimensionally correct, though the correct equation, s = ut + 1/2 at².

2. Deriving the correct relationship between different physical quantities:
The principle of homogeneity of dimensions also helps to derive a relationship between the different physical quantities involved. This method is known as dimensional analysis.
Example :
Question 1.
Deduce an expression for the period of oscillation of a simple pendulum.
The period of the simple pendulum may possibly depend upon

• The mass of the bob, m
• The length of the pendulum, I
• Acceleration due to gravity, g
• The angle of swing, q

Let us write the equation for the time period as t = km^a l^b g^c θ^d
where, k is a constant having no dimensions; a, b, c are to be found out. ’
The dimensions of, t = T¹
Dimensions of m = M¹
Dimensions of, l = L¹
Dimensions of, g = L¹T^-2
Angle q has no dimensions (since, q = arc/radius = L/L) Equating the dimensions of both sides of the equation, we get,
T¹ = M^aL^b (L¹T^-2)^c
ie. T¹ = M^aL^b+cT^-2c
The dimensions of the terms on both sides must be the same. Equating the powers of M, L and T.
a = 0; b + c = 0; -2c = 1
∴ c = −12−12, b = ^–c = \(\frac{1}{2}{/latex]
Hence, the equation becomes,
t = kl^1/2, 2g^-1/2
ie, t = k[latex]\sqrt{l/g}\)
Experimentally, the value of k is found to be 2p.
Limitations of Dimensional Analysis:
The method of dimensional analysis has the following limitations:

• It gives no information about the dimensionless constant involved in the equation.
• The method is not applicable to equations involving trigonometric and exponential functions.
• This method cannot be employed to derive the exact form of the relationship, if it contains sum
  of two, or more terms.
• If the given physical quantity depends on more than three unknown quantities, the method fails.

3. Conversion of one system of units to another:
Suppose we have a physical quantity of dimensions a, b and c in mass, length and time. The dimensional formula for the quantity is therefore, M^aL^bT^c. Let its numerical value be n, in one system in which the fundamental units of mass, length and time are M₁, L₁ and T₁ respectively. Then, the magnitude of the physical quantity
= n₁ M₁^aL₁^bT₁^c
Also, let the numerical value of the same quantity be n₂ in another system where the fundamental units of mass, length and time are M₂, L₂ and T₂respectively. Then the magnitude of the quantity
= n₂ M₂^aL₂^bT₂^c
Equating, n₂ M₂^aL₂^bT₂^c =
n₁ M₁^aL₁^bT₁^c

Example :
Question 1.
Find the number of dynes in one newton.
Answer:
Dyne is the unit of force in the C.G.S. system and newton is the S.I.unit. The dimensional formula for force is M¹L¹T^-2. In eqn. (1) let the suffix 1 refer to quantities in S.I and 2 those in the C.G.S. system.
Here, a = 1, b = 1 and c = 2

and n₁ = 1 (ie. one Newton)
By eqn. (1),
n₂ = 1 (1000)¹ (100)¹ (1)^-2 = 10⁵
ie. 1 newton = 10⁵ dynes.

Physical World Class 11 Notes Physics Chapter 1

▶What is Physics?

Physics is the study of nature and its laws. There are so many different events in nature which are taking place and we expect that all these different events in nature are taking place according to some basic law and revealing these laws of nature from the observed events is physics.

Humans have always been curious about the world around them. The night sky with its bright celestial objects has fascinated humans since time immemorial. The regular repetitions of the day and night, the annual cycle of seasons, the eclipses, the tides, the volcanoes, the rainbow have always been a source of wonder.

The word Science originates from the Latin verb Scientia meaning ‘to know’. The Sanskrit word Vigyan and the Arabic word ilm convey similar meaning, namely ‘knowledge’. Science is a systematic attempt to understand natural phenomena in as much detail and depth as possible, and use the knowledge so gained to predict, modify and control phenomena. Science is exploring, experimenting and predicting from what we see around us.

▶Role of Mathematics in Physics

Description of all natural phenomenon is made easy by the help of mathematics. Thus, we can say that mathematics is the language of physics, by the help of mathematics we explain and understand basic law of physics in more better way. For example, gravitational force of attraction between two point masses m₁ and m₂ can be written by mathematical equation

F=Gm1m2r2F=Gm1m2r2

Scope and Expansion of Physics

Various branches of physics are mainly divided into two parts- (A) Classical Physics and (B) Modern Physics.

