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MOTION IN A STRAIGHT LINE

1. The area under the velocity­time curve and time axis gives the displacement of the object for given interval of time.

2. If a body falls freely, the distance covered by it in each subsequent second starting from first second will be in the ratio 1 : 3 : 5 : 7, etc.

3. If a body is thrown vertically up with an initial velocity u, it takes u/g second to reach maximum height and u/g second to return, if air resistance is negligible.

4. If air resistance acting on a body is considered, the time taken by the body to reach maximum height is less than the time to fall back the same height.

5. For a particle having zero initial velocity if s t α ∝ , where 2, α > then particle’s acceleration increases with time.

6. For particle having zero initial velocity if s t α ∝ , where

0, α < then particle ‘s acceleration decreases with time.

7. Kinematic equations:

2 2 v u at;v u 2as = + = + , 2 1 s ut at 2

= +

aplicable only when particles moves with constant acceleration

8. If acceleration is varibale use calculus approach.

9. Relative velocity BA B A = − r r r v v v

MOTION IN A PLANE

1. If T is the time of flight, h maximum height, R horizontal range of a projectile, α its angle of projection, then the relations among these quantities.

2 T h 8

= g .......... (1); 2 T 2R tan = α g ....... (2);

R tan 4h α = ............(3) 2. For a given initial velocity, to get the same horizontal range,

there are two angles of projection α and 90º – α 3. The equation to the parabola traced by a body projected

horizontally from the top of a tower of height y, with a velocity u is y = gx 2 /2u 2 , where x is the horizontal distance covered by it from the foot of the tower.

4. At any instant if v is the velocity of projectile making angle β with the horizontal, then

x y v cos cos and sin sin gt = β = θ = β = θ − v u v v u

5. Equation of trajectory is 2

2 2

gx y x tan , 2u cos

= θ − θ which

is parabola. 6. Maximum height is equal to n times the range when the

projectile is launched at an angle 1 tan (4n) − θ = .

7. In a uniform circular motion, velocity and acceleration are constant only in magnitude. Their directions change.

8. In a uniform circular motion, the kinetic energy of the body

is a constant. W 0, a 0, p = ≠ ≠ r r

constant, L = r

constant

9. 2

2 r a r

r = ω = = ω v v (Always applicable)

2 2 2

r 2

4 a 4 n r r T π

= π = (Applicable in uniform circular

motion) n = frequency of rotation, T = time period of ratation.

r a v = ω× r r r

LAWS OF MOTION

1. Newton’s second law : F ma, F dp / dt = = r r r r

2. Impulse : 2

1

t

2 1 t

p F t,p p F dt ∆ = ∆ − = ∫ r r r r

3. Newston’s third law : 12 21 F F = − r r

4. Frictional force s s max s k k ( ) µ ; µ ≤ = = f f R f R

5. Circular motion with variable speed. For complete circle,

the string must be taut in the highest position, 2 u 5g ≥ l where l is the length of string.

Circular motion ceases at the instant when the string becomes slack, i.e. whent T = 0, range of values of u for

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which the string does go slack is 2g u 5g < < l l .

6. Conical pendulum : g / h ω = where h is height of a point of suspension form the centre of circular motion.

7. The acceleration of a lift

actual weight apparent weight a mass −

= , where the

weights is in N. if ‘a’ is positive the lift is moving down, and if it is negative the lift is moving up.

WORK, ENERGY AND POWER

1. velocity of separation of colliding bodies velocity of approach of colliding bodies

e = −

2. The total momentum of a system of particles is a constant in the absence of external forces.

SYSTEM OF PARTICLES AND ROTATIONAL MOTION

1. The centre of mass of a system of particles is defined as

the point whose position vector is i i m r M

R Σ = The centre

of gravity of an extended body is that point where the total gravitational torque on the body is zero. The centre of gravity of a body conincides with its centre of mass only if the gravitational field does not vary from one part of the body to the other.

2. The angular momentum of a system of n particles about

the origin is n

i i i 1

r p L =

× = ∑ The torque or moment of force on a system of n particles

about the origin is i i i

r F × τ = ∑ 3. The moment of inertia of a rigid body about an axis is

defined by the formula 2 i i r I m = Σ where r i is the

perpendicular distance of the i th point of the body from the

axis. The kinetic energy of rotation is 2 1 I2

K ω =

4. The theorem of parallel axes : I z = I z + Ma 2 ,. Theorem of perpendicular Axes : I z = I x + I y .

5. For rolling motion without slipping v cm = R , ω where v cm is the velocity of translation (i.e., of the centre of mass ), R is the radius and m is the mass of the body. The kinetic energy of such a rolling body is the sum of kinetic energies

of translation and rotation : 2 2 cm

1 1 mv I 2 2

K + ω =

6. A rigid body is in mechanical equilibrium if

(a) It is in translational equilibrium if the total external

force on it is zero : i 0. F = Σ (b) It is in rotational equilibrium i.e., the total external torque on it is zero : i i i r F 0 = Σ × = Σ .

GRAVITATION

2. The acceleration due to gravity. (a) At a height ha above the Earth’s surface

E E E 2 2

E E E

GM GM 2h g(h) 1 for h R (R h) R R

= = − << +

(b) at depth d below the Earth’s surface is

E 2 E E

G M d 1 R R

g ( d ) −

=

3. The gravitational potential energy

1 2 Gm m r

V + = − constnat

4. The escape speed from the surface of the Earth is

E e E

E

2GM 2gR R

v = = and has value of 11.2 km s –1 .

5. A geostationary (geosynchronous communication) satellite moves in a circular orbit in the equatorial plane at a approximate distance of 4.22 × 10 4 km from the Earth’s centre.

6. Whenever force responsible for orbital motion obeys inverse square law, then only square of time pariod is directly proportional to cube of average distance between planet and sun.

2 3 2 3 1 1

2 3 2 2

T a a ; T a

T ∝ =

Applicable only when both planets revolve around same mass. Length of semi major axis is the average distance between sun and planet during its complete orbital motion.

MECHANICAL PROPERTIES OF FLUIDS

1. Pascal’s law :A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.

2. Bernoulli’s principle states that as we move along a streamline, the sum of the pressure (P), the kinetic energy

per unit volume 2 / 2) ( v ρ and the potential energy per

unit volume ) ( gy ρ remains a constant.

2 / 2 gy P v + ρ + ρ = constant

3. Surface tension is a force per unit length (or surface energy per unit area) acting in the plane of interface between the

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liquid and the bounding surface. 4. Stokes’ law state that the viscous drag froce F on a sphere

of radius a moving with velocity v through a fluid of viscosity η is, F 6 av = πη .

5. The surface tension of a liquid is zero at boiling point. the surface tension is zero at critical temperature.

6. If a drop of water of radius R is broken into n identical

drops, the work done in the process is 2 1/3 S(n 1) 4 R − π 7. Two capillary tubes each of radius r are joined in parallel.

The rate of flow of liquid is Q. If they are replaced by single capillary tube of radius R for the same rate of flow then R = 2 1/4 r.

8. If radius of a drop is doubled its terminal velocity increases to four times.

THERMAL PROPERTIES OF MATTER

1. The coefficient of linear expansion ( ) α l , superficial

expansion ( ) β and volume expansion V ( ) α are defined by the relations:

V A V T; T; T A V

∆ ∆ ∆ = α ∆ = β∆ = α ∆

l l

l

where and V ∆ ∆ l denote the change in length l and volume V due to change of temperature T ∆ . The realtion between them is :

v 3 ; 2 α = α β = α l l

2. In conduction, heat is transfered between neighbouring parts of a body through molecular collisions, without any flow of matter. For a bar of length L and uniform cross section A with its ends maintained at temperature T C and T D , then rate of flow of heat H is :

C D T T K KA L −

= , where K is the thermal conductivity of

the material of the bar. 3. Convection involves flow of matter within a fluid due to

unequal temperatures of its parts. 4. Stefan’s law of radiation : The energy emitted by a black

body per unit area per second is directly proportional to the fourth power of its Kelvin temperature. 4 E T = σ , where the constant σ is known as Stefan’s constant.

5. Newton’s Law of cooling : 2 1 1 dQ k(T T );whereT dt

= − − is

the temperature of the surrounding medium and T 2 is the temperatures of the body.

THERMODYNAMICS

1. The fist law of thermodynamics : It states that

Q U W, ∆ = ∆ + ∆ where Q ∆ is the heat supplied to the system, W ∆ is the work done by the system and U ∆ is the change in internal energy of the system. If Q > 0, heat is added to the system, if Q < 0, heat is removed to the system, if W > 0, Work is done by the system, if W < 0, Work is done on the system quantity.

2. In an isothermal expansion of an ideal gas from volume V 1 to V 2 at temerature T the heat absorbed (Q) equals the work done (W) by the gas, each given by

Q = W = nRT ln 2

1

V V

.

3. In an adiabatic process of an ideal gas PV γ = constant,

where p

v

C C

γ = .

Work done by an ideal gas in an adiabatic change of state

from (P 1 , V 1 , T 1 ) to (P 2 , V 2 , T 2 ) is 1 2 nR(T T ) W

− =

γ −1

4. The efficiency of a carnot engine is given by 2

1

T 1 T

η = −

KINETIC THEORY

1. Kinetic theory of an ideal gas gives the relation 2 1 P nmv

3 = , where n is number density of molecules, m

the mass of the molecule and 2 v is the mean of squared speed. Combined with the ideal gas equation it yields a kinetic interpretation of temperature.

2 2 1/2 B B rms

1 3 3k T nmv k T, v (v ) 2 2 m

= = =

2. The law of equipartition of energy is stated thus: the energy for each degree of freedom in thermal equilibrium is 1/2 (k B T)

3. The translational kinetic energy B 3 E k NT. 2

= This leads

to a realtion PV = 2 E. 3

4. Speed of sound in a gas s s

rms

RT v v ,

M v 3 γ γ

= = i.e.,

v s v rtms OSCILLATIONS

1. The particle velocity and acceleration during SHM as function of time are given by

v(t) Asin( ) = −ω ω + φ (velocity),

2 2 a(t) A cos ( t ) x(t) = −ω ω + φ = −ω (acceleration)

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where x(t) = A cos ( t ) ω + φ

Velocity amplitude v m A. = ω and acceleration amplitude 2

m a A = ω

2. A particle of mass m oscillating under the influence of a Hook’s law restoring force given by F = – k x exhibits

simple harmonic motion with k m

ω = (angular

frequency),

m T 2 k

= π (period)

Such a system is also called a linear oscillator. 3. A body of mass M is suspended from a spring whose force

constant is K and mass is m. The time period of this system

will be (M m / 3) 2

k +

π

4. Time period for conical pendulum cos T 2 g

θ = π

l

where θ is angle between string & vertical.

