MATHS FORMULA - POCKET BOOK
QUADRATIC EQUATION & EXPRESSION
1.
MATHS FORMULA - POCKET BOOK
4.
Conjugate roots :
Irrational roots and complex roots occur in conjugate pairs
i.e.
Quadratic expression :
A polynomial of degree two of the form ax2 + bx + c, a ≠ 0 is
called a quadratic expression in x.
if one root α + iβ, then other root α – iβ
if one root α +
2.
β , then other root α –
β
Quadratic equation :
An equation ax2 + bx + c = 0, a ≠ 0, a, b, c ∈ R has two and
only two roots, given by
α=
3.
−b + b2 − 4ac
2a
and
β=
5.
Sum of roots :
−b − b2 − 4ac
2a
S=α+β=
−Coefficient of x
−b
=
Coefficient of x2
a
Product of roots :
Nature of roots :
Nature of the roots of the given equation depends upon the
nature of its discriminant D i.e. b2 – 4ac.
P = αβ =
cons tant term
c
=
Coefficient of x2
a
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
Formation of an equation with given roots :
x2 – Sx + P = 0
roots are imaginary and unequal i.e.
⇒
non real complex numbers.
Suppose a, b, c ∈ Q a ≠ 0 then
If D > 0 and D is a perfect square ⇒ roots are rational
(i)
6.
⇒
7.
Roots under particular cases :
For the equation ax2 + bx + c = 0, a ≠ 0
& unequal
(ii)
x2 – (Sum of roots) x + Product of roots = 0
If b = 0 ⇒ roots are of equal magnitude but of opposite
(i)
If D > 0 and D is not a perfect square ⇒ roots are
sign.
irrational and unequal.
For a quadratic equation their will exist exactly 2 roots real
or imaginary. If the equation ax 2 + 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.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
(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.
PAGE # 1
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 2
MATHS FORMULA - POCKET BOOK
(v)
MATHS FORMULA - POCKET BOOK
(vi) α4 + β4 = (α2 + β2)2 – 2α2β2
If a > 0, c < 0 or a < 0, c > 0 ⇒ roots are of opposite
signs
2
(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 ⇒ both
={(α + β)2 –2αβ}2 – 2α2β2
roots are –ve
(vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0
If roots of quadratic equation ax2 + bx + c, a ≠ 0 are α and β,
then
(α – β) =
(ii)
α2 + β2 = (α + β)2 – 2αβ =
(iii)
α2 – β2 = (α + β)
(α + β) − 4αβ = ±
2
b2 − 4ac
a
b2 − 2ac
a2
(α + β) − 4αβ =
2
−b b2 − 4ac
a2
9.
=
D U
C
A T
I O
N
S
2c2
a2
−b(b2 − 2ac) b2 − 4ac
a4
(viii) α2 + αβ + β2 = (α + β)2 – αβ =
b2 + ac
a2
(ix)
α
α 2 + β2
(α + β)2 − 2αβ
β
+
=
=
β
αβ
αβ
α
(x)
FG α IJ
H βK
2
+
FG β IJ
H αK
2
=
α4 + β4
[(b2 − 2ac)2 − 2a2c2 ]
=
α 2 β2
a2c 2
b1c2 − b2c1
c1a2 − c2a1
c1a2 − c2a1 = a1b2 − a2b1
(i)
One common root if
(ii)
a1
b1
c1
Both roots common if a = b = c
2
2
2
(α + β)2 − 4αβ [α2+ β2 – αβ]
(b2 − ac) b2 − 4ac
a3
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
–
The equations a1 x2 + b1 x + c1 = 0 and a2x2 + b2x + c2 = 0
have
α 3 – β3 = (α – β) [α2+ β2 – αβ]
=
2
Condition for common roots :
−b(b2 − 3ac)
(iv) α3 + β3 = (α + β)3 – 3(α + β) αβ =
a3
(v)
I
JK
(vii) α4 – β4 =(α2 + β2) (α2 – β2) =
Symmetric function of the roots :
(i)
− 2ac
a2
⇒ both
roots are +ve.
8.
Fb
=G
H
PAGE # 3
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 4
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
10. Maximum and Minimum value of quadratic expression :
2
In a quadratic expression ax
LF b I
+ bx + c = a MGH x + 2a JK
MN
2
−
(v)
OP
PQ
D
4a2 ,
D ≥ 0, a.f(k1) > 0, a.f(k2) > 0, k1 <
If a > 0, quadratic
a.f(k1) < 0, a.f(k2) < 0
expression has minimum value
−b
4ac − b2
at x =
and there is no maximum value.
2a
4a
(ii)
(vii) λ will be the repeated root of f(x) = 0 if
f(λ) = 0 and f'(λ) = 0
If a < 0, quadratic expression has maximum value
−b
4ac − b2
at x =
and there is no minimum value.
2a
4a
12. For cubic equation ax3 + bx2 + cx + d = 0 :
We have α + β + γ =
11. Location of roots :
If k lies between the roots then a.f(k) < 0
(necessary & sufficient)
(ii)
13. For biquadratic equation ax 4 + bx3 + cx2 + dx + e = 0 :
If between k1 & k2 their is exactly one root of k1, k2
themselves are not roots
f(k1) . f(k2) < 0
(iii)
We have α + β + γ + δ = –
−d
b
, αβγ + βγδ + γδα + γδβ =
a
a
(necessary & sufficient)
If both the roots are less than a number k
D ≥ 0, a.f(k) > 0,
−b
c
−d
, αβ + βγ + γα =
and αβγ =
a
a
a
where α, β, γ are its roots.
Let f(x) = ax2 + bx + c, a ≠ 0 then w.r.to f(x) = 0
(i)
−b
< k2
2a
(vi) If k1, k2 lies between the roots
Where D = b2 – 4ac
(i)
If both the roots lies in the interval (k1, k2)
−b
<k
2a
αβ + αγ + αδ + βγ + βδ + γδ =
c
e
and αβγδ =
a
a
(necessary & sufficient)
(iv) If both the roots are greater than k
D ≥ 0, a.f(k) > 0,
−b
>k
2a
(necessary & sufficient)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 5
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 6
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
COMPLEX NUMBER
1.
Complex Number :
A number of the form z = x + iy (x, y ∈ R, i =
−1 ) is called
a complex number, where x is called a real part i.e. x = Re(z)
and y is called an imaginary part i.e. y = Im(z).
Modulus |z| =
y
.
x
*
z z = |z|2
*
z1 + z2 +....+zn = z 1 + z 2 + .......... + z n
*
z1 − z2 = z 1 – z 2
*
z1z2 = z 1 z 2
*
FG z IJ
Hz K
x = r cosθ, y = r sinθ, r = |z| =
1
x2 + y2
Exponential form :
z = reiθ , where r = |z|, θ = amp.(z)
→
(provided z2 ≠ 0)
ez j
= ( z )n
*
c zh
= z
*
If α = f(z), then α = f( z )
n
*
z + z = 0 or z = – z ⇒ z = 0 or z is purely imaginary
*
z= z
⇒
z is purely real
Integral Power of lota :
i=
4.
2
3
4
−1 , i = –1, i = –i , i = 1
Modulus of a complex number :
Magnitude of a complex number z is denoted as |z| and is
defined as
Hence i4n+1 = i, i4n+2 = –1, i4n+3 = –i, i4n or i4(n+1) = 1
3.
|z| =
Complex conjugate of z :
C
A T
(Re(z))2 + (Im(z))2 , |z| ≥ 0
If z = x + iy, then z = x – iy is called complex conjugate
of z
(i)
z z = |z|2 = | z |2
*
z is the mirror image of z in the real axis.
(ii)
z–1 =
*
|z| = | z |
*
z + z = 2Re(z) = purely real
(iii)
|z1 ± z2|2 = |z1|2 + |z2|2 ± 2 Re (z1 z2 )
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
2
Where α = f(z) is a function in a complex variable
with real coefficients.
P(x, y) then its vector representation is z = OP
E
1
*
(iii) Vector representation :
2.
F z IJ
GH z K
=
2
Polar representation :
(ii)
z – z = 2i Im(z) = purely imaginary
x2 + y2 ,
amplitude or amp(z) = arg(z) = θ = tan–1
(i)
*
I O
N
S
PAGE # 7
z
|z|2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 8
MATHS FORMULA - POCKET BOOK
6.
(iv) |z1 + z2|2 + |z1 – z2|2 = 2 [|z1|2 + |z2|2]
(v)
MATHS FORMULA - POCKET BOOK
Square root of a complex no.
|z1 ± z2| ≤ |z1| + |z2|
a + ib = ±
(vi) |z1 ± z2| ≥ |z1| – |z2|
5.
Argument of a complex number :
= ±
Argument of a complex number z is the ∠ made by its radius
vector with +ve direction of real axis.
arg z
= θ,
z ∈ 1st quad.
7.
= π – θ , z ∈ 2nd quad.
= –θ,
arg (any real – ve no.) = π
(iii)
arg (z – z ) = ± π/2
and (cosθ + isinθ)–n = cos nθ – i sin nθ
Euler's formulae as z = reiθ, where
8.
eiθ = cosθ + isinθ and e–iθ = cosθ – i sinθ
(iv) arg (z1.z2) = arg z1 + arg z2 + 2 k π
(v)
arg
FG z IJ
Hz K
1
2
∴
FG 1 IJ
H zK
eiθ + e–iθ = 2cosθ and eiθ – e–iθ = 2 isinθ
nth roots of complex number z1/n
9.
= arg z1 – arg z2 + 2 k π
(vi) arg ( z ) = –arg z = arg
OP
PQ
|z|+a
|z|−a
−i
, for b < 0
2
2
(cosθ + isinθ)n = cosθ + isin nθ
= θ – π , z ∈ 4 quad.
(ii)
LM
MN
It states that if n is rational number, then
th
arg (any real + ve no.) = 0
OP
PQ
|z|+a
|z|−a
+i
, for b > 0
2
2
De-Moiver's Theorem :
z ∈ 3rd quad.
(i)
LM
MN
LM FG 2mπ + θ IJ + i sinFG 2mπ + θ IJ OP
N H n K H n KQ ,
= r1/n cos
, if z is non real
where m = 0, 1, 2, ......(n – 1)
= arg z, if z is real
(vii) arg (– z) = arg z + π, arg z ∈ (– π , 0]
= arg z – π, arg z ∈ (0, π ]
(viii) arg (zn) = n arg z + 2 k π
(ix)
(i)
Sum of all roots of z1/n is always equal to zero
(ii)
Product of all roots of z1/n = (–1)n–1 z
10. Cube root of unity :
arg z + arg z = 0
cube roots of unity are 1, ω, ω2 where
argument function behaves like log function.
ω=
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 9
−1 + i 3
and 1 + ω + ω2 = 0, ω3 = 1
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 10
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
11. Some important result :
If z = cosθ + isinθ
(i)
z+
(ii)
z–
(iii)
zn +
1
= 2cosθ
z
1
= 2 isinθ
z
1
= 2cosnθ
zn
(c)
2
or
z−
z1 + z 2
2
=
|z1 − z 2 |
2
|z – z1|2 + |z – z2|2 = |z1 – z2|2
Where z1, z2 are end points of diameter and z is any
point on circle.
1
1
1
+
+
=0
y
x
z
2
z − z1
z − z1
z − z2 + z − z2 = 0
or
(iv) If x = cosα + isinα , y = cos β + i sin β & z = cosγ + isinγ
and given x + y + z = 0, then
(a)
or
13. Some important points :
(i)
Distance formula PQ = |z2 – z1|
(ii)
Section formula
(b) yz + zx + xy = 0
2
3
x +y +z =0
3
For internal division =
3
(d) x + y + z = 3xyz
m1z2 + m2 z1
m1 + m2
MATHS FORMULA - POCKET BOOK MATHS FORMULA - P
12. Equation of Circle :
*
|z – z1| = r represents a circle with centre z 1 and
radius r.
*
|z| = r represents circle with centre at origin.
*
|z – z1| < r and |z – z1| > r represents interior and
exterior of circle |z – z1| = r.
*
z z + a z + a z + b = 0 represents a general circle
where a ∈ c and b ∈ R.
*
Let |z| = r be the given circle, then equation of
tangent at the point z1 is z z 1 + z z1 = 2r2
*
diametric form of circle :
arg
Fz − z I
GH z − z JK
1
2
D U
C
A T
I O
N
S
(iii)
m1z2 − m2 z1
m1 − m2
Equation of straight line.
* Parametric form z = tz1 + (1 – t)z2 where t ∈ R
* Non parametric form
z
z1
z2
z 1
z1 1
z2 1
= 0.
* Three points z1, z2, z3 are collinear if
π
= ±
,
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
For external division =
z1
z2
z3
z1 1
z2 1 = 0
z3 1
or slope of AB = slope of BC = slope of AC.
PAGE # 11
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 12
MATHS FORMULA - POCKET BOOK
z − z1
(iv) The complex equation z − z
2
MATHS FORMULA - POCKET BOOK
(xii) If z1, z2, z3 be the vertices of a triangle, then the
triangle is equilateral
= k represents a circle
iff (z1 – z2)2 + (z2 – z3)2 + (z3 – z1)2 = 0.
if k ≠ 1 and a straight line if k = 1.
(v)
(xiii) If z1, z2, z3 are the vertices of an isosceles triangle,
right angled at z2,
The triangle whose vertices are the points represented
by complex numbers z1, z2, z3 is equilateral if
then z12 + z22 + z32 = 2z2 (z1 + z3).
(xiv) z1, z2, z3. z4 are vertices of a parallelogram then
1
1
1
z2 − z3 + z3 − z1 + z1 − z2 = 0
z1+ z3 = z2 + z4
i.e. if z12 + z22 + z32 = z1z2 + z2z3 + z1z3.
(vi) |z – z1| = |z – z2| = λ , represents an ellipse if
|z1 – z2| < λ , having the points z1 and z2 as its foci
and if |z1 – z2| = λ , then z lies on a line segment
connecting z1 & z2
(vii) |z – z1| ~ |z – z2| = λ represents a hyperbola if
|z1 – z2| > λ , having the points z1 and z2 as its foci,
and if |z1 – z2| = λ , then z lies on the line passing
through z1 and z2 excluding the points between z1 & z2.
(viii) If four points z1, z2, z3, z4 are concyclic,
then
FG z
Hz
1
1
− z2
− z4
IJ FG z
K Hz
3
3
− z4
− z2
IJ
K
is purely real.
(ix)
If three complex numbers are in A.P., then they lie on
a straight line in the complex plane.
(x)
If z1, z2, z3 be the vertices of an equilateral triangle
and z0 be the circumcentre,
then z12 + z22 + z32 = 3z02.