(A). Classical Physics

(i). Mechanics:- Under this subject the systematic motion of objects is studied. One of its branches is Fluid Mechanics, in which the dynamic behavior of liquids is studied.

(ii). Thermodynamics:- Under this subject the motion in a system made of heat, heat and fine particles is studied.

(iii). Electromagnetism:- Under this subject the theory of electromagnetism and electromagnetic waves is studied.

(iv). Classical Wave Mechanics and Sound:- Under this subject, vibrations and progressive and progressive waves are studied.

(v). Optics:- Under this subject the nature and transmission of light is studied. To understand the Images, refraction, reflection, interference, diffraction and polarization formed through lenses and mirrors, it is necessary to have knowledge of this subject.

(B). Modern Physics

(i). Relativity:- Under this subject the motion of those bodies is studied which move with a velocity equal to the speed of light. In fact it is a theory of relativism in nature.

(ii). Quantum Mechanics:- Under this subject the principles of modern physics, the dual nature of light and matter are studied. It acts as a bridge between classical physics and modern physics.

(iii). Atomic Physics:- Under this subject, atomic structure and properties of atoms are studied.

(iv). Nuclear Physics:- Under this subject, the nucleus of an atom and its properties are studied, apart from this some other subjects are as follows-

1. Solid State Physics
2. Plasma Physics
3. High Energy Physics
4. Electronics
5. Engineering Physics
6. Medical Physics
7. Cosmology
8. Bio Physics
9. Chemical Physics
10. Geo Physics

Contribution of Physics in Technology and Society

Physics is an important branch of science, without whose knowledge the development of other branches of science is not possible. Physics has an important contribution in the development of all branches of science and the upliftment of society.

(i). Importance of Physics in Chemistry:- The study of the chemical composition of matter, types of bonds, etc. has become possible on the basis of intermolecular forces found between molecules. On the basis of the diffraction of X-rays, the structure of the atom, the radioactivity, the detailed study of the structures of many solids has become possible.

(ii). Importance of Physics in Biology:- Many biological specimens are studied with the help of light microscope. The study of many physical structures became possible with the creation of the electron microscope.

(iii). Importance of Physics in Astronomy:- With the help of optical telescope, the study of motion of various planets and celestial bodies has become possible.

(iv). Importance of Physics in Mathematics:- The development of many activities has been made possible by the principles of physics. Technological development is particularly concerned with the application of physics. Examples of some new techniques based on the application of physics are as follows-

1. Power generation is based on the principle of electromagnetic induction.
2. Diesel engine, petrol engine, steam engine etc. are based on the laws of thermodynamics.
3. Radio, Television, S.T.D., I.S.D., Fax, Wireless etc. are based on the principle of electromagnetic waves.
4. The development of the atomic furnace and atomic bomb is based on nuclear fission.
5. Rocket propulsion is based on Newton’s second and third laws of motion.
6. The flight of air vehicles is based on the Bernoulli principle.

▶Purpose and Excitement of Physics

We can get some idea of the scope of physics by looking at its various sub-disciplines. Basically, there are two domains of interest : macroscopic and microscopic. The macroscopic domain includes phenomena at the laboratory, terrestrial and astronomical scales. The microscopic domain includes atomic, molecular and nuclear phenomena.  deals mainly with macroscopic phenomena and includes subjects like Mechanics, Electrodynamics, Optics

You can now see that the scope of physics is truly vast. It covers a tremendous range of magnitude of physical quantities like length, mass, time, energy, etc. At one end, it studies phenomena at the very small scale of length (10^-14 m or even less) involving electrons, protons, etc.; at the other end, it deals with astronomical phenomena at the scale of galaxies or even the entire universe whose extent is of the order of 10²⁶ m.

▶Fundamental Forces in Nature

Mainly, four types of force are exist in our nature that are described bellow:

▶(i) Gravitational Force

It is the force of mutual attraction between any two objects due to their masses. All bodies on the Earth experience this force due to the Earth. Gravity governs the motion of moons around the Earth, motion of planets around the sun. It plays a main role in formation and evolution of stars, galaxies and galactic clusters newtons law of gravitation gives the magnitude of force exerted by a particle of point mass m₁ on another particle of point mass m₂ at a distance r from it as

F=Gm1m2r2F=Gm1m2r2

where G is universal gravitational constant G = 6.67 × 10^–11 Nm²/kg². This force acts along the line joining the two particles.

