Jr.intermediatr MPC CDF

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YEVA Ja

S R A M

YAY

AO

NARAYANA

YEARS

OF EXCELLENCE

JUNIOR COLLEGES

Jr. Intermediate - MPC 2+2 Sina8&8

%12 X)

CDF HOW TO USe

Jr. INTERMEDIATE INDEX Page No.

NAME OF THE SUBJECT

1 - 16

MATHEMATICS-IA

MATHEMATICcs-IB

17- 33

PHYSICS

34 - 63

64- 106

CHEMISTRY

NARAYANA JUNIOR COLLEGES

2019-20

MPC

CDF

ANDHRA PRADESH

lay1N CDF 2019 MPC JR

&% TELANGANA NARAYANA

JR MATHS-IA

CDF MATERIAL

FUNCTIONS SET:

A well defined collection of objects is called a set.

2

FUNCTION

Aand B are two non-empty sets. Then a relation f from A to B, which associates every clement of A to a unique element of B is called a Function.

3

DOMAIN, CO-DOMAIN :

4.

IMAGE , PRE-IMAGE

f:A’ B is a function. Then A = Domain and B = Co-domain

:f:A’ Bis afunction if f(a) =b for ae A,be B, then "6' is im age of 'a'and 'a' is pre-image of 'b'. The set of all images of the elements ofA in B is called a Range.

RANGE

Range of fccodomain of f. 6

INJECTION (OR) ONE ONE FUNCTION

:f:A’ Bis one -one

a,a,EA,f(a)=f(4)’a =4 7.

SURJECTION (OR)

ONTOFUNCTION

:f:AB is onto if VbeB 2 ae ATf(a)=b or -f:A’B is onto Range off= codomain of f (Mar-2017) (May-2016) ( TS Mar-2016) (TS Mar-2017)

8.

BIJECTION (OR) ONE-ONE ONTO

: If f:A’ Bis both one-one and onto, then f is called aBijection.

FUNCTION

9

10.

IDENTITY

Afunctionf:A’A is defined by f()=x,VxeA is called

FUNCTION

Identity function and is denoted by 1,

INVERSE FUNCTION

: If

f:A’B is a bijection, then f:B’A defined

as(b)=u 11,

CONSTANT FUNCTION :

f(a)=b, VbeB is called inverse function.

Afunction f:A’ Bis said to be constant if f(x)=C (a constant) VreA

12. lay2N CDF.

EVEN AND ODD FUNCTIONS : Let f:A’R be afunction i) If f(-r)= f(x), xeA then f is called an even fucntion ii) If f(-x)=-f(), VreA then f is called an odd fucntion

2019 MPC

13. GREATEST INTEGER FUNCTION

For any real number x, we denote by [x] the greatest integer less than or equal to x

JR

ie, [x]=n, if nab,then ax?+ 2hxy + by'= 0(second degree homogenous equation in x and y) represents a pair

of lines passingthrough the origin, which are ax+ (h+h'-ab y=0 and

ax +(h-Vh-ab y=0 2.

It m, and m, are the slopes of the lines represented by ax? +2hxy + by² -0 then -2h

i) Sum of the slopes : m,tm, 3

-2h

, m,m, =

ax +2hxy + by =(1,x +my)(1,x +m,y) Comparing the coefficients of like terms on either side ’1, = a, m,m, = b,

5.

Ja+b|

Ja-b)' +4h?

tan=

atb

V(a-b)' +4h'

2yh'-ab sin = 2/h'-ab la+b|

Ja-b)+44

The angle betwcen the pair of lines represented by the equation ax? +2hxy + by' +2gx +2fy +c=0 is a+b

cos=t

J(a-b) +4h' 1.

