55099833 math formula sheet aieee
TRANSCRIPT
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
QUADRATIC EQUATION & EXPRESSION
1. Quadratic expression :
A polynomial of degree two of the form ax2 + bx + c, a 0 iscalled a quadratic expression in x.
2. Quadratic equation :
An equation ax2 + bx + c = 0, a 0, a, b, c R has two and
only two roots, given by
= + b b ac
a
2 4
2and =
b b ac
a
2 4
2
3. Nature of roots :
Nature of the roots of the given equation depends upon the
nature of its discriminant D i.e. b2 4ac.
Suppose a, b, c R, a 0 then
(i) If D > 0 roots are real and distinct (unequal)
(ii) If D = 0 roots are real and equal (Coincident)
(iii) If D < 0 roots are imaginary and unequal i.e.
non real complex numbers.
Suppose a, b, c Q a 0 then
(i) If D > 0 and D is a perfect square roots are rational
& unequal
(ii) If D > 0 and D is not a perfect square roots are
irrational and unequal.
or a quadratic equation their will exist exactly 2 roots real
or imaginary. If the equation ax2 + bx + c = 0 is satisfied for
more than 2 distinct values of x, then it will be an identity &
will be satisfied by all x. Also in this case a = b = c = 0.
PAGE # 2PAGE # 1
4. Conjugate roots :
Irrational roots and complex roots occur in conjugate pairs
i.e.if one root + i, then other root i
if one root + , then other root
5. Sum of roots :
S = + = ba
= Coefficient of xCoefficient of x2
Product of roots :
P = =c
a=
cons t term
Coefficient of x
tan2
6. ormation of an equation with given roots :
x2 Sx + P = 0
x2 (Sum of roots) x + Product of roots = 0
7. Roots under particular cases :
or the equation ax2 + bx + c = 0, a 0
(i) If b = 0 roots are of equal magnitude but of opposite
sign.
(ii) If c = 0 one root is zero and other is b/a
(iii) If b = c = 0 both roots are zero
(iv) If a = c roots are reciprocal to each other.
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 4PAGE # 3
(v) If a > 0, c < 0 or a < 0, c > 0 roots are of opposite
signs
(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 both
roots are ve
(vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0 both
roots are +ve.
8. Symmetric function of the roots :
If roots of quadratic equation ax2 + bx + c, a 0 are and ,then
(i) ( ) = ( ) + 2 4 = b ac
a
24
(ii) 2 + 2 = ( + )2 2 =b ac
a
2
2
2
(iii) 2 2 = ( + ) ( ) + 2 4 = b b ac
a
2
2
4
(iv) 3 + 3 = ( + )3 3( + ) = b b ac
a
( )2
3
3
(v) 3 3 = ( ) [2+ 2 ]
= ( ) + 2 4 [2+ 2 ]
=( )b ac b ac
a
2 2
3
4
(vi) 4 + 4 = (2 + 2)2 222
={( + )2 2}2 222 = b aca
2
2
2
2.HG IKJ 22
2ca
(vii) 4 4 =(2 + 2) (2 2) = b b ac b ac
a
( )2 2
4
2 4
(viii) 2 + + 2 =( + )2 =b ac
a
2
2
+
(ix)
+
=
2 2+=
( )
+ 2 2
(x)
.
HG
I
KJ
2
+
.
HG I
KJ
2
=
4 4
2 2
+=
[( ) ]b ac a c
a c
2 2 2 2
2 2
2 2
9. Condition for common roots :
The equations a1
x2 + b1
x + c1= 0 and a
2x2 + b
2x + c
2= 0
have
(i) One common root ifb c b cc a c a
1 2 2 1
1 2 2 1
=
c a c aa b a b
1 2 2 1
1 2 2 1
(ii) Both roots common ifa
a1
2=
b
b1
2=
c
c1
2
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 PAGE # 6PAGE # 5
10. Maximum and Minimum value of quadratic expression :
In a quadratic expression ax2 + bx + c = a xb
a
D
a+.HG
IKJ
L
NMMO
QPP2 4
2
2 ,
Where D = b2 4ac
(i) If a > 0, quadratic expression has minimum value
4
4
2ac b
a
at x =
b
a2and there is no maximum value.
(ii) If a < 0, quadratic expression has maximum value
4
4
2ac b
a
at x =
b
a2and there is no minimum value.
11. Location of roots :
Let f(x) = ax2+ bx + c, a 0 then w.r.to f(x) = 0
(i) If k lies between the roots then a.f(k) < 0
(necessary & sufficient)
(ii) If between k1
& k2
their is exactly one root of k1, k
2
themselves are not roots
f(k1) . f(k
2) < 0 (necessary & sufficient)
(iii) If both the roots are less than a number k
D 0, a.f(k) > 0,b
a2< k (necessary & sufficient)
(iv) If both the roots are greater than k
D 0, a.f(k) > 0,b
a2> k (necessary & sufficient)
(v) If both the roots lies in the interval (k1, k
2)
D 0, a.f(k1) > 0, a.f(k2) > 0, k1 0, xy > 1
(iii) tan1x tan1y = tan1x y
xy
+
.HG
IKJ1 ; if xy > 1
(iv) tan1x tan1y = + tan1x y
xy
+
.HG
IKJ1 ; if x > 0, y < 0, xy < 1
(v) tan1x + tan1y + tan1z = tan1x y z xyz
xy yz zx
+ +
.HG
IKJ1
(vi) sin1x sin1y = sin1 x y y x1 12 2
L
NM
O
QP;
if x,y 0 & x2 + y2 1
(vii) sin1x sin1y = sin1 x y y x1 12 2 LNM
OQP ;
if x,y 0 & x2 + y2 > 1
(viii) cos1x cos1y = cos1 xy x ym 1 12 2
LNMOQP ;
if x,y > 0 & x2 + y2 1
(ix) cos1x cos1y = cos1 xy x ym 1 12 2
LNMOQP ;
if x,y > 0 & x2 + y2 > 1
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
6. Inverse trigonometric ratios of multiple angles
(i) 2sin1x = sin1(2x 1 2 x ), if 1 x 1
(ii) 2cos1
x = cos1
(2x2
1), if 1 x 1
(iii) 2tan1x = tan12
1 2x
x
.HG
IKJ= sin1
2
1 2x
x+
.HG
IKJ = cos1
1
1
2
2
+
.HG
IKJ
x
x
(iv) 3sin1x = sin1(3x 4x3)
(v) 3cos1x = cos1(4x3 3x)
(vi) 3tan1x = tan13
1 3
3
2
x x
x
.
HG
I
KJ
PROPERTIES & SOLUTION O. TRIANGLE
Properties of triangle :
1. A triangle has three sides and three angles.In any ABC, we write BC = a, AB = c, AC = b
BB
CC
A
A
a
bc
and BAC = A, ABC = B, ACB = C
2. In ABC :
(i) A + B + C =
(ii) a + b > c, b +c > a, c + a > b
(iii) a > 0, b > 0, c > 0
3. Sine formula :
a
Asin=
b
Bsin=
c
Csin= k(say)
orsinA
a=
sinB
b=
sinC
c= k (say)
4. Cosine formula :
cos A =b c a
bc
2 2 2
2
+
cos B =c a b
ac
2 2 2
2
+
cos C =
a b c
ab
2 2 2
2
+
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
5. Projection formula :
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
6. Napier's Analogies :
tanA B
2=
a b
a b
+cot
C
2
tanB C
2=
b c
b c
+cot
A
2
tanC A
2=
c a
c a
+cot
B
2
7. Half angled formula - In any ABC :
(a) sinA
2=
( ) ( )s b s c
bc
sinB
2=
( ) ( )s c s a
ca
sinC
2=
( ) ( )s a s b
ab
where 2s = a + b + c
(b) cosA2
=s s a
bc
( )
cosB
2=
s s b
ca
( )
cosC
2=
s s c
ab
( )
(c) tanA
2=
( ) ( )
( )
s b s c
s s a
tanB
2=
( ) ( )
( )
s c s a
s s b
tanC
2=
( ) ( )
( )
s b s a
s s c
8. , Area of triangle :
(i) =1
2ab sin C =
1
2bc sin A =
1
2ca sin B
(ii) = s s a s b s c( ) ( ) ( )
9. tan A2
tan B2
= s cs
tanB
2tan
C
2=
s a
s
tanC
2tan
A
2=
s b
s
10. Circumcircle of triangle and its radius :
(i) R =a
A2sin=
b
B2sin=
c
C2sin
(ii) R =abc
4Where R is circumradius
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
11. Incircle of a triangle and its radius :
(iii) r =
s
(iv) r = (s a) tanA
2= (s b) tan
B
2= (s c) tan
C
2
(v) r = 4R sinA
2sin
B
2sin
C
2
(vi) cos A + cos B + cos C = 1 +
r
R
(vii) r =
aB C
A
sin sin
cos
2 2
2
=b
A C
B
sin sin
cos
2 2
2
=c
B A
C
sin sin
cos
2 2
2
12. The radii of the escribed circles are given by :
(i) r1
=
s a, r
2=
s b, r
3=
s c
(ii) r1
= s tanA
2, r
2= s tan
B
2, r
3= s tan
C
2
(iii) r1 = 4R sin
A
2 cos
B
2 cos
C
2 ,
r2
= 4R cosA
2sin
B
2cos
C
2,
r3
= 4R cosA
2cos
B
2sin
C
2
(iv) r1
+ r2
+ r3
r = 4R
(v)1
1r+
1
2r+
1
3r=
1
r
(vi)1
12r
+1
22r
+1
32r
+12r
=a b c2 2 2
2
+ +
(vii)1
bc+
1
ca+
1
ab=
1
2Rr
(viii) r1r2 + r2r3 + r3r1 = s
2
(ix) = 2R2 sin A sin B sin C = 4Rr cosA
2cos
B
2cos
C
2
(x) r1
=
aB C
A
cos cos
cos
2 2
2
, r2
=
bC A
B
cos cos
cos
2 2
2
,
r3
=
cA B
C
cos cos
cos
2 2
2
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
(ii) d = h (cot cot)
d
h
HEIGHT AND DISTANCE
1. Angle of elevation and depression :
If an observer is at O and object is at P then XOP is
called angle of elevation of P as seen from O.