WAVES

1. The speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with

tension T and linear mass density µ v T / µ = .

2. Sound waves are longitudinal mechanical waves that can travel through solids, liquids, or gases. The speed v of sound wave in a fluid having bulk modulus B and density

ρ is v B / = ρ .

The speed of longitudinal waves in a metallic bar (stretched

wire) is v Y / = ρ

For gases, since B = γ P (Adiabatic bulk modulus of elasticity), the speed of sound is v p / = γ ρ

3. Beats arise when two waves having slightly different frequencies, f 1 and f 2 and comparable amplitudes, are superposed. The beat frequency is f beat = f 1 – f 2

4. The Doppler effect is a change in the observed frequency of a wave when the source S and the observer O moves relative to the medium. For sound the observed frequency f is given in term of the source frequency f 0 by

0 0

s

v v f f v v

± = ±

Here v is the speed of sound through the medium, v 0 is the velocity of observer relative to the medium, and v s is the source velocity relative to the medium. In using this formula, velocities in the direction OS should be treated

as positives and those opposite to it should be taken to be negative.

ELECTROSTATICS

1. Coulomb’s Law : 21 F r

= force on q 2 due to 1 2

1 21 2 21

k(q q ) q r r

=

where 21 ˆ r is a unit vector in the direction from q 1 to q 2 and

0

1 k 4

= πε is the constant of proportionality. .

2. Electric field due to a point charge q has a magnitude 2

0 | q | /4 r πε it is radially outwards from q, if q is positive, and radially inwards if q is negative. Like coulomb force, electric field also satisfies superposition principle.

3. An electric dipole is a pair of equal and opposite charges q and – q separated by some distance 2a. Its dipole moment vector p

r has magnitude 2qa and is in the direction of the

dipole axis from – q to q. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distance r from the centre.

2 2 3/2 3 0 0

p 1 p E . 4 (a r ) 4 r − −

= = πε + πε

r r r for r >> a

Dipole electric field on the axis at a distance r from the centre:

2 2 2 3 0 0

2pr 2p E for r a 4 (r a ) 4 r

= = >> πε − πε

r r r

The 1/r 3 dependence of dipole electric fields should be noted in contrast to the 2 1/ r dependence of electric field

due to a point charge. In a uniform electric field E r , a

dipole experiences a torque τ r given by p E τ = ×

r r r but experiences zero net force

4. The flux E S ∆φ = ∆ r r

5. Gauss’s law : The flux of electric field passing through any closed surface S is 0 1/ ε times the total charge enclosed by S. the law is especially useful in determining electric field

E, r

when the source distribution has simple sysmmetry. .

(a) Thin infinitely long straight wire of uniform linear charge

0

ˆ : E n 2 r

λ λ =

πε

r

where r is the perpendicular distance of the point from the wire and ˆ n is the radial unit vector in the plane normal to the wire passing through the point.

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(b) Infinite thin plane sheet of uniform surface charge

density 0

ˆ : E n 2 σ

σ = ε

r

(c) Thin spherical shell of uniform surface charge density

2 0

ˆ : E r 4 r

σ σ =

πε

r (r R);E 0 (r R) ≥ = <

r

6. Potential 0

1 Q :V(r) 4 r

= πε

r .

For a charge configuration q 1 , q 2 , ... q n with position vectors r 1 , r 2 , .... r n , the potential at a point P is tiven by the

superposition principle 1 2 n

0 1 2 np

1 q q q V .... 4 r p r p r

= + + + πε

where r 1p is the distance between q 1 and P, and so on. 7. The electrostatic potential at a point with position vector

r r due to point dipole of dipole moment p

r placed at the

origin is 2 0

1 p.r V(r) 4 r

= πε

r r

8. An equipotential surface is a surface over which potential has a constant value. For a point charge, concentric spheres centred at a location of the charge are equipotential surfaces. The electric field E

r at a poinnt is perpendicular

to the equipotential surface through the point. E r is in the

direction of the steepest decrease of potential. 9. Potential energy stored in a system of charges is the work

done (by an extenal agency) in assembling the charges at their locations. Potential energy of two charges q 1 , q 2 at r

distance is given by 1 2

0

1 q q U 4 r

= πε , where r is distance

between q 1 and q 2 . 10. Capacitance is defined by C = Q/V, where Q is the charge

on positive plate and V is the potential difference between plates. C is determined purely geometrically, by the shapes, sizes and relative positions of the two plates. The unit of capacitance is farad;:, 1 F = 1 C V –1 . For a parallel plate capacitor (with vacuum between the plates),

0 A C d

= ε , where A is the area of each plate and d the

separation between them. 11. The energy U stored in a capacitor of capacitance C, with

charge Q and voltage V is 2

2 1 1 1 Q U QV CV 2 2 2 C

= = =

12. For capacitors in the series combination, the total capacitance

C is given by 1 2 3

1 1 1 1 ........ C C C C

= + + +

In the parallel combination, the total capacitance C is given by C = C 1 + C 2 C 3 +... where C 1 , C 2 , C 3 ..... are individual capacitances.

CURRENT ELECTRICITY

1. Current density j gives the amount of charge flowing per second per unit area normal to the flow, J

r = nqv d

2. Equation E J = ρ r r

another statement of Ohm’s law, i.e., a conducting material obeys Ohm’s law when the resistivity of the material does not depend on the magnitude and direction of applied electric field.

3. (a) Total resistance R of n resistors connected in series is given by R = R 1 + R 2 + ..... + R n (b) Total resistance R of n resistors connected in parallel

is given by 1 2 n

1 1 1 1 ... R R R R

= + + + .

where R 1 , R 2 ..... R n are individual resistance. 4. Kirchhoff’s Rules – (a) Junction Rule : At any junction of

circuit element, the sum of currents entering the junction is equal to the sum of currents leaving it. (b) Loop Rule: the algebraic sum of the changes in potential in any closed loop is zero.

5. The Wheatstone bridge is an arrangement of four resistances – R 1 , R 2 , R 3 , R 4 . The null­point condiction is

given by 1 3

2 4

R R R R

= using which the value of one resistance

can be determined, knowing the other three resistance. 6. The potentiometer is a device to compare potential

differences. Since the method involves a condition of no current flow, the device can be used to measure potential difference; internal resistance of a cell and compare emf s

of two sources. 1

2

r R 1

= −

l

l

7. RC circuit: During charging: q = CE (1 – e –t/RC ) During discharging : q = q 0 e

–t/RC

MAGNETISM

1. The total force on a charge q moving with velocity v in the presence of electric and magnetic field E

r and B r ,

respectively is called the Lorentz force. It is given by the

expression : F q E (v B) = + × r r r r

.

2. A straight conductor of length and carrying a steady current I experiences a force F

r in a uniform external

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magnetic field B , F I B = × r r r r

l , the direction of l is given by the direction of the current.

3. The Biot­Savart law asserts that the magnetic field dB r

due to an element d r l carrying a steady current I at a point

P at a distance r from the current element is

0 3

µ d r dB I 4 r

× =

π

r r r l .

The obtain the total field at P, we must integrate this vector expression over the entire length of the conductor.

4. Magnetic field due to straight current carring conductor

0 1 2

µ I B (sin sin ) 4 a

= θ + θ π

r , where 1 2 and θ θ are the angles

between the line joining the point to the ends of conductor and perpendicular through the point to the conductor.

5. The magnitude of the magnetic field due to a circular coil of radius R carrying a current I at an axial distance x from

the centre is 2

0 2 2 3/2

µ IR B 2(x R )

= + .

6. The magnitude of the field B inside a long solenoid carrying a current I is: B = µ 0 nI, where n is the number of

turns per unit length. For a toroid, 0 µ NI B , 2 r

= π

where N is

the total numer of turns and r is the mean radius. 7. Ampere’s Circuital Law: Let an open surface S be bounded

by a loop C. Then the Ampere’s law states that

∫ C 0 B. d µ I, =

r r l where I refers to the current passing through

S. 8. force per unit length between two long parallel wires

carrying currents I 1 , I 2 and separated by distance a in a

free space or air 1 0 1 2 µ I I F Nm 2 a

− = π

.

The force is attractive if currents are in the same direction and repulsive currents are ion the opposite direction.

9. For current carrying coil M NIA : = r r

torque M B = τ = × r r r

10. Consider a material placed in an external magnetic

field 0 B r .The magnetic intensity is defined as, 0

0

B H µ

= r

r .

The magnetisation M r of the material is its dipole moment

per unit volume. The magnetic field B in the material is,

0 B µ (H M) = + r r r

11. For a linear material m M H. = χ r r

So that B µH = r r

and m χ is is called the magnetic susceptibility of the material. The three quantities, m χ , the relative magnetic permeability µ r , and the magnetic permeability µ are related as follows.

µ = µ 0 µ r ; µ r = 1 + m χ .

ELECTRO MAGNETIC INDUCTION 1. The magnetic flux through a surface of area A placed in a

uniform magnetic field B r is defined as

B B.A BA cos φ = = θ r r

, where θ is the angle between

B and A r r .

2. Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of

flux through it B d N dt φ

ε = −

3. When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is B v ε = l

4. When an current in a coil changes, it induces a back emf in the same coil. The self­induced emf is given by,

dI L dt

ε = − L is the self inductance of the coil, its a mesure

of the intertia of the coil against the change of current through it.

5. A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1). This relation is given by,

2 1 12

dI M , dt

ε = − The quantity M 12 is called mutual

inductance of coil 1 with respect to coil 2. 12 1 2 M K L L = .

6. LR circuit: For growth current, Rt /L 0 i i [1 e ] − = −

For decay of current, i = i 0 e –Rt/L

ALTERNATING CURRENT

1. For an alternating current i = i m sin t ω passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is (1/2)i 2 m R . To express it in the same form as the dc power (P = I 2 R), a special value of current is used. It is called root mean square (rms) current and is

denoted by m m

i I 0.707 i 2

= =

The average power loss over a complete cycle is given by P = V I cos φ . the term cos φ is caled the power factor. .

2. An ac voltage v = v m sin t ω applied to a pure inductor L,

drive s a current in the inductor i = i m sin ( t / 2), ω − π where i m = v m /X L . X L = L ω is called inductive reactance. The current in the inductor lags the voltage by / 2 π . The average power supplied to an inductor over one complete cycle is zero.

An ac voltage v = v m sin t ω applied to a capacitor drives

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a current in the capacitor: i = i m sin ( t / 2).Here, ω + π

m m C

C

v 1 i ,X X C

= = ω is called capacitive reactance.

3. An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e., the amplitude of the current is maximum at the resonant

frequency 0 L C 1 (X X ) LC

ω = = .

The equality factor Q defined by 0

0

L 1 Q R CR

ω = =

ω is an

indictor of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.