▶(ii) Electromagnetic Force

It is the force between charge particals . It includes electric and magnetic forces. If two static point charges q₁ and q₂ are kept at a distance r, then the electrostatic force between them is given as

F=14πε0q1q2r2F=14πε0q1q2r2

This is known as the Coulomb force.

▶(iii) Strong Nuclear Force

It is the force that binds nucleons (protons and neutrons) in a nucleus. The nucleus contains positively charged protons and electrically neutral neutrons. The repulsive electrostatic force between protons should make a nucleus unstable. There should be a strong attractive force that counteracts the repulsive force to keep a nucleus stable.

We know that gravitational force is negligible as compared to electrostatic force. So, we have a new basic force, i.e., strong nuclear force, which is the strongest of all fundamental forces, about 100 times the electromagnetic force. It is the same between a proton and a neutron, a proton and a proton, a neutron and a neutron. It is an extremely short-ranged force (≈ 10^–15 m). It keeps the nucleus stable. It does not depend upon charge. An electron does not experience this force.

▶(iv) Weak Nuclear Force

It appears in some nuclear processes like of the nucleus, in which the nucleus emits an electron and an uncharged particle called the neutrino. It is also responsible for the decay of many unstable particles (muons into electrons, pions into muons, and so on). It is not as weak as the gravitational force, but much weaker than the strong nuclear and electromagnetic force. They are exceedingly short-ranged forces, of the order of 10^–16 m.

▶Conservation Laws in Physics

A remarkable fact about any physical phenomenon is the invariance of some special physical quantities. They are the conserved quantities of nature, i.e., energy, mass, charge, linear momentum, and angular momentum. In classical physics, we have the following conservation laws:

▶(i) Law of Conservation of Energy

According to this law, the sum of energy of all kinds in this universe or of an ideal isolated system remains constant. Energy can neither be created nor be destroyed. It can only be transformed from one form to the other.

▶(ii) Law of Conservation of Linear Momentum

According to this law, the linear momentum of a system remains unchanged in the absence of an external force. It is denoted by P and expression is given by P = mv.

▶(iii) Law of Conservation of Angular Momentum

A rotating body has inertia, so it also possesses momentum associated with its rotation. This momentum is called ‘angular momentum‘.

Angular momentum = Moment of inertia × Angular speed

L = I × ω

According to this law, the angular momentum of the system remains constant if the total external torque acting on it is zero. e.g., Planets revolving around the sun in an elliptical orbit with constant angular momentum.

▶(iv) Law of Conservation of Charge

According to this law, charges (in the form of electrons) are neither created nor destroyed but are simply transferred from one body to another.

Class 11 Maths Notes Chapter 16 Probability

Random Experiment
An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.

Outcome
A possible result of a random experiment is called its outcome.

Sample Space
A sample space is the set of all possible outcomes of an experiment.

Events
An event is a subset of a sample space associated with a random experiment.

Types of Events
Impossible and sure events: The empty set Φ and the sample space S describes events. Intact Φ is called the impossible event and S i.e. whole sample space is called sure event.

Simple or elementary event: Each outcome of a random experiment is called an elementary event.

Compound events: If an event has more than one outcome is called compound events.

Complementary events: Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.

Mutually Exclusive Events
Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously and thus P(A ∩ B) = 0.

Exhaustive Events
If E₁, E₂,…….., E_n are n events of a sample space S and if E₁ ∪ E₂ ∪ E₃ ∪………. ∪ E_n = S, then E₁, E₂,……… E₃ are called exhaustive events.

Mutually Exclusive and Exhaustive Events
If E₁, E₂,…… E_n are n events of a sample space S and if
E_i ∩ E_j = Φ for every i ≠ j i.e. E_i and E_j are pairwise disjoint and E₁ ∪ E₂ ∪ E₃ ∪………. ∪ E_n = S, then the events
E₁, E₂,………, E_n are called mutually exclusive and exhaustive events.

Probability Function
Let S = (w₁, w₂,…… w_n) be the sample space associated with a random experiment. Then, a function p which assigns every event A ⊂ S to a unique non-negative real number P(A) is called the probability function.
It follows the axioms hold

• 0 ≤ P(w_i) ≤ 1 for each W_i ∈ S
• P(S) = 1 i.e. P(w₁) + P(w₂) + P(w₃) + … + P(w_n) = 1
• P(A) = ΣP(w_i) for any event A containing elementary event w_i.

Probability of an Event
If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of occurrence of A is defined as

The odd in favour of occurrence of the event A are defined by m : (n – m).
The odd against the occurrence of A are defined by n – m : m.
The probability of non-occurrence of A is given by P(A¯) = 1 – P(A).