If an observer is at P and object is at O, then QPO is
called angle of depression of O as seen from P.
2. Some useful result :
(i) In any triangle ABC if AD : DB = m : n
ACD = , BCD = & BDC =
then (m + n) cot = m cot ncot
C
B
B
A
A
Dm n
= ncotA mcotB [m n Theorem]
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MATHS FORMULA POCKET BOOK
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
POINT
1. Distance formula :
Distance between two points P(x1, y
1) and Q(x
2, y
2) is
given by d(P, Q) = PQ
= ( ) ( )x x y y2 12
2 12 +
= ( ) ( )Difference of x coordinate Difference of ycoordinate2 2+
Note : (i) d(P, Q) 0
(ii) d(P, Q) = 0 P = Q
(iii) d(P, Q) = d(Q, P)
(iv) Distance of a point (x, y) from origin
(0, 0) = x y2 2+
2. Use of Distance ormula :
(a) In Triangle :
Calculate AB, BC, CA
(i) If AB = BC = CA, then is equilateral.
(ii) If any two sides are equal then is isosceles.
(iii) If sum of square of any two sides is equal tothe third, then is right triangle.
(iv) Sum of any two equal to left third they do notform a triangle
i.e. AB = BC + CA or BC = AC + AB
or AC = AB + BC. Here points are collinear.
(b) In Parallelogram :Calculate AB, BC, CD and AD.
(i) If AB = CD, AD = BC, then ABCD is a parallelo-gram.
(ii) If AB = CD, AD = BC and AC = BD, then ABCDis a rectangle.
(iii) If AB = BC = CD = AD, then ABCD is a rhombus.
(iv) If AB = BC = CD = AD and AC = BD, then ABCD
is a square.
(C) or circumcentre of a triangle :
Circumcentre of a triangle is equidistant from verticesi.e. PA = PB = PC.
Here P is circumcentre and PA is radius.
(i) Circumcentre of an acute angled triangle is in-side the triangle.
(ii) Circumcentre of a right triangle is mid point ofthe hypotenuse.
(iii) Circumcentre of an obtuse angled triangle isoutside the triangle.
3. Section formula :
(i) Internally :
AP
BP=
m
n= , Here > 0
A(x , y )1 1 B(x , y )2 2P
m n
Pmx nx
m n
my ny
m n2 1 2 1+
+
+
+
.
H
G I
K
J,
(ii) Externally :
AP
BP=
m
n=
A(x , y )1 1
B(x , y )2 2
P
nm
Pmx nx
m n
my ny
m n2 1 2 1
.HG
IKJ,
(iii) Coordinates of mid point of PQ arex x y y1 2 1 2
2 2
+ +.HG
IKJ,
(iv) The line ax + by + c = 0 divides the line joining thepoints
(x1, y
1) & (x
2, y
2) in the ratio =
( )
( )
ax by c
ax by c1
2 2
+ +
+ +
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MATHS FORMULA POCKET BOOK MATHS FORMULA POCKET BOOK
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
(v) or parallelogram midpoint of diagonal AC = mid pointof diagonal BD
(vi) Coordinates of centroid G
x x x y y y1 2 3 1 2 3
3 3
+ + + +.
HG
I
KJ,
(vii) Coordinates of incentre I
ax bx cx
a b c
ay by cy
a b c
1 2 3 1 2 3+ +
+ +
+ +
+ +
.HG
IKJ,
(viii) Coordinates of orthocentre are obtained by solving theequation of any two altitudes.
4. Area of Triangle :
The area of triangle ABC with vertices A(x1, y
1), B(x
2, y
2)
and C(x3, y
3).
=1
2
x y
x y
x y
1 1
2 2
3 3
1
1
1
(Determinant method)
=1
2
x y
x y
x y
x y
1 1
2 2
3 3
1 1
=1
2[x
1y
2+ x
2y
3+ x
3y
1 x
2y
1 x
3y
2 x
1y
3]
[Stair method]Note :
(i) Three points A, B, C are collinear if area of triangleis zero.
(ii) If in a triangle point arrange in anticlockwise thenvalue of be +ve and if in clockwise then will beve.
5. Area of Polygon :
Area of polygon having vertices (x1, y
1), (x
2, y
2), (x
3, y
3)
........ (xn, y
n) is given by area
=1
2
x y
x y
x y
x y
x y
n n
1 1
2 2
3 3
1 1
M M. Points must be taken in order.
6. Rotational Transformation :
If coordinates of any point P(x, y) with reference to newaxis will be (x', y') then
xB yBx' cos sin
y' sin cos
7. Some important points :
(i) Three pts. A, B, C are collinear, if area of triangle iszero
(ii) Centroid G ofABC divides the median AD or BE or C inthe ratio 2 : 1
(iii) In an equilateral triangle, orthocentre, centroid,
circumcentre, incentre coincide.(iv) Orthocentre, centroid and circumcentre are always
collinear and centroid divides the line joining orthocentreand circumcentre in the ratio 2 : 1
(v) Area of triangle formed by coordinate axes & the line
ax + by + c = 0 isc
ab
2
2.
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
4. Position of a point with respect to a straight line :
The line L(xi, y
i) i = 1, 2 will be of same sign or of opposite
sign according to the point A(x1, y
1) & B (x
2, y
2) lie on same
side or on opposite side of L (x, y) respectively.
5. Equation of a line parallel (or perpendicular) to the line
ax + by + c = 0 is ax + by + c' = 0 (or bx ay + = 0)
6. Equation of st. lines through (x1,y
1) making an angle
with y = mx + c is
y y1=
m
m
tan
tan
1 m(x x
1)
7. length of perpendicular from (x1, y1) on ax + by + c = 0
is| |ax by c
a b
1 1
2 2
+ +
+
8. Distance between two parallel lines ax + by + ci= 0,
i = 1, 2 is| |c c
a b
1 2
2 2
+
9. Condition of concurrency for three straight lines
Li a
ix + b
iy + c
i= 0, i = 1, 2, 3 is
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= 0
10. Equation of bisectors of angles between two lines :
a x b y c
a b
1 1 1
12
12
+ +
+ =
a x b y c
a b
2 2 2
22
22
+ +
+
11. amily of straight lines :
The general equation of family of straight line will be written
in one parameter
The equation of straight line which passes through point ofintersection of two given lines L
1and L
2can be taken as
L1 + L2 = 0
12. Homogeneous equation : If y = m1x and y = m
2x be the
two equations
represented by ax2 + 2hxy + by2 = 0 , then m1+ m
2= 2h/b
and m1m
2= a/b
13. General equation of second degree :
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent a pair of
straight line if
a h g
h b f
g f c= 0
If y = m1
x + c & y = m2
x + c represents two straight lines
then m1+ m
2=
2h
b, m
1m
2=
a
b.
14. Angle between pair of straight lines :
The angle between the lines represented by
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 or ax2 + 2hxy + by2 = 0
is tan = 22
h aba b
+( )
(i) The two lines given by ax2 + 2hxy + by2 = 0 are
(a) Parallel and coincident iff h2 ab = 0
(b) Perpendicular iff a + b = 0
(ii) The two line given by ax2 + 2hxy + by2 + 2gx + 2fy + c
= 0 are
(a) Parallel if h2 ab = 0 & af2 = bg2
(b) Perpendicular iff a + b = 0
(c) Coincident iff g2 ac = 0
13. Combined equation of angle bisector of the angle between
the lines ax2 + 2hxy + by2 = 0 is
x y
a b
2 2
=
xy
h
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MATHS FORMULA POCKET BOOK MATHS FORMULA POCKET BOOK
E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510E D U C A T I O N S
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
CIRCLE
1. General equation of a circle :x2 + y2 + 2gx + 2fy + c = 0
where g, f and c are constants(i) Centre of the cirle is (g, f)
i.e. .
HGIKJ
1
2
1
2coeff of x coeff of y. , .