RAY OPTICS

1. For a prism of the angle A, of refractive index n 2 placed in a medium of refractive index n 1 ,

2 m 21

1

n sin[(A D ) / 2] n , n sin(A / 2)

+ = = where D m is the angle of

minimum deviation. Dispersion is the splitting of light into its constituent colours. The deviation is maximum for violet and minimum for red. Dispersive power ω is the ratio of angular dispersion v r ( ) δ − δ to the mean deviation δ .

v r δ − δ ω =

δ , where v r , δ δ are deviation of violet and red

respectively and δ the deviation of mean ray (usually yellow).

2. for refraction through a spherical interface (from medium 1 to 2 of refractive index n 1 and n 2 respectively)

2 1 2 1 n n n n . v v R

− − = Thin lens formula

1 1 1 ,Lens v u f

− =

maker’s formula: 2 1

1 1 2

1 (n n ) 1 1 f n R R

− = −

The power of a lens P = 1/f. The SI unit for power of a lens is dioptre (D): 1D =1 m –1 . If several thin lenses of focal length f 1 , f 2 , f 3 .. are in contact, the effective focal length of their combination, is given by

1 2

1 1 1 = +

f f f +........

The total power of a combination of several lenses is P = P 1 + P 2 + P 3 + .... If distance between lens is d then power of combination = P 1 + P 2 – d × P 1 P 2 Chromatic aberration is the colouring of image produced by lenses. This can be avoided by combining a covex and a concave lens of focal length f 1 and f 2 and dispersive

powers 1 2 , ω ω respectively satisfying the equation

1 2

1 2

0 f f ω ω

+ = or in terms of powers 1 1 2 2 P P 0 ω + ω = .

WAVE OPTICS

1. Young’s double slit of separation d gives equally spaced fringes of angular separation / d. λ The source, mid­point of the slits, and central bright fringe lie in a straight line. An extended source will destroy the fringes if it subtends angle more than / d λ at the slits. The resultant intensity

of two waves of intensity I 0 /4 of phase difference φ at any points is given by

2 0 I I cos ,

2 φ =

where I 0 is the maximum intensity. .

Condition for dark band : (2n 1) ,2 λ

δ = −

For bright band : n , δ = λ Fringe width D d λ

β =

MODERN PHYSICS

1. Einstein’s photoclectric equation is 2 max 0 0 0

1 mv v e hv h(v v ) 2

= = − φ = −

2. The nuclear mass M is always less than the total mass, m, ∑ of its constituents. The difference in mass of a

nucleus and its constituents is called the mass defect, 2

p n b M (Zm (A Z)m ) M; E Mc . ∆ = + − − ∆ = ∆

1 amu = 931 MeV

3. E n = 2

2

Z 13.6eV n

− × (for hydrogen like atom)

4. Bragg’s law : 2d sin n θ = λ

5. Law of radioactive decay : t 0 N N e −λ = .

Activity dN N N

= = −λ (unit is Becquerel)

6. Half time period, T 1/2 = 0.693

λ

7. X­ray : min 12400 Å V

λ =

Characteristics X­ray: K L α α λ < λ

Moseley law : v = a (Z – b) 2

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ATOMIC STRUCTURE

1. 2 2 2

2 n n 1 2 2

n h n r 0.529 Å, r n r 4 mZe Z

= = = × π

2. 2

T 2

PE Z E KE 13.6 eV, 2 n

= − = = −

3. 2 4

2 2 2 1 2

hc 2 me 1 1 E h n n

π ∆ = = − λ

4. 2 7 1

2 2 1 2

1 1 1 RZ [R 1.0968 10 m ] n n

− υ = = − = × λ

5. Total no. of spectrum lines = n(n 1)

2 −

6. Heisenberg Uncertainty Principle ( x)( p) h / 4 ∆ ∆ ≥ π

7. Moseley’s law “ n 2

13.7 v a(Z b),E eV / atom n

= − = −

8. Nodes (n ­1) = total nodes, = l angular nodes,

(n 1) − − = l Radial nodes

9. Photoelectric effect : 2 0

1 hv hv mv 2

= +

10. Orbital angular momentum: h ( 1) 2

+ π

l l

CHEMICAL BONDING

1. % ionic character = Actual dipole moment 100

Calculated dipole moment ×

2. Fajan’s Factors: Following factors are helpful in inducing covalent character in Ionic compounds (a) Small cation (b) Big anion (c) High charge on cation (d) High charge on anion (e) Cation having pseudo inert gas configuration (ns 2 p 6 d 10 )

PHYSICAL CHEMISTRY

e.g. Cu + , Ag + , Zn +2 , Cd +2

3. f S d EG L 1 H H H IE H E 2

−∆ = + + + ∆ −

4. M.O. theory: (a) Bond order = ½(N b – N a ) (b) Higher the bond order, higher is the bond dissociation

energy, greater is the stability, shorter is the bond length. (c) Species Bond order Magnetic properties

H 2 1 Diamagnetic H 2

+ 0.5 Paramagnetic Li 2 1 Diamagnetic

5. 1 Q [V SA ( q)] 2

= + − ±

6. Formal charge = 1 V L S 2

− +

7. VSEPR theory (a) (LP­LP) repulsion > (LP­BP) > (BP­BP) (b) For NH 3 → Bond Angle 106º45 in case of water it decreases to → 104º 27’ because H 2 O molecule contains 2LP and 2BP whereas NH 3 has 1LP and 3BP.

8. Bond angle: Decrease in bond angle down the gp is due to LP­BP repulsion (a) NH 3 > PH 3 > AsH 3 (b) H 2 O > H 2 S > H 2 Se

CHEMICAL EQUILIBRIUM

1. K P = K c ( ) g n RT ∆ where g p R n n n ∆ = −

2. Free Energy change ( G) ∆

(a) If G ∆ = 0 then reversible reaction would be in equilibrium.

(b) If G ∆ = (+) ve the equilibrium will be displace in

backward direction.

(c) If G ∆ = (–) ve then equilibrium will displace in forward direction.

3. (a) K c unit → (moles/lit) n ∆ ,

(b) K P unit → (atm) n ∆

(c) total moles at equilibrium = [total initial moles + n ∆ ]

(d) time required to establish equilibrium c 1/ K ∝

(e) If in any heterogenous equilibrium solid substance is also present then its active mass & partial pressure is assumed 1.

4. Le chatelier’s princple (i) Increase of reactant conc. (Shift forward) (ii)Decrease of reactant conc. (shift backward) (iii) Increase of pressure (from more moles to less moles) (iv) Decrease of pressure (from less moles to more moles) (v) For exothermic reaction decrease in temp. (Shift forward) (vi) for endothermic increase in temp. (“Shift forward)

ACID BASE

1. (a) Lewis Acid (e – pair acceptor) → CO 2 BF 3 , AlCl 3 , ZnCl 2 , FeCl 3 , PCl 3 , PCl 5 , SiCl 4 , SF 6 , normal cation (b) Lewis Base (e – pair donor) NH 3 , ROH, ROR, H 2 O, RNH 2 , R 2 NH, R 3 N, normal anion

2. Dissociation of Weak Acid & Weak Base → (a) Weak Acid → K a = Cx

2 /(1 – x) or K a = Cx 2

(b)Weak Base → K b = Cx 2 /(1 – x) or K b = Cx

2

3. Buffer solution: (a) Acidic → pH = pK a + log Salt/Acid for Maximum Buffer action pH = pK a Range of Buffer pH = pK a + 1 (b) Alkaline → pOH = pK b + log Salt/Base for max.

Buffer range for basic buffer = pK b + 1

(c) Buffer Capacity = Moles / lit of Acid or BaseMixed

change in pH

dCBOH dCHB B dpH dpH

= = −

4. Necessary condition for showing neutral colour of Indicator pH = pKln or [HIn]=[In – ] or [InOH] = [In + ]

IONIC EQUILIBRIUM

1. Relation between ionisation constant (K i ) & degree of ionisation ( ) α :­

2 2

i C K

(1 )V (1 ) α α

= = − α − α (Ostwald’s dilution law)

It is applicable to weak electrolytes for which 1 α << then

i i

K K V C

α = = or V C ↑ ↓ α ↑

2. Common ion effect : By addition of X mole/L of a common ion, to a weak acid (or weak base) α becomes equal to

a b K K or X X

[where α = degree of dissociation]

3. (A) If solubility product = ionic product then the solution saturates. (B)If solubility product > ionic product then the solution is unsaturated and more of the substance can be dissolved in it. (C)If ionic product > solubility product the solution is super saturated (principle of precipitation).

4. Salt of weak acid and strong base: pH = 0.5 (Pk W + pK a + log c): Salt of weak base and strong acid: pH = 0.5 (pK W – pK b – log c) Salt of weak acid and weak base : pH = 0.5 (pK w + pK a – pK b )

CHEMICAL KINETICS

1. Unit of Rate constant: 1 n n 1 1 K mol lit sec −∆ ∆ − − =

2. First Order reaction:

10 ½ 2.303 a 0.693 K log & t t (a x) K

= = −

kt t 0 [A] [A] e − =

3. Second Order Reaction: When concentration of A and B taking same.

2 1 x K t a(a x)

= −

when concentration of A and B are taking different.

2 2.303 b(a x) K log t(a b) a(b x)

− =

− −

4. Zero Order Reaction:

0 t a a K t −

=

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0 1/2

a x Kt & t 2K

= =

The rate of reaction is independent of the concentration of the reacting substance.

5. Arrhenius equation:

a E /RT a E k Ae & slope 2.303R

− − = =

when ( ) a E /RT T , then K A e 1 − → ∞ = = Q

6. log 2 a 2 1

1 1 2

k E T T k 2.303R TT

− =

OXIDATION-REDUCTION

1. Oxidant itself is reduced (gives O 2 )

Or Oxidant → e – (s) Acceptor

Reductant itself is oxidised (given H 2 )

Or reductant → e – (s) Donor

2. (i) Strength of acid O.N ∝

(ii)Strenght of base 1 / O.N ∝

3. (a) Electro Chemical Series:­ Li, K, Ba, Sr, Ca, Na, Mg, Al, Mn, Zn, Cr, Fe, Cd, Co, Ni, Sn, Pb, H 2 , Cu, I 2 , Hg, Ag, Br 2 , Cl 2 , Pt, Au, F 2 .