I,m, +l, m, = 2h

i)If '9 is angle between the lines represented by ax' +2hxy +by' =0 then cos=+

ii) If '9 is acute then Cost

6

b

ax +2hxy +hy' =(y-m,r)(y-m,x) Comparing thecoefficients oflike terms on cither side m+ m, =

4

ii) Product of the slopes : m m,

b

tan=t

2Vh'-ab sin = + a+b

2yh-ab

Va-b)' +44

(EAMCET-99)

The pair of lines ar + 2hxy +by² =0 represents a pair of perpendicular lines then coefficient of x +coefficient of y' =0(i.e, a +b= 0) If the pair of lines ax + 2hry+ by =0represents coincident lines then h = ab

9

The equation ofbisectors ofangle between the lines a,x+hy+c =0, a,x+ b,y +c, =0is a,x +bytc

a,X+b,y +c

Ja; +b; 10.

If theequat ion ax' + 2hxy +by' =0represents a pair of distinct (i.e., intersecting) lines then the

combined eqation ofthe pair of bisectors of the angles between the lines is A(r-y')= (a-b)*y 11.

The equation of the pair of lines passing through (x,, y,) and parallel to ar' +2hxy +by* =0

isa(x-x,' +2h(x-x,)(y-y,)+b(y-}=0 NARAYANA

Page.No.23

JR MATHS-IB

12. 13.

CDF MATERIAL

The equation ofthe pair of lines passing through (X ) and perpendicular toax' +2hxy +by' =0 is b{r-x,)- 2h(x-x,)(y-y%)+ a(y- y,) =0 ax* +2hxy +by' +2gx + 2 fy +c =(,x+ my+ n)(x+ m,y+n,)

Conparing the cocficients of ike terns on either side 1,1, =a

m,m, =b ; nn, =c; Im, +I,m, = 2h ; In, +l,n, =2g ; m,n, + m,n, =2/

14.

ax +2hxy +by' +2gr+ 2fy +c =(L+my +n, )(x +my +n,) Comparing the coefficients of like terms on either side I' =a,m' =b,n,n, = c, 2im =2h,1(n, +n,) = 2g, m{n, + n,)=2/

1s.

If the second degree equation S= ax? + 2hxy + by² + 2gx + 2fy + c-0 in two variables x and y

represents a pair of straight lines then

i) A= abe +2fgh

-af'-bg- ch' =0

(EAMCET-82,93,96,2000, 08)

and i) h'> ab,g' ac andf' be

16. If ar + 2hry +by' +2g +2fy +c=0 represents a pair of intersecting lines, then their point of intersection is

hf-bg gh-af (EAMCET-97,2000, 02)

ab-h ab-h?

DIRECTION COSINES AND DIRECTION RATIOS 1.

If a given directed line makes angles a, B,y with positive direction of the axes of x, y and z

respectively then cos a, cos B, cos y are called direction cosines (d.c's) of the line and these are usually denoted by l, m, n. 2.

If the direction cosines of a directed line AB are l, m, n then the direction cosines of the directed line BA are -I, -m, -n.

3.

Direction cosines of

) x-axis are (1, 0, 0)

4.

i) y - axis are (0, 1, 0) ii) z-axis are (0, 0, 1). If (. m, n) are the direction cosines of aline then P+ m+ n= 1, i.e cos'a + cos²B +cos²y =1.

5.

If a. B.y are the angle made by a directed line with the positive direction of the coordinate axes, then

6.

7.

(EAMCET-2001)

sin'a +sin'ß +sin'y=2. Direction ratios of the line joining the points A(x, y,, Z), B(X, Y, 2) are (x,-X Y,y, 4-2) If (a, b, c) are direction ratios of a line then direction cosines of the line.

8.

ta

tb

te

Va'+b' +e' va' +b' +c' va' +h' +'

are

If(U.. m,. n,) and (U, m, n,) are direction cosines of two lines and ''is an angle between them, then

NARAYANA

Page. No.24

JR MATHS-IB

CDF MATERIAL

cosd +(I, +m,m, tn,n,). 9.