Addition Rule of Probabilities
If A and B are two events associated with a random experiment, then
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Similarly, for three events A, B, and C, we have
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

Note: If A andB are mutually exclusive events, then
P(A ∪ B) = P(A) + P(B)

 Class 11 Maths Notes Chapter 15 Statistics

Measure of Dispersion
The dispersion is the measure of variations in the values of the variable. It measures the degree of scatteredness of the observation in a distribution around the central value.

Range
The measure of dispersion which is easiest to understand and easiest to calculate is the range.
Range is defined as the difference between two extreme observation of the distribution.
Range of distribution = Largest observation – Smallest observation.

Mean Deviation
Mean deviation for ungrouped data
For n observations x₁, x₂, x₃,…, x_n, the mean deviation about their mean x¯ is given by

Mean deviation about their median M is given by

Mean deviation for discrete frequency distribution
Let the given data consist of discrete observations x₁, x₂, x₃,……., x_n occurring with frequencies f₁, f₂, f₃,……., f_n respectively in case

Mean deviation about their Median M is given by

Mean deviation for continuous frequency distribution

where x_i are the mid-points of the classes, x¯ and M are respectively, the mean and median of the distribution.

Variance
Variance is the arithmetic mean of the square of the deviation about mean x¯.
Let x₁, x₂, ……x_n be n observations with x¯ as the mean, then the variance denoted by σ², is given by

Standard deviation
If σ² is the variance, then σ is called the standard deviation is given by

Standard deviation of a discrete frequency distribution is given by

Standard deviation of a continuous frequency distribution is given by

Coefficient of Variation
In order to compare two or more frequency distributions, we compare their coefficient of variations. The coefficient of variation is defined as

Note: The distribution having a greater coefficient of variation has more variability around the central value, then the distribution having a smaller value of the coefficient 0f variation.

Class 11 Maths Notes Chapter 14 Mathematical Reasoning

Statements
A statement is a sentence which is either true or false, but not both simultaneously.

Note:
No sentence can be called a statement if

• It is an exclamation.
• It is an order or request.
• It is a question.

Simple Statements
A statement is called simple if it cannot be broken down into two or more statements.

Compound Statements
A compound statement is the one which is made up of two or more simple statement.

Connectives
The words which combine or change simple statements to form new statements or compound statements are called connectives.

Conjunction
If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q is written in symbolic form as “p ∧ q”.

Note:

• The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T.
• The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.

Disjunction
If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.

Note:

• The statement p ∨ q has the truth value F whenever both p and q have the truth value F.
• The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.

Negation
An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement p in symbolic form is written as “~p”.

Note:

• ~p has truth value T whenever p has truth value F.
• ~p has truth value F whenever p has truth value T.

Negation of Conjunction
The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q.
Equivalently we write ~ (p ∧ q) = ~p ∨ ~q.

Negation of Disjunction
The negation of a disjunction p v q is the conjunction of negation of p and the negation of q.
Equivalently, we write ~(p ∨ q) = ~p ∧ ~q.

Negation of Negation
Negation of negation of a statement is the statement itself.
Equivalently, we write ~(~p) = p

The Conditional Statement
If p and q are any two statements, then the compound statement “if p then g” formed by joining p and q by a connective ‘if-then’ is called a conditional statement or an implication and is written in symbolically p → q or p ⇒ q, here p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q).

Contrapositive of Conditional Statement
The statement “(~q) → (~p) ” is called the contrapositive of the statement p → q.

Converse of a Conditional Statement
The conditional statement “q → p” is called the converse of the conditional statement “p → q”.

Inverse of Conditional Statement
The Conditional statement “q → p” is called inverse of p → q.

The Biconditional Statement
If two statements p and q are connected by the connective ‘if and only if’, then the resulting compound statement “p if and only if q” is called biconditional of p and q and is written in symbolic form as p ⇔ q.

Quantifier
(i) For all or for every is called universal quantifier.
(ii) There exists is called existential quantifier.

Validity of Statements
A statement is said to valid or invalid according to as it is true or false.
If p and q are two mathematical statements, then the statement
(i) “p and q” is true if both p and q are true.
(ii) “p or g” is true if p is false
⇒ q is true orq is false ⇒ p is true.
(iii) “If p, then q” is true p is true ⇒ q is true
or
q is false
⇒ p is false
or
p is true and q is false less us to a contradiction,
(iv) “p if and only if q” is true, if
(a) p is true ⇒ q is true and
(b) q is true ⇒ p is true.