(b) As we move from top to bottom in this series

(1) Standard Reduction Potential ↑

(2) Standard Oxidation Potential ↓

(3) Reducing Capacity ↓

(4) Ip ↑

(5) Reactivity ↓ 4. (a) Formal charge = Group No. – [No of bonds

+ No. of non­bonded e – s] (b) At anode → Oxidation, Cathode → Reduction

VOLUMETRIC ANALYSIS

1. Equivalent weight of element =

Atomic wt of the element n factor

2. Equivalent weight of compound

Formula wt of the compound n factor

=

3. Equivalent wt. of an ion = formula wt (or At. Wt.)of ion

itsvalency

4. The law Dulong and Petit atomic wt. × specific heat = 6.4

5. Normality (N) = number of equivalent of solute volume of solution in litres

6. Molarity (M) number of moles of solute volume of solution in litres

7. When a solution is diluted N 1 × V 1 = N 2 × V 2 (before dilution) (after dilution)

8. Common acid­base indicators

Indicator Colour in Colour in pH range acidic alkaline medium medium

Methyl orange Pink Yellow 3.0 – 4.4 Methyl red Red Yellow 4.2 – 6.2 Litmus Red Blue 5.5 – 7.5 Phenolphthalein Colourless Pink 8.3 – 9.8

MOLE CONCEPT

1. Mole concept

GAM ≡ 1 gm atom ≡ 6.02 × 10 23 atom,

GMM = 1 gm molecule ≡ 6.02 × 10 23 molecules

N A = 6.02 × 10 23

2. Moles (gases) at NTP = volume(L)

22.4

3. Molecular mass = 2 × vapour density

CHEMICAL ENERGETICS

1. First Law : E Q W ∆ = +

Expression for pressure volume work W = – P V ∆

Maximum work in reversible expansion:

2 1

1 2

V P W 2.30nRTlog 2.303nRTlog V P

= − = −

2. Enthalpy and heat content : H E P V ∆ = ∆ + ∆

(p) ( v) g [q q n RT] = + ∆ g H E n RT ∆ = ∆ + ∆

g p(g) r(g) [ n n n ] ∆ = −

3. Kirchoff’s equation:

2 1 T T V 2 1 E E C (T T )[cons tan t V] ∆ = ∆ + ∆ −

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2 l T T P 2 1 H H C (T T ) ∆ = ∆ + ∆ − [constant P]

4. Entropy (s) : Measure of disorder or randomness

P R S S S ∆ = ∑ − ∑

rev 2 1

1 2

q V P S 2.303 nR log 2.030 nR log T V P

∆ = = =

5. Free energy change; G H T S ∆ = ∆ − ∆

G 0 ∆ < (spontaneous) [–ve] G 0 ∆ = (equilibrium)

G 0 ∆ > (non­spontaneous) [+ ve]

G W −∆ = (maximum) P V − ∆

ELECTRO CHEMISTRY

1. m = Z.I.t

2. Degree of dissociation: eq

eq ∞

λ α =

λ

3. Kohlrausch’s law : 0 0 0 m A B x y Λ = λ + λ

4. Nernst Equation

10 0.0591 [Products] E Eº log n [Reac tan ts]

= −

o o o Cell anode cathode eq.

nFEº & E E E &K anti log 0.0591

= + =

o cell cell G nFE & Gº nFE ∆ = − ∆ = −

c Gº 2.303RTlogK −∆ =

& max G W nFEº& G H T T

∂∆ = + ∆ = ∆ + ∂

5. Calculation of pH of an electrolyte by using a calomel

electrode: cell E 0.2415 PH 0.0591

− =

SOLUTION AND COLLIGATIVE PROPERTIES

1. Raoult’s law

0 0 A B A A B B P p p p X p X = + = +

= (1 – X B ) p 0 A + p

0 B X B = (p

0 B – p

0 A ) X B + p

0 A

0 s

0

P P n P n N −

= +

0 s

0

P P w.m & P W.M −

=

2. Colligative ∝ Number of particles properties ∝ Number of molecules (in case of

nonelectrolytes)

∝ Number of ions (in case of

electrolytes)

∝ Number of moles of solute

∝ Mole fraction of solute

3. Depression of freezing point, f f T K m ∆ =

4. Elevation in boiling point with relative lowering of vapour pressure

0 b

b 0 1

1000k p p T M p

− ∆ =

(M 1 = mol. wt. of solvent)

5. Osmotic pressure (P) with depression in freezing point

f T ∆ f f

dRT P T 1000K

= ∆ ×

6. Normal molar mass Observed colligative property i Observed molar mass Normal colligative property

= =

Observed osmotic pressure i Normal osmotic pressure

=

Actual number of particles No.of particles for no. ionisation

=

degree of association (a) = (1 – i) n

n 1 −

& degree of dissociation i 1 ( ) n 1

− α =

−

NUCLEAR CHEMISTRY

1. Radius of the nucleus : R = R 0 A 1/3

2. The amount N of the radioactive substance left after ‘n’

half lives = 0 n

N (initial amount) 2

3. Half­life period 1/2 0.693 t =

λ

4. Rate of disintegration:

t 0 10 0

dN 2.303 N .N & log or N N e dt t N

−λ − = λ λ = =

5. Average life AV Total life time of all the atoms (t )

Total number of atoms =

0 1/2

0

tdN 1 1.44 t N

∞

= = = λ

∫

GASEOUS STATE

1. Ideal gas equation: PV = nRT (i) R =0.0821 litre atm. K –1 mol –1

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(ii)R = 62.4 litres mm Hg K –1 . mol –1

(iii) R = 8.314 × 10 7 ergs K –1 . mol –1

(iv) R = 2 cals K –1 mol –1

(v) R = 8.314 JK –1 mol –1

2. Velocities related to gaseous state

RMS velocity 3PV 3RT 3P C M M d

= = =

Average speed = 8RTM π

& Most probable speed =

2RT M

Average speed = 0.9213 × RMS speed RMS speed = 1.085 × Average speed MPS = .816 × RMS; RMS = 1.224 MPS MPS;A.V. speed : RMS = 1 : 1.128 : 1.224

3. Rate of diffusion 1

density of gas ∝

4. Van der Waal’s equation

2

2

n a P (V nb) nRT V

+ − =

for n moles

5. Z (compressibility factor ) PV ;Z 1 nRT

= = for ideal gas

SOLID AND LIQUID STATE

1. Available space filled up by hard spheres (packing fraction)

Simple cubic = 0.52 6 π

=

3 bcc 0.68 8

π = = 2 fcc. 0.74

6 π

= =

2 hcp 0.74 6

π = = diamond = 3 0.34

6 π

=

2. Radius ratio and co­ordination number (CN)

Limiting radius ratio CN Geometry [0.155 – 0.225] 3 [plane triangle] [0.255 –0.414] 4 [tetrahedral] [0.414–0.732] 6 [octahedral] [0.732–1] 8 [bcc]

3. Atomic radius r rand the edge of the unit cell. Pure elements:

Simple cubic = a r 2

= 3a bcc r 4

= 2 a fcc 4

=

4. Relationship between radius of void (r) and the radius of the spere (R) : r (tetrahedral) = 0.225 R; r(octahedral) = 0.414 R

5. Paramagnetic : Presence of unpaired electrons [attracted by magnetic field]

6. Paramagnetic: Permanent magnetism ↑ ↑ ↑↑ 7. Antiferromagnetic: net magnetic moment is zero

↑ ↓ ↑ ↓

8. Ferrimagnetic: net magnetic moment is three ↑ ↓ ↓ ↑ ↑

INORGANIC CHEMISTRY PERIODIC TABLE

1. General electronic configuration (of outer orbits) s­block ns 1–2 p­block ns 2 np 1–6

d­block (n– 1) d 1–10 ns 1–2

f­block (n – 2) s 2 p 6 d 10 f 1–14 (n – 1)s 2 p 6 d 0 or 1 ns 2

2. Properties Pr(L TO R) Gr(T to B)

(a) atomic radius ↓ ↑ (b) ionisation potential ↑ ↓ (c) electron affinity ↑ ↓ (d) electro negativity ↑ ↓ (e) metallic character or ↓ ↑

electropositive character

3. 1 EA size

∝ ∝ nuclear charge.

Second electron affinity is always positive. Electron affinity of chlorine is greater than fluorine.

4. The first element of a group has similar propertis with the second element of the next group. This is called diagonal relationship.

EXTRACTIVE METALLUGRY

1. Floatation is a physical method of separating a mineral from the gangue depending on differences in their wettabilities by a liquid solution.

2. Roasting is the process of heating a mineral in the presence of air.

3. Calcination is the process of heating the ore in the absence of air.

4. Electrolytic reduction: Highly electropositive metals are extracted by the electolysis of their oxides and hydroxides.

s-BLOCK ELEMENTS

1. Atomic radii : Li < Na < K < Rb < Cs 2. Ionic radii : Li + < Na + <K + < Rb + < Cs + 3. Electronegativity: Li > Na > K > Rb > Cs

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4. First ionization potential : Li > Na > K > Rb > Cs 5. Melting point Li > Na > K > Rb > Cs 6. Density : Li > Na < K > Rb > Cs 7. Colour of the flame Li ­red, Na­Golden, K­Violet, Rb­

Red, Cs ­ Blue, Ca ­ Brick red, Sr­Blood red, Ba­ Apple green

8. Rb and Cs show photoelectric effect. 9. Stability of hydrides : LiH > NaH > KH > RbH > CsH 10. Basic nature of hydroxides: [LiOH < NaOH < KOH <

RbOH < CsOH]

BORON FAMILY 1. Stability of + 3 oxidation state : B > Al > ga > ln > Tl 2. Stability of + 1 oxidation state : Ga < ln < Tl 3. Raducing nature : Al > Ga > ln > Tl 4. Basic nature of the oxides and hydroxides :

B < Al < ga < ln < Tl 5. Relative strength of Lewis acid: BF 3 < BCl 3 < BBr 3 < BI 3

CARBON FAMILY

1. Reactivity : C < Si < Ge < Sn < Pb 2. Metallic character : C < Si < Ge < Sn < Pb 3. Acidic character of the oxides:

CO 2 > SiO 2 > GeO 2 > SnO 2 > PbO 2 Weaker acidic (amphoteric)

4. Thermal stability and volatility of hydrides: CH 4 > SiH 4 > GeH 4 > SnH 4 > PbH 4

5. Reducing nature of hydrides < CH 4 < SiH 4 < GeCl 4 < SnH 4 < PbH 4

6. Thermal stability of tetrahalides CCl 4 > SiCl 4 > GeCl 4 > SnCl 4 > PbCl 4

7. Thermal stability and volatility of tetrahalide with a common central atom MF 4 >MCl 4 > MBr 4 > MI 4

8. Oxidising character of M +4 species GeCl 4 < SnCl 4 < PbCl 4

9. Ease of hydrolysis of tetrahalides SiCl 4 > GeCl 4 > SnCl 4 > PbCl 4

10. Reducing character of dihalides GeCl 2 > SnCl 2 > PbCl 2

NITROGEN FAMILY

1. Acidic strength of trioxides : N 2 O 3 > P 2 O 3 > As 2 O 3 2. Acidic strength of pentoxides

N 2 O 5 > P 2 O 5 > As 2 O 5 > Sb 2 O 5 > Bi 2 O 5 3. Acideic strength of oxides of nitrogen