(ii) Radius is g f c2 2+
2. Central (Centre radius) form of a circle :(i) (x h)2 + (y k)2 = r2 , where (h, k) is circle centre and
r is the radius.
(ii) x2 + y2 = r2 , where (0, 0) origin is circle centre and r is
the radius.
3. Diameter form : If (x1, y
1) and (x
2, y
2) are end pts. of a
diameter of a circle, then its equation is
(x x1) (x x2) + (y y1) (y y2) = 0
4. Parametric equations :
(i) The parametric equations of the circle x2 + y2 = r2 are
x = rcos, y = r sin ,
where point (r cos , r sin )
(ii) The parametric equations of the circle
(x h)2 + (y k)2 = r2 are x = h + rcos, y = k + rsin(iii) The parametric equations of the circle
x2 + y2 + 2gx + 2fy + c = 0 are
x = g + g f c2 2+ cos, y = f + g f c2 2+ sin
(iv) or circle x2 + y2 = a2, equation of chord joining 1 & 2 is
xcos 1 2
2
++ ysin
1 2
2
+= r cos
1 2
2
.
5. Concentric circles : Two circles having same centre C(h, k)
but different radii r1
& r2
respectively are called concentric
circles.
6. Position of a point w.r.t. a circle : A point (x1, y
1) lies
outside, on or inside a circle
S x2 + y2 + 2gx + 2fy + c = 0 according as
S1 x1
2 + y12 + 2gx
1+ 2fy
1+ c is +ve, zero or ve
7. Chord length (length of intercept) = 2 r p2 2
8. Intercepts made on coordinate axes by the circle :
(i) x axis = 2 g c2
(ii) y axis = 2 f c2
9. Length of tangent = S1
10. Length of the intercept made by line : y = mx + c with the
circle x2 + y2 = a2 is
2a m c
m
2 2 2
2
1
1
( )+
+or (1 + m2) |x
1 x
2|
where |x1 x
2| = difference of roots i.e.
D
a
11. Condition of Tangency : Circle x2 + y2 = a2 will touch the
line y = mx + c if c = a 1 2+ m
PAGE # 63 PAGE # 64
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12. Equation of tangent, T = 0 :
(i) Equation of tangent to the circle
x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y
1) is
xx1
+ yy1
+ g(x + x1) + f(y + y
1) + c = 0
(ii) Equation of tangent to the circle x2 + y2 = a2at any
point (x1, y
1) is xx
1+ yy
1= a2
(iii) In slope form : rom the condition of tangency for
every value of m.
The line y = mx a 1 2+ m is a tangent to the circle
x2 + y2 = a2 and its point of contact is
+
+
.HG
IKJ
am
m
a
m1 12 2,
(iv) Equation of tangent at (a cos , a sin ) to the
circle x2 + y2 = a2 is x cos + y sin = a.
13. Equation of normal :
(i) Equation of normal to the circle
x2 + y2 + 2gx + 2fy + c = 0 at any point P(x1, y
1) is
y y1 =
y f
x g
1
1
+
+ (x x1)
(ii) Equation of normal to the circle x2 + y2 = a2at any
point (x1, y
1) is xy
1 x
1y = 0
14. Equation of pair of tangents SS1= T2
15. The point of intersection of tangents drawn to the circle x2
+ y2 = r2 at point 1 & 2 is given as
r rcos
cos
,sin
cos
1 2
1 2
1 2
1 2
2
2
2
2
+
+
.
H
GGG
I
K
JJJ
16. Equation of the chord of contact of the tangents drawn
from point P outside the circle is T = 0
17. Equation of a chord whose middle pt. is given by T = S1
18. Director circle : Equation of director circle for x2 + y2 = a2 is
x2 + y2 = 2a2. Director circle is a concentric circle whose
radius is 2 times the radius of the given circle.
19. Equation of polar of point (x1, y
1) w.r.t. the circle S = 0 is T = 0
20. Coordinates of pole : Coordinates of pole of the line
lx + my + n = 0 w.r.t the circle x2 + y2 = a2 are .
HGIKJ
a l
n
a m
n
2 2
,
21. amily of Circles :
(i) S +S' = 0 represents a family of circles passing through
the pts. of intersection of
S = 0 & S' = 0 if 1
(ii) S + L = 0 represent a family of circles passing through
the point of intersection of S = 0 & L = 0
(iii) Equation of circle which touches the given straight line
L = 0 at the given point (x1, y
1) is given as
(x x1)2+ (y y1)2 + L = 0.
PAGE # 65 PAGE # 66
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(iv) Equation of circle passing through two points A(x1, y
1)
& B(x2, y
2) is given as
(x x1) (x x
2) + (y y
1) (y y
2) +
x yx y
x y
11
1
1 1
2 2
= 0.
22. Equation of Common Chord is S S1= 0.
23. The angle of intersection of two circles with centres C1
& C2and radii r
1& r
2is given by
cos =r r d
r r
12
12 2
1 22
+ , where d = C
1C
2
24. Position of two circles : Let two circles with centres C1, C
2
and radii r1, r
2.
Then following cases arise as
(i) C1C
2> r
1+ r
2 do not intersect or one outside the
other, 4 common tangents.
(ii) C1C
2= r
1+ r
2 Circles touch externally, 3 common
tangents.
(iii) |r1 r2| < C1 C2 < r1 + r2 Intersection at two realpoints, 2 common tangents.
(iv) C1C
2= |r
1 r
2| internal touch, 1 common tangent.
(v) C1
C2< |r
1+ r
2| one inside the other, no tangent.
Note : Point of contact divides C1
C2
in the ratio r1: r
2
internally or externally as the case may be
PAGE # 67 PAGE # 68
25. Equation of tangent at point of contact of circle is
S1 S
2= 0
26. Radical axis and radical centre :
(i) Equation of radical axis is S S1= 0
(ii) The point of concurrency of the three radical axis of
three circles taken in pairs is called radical centre of
three circles.
27. Orthogonality condition :
If two circles S x2
+ y2
+ 2gx + 2fy + c = 0and S' = x2 + y2 + 2g'x + 2f'y + c' = 0 intersect each other
orthogonally, then 2gg' + 2ff' = c + c'.
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PARABOLA
1. Standard Parabola :
Imp. Terms y2 = 4ax y2 = 4ax x2 = 4ay x2 = 4ay
Vertex (v) (0, 0) (0, 0) (0, 0) (0, 0)
ocus (f) (a, 0) (a, 0) (0, a) (0, a)
Directrix (D) x = a x = a y = a y = a
Axis y = 0 y = 0 x = 0 x = 0
L.R. 4a 4a 4a 4a
ocal x + a a x y + a a y
distance
Parametric (at2, 2at) ( at2, 2at) (2at, at2) (2at, at2)
Coordinates
Parametric x = at2 x = at2 x = 2at x = 2at
Equations y = 2at y = 2at y = 2at2 y = at2
y2 = 4ax
y2 = 4ax
x2 = 4ay
x2
= 4ay
PAGE # 69 PAGE # 70
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2. Special orm of Parabola
* Parabola which has vertex at (h, k), latus rectum l
and axis parallel to x-axis is
(y k)2 = l (x h)
axis is y = k and focus at h k+.HG
IKJ
l
4,
* Parabola which has vertex at (h, k), latus rectum l and
axis parallel to y-axis is
(x h)2 = l (y k)
axis is x = h and focus at h k, +
.HG
IKJ
l
4
* Equation of the form ax2 + bx + c = y represents
parabola.
i.e. y 4
4
2ac b
a
= a x
b
a+
.HG
IKJ22
,with vertex
.
HGIKJ
b
a
ac b
a2
4
4
2
, and axes parallel to y-axis
Note : Parametric equation of parabola (y k)2
= 4a(x h) are x = h + at2, y = k + 2at
3. Position of a point (x1, y
1) and a line w.r.t. parabola
y2 = 4ax.
* The point (x1, y1) lies outside, on or inside theparabola y2 = 4ax
according as y12 4ax
1>, = or < 0
* The line y = mx + c does not intersect, touches,
intersect a parabola y2 = 4ax according as
c > = < a/m
Note : Condition of tangency for parabola y2 = 4ax, we
have c = a/m and for other parabolas check disc. D = 0.
4. Equations of tangent in different forms :
(i) Point orm / Parametric form
Equations of tangent of all other standard parabolas
at (x1, y1) / at t (parameter)Equation Tangent at Parametric Tangent of 't'
of parabola (x1, y
1) coordinates't'
y2=4ax yy1=2a(x+x
1) (at2, 2at) ty=x+at2
y2=4ax yy1=2a(x+x
1) (at2, 2at) ty=x+at2
x2=4ay xx1=2a(y+y
1) (2at, at2) tx=y + at2
x2=4ay xx1=2a(y+y
1) (2at, at2) tx =y+at2
(ii) Slope form
Equations of tangent of all other parabolas in slope
form
Equation Point of Equations Condition of
of contact in of tangent Tangency
parabolas terms of in terms of
slope(m) slope (m)
y2 = 4ax am
am22,.HG IKJ y = mx + am c = am
y2 = 4ax .HG
IKJ
a
m
a
m22
, y = mx a
mc =
a
m
x2 = 4ay (2am, am2) y = mx am2 c = am2
x2 = 4ay (2am, am2) y = mx + am2 c = am2
5. Point of intersection of tangents at any two points
P(at12, 2at
1) and Q(at
22, 2at
2) on the parabola y2 = 4ax
is (at1t2, a(t
1+ t
2)) i.e. (a(G.M.)2, a(2A.M.))