Class 11 Maths Notes Chapter 13 Limits and Derivatives

Limit
Let y = f(x) be a function of x. If at x = a, f(x) takes indeterminate form, then we consider the values of the function which is very near to a. If these value tend to a definite unique number as x tends to a, then the unique number so obtained is called the limit of f(x) at x = a and we write it as limx→af(x).

Left Hand and Right-Hand Limits
If values of the function at the point which are very near to a on the left tends to a definite unique number as x tends to a, then the unique number so obtained is called the left-hand limit of f(x) at x = a, we write it as

Existence of Limit

Some Properties of Limits
Let f and g be two functions such that both limx→af(x) and lim limx→ag(x) exists, then

Some Standard Limits

Derivatives
Suppose f is a real-valued function, then

Fundamental Derivative Rules of Function
Let f and g be two functions such that their derivatives are defined in a common domain, then

Some Standard Derivatives

Class 11 Maths Notes Chapter 12 Introduction to Three Dimensional Geometry

Coordinate Axes
In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. These axes are called the X, Y and Z axes.

Coordinate Planes
The three planes determined by the pair of axes are the coordinate planes. These planes are called XY, YZ and ZX plane and they divide the space into eight regions known as octants.

Coordinates of a Point in Space
The coordinates of a point in the space are the perpendicular distances from P on three mutually perpendicular coordinate planes YZ, ZX, and XY respectively. The coordinates of a point P are written in the form of triplet like (x, y, z).
The coordinates of any point on

• X-axis is of the form (x, 0,0)
• Y-axis is of the form (0, y, 0)
• Z-axis is of the form (0, 0, z)
• XY-plane are of the form (x, y, 0)
• YZ-plane is of the form (0, y, z)
• ZX-plane are of the form (x, 0, z)

Distance Formula
The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by

The distance of a point P(x, y, z) from the origin O(0, 0, 0) is given by
OP = x2+y2+z2−−−−−−−−−−√

Section Formula
The coordinates of the point R which divides the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally or externally in the ratio m : n are given by

The coordinates of the mid-point of the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are

The coordinates of the centroid of the triangle, whose vertices are (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) are

Class 11 Maths Notes Chapter 11 Conic Sections

Circle
A circle is the set of all points in a plane, which are at a fixed distance from a fixed point in the plane. The fixed point is called the centre of the circle and the distance from centre to any point on the circle is called the radius of the circle.
The equation of a circle with radius r having centre (h, k) is given by (x – h)² + (y – k)² = r².

The general equation of the circle is given by x² + y² + 2gx + 2fy + c = 0 , where, g, f and c are constants.

• The centre of the circle is (-g, -f).
• The radius of the circle is r = g2+f2−c−−−−−−−−−√

The general equation of the circle passing through origin is x² + y² + 2gx + 2fy = 0.

The parametric equation of the circle x² + y² = r² are given by x = r cos θ, y = r sin θ, where θ is the parametre and the parametric equation of the circle (x – h)² + (y – k)² = r² are given by x = h + r cos θ, y = k + r sin θ.

Note: The general equation of the circle involves three constants which implies that at least three conditions are required to determine a circle uniquely.

Parabola
A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distance from a fixed line l in the plane. The fixed point F is called focus and the fixed line l is the directrix of the parabola.

Main Facts About the Parabola

Forms of parabola	y2= 4ax	y2 = -4ax	x2 = 4ay	x2 = -4ay
Axis of parabola	y = 0	y = 0	x = 0	x = 0
Directrix of parabola	x = -a	x = a	y = -a	y = a
Vertex	(0, 0)	(0, 0)	(0, 0)	(0, 0)
Focus	(a, 0)	(-a, 0)	(0, a)	(0, -a)
Length of latus rectum	4a	4a	4a	4a
Focal length	|x + a|	|x – a|	|y + a|	|y – a|

Ellipse
An ellipse is the set of all points in a plane such that the sum of whose distances from two fixed points is constant.
or
An ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant ratio, less than to their distance from a fixed point in the plane. The fixed point is called focus, the fixed line a directrix and the constant ratio(e) the eccentricity of the ellipse. We have two standard forms of ellipse i.e.

Main Facts about the Ellipse

Hyperbola
A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio, generally denoted bye, is known as the eccentricity of the hyperbola.
We have two standard forms of hyperbola i.e.

Main Facts About Hyperbola