N 2 O < NO < N 2 O 3 < N 2 O 4 < N 2 O 5 4. The stability of pentoxides

P 2 O 5 < As 2 O 5 > Sb 2 O 5 > N 2 O 5 > Bi 2 O 5 5. Basic nature, bond angle, thermal stability and dipole

moment of hydrides NH 3 > PH 3 > AsH 3 > SbH 3 > BiH 3

6. Reducing power, covalent nature of hydrides : NH 3 < PH 3 < AsH 3 < SbH 3 < BiH 3

7. Stability of trihalides of nitrogen : NF 3 > NCl 3 > NBr 3 8. Ease of hydrolysis of trichlorides

NCl 3 > PCl 3 > AsCl 3 > SbCl 3 > BiCl 3 9. Lewis acid strength of trihalides of P, As and Sb

PCl 3 > AsCl 3 > SbCl 3 10. Lewis acid strength among phosphorus trihalides

PF 3 > PCl 3 > PBr 3 > PI 3 11. Bond angle, among the halides of phosphorus

PF 3 < PCl 3 < PBr 3 < PI 3

OXYGEN FAMILY

1. Melting and boiling point of hydrides H 2 O > H 2 Te > H 2 Se > H 2 S

2. Volatility of hydrides H 2 O < H 2 Te < H 2 Se < H 2 S

3. Thermal stability of hydrides H 2 O > H 2 S > H 2 Se > H 2 Te

4. Reducing nature of hydrides H 2 S < H 2 Se < H 2 Te

5. Covalent character of hydrides H 2 O < H 2 S < H 2 Se < H 2 Te

6. Bond angle & Diole moment of hydrides H 2 O > H 2 S > H 2 Se > H 2 Te (104º) (92º) (91º) (90º)

7. Ease of hydrolysis of hexahalides: SF 6 > SeF 6 > TeF 6 8. The acidic character of oxides (elements in the same

oxidation state) SO 2 > SeO 2 > TeO 2 > PoO 2 SO 3 > SeO 3 > TeO 3

9. Acidic character of oxide of a particular element (e.g.S) SO 2 < SO 3

10. Stability of dioxides SO 2 > TeO 2 > SeO 2 > PoO 2

HALOGEN FAMILY

1. Bond energy of halogens : Cl 2 > Br 2 > F 2 > I 2 2. Bond length in X 2 molecule : F 2 < Cl 2 < Br 2 < I 2 3. Solubility of halogen in water : F 2 > Cl 2 > Br 2 > I 2

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Revision Capsule ­ Chemistry 4. Oxidising power : F 2 > Cl 2 > Br 2 > I 2 5. Enthalpy of hydration of X – ion: F – > Cl – > Br – > I –

6. Reactivity of halogens : F > Cl > Br > I 7. Ionic character of M – X bond in halides

M – F > M – Cl > M – Br > M – I 8. Reducing character of X – ion : I – > Br – > Cl – > F –

9. Thermal stability of hydrides : HF > HCl > HBr > HI 10. Acidic strength of halogen acids: HI > HBr > HCl > HF 11. Conjugate base strength of halogen acids

I – < Br – < Cl – < F –

12. Reducing power of hydrogen halides HF < HCl < HBr < HI

13. Dipole moment of hydrogen halides HF > HCI > HBr > HI

14. Oxidising power of oxides of chlorine Cl 2 O > ClO 2 > Cl 2 O 6 > cl 2 O 7

15. Acidic character of oxyacids of chlorine HClO < HClO 2 < HClO 3 < HClO 4

16. Strength of conjugate bases of oxyacids of chlorine ClO – > ClO 2 > ClO 3

– > ClO 4 –

17. Oxidising power of oxyacids of chlorine HClO > HClO 2 > HClO 3 > HClO 4

18. thermal stability of oxyacids of chlorine HClO < HClO 2 < HClO 3 < HClO 4

19. Stability of anions of oxyacids of chlorine ClO – < ClO 2

– < ClO 3 – < ClO 4

–

TRANSITION ELEMENTS (D-BLOCK ELEMENTS)

1. The element with exceptional configuration are Cr 24 [Ar] 3d 5 4s 1 , Cu 29 [Ar] 3d 10 4s 1

Mo 42 [Kr] 4d 5 5s 1 , Pd 46 [Kr] 4d 10 5s 0

Ag 47 [Kr] 4d 10 5s 1 , Pt 78 [Xe] 4f 14 5d 10 6s 0

Au 97 [Xe] 5f 14 5d 10 6s 1

2. Frerromagnetic substances are those in which there are large number of electrons with unpaired spins and whose magnetic moments are aligned in the same direction.

ORGANIC CHEMISTRY GOC

1. The order of decreasing electronegativity of hybrid orbitals is sp > sp 2 > sp 3 .

2. conformational isomers are those isomers which arise due to rotation around a single bond.

3. A meso compound is optically inactive, even though it

has asymmetric centres (due to internal compensation of rotation of plane polarised light)

4. An equimolar mixture of enantiomers is called racemic mixture, which is optically inactive.

5. Tautomerism is the type of isomerism arising by the migration of hydrogen.

6. Reaction intermediates and reagents: Homolytic fission → Free radicals Heterolytic fission → Carbocation and carbanion

7. Nucleophiles – electron rich Two types : 1. Anions 2. Neutral molecules with lone pair of electrons (Lewis bases) Electrophiles: electron deficient. Two types : 1. Cations 2. Neutral molecules with vacant orbitals (Lewis acids).

8. Inductive effect is due to σ electron displacement along a chain and is permanent effect.

9. + I (inductive effect) increaes basicity, – I effect increases acidity of compounds.

10. Resonance is a phenomenon in which two or more structures can be written for the same compound but none of them actually exists.

ALKANES

1. Pyrolytic cracking is a process in which alkane decomposes to a mixture of smaller hydrocarbons, when it is heated strongly, in the absence of oxygen,

2. Combustion is a process in which hydrocarbons form carbon dioxide and H 2 O ( l ) when they are completely burnt in air/O 2 .

ALKENES

1. In dehydration and dehydrohalogenation the preferential order for removal of hydrogen is 3º > 2º > 1º (Saytzeff’s rule).

2. The lower the h H ∆ (heat of hydrogenation) the more stable the alkene is.

3. alkenes undergo anti­Markonikov addition only with HBr in the presence of peroxides.

ALKYNES

1. Alkynes add water molecule in presence of mercuric sulphate and dil. H 2 SO 4 and form carbonyl compounds.

2. Terminal alkynes have acidic H­atoms, so they form metal alkynides with Na, ammonical cuprous chloride solution and ammonical silver nitrate solution.

3. Alkynes are acidic because of H­atoms which are attached to sp hybridised ‘C’ atom has more electronegativity as it has more ‘s’ character than sp 2 and sp 3 hybridised ‘C’ atoms.

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ARENES

1. All o and p­directing groups are ring activating groups (except – X) The are : – OH, – NH 2 , – X, – R, – OR etc.

2. All m­directing groups are ring deactivating groups.

They are : 2 3 CHO,–COOH,–NO ,–CN,– NR

+ etc.

HALOGEN COMPOUNDS

1. The order of reactivity is (a) RI > RBr > RCl > RE (b) allyl halide > Alkyl halide > vinyl halide (c) Alkyl halide > Aryl halide

2. S N 1 reaction: Mainly 3º alkyl halides undergo this reaction and form racemic mixture. S N 1 is favoured by polar solvent and low concentration of nucleophile.

3. S N 2 reaction: Mainly 1º alkyl halides undergo this substitution. S N 2 reaction is preferred by non­polar solvents and high concentration of nucleophile.

ALCOHOLS

1. Alkenes are converted to alcohol in different ways as follows Reagent types of addition dil H 2 SO 4 – Markovnikow B 2 H 6 and H 2 O 2 , OH

– – Anti­Markovnikow Oxymercuration demercuration –Markovnikow

2. Oxidation of

1º alcohol → aldehyde → carboxylic acid

(with same no. (with same no. of of C atom C atom)

2º alcohol → ketone → carboxylic acid

(with same no. (with less no. of of C atom) C atom)

3º alcohol → ketone → carboxylic acid

(with less no. (with less no. of of C atom) C atom)

PHENOLS

1. Phenol 3 CHCl /OH Θ

→ Salicylaldehyde

(Reimer­Tieman reaction)

2. Phenol 2 CO ∆ → Salicyclic acid (Kolbe reaction)

3. Acidity of phenols (a) Increases by electron withdrawing substituents like

2 3 NO , CN, CHO, COOH, X, NR +

− − − − − −

(b) Decrease by electron relasing substituents like

2 2 R, OH, NH , NR , OR − − − − −

ETHERS

1. 2 3 Al O 2 250ºC 2ROH R O R H O → − − +

2. RONa X R ' ROR ' NAaX + − → +

(Williamson’s synthesis)

3. 2 4 4 2 (conc) ROR 2H SO 2RHSO H O ∆ + → +

4. 2 4 dil.H SO 2 ROR H O 2ROH ∆ + →

CARBONYL COMPOUNDS

1. Formation of alcohols using RMgX

(a) Formaldehyde + Hydrolysis RMgX 1ºalcohol →

(b) aldehyde + RMgX Hydrolysis 2ºalcohol →

(other than HCHO)

(c) Ketone + RMgX Hydrolysis 3º → alcohol

2. Cannizzaro reaction (Disproportionation)

Aldehyde HOtconc. alkali →Alcohol + Salt of Acid

(no α H­atom) Crossed ­Cannizzaro reaction gives alcohol with aryl group or bigger alkyl group.