6. Combined equation of the pair of tangents drawn from a
point to a parabola is SS' = T2, where S = y2 4ax,
S' = y12 4ax
1and T = yy
1 2a(x + x
1)
PAGE # 71 PAGE # 72
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7. Equations of normal in different forms
(i) Point orm / Parametric form
Equations of normals of all other standard parabolas
at (x1, y
1) / at t (parameter)
Eqn. of Normal Point Normals
parabola at (x1, y
1) 't' at 't'
y2 = 4ax yy1
=y
a1
2(xx
1) (at2, 2at) y+tx = 2at+at3
y2 = 4ax yy1
= ya1
2(xx
1) (at2, 2at) ytx = 2at+at3
x2 = 4ay yy1
= 2
1
a
x (xx1) (2at, at2) x+ty = 2at+at3
x2 = 4ay yy1
=2
1
a
x (xx1) (2at, at2) xty = 2at+at3
(ii) Slope form
Equations of normal, point of contact, and condition
of normality in terms of slope (m)
Eqn. of Point of Equations Condition of
parabola contact of normal Normality
y2 = 4ax (am2, 2am) y = mx2amam3 c = 2amam3
y2 = 4ax (am2, 2am) y = mx+2am+am3 c = am+am3
x2 = 4ay .HG
IKJ
22
a
m
a
m, y = mx+2a+
a
m2c = 2a+
a
m2
x2 = 4ay2
2
a
m
a
m,
.HG
IKJ y = mx2a
a
m2c = 2a
a
m2
Note :
(i) In circle normal is radius itself.
(ii) Sum of ordinates (y coordinate) of foot of normals
through a point is zero.
(iii) The centroid of the triangle formed by taking the foot
of normals as a vertices of concurrent normals of
y2 = 4ax lies on x-axis.
8. Condition for three normals from a point (h, 0) on x-axis
to parabola y2 = 4ax
(i) We get 3 normals if h > 2a
(ii) We get one normal if h 2a.
(iii) If point lies on x-axis, then one normal will be x-axis
itself.
9. (i) If normal of y2 = 4ax at t1
meet the parabola again
at t2
then t2
= t1
2
1t
(ii) The normals to y2
= 4ax at t1 and t2 intersect eachother at the same parabola at t
3, then
t1t2
= 2 and t3
= t1
t2
10. (i) Equation of focal chord of parabola y2 = 4ax at t1
is
y =2
1
1
12
t
t (x a)
If focal chord of y
2
= 4ax cut (intersect) at t1 andt2
then t1t2
= 1 (t1
must not be zero)
(ii) Angle formed by focal chord at vertex of parabola is
tan =2
3|t
2 t
1|
(iii) Intersecting point of normals at t1
and t2
on the
parabola y2 = 4ax is
(2a + a(t1
2
+ t2
2 + t1
t2
), at1
t2
(t1
+ t2
))
PAGE # 73 PAGE # 74
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11. Equation of chord of parabola y2 = 4ax which is bisected
at (x1, y
1) is given by T = S
1
12. The locus of the mid point of a system of parallel chords
of a parabola is called its diameter. Its equation is y =2a
m.
13. Equation of polar at the point (x1, y
1) with respect to
parabola y2 = 4ax is same as chord of contact and is given
by
T = 0 i.e. yy1
= 2a(x + x1
)
Coordinates of pole of the line l x + my + n = 0 w.r.t. the
parabola y2 = 4ax isn am
l l,
.HG
IKJ
2
14. Diameter : It is locus of mid point of set of parallel chords
and equation is given by T = S1
15. Important results for Tangent :
(i) Angle made by focal radius of a point will be twice
the angle made by tangent of the point with axis of
parabola
(ii) The locus of foot of perpendicular drop from focus to
any tangent will be tangent at vertex.(iii) If tangents drawn at ends point of a focal chord are
mutually perpendicular then their point of intersection
will lie on directrix.
(iv) Any light ray travelling parallel to axis of the parabola
will pass through focus after reflection through
parabola.
PAGE # 75 PAGE # 76
(v) Angle included between focal radius of a point and
perpendicular from a point to directrix will be bisected
of tangent at that point also the external angle will
be bisected by normal.
(vi) Intercepted portion of a tangent between the point
of tangency and directrix will make right angle at
focus.
(vii) Circle drawn on any focal radius as diameter will
touch tangent at vertex.
(viii) Circle drawn on any focal chord as diameter will touch
directrix.
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ELLIPSE
1. Standard Ellipse (e < 1)
x
a
y
b
2
2
2
21+ =
RS|T|
UV|W|
or a > b or b > a
Centre (0, 0) (0, 0)
Vertices (a, 0) (0, b)
Length of major axis 2a 2b
Length of minor axis 2b 2a
oci (ae, 0) (0, be)
Equation of directrices x = a/e y = b/e
Relation in a, b and e b2 = a2(1 e2) a2 = b2(1 e2)
Length of latus rectum 2b2/a 2a2/b
Ends of latus rectum .HG
IKJ
aeb
a,
2
.HG
IKJ
a
bbe
2
,
Parametric coordinates (a cos , b sin ) (a cos , b sin )
0 < 2
ocal radii SP = a ex1 SP = b ey1
S'P = a + ex1
S'P = b + ey1
Sum of focal radii SP + S'P = 2a 2b
Distance btn foci 2ae 2be
Distance btn directrices 2a/e 2b/e
Tangents at the vertices x = a, x = a y = b, y = b
Note : If P is any point on ellipse and length of perpendiculars
from to minor axis and major axis are p1
& p2, then |x
p|
= p1
, |yp| = p
2
p
a
p
b
12
222
2+ = 1
a > b
b > a
PAGE # 77 PAGE # 78
Imp. terms
Ellipse
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2. Special form of ellipse :
If the centre of an ellipse is at point (h, k) and the
directions of the axes are parallel to the coordinate axes,
then its equation isx h
a
c h22
+( )y k
b
2
2 = 1.
3. Auxillary Circle : The circle described by taking centre of
an ellipse as centre and major axis as a diameter is called
an auxillary circle of the ellipse.
If xa
2
2 +yb
2
2 = 1 is an ellipse then its auxillary circle is
x2 + y2 = a2.
Note : Ellipse is locus of a point which moves in such a
way that it divides the normal of a point on diameter of
a point of circle in fixed ratio.
4. Position of a point and a line w.r.t. an ellipse :
* The point lies outside, on or inside the ellipse if
S1
=x
a
12
2 +y
b
12
2 1 > , = or < 0
* The line y = mx + c does not intersect, touches,
intersect, the ellipse if
a2m2+ b2 < = > c2
5. Equation of tangent in different forms :
(i) Point form : The equation of the tangent to the
ellipsex
a
2
2 +y
b
2
2 = 1 at the point (x1, y1) is
xx
a
1
2 +yy
b
1
2 = 1.
(ii) Slope form : If the line y = mx + c touches the
ellipsex
a
2
2 +y
b
2
2 = 1, then c2 = a2m2 + b2. Hence, the
straight line y = mx a m b2 2 2+ always represents
the tangents to the ellipse.
Point of contact :
Line y = mx a m b2 2 2+ touches the ellipse
xa
2
2 +yb
2
2 = 1 at
+
+.HG IKJ
a ma m b
ba m b
2
2 2 2
2
2 2 2, .
(iii) Parametric form : The equation of tangent at any
point (a cos , b sin ) is
x
acos +
y
bsin = 1.
6. Equation of pair of tangents from (x1, y
1) to an ellipse
x
a
2
2 +y
b
2
2 = 1 is given by SS1 = T2
7. Equation of normal in different forms :
(i) Point form : The equation of the normal at (x1, y1)
to the ellipsex
a
2
2 +y
b
2
2 = 1 is
a x
x
2
1
b x
y
2
1= a2 b2.
PAGE # 79 PAGE # 80
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(ii) Parametric form : The equation of the normal to the
ellipsex
a
2
2 +y
b
2
2 = 1 at (a cos , b sin ) is
ax sec by cosec = a2 b2.
(iii) Slope form : If m is the slope of the normal to the
ellipsex
a
2
2 +y
b
2
2 = 1, then the equation of normal
is y = mx m a b
a b m
( )2 2
2 2 2
+.
The co-ordinates of the point of contact are
+
+
.HG
IKJ
a
a b m
mb
a b m
2
2 2 2
2
2 2 2, .
Note : In general three normals can be drawn from a point
(x1, y
1) to an ellipse
x
a
2
2 +y
b
2
2 = 1.
8. Properties of tangents & normals :
(i) Product of length of perpendicular from either focii to
any tangent to the ellipse will be equal to square of
semi minor axis.