(xi)
If z1, z3, z3 ....... zn be the vertices of a regular
polygon of n sides & z0 be its centroid, then
z12 + z22 + ......... + zn2 = nz02.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 13
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 14
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(iv) If out of n objects, 'a' are alike of one kind, 'b' are alike
of second kind and 'c' are alike of third kind and the
rest distinct, then the number of ways of permuting
PERMUTATION & COMBINATION
1.
Factorial notation The continuous product of first n natural numbers is called
factorial
i.e. n or n! = 1. 2. 3........(n – 1).n
n! = n(n – 1)! = n(n – 1)(n – 2)! & so on
the n objects is
4.
Restricted Permutations -
n!
n (n – 1)......... (n – r + 1) =
(n − r)!
or
Here 0! = 1 and (–n)! = meaningless.
2.
Fundamental principle of counting (i)
Addition rule : If there are two operations such that
they can be done independently in m and n ways
respectively, then either (any one) of these two
operations can be done by (m + n) ways.
Addition ⇒ OR (or) Option
(ii) Multiplication rule : Let there are two tasks of an
operation and if these two tasks can be performed in m
and n different number of ways respectively, then the
two tasks together can be done in m × n ways.
Multiplication ⇒ And (or) Condition
(iii)
3.
(iii)
D U
C
A T
I O
N
S
The number of permutations of n dissimilar things taken
r at a time, when m particular things always occupy
definite places = n–mpr–m
(ii)
The number of permutations of n different things taken
r at a time, when m particular things are always to be
excluded (included)
n–m
Pr (n–mCr–m × r!)
Circular Permutations When clockwise & anticlockwise orders are treated as
different.
(i)
The number of circular permutations of n different things
n
Pr
r
taken r at a time
(ii)
The number of circular permutations of n different things
n
taken altogether
Pn
= (n – 1)!
n
When clockwise & anticlockwise orders are treated
as same.
(i)
The number of circular permutations of n different things
n!
(n − r)!
n
taken r at a time
The number of permutations of n dissimilar things taken
all at a time is npn = n!
The number of permutations of n distinct objects taken
r at a time, when repetition of objects is allowed is nr.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
5.
Permutations (Arrangement of objects) (i)
The number of permutations of n different things taken
(ii)
(i)
=
Bijection Rule : Number of favourable cases
= Total number of cases
– Unfavourable number of cases.
r at a time is npr =
n!
a! b! c!
(ii)
Pr
2r
The number of circular permutations of n different things
n
taken all together
PAGE # 15
1
Pn
=
(n – 1)!
2
2n
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 16
MATHS FORMULA - POCKET BOOK
6.
MATHS FORMULA - POCKET BOOK
Combination (selection of objects) -
(iv) Total number of selections of zero or more objects
from n identical objects is n + 1.
The number of combinations of n different things taken r at
a time is denoted by nCr or C (n, r)
n
Cr =
n!
r !(n − r)!
(v)
n
=
(i)
n
(ii)
n
(iii)
n
(iv)
n
(v)
n
(vi)
n
n
r
(vii)
n
1
(n – r + 1) nCr–1
r
Pr
r!
=
Cr + nCr–1 = n+1Cr
⇒
r = s or r + s = n
n
C0 = Cn = 1
Cr =
n–1
Cr–1
Division and distribution (i)
The number of ways in which (m + n + p) different
objects can be divided into there groups containing m,
The number of combinations of n distinct objects taken r at
a time, when k particular objects are always to be
8.
(i)
included is n–kCr–k
(ii)
excluded is n–kCr
(iii)
included and s particular things are to be excluded is
n, & p different objects respectively is
(i)
The number of selections of n identical objects, taken
at least one = n
(ii)
The number of selections from n different objects, taken
at least one
C
A T
I O
N
S
(iii)
n
Cn = 2n – 1
The total number of ways in which n different objects
are to be divided into r groups of group sizes n1, n2, n3,
............. nr respectively such that size of no two groups
The total number of ways in which n different objects
are to be divided into groups such that k1 groups have
group size n1, k2 groups have group size n2 and so on,
kr groups have group size nr, is given as
n!
The number of selections of r objects out of n identical objects is 1.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
(m + n + p)!
m! n! p!
n!
is same is n ! n !............n ! .
r
1
2
Total number of combinations in different cases -
(iii)
E
(ii)
Cr–k
= nC1 + nC2 + nC3 + ....... +
Cn = 2n
[(a1 + 1) (a2 + 1) (a3 + 1) .......... (an + 1)] 2k – 1
Restricted combinations -
n–k–s
n
(vii) The number of selections taking atleast one out of
a1 + a2 + a3 + ....... + an + k objects when a1 are
alike (of one kind), a2 are alike (of second kind),
........ an are alike (of kth kind) and k are distinct is
9.
7.
C0 + nC1 + nC2 + nC3 + ....... +
[(a1 + 1) (a2 + 1) (a3 + 1) + ...... + (an + 1)] – 1
C1 = nCn–1 = n
Cr =
n
(vi) The total number of selections of at least one out of
a1 + a2 + ...... + an objects where a1 are alike (of
one kind), a2 are alike (of second kind), ......... an are
alike (of nth kind) is
Cr = nCn–r
Cr = nCs
Total number of selections of zero or more objects
out of n different objects
k1
(n1 !) (n2 !) .............(nr !)k r k1 ! k 2 !............ k r !
PAGE # 17
k2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
.
PAGE # 18
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(iv) The total number of ways in which n different objects
are divided into k groups of fixed group size and are
distributed among k persons (one group to each) is
given as
(b)
Number of total triangles formed by joining the n
points on a plane of which m(< n) are collinear is
n
C3 –
(c)
(number of ways of group formation) × k!
m
C3.
Number of diagonals in a polygon of n sides is
n
C2 – n.
10. Selection of light objects and multinomial theorem (i)
The coefficient of xn in the expansion of (1 – x–r) is
equal to n + r – 1Cr – 1
(ii)
The number of solution of the equation x1 + x2 + ..........
+ xr = n, n ∈ N under the condition n1 ≤ x1 ≤ n'1,
(d)
If m parallel lines in a plane are intersected by a
family of other n parallel lines. Then total number of
parallelogram so formed is mC2 × nC2.
(e)
Given n points on the circumference of a circle, then
number of straight lines nC2
number of triangles nC3
n2 ≤ x2 ≤ n'2 , ................ nr ≤ xr ≤ n'r
number of quadrilaterals nC4
where all x'is are integers is given as
Coefficient of xn is
LMex
N
n1
+x
n1 +1
+...+x
n'1
j ex
n2
+x
n2 +1
+...+x
n'2
j...ex
nr
+x
nr +1
+...+ x
n'r
If n straight lines are drawn in the plane such that
no two lines are parallel and no three lines are
concurrent. Then the number of part into which these
lines divide the plane is = 1 + Σn
(g)
Number of rectangles of any size in a square of n × n
jOQP
11. Derangement Theorem (i)
(f)
If n things are arranged in a row, then the number of
ways in which they can be rearranged so that no one
of them occupies the place assigned to it is
n
n
is
(h)
L 1 + 1 − 1 + 1 −....+(−1) 1 OP
= n! M1 −
n!Q
N 1! 2! 3! 4!
∑ r3
r =1
and number of squares of any size is
∑ r2 .
r =1
Number of rectangles of any size in a rectangle of
n
(ii)
n × p is
If n things are arranged at n places then the number of
ways to rearrange exactly r things at right places is
n!
=
r
LM1 − 1 + 1 − 1 + 1 +....+(−1)
N 1! 2! 3! 4!
n− r
1
(n − r)!
np
(n + 1) (p + 1) and number of squares
4
n
of any size is
OP
Q
∑
r =1
(n + 1 – r) (p + 1 – r).
12. Some Important results (a)
Number of total different straight lines formed by joining
the n points on a plane of which m(<n) are collinear is
n
C2 – mC2 + 1.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 19
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 20
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
PROBABILITY
1.
(vi) P(AB) ≤ P(A) P(B) ≤ P(A + B) ≤ P(A) + P(B)
(vii) P(Exactly one event) = P(A B ) + P( A B)
Mathematical definition of probability :
(viii) P( A + B ) = 1 – P(AB) = P(A) + P(B) – 2P(AB)
Probability of an event
=
Note :
2.
= P(A + B) – P(AB)
No. of favourable cases to event A
Total no. of cases
(ii)
0 ≤ P (A) ≤ 1
Probability of an impossible event is zero
(iii)
Probability of a sure event is one.
(iv)
P(A) + P(Not A) = 1 i.e. P(A) + P( A ) = 1
(i)
(ix)
P(neither A nor B) = P ( A B ) = 1 – P(A + B)
(x)
When a coin is tossed n times or n coins are tossed
once, the probability of each simple event is
(xi)
Odds for an event :
If P(A) =
1
.
6n
(xii) When n cards are drawn (1 ≤ n ≤ 52) from well shuffled
deck of 52 cards, the probability of each simple event
P(A)
m
=
P(A)
n−m
Then odds in favour of A =
and odds in against of A =
When a dice is rolled n times or n dice are rolled once,
the probability of each simple event is
m
n−m
and P( A ) =
n
n
is
P(A)
n−m
=
P(A)
m
1
.
Cn
52
(xiii) If n cards are drawn one after the other with replacement, the probability of each simple event is
3.
(xiv) P(none) = 1 – P (atleast one)
(i)
P(A + B) = 1 – P( A B )
(xv) Playing cards :
(a)
Total cards : 52 (26 red, 26 black)
(ii)
P(AB)
P(A/B) =
P(B)
(b)
Four suits : Heart, diamond, spade, club (13 cards
each)
(iii)
P(A + B) = P(AB) + P( A B) + P(A B )
(c)
Court (face) cards : 12 (4 kings, 4 queens, 4
jacks)
(d)
Honour cards : 16 (4 Aces, 4 kings, 4 queens, 4
Jacks)
(v)
P( AB ) = P(B) – P(AB)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
C
A T
1
.
(52)n
Set theoretical notation of probability and some important results :
(iv) A ⊂ B ⇒ P(A) ≤ P(B)
E
1
.
2n
I O
N
S
PAGE # 21
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 22
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
5.
(xvi) Probability regarding n letters and their envelopes :
Conditional probability :
P(A/B) = Probability of occurrence of A, given that B has
If n letters corresponding to n envelopes are placed in
the envelopes at random, then
(a)
Probability that all the letters are in right envelopes =
(b)
already happened =
P(B/A) = Probability of occurrence of B, given that A has
1
n!
already happened =
Probability that all letters are not in right envelopes = 1 –
Note :
1
n!
Probability that no letters are in right envelope
1
1
1
1
–
+
.... + (–1)n
=
2!
3!
4!
n!
(d)
LM 1 − 1 + 1 +.....+(−1)
N2! 3! 4!
n− r
1
(n − r)!
No. of sample pts. in A ∩ B
.
No. of pts. in B
(i)
If A and B are independent event, then P(A/B) = P(A)
and P(B/A) = P(B)
(ii)
Multiplication Theorem :
Probability that exactly r letters are in right
1
envelopes =
r!
P(A ∩ B)
P(A)
If the outcomes of the experiment are equally
likely, then P(A/B) =
(c)
P(A ∩ B)
P(B)
P(A ∩ B) = P(A/B). P(B), P(B) ≠ 0
or
OP
Q
P(A ∩ B) = P(B/A) P(A), P(A) ≠ 0
Generalized :
P(E1 ∩ E2 ∩ E3 ∩ ............... ∩ En)
4.
Addition Theorem of Probability :
= P(E1) P(E2/E1) P(E3/E1 ∩ E2) P(E4/E1 ∩ E2 ∩ E3) .........
(i)
When events are mutually exclusive
If events are independent, then
i.e. n (A ∩ B) = 0
P(E1 ∩ E2 ∩ E3 ∩ ....... ∩ En) = P(E1) P(E2) ....... P(En)
⇒
P(A ∩ B) = 0
∴ P(A ∪ B) = P(A) + P(B)
(ii)
When events are not mutually exclusive i.e.
6.
Probability of at least one of the n Independent events :
P(A ∩ B) ≠ 0
(iii)
∴
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
or
P(A + B) = P(A) + P(B) – P(AB)
C
A T
I O
N
S
P(A1 + A2 + ... + An) = 1 – P ( A 1 ) P ( A 2 ) .... P( A n )
or
P(A + B) = P(A) + P(B) – P(A) P(B)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
1 – [(1 – P1) (1 – P2)......(1 – Pn)]
When events are independent i.e. P(A ∩ B) = P(A) P (B)
∴
E
If P1, P2, ....... Pn are the probabilities of n independent
events A, A2, .... An then the probability of happening of at
least one of these event is.
PAGE # 23
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 24
MATHS FORMULA - POCKET BOOK
7.
MATHS FORMULA - POCKET BOOK
Total Probability :
(a)
Let A1, A2, ............. An are n mutually exclusive & set of
exhaustive events and event A can occur through any one
of these events, then probability of occurence of A
(b)
P(A) = P(A ∩ A1) + P(A ∩ A2) + ............. + P(A ∩ An)
8.
∑
r =1
(i)
P(Ar) P(A/Ar)
(ii)
P(B / A i)P(A i )
P(Ai/B) = P(B / A )P(A ) + P(B / A )P(A ) + P(B / A )P(A ) ,
1
1
2
2
3
3
(d)
Standard deviation = npq
If A and B both assert that an event has occurred,
probability of occurrence of which is α then the probability that event has occurred.
Given that the probability of A & B speaking truth is p1, p2.
αp1p2
αp1p2 + (1 − α) (1 − p1) (1 − p2 )
i = 1, 2, 3
Probability distribution :
(iii)
If a random variable x assumes values x1, x2, ......xn
with probabilities P1, P2, ..... Pn respectively then
(ii)
Variance E(x2) – (E(x))2 = npq
p1p2
p1p2 + (1 − p1 ) (1 − p2 ) .
Let A1, A2, A3 be any three mutually exclusive & exhaustive
events (i.e. A1 ∪ A2 ∪ A3 = sample space & A1 ∩ A2 ∩ A3 = φ)
an sample space S and B is any other event on sample space
then,
(i)
(c)
If two persons A and B speaks truth with the probability p1 & p2 respectively and if they agree on a statement, then the probability that they are speaking truth
will be given by
Baye's Rule :
9.
E (x2) = npq + n2 p2
10. Truth of the statement :
n
=
mean E(x) = np
(a)
P1 + P2 + P3 + ..... + Pn = 1
(b)
mean E(x) = Σ Pixi
(c)
Variance = Σx Pi – (mean) = Σ (x ) – (E(x))
2
2
2
If in the second part the probability that their lies
(jhuth) coincides is β then from above case required
probability will be
αp1p2
+
−
αp1p2 (1 α) (1 − p1) (1 − p2 ) β .