3. Aldol condensation:

Carbonyl compound + dil. Alkali → β ­hydroxy(carbonyl)

(with α H­atom) compound 4. Benzoin condensation

Benzaldehyde ethanolic NaCN → Benzoin

NITROGEN COMPOUNDS 1. Order of basicity : (R = – CH 3 or – C 2 H 5 )

2º > 1º > 3º > NH 3 2. Hofmann degradation

Amides 2 Br /KOH lº → amine 3. The basicity of amines is

(a) decreased by electron withdrawing groups (b) increased by electron releasing groups

4. Reduction of nitrobenzone in different media gives different products Medium Product Acidic Aniline

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Basic Azoxy, Azo and finally hydrazobenzene Neutral Phenyl hydroxylamine Tests to differentiate: 1º, 2º and 3º alcohols Lucas test

Victormayer’s test 1º, 2º and 3º amines Hinsberg test 1º, 2º and 3º nitro compounds Test with HNO 2 and KOH Aryl halides and alkyl halides Test with AgNO 3

Solution Aldehydes and ketones Tollen’s Test/Fehlings test Aromatic aldehdes and Fehling’s Test Aliphatic aldehydes

IMPORTANTREAGENT 1. Dil H 2 SO 4 [or conc. H 2 SO 4 + H 2 O]

Use → Dehydrating agent (+HOH)

(a) 2 4 dil.H SO 2 2 3 2 CH CH CH CH OH = → − −

(b) 2 4 dil.H SO 2 5 2 5 2 5 C H OC H 2C H OH →

2. Alc. KOH or 2 NaNH ( se HX) → − U

alc KOH 3 2 2 2 HCl CH CH Cl CH CH − → =

3. Cu or ZnO/300ºC " " 1º alc ald, 2º alc ketone. → → " 3º alc alkene (exception) →

4. Lucas reagent ZnCl 2 + Conc. HCl Use → for distinction between 1º , 2º & 3º alcohol

5. Tilden Reagent NOCl (Nitrosyl chloride) NOCl

2 5 2 2 5 C H NH C H Cl → 6. Alkaline KMnO 4 (Strong oxidant)

Toluene → Benzoic acid 7. Bayer’s Reagent

1 % alkaline KMnO 4 (Weak oxidant)

Use → For test of C C or C C > = < − ≡ −

2 2 2 2 2 CH CH H O [O] CH OH CH OH = + + → −

8. Acidic K 2 Cr 2 O 7 (Strong oxidant)

[O] 2 RCH OH RCHO →

9. SnCl 2 /HCl or Sn/HCl use → for reduction of nitrobenzene in acidic medium.

2 SnCl /HCl 6 5 2 6 5 2 6H C H NO C H NH →

10. Lindlar’s Catalyst = Pb/CaCO 3 + in small quantity (CH 3 COO) 2 Pb

" 2 2 – butyne H Cis 2butene + → −

(main product) 11. Ziegler – Natta Catalyst (C 2 H 5 ) 3 Al + TiCl 4

Use – Ln Addition polymerisation

Propene " → Poly propene

MAIN USE OF COMPOUNDS Alkane → Fuel, alkene → polymer, Alkyne → Solvent making westron, Westrosol, General alkyl halide → as solvents, CHCl 3 → Anaesthetic, Germicide, CCl 4 → Pyrene & Fire extinguisher, CH 3 OH → Antifreeze, deforming of alcohol, C 2 H 5 OH → Tonic, wine preparation, Power alcohol, C 2 H 5 – O– C 2 H 5 → Antiseptic, Natellite, HCHO → Formamint medicine, CH 3 CHO → Antiseptic, CH 3 COCH 3 → as solvent, CH 3 COOC 2 H 5 → Artificial silk & flavour, CH 3 NH 2 → Refrigerating agent, C 2 H 5 NH 2 → in development of photography.

IDENTIFICATIONTESTS (a) Unsaturated compound (Bayer’s reagent)

Decolourising the reagent (b) Alcohols (Certic ammonium nitrate solution)

Red colouration (c) Phenols (neutral FeCl 3 solution)

Violet/deep blue colouration (d) Aldehydes and ketones (2, 4­D.N.P.)

Orange precipitate (e) Acids (NaHCO 3 solution)

Brisk effervescence (CO 2 is evolved) (f) 1º amine (CHCl 3 + KOH)

Foul smell (isocyanide) (g) 2º amine (NaNO 2 + HCl)

Yellow oily liquid (Nitrosoamine)

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MATHEMATICS RELATIONSAND FUNCTIONS

1. A function f:X Y → is one­one (or injective) if

f(x 1 ) = f(x 2 ) ⇒ x 1 = x 2 1 2 x ,x X. ∀ ∈

2. A function f :X Y → is onto (or surjective ) if given any y Y, x X ∈ ∃ ∈ such that f(x) = y. [codomain = range]

3. A function f: X Y → biective if is both one­one and onto.

4. The composition of function f: A B → and g: B C → is the function gof : A C → given by gof (x) = g (f(x)) x A ∀ ∈ .

5. A function f: X Y → is invertible if g :Y X ∃ → such that gof = 1 x and fog = I Y .

6. A function f: X Y → is invertible if and only if f is one­ one and onto.

TRIGONOMETRIC FUNCTION

1. sin( ) sin cos cos sin α ± β = α β ± α β

2. cos( ) cos cos sin sin α±β = α β α β m

3. tan tan tan( ) 1 tan tan

α± β α±β =

α β m

4. A B A B sinA sin B 2sin cos 2 2 + −

+ =

5. A B A B sinA sinB 2cos sin 2 2 + −

− =

6. cos A + cos B = 2 cos A B A B cos 2 2 + −

7. A B A B cosA cosB 2sin sin 2 2 + −

− = −

8. sin sin( ) sin( 2 ) ... α + α + β + α + β + ..... to n terms

n 1 n sin sin 2 2

sin 2

− β α + β = β

; 2n β ≠ π

9. cos cos( ) cos( 2 ) .... α + α + β + α + β + to n terms

n 1 n cos sin 2 2 ; 2n sin

2

− β α + β = ≠ π β

10. Sine rule : a b c 2R

sinA sinB sinC = = =

11. Consine Rule : cos 2 2 2 b c a A 2bc

+ − = ,

2 2 2 a c b cosB 2ac

+ − = ,

2 2 2 a b c cosC 2ab

+ − = .

12. B C b c A tan cot 2 b c 2 − − = +

13. A (s b)(s c) sin 2 bc

− − =

14. A (s b)(s c) tan 2 s(s a)

− − = −

15. 1 1 1 abc absinC bc sin A acsinB rs 2 2 2 4R

∆ = = = = = . R =

circum radius, r = in­radius

16. s(s a)(s b)(s c) ∆ = − − −

17. A B C r 4Rsin sin . sin 2 2 2

=

18. a ccosB bcosC = + b = c cos A + a cos C

INVERSETRIGONOMETRIC FUNCTIONS 1. Domain and range of inverse trigonometric finction

Function Domain Range

y = sin –1 x [ 1,1] − ,2 2 π π −

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y = cos –1 x [ 1,1] − [0, ] π

1 y tan x − = ( , ) −∞ ∞ ,2 2 π π −

y = cot –1 x ( , ) −∞ ∞ [0, ] π

y = sec –1 x ( , 1] [1, ) −∞ − ∞ U 0, , 2 2 π π π

U

1 y cosec − = x ( , 1] [1, ) −∞ − ∞ U , 0 0, 2 2 π π −

2. Some properties of inverse trigonometric function:

1 1 sin x cos x ;2

− − π + = x [ 1,1] ∈ −

1 1 tan x cot x 2

− − π + = ; x R ∈

1 1 sec x cosec x 2

− − π + = x ( , 1] [1, ) ∈ −∞ − ∞ U

1 2

1 1 2

1 2

1 1 sin (2x 1 x ), if x 2 2 1 2sin x sin (2x 1 x , if x 1 ` 2

1 sin (2x 1 x , if 1 x 2

−

− −

−

− − ≤ ≤ = π − − ≤ ≤

−π − − − ≤ ≤ −

1 2

1 1 2

1 2

2x tan , if 1 x 1 1 x

2x 2 tan x tan , if x 1 1 x 2x tan ifx 1

1 x

−

− −

−

− < < − = π + > − −π + < − −

QUADRATIC EQUATIONSAND INEQUALITIES 1. For the quadratic equation ax 2 + bx + c = 0

2 b b 4ac x 2a

− ± − =

b c a a

− α + β = αβ =

(i) If b = 0 ⇒ roots are of equal magnitude but of opposite sign (ii) If a = c ⇒ roots are reciprocal to each other

(iii) a 0, c 0 a 0, c 0

> < < >

⇒ Roots are of opposite signs.

(iv) If a 0,b 0,c 0 a 0,b 0,c 0

> > > ⇒ < < < both roots are negative

(vii) If a 0, b 0, c 0 a 0,b 0,c 0

> < > ⇒ < > <

both roots are positive

(viii) If gisn of a = sing of b ≠ sign of c⇒ Greater root in magnitude is negative

2. If p + iq (p and q being real ) is a root of the quadratic equation, where i 1 = − , then p – iq is also a root of the quadratic euquation.

3. Envery equation of n th degree (n 1) ≥ has exactly n roots and if the equation has more than n roots, it is an identity.

4. An inequality of the form log a f(x) > b is equivalent to the followig systems of inequalities: (a) f(x) > 0, f(x) > a b for a > 1 (b) f(x) > 0, f(x) < a b for a < 1, a > 0

5. Irrational roots occures in pairs ( ) ( ) p q and p q + −

COMPLEX NUMBERS 1. For any integer k, i 4k = 1, i 4k+1 = i, i 4K+2 = – 1, i 4k+3 = –i

2. |z – z 1 | + |z – z 2 | = λ , represents an ellipse if

|z 1 – z 2 | < λ , having the points z 1 and z 2 as it foci. And if

1 2 | z z | − =λ , then z lies on a line segment connecting z 1 and z 2

3. Cube roots of unity ( ) 2 1, , ω ω

(i) 3 1 ω = (ii) 2 1 0 + ω+ ω = The three cube roots of unity are

2 1 1 1, ( 1 3i), ( 1 3i) 2 2

ω= − + ω = − − which are the same as

2 2 4 4 1,cos i sin and cos i sin 3 3 3 3 π π π π + +

4. De Moivre’s theorem: For all real values of n,

n (cos i sin ) cos n i sin n θ + θ = θ + θ

If z cos i sin = θ + θ using De­Moivre’s theorem

n n n n

1 1 z 2 cos n ; z 2 i sin n z z

+ = θ − = θ

PERMUTATIONSAND COMBINATIONS 1. The number of permutations of n different things taken r

at a time, where repetition is not allowed, is denoted by

n P r and is given by n

r n! P

(n r)! =

− , where 0 r n ≤ ≤

2. The number of permutations of n different things, taken r at a time, where repetition is allowed, is n r .

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3. The number of permutations of n objects taken all at a time, where p 1 objects are of first kind, p 2 obects are of the second kind, ... p k objects are of the k

th kind and rest, if

any are all different is 1 2 k

n! p !p !....p !

4. The number of combination of n different things taken r at a time, denoted by n C r , is givne by

n r

n! C ,0 r n r!(n r)!

= ≤ ≤ −

5. Number of circular permutatios of n things when p alike and the rest different taken all at a time distinguish

clockwise and anticlockwise arrangement is (n 1)! p! −

BINOMIALTHEOREM 1. The expansion of a binomial for any positive integral n is

given by Binomial Theorem, which is (a + b) n = n C 0 a

n + n C 1 a n–1 b + n C 2 a

n – 2 b 2 + ... + n C n–1 a.b n –1

+ n C n b n

2. The general term of an expansion (a + b) n is T r + 1 =

n C r a n –r . b r .