(ii) The locus of foot of perpendicular drawn from either
focii to any tangent lies on auxillary circle.
(iii) The circle drawn on any focal radius as diameter will
touch auxillary circle.
(iv) The protion of the tangent intercepted between the
point and directrix makes right angle at corresponding
focus.
(v) Sum of square of intercept made by auxillary circle on
any two perpendicular tangents of an ellipse will be
constant.
(vi) If a light ray originates from one of focii, then it willpass through the other focus after reflection from
ellipse.
9. Equation of chord of contact of the tangents drawn from
the external point (x1, y
1) to an ellipse is given by
xx
a12 +
yy
b12 = 0 i.e. T = 0.
10. The equation of a chord of an ellipsex
a
2
2 +y
b
2
2 = 1 whose
mid point is (x1, y
1) is T = S
1.
11. Equation of chord joining the points (a cos , b sin ) and
(a cos , b sin ) on the ellipsex
a
2
2 +y
b
2
2 = 1 is
x
a
cos +
2
+y
b
sin +
2
= cos
2
(i) Relation between eccentric angles of focal chord
tan12
, tan22
=
e
e
1
1
(ii) Sum of feet of eccentric angles is odd .
i.e. 1 + 2 + 3 + 4 = (2n + 1) .
PAGE # 81 PAGE # 82
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12. Equation of polar of the point (x1, y
1) w.r.t. the ellipse
x
a
2
2 +y
b
2
2 = 1 is given byxx
a
1
2 +yy
b
1
2 = 0 i.e. T = 0.
The pole of the line l x + my + n = 0 w.r.t. the ellipse
x
a
2
2 +y
b
2
2 = 1 is .
HGIKJ
a
n
b n
n
2 2l, .
13. Eccentric angles of the extremities of latus rectum of the
ellipse
x
a
2
2 +
y
b
2
2 = 1 are tan1
.HG
IKJ
b
ae .
14. (i) Equation of the diameter bisecting the chords of
slope in the ellipsex
a
2
2 +y
b
2
2 = 1 is
y = b
a m
2
2 x
(ii) Conjugate Diameters : The straight lines y = m1x,
y = m2x are conjugate diameters of the ellipse
x
a
2
2 +y
b
2
2 = 1 if m1m2 = b
a
2
2 .
(iii) Properties of conjugate diameters :
(a) If CP and CQ be two conjugate semi-diameters
of the ellipse xa
2
2 + yb
2
2 = 1, then
CP2 + CQ2 = a2 + b2
(b) If and are the eccentric angles of the
extremities of two conjugate diameters, then
=
2
(c) If CP, CQ be two conjugate semi-diameters of
the ellipsex
a
2
2 +y
b
2
2 = 1 and S, S' be two foci
of the ellipse, then SP.S'P = CQ2
(d) The tangents at the ends of a pair of conjugate
diameters of an ellipse form a parallelogram.
15. The area of the parallelogram formed by the tangents at
the ends of conjugate diameters of an ellipse is constant
and is equal to the product of the axis i.e. 4ab.
16. Length of subtangent and subnormal at p(x1, y
1) to the
ellipse
x
a
2
2 +y
b
2
2 = 1 isa
x
2
1 x
1& (1 e2) x
1
PAGE # 83 PAGE # 84
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
HYPERBOLA
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HYPERBOLA
1. Standard Hyperbola :
x
a
2
2 y
b
2
2 = 1 x
a
2
2 +y
b
2
2 = 1
orx
a
2
2 y
b
2
2 = 1
Centre (0, 0) (0, 0)
Length of
transverse axis 2a 2bLength of
conjugate axis 2b 2a
oci (ae, 0) (0, be)
Equation of
directrices x = a/e y = b/e
Eccentricity e =
a b
a
2 2
2
+.
HG
I
KJ e =a b
b
2 2
2
+.
HG
I
KJLength of L.R. 2b2/a 2a2/b
Parametric
co-ordinates (a sec , b tan ) (b sec , a tan )
0 < 2 0 < 2
ocal radii SP = ex1
a SP = ey1
b
S'P = ex1 + a S'P = ey1 + bS'P SP 2a 2b
Tangents at
the vertices x = a, x = a y = b, y = b
Equation of the y = 0 x = 0
transverse axis
Equation of the x = 0 y = 0
conjugate axis
Hyperbola
Conjugate Hyperbola
PAGE # 85 PAGE # 86
Imp. terms
Hyperbola
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
f f
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2. Special form of hyperbola :
If the centre of hyperbola is (h, k) and axes are parallel
to the co-ordinate axes, then its equation is
( )x h
a
22
( )y k
b
22 = 1.
3. Parametric equations of hyperbola :
The equations x = a sec and y = b tan are known
as the parametric equations of hyperbolax
a
2
2
y
b
2
2= 1
4. Position of a point and a line w.r.t. a hyperbola :
The point (x1, y
1) lies inside, on or outside the hyperbola
x
a
2
2 y
b
2
2 = 1
according asx
a
12
2 y
b
12
2 1 is +ve, zero or ve.
The line y = mx + c does not intersect, touches, intersect
the hyperbola
according as c2 a2m2 b2.
5. Equations of tangents in different forms :
(a) Point form : The equation of the tangent to the
hyperbolax
a
2
2 y
b
2
2 = 1
at (x1, y
1) is
xx
a
1
2 yy
b
1
2 = 1.
(b) Parametric form : The equation of tangent to the
hyperbola
x
a
2
2
y
b
2
2 = 1 at (a sec , b tan ) is
x
asec
y
btan = 1.
(c) Slope form : The equations of tangents of slope m
to the hyperbola
x
a
2
2
y
b
2
2 = 1 are y = mx a m b2 2 2 and the
co-ordinates of points of contacts are
.HG
IKJ
a m
a m b
b
a m b
2
2 2 2
2
2 2 2, .
6. Equation of pair of tangents from (x1, y1) to the hyperbola
x
a
2
2 y
b
2
2 = 1 is given by SS1 = T2
7. Equations of normals in different forms :
(a) Point form : The equation of normal to the hyperbola
xa
2
2 yb
2
2 = 1 at (x1, y1) isa xx
2
1+ b y
y
2
1= a2 + b2.
(b) Parametric form : The equation of normal at
(a sec , b tan ) to the hyperbola
x
a
2
2 y
b
2
2 = 1 is ax cos + by cot = a2+ b2
PAGE # 87 PAGE # 88
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
( ) Sl f Th i f h l h
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(c) Slope form : The equation of the normal to the
hyperbolax
a
2
2 y
b
2
2 = 1 in terms of the slope m of
the normal is y = mx m m a b
a b m
( )2 2
2 2 2
+
(d) Condition for normality : If y = mx + c is the normal
ofx
a
2
2 y
b
2
2 = 1,
then c = m m a b
a b m
( )2 2
2 2 2
+
or c2 =
m a b
a m b
( )
( )
2 2 2
2 2 2
+
, which
is condition of normality.
(e) Points of contact : Co-ordinates of points of contact
are
.HG
IKJ
a
a b m
mb
a b m
2
2 2 2
2
2 2 2, m .
8. The equation of director circle of hyperbola
x
a
2
2 y
b
2
2 = 1 is x2 + y2
= a2 b2.
9. Equation of chord of contact of the tangents drawn from
the external point (x1, y
1) to the hyperbola is given by
xx
a
1
2 yy
b
1
2 = 1.
10. The equation of chord of the hyperbolax
a
2
2 y
b
2
2 = 1
whose mid point is (x1, y
1) is T = S
1.
11. Equation of chord joining the points P(a sec 1, b tan
1)
and Q(a sec 2, b tan
2) is
xa
cos 1 22.HG IKJ yb sin
1 22+.HG IKJ = cos
1 2
2+.HG IKJ.
12. Equation of polar of the point (x1, y
1) w.r.t. the hyperbola
is given by T = 0.
The pole of the line l x + my + n = 0 w.r.t.
xa
2
2 yb
2
2 = 1 is .HG IKJan b mn2 2l,
13. The equation of a diameter of the hyperbola
x
a
2
2 y
b
2
2 = 1 is y =b
a m
2
2 x.
14. The diameters y = m1x and y = m
2x are conjugate if
m1m
2=
b
a
2
2
15. Asymptotes of a hyperbola :
* The equations of asymptotes of the hyperbola
x
a
2
2 y
b
2
2 = 1 are y = b
ax.
Asymptote to a curve touches the curve at infinity.
* The asymptote of a hyperbola passes through the
centre of the hyperbola.
PAGE # 89 PAGE # 90
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
* The combined equation of the asymptotes of the * Equation of tangent at (x y ) to xy
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The combined equation of the asymptotes of the
hyperbolax
a
2
2 y
b
2
2 = 1 isx
a
2
2 y
b
2
2 = 0.
* The angle between the asymptotes of
x
a
2
2 y
b
2
2 = 1 is 2 tan1
y
b
2
2 or 2 sec1 e.
* A hyperbola and its conjugate hyperbola have the
same asymptotes.
* The bisector of the angles between the asymptotes
are the coordinate axes.