2
Binomial distribution : If an experiment is repeated n
times, the successive trials being independent of one
another, then the probability of r success is nCr Pr qn–r
n
atleast r success is
∑
k =r
n
Ck Pk qn–k
where p is probability of success in a single trial, q = 1 – p
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 25
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 26
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
PROGRESSION AND SERIES
1.
(iv) If A1, A2,...... An are n A.M's between a and b,
then A1 = a + d, A2 = a + 2d,...... An = a + nd,
Arithmetic Progression (A.P.) :
(a)
where d =
General A.P. — a, a + d, a + 2d, ...... , a + (n – 1) d
b−a
n+1
where a is the first term and d is the common difference
(b)
(v)
General (nth) term of an A.P. —
Tn = a + (n – 1)d [nth term from the beginning]
Sum of n A.M's inserted between a and b is
(vi) Any term of an A.P. (except first term) is equal to
the half of the sum of term equidistant from the
If an A.P. having m terms, then nth term from end
= a + (m – n)d
(c)
term i.e. an =
Sum of n terms of an A.P. —
Sn
=
n
n
[2a + (n – 1)d] = [a + Tn]
2
2
2.
where Sn–1 is sum of (n – 1) terms.
(e)
Supposition of terms in A.P. —
(i)
Three terms as a - d, a, a + d
(ii)
Four terms as a – 3d, a – d, a + d, a + 3d
(iii)
Five terms as a – 2d, a – d, a, a + d, a + 2d
A.M. of n numbers A1, A2, ................ An is defined
as
A.M. =
(ii)
(iii)
For an A.P., A.M. of the terms taken symmetrically
from the beginning and from the end will always
be constant and will be equal to middle term or
A.M. of middle term.
a+b
i.e.
2
C
A T
I O
N
S
a − Tnr
a(1 − r n )
=
, r<1
1−r
1−r
Tnr − a
a(r n − 1)
=
,r>1
r −1
r −1
(d)
Sum of an infinite G.P. — S ∞ =
(e)
Supposition of terms in G.P. —
If A is the A.M. between two given nos. a and b,
then
2A = a + b
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
=
=
A1 + A 2 +.........+ A n
ΣA i
Sum of numbers
=
=
n
n
n
A=
E
Sn
Arithmetic mean (A.M.) :
(i)
1
(a + an+r), r < n
2 n–r
Geometric Progression (G.P.)
(a) General G.P. —
a, ar, ar2 , ......
where a is the first term and r is the common ratio
(b) General (nth) term of a G.P. — Tn = arn–1
If a G.P. having m terms then nth term from end = arm–n
(c) Sum of n terms of a G.P. —
Note : If sum of n terms i.e. Sn is given then Tn = Sn – Sn–1
(d)
n
(a + b)
2
PAGE # 27
a
,
1−r
a
, a, ar
r
(i)
Three terms as
(ii)
Four terms as
a a
, , ar, ar3
r3 r
(iii)
Five terms as
a a
, , a, ar, ar2
r2 r
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
|r|<1
PAGE # 28
MATHS FORMULA - POCKET BOOK
(f)
MATHS FORMULA - POCKET BOOK
Geometric Mean (G.M.) —
(i)
Geometrical mean of n numbers x1, x2, .......... xn
is defined as
G.M. = (x1 x2 ............... xn)1/n.
(i)
If G is the G.M. between two given numbers a
and b, then
G2 = ab ⇒ G =
(ii)
ab
5.
G1 = ar, G2 = ar ,..... Gn
(iii)
F bI
= ar , where r = G J
H aK
(d)
Σa = a + a + .... + (n times) = na
(e)
Σ(2n – 1) = 1 + 3 + 5 + .... (2n – 1) = n2
(f)
Σ2n = 2 + 4 + 6 + .... + 2n = n (n + 1)
1/n+1
n
General (nth) term — Tn = [a + (n – 1) d] rn–1
(c)
Sum of n terms of an A.G.P — Sn =
(d)
Sum of infinite terms of an A.G.P.
S∞ =
=
Σn2 = 12 + 22 + 32 + ..... + n2 =
D U
C
A T
I O
N
S
(ii)
If H1, H2,......,Hn are n H.M's between a and b,
ab(n + 1)
ab(n + 1)
, ....., Hn =
bn + a
na + b
1
1
&
, then their
a
b
Relation Between A.M., G.M. and H.M.
(i)
AH = G2
(ii)
A ≥ G ≥ H
If A and G are A.M. and G.M. respectively between two
+ve numbers, then these numbers are
(iii)
n(n + 1)(2n + 1)
6
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
If H is the H.M. between a and b, then H =
reciprocal will be required H.M's.
n(n + 1)
2
(b)
2ab
a+b
(i)
or first find n A.M.'s between
Sum standard results :
Σn = 1 + 2 + 3 + ..... + n =
Harmonic Mean (H.M.)
then H1 =
dr
a
+
(1 − r)2
1−r
(a)
1
1
= th
n term coresponding to A.P.
a + (n − 1)d
d(1 − r n−1)
a
+ r.
(1 − r)2
1−r
6.
4.
1
1
1
,
,
+......
a a + d a + 2d
General (nth term) of a H.P. — Tn
(a + 2d) r2, .............
(b)
3
(b)
Arithmetico - Geometric Progression (A.G.P.) :
General form — a, (a + d)r,
3
General H.P. —
Product of the n G.M.'s inserted between a & b is
(ab)n/2
(a)
3
(a)
(c)
3.
2
Σn = 1 + 2 + 3 + .... + n
3
Harmonic Progression (H.P)
If G1, G2, .............. Gn are n G.M's between a
and b, then
2
Ln(n + 1) OP
= M
N 2 Q
(c)
3
A±
PAGE # 29
A 2 − G2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 30
MATHS FORMULA - POCKET BOOK
BINOMIAL THEOREM
1.
MATHS FORMULA - POCKET BOOK
3.
Properties of Binomial coefficients :
n
(x + a) n = nC0 xn + nC1 xn–1 a + nC2 xn–2 a2 + .......
+ nCr xn–r ar + .... + nCn an
∑
n
Cr xn–r ar
r=0
General term - Tr+1 = nCr xn–r ar is the (r + 1)th term from
beginning.
(i)
th
(m + 1) term from the end = (n – m + 1) from beginning = Tn–m+1
(iii)
middle term
Fn I
(a) If n is even then middle term = G + 1J
H2 K
(b) If n is odd then middle term =
*
C0 + C1 + C2 + ..... + Cn = 2n
*
C0 – C 1 + C2 – C3 + ..... + C n = 0
*
C0 + C 2 + C4 + ..... = C1 + C 3 + C5 + .... = 2n–1
*
n
*
2n
FG n + 1IJ
H 2 K
*
n
*
C1 + 2C2 + 3C3 + ... + nCn = n.2n–1
*
C1 – 2C2 + 3C3 ......... = 0
*
C0 + 2C 1 + 3C2 + ......+ (n + 1)Cn = (n + 2)2n–1
*
C02 + C 12 + C22 + ..... + Cn2 =
and
term
To determine a particular term in the given expasion :
α
±
1
xβ
IJ
K
*
C02 – C 12 + C22 – C32 + .....
n
, if xn occurs in
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
I O
N
S
n n −1
nn−1
Cr −1 =
r
r r −1
Cn + r =
n−2
Cr −2 and so on ...
2n!
n−r ! n+r !
c
hc
h
Cr +
n
Cr − 1 =
n+1
Cr
=
and for x0, n α – r (α + β) = 0
C
Cr =
th
Tr+1 (r + 1)th term then r is given by n α – r (α + β) = m
D U
Cr ..... n Cn are usually denoted by C0, C1 ,.....
term
th
F
Let the given expansion be G x
H
E
C1 ,
th
Binomial coefficient of middle term is the greatest binomial coefficient.
2.
n
n
th
(ii)
FG n + 3IJ
H 2 K
C2 .....
C0 ,
C r .......... Cn respectively.
n
=
n
For the sake of convenience the coefficients
Binomial Theorem for any +ve integral index :
PAGE # 31
|RS
T|c−1h
c2nh!
cn!h
2
D U
C
A T
I O
N
S
2n
Cn
if n is odd
0,
n/2 n
Cn/2 , if n is even
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
=
PAGE # 32
MATHS FORMULA - POCKET BOOK
Note :
2n + 1
4.
2n + 1
2n + 1
C0 +
2n + 1
Cn + 2 + .....
C1 + .... +
C2n + 1 = 2
2n + 1
Cn =
2n + 1
MATHS FORMULA - POCKET BOOK
Cn + 1 +
(ii)
(x + y + z)n =
∑
r + s + t =n
n!
xr ys z t
s! r !t !
2n
Generalized (x1 + x2 +..... xk)n
*
C0 +
C1
C2
Cn
2n+1 − 1
+
+ ..... +
=
2
3
n+1
n+1
*
C0 –
C1
C2
C3
(−1)n C n
+
–
.... +
2
3
4
n+1
=
=
1
n+1
∑
r1 +r2 +...rk =n
n!
r1 r2
rk
r1 ! r2 !....rk ! x1 x2 ..... xk
Total no. of terms in the expansion (x1 + x2 +... xn)m is
m+n–1
C n–1
6.
Greatest term :
(i)
If
(n + 1)a
∈ Z (integer) then the expansion has two
x+a
greatest terms. These are kth and (k + 1)th where x & a
are +ve real nos.
(ii)
If
(n + 1)a
∉ Z then the expansion has only one greatx+a
est term. This is (k + 1)th term k =
LM(n + 1)aOP ,
N x+a Q
{[.] denotes greatest integer less than or equal to x}
5.
Multinomial Theorem :
n
(x + a)n = ∑
(i)
r =0
n
= ∑
r =0
n
Cr xn–r ar, n ∈N
n!
n!
xn–r ar = r +∑
x s ar ,
s =n s ! r !
(n − r)! r !
where s = n – r
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 33
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 34
MATHS FORMULA - POCKET BOOK
TRIGONOMETRIC RATIO AND IDENTITIES
1.
Trigonometric identities :
(i)
sin2θ + cos2θ = 1
(ii) cosec2θ – cot2θ = 1
(iii) sec2θ – tan2θ = 1
4.
Sign convention :
Arc length AB = r θ
Area of circular sector =
(ii)
1 2
r θ
2
y
π
n
(a)
Internal angle of polygon = (n – 2)
(b)
Sum of all internal angles = (n – 2) π
(c)
Radius of incircle of this polygon r =
(d)
Radius of circumcircle of this polygon R
x'
III quadrant
tan & cot
are +ve
1
na2 cot
4
(f)
Area of triangle =
(g)
F a πI
Area of incircle = π G cot J
H 2 nK
FG π IJ
H nK
π
1 2
a cos
4
n
Area of circumcircle = π
2
FG a cos ec π IJ
H2
nK
I O
N
S
IV quadrant
cos & sec
are +ve
sinθ
–sinθ
cosθ
cosθ
cosθ
tanθ
–tanθ
m sinθ
m cotθ
cotθ
–cotθ
secθ
cosecθ
2
6.
m sinθ
–cosθ
±tanθ
–cosθ
±sinθ
±sinθ
cosθ
m cotθ
m tanθ
±tanθ
±cotθ
±cotθ
m tanθ
secθ
±cosecθ secθ
m cosecθ –secθ
±cosecθ
–cosecθ secθ
m cosecθ –sec θ
Sum & differences of angles of t-ratios :
(i)
sin(A ± B) = sinA cosB ± cosA sinB
(ii) cos(A ± B) = cosA cosB ± sinA sinB
& π radian = 1800
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
x
T-ratios of allied angles : The signs of trigonometrical ratio
in different quadrant.
Allied∠ of (–θ)
900 ± θ 1800 ± θ 2700 ± θ 3600 ± θ
T-ratios
Relation between system of measurement of angles :
C
O
π
a
cot
2
n
5.
Area of the polygon =
D
G
2C
=
=
π
90
100
I quadrant
All +ve
y'
(e)
(h)
2.
II quadrant
sin & cosec
are +ve
For a regular polygon of side a and number of sides n
a
π
=
cosec
2
n
D U
3.
Some important results :
(i)
E
MATHS FORMULA - POCKET BOOK
(iii)
PAGE # 35
tan (A ± B) =
tan A ± tanB
1 m tan A tan B
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 36
MATHS FORMULA - POCKET BOOK
(iv) cot (A ± B) =
MATHS FORMULA - POCKET BOOK
7.
cot A cot B m 1
cot B ± cot A
Formulaes for product into sum or difference and viceversa :
(v) sin(A + B) sin(A – B) = sin2A – sin2B = cos2 B – cos2 A
(vi) cos(A + B) cos (A – B) = cos2A – sin2B = cos2B – sin2A
(vii) tan(A + B + C)
=
S1 − S3 + S5 − S7 +......
= 1 − S + S − S + S −......
2
4
6
8
S1 = Σ tan A
S2 = Σ tan A tan B,
S3 = Σ tan A tan B tan C & so on
(viii) sin (A + B + C) = Σ sin A cos B cos C – Π sin A
= Π cos A (Numerator of tan (A + B + C))
cos (A + B + C) = Π cos A – Σ sin A sin B cos C
= Π cos A (Denominator of tan (A + B + C))
for a triangle A + B + C = π
Σ tan A = Π tan A
Σ sin A = Σ sin A cos B cos C
1 + Π cos A = Σ sin A sin B cos C
0
(viii) sin75 =
(ix)
cos750 =
= cos15
2 2
D U
C
A T
tanA + tanB =
(i)
sin(A + B)
cos A cos B
sin2A = 2sinA cosA =
2 tan A
1 + tan2 A
(xi)
cot750 = 2 –
0
3 = tan15
PAGE # 37
2 tan A / 2
1 + tan2 A / 2
cos2A = cos2A – sin2A = 2cos2A – 1
= 1 – 2sin2A =
0
3 = cot15
S
(ix)
(ii)
tan75 = 2 +
N
sinC + sinD = 2sin
T-ratios of multiple and submultiple angles :
= sin150
(x)
I O
8.