The total number of terms in the expansion of (a + b) n is n + 1.

3. In the expansion (a + b) n , if n is even, then the middle term

is the th n 1

2 +

term. If n is odd, then the middle terms

are

th th n 1 n 3 and 2 2 + +

terms.

SEQUENCEAND SERIES 1. The general term or the n th term of the A.P. is given by a n =

a + (n –1)d. The sum S n of the first n terms of an A.P. is given by

n n n S [2a (n –1)d] (a ) 2 2

= + = + l

2. The sum S n of the first n terms of G.P. is given by

n n

n a(r 1) a(1 r ) S or , if r 1 r 1 1 r

− − = ≠

− − ,

a S 1 r ∞ =

− 3. A series whose each term is formed, by multiplying

corresponding terms of an A.P. and a G.P. is called ar Arithmetic­geometric series. Summation of n terms: \

n 1 n

n 2

a dr(1 r ) [a (n 1)d] S r 1 r (1 r) 1 r

− − + − = + −

− − −

4. Harmonical progression is defined as a series in which reciprocal of its terms are in A.P.

The standard for of a H.P. is 1 1 1 ... a a d a 2d

+ + + + +

STRAIGHT LINES

1. An acute angle (say θ ) between lines L 1 and L 2 with slopes

m 1 and m 2 is given by 2 1

1 2 1 2

m m tan ,1 m m 0 1 m m

− θ = + ≠

+

2. The perpendicular distance (d) of a line Ax + By + C = 0

from a point (x 1 , y 1 ) is givne by 1 1

2 2

| Ax By C | d A B

+ + =

+ .

3. Distance between the parallel lines Ax + By + C 1 = 0 and

2 Ax By C 0,is + + = given by 1 2 2 2

| C C | d A B

− =

+

CONIC SECTIONS 1. The equation of a circle with centre (h, k) and the radius r

is (x – h) 2 + (y – k) 2 = r 2 . 2. Equation of tangent :xx 1 + yy 1 + g(x + x 1 ) + f(y + y 1 ) +

c = 0 3. The equation of the parabola with focus at (a, 0) a > 0

directrix x = – a is y 2 = 4ax. 4. Latus rectum of a parabola is a line segment perpendicular

to the axis of the parabola, through the focus and whose end points lie on the parabola.

5. Length of the latus rectum of the parabola y 2 = 4ax is 4a. 6. The parametric equation of the parabola is x = at 2 , y = 2at 7. An ellipse is the set of all points in a plane, the sum of

whose distance from two fixed points in the plane is a constant.

8. The equations of an ellipse with foci on the x­axis is 2 2

2 2

x y 1 a b

+ = ; its parametric equation is x = a cos θ ; y = b

sin θ

9. Latus rectum of the ellispe 2 2

2 2

x y 1 a b

+ = is 2 2b

a

10. The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.

11. A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

12. The equation of hperbola with foci on the x­axis is

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2 2

2 2

x y 1; a b

− = Two asymptotes: 2 2

2 2

x y 0 a b

− =

13. Letus rectum of the hyperbola : 2 2

2 2

x y 1 a b

− = is 2 2b

a

THEREE DIMENSIONALGEOMETRY 1. The coordinates of the point R which divides the line

segment joining two points P (x 1 , y 1 , z 1 ) and Q(x 2 ,y 2 ,z 2 ) internally and externally in the ratio m : n are given by

2 1 2 1 2 1 mx nx my ny mz nz , , m n m n m n

+ + + + + +

and 2 1 2 1 2 1 mx nx my ny mz nz , , m n m n m n

− − − − − −

, respectivley. .

2. The coordinates of the centroid of the triangle, whose vertices are (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ) and (x 3 , y 3 , z 3 ) are

1 2 3 1 2 3 1 2 3 x x x y y y z z z , , 3 3 3

+ + + + + +

3. If l m, n are the direction cosines and a, b, c are the direction ratios of a line then

2 2 2 2 2 2 2 2 2

a b c ,m ,n

a b c a b c a b c ± ± ±

= = = + + + + + +

l

4. If 1 1 1 ,m n l and 2 2 2 , m ,n l are the direction cosines of two lines; and θ is the acute angle between the two lines; then

1 2 1 2 1 2 cos | m m n n | θ = + + l l

5. Equation of a line through a point (x 1 , y 1 , z 1 ) and having

direction cosines , m, n l is 1 1 1 x x y y z z

m n − − −

= = l

6. Shortest distance between 1 1 r a b = + λ r r r and 2 2 r a µb = +

r r r

is 1 2 2 1

1 2

(b b ) . (a a ) b b

× −

×

r r r r

r r

7. The equation of a plane through a point whose position vector is a r and perpendicular to the vector N

r is

(r a).N 0 − = r r r

8. Vector equation of a plane that passes through the line of intersection of planes 1 1 r .n d =

r r and 2 2 r .n d = r r is

1 2 1 2 r. (n n ) d d . + λ = + λ r r r

where λ is any nonzero constant.

9. The distance of a point whose position vector is a r from

the plane ˆ r .n d = r is ˆ | d a .n | −

r .

10. Skew line: Two straight lines are said to be skew lines if

they are neither parallel nor intersecting.

Shortest distance: 2 1 1 2 2 1

2 1 2 2 1

(x x )(m n m n ) (m n m n )

Σ − −

Σ −

DIFFERENTIAL CALCULUS LIMIT 1. A function f has the limit L as X approaches a if the limit

from the left exists and the limit from the right exists and both limit

are L that is, x a x a x a lim f (x) lim f (x) lim f (x)

− → → → = =

Remember x a Limit x a

Let x a Lim f (x)

→ = l and

x a Lim g(x) m.

→ = If l and m are finite

then:

(i) x a Lim(f (x) g(x)) l m.

→ ± = ±

(ii) x a Limf (x). g(x) .m.

→ = l

(iii) x a

f (x) Lim , g(x) m →

= l

Provided m 0 ≠

(iv) x a x a Lim k f (x) k Lim f (x) k ;

→ → = = l where k is a constant.

(v) x a x a Lim[f (x) k] Limf (x) k,

→ → + = + where k is a constant.

(vi) If x a x a

f (x) g(x), then Limf (x) Limg(x). → →

≤ ≤

2. Limit in case of Composite Function:

( ) x a x a Limf[g(x)] f Limg(x) ;

→ → = provided f is continuous at

x.

3. 1

1 x 0 x 0 x 0 x 0

sin x x sin x x Lim Lim Lim Lim 1

x sin x x sin x

−

− → → → → = = = =

(where x is mensured in radian)

4. x

1/x

x x 0

1 Lim 1 Lim(1 x) e. x →∞ →

+ = + =

x ax a /x a

x x 0 x

a 1 l im 1 lim(1 x) lim 1 e x x →∞ → →∞

+ = + = + =

CONTINUITY OF FUNCTIONS 1. All polynomials, Trigonometrical, exponential and

Logarthmic functions are continuous in their domain. 2. If f(x) is continuous and g(x) is discontinuous at x = a then

the product function φ (x) = f(x) g(x) is not necessarily be

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discontinuous at x = a. e.g. f(x) & g(x) = sin , x 0

x0 x 0

π ≠

=

3. For any positive integer n and any continuous function f,

[f (x)] n and n f (x) are continuous when n is even the input

of f in n f (x) restricted to inputs x for which f(x) 0 ≥ .

4. If f(x) and g(x) are continuous, then so are f(x) + g(x). f(x) – g(x), and f(x) . g(x)

5. If f(x) and g(x) are continuous, so is g(x) /f(x), so long as the inputs x do not yield outputs f(x) = 0. DIFFERENTIATION AND APPLICATION

1. Interpretation of the Derivative: If y = f(x) then, (a) m = f'(a) is the slope of the tangent line to y = f(x) at x = a and the equation of the tangent line at x = a is given by y = f(a) + f'(a) (x – a). (b) f'(a) is the instantaneous rate of changes of f(x) at x = a. (c) If f(x) is the position of an object at time x then f'(a) is the velocity of the object at x = a.

2. Basic Propertices and Formulas : If f(x) and g(x) are differentiable function s (the derivative exists), c and n are any real numbers

1. (cf)' = cf'(x) 2. (f g) f (x) g (x) ′ ′ ′ ± = ±

3. (fg)' = f' g+ fg' 4. 2

f f g fg g g

′ ′ ′ − =

5. n n 1 d (x ) nx dx

− =

6. ( ) d (f (g(x))) f ' g(x) g '(x) dx

=

3. Increasing/Decreasing: (i) If f'(x) > 0 for all x in an interval I then f(x) is increasing

on the interval I. (ii) If f'(x) < 0 for all x in an interval I then f(x) is decreasing

on the interval I. (iii) If f'(x) = 0 for all x in an interval I then f(x) is constant

on the interval I. 4. Concave Up/Concave Down:

(i) If f''(x) > 0 for all x in an interval I then f(x) is concave up on the interval I. (ii) If f''(x) < 0 for all x in an interval I then f(x) is concave down on the interval I.

5. 1 st Derivative Test: If x = c is a critical point of f(x) then x = c is 1. a rel. max. of f(x) if f'(x) > 0 to the left of x = c and

f’(x) < 0 to the right of x = c. 2. a rel. min. of f(x) if f'(x) < 0 to the left of x = c and f'(x)

> 0 to the right of x = c. 3. not a relative extrema of f(x) if f'(x) is the same sign on

both sides of x = c. 2nd Derivative Test: If x = c is critical point of f(x) such that f'(c) = 0 then x = c

1. is a relative maximum of f(x) if f''(c) < 0 2. is a relative minimum of f(x) if f''(c) > 0 3. f(x) may have a relative maximum, relative minimum,

or neither if f''(c) = 0 6. Mean value theorem: If f(x) is continuous on the closed

interval [a, b] and differentiable on the open interval (a, b)

then there is a number a < c < b such that f'(c) = f (b) f (a)

b a − −

7. Rolle’s theorem: If a function f(x) is continuous onthe closed interval [a, b] and differentiable in an interval (a, b) and also f(a) = f(b), then there exist at least one value c of x in the interval (a, b) such that f'(c) = 0.

8. L'Hospital’s rule: x a

f (x) f (a) f (a) lim (x) (a) (a) →

′ ′′ = =

′ ′′ φ φ φ

if f(a) = 0 or ∞ , (a) 0 or , f (a) 0 or , (a) 0 or ′ ′ φ = ∞ = ∞ φ = ∞

9. Length of sub­tangent = 1 1

1 ( x ,y )

dx y dy

; Sub­normal

1 1

1 (x ,y )

dy y dx

=

Length of tangent = 1 1

2

1 (x ,y )

dx y 1 dy

+

Length of normal 1 1

2

1 (x ,y )

dy y 1 dx

= +

10. Orthogonal trajectory: Any curve which cuts every member of a given family of curves at right angle, is called an orthogonal trajectory of the family.