(EAMCET- 01, 02, 03)

Let (, m, n) and (/, m,, n,) be d.c's of two lines. Then )The lines are paralle) >I=

m,

n,

ii) The line are perpendicular |, tm,m, +n,n, = 10. If0'is an angle betwcen two lincs whose d.r. 's are (a,, b,, c,) and (a,, b,, C,) then a,a, +bb, +e,c, i) cos =t

Va +bË+ Va; +bË +eË iü) The lines are parallel

a,

ii) The lines,are perpendicular 11.

b, a, a, +bb, +c, C, =U.

If A(X,, y,, z), B(x,, y, z,) then the projection of AB on a) i) x - axis is |x-x, | ii) y- axis isy,-y, b) i)) xy plane is

ii) z axis is |Z,- Zl

V,-)+(W;-)' ) yz plane is

i) x plane is J(z,-) +{x,-}

1.

2. 3.

3D-GEOMETRY The distance between twopoints A(4M,4), B(ay,z,) is AB=(-x}+(y-} +{3-3} units

Midpoint of 4(43:4)&B(xz},)is

2

,

2

If P°divides the line segment joining A(,.,z,), B(*,, V,,2,) intheratio /:m internally then P = tmz, y, tmy, lz, + mz, I+ m

4

1+ m

1+ m

If P divides the line segment joining A(x,, y,,z,), B(x,, J,,z,) in the ratio /:m externaly

- mx, ly,- my b, - mz, then P= Lx,|-m |- m |-m 5.

YZ-plane divides the line joining A(4}2), B(x, ),»2,) in the ratio -x :

6.

XY-plane divides the line joining A(x,, y,, z,), B(x,, ',,z, )in the ratio - z, :2,

7

ZX-plane divides the line joining A4,,},,4), B(x,, y,,z,) in the ratio -yi P(x,y,z) diides the line segment A(, y,2).B(,,y,,4]inthe ratio x, -x:-x,

8

NARAYANA

Page.No.25

MATERIAL CDF

3

3

Xyz) 2,).C((

y,, B(M,, z), A(x,V, are vertices whose triangle ofa centroid

JR MATHS-IB 3

The 9

then

C(K,.),z,) D[X,Ya4)

B(,,y,,*4,), A(.J.;),

fourth The

11.

is

to

z)changed

When(X,

12.

are C B, A, If

13.

i)

The

2.

i)

is k where

3.

The

4.

is a,b,c

5.

The

6.

b a

and origin

The whose ii) i)

7.

8.

9

Page.No.26

are is i s (EAMCET-02) k=0, +d=0 'c' the d.r's ).2, from is y, whose intercept =0 + C(x,y} (x, cz cz BC units + +d + of ax+by line -z +b' +e' Va' cz k by co-ordinates= 0 2,), AB 'p' = + the = and + y y V, bv to + of =p. is ax is B(x, to b' CA form distance zx-plane zx-plane + parallel perpendicular nz ax or z,), intercept the PLANE THE the + =0. plane -Z,), AB +my axes a ii) of nz ii) and is A(x, = at is+d=0 (x,,J,,z,) +d=0 the +2, CA is Ix + y are of and are to which my translation BC+ 'a', 4 i s parallelogram (x,,Z) Ix+ ) vertices 0 k +y,-y,, (,m,n) Z+I z, cz intercept = = form or , x + x tcz is through (,V, CA by is is -y,)+e(z-z)=0 a(x-x,)+B(y origin by +e(z-z,) =0 a(x-x,)+b(y-y,) by = whose Y+k,z normal yz-plane yz-plane cosines + from to parallel ax+ O'(h,k,i) =BC ax through -x +X,-X,,)} passing ofa the having plane tetrahedron4 to the direction = + perpendicular through ii) ii) parallel +h,y AB vertices in the plane passing plane plane iff to plane =(x, shifted X to the plane origin plane consecutive the x= collinear +d the a plane the 0z= the k D of Ja'+b'+c' of has = the then to of of The equation to isz the i s vertex Centroid equation from of tcz, ofa origin equation aof normal is equation equation Equation of Y,Z) xy-plane xy-plane Equation Equation Distancelength +by, constant. Ifthree NARAYANA The ax, 10.