(v)
0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
2cosA cosB = cos(A + B) + cos(A – B)
⇒ sinA = 2sinA/2 cosA/2 =
3 −1
0
(iii)
= (sin A + cos A)2 – 1 = 1 – (sin A – cos A)2
3 +1
2 2
2cosA sinB = sin(A + B) – sin(A – B)
FG C + D IJ cos FG C − D IJ
H 2 K H 2 K
F C + DI F C − DI
(vi) sinC – sinD = 2cos GH 2 JK sin GH 2 JK
F C + D IJ cos FG C − D IJ
(vii) cosC + cosD = 2cos GH
H 2 K
2 K
F C + D IJ sin FG D − C IJ
(viii) cosC – cosD = 2sin GH
H 2 K
2 K
Generalized tan (A + B + C + ......... )
(ix)
2sinA cosB = sin(A + B) + sin(A – B)
(ii)
(iv) 2sinA sinB = cos(A – B) – cos(A + B)
S1 − S3
tan A + tanB + tan C − tan A tan B tan C
= 1−S
1 − tan A tan B − tanB tan C − tan C tan A
2
Where
(i)
1 − tan2 A
1 + tan2 A
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 38
MATHS FORMULA - POCKET BOOK
(iii)
2 tan A
tan2A =
1 − tan2 A
MATHS FORMULA - POCKET BOOK
2 tan A / 2
⇒ tanA =
1 − tan2 A / 2
(iii)
sinA + sinB + sinC = 4cosA/2 cosB/2 cosC/2
(v)
sin2A + sin2B + sin2C = 1 – 2sinA sinB cosC
= sin θ (2 cos θ – 1) (2 cos θ + 1)
(vi) cos2A + cos2B + cos2C = 1 – 2cosA cosB cosC
cos3θ = 4cos3θ – 3cosθ
(vii) tanA + tanB + tanC = tanA tanB tanC
= 4cos(60 – θ ) cos(60 + θ ) cos θ
(viii) cotB cotC + cotC cotA + cotA cotB = 1
= cos θ (1 – 2 sin θ ) (1 + 2 sin θ )
(ix)
0
0
3 tan A − tan3 A
(vi) tan3A =
1 − 3 tan2 A
= tan(600 – A) tan(600 + A)tanA
(vii) sinA/2 =
(viii) cosA/2 =
(ix)
cos2A + cos2B + cos2C = –1– 4cosA cosB cosC
(iv) cosA + cosB + cosC = 1 + 4 sinA/2 sinB/2 sinC/2
(iv) sin3θ = 3sinθ – 4sin3θ = 4sin(600 – θ ) sin(600 + θ ) sin θ
(v)
(ii)
tanA/2 =
(x)
Σ tan A/2 tan B/2 = 1
Σ cot A cot B = 1
(xi)
Σ cot A/2 = Π cot A/2
11. Some useful series :
sinα + sin(α + β) + sin(α + 2β) + .... + to nterms
(i)
1 − cos A
2
LM FG n − 1IJβOP sinLnβ O
N H 2 K Q MN 2 PQ , β ≠ 2nπ
sin α +
1 + cos A
2
=
1 − cos A
1 − cos A
=
, A ≠ (2n + 1)π
sin A
1 + cos A
(ii)
sin
β
2
cosα + cos(α + β) + cos(α + 2β) + .... + to nterms
LM FG n − 1IJ βOP sin nβ
N H 2 K Q 2 β ≠ 2nπ
cos α +
9.
Maximum and minimum value of the expression :
acosθ + bsinθ
=
sin
β
2
Maximum (greatest) Value = a2 + b2
Minimum (Least) value = – a2 + b2
(iii)
sin 2n α
, α ≠ nπ
2n sin α
= 1 , α = 2kπ
10. Conditional trigonometric identities :
= –1 , α = (2k+1)π
If A, B, C are angles of triangle i.e. A + B + C = π, then
(i)
cosα .cos2α . cos22α ....cos(2n–1 α) =
sin2A + sin2B + sin2C = 4sinA sinB sinC
i.e. Σ sin 2A = 4 Π (sin A)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 39
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 40
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
TRIGONOMETRIC EQUATIONS
1.
Thus the equation reduces to form
cos(θ – α) =
General solution of the equations of the form
(i)
sinθ = 0
(ii)
cosθ = 0
θ = nπ,
n∈I
⇒
θ = (2n + 1)
⇒
θ = nπ,
(iv) sinθ = 1
⇒
θ = 2nπ +
(v)
cosθ = 1
⇒
θ = 2πn
(vi) sinθ = –1
⇒
θ = 2nπ –
(vii) cosθ = –1
⇒
θ = (2n + 1)π
(viii) sinθ = sinα
⇒
θ = nπ + (–1)nα
a + b2
= cosβ(say)
now solve using above formula
π
, n∈I
2
tanθ = 0
(iii)
2.
⇒
c
2
3.
Some important points :
n∈I
π
2
(i)
If while solving an equation, we have to square it, then
the roots found after squaring must be checked
wheather they satisfy the original equation or not.
(ii)
If two equations are given then find the common values of θ between 0 & 2π and then add 2nπ to this
common solution (value).
π
3π
or 2nπ +
2
2
θ = 2nπ ± α
(ix)
cosθ = cosα
⇒
(x)
tanθ = tanα ⇒
θ = nπ + α
(xi)
sin2θ = sin2α
⇒
θ = nπ ± α
(xii) cos2θ = cos2α
⇒
θ = nπ ± α
(xiii) tan2θ = tan2α
⇒
θ = nπ ± α
For general solution of the equation of the form
a cosθ + bsinθ = c, where c ≤
a2 + b2 , divide both side by
a2 + b2
a
and put
a +b
2
b
2
= cosα,
a + b2
2
= sinα.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 41
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 42
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
INVERSE TRIGONOMETRIC FUNCTIONS
If y = sin x, then x = sin–1 y, similarly for other inverse Tfunctions.
1.
2.
(ii)
Range (R)
– 1 ≤ x ≤ 1
π
–
2
cos–1 x
– 1 ≤ x ≤ 1
0 ≤ θ ≤ π
tan–1 x
– ∞ < x < ∞
–
cot–1 x
– ∞ < x < ∞
0 < θ < π
sec–1 x
x ≤ – 1, x ≥ 1
0 ≤ θ ≤ π, θ ≠
x ≤ – 1, x ≥ 1
π
π
–
, θ ≠ 0
≤ θ ≤
2
2
sin
x
cosec–1 x
≤ θ ≤
π
2
sin (sin–1 x) = x provided – 1 ≤ x ≤ 1
cos (cos–1 x) = x provided – 1 ≤ x ≤ 1
Domain (D)
–1
≤ θ < 0
or 0 < θ ≤
Domain and Range of Inverse T-functions :
Function
π
2
cosec–1 (cosec θ ) = θ provided –
tan (tan–1 x) = x provided – ∞ < x < ∞
π
2
cot (cot–1 x) = x provided – ∞ < x < ∞
sec (sec–1 x) = x provided – ∞ < x ≤ – 1 or 1 ≤ x < ∞
π
π
< θ <
2
2
cosec (cosec–1 x) = x provided – ∞ < x ≤ – 1
or 1 ≤ x < ∞
(iii)
π
2
sin–1 (– x) = – sin–1 x,
cos–1 (– x) = π – cos–1 x
tan–1 (– x) = – tan–1 x
cot–1 (– x) = π – cot–1 x
cosec–1 (– x) = – cosec–1 x
3.
Properties of Inverse T-functions :
sin–1 (sin θ ) = θ provided –
(i)
sec–1 (– x) = π – sec–1 x
π
2
π
≤ θ ≤ 2
cos–1 (cos θ ) = θ provided θ ≤ θ ≤ π
tan–1 (tan θ ) = θ provided –
cot
–1
C
A T
I O
N
S
∀ x ∈ [– 1, 1]
tan–1 x + cot–1 x =
π
,
2
∀ x ∈ R
(cot θ ) = θ provided 0 < θ < π
π
π
or
< θ ≤ π
2
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
π
,
2
π
π
< θ <
2
2
sec–1 (sec θ ) = θ provided 0 ≤ θ <
E
(iv) sin–1 x + cos–1 x =
sec–1 x + cosec–1 x =
PAGE # 43
π
,
x ∈ (– ∞ , – 1] ∪ [1, ∞ )
2 ∀
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 44
MATHS FORMULA - POCKET BOOK
4.
Value of one inverse function in terms of another
inverse function :
sin–1 x = cos–1
(i)
1−x
1
= sec–1
1−x
2
= tan–1
cos–1 x = sin–1 1 − x2 = tan–1
(ii)
= sec–1
1
1
= cosec–1
x
1 − x2
1 − x2
1 − x2
= cot–1
x
tan–1 x = sin–1
1 − x2
x
x
= cos–1
1 + x2
–1
(iii)
1 − x2
(iv)
= cot–1
(v)
1
x
(vi)
= sec–1
(iv) sin–1
(v)
cos
–1
(vi) tan–1
FG 1 IJ
H xK
–1
1 + x2 = cosec
1 + x2
, x ≥ 0
x
D U
C
A T
I O
N
S
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
2
–1
2
if x,y ≥ 0 & x2 + y2 ≤ 1
OP
Q
LM
N
2
2
(vii) sin–1x ± sin–1y = π – sin–1 x 1 − y ± y 1 − x ;
= cosec–1 x , ∀ x ∈ (– ∞ , 1] ∪ [1, ∞ )
FG 1 IJ
H xK
= sec
FG 1 IJ
H xK
=
–1
if x,y ≥ 0 & x2 + y2 > 1
RS cot x
|T− π + cot x
−1
−1
OP
Q
LM
N
2
2
(viii) cos–1x ± cos–1y = cos–1 xy m 1 − x 1 − y ;
x, ∀ x ∈ (– ∞ , 1] ∪ [1, ∞ )
if x,y > 0 & x2 + y2 ≤ 1
LM
N
for x > 0
OP
Q
2
2
(ix) cos–1x ± cos–1y = π – cos–1 xy m 1 − x 1 − y ;
for x < 0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
FG x + y IJ ; if x > 0, y > 0, xy < 1
H 1 − xy K
F x + y IJ ; if x > 0, y > 0, xy > 1
tan x + tan y = π + tan G
H 1 − xy K
F x−yI
tan x – tan y = tan GH 1 + xy JK ; if xy > –1
F x − y IJ ; if x > 0, y < 0, xy < –1
tan x – tan y = π + tan G
H 1 + xy K
F x + y + z − xyz IJ
tan x + tan y + tan z = tan G
H 1 − xy − yz − zx K
O
L
sin x ± sin y = sin Mx 1 − y ± y 1 − x P ;
Q
N
tan–1x + tan–1y = tan–1
(i)
(ii)
1
1 + x2
Formulae for sum and difference of inverse trigonometric function :
, 0 ≤ x ≤ 1
x
(iii)
= cot–1
5.
1
, 0 ≤ x ≤ 1
x
= cosec–1
2
x
MATHS FORMULA - POCKET BOOK
if x,y > 0 & x2 + y2 > 1
PAGE # 45
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 46
MATHS FORMULA - POCKET BOOK
6.
MATHS FORMULA - POCKET BOOK
PROPERTIES & SOLUTION OF TRIANGLE
Inverse trigonometric ratios of multiple angles
1 − x2 ), if –1 ≤ x ≤ 1
(i)
2sin–1x = sin–1(2x
(ii)
2cos–1x = cos–1(2x2 –1), if –1 ≤ x ≤ 1
Properties of triangle :
1.
A triangle has three sides and three angles.
In any ∆ABC, we write BC = a, AB = c, AC = b
FG 2x IJ = sin FG 2x IJ = cos FG 1 − x IJ
H1 − x K
H1 + x K
H1 + x K
2
(iii)
2tan–1x = tan–1
–1
2
2
–1
A
2
A
(iv) 3sin–1x = sin–1(3x – 4x3)
(v)
–1
–1
c
b
3
3cos x = cos (4x – 3x)
B
F 3x − x I
GH 1 − 3x JK
B
3
(vi) 3tan–1x = tan–1
C
a
C
and ∠BAC = ∠A, ∠ABC = ∠B, ∠ACB = ∠C
2
In ∆ABC :
(i)
A + B + C = π
(ii) a + b > c, b +c > a, c + a > b
2.
(iii)
3.
a > 0, b > 0, c > 0
Sine formula :
a
b
c
=
=
= k(say)
sin A
sinB
sin C
sin A
sinB
sinC
=
=
= k (say)
a
b
c
or
4.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
Cosine formula :
cos A =
b2 + c2 − a2
2bc
cos B =
c2 + a2 − b2
2ac
cos C =
a2 + b2 − c2
2ab
PAGE # 47
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 48
MATHS FORMULA - POCKET BOOK
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.
(c)
Napier's Analogies :
A −B
a−b
=
cot
2
a+b
C
2
tan
B−C
b−c
=
cot
2
b+c
A
2
tan
C−A
c−a
=
cot
2
c+a
B
2
tan
(a)
sin
sin
sin
(b)
A
=
2
B
=
2
C
=
2
(s − a) (s − b)
ab
A
=
2
s (s − a)
bc
B
cos
=
2
s (s − b)
ca
C
=
2
s (s − c)
ab
cos
cos
9.
C
A T
I O
N
S
(s − b) (s − c)
s (s − a)
tan
B
=
2
(s − c) (s − a)
s (s − b)
tan
C
=
2
(s − b) (s − a)
s (s − c)
1
1
1
∆ = 2 ab sin C = 2 bc sin A = 2 ca sin B
(ii)
∆ =
s(s − a) (s − b) (s − c)
tan
A
B
s −c
tan
=
2
2
s
tan
B
C
s −a
tan
=
2
2
s
tan
C
tan
2
where 2s = a + b + c
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
A
=
2
(i)
(s − b) (s − c)
bc
(s − c) (s − a)
ca
tan
∆, Area of triangle :
8.
Half angled formula - In any ∆ABC :
7.
E
MATHS FORMULA - POCKET BOOK
A
s −b
=
2
s
10. Circumcircle of triangle and its radius :
(i)
R=
a
b
c
=
=
2sin A
2sinB
2sin C
(ii)
R=
abc
4∆
PAGE # 49
Where R is circumradius
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 50
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
11. Incircle of a triangle and its radius :
∆
s
(iii)
r=
(iv)
r = (s – a) tan
(v)
(iv) r1 + r2 + r3 – r = 4R
(v)
r = 4R sin
A
B
C
= (s – b) tan
= (s – c) tan
2
2
2
1
(vi)
A
B
C
sin
sin
2
2
2
(vii)
(vi) cos A + cos B + cos C = 1 +
1
1
1
1
r1 + r2 + r3 = r
r
R
r12
1
+
(ix)
A
B
, r2 = s tan
, r3 = s tan
2
2
r1 = s tan
(iii)
r1 = 4R sin
A
B
C
cos
cos
,
2
2
2
r2 = 4R cos
A
B
C
sin
cos
,
2
2
2
r3 = 4R cos
A
B
cos
sin
2
2
D U
C
A T
I O
N
S
r2
=
a2 + b2 + c2
∆2
∆ = 2R2 sin A sin B sin C = 4Rr cos
A
B
C
cos
cos
2
2
2
B
C
C
A
b cos cos
cos
2
2
2
2
, r2 =
,
A
B
cos
cos
2
2
(x)
r1 =
A
B
cos
2
2
C
cos
2
c cos
r3 =
C
2
C
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
1
a cos
∆
∆
∆
, r2 =
, r3 =
s−a
s −b
s−c
(ii)
+
1
1
1
1
+
+
=
2Rr
bc
ca
ab
12. The radii of the escribed circles are given by :
r1 =
r32
(viii) r1r2 + r2r3 + r3r1 = s2
B
C
A
C
B
A
a sin sin
b sin sin
c sin sin
2
2
2
2
2
2
(vii) r =
=
=
A
B
C
cos
cos
cos
2
2
2
(i)
1
+
r22
PAGE # 51
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 52
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
HEIGHT AND DISTANCE
1.