* Equation of hyperbola Equation of asymptotes =
Equation of asymptotes Equation of conjugate
hyperbola = constant.
16. Rectangular or Equilateral Hyperbola :
* A hyperbola for which a = b is said to be rectangular
hyperbola, its equation is x2 y2 = a2
* xy = c2 represents a rectangular hyperbola with
asymptotes x = 0, y = 0.
* Eccentricity of rectangular hyperbola is 2 and angle
between asymptotes of rectangular hyperbola is 90.
* Parametric equation of the hyperbola xy = c2
are
x = ct, y =c
t, where t is a parameter.
* Equation of chord joining t1, t
2on xy = c2 is
x + y t1t2
= c(t1
+ t2)
* Equation of tangent at (x1, y
1) to xy
= c2 isx
x1+
y
y1= 2.
Equation of tangent at t is x + yt2 = 2ct
* Equation of normal at (x1, y
1) to xy = c2 is
xx1
yy1
= x12 y
12
* Equation of normal at t on xy = c2 is
xt3 yt ct4 + c = 0.
(This results shows that four normal can be drawn
from a point to the hyperbola xy = c2)
* If a triangle is inscribed in a rectangular hyperbola
then its orthocentre lies on the hyperbola.
* Equation of chord of the hyperbola xy = c2 whose
middle point is given is T = S1
* Point of intersection of tangents at t1
& t2
to the
hyperbola xy = c2 is2 21 2
1 2 1 2
c t t
t t
c
t t+ +
.HG
IKJ
,
PAGE # 91 PAGE # 92
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
MEASURES O. CENTRAL TENDENCY 3 Geometric Mean :
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MEASURES O. CENTRAL TENDENCY
AND DISPERSION
1. Arithmetic mean :(i) or ungrouped data (individual series) x =
x x x
n no of termsn1 2+ + +......
( . )=
i
n
ix
n
=1
(ii) or grouped data (continuous series)
(a) Direct method x =
i
n
i i
i
n
i
fx
f
=
=
1
1
, where xi, i = 1 .... n
be n observations and fibe their corresponding
frequencies
(b) short cut method : x = A +
fd
f
i i
i
,
where A = assumed mean, di= x
i A = deviation
for each term
2. Properties of A.M.
(i) In a statistical data, the sum of the deviation of items
from A.M. is always zero.
(ii) If each of the n given observation be doubled, thentheir mean is doubled
(iii) If x is the mean of x1, x2, ...... xn. The mean of ax1, ax2.....ax
nis a x where a is any number different from
zero.
(iv) Arithmetic mean is independent of origin i.e. it is x
effected by any change in origin.
3. Geometric Mean :
(i) or ungrouped data
G.M. = (x1x
2x
3.....x
n)1/n
or G.M. = antilog1
1n
xii
n
log=
.HGIKJ
(ii) or grouped data
G.M. = x x xf f nf Nn
1 2
1
1 2....e j , where N = fii
n
=
1
= antilog
f x
f
i
i
n
i
i
i
n
=
=
.
H
GGGG
I
K
JJJJ
1
1
log
4. Harmonic Mean - Harmonic Mean is reciprocal of arith-
metic mean of reciprocals.
(i) or ungrouped data H.M. =n
xii
n1
1=
(ii) or grouped data H.M. =
f
f
x
i
i
n
i
ii
n
=
=
.HG IKJ1
1
5. Relation between A.M., G.M and H.M.
A.M. G.M. H.M.
Equality holds only when all the observations in the series
are same.
PAGE # 93 PAGE # 94
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
6. Median : (ii) or series in descending order
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6. Median :
(a) Individual series (ungrouped data) : If data is raw,
arrange in ascending or descending order and n be
the no. of observations.
If n is odd, Median = Value ofn
th+.
HGIKJ
1
2observation
If n is even, Median =1
2[Value of
nth
2
.HG
IKJ + value of
n
th
2 1+.HG
IKJ ] observation.
(b) Discrete series : irst find cumulative frequencies of
the variables arranged in ascending or descending
order and
Median =n
th+.
HG I
KJ1
2
observation, where n is cumulative
frequency.
(c) Continuous distribution (grouped data)
(i) or series in ascending order
Median = l +
NC
f
2
.HG
IKJ
i
Where l = Lower limit of the median class.
f = requency of the median class.
N = Sum of all frequencies.
i = The width of the median class
C = Cumulative frequency of the
class preceding to median class.
g
Median = u -
NC
f
2
.HG
IKJ
i
where u = upper limit of median class.
7. Mode :
(i) or individual series : In the case of individual series,
the value which is repeated maximum number of times
is the mode of the series.
(ii) or discrete frequency distribution series : In the caseof discrete frequency distribution, mode is the value of
the variate corresponding to the maximum frequency.
(iii) or continuous frequency distribution : first find the
model class i.e. the class which has maximum frequency.
or continuous series
Mode = l 1 +
f f
f f f1 0
1 0 22
LNM
OQP i
Where l 1 = Lower limit of the model class.
f1
= requency of the model class.
f0
= requency of the class preceding model
class.
f2
= requency of the class succeeding model
class.i = Size of the model class.
8. Relation between Mean, Mode & Median :
(i) In symmetrical distribution : Mean = Mode = Median
(ii) In Moderately symmetrical distribution : Mode = 3 Me-
dian 2 Mean
PAGE # 95 PAGE # 96
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
Measure of Dispersion : fi= requency of the corresponding x
i
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The degree to which numerical data tend to spread about
an average value is called variation or dispersion.
Popular methods of measure of dispersion.
1. Mean deviation : The arithmetic average of deviations
from the mean, median or mode is known as mean deviation.
(a) Individual series (ungrouped data)
Mean deviation =| |x S
n
Where n = number of terms, S = deviation of variate
from mean mode, median.(b) Continuous series (grouped data).
Mean deviation =
f x s
f
| |=
f x s
N
| |
Note : Mean deviation is the least when measured from the
median.
2. Standard Deviation :S.D. () is the square root of the arithmetic mean of the
squares of the deviations of the terms from their A.M.
(a) or individual series (ungrouped data)
=( )x x
N
2where x = Arithmetic mean of
the series
N = Total frequency
(b) or continuous series (grouped data)
(i) Direct method =f x x
Ni i( )
2
Where x = Arithmetic mean of series
xi
= Mid value of the class
i i
N = f = Total frequency
(ii) Short cut method
= fd
N
fd
N
2 2
.HGIKJ or =
d
N
d
N
2 2
.HGIKJ
Where d = x A = Derivation from assumed mean A
f = requency of item (term)
N = f = Total frequency.
Variance Square of standard direction
i.e. variance = (S.D.)2 = ()2
Coefficient of variance = Coefficient of S.D. 100
=
x 100
PAGE # 97 PAGE # 98
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
MATRICES AND DETERMINANTS 4. Trace of a matrix : Sum of the elements in the principal
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MATRICES AND DETERMINANTS
MATRICES :
1. Matrix - A system or set of elements arranged in a rectan-gular form of array is called a matrix.
2. Order of matrix : If a matrix A has m rows & n columns then
A is of order m n.
The number of rows is written first and then number of col-
umns. Horizontal line is row & vertical line is column
3. Types of matrices : A matrix A = (aij)
mn
A matrix A = (aij)
mxnover the field of complex numbers is
said to be
Name Properties
A row matrix if m = 1
A column matrix if n = 1
A rectangular matrix if m n
A square matrix if m = n
A null or zero matrix if aij= 0 i j. It is denoted by O.
A diagonal matrix if m = n and aij
= 0 for i j.
A scalar matrix if m = n and aij
= 0 for i j
= k for i = j
i.e. a11
= a22
....... = ann
= k (cons.)
Identity or unit matrix if m = n and aij
= 0 for i j
= 1 for i = j
Upper Triangular matrix if m = n and aij
= 0 for i > j
Lower Triangular matrix if m = n and aij
= 0 for i < j
Symmetric matrix if m = n and aij
= aji
for all i, j
or AT = A
Skew symmetric matrix if m = n and aij
= aji i, j
or AT = A
p p
diagonal is called the trace of a matrix.
trace (A B) = trace A trace B
trace kA = k trace A
trace A = trace AT
trace In= n when I
nis identity matrix.
trace On= O O
nis null matrix.
trace AB trace A trace B.
5. Addition & subtraction of matrices : If A and B are twomatrices each of order same, then A + B (or A B) is defined
and is obtained by adding (or subtracting) each element of B
from corresponding element of A
6. Multiplication of a matrix by a scalar :
KA = K (aij)
mn= (Ka)
mnwhere K is constant.
Properties :
(i) K(A + B) = KA + KB
(ii) (K1K
2)A = K
1(K
2A) = K
2(K
1A)
(iii) (K1+ K
2)A = K
1A + K
2A
7. Multiplication of Matrices : Two matrices A & B can be
multiplied only if the number of columns in A is same as the
number of rows in B.
Properties :
(i) In general matrix multiplication is not commutative i.e.
AB BA.