1.

JR MATHS-IB CDF MATERIAL 10.

ld,-d, The distance between parallel planes ax + hv+cz+d. =0, ax + by +cz +d, =0 S Va' +b'+?

11.

1) The angle between any two

intersecting planes is the angle between their normals ii) Angle between the planes a,x+ bv+cz+ d, =0 and a,x +b,y +c,z +, 0 cos

a,a, +b,b, tcc,

=+

Ja' tb, +e a,'+b, te,' 12.

)The equations a,x + b,y + c,z+ d, = 0 and a,x + b,y +c,z + d, =0 represent the same plane ift a, :b, : c, :d, = a, :b, : ,:d,. i) parallel a, :b, :, = a, :b, i C; i) perpendicular a, a, +b, b, +c, c,=0.

LIMITS& CONTINUITY lim:

x-a"

3.

lim

5

lim

sin x

4. lim X’0

=1

6. km X0

7.

8. lim

X’0

Sin ax =a

tan ac

a-1

= log, a

10. lin1+

9

lim

11.

m-n

n

I’0

tan x

m

2. lim

X-a

=e

log.(1+x) =1 .

Continuity of afunction f(ox):

12.

Afunction f(x) is said to be continuous at x=aif ’

L j()exists and is equal to fia)

Ltf(x) = f(a) = Lt f(x)

DIFFERENTIATION First principle: If f is differentiable at x'then f'(x) = Lt

f(x+ h)-f(x)

h’0

1.

Thederivative of a function f" at x = a is defined by

f'(a) =lim h’0

NARAYANA

f(a + h) -f(a) h

= lim

f(x) - f(a) X - a

Page.No.27

CDF MATERIAL

JR MATHS-IB 2.

(k is constant) d

3.

dx

iv)

-(kx) k=klk = (kis constant)

4.

dx 5.

6.

dx

d

dd (og,1)=l X

7.

-(a')=a' log,a

dx

Derivatives of trignometric functions_: d 8.

dx d

9.

dz d

10.

11.

dx

dx d

12.

dx d

13.

(sinx) =cosx

(coSx) =-sinx (tanx) = sec'x -(cotx) = -cOsec'x -(secx) = secx tanx

-(cosec x) = -cosecx cotx

dx

Derivatives of inverse trignometric functions: 1

14.

dx

(Sin'x)J -1

15.

16.

(EAMCET-99)

dx

dx

Tan'x)=1t -1

17.

(Cot'x+r

dx

NARAYANA

Page.No.28

JR MATHS-IB d

18.

dx

19.

dx

-(Cose'x)

dx d

21.

22.

23.

25.

of

hyperbolic functions

:

(sinhx)=coshx (coshx) =sinhx

d dx

(tanhx)=sech'x

dx

(sechx)=-sechx tanhx

d 24.

1

-(Se'x)

Derivatives 20.

CDFMATERIAL

dx

cosechx) =-cosechx cÑthí

dx

(coth x)-cosech'x

Derivatives of inverse hyperbolic functions : 26.

1

dx (Sinh'x)=

27. d

28.

dx d

29.

dx

(Tanh'x) -

1

-1

(Sechx)* -* -1

30.

dx

(Coscch')i 1

31.

dx

(Coth 's)F

Let u, V, w be the functions of x and whose derivatives exists : d

32.

33.

-(u±v)= dx -(u) dx

(uv)=

NARAYANA

d dx

(v)

d(u) dx

dx

(or) (u.v) =u.v + v.u'

Page.No.29

JR MATHS-IB 34.