(ii)
d = h (cotα – cotβ)
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.
h
α
β
d
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 = α , ∠ BCD = β & ∠ BDC = θ
then (m + n) cotθ = m cotα – ncot β
C
α β
θ
A
A
m
D
B
n
B
= ncotA – mcotB [m – n Theorem]
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 53
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 54
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(C)
POINT
1.
Distance formula :
Distance between two points P(x1, y1) and Q(x2, y2) is
given by d(P, Q) = PQ
=
(x 2 − x1)2 + (y2 − y1 )2
(Difference of x coordinate)2 + (Difference of y coordinate)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) =
2.
x2 + y2
AP
m
=
= λ , Here λ > 0
BP
n
C
A T
I O
N
S
n
m
A(x 1 , y1 )
P
(ii)
FG mx + nx
H m+n
2
1
,
B(x 2 , y2 )
P
my 2 + ny1
m+n
IJ
K
Externally :
m
AP
m
=
= λ
BP
n
P
In Parallelogram :
Calculate AB, BC, CD and AD.
(i)
If AB = CD, AD = BC, then ABCD is a parallelogram.
(ii) If AB = CD, AD = BC and AC = BD, then ABCD
is 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.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
Section formula :
(i)
Internally :
Use of Distance Formula :
(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 to
the third, then ∆ is right triangle.
(iv) Sum of any two equal to left third they do not
form a triangle
i.e. AB = BC + CA or BC = AC + AB
or AC = AB + BC. Here points are collinear.
(b)
E
3.
For circumcentre of a triangle :
Circumcentre of a triangle is equidistant from vertices
i.e. PA = PB = PC.
Here P is circumcentre and PA is radius.
(i)
Circumcentre of an acute angled triangle is inside the triangle.
(ii) Circumcentre of a right triangle is mid point of
the hypotenuse.
(iii) Circumcentre of an obtuse angled triangle is
outside the triangle.
(iii)
FG mx − nx
H m−n
2
1
,
n
P
B(x 2, y 2 )
A(x 1, y 1 )
my 2 − ny1
m−n
IJ
K
Coordinates of mid point of PQ are
FG x
H
1
+ x2 y1 + y2
,
2
2
IJ
K
(iv) The line ax + by + c = 0 divides the line joining the
points
(ax1 + by + c)
(x1, y1) & (x2, y2) in the ratio = – (ax + by + c)
2
2
PAGE # 55
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 56
MATHS FORMULA - POCKET BOOK
(v)
For parallelogram – midpoint of diagonal AC = mid point
of diagonal BD
(vi) Coordinates of centroid G
FG x
H
1
+ x 2 + x 3 y1 + y 2 + y 3
,
3
3
MATHS FORMULA - POCKET BOOK
5.
Area of Polygon :
Area of polygon having vertices (x 1, y1), (x2, y2), (x3, y3)
........ (xn, yn) is given by area
IJ
K
(vii) Coordinates of incentre I
FG ax + bx + cx
H a+b+c
1
2
3
ay + by 2 + cy 3
, 1
a+b+c
IJ
K
=
(viii) Coordinates of orthocentre are obtained by solving the
equation of any two altitudes.
4.
Area of Triangle :
The area of triangle ABC with vertices A(x1, y1), B(x2, y2)
and C(x3, y3).
∆=
=
1
2
x1
y1 1
x2
y2 1
x3
y3 1
x1
y1
x2
1
2 x3
y2
x1
y1
y3
6.
1
[x y + x2y3 + x3y1 – x2y1 – x3y2 – x1y3]
2 1 2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
A T
I O
N
S
y2
x3
y3
M
M
xn
yn
x1
y1
. Points must be taken in order.
B
y
B
x' → cosθ
sinθ
y' → –sinθ
cosθ
Some important points :
(i)
Three pts. A, B, C are collinear, if area of triangle is
zero
(ii) Centroid G of ∆ABC divides the median AD or BE or CF in
the 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 orthocentre
and circumcentre in the ratio 2 : 1
(v) Area of triangle formed by coordinate axes & the line
Note :
(i)
Three points A, B, C are collinear if area of triangle
is zero.
(ii) If in a triangle point arrange in anticlockwise then
value of ∆ be +ve and if in clockwise then ∆ will be
–ve.
D U
x2
x
(Determinant method)
[Stair method]
E
y1
Rotational Transformation :
If coordinates of any point P(x, y) with reference to new
axis will be (x', y') then
7.
=
1
2
x1
ax + by + c = 0 is
PAGE # 57
c2
.
2ab
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 58
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(ix)
STRAIGHT LINE
1.
(x)
making an angle θ , 0 ≤ θ ≤ π , θ ≠
a
b
(xi)
(vi) Slope of two parallel lines are equal.
(vii) If m1 & m2 are slopes of two ⊥ lines then m1m2 = – 1.
2.
Standard form of the equation of a line :
(i)
Equation of x-axis is y = 0
(ii) Equation of y-axis is x = 0
(iii) Equation of a straight line || to x-axis at a distance
b from it is y = b
(iv) Equation of a straight line || to y-axis at a distance
a from it is x = a
(v) Slope form : Equation of a line through the origin and
having slope m is y = mx.
(vi) Slope Intercept form : Equation of a line with slope m
and making an intercept c on the y-axis is y = mx + c.
(vii) Point slope form : Equation of a line with slope m
and passing through the point (x1, y1) is
y – y1 = m(x – x1)
(viii) Two point form : Equation of a line passing through
the points (x1, y1) & (x2, y2) is
3.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
C
A T
I O
N
S
x − x1
y − y1
=
= r
cos θ
sinθ
x = x1 + r cos θ , y = y1 + r sin θ
⇒
Where r is the distance of any point P(x, y) on the
line from the point (x1, y1)
Normal or perpendicular form : Equation of a line
such that the length of the perpendicular from the
origin on it is p and the angle which the perpendicular
makes with the +ve direction of x-axis is α , is
x cos α + y sin α = p.
Angle between two lines :
(i)
Two lines a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0 are
y − y1
x − x1
y2 − y1 = x2 − x1
E
π
with the
2
+ve direction of x-axis is
y2 − y1
B(x2, y2) is x − x .
2
1
Slope of the line ax + by + c = 0, b ≠ 0 is –
x
y
+
= 1.
a
b
Parametric or distance or symmetrical form of the
line : Equation of a line passing through (x1, y1) and
and b respectively on x-axis and y-axis is
Slope of a Line : The tangent of the angle that a line
makes with +ve direction of the x-axis in the anticlockwise
sense is called slope or gradient of the line and is generally
denoted by m. Thus m = tan θ .
(i)
Slope of line || to x-axis is m = 0
(ii) Slope of line || to y-axis is m = ∞ (not defined)
(iii) Slope of the line equally inclined with the axes is 1
or – 1
(iv) Slope of the line through the points A(x1, y1) and
(v)
Intercept form : Equation of a line making intercepts a
(a)
a1
b1
c1
Parallel if a = b ≠ c
2
2
2
(b)
Perpendicular if a1a2 + b1b2 = 0
(c)
a1
b1
c1
Identical or coincident if a = b = c
2
2
2
(d)
a2b1 − a1b2
If not above three, then θ = tan–1 a a − b b
1 2
1 2
(ii)
(a)
(b)
Two lines y = m1 x + c and y = m2 x + c are
Parallel if m1 = m2
Perpendicular if m1m2 = –1
(c)
m1 − m2
If not above two, then θ = tan–1 1 + m m
1 2
PAGE # 59
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 60
MATHS FORMULA - POCKET BOOK
4.
Position of a point with respect to a straight line :
The line L(xi, yi) i = 1, 2 will be of same sign or of opposite
sign according to the point A(x1, y1) & B (x2, y2) 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 + λ = 0)
6.
Equation of st. lines through (x1,y1) making an angle α
with y = mx + c is
12. Homogeneous equation : If y = m1x and y = m2x be the
two equations
represented by ax2 + 2hxy + by2 = 0 , then m1 + m2 = –2h/b
and m1m2 = a/b
13. General equation of second degree :
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
represent a pair of
a h g
h
b f =0
straight line if ∆ ≡
g f c
m ± tan α
(x – x1)
1 m m tan α
y – y1 =
7.
MATHS FORMULA - POCKET BOOK
If y = m1x + c & y = m2x + c represents two straight lines
length of perpendicular from (x1, y1) on ax + by + c = 0
|ax1 + by1 + c|
is
8.
then m1 + m2 =
a2 + b2
Distance between two parallel lines ax + by + ci = 0,
i = 1, 2 is
9.
|c1 − c 2|
a2 + b2
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
2 h2 − ab
is tanθ =
(a + b)
Condition of concurrency for three straight lines
Li ≡ ai x + bi y + ci = 0, i = 1, 2, 3 is
a1 b1
a2 b2
a3 b3
(i)
c1
c2 = 0
c3
(ii)
10. Equation of bisectors of angles between two lines :
a1x + b1y + c1
a +b
2
1
2
1
a2 x + b2 y + c 2
=±
a22 + b22
11. Family 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 of
intersection of two given lines L1 and L2 can be taken as
L1 + λ L2 = 0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
−2h
a
, m1m2 = .
b
b
The two lines given by ax2 + 2hxy + by2 = 0 are
(a) Parallel and coincident iff h2 – ab = 0
(b) Perpendicular iff a + b = 0
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
2
x −y
a−b
PAGE # 61
2
=
xy
h
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 62
MATHS FORMULA - POCKET BOOK
CIRCLE
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)
1.
i.e.
(ii)
2.
4.
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, y1 ) lies
outside, on or inside a circle
FG − 1 coeff. of x, −1 coeff. of yIJ
K
H 2
2
Radius is
S ≡ x2 + y2 + 2gx + 2fy + c = 0 according as
S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c is +ve, zero or –ve
g2 + f 2 − c
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.
MATHS FORMULA - POCKET BOOK
7.
Chord length (length of intercept) = 2 r2 − p2
8.
Intercepts made on coordinate axes by the circle :
(i)
x axis = 2 g2 − c
(ii)
y axis = 2 f 2 − c
Diameter form : If (x1, y1) and (x2, y2) are end pts. of a
diameter of a circle, then its equation is
(x – x1) (x – x2) + (y – y1) (y – y2) = 0
9.
Parametric equations :
(i)
The parametric equations of the circle x2 + y2 = r2 are
x = rcosθ, y = r sinθ ,
10. Length of the intercept made by line : y = mx + c with the
circle x2 + y2 = a2 is
Length of tangent = S1
where point θ ≡ (r cos θ , r sin θ )
(ii)
(iii)
The parametric equations of the circle
(x – h)2 + (y – k)2 = r2 are x = h + rcosθ, y = k + rsinθ
The parametric equations of the circle
x2 + y2 + 2gx + 2fy + c = 0 are
x = –g +
(iv)
g2 + f 2 − c cosθ, y = –f +
D U
C
A T
I O
N
S
(1 + m2) |x1 – x2|
where |x1 – x2| = 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
θ1 + θ2
θ + θ2
θ − θ2
+ y sin 1
= r cos 1
.
2
2
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
or
g2 + f 2 − c sinθ
For circle x2 + y2 = a2, equation of chord joining θ 1 & θ 2 is
x cos
a2 (1 + m2 ) − c2
1 + m2
2
PAGE # 63
1 + m2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 64
MATHS FORMULA - POCKET BOOK
15. The point of intersection of tangents drawn to the circle x2
+ y2 = r2 at point θ 1 & θ 2 is given as
12. Equation of tangent, T = 0 :
(i)
MATHS FORMULA - POCKET BOOK
Equation of tangent to the circle
F r cos θ + θ
GG
2
θ −θ
GH cos 2
x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y1) is
1
xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
(ii)
Equation of tangent to the circle x2 + y2 = a2 at any
point (x1, y1) is xx1 + yy1 = a2
(iii)
In slope form : From the condition of tangency for
every value of m.
1
2
2
θ1 + θ 2
2
,
θ1 − θ2
cos
2
r sin
I
JJ
JK
16. Equation of the chord of contact of the tangents drawn
from point P outside the circle is T = 0
The line y = mx ± a 1 + m2 is a tangent to the circle
17. Equation of a chord whose middle pt. is given by T = S1
x2 + y2 = a2 and its point of contact is
F
GH
±am
1 + m2
,
±a
1 + m2
I
JK
18. Director circle : Equation of director circle for x2 + y2 = a2 is
x2 + y2 = 2a2. Director circle is a concentric circle whose
radius is
(iv) Equation of tangent at (a cos
θ , a sin
θ ) to the
circle x + y = a is x cos θ + y sin θ = a.
2
2
2
19.
Equation of polar of point (x1, y1) w.r.t. the circle S = 0 is T = 0
20. Coordinates of pole : Coordinates of pole of the line
13. Equation of normal :
(i)
F −a l −a mI
are G n , n J
K
H
2
Equation of normal to the circle
2
y1 + f
y – y1 = x + g (x – x1)
1
2
2
lx + my + n = 0 w.r.t the circle x + y = a
x2 + y2 + 2gx + 2fy + c = 0 at any point P(x1, y1) is
(ii)
2 times the radius of the given circle.
2
21. Family of Circles :
S + λS' = 0 represents a family of circles passing through
the pts. of intersection of
(i)
2
2
2
Equation of normal to the circle x + y = a
point (x1, y1) is xy1 – x1y = 0
at any
S = 0 & S' = 0 if λ ≠ –1
14. Equation of pair of tangents SS1 = T2
(ii)
S + λ 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, y1) is given as
(x – x1)2 + (y – y1)2 + λL = 0.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 65
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 66
MATHS FORMULA - POCKET BOOK
(iv) Equation of circle passing through two points A(x1, y1)
& B(x2, y2) is given as
(x – x1) (x – x2) + (y – y1) (y – y2) + λ
x
y
x1
y1 1
x2
1
MATHS FORMULA - POCKET BOOK
25. Equation of tangent at point of contact of circle is
S 1 – S2 = 0
26. Radical axis and radical centre :
= 0.
y2 1
22. Equation of Common Chord is S – S1 = 0.
(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 = 0
23. The angle θ of intersection of two circles with centres C1
& C2 and radii r1 & r2 is given by
and S' = x2 + y2 + 2g'x + 2f'y + c' = 0 intersect each other
orthogonally, then 2gg' + 2ff' = c + c'.
r12 + r12 − d2
, where d = C1C2
cosθ =
2r1r2
24. Position of two circles : Let two circles with centres C1, C2
and radii r1, r2 .