(ii) A(BC) = (AB)C [Associative law]
PAGE # 99 PAGE # 100
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
(iii) A.(B + C) = AB + AC [Distributive law] DETERMINANT:
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( ) ( ) [ ]
(iv) If AB = AC / B = C
(v) If AB = 0, then it is not necessary A = 0 or B = 0
(vi) AI = A = IA
(vii) Matrix multiplication is commutative for +ve integral
i.e. Am+1 = Am A = AAm
8. Transpose of a matrix :
A' or A
T
is obtained by interchanging rows into columns orcolumns into rows
Properties :
(i) (AT)T = A
(ii) (A B)T = AT BT
(iii) (AB)T = BTAT
(iv) (KA)T
= KAT
(v) IT = I
9. Some special cases of square matrices :A square matrix
is called
(i) Orthogonal matrix : if AAT = In= ATA
(ii) Idempotent matrix : if A2
= A(iii) Involutory matrix : if A2 = I or A1 = A
(iv) Nilpotent matrix : if p N such that Ap = 0
(v) Hermitian matrix : if A = A i.e. aij= a
ji
(vi) Skew - Hermitian matrix : if A = A
DETERMINANT :
1. Minor & cofactor : If A = (aij)
33, then minor of a
11is
M11
= a aa a22 23
32 33and so.
cofactor of an element aijis denoted by C
ijor
ijand is equal
to (1)i+j Mij
or Cij= M
ij, if i = j
= Mij, if i j
Note : |A| = a1111 + a12 12 + a13 13
and a11
21
+ a12
22
+ a13
23
= 0
2. Determinant : if A is a square matrix then determinant of
matrix is denoted by det A or |A|.
expansion of determinant of order 3 3
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
= a1
b c
b c
2 2
3 3b
1a c
a c
2 2
3 3+ c
1a b
a b
2 2
3 3
or = a2b c
b c
1 1
3 3+ b
2a c
a c
1 1
3 3 c
2a ba b
1 1
3 3
Properties :(i) |AT| = |A|
(ii) By interchanging two rows (or columns), value of de-
terminant differ by ve sign.
(iii) If two rows (or columns) are identical then |A| = 0
(iv) |KA| = Kn det A, A is matrix of order n n
PAGE # 101 PAGE # 102
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
(v) If same multiple of elements of any row (or column) of Properties :
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a determinant are added to the corresponding elements
of any other row (or column), then the value of the
new determinant remain unchanged.
(vi) Determinant of :
(a) A nilpotent matrix is 0.
(b) An orthogonal matrix is 1 or 1
(c) A unitary matrix is of modulus unity.
(d) A Hermitian matrix is purely real.
(e) An identity matrix is one i.e. |In| = 1, where In is aunit matrix of order n.
(f) A zero matrix is zero i.e. |0n| = 0, where 0
nis a zero
matrix of order n
(g) A diagonal matrix = product of its diagonal elements.
(h) Skew symmetric matrix of odd order is zero.
3. Multiplication of two determinants :
Multiplication of two second order determinants is defined as
follows.
a b
a b
1 1
2 2
l
l
1 1
2 2
m
m =a b a m b ma b a m b m
1 1 1 2 1 1 1 2
2 1 2 2 2 1 2 2
l l
l l
+ ++ +
If order is different then for their multiplication, express themfirstly in the same order.
MATRICES AND DETERMINANTS :
1. Adjoint of a matrix :
adj A = (Cij)T , where C
ijis cofactor of a
ij
(i) A(adj A) = (adjA) A = |A|In
(ii) |adj A| = |A|n1
(iii) (adjAB) = (adjB) (adjA)
(iii) (adj AT) = (adjA)T
(iv) adj(adjA) = |A|n2
(v) (adj KA) = Kn1(adj A)
2. Inverse of a matrix :
(i) A1 exists if A is non singular i.e. |A| 0
(ii) A1 =adjA
A| |, |A| 0
(iii) A1A = In= A A1
(iv) (AT)1 = (A1)T
(v) (A1
)1
= A
(vi) |A1| = |A|1 =1
| |A
(vii) If A & B are invertible square matrices then
(AB)1 = B1 A1
3. Rank of a matrix :
A non zero matrix A is said to have rank r, if
(i) Every square sub matrix of order (r + 1) or more is
singular
(ii) There exists at least one square submatrix of order r
which is non singular.
PAGE # 103 PAGE # 104
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
4. Homogeneous & non homogeneous system of linear(b) Solution of homogeneous system of linear equa-
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equations :
A system of equations Ax = B is called a homogeneous sys-
tem if B = 0. If B 0, then it is called non homogeneous
system equations.
5. (a) Solution of non homogeneous system of linear
equations :
(i) Cramer's rule : Determinant method
The non homogeneous system Ax = B, B 0 of n
equations in n variables is -
Consistent (with unique solution) if |A| 0 and
for each i = 1, 2, ........ n,
xi=
det
det
A
Ai, where A
iis the matrix obtained
from A by replacing ith column with B.
Inconsistent (with no solution) if |A| = 0 and
at least one of the det (A i) is non zero.Consistent (With infinite many solution), if
|A| = 0 and all det (Ai) are zero.
(ii) Matrix method :
The non homogeneous system Ax = B, B 0 of
n equations in n variables is -
Consistent (with unique solution) if |A| 0 i.e.if A is non singular, x = A1 B.
Inconsistent (with no solution), if |A| = 0 and
(adj A) B is a non null matrix.
Consistent (with infinitely many solutions), if
|A| = 0 and (adj A) B is a null matrix.
PAGE # 105 PAGE # 106
(b) Solution of homogeneous system of linear equa
tions :
The homogeneous system Ax = B, B = 0 of n equations
in n variables is
(i) Consistent (with unique solution) if |A| 0 and
for each i = 1, 2, ......... n
xi= 0 is called trivial solution.
(ii) Consistent (with infinitely many solution),
if |A| = 0
(a) |A| = |Ai| = 0 (for determinant method)
(b) |A| = 0, (adj A) B = 0 (for matrix method)
NOTE : A homogeneous system of equations is never
inconsistent.
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
.UNCTIONProperties :
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1. Modulus function :
|x| =
x xx xx
,,
,
> 0
|x| = a x = a
|x| = a no solution
|x| > a x < a or x > a
|x| a a x a
|x| < a No solution.
|x| > a x R
2. Logarithmic unction :
(i) logba to be defined a > 0, b > 0, b 1
(ii) logab = c b = ac
(iii) logab > c
b > ac, a > 1
or b < ac, 0 < a < 1
(iv) logab > log
ac
b > c, if a > 1
or b < c, if 0 < a < 1
(i) loga1 = 0
(ii) logaa = 1
(iii) a a blog = b
if k > 0, k = b bklog
(iv) logab
1+ log
ab
2+ ...... + log
ab
n= log
a(b
1b
2........b
n)
(v) loga
b
c
.HG
IKJ = loga b loga c
(vi) Base change formulae
logab =
log
logc
c
b
a or loga b =1
logb a
(vii) logam
bn =n
mlog
ab
(viii) loga
1
b
.HG
IKJ = loga b = log1/a b
(ix) log1/a
b
c
.HG
IKJ = loga
c
b
.HG
IKJ
(x)a
b clog =c
b alog
3. Greatest Integer function :
f(x) = [x], where [.]denotes greatest integer function equal
or less than x.
i.e., defined as [4.2] = 4, [4.2] = 5
Period of [x] = 1
PAGE # 107 PAGE # 108
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
Properties :
(i) 1 [ ]
6. Definition :
d b d f h l
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(i) x 1 < [x] x
(ii) [x + I] = [x] + I
[x + y] [x] + [y]
(iii) [x] + [x] = 0, x I
= 1, x I
(iv) [x] = I, where I is an integer x [I, I + 1)
(v) [x] I, x [I, )
(vi) [x] I, x ( , I + 1]
(vii) [x] > I, [x] I + 1, x [I + 1, )
(viii) [x] < I, [x] I 1, x ( , I)
4. ractional part function :
f(x) = {x} = difference between number & its integral part
= x [x].
Properties :
(i) {x}, x [0, 1)
(ii) {x + I} = {x}
{x + y}
{x} + {y}
(iii) {x} + {x} = 0, x I
= 1, x I
(iv) [{x}] = 0, {{x}} = {x}, {[x]} = 0
5. Signum function :
f(x) = sgn (x) =
=
R
S||T||
+
1
0 0
1
,
,
,
x R
x
x R
or f(x) =| |x
x, x 0
= 0, x = 0
Let A and B be two given sets and if each element a A is
associated with a unique element b B under a rule f, then
this relation (mapping) is called a function.
Graphically - no vertical line should intersect the graph of
the function more than once.
Here set A is called domain and set of all f images of the
elements of A is called range.
i.e., Domain = All possible values of x for which f(x) exists.
Range = or all values of x, all possible values of f(x).