INTEGRAL CALCULUS 1. Some properties of definite integral:

(a) If an interval [a, b] (a < b), the function f(x) and φ (x)

satisfy the condition f (x) (x), then ≤ φ

a b

b a

f (x)dx (x) dx ≤ φ ∫ ∫

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(b) If m and M are the smallest and greatest values of a function f(x) on an interval [a, b] and a b, ≤ then

a

b

m(b a) f (x)dx M(b a) − ≤ ≤ − ∫

(c) b c b

a a c

f (x)dx f (x)dx f (x), = + ∫ ∫ ∫ where a < c < b

(d) b b

a a

f (x)dx f (a b x)dx = + − ∫ ∫

(e) na a

0 0

f (x)dx n f (x) dx = ∫ ∫ where a is the period of the

function and n I ∈

(f)

a a

0 a

2 f (x) dx, If f (x) is even function i.e.f ( x) f (x) f (x) dx

0, If f (x) is odd function, i.e.f (x) f (x) −

− = =

= −

∫ ∫

(g)

a 2a

0 0

2 f (x)dx, If f (2a x) f (x) f (x)dx

0, If (2a x) f (x)

− = =

− = −

∫ ∫

2. Leibnitz rule :

g(x )

f (x )

d F(t) dt g (x) g (x)F(g(x)) f (x)F(f (x)) dx

′ ′ ′ = = − ∫

3. If aseires can be put in the form

r n 1 r n

r 0 r 1

1 r 1 r f or f , n n n n

= − =

= =

∑ ∑ then its limits as

1

0

n is f (x) dx → ∞ ∫

PROBABILITY 1. Probability of an event : For a finite sample space with

equally likely outcomes Probability of an event

n(A) P(A) , n(S)

= where n(A) = number of elements in the

set A, n(S) = number of elements in the set S. 2. f A and B are any two events, then

P (A or B) = P(A) + P(B) – P(A and B)

equivalently, P(A B) P(A) P(B) P(A B) = + − U I

3. If A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

4. If A is any event, the P (not A) = 1 – P(A)

5. The conditional probability of an event E, given the occurrence of the event F is given by

P(E F) P(E | F) ,P(F) 0 P(F)

= ≠ I

6. Theorem of total probability: Let E 1 , E 2 , ....E n ) be a partition of a sample space and suppose that each of E 1 , E 2 ,... E n has nonzero probability. Let A be any event associated with S, then P(A) = P(E 1 ) P(A|E 1 ) + P(E 2 ) P(A|E 2 ) + .... + P(E n ) P(A|E n )

7. Bayes’ theorem: If E 1 , E 2 .... E n are events which constitute. a partition of sample space S, i.e., E 1 , E 2 .... E n are pairwise disjoint and 1 2 n E E .... E S and A = U U U be any event with nonzero probability, then

1 1 1 n

1 j j 1

P(E )P(A | E ) P(E | A)

P(E )P(A | E ) =

=

∑

8. Let X be a random variable whose possible values x 1 ,. x 2 x 3 ....x n occur with probabilities P 1 , P 2 , P 3 .... P n respectively.

The mean of X, denoted by µ, is the number n

i i i 1 x p

= ∑

The mean of a random variable X is also called the expectation of X, denoted by E(X).

9. Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions: (a) There should be a finite number of trials, (b) The trials should be independent. (c) Each trial has exactly two outcomes: success or failure. (d) The probability of success remains the same in each trial. For Binomial distribution B(n, p), P(X = x) = n C x q

n–x p x , x = 0, 1,.... n(q = 1 – p) MATRICES

1. Positive Integral Powers of a Matrix: For any positive integers m, n (i) A m A n = A m+n

(ii) (A m ) n = A mn = (A n ) m

(iii) I n = I m = I (iv) Aº = I n whnere A is a square matrices of order n.

2. Transpose of a matrix:The matrix obtained from a given matgrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by A T or A'. From the definition it is obvious that if order of A is m × n, then order of A T is n × m. Properties of Transose

(i) (A T ) T = A (ii) T T T (A B) A B ± = ±

(iii) T T T (AB) B A = (iv) T T (kA) k(A) K = is scalar

(v) I T = I (vi) tr (A) = tr (A) T

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(vii) T 1 2 3 n 1 n (A A A ...A A ) − = A A n

T A n–1 T ...... A 3

T A 2 T A 1

T

3. Inverse of a matrix : If A and B are two matrices such that AB = I = BA the B is called the inverse of A and it is denoted by A –I , thus A –1 = B⇔ AB = I = BA To find inverse matrix of a given matrix A we use following

formula 1 adj.A A

| A | − =

Thus A –1 exists | A | 0 ⇔ ≠

DETERMINANTS 1. Minor :The determinant that is left by cancelling the row

and column intersecting at a particular element is called the minor of that element.

2. Cofactor: The cofactor of an element a ij is denoted by F ij and is equal to (–1) i+j M ij where M is a minor of element a ij

3. Multiplication of two diterminants: Multiplication of two third order determinants is defined as follows:

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

a b c m n a b c m n a b c m n

× =

l

l

l

1 2 1 1 3 1 1 1 2 1 3 1 1 1 2 1 3

2 1 2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

3 1 3 2 3 3 3 1 3 2 3 3 3 1 3 2 3 3

a b c a m b m c m a n b n c n a b c a m b m c m a n b n c n a b c a m b m c m a n b n c n

+ + + + + + + + + + + + + + + + + +

l l

l l l

l l l

4. System of linear equation in three unknowns: Using Crammer’s rule of determinant we get

3 1 2

1 2 3

x y z 1 i.e. x , y , z ∆ ∆ ∆

= = = = = = ∆ ∆ ∆ ∆ ∆ ∆ ∆

Case ­ I : If 0 ∆ ≠

Then 3 1 2 x , y ,z ∆ ∆ ∆

= = = ∆ ∆ ∆

∴ The system is consistent and has unique solution.

Case ­ II if 0 ∆ = and

(i) If a least one of 1 2 3 , , ∆ ∆ ∆ is not zero then the system of equations is inconsistent i.e. has no solution.

(ii) If d 1 = d 2 = d 3 = 0 1 2 3 or , , ∆ ∆ ∆ are all zero then the system is consistent and has infinitely many solutions.

VECTORS 1. Triangle law of vector addition of two vector

Magnitude of 2 2 R: R A B 2ABcos = + + θ r r

2. Scalar Product : A. B AB cos = θ r r

(here θ is the angle between the vector)

3. Vector product : ˆ C A B AB sin n = × = θ r r r

4. Given vectors 1 1 1 x a y b z c, + + r r r , 2 2 2 x a y b z c + +

r r r ,

3 3 3 x a y b z c + + r r r where a,b,c

r r r are non­coplanar vectors,

will be coplanar if and only if

1 1 1

2 2 2

3 3 3

x y z x y z 0 x y z

=

5. Scalar triple product:

(a) If 1 2 3 1 2 3 ˆ ˆ ˆ ˆ ˆ ˆ a a i a j a k,b b i b j b k = + + = + +

r r and

1 2 3 ˆ ˆ ˆ c c i c j c k = + +

r then

(a b).c [a bc] × = = r r r r r r

1 2 3

1 2 3

1 2 3

a a a b b b c c c

(b) [a b c] = volume of the parallelopiped whose

coterminous edges are formed by a ,b,c r r r

(c) a , b,c r r r are coplanar if an only if [a ,b c] 0 =

r r r

(d) Four points A, B, C, D with position vectors

a, b c, d r r r r respectively are coplanar if and only if

[ABAC AD] uuur uuur uuur

= 0 i.e. if and only if

[b a c a d a] 0 − − − = r r r r r r

(e) Volume of a tetrahedron with three coterminous edges

1 a , b, c [a b c] 6

= r r r r r r

(f) Volume of prism on a triangular base with three

coterminous edges 1 a ,b c [a b c] 2

= r r r r r r

6. Lagrange’s identity:

a.c a.d (a b) .(c d) (a.c)(b.d) (a . d)(b.c)

b.c b.d × × = = −

r r r r r r r r r r r r r r r r

r r r r

7. Reciprocal system of vectors:

If a ,b,c r r r be any three non coplanar vectors so that

[a b c] 0 ≠ r r r then the three vectors a ' b ' c '

r r r defined by the

equations b c c a a b a ' , b , c [a b c] [a b c] [a b c]

× × × ′ = = = r r r r r r r r r

r r r r r r r r r are called

the reciprocal system of vectors to the given vectors

a ,b, c r r r .

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8. Application of vector in geometry: (a) Vector equation of a straight line passing through two

points a and b r r is r a t(b a) = + −

r r r r

(b) Vector equation of a plane passing through the point

a, b,c is r (1 s t)a sb tc = − − + + r r r r r r r

or r. (b c c a a b) [a b c] × + × + × = r r r r r r r r r r

(c) Vector equation of a plane passing through the point a r

and perpendicular to n is r .n a .n = r r r r r

(d) Perpendicular distance of a point P(r) from a line passing through a r and parallel to b

r is given by

1/2 2

2 | ( r a) b | ( r a) .b PM (r a) | b | | b |

− × − = = − −

r r r r r r r r

r r

(e) Perpendicualr distance of a point P(r) from a plane

passing through a r and parallel to b and c r r is given by

( ) ( ) r a . b c P M

b c

− × =

×

r r r r

r r

STATISTICS 1. Mean deviation for ungrouped data

i | x x | M.D.(x)

n

− =

∑ r , i | x M |

M.D.(M) n

− =

∑

2. Mean deviation for grouped data

i i f | x x | M.D.(x)

N

− =

∑

i i

i

f | x M | M.D.(M) , where N f

N

− = =

∑ ∑

3. Variance and standard deviation for ungrouped data

2 2 2 i i

1 1 (x x) , (x x) n n

σ = − σ = − ∑ ∑

4. Variance and standard deviation of a discrete frequency distribution

2 2 2 i i i i

1 1 f (x x) , f (x x) N N

σ = − σ = − ∑ ∑

5. Variance and standard deviation of a continuous frequency distibution

2

i i 2 2 2

i i i i

f x 1 1 f (x x) , f x N N N

σ = − σ = −

∑ ∑ ∑

6. Coefficient of variation (C.V.) = 100,x 0 x σ

× ≠

For series with equal means, the series with lesser standard deviation is more consistent or less scattered.Q function of x or constant

ORIGINATIVE