Then following cases arise as
(i)
C1 C2 > r1 + r2 ⇒ do not intersect or one outside the
other, 4 common tangents.
(ii)
C1 C2 = r1 + r2 ⇒ Circles touch externally, 3 common
tangents.
(iii)
|r1 – r2| < C1 C2 < r1 + r2 ⇒ Intersection at two real
points, 2 common tangents.
(iv) C1 C2 = |r1 – r2| ⇒ internal touch, 1 common tangent.
(v)
C1 C2 < |r1 + r2| ⇒ one inside the other, no tangent.
Note : Point of contact divides C1 C2 in the ratio r1 : r2
internally or externally as the case may be
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 67
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 68
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
PARABOLA
1.
Standard Parabola :
Imp. Terms
y2 = 4ax y2 = – 4ax x2 = 4ay
x2 = – 4ay
Vertex (v)
(0, 0)
(0, 0)
(0, 0)
(0, 0)
Focus (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
Focal
x + a
a – x
y + a
a – y
(at2, 2at) (– at2, 2at)
(2at, at2)
(2at, – at2)
Parametric
x = at2
x = – at2
x = 2at
x = 2at
Equations
y = 2at
y = 2at
y = 2at2
y = – at2
y2 = – 4ax
distance
Parametric
Coordinates
x2 = 4ay
y2 = 4ax
x2 = – 4ay
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 69
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 70
MATHS FORMULA - POCKET BOOK
2.
Special Form 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
MATHS FORMULA - POCKET BOOK
4.
Equations of tangent in different forms :
(i)
Equations of tangent of all other standard parabolas
at (x1, y1) / at t (parameter)
FGh + l , kIJ
H 4 K
Equation
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
FGh, k + l IJ
H
4K
i.e.
FG
H
b
4ac − b2
= a x+
y –
2a
4a
F − b , 4ac − b I
GH 2a 4a JK
IJ
K
y 2=4ax
yy1 =2a(x+x1 )
(at2, 2at)
y 2 =–4ax
yy 1=–2a(x+x 1 ) (–at2, 2at)
xx1 =2a(y+y1 )
x 2 =–4ay
xx 1=–2a(y+y 1 ) (2at, –at2)
Equation
of
parabolas
,with vertex
and axes parallel to y-axis
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
I O
N
S
tx=y + at2
(2at, at )
tx =–y+at2
Point of
Equations
Condition of
contact in
terms of
of tangent
in terms of
Tangency
slope(m)
y2 = 4ax
y2 = – 4ax
x
5.
2
= – 4ay
FG a
Hm
2
FG − a
H m
,
2
,
2a
m
slope (m)
IJ
K
2a
m
IJ
K
(2am, am2)
2
(–2am, –am )
y = mx +
a
m
y = mx –
a
m
c =
a
m
c = –
y = mx – am2
2
y = mx + am
a
m
c = –am2
c = am2
Point of intersection of tangents at any two points
P(at12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax
is (at1t2, a(t1 + t2)) 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,
Note : Condition of tangency for parabola y2 = 4ax, we
have c = a/m and for other parabolas check disc. D = 0.
A T
ty=–x+at2
Equations of tangent of all other parabolas in slope
form
2
Position of a point (x1, y1) and a line w.r.t. parabola
y2 = 4ax.
*
The point (x1, y1) lies outside, on or inside the
parabola y2 = 4ax
according as y12 – 4ax1 >, = or < 0
*
The line y = mx + c does not intersect, touches,
intersect a parabola y2 = 4ax according as
c > = < a/m
C
ty=x+at2
2
x =4ay
x2 = 4ay
D U
Tangent of 't'
(ii) Slope form
Note : Parametric equation of parabola (y – k)2
= 4a(x – h) are x = h + at2, y = k + 2at
E
Parametric
coordinates't'
2
3.
Tangent at
of parabola (x 1, y1)
2
Equation of the form ax2 + bx + c = y represents
parabola.
*
Point Form / Parametric form
S' = y12 – 4ax1 and T = yy1 – 2a(x + x1)
PAGE # 71
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 72
MATHS FORMULA - POCKET BOOK
7.
MATHS FORMULA - POCKET BOOK
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.
Equations of normal in different forms
(i)
Point Form / Parametric form
Equations of normals of all other standard parabolas
at (x1, y1) / at t (parameter)
Eqn. of
Normal
Point
Normals
parabola
at (x1, y1)
't'
at 't'
y2 = 4ax
y–y1 =
−y1
(x–x1) (at2, 2at)
2a
y+tx = 2at+at3
y2 = –4ax
y–y1 =
y1
(x–x1)
2a
y–tx = 2at+at3
x2 = 4ay
x2 = –4ay
(ii)
y–y1 = –
y–y1 =
2a
x1
2a
x1
(x–x1)
(–at2, 2at)
(2at, at2)
x+ty = 2at+at3
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
(iii)
We get one normal if h ≤ 2a.
If point lies on x-axis, then one normal will be x-axis
itself.
(i)
If normal of y2 = 4ax at t1 meet the parabola again
(ii)
9.
at t2 then t2 = – t1 –
(x–x1)
(2at, –at2)
x–ty = 2at+at3
(ii)
The normals to y2 = 4ax at t1 and t2 intersect each
other at the same parabola at t3, then
t1t2 = 2 and t3 = – t1 – t2
10. (i)
Equation of focal chord of parabola y2 = 4ax at t1 is
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 = mx–2am–am3 c = –2am–am3
y
2
y =
(ii)
= – 4ax (–am2, 2am) y = mx+2am+am3 c = am+am3
x2 = 4ay
FG − 2a , a IJ
H m mK
2
y = mx+2a+
a
2
m
c = 2a+
a
FG 2a , − a IJ
Hm m K
2
y = mx–2a–
a
2
m
m
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
c = –2a–
2t1
t12
−1
a
m2
PAGE # 73
(x – a)
If focal chord of y 2 = 4ax cut (intersect) at t1 and
t2 then t1t2 = – 1 (t1 must not be zero)
Angle formed by focal chord at vertex of parabola is
tan θ =
2
(iii)
x2 = –4ay
2
t1
2
|t2 – t1|
3
Intersecting point of normals at t1 and t2 on the
parabola y2 = 4ax is
(2a + a(t12 + t22 + t1t2), – at1t2 (t1 + t2))
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 74
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
11. Equation of chord of parabola y2 = 4ax which is bisected
at (x1, y1) is given by T = S1
(v)
12. The locus of the mid point of a system of parallel chords
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.
2a
.
m
(vi) Intercepted portion of a tangent between the point
of tangency and directrix will make right angle at
focus.
13. Equation of polar at the point (x1, y1) with respect to
parabola y2 = 4ax is same as chord of contact and is given
by
(vii) Circle drawn on any focal radius as diameter will
touch tangent at vertex.
of a parabola is called its diameter. Its equation is y =
(viii) Circle drawn on any focal chord as diameter will touch
directrix.
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 is
FG n , −2amIJ
Hl l K
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.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 75
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 76
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
Note : If P is any point on ellipse and length of perpendiculars
from to minor axis and major axis are p1 & p2, then |xp|
= p1 , |yp| = p2
ELLIPSE
1.
Standard Ellipse (e < 1)
R| x
S| a
T
2
Ellipse
2
Imp. terms
+
y
2
b2
U|
= 1V
W|
⇒
For a > b
For b > a
Centre
(0, 0)
(0, 0)
Vertices
(±a, 0)
(0, ±b)
Length of major axis
2a
2b
Length of minor axis
2b
2a
Foci
(±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
F ±ae, ± b I
GH
a JK
2
Ends of latus rectum
Parametric coordinates
p12
a2
+
p22
b2
= 1
a > b
F ± a , ± beI
GH b
JK
2
(a cos φ , b sin φ ) (a cos φ , b sin φ )
0 ≤ φ < 2π
Focal 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 bt
2ae
2be
Distance btn directrices
2a/e
2b/e
Tangents at the vertices
x = –a, x = a
y = b, y = – b
n
foci
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
b > a
PAGE # 77
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 78
MATHS FORMULA - POCKET BOOK
2.
MATHS FORMULA - POCKET BOOK
Special form of ellipse :
(ii)
If the centre of an ellipse is at point (h, k) and the
directions of the axes are parallel to the coordinate axes,
cx − hh
2
then its equation is
3.
a2
+
(y − k)2
b2
ellipse
= 1.
If
a2
+
y2
x2
a2
(iii)
*
x12
2
+
a
y12
2
x2
a2
The line y = mx + c does not intersect, touches,
intersect, the ellipse if
(i)
7.
±b2
a2m2 + b2
I
JK .
Parametric form : The equation of tangent at any
xx1
a2
+
x
2
a2
+
(i)
yy1
b2
y
S
= 1 is given by SS1 = T2
to the ellipse
= 1 at the point (x1, y1) is
x2
2
a
+
y2
b2
= 1 is
a2 x
b2x
–
= a2 – b2.
y1
x1
= 1.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
N
b2
Point form : The equation of the normal at (x1, y1)
2
b2
+
y2
Equation of normal in different forms :
Point form : The equation of the tangent to the
ellipse
I O
a2m2 + b2
,
Equation of pair of tangents from (x 1, y1) to an ellipse
Equation of tangent in different forms :
A T
= 1 at
±a2m
x
y
cos φ +
sin φ = 1.
a
b
– 1 > , = or < 0
b
a2m2 + b2 < = > c2
C
b2
F
GH
The point lies outside, on or inside the ellipse if
S1 =
D U
y2
a2m2 + b2 touches the ellipse
point (a cos φ , b sin φ ) is
6.
E
+
Position of a point and a line w.r.t. an ellipse :
5.
= 1, then c2 = a2m2 + b2. Hence, the
b2
Line y = mx ±
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.
*
+
Point of contact :
x2 + y2 = a2.
4.
a2
y2
a2m2 + b2 always represents
the tangents to the ellipse.
= 1 is an ellipse then its auxillary circle is
b2
x2
straight line y = mx ±
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.
x2
Slope form : If the line y = mx + c touches the
PAGE # 79
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 80
MATHS FORMULA - POCKET BOOK
(ii)
MATHS FORMULA - POCKET BOOK
(v)
Parametric form : The equation of the normal to the
ellipse
x2
a2
+
y2
b2
= 1 at (a cos φ , b sin φ ) is
(vi) If a light ray originates from one of focii, then it will
pass through the other focus after reflection from
ellipse.
ax sec φ – by cosec φ = a – b .
2
(iii)
Sum of square of intercept made by auxillary circle on
any two perpendicular tangents of an ellipse will be
constant.
2
Slope form : If m is the slope of the normal to the
ellipse
x2
a2
+
is y = mx ±
y2
b2
= 1, then the equation of normal
m (a2 − b2 )
a2 + b2m2
9.
Equation of chord of contact of the tangents drawn from
the external point (x1, y1) to an ellipse is given by
xx1
.
2
a
yy1
+
b2
= 0 i.e. T = 0.
The co-ordinates of the point of contact are
F
GH
±a2
a2 + b2m2
,
±mb2
a2 + b2m2
I
JK .
(x1, y1) to an ellipse
a2
+
y2
b2
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.
x
cos
a
(ii)
The locus of foot of perpendicular drawn from either
focii to any tangent lies on auxillary circle.
(i)
(iii)
The circle drawn on any focal radius as diameter will
touch auxillary circle.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
C
A T
I O
N
S
θ+φ
y
+
sin
b
2
b2
= 1 whose
x2
a2
θ+φ
= cos
2
+
(ii)
PAGE # 81
C
A T
I O
N
= 1 is
θ−φ
2
tan
θ1
, tan
2
θ2
±e − 1
=
2
1±e
θ 1 + θ 2 + θ 3 + θ 4 = (2n + 1) π .
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
b2
Sum of feet of eccentric angles is odd π.
i.e.
E
y2
Relation between eccentric angles of focal chord
⇒
(iv) The protion of the tangent intercepted between the
point and directrix makes right angle at corresponding
focus.
E
y2
11. Equation of chord joining the points (a cos θ , b sin θ ) and
= 1.
(a cos φ , b sin φ ) on the ellipse
8.
a2
+
mid point is (x1, y1) is T = S1.
Note : In general three normals can be drawn from a point
x2
x2
10. The equation of a chord of an ellipse
S
PAGE # 82
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
12. Equation of polar of the point (x1, y1) w.r.t. the ellipse
x2
a2
y2
+
= 1 is given by
b2
xx1
yy1
+
a2
b2
(c)
= 0 i.e. T = 0.
the ellipse
+
a2
= 1 is
b2
(d)
F −a l , −b nI
GH n n JK .
2
y2
2
13. Eccentric angles of the extremities of latus rectum of the
ellipse
14. (i)
x2
+
2
a
y2
2
b
FG ± b IJ .
H aeK
= 1 are tan–1
Equation of the diameter bisecting the chords of
slope in the ellipse
y = –
(ii)
b2
a2m
x2
a2
+
y2
b2
x2
a2
+
y2
b2
= 1 and S, S' be two foci
of the ellipse, then SP.S'P = CQ2
The pole of the line l x + my + n = 0 w.r.t. the ellipse
x2
If CP, CQ be two conjugate semi-diameters of
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(x 1, y1) to the
ellipse
x2
= 1 is
a2
+
y2
b2
a2
= 1 is
– x1 & (1 – e2) x1
x1
x
Conjugate Diameters : The straight lines y = m1x,
y = m2x are conjugate diameters of the ellipse
x2
a2
(iii)
+
y2
b2
= 1 if m1m2 = –
b2
a2
.
Properties of conjugate diameters :
(a) If CP and CQ be two conjugate semi-diameters
x2
of the ellipse
a2
+
y2
b2
= 1, then
CP2 + CQ2 = a2 + b2
(b)
If θ and φ are the eccentric angles of the
extremities of two conjugate diameters, then
θ – φ = ±
π
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 83
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 84
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
HYPERBOLA
1.
Standard Hyperbola :
Hyperbola
x2
a2
–
y2
b2
= 1
Imp. terms
–
x2
a2
or
(0, 0)
2a
2b
2b
(±ae, 0)
2a
(0, ±be)
x = ±a/e
y = ± b/e
= – 1
e =
Length of L.R.