Table :Domain and Range of some standard functions -
unctions Domain Range
Polynomial function R R
Identity function x R R
Constant function K R (K)
Reciprocal function
1
x R0 R0
x2, |x| (modulus function) R R+{x}
x3, x|x| R R
Signum function| |x
xR {-1, 0, 1}
x +|x| R R+{x}
x -|x| R R-{x}
[x] (greatest integer function) R 1
x - {x} R [0, 1]
x [0, ) [0, ]
ax (exponential function) R R+
log x (logarithmic function) R+ R
PAGE # 109 PAGE # 110
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
Trigonometric Domain Range7. Kinds of functions :
(i) One one (injection) function f : A B is one one if
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unctions
sin x R [-1, 1]
cos x R [-1, 1]
tan x R- RST
UVW
2
3
2, ,... R
cot x R- {0, , 2 ,...} R
sec x R -
R
ST
U
VW
2
3
2, ,...
R - (-1,1)
cosec x R- {0, , 2} R - (-1,1)
Inverse Domain Range
Trigo unctions
sin -1 x (-1, 1] LNM OQP
2 2,
cos-1 x [-1,1] [0, ]
tan-1 x R.
HGIKJ
2 2,
cot-1 x R (0, )
sec-1 x R -(-1,1) [0, ]-
2
RSTUVW
cosec-1 x R - (-1,1) LNM
OQP
2 2, - {0}
(i) One-one (injection) function - f : A B is one-one if
f(a) = f(b) a = b
or a b f(a) f(b), a, b A
Graphically-no horizontal line intersects with the graph
of the function more than once.
(ii) Onto function (surjection) - f : A B is onto if
R (f) = B i.e. if to each y B x A s.t. f(x) = y
(iii) Many one function : f : A B is a many one function
if there exist x, y A s.t. x y
but f(x) = f(y)
Graphically - atleast one horizontal line intersects with
the graph of the function more than once.
(iv) Into function : f is said to be into function if R(f) < B
(v) One-one-onto function (Bijective) - A function which
is both one-one and onto is called bijective function.
8. Inverse function : f1 exists iff f is one-one & onto both
f1 : B A, f1(b) = a f(a) = b
9. Transformation of curves :
(i) Replacing x by (x a) entire graph will be shifted parallel
to x-axis with |a| units.If a is +ve it moves towards right.
a is ve it moves toward left.
Similarly if y is replace by (y a), the graph will be
shifted parallel to y-axis,
upward if a is +ve
downward if a is ve.
PAGE # 111 PAGE # 112
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MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
unction Period 15. Composite function :
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sin x, cos x 2
sec x, cosec x
tan x, cot x
sin (x/3) 6
tan 4x /4
cos 2x 1
|cos x|
sin4 x + cos4 x /2
2 cosx .
HGIKJ
36
sin3 x + cos3 x 2/3
sin3 x + cos4 x 2
sinsin
xx5
2
tan2 x cot2 x
x [x] 1
[x] 1
NON PERIODIC UNCTIONS :
x , x2, x3, 5
cos x2
x + sin x
x cos x
cos x
If f : X Y and g : Y Z are two function, then the
composite function of f and g, gof : X Z will be defined as
gof(x) = g(f(x)), x X
In general gof fog
If both f and g are bijective function, then so is gof.
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
LIMIT
lim
5. Limit of the greatest integer function :
Let c be any real number
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1. Limit of a function : limx a f(x) = l (finite quantity)
2. Existence of limit : limx a f(x) exists ifflim
x a f(x) = lim
x a +f(x) =l
3. Indeterminate forms :0
0,
, , 0, 0 , 0 , 1
4. Theorems on limits :
(i) limx a (k f(x)) = klimx a f(x), k is a constant.
(ii) limx a
(f(x) g(x)) = limx a
f(x) Limx a
g(x)
(iii) limx a f(x).g(x) =limx a f(x).
Limx a g(x)
(iv) limx af x
g x
( )
( )=
lim ( )
lim ( )
x a
x a
f x
g x
, provided limx a g(x) 0
(v) limx a f(g(x)) = flim ( )x a
g x
.H
IK, provided value of
g(x) function f(x) is continuous.
(vi) limx a [f(x) + k] = limx a f(x) + k
(vii) limx a log(f(x)) = loglim ( )x a
f x
.H
IK
(viii) limx a (f(x))g(x) = lim ( )
lim ( )
x a
g x
f xx a
LNM
OQP
Case I : If c is not an integer, then limx c [x] = [c]
Case II: If c is an integer, then limx c [x] = c 1, limx c + [x] = c
and limx c
[x] = does not exist
6. Methods of evaluation of limits :
(i) actorisation method : If limx a
f x
g x
( )
( )is of
0
0form
then factorize num. & devo. separately and cancel the
common factor which is participating in making0
0form.
(ii) Rationalization method : If we have fractional powers
on the expression in num, deno or in both, we rationalize
the factor and simplify.
(iii) When x :Divide num. & deno. by the highest power
of x present in the expression and then after removing
the indeterminate form, replace1
x,
12x
,.. by 0.
(iv) limx a
x a
x a
n n
= nan1
(v) By using standard results (limits) :
(a) limx 0 sinx
x= 1 =
limx 0
x
xsin
(b) limx 0 tanx
x= 1 = limx 0
x
xtan
(c) limx 0 sinx = 0
(d) limx 0 cosx =limx 0
1
cosx= 1
PAGE # 117 PAGE # 118
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
(e) limx 0
sinx
x
0
=
180
(vi) By substitution :
(a) If x a, then we can substitute
+ t t
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x 180
(f) limx 0
sin1x
x= 1 = lim
x 0
x
xsin1
(g) limx 0
tan
1x
x= 1 = lim
x 0
x
xtan1
(h) limx 0 a
x
x 1
= logea
(i)limx 0
e
x
x 1
= 1
(j) limx 0
log( )1 + x
x= 1
(k) limx0
log ( )a x
x
1 +=
1
loga
(l) limx 0( )1 1+ x
x
n
= n
(m) limx sinx
x= limx
cosx
x= 0
(n) limx
sin1
1x
x
= 1
(o) limx0
(1 + x)1/x = e = limx
11
+.HG
IKJx
x
(p) limx 0 (1 + ax)1/x = ea = limx 1 +
.HG
IKJ
a
x
x
x = a + t t = x a
If x a, t 0.
(b) When x substitute x = t t
(c) When x substitute t =1
x t 0+
(vii) By using some expansion :
ex = 1 + x +x2
2!+
x3
3!+ .....
e
x
= 1 x +
x2
2!
x3
3! + .....
log(1 + x) = x x
2
2+
x3
3 ......
log(1 x) = x x2
2
x3
3.....
ex ln a = ax = 1 + xlogea +
( log )
!
x ae2
2
+( log )
!
x ae3
3
+ ......
sinx = x
x3
3!+
x5
5!.......
cosx = 1 x
2
2!+
x4
4!......
tanx = x +x3
3+
2
15x5 + .....
(1 + x)n = 1 + nx +n n( )
!
1
2x2 + .....
7. Sandwich Theorem : In the neighbour hood of x = a
f(x) < g(x) < h(x)
limx a
f(x) = limx a
h(x) = l, then limx a
g(x) = l.
l < limx a g(x) < l.
PAGE # 119 PAGE # 120
MATHS FORMULA - POCKET BOOK MATHS FORMULA - POCKET BOOK
DI..ERENTIATION 2. UNDAMENTAL RULES OR DIERENTIATION :
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1. SOME STANDARD DIERENTIATION :
unction Derivative unction Derivative
A cons. (k) 0 xn nxn 1
loga
x1
x aeloglog
ex
1
x
ax ax loge
a ex ex
sin x cos x cos x sin x
tan x sec2 x cot x cosec2 x
cosec x cosec x cot x sec x sec x tan x
sin1 x1
1 2 x,1
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i.e if y = un
dy
dx = nun1
du
dx
OR
(viii) Differentiation of composite functions
Suppose a function is given in form of fog(x) or
f[g(x)], then differentiate applying chain rule
i.e.,d
dx
f[g(x)] = f'g(x) . g'(x)
(ix)d
dx
1
u
.HG
IKJ =
12u
du
dx, u 0
(x)d
dx|u| =
u
u| |
du
dx, u 0
(xi) Logarithmic Differentiation : If a functionis in the
form (f(x))g(x) orf x f x
g x g x1 2
1 2
( ) ( )....
( ) ( ).... We first take log on
both sides and then differentiate.
(a) loge
(mn) = logem + log
en
(b) logem
n= logm log
en
(c) loge(m)n = nlog
em (d) log
nm log
mn = 1
(e) logan
xm =m
nlog
ax (f) aloga x = x
(g) logee = 1 (h) log
nm =
log
loge
e
m
n
dy
dx
at one side and finddy
dx
.
[The relation f(x, y) = 0 in which y is not expressible
explicitly in terms of x are called implicit functions]
(xiii) Differentiation of parametric functions : If x = f(t)
and y = g(t), where t is a parameter, then
dy
dx=
dy
dtdx
dt
=g t
f t
'( )
'( )
(xiv) Differentiation of a function w.r.t. another func-
tion : Let y = f(x) and z = g(x), then differentiationof y w.r.t. z is
dy
dz=
dy dx
dz dx
/
/=
f x
g x
'( )
'( )
(xv) Differentiation of inverse Trigonometric functions
using Trigonometrical Transformation : To sol