MATHEMATICS
Parabola
1. Distance of any point P on the parabola from the focus S is always equal to perpendicular
distance of P from the directrix i.e. SP=PM.
2. Parametric equation of parabola y2=4ax is x = at2 , y = 2at. coordinates of any point (t) is (at2,
2at).
3. Different types of standard parabola
Sr.
No.
1.
2.
3.
4.
Parabola
Focus
directrix
y2=4ax
y2=-4ax
x2=4by
x2=-4by
(a,0)
(-a,0)
(0,b)
(0,-b)
x = -a
x=a
y = -b
y = -b
1.
Latus
rectum
4a
4a
4b
4b
Axis of Parabola
(axis of symmetry)
y=0
y=0
x=0
x=0
2.
3.
4.
2
4. For the parabola y = 4ax.
a.
b.
c.
d.
e.
Equation of tangent at (x1,y1) is yy1 = 2a (x + x1)
Parametric equation of tangent (at21,2at1) is yt1 = x + at21
Tangent in term of slope m is y = mx + a/m and its point of contact is (a/m2, 2am)
If P(t1) and Q(t2) are the ends of a focal chord then t1t2 = -1
Focal distance of a point P(x1, y1) is x1 + a.
MATHEMATICS
Trigonometry
28.
29.
30.
31.
32.
33.
0
30o
o
45o
90o
60o
sin
0
1
cos
1
0
tan
0
1
34. sin (-θ) = - sin θ;
cos (-θ) = cos θ ;
120o 135o 150o 180o
0
-
-
-1
-
-1
0
tan (-θ ) = - tan θ
35.
sin (90o-θ) = cos θ
cos (90o-θ)= sin θ
tan (90o-θ)= cot θ
cot (90o-θ) = tan θ
cosec (90o-θ)=sec θ
sec (90o-θ) = cosec θ
sin (90o+ θ) = cos θ
cos (90o + θ)= - sin θ
tan (90o + θ)= - sin θ
cot (90o + θ) = - tan θ
cosec (90o + θ)=sec θ
sec (90o + θ) = - cosec θ
36. Sin (A + B) = Sin A Cos B + Cos A Sin B
Sin (A - B) = Cos A Sin B - Sin A Cos B
Cos (A + B) = Cos A Cos B - Sin A Sin B
Cos (A - B) = Cos A Cos B + Sin A Sin B
37.
sin (180o + θ) = - cos θ
cos (180o - θ) = - cos θ
tan (180o-θ)= - tan θ
cot ( 180o-θ) = - cot θ
cosec (180o-θ) = cosec θ
sec (180o- θ) = - sec θ
sin(180o+θ) = - sin θ
cos (180o+θ)= -cos θ
tan (180o+θ) = tan θ
cot (180o+θ) = cot θ
sec (180o+θ) = - sec θ
cosec (180o+θ) = -cosec θ
38. 2
2
2
2
sin A cos B = sin ( A+ B) + sin ( A - B)
cos B sin B = sin ( A+ B) - sin ( A - B)
cos A cos B = cos (A + B) + cos (A - B)
sin A sin B = cos (A - B) - cos (A + B)
39. cos (A + B).cos (A - B) = cos 2A - sin 2B
sin (A + B).sin (A - B) = sin 2A - sin 2B
40.
sin 2θ = 2 sin θ cos θ =
41.
cos 2θ = cos2θ-sin2θ = 2 cos2 θ-1 = 1 - 2 sin2θ =
42. 1 + cos 2θ = 2 cos2θ; 1 - cos 2θ = 2 sin2 θ
44.
=
45. sin 3θ = 3 sin θ - 4 sin3 θ ; cos 3θ = 4 cos 3θ - 3 cos θ ;
=
46.
47.
48. a = b cos C + c cos B; b = c cosA + a cos C; c= a cos B + b cos A
49.
Area of triangle ABC =
50.
51.
52.
53.
XII - Science - Formula Book
MATHEMATICS
Ellipse
1.
Distance of any point on an ellipse from the focus = e (perpendicualr distance of the point from
the corresponding directrix) i.e. SP= e PM.
2.
Different types of ellipse.
Ellipse.
Foci
Directrices
Latus
rectum
Equation of
axis
Ends of L.R
major Axis
y=0
(± ae , 0)
(a > b)
minor Axis
x=0
major Axis
x=0
(0 , ±be)
(a < b)
minor Axis
y=0
1.
2.
3.
(a > b) is x = aCosθ and y = b Sinθ where θ is
Parametric equation of ellipse
called the eccentric angle.
4.
For the ellipse
, (a > b) , b2 = a2 (1 - e2) and
, (a < b ) , a2 = b2 ( 1- e2)
5.
For the ellipse
(a > b) .
Equation of tangent at (x1, y1) is
.
Equation of tangent in terms of its slope m is
Tangent at (aCosθ , bSinθ) is
6.
.
.
Focal distance of P(x1,y1) are SP= | a - ex1 | and S'P = | a + ex1 |
MATHEMATICS
Hyperbola
1.
Distance of a point on the hyperbola from the focus e (perpendicular distance of the point
from the corresponding directrix ) ie. SP = e PM
2.
Different types of Hyperbola
Hyperbola
Foci
Directrices
L.R.
End of L.R.
Eqn. of axis.
Transverse
axis y=0
(±ae,o)
Conjugate
axis x = 0
Transverse
axis x = 0
(0, ±be)
Conjugate
axis y = 0
1.
2.
3.
For the Hyperbola
and for
4.
Parametric equations of hyperbola
called the eccentric angle.
, are x = a sec θ, y= b tan θ. Where θ is
5.
For the hyperbola
(a)
Equation of tangent at
are
(b)
Equation of tangent in terms of its slope m is
(c)
Equation of tangent at (a sec θ, b tan θ ) is
(d)
Focal distances of P ( x1,y1) are SP = l ex1 - a l and SP = l ex1 + a l
MATHEMATICS
Probability
1.
Probability of an event A is
2.
if A B are mutually exclusive then
3.
P(A') = 1 - P(A) or P(A) = 1 - P(A' )
4.
If A and B are independent events
5.
6.
7.
Where θ is meadured in radians
8.
=