Parametric
2b2/a
2a2/b
co-ordinates
(a sec φ , b tan φ )
(b sec φ , a tan φ )
0 ≤ φ < 2π
0 ≤ φ < 2π
SP = ex1 – a
S'P = ex1 + a
2a
SP = ey1 – b
S'P = ey1 + b
2b
Fa + b I
GH a JK
2
2
I O
N
S
Hyperbola
Fa + b I
GH b JK
2
e =
2
2
y = – b, y = b
x = 0
Conjugate Hyperbola
y = 0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
b2
(0, 0)
S'P – SP
Tangents at
the vertices
x = – a, x = a
Equation of the y = 0
transverse axis
Equation of the x = 0
conjugate axis
C
y2
–
Eccentricity
Focal radii
D U
a2
= 1
b2
Centre
Length of
transverse axis
Length of
conjugate axis
Foci
Equation of
directrices
2
E
x2
y2
+
PAGE # 85
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 86
MATHS FORMULA - POCKET BOOK
2.
MATHS FORMULA - POCKET BOOK
Special form of hyperbola :
(b)
If the centre of hyperbola is (h, k) and axes are parallel
to the co-ordinate axes, then its equation is
(x − h)
2
a2
3.
(y − k)
Parametric form : The equation of tangent to the
hyperbola
x2
2
–
a2
= 1.
b2
y2
b2
= 1 at (a sec φ , b tan φ ) is
x
y
sec φ –
tan φ = 1.
b
a
Parametric equations of hyperbola :
(c)
The equations x = a sec φ and y = b tan φ are known
as the parametric equations of hyperbola
–
x2
2
a
–
y2
2
b
Slope form : The equations of tangents of slope m
to the hyperbola
x2
= 1
a2
–
y2
b2
= 1 are y = mx ±
a2m2 − b2 and the
co-ordinates of points of contacts are
4.
Position of a point and a line w.r.t. a hyperbola :
F±
GH
The point (x1, y1) lies inside, on or outside the hyperbola
x2
–
a2
y2
b2
6.
x12
–
a2
y12
b2
7.
(a)
x2
a2
y2
–
b2
xx1
2
a
–
2
b
(b)
I O
N
S
= 1 is given by SS1 = T2
b2
Point form : The equation of normal to the hyperbola
–
y2
b2
= 1 at (x1, y1) is
a2 x
b2y
+
= a2 + b2.
y1
x1
Parametric form : The equation of normal at
(a sec θ , b tan θ ) to the hyperbola
x2
= 1.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
y2
a2
= 1
yy1
–
x2
Point form : The equation of the tangent to the
at (x1, y1) is
C
a2m2 − b2
I
JK .
Equations of normals in different forms :
Equations of tangents in different forms :
hyperbola
D U
x2
a2
according as c2 <, =, > a2m2 – b2.
E
b2
Equation of pair of tangents from (x1, y1) to the hyperbola
– 1 is +ve, zero or –ve.
The line y = mx + c does not intersect, touches, intersect
the hyperbola
(a)
a2m2 − b2
,±
= 1
according as
5.
a2m
a2
PAGE # 87
–
y2
b2
= 1 is ax cos θ + by cot θ = a2 + b2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 88
MATHS FORMULA - POCKET BOOK
(c)
Slope form : The equation of the normal to the
hyperbola
x2
2
a
–
y2
b
–
a2
y2
b2
are
8.
a2 − b2m2
a2
a2 − b2m2
or c2 =
m(a2 + b2 )2
–
a2
9.
y2
b2
2
(a2 − m2b2)
,m
mb2
a2 − b2m2
2
2
a2
–
–
I
JK .
x2
a2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
I O
N
S
y
sin
b
FG φ
H
1
+ φ2
2
IJ
K
= cos
FG φ
H
1
IJ
K
+ φ2
.
2
–
y2
= 1 is y =
b2
*
2
b2
a2m
x.
–
y2
b2
b2
a2
The equations of asymptotes of the hyperbola
x2
a2
F − a l , b mI
GH n n JK
15. Asymptotes of a hyperbola :
= 1.
x2
= 1 is
b2
m1m2 =
–
y2
b2
= 1 are y = ±
b
x.
a
Asymptote to a curve touches the curve at infinity.
= 1
*
The asymptote of a hyperbola passes through the
centre of the hyperbola.
whose mid point is (x1, y1) is T = S1.
C
–
14. The diameters y = m1x and y = m2x are conjugate if
2
10. The equation of chord of the hyperbola
D U
IJ
K
2
y2
a2
E
− φ2
2
13. The equation of a diameter of the hyperbola
= 1 is x + y = a – b .
yy1
b2
a2
, which
Equation of chord of contact of the tangents drawn from
the external point (x1, y1) to the hyperbola is given by
xx1
1
12. Equation of polar of the point (x1, y1) w.r.t. the hyperbola
is given by T = 0.
x2
The equation of director circle of hyperbola
x2
FG φ
H
The pole of the line l x + my + n = 0 w.r.t.
is condition of normality.
Points of contact : Co-ordinates of points of contact
F±
GH
φ 2, b tan φ 2) is
a2 − b2m2
= 1,
m (a2 + b2 )
then c = m
(e)
x
cos
a
2
Condition for normality : If y = mx + c is the normal
x2
and Q(a sec
m (a + b )
the normal is y = mx m
of
11. Equation of chord joining the points P(a sec φ 1, b tan φ 1)
= 1 in terms of the slope m of
2
2
(d)
MATHS FORMULA - POCKET BOOK
PAGE # 89
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 90
MATHS FORMULA - POCKET BOOK
*
MATHS FORMULA - POCKET BOOK
The combined equation of the asymptotes of the
hyperbola
*
x2
a2
–
y2
b2
x2
= 1 is
a2
–
y2
b2
*
Equation of tangent at (x1, y1) to xy
= 0.
x
y
= c2 is x + y = 2.
1
1
The angle between the asymptotes of
Equation of tangent at t is x + yt2 = 2ct
x
2
2
a
*
–
2
y
2
b
= 1 is 2 tan–1
y
2
b2
or 2 sec–1 e.
Equation of normal at (x1, y1) to xy = c2 is
*
xx1 – yy1 = x12 – y12
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.
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.
16. Rectangular or Equilateral Hyperbola :
*
A hyperbola for which a = b is said to be rectangular
hyperbola, its equation is x2 – y2 = a2
*
Equation of chord of the hyperbola xy = c2 whose
middle point is given is T = S1
*
xy = c2 represents a rectangular hyperbola with
asymptotes x = 0, y = 0.
*
Point of intersection of tangents at t1 & t2 to the
hyperbola xy = c2 is
2 and angle
between asymptotes of rectangular hyperbola is 90º.
*
Eccentricity of rectangular hyperbola is
*
Parametric equation of the hyperbola xy = c2 are
x = ct, y =
F 2c t t
GH t + t
1 2
1
2
,
2c
t1 + t 2
I
JK
c
, where t is a parameter.
t
Equation of chord joining t1, t2 on xy = c2 is
*
x + y t1t2 = c(t1 + t2)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 91
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 92
MATHS FORMULA - POCKET BOOK
MEASURES OF CENTRAL TENDENCY
AND DISPERSION
1.
MATHS FORMULA - POCKET BOOK
3.
Geometric Mean :
(i)
For ungrouped data
G.M. = (x1 x2 x3 .....xn)1/n
Arithmetic mean :
(i)
n
For ungrouped data (individual series)
or
x =
n
(ii)
x1 + x2 +......+ xn
Σ xi
= i=1
n(no. of terms)
n
(ii)
Direct method x =
i =1
n
Σ fi
For grouped data
e
j
1
N
n
, where N =
∑f
i
i= 1
F f log x I
GG ∑
JJ
= antilog G
GH ∑ f JJK
n
Σ fixi
i
i =1
G.M. = x1f1 x2f2 .... xnfn
For grouped data (continuous series)
(a)
F1
I
G.M. = antilog G n ∑ log x J
H
K
n
, where xi , i = 1 .... n
i
i
i =1
n
i=1
i
i= 1
be n observations and fi be their corresponding
frequencies
(b)
Σfidi
short cut method : x = A + Σf ,
i
4.
Harmonic Mean - Harmonic Mean is reciprocal of arithmetic mean of reciprocals.
where A = assumed mean, di = xi – A = deviation
for each term
(i)
n
For ungrouped data H.M. =
n
∑x
1
i=1
2.
(i)
In a statistical data, the sum of the deviation of items
from A.M. is always zero.
(ii)
n
If x is the mean of x1, x2, ...... xn. The mean of ax1, ax2
.....axn is a x where a is any number different from
zero.
(ii)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
A T
I O
N
S
For grouped data H.M. =
i= 1
n
Ff I
∑ GH x JK
i
i =1
5.
i
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.
(iv) Arithmetic mean is independent of origin i.e. it is x
effected by any change in origin.
D U
∑f
i
If each of the n given observation be doubled, then
their mean is doubled
(iii)
E
i
Properties of A.M.
PAGE # 93
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 94
MATHS FORMULA - POCKET BOOK
6.
MATHS FORMULA - POCKET BOOK
(ii)
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 of
FG n + 1IJ
H 2 K
Median = u -
th
7.
Mode :
(i)
For individual series : In the case of individual series,
the value which is repeated maximum number of times
is the mode of the series.
(ii)
For discrete frequency distribution series : In the case
of discrete frequency distribution, mode is the value of
the variate corresponding to the maximum frequency.
(iii)
For continuous frequency distribution : first find the
model class i.e. the class which has maximum frequency.
th
Discrete series : First find cumulative frequencies of
the variables arranged in ascending or descending
order and
(c)
FG n + 1IJ
H 2 K
observation, where n is cumulative
Where
f
C
A T
I O
N
S
LM f − f OP
N2f − f − f Q
1
1
0
0
2
× i
Where l 1 = Lower limit of the model class.
f1 = Frequency of the model class.
f0 = Frequency of the class preceding model
class.
× i
f2 = Frequency of the class succeeding model
class.
l = Lower limit of the median class.
f = Frequency 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.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
Mode = l 1 +
frequency.
Continuous distribution (grouped data)
(i)
For series in ascending order
Median = l +
E
For continuous series
th
FG N − CIJ
H2 K
× i
f
where u = upper limit of median class.
observation
FG nIJ + value of
H 2K
FG n + 1IJ ] observation.
H2 K
Median =
FG N − CIJ
H2 K
th
1
If n is even, Median =
[Value of
2
(b)
For series in descending order
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 Median – 2 Mean
PAGE # 95
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 96
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
Measure of Dispersion :
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.
fi = Frequency of the corresponding xi
(ii)
σ =
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 =
(b)
FG IJ
H K
Σfd2
Σfd
−
N
N
2
or
σ=
FG IJ
H K
Σd2
Σd
−
N
N
2
Where d = x – A = Derivation from assumed mean A
f = Frequency of item (term)
N = Σf = Total frequency.
Σ|x − S|
n
Where n = number of terms, S = deviation of variate
from mean mode, median.
Continuous series (grouped data).
Mean deviation =
N = Σ f = Total frequency
Short cut method
Variance – Square of standard direction
i.e. variance = (S.D.)2 = (σ)2
Coefficient of variance = Coefficient of S.D. × 100
Σ f | x − s|
Σf |x − s|
=
Σf
N
=
σ
× 100
x
Note : Mean deviation is the least when measured from the
median.
2.
Standard Deviation :
S.D. (σ) is the square root of the arithmetic mean of the
squares of the deviations of the terms from their A.M.
(a) For individual series (ungrouped data)
σ =
Σ(x − x)2
N
where x = Arithmetic mean of
the series
(b)
N = Total frequency
For continuous series (grouped data)
(i)
Direct method σ =
Where
Σfi (xi − x)2
N
x = Arithmetic mean of series
xi = Mid value of the class
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 97
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 98
MATHS FORMULA - POCKET BOOK
MATRICES AND DETERMINANTS
1.
2.
MATHS FORMULA - POCKET BOOK
4.
Trace of a matrix : Sum of the elements in the principal
diagonal is called the trace of a matrix.
MATRICES :
trace (A ± B) = trace A ± trace B
Matrix - A system or set of elements arranged in a rectangular form of array is called a matrix.
trace kA = k trace A
trace A = trace AT
trace In = n when In is identity matrix.
Order of matrix : If a matrix A has m rows & n columns then
A is of order m × n.
trace On = O
trace AB ≠ trace A trace B.
The number of rows is written first and then number of columns. Horizontal line is row & vertical line is column
3.
Types of matrices : A matrix A = (aij)m×n
A matrix A = (aij)mxn over the field of complex numbers is
said to be
On is null matrix.
5.
Addition & subtraction of matrices : If A and B are two
matrices 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 :
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
Properties :
A null or zero matrix
if aij = 0 ∀ i j. It is denoted by O.
if m = n and aij = 0 for i ≠ j.
(i)
K(A + B) = KA + KB
A diagonal matrix
A scalar matrix
if m = n and aij = 0 for i ≠ j
(ii)
(K1 K2)A = K1(K2 A) = K2(K1A)
(iii)
(K1 + K2)A = K1A + K2A
KA = K (aij)m×n
= k for i = j
= (Ka)m×n where K is constant.
i.e. a11 = a22 ....... = ann = k (cons.)
Identity or unit matrix
if m = n and aij = 0 for i ≠ j
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.
= 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
Properties :
Symmetric matrix
if m = n and aij = aji for all i, j
or AT = A
(i)
Skew symmetric matrix
if m = n and aij = – aji ∀ i, j
AB ≠ BA.
(ii)
or AT = – A
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
In general matrix multiplication is not commutative i.e.
PAGE # 99
A(BC) = (AB)C
[Associative law]
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 100
MATHS FORMULA - POCKET BOOK
(iii)
A.(B + C) = AB + AC
[Distributive law]
DETERMINANT :
1.
/ B=C
(iv) If AB = AC ⇒
(v)
MATHS FORMULA - POCKET BOOK
Minor & cofactor : If A = (aij)3×3, then minor of a11 is
If AB = 0, then it is not necessary A = 0 or B = 0
a22
M11 = a
32
(vi) AI = A = IA
(vii) Matrix multiplication is commutative for +ve integral
i.e. Am+1 = Am A = AAm
cofactor of an element aij is denoted by Cij or Fij and is equal
to (–1)i+j Mij
or
8.