CDF MATERIAL

du

35,

dx

V.u

dy

(or)

(uvw)uyd(w) + vw d{u) + wu d(v) dx

dx

(or) (u.v.w) u.v.w + v.wu + w.u.v

dy

36. If x= f(),

y=g() then

dy dt dx dx dt

ERRORS AND APPROXIMATIONS 1.

If y =f(r) be afunction then i) change (error) in y = Ay

.Ay =f(x+ Ar)-f() ii) Differential of y =dy

:.dy =f' (x)Ar iii) Relative errorin

Ay

iv) Percentage error in y =x100

v) Approximationformula flx+Ax) f(r)+f()Ar 2.

i)Area of the square

, where 'x' is side ofa square

i) volume of the cube =, where '*' is side ofa cube iü) Surface area ofthe sphere 4', where »r' is radius ofa sphere 4

iv)Volume ofthe sphere =r , where 'r' is radius ofasphere 3 v) Volume of thecone =;r'h, where 'r' is the radius, 'h' is the height ofcone vi) Volume ofcylinder =zr'h,where 'r' is base radius,'h' is the height ofcylinder vi) Surface area ofcylinder = 2Irh, where 'r' is base radius, 'h' is the height ofcylinder

TANGENTS AND NORMALS Equations of the tangent and normal to the curve y = f (x) : 1.

The slope of the tangent to the curve y = f(x) at a point P (x,, y, ) is m=

2.

The equation of the tangent at P(x,, y) is y-y, =m(x-x)

3.

The equation of the normal at P (x, y,) is

NARAYANA

dy

dx r()

Page. No.30

CDFMATERIAL

JR MATHS-IB Angle betwecn the two curves : 4.

Let y = fx), y* g(x) be two curves intersecting at a point "P' and let m,, m, bhe the siopes tagens to the twO curves at P' respectively, IrA'is theacute angle between the curves at P then

m, -m,

tan =

|1+m,m,| i)) If m, = m, then the two curves touch cach other at P

i) If mm, = -1 then the two curves cuts orthogonally at 'P'" y=f (x),

P(x,y,) TANGENT

NORMAL

T

5.

N

sub-normal

sub-tangent

P meets the Let P (x,, y) be the any point on the curve y=f(x). Let the tangent and normal at

X-axis at M and N respectively. Draw PO perpendicular to the x-axis Then i) The length of the tangent PT=

y, V1 +m m

i) The length of the normal PN=|y, V1+m' ii) The length of the sub tangent

m

iv) The length of the sub normal QN =|y,.m

Where

RATE MEASURE 1.

The approximate value of

Ay

When Lt Ay exists it is denoted by Ar-0 Ar

2

tends to zero, it is called the rate of change of y with respect to x.

dy dx

is the rate of change of y with respect to x

iedr-= Lt

Ay

Velocity : Consider a particle moving along a straight line OX. Let OP =s be the distance travelled by the particle in time t' measured from a fixed point '0' along OX. Let the particle further travel a small

NARAYANA

Page.No.31

CDF MATERIAL

v. by

denotcd

is and 0

the is '1' time

V=

at velocity

Lt t= time

unit/sec A A time -as As A of ds dt As interval is of A At interval limit As A-s0 snall the a in in AS speed PO=

at

Ihe

JR MATHS-IB average velocity distance The

..

/ser2 unide?t d's

THEOREM conditions following VALUE =0

that such b) (a, conditionsce

the LAGRANGE'SMEAN satisfying f'(c) sec? /radians

d'0dt

point that function such followingone b) atleast ce(a,: abe THEOREM the exists f:la.b]-’R one satisfying there atleast

VALUE function then

exists

f(b) ROLLE'S ROLLES =

is is fla) f f i)) ii) im)

2.

f:(a, If

is f i)

is f ü)

an

abe

el el ,, x,,x, l el all e all for x,x, for increasing functionfunction X,X, f()