Transpose of a matrix :
A' or A T is obtained by interchanging rows into columns or
columns into rows
Properties :
(AT)T = A
(ii)
(A ± B)T = AT ± BT
(iii)
T
T
9.
expansion of determinant of order 3 × 3
(AB) = B A
a1 b1
a2 b2
⇒
a3 b3
IT = I
or
AAT = In = ATA
(i)
Orthogonal matrix : if
(ii)
Idempotent matrix : if A2 = A
(iii)
Involutory matrix : if A = I
2
or A
=A
Hermitian matrix : if Aθ = A i.e. aij = a ji
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
A T
I O
N
S
c1
c2
c3
b1
= –a2 b
3
b2
= a1 b
3
c2
a2
–b
1
c3
a3
c1
a1
+
b
2
c3
a3
c2
a2 b2
+
c
1
c3
a3 b3
c1
a1 b1
c3 – c2 a3 b3
Properties :
–1
(vi) Skew - Hermitian matrix : if A = –Aθ
D U
if i ≠ j
T
(iv) Nilpotent matrix : if ∃ p ∈ N such that Ap = 0
E
= –Mij,
Determinant : if A is a square matrix then determinant of
matrix is denoted by det A or |A|.
Some special cases of square matrices : A square matrix
is called
(v)
if i = j
and a11 F21 + a12 F22 + a13 F23 = 0
(iv) (KA)T = KAT
(v)
Cij = Mij,
Note : |A| = a11F11 + a12 F12 + a13 F13
2.
(i)
a23
a33 and so.
(i)
|AT| = |A|
(ii)
By interchanging two rows (or columns), value of determinant 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
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 102
MATHS FORMULA - POCKET BOOK
(v)
MATHS FORMULA - POCKET BOOK
Properties :
If same multiple of elements of any row (or column) of
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 :
(i)
A(adj A) = (adjA) A = |A|In
(ii)
|adj A| = |A|n–1
(iii)
(adjAB) = (adjB) (adjA)
(iii)
(adj AT) = (adjA)T
(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 a
unit matrix of order n.
(i)
A–1 exists if A is non singular i.e. |A| ≠ 0
(f)
A zero matrix is zero i.e. |0n| = 0, where 0n is a zero
matrix of order n
(ii)
A–1 =
(iii)
A–1A = In = A A–1
(iv) adj(adjA) = |A|n–2
(v)
2.
(g)
A diagonal matrix = product of its diagonal elements.
(h)
Skew symmetric matrix of odd order is zero.
Inverse of a matrix :
a1 b1
l1 m1
a1l1 + b1l2
×
a2 b2
l2 m2 = a2 l1 + b2 l2
(vi) |A–1| = |A|–1 =
(AB)–1 = B–1 A–1
3.
Rank of a matrix :
A non zero matrix A is said to have rank r, if
MATRICES AND DETERMINANTS :
1.
Adjoint of a matrix :
(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.
adj A = (Cij)T , where Cij is cofactor of aij
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
A T
I O
N
S
1
| A|
(vii) If A & B are invertible square matrices then
a1m1 + b1m2
a2m1 + b2m2
If order is different then for their multiplication, express them
firstly in the same order.
D U
(A–1)–1 = A
Multiplication of two determinants :
Multiplication of two second order determinants is defined as
follows.
E
adjA
, |A| ≠ 0
| A|
(iv) (AT)–1 = (A–1)T
(v)
3.
(adj KA) = Kn–1(adj A)
PAGE # 103
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 104
MATHS FORMULA - POCKET BOOK
4.
MATHS FORMULA - POCKET BOOK
Homogeneous & non homogeneous system of linear
equations :
(b) Solution of homogeneous system of linear equations :
A system of equations Ax = B is called a homogeneous system if B = 0. If B ≠ 0, then it is called non homogeneous
system equations.
The homogeneous system Ax = B, B = 0 of n equations
in n variables is
(i)
5.
(a) Solution of non homogeneous system of linear
equations :
(i)
Consistent (with unique solution) if |A| ≠ 0 and
for each i = 1, 2, ......... n
xi = 0 is called trivial solution.
Cramer's rule : Determinant method
(ii)
The non homogeneous system Ax = B, B ≠ 0 of n
equations in n variables is -
Consistent (with infinitely many solution),
if |A| = 0
(a) |A| = |Ai| = 0
Consistent (with unique solution) if |A| ≠ 0 and
for each i = 1, 2, ........ n,
(for determinant method)
(b) |A| = 0, (adj A) B = 0
(for matrix method)
NOTE : A homogeneous system of equations is never
inconsistent.
det A i
xi =
, where Ai is the matrix obtained
det A
from A by replacing ith column with B.
Inconsistent (with no solution) if |A| = 0 and
at least one of the det (Ai) 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 = A–1 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.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 105
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 106
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
Properties :
FUNCTION
1.
Modulus function :
R|x, x > 0
|x| = S−x, x < 0
|T0, x = 0
(i)
loga 1 = 0
(ii)
loga a = 1
(iii)
aloga b = b
if k > 0,
Properties :
|x| ≠ ± x
|xy| = |x||y|
(i)
(ii)
(iii)
(iv) loga b1 + loga b2 + ...... + loga bn = loga (b1 b2 ........bn)
(v)
x
|x|
=
y
|y|
|x| = a
⇒ x=±a
|x| = –a
⇒ no solution
|x| > a
⇒ x < – a or x > a
|x| ≤ a
|x| < –a
⇒ –a ≤ x ≤ a
⇒ No solution.
|x| > –a
⇒ x∈R
⇒
b=a
b > ac,
⇒
or
b < ac,
(iv) loga b > loga c
b > c,
⇒
a>1
loga b = c
(iii)
loga b > c
or
b < c,
D U
FG IJ
H K
1
(viii) loga b
C
A T
I O
N
S
3.
or
1
loga b = log a
b
n
loga b
m
= – loga b = log1/a b
FG bIJ
H cK
= loga
(ix)
log1/a
(x)
alogb c = clogb a
c
FG c IJ
H bK
Greatest Integer function :
0<a<1
f(x) = [x], where [.]denotes greatest integer function equal
or less than x.
if a > 1
i.e., defined as [4.2] = 4, [–4.2] = –5
if 0 < a < 1
Period of [x] = 1
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
= loga b – loga c
(vii) logam bn =
logb a to be defined a > 0, b > 0, b ≠ 1
(ii)
FG bIJ
H cK
logc b
loga b = log a
c
Logarithmic Function :
(i)
loga
(vi) Base change formulae
(iv) |x + y| ≤ |x| + |y|
(v) |x – y| ≥ |x| – |y| or ≤ |x| + |y|
(vi) ||a| – |b|| ≤ |a – b| for equality a.b ≥ 0.
(vii) If a > 0
2.
k = blogb k
PAGE # 107
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 108
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
6.
Properties :
(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, ∞ )
Definition :
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.
(vi) [x] ≤ I, x ∈ (– ∞ , I + 1]
(vii) [x] > I, [x] ≥ I + 1, x ∈ [I + 1, ∞ )
(viii) [x] < I, [x] ≤ I – 1, x ∈ (– ∞ , I)
4.
Table : Domain and Range of some standard functions -
or
f(x)
=
|x|
,
x
= 0,
A T
I O
N
S
Domain
Range
Polynomial function
R
R
Identity function x
R
R
Constant function K
R
(K)
R0
R0
x2, |x| (modulus function)
R
R+ ∪{x}
x3, x|x|
R
R
R
{-1, 0, 1}
Signum function
1
x
|x|
x
, x ∈R−
x +|x|
R
R+ ∪{x}
x=0
x -|x|
R
R- ∪{x}
[x] (greatest integer function)
R
1
x - {x}
R
[0, 1]
[0, ∞ )
[0, ∞ ]
ax (exponential function)
R
R+
log x (logarithmic function)
R+
R
,
, x ∈R+
x
x ≠ 0
x=0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
Functions
Reciprocal function
Signum function :
R|−1
|S 0
f(x) = sgn (x) = |
|T 1
D U
Range = For all values of x, all possible values of f(x).
Fractional 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.
E
i.e., Domain = All possible values of x for which f(x) exists.
PAGE # 109
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 110
MATHS FORMULA - POCKET BOOK
Trigonometric
Domain
MATHS FORMULA - POCKET BOOK
7.
Range
Kinds of functions :
sin x
R
[-1, 1]
cos x
R
[-1, 1]
tan x
R-
RS± π , ± 3π ,...UV
T2 2 W
R
cot x
R- {0, ± π , ± 2 π ,...}
R
sec x
R -
RS± π , ± 3π ,...UV
T2 2 W
R - (-1,1)
cosec x
R- {0, ± π , ± 2 π }
R - (-1,1)
Domain
Range
Inverse
or
-1
x
Graphically-no horizontal line intersects with the graph
of the function more than once.
(ii)
(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)
(iv) Into function : f is said to be into function if R(f) < B
LM −π , π OP
N 2 2Q
[0, π ]
tan-1 x
R
FG −π , π IJ
H 2 2K
cot-1 x
R
(0, π )
sec-1 x
R -(-1,1)
π
[0, π ]2
One-one-onto function (Bijective) - A function which
is both one-one and onto is called bijective function.
Inverse function : f–1 exists iff f is one-one & onto both
8.
[-1,1]
R - (-1,1)
i.e. if to each y ∈ B ∃ x ∈ A s.t. f(x) = y
Graphically - atleast one horizontal line intersects with
the graph of the function more than once.
cos-1 x
cosec-1 x
Onto function (surjection) - f : A → B is onto if
R (f) = B
(v)
(-1, 1]
a≠b
⇒ f(a) ≠ f(b), a, b ∈ A
Trigo Functions
sin
One-one (injection) function - f : A → B is one-one if
f(a) = f(b) ⇒ a = b
(i)
Functions
f–1 : B → A, f–1(b) = a
9.
⇒ f(a) = b
Transformation of curves :
(i)
Replacing x by (x – a) entire graph will be shifted parallel
to x-axis with |a| units.
If
RS UV
TW
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,
LM− π , π OP- {0}
N 2 2Q
upward if a is +ve
downward if a is –ve.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 111
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 112
MATHS FORMULA - POCKET BOOK
(ii)
(iii)
MATHS FORMULA - POCKET BOOK
Replacing x by –x, take reflection of entire curve is yaxis.
(g)
Zero function i.e. f(x) = 0 is the only function which
is even and odd both.
Similarly if y is replaced by –y then take reflection of
entire curve in x-axis.
(h)
If f(x) is odd (even) function then f'(x) is even (odd)
function provided f(x) is differentiable on R.
Replacing x by |x|, remove the portion of the curve
corresponding to –ve x (on left hand side of y-axis)
and take reflection of right hand side on LHS.
(i)
A given function can be expressed as sum of even
& odd function.
i.e.
(iv) Replace f(x) by |f(x)|, if on L.H.S. y is present and
mode is taken on R.H.S. then portion of the curve below
x-axis will be reflected above x-axis.
(v)
Replace x by ax (a > 0), then divide all the value on xaxis by a.
Similarly if y is replaced by ay (a > 0) then divide all the
values of y-axis by a.
Even function if f(–x) = f(x) and
(ii)
Odd function if f(–x) = –f(x).
= even function + odd function.
12. Increasing function : A function f(x) is an increasing function
in the domain D if the value of the function does not decrease
by increasing the value of x.
14. Periodic function: Function f(x) will be periodic if a +ve real
number T exist such that
11. Properties of even & odd function :
C
A T
∀ x ∈ Domain.
The graph of an even function is always symmetric
about y-axis.
There may be infinitely many such T which satisfy the above
equality. Such a least +ve no. T is called period of f(x).
(b)
The graph of an odd function is always symmetric
about origin.
(i)
(d)
Sum & difference of two even (odd) function is an
even (odd) function.
(e)
Product of an even or odd function is an odd function.
(f)
Sum of even and odd function is neither even nor
odd function.
I O
N
S
If a function f(x) has period T, then
Period of f(xn + a) = T/n and
Period of (x/n + a) = nT
Product of two even or odd function is an even
function.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
f(x + T) = f(x),
(a)
(c)
E
1
1
[f(x) + f(–x)] +
[f(x) – f(–x)]
2
2
13. Decreasing function : A function f(x) is a decreasing function
in the domain D if the value of function does not increase by
increasing the value of x.
10. Even and odd function : A function is said to be
(i)
f(x) =
(ii)
If the period of f(x) is T1 & g(x) has T2 then the period
of f(x) ± g(x) will be L.C.M. of T1 & T2 provided it
satisfies definition of periodic function.
(iii)
If period of f(x) & g(x) are same T, then the period of
af(x) + bg(x) will also be T.
PAGE # 113
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 114
MATHS FORMULA - POCKET BOOK
Function
Period
sin x, cos x
2π
MATHS FORMULA - POCKET BOOK
15. Composite function :
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
sec x, cosec x
tan x, cot x
π
In general gof ≠ fog
sin (x/3)
6π
If both f and g are bijective function, then so is gof.
tan 4x
π/4
cos 2πx
1
π
|cos x|
4
4
sin x + cos x
2 cos
FG x − π IJ
H 3 K
sin3 x + cos3 x
3
4
π/2
6π
2π/3
sin x + cos x
2π
sin x
sin5x
2π
tan2 x – cot2 x
π
x – [x]
1
[x]
1
NON PERIODIC FUNCTIONS :
2
3
x, x , x , 5
cos x2
x + sin x
x cos x
cos x
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 115
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 116
MATHS FORMULA - POCKET BOOK
LIMIT
1.
Limit of a function : xlim
→ a f(x) = l (finite quantity)
2.
lim
lim
Existence of limit : xlim
→ a f(x) exists iff x → a− f(x) = x → a+ f(x) = l
3.
Indeterminate forms :
0 ∞
,
, ∞ – ∞ , ∞ × 0, ∞ 0 , 0∞ , 1∞
0 ∞
MATHS FORMULA - POCKET BOOK
5.
Limit of the greatest integer function :
Let c be any real number
Case I : If c is not an integer, then xlim
→ c [x] = [c]
Case II: If c is an integer, then xlim
[x] = c – 1, xlim
[x] = c
→ c−
→ c+
and xlim
[x] = does not exist
→c
6.
Methods of evaluation of limits :
f(x)
0
is of
form
g(x)
0
then factorize num. & devo. separately and cancel the
Factorisation method : If xlim
→a
(i)
4.
Theorems on limits :
lim (k f(x)) = k lim f(x),
x→a
(i)
x→a
(ii)
x→a
(iii)
x→a
(iv)
lim
(v)
k is a constant.
lim (f(x) ± g(x)) = lim f(x) ± Lim g(x)
x→a
x→a
(ii)
lim f(x).g(x) = lim f(x). Lim g(x)
x→a
x→a
(iii)
x→a
lim f(x)
x→a
f(x)
= lim g(x) , provided xlim
→ a g(x) ≠ 0
g(x)
x→a
FH
the indeterminate form, replace
(iv)
IK
lim f(g(x)) = f lim g(x) , provided value of
x→a
x→a
0
form.
0
Rationalization method : If we have fractional powers
on the expression in num, deno or in both, we rationalize
the factor and simplify.
When x → ∞ : Divide num. & deno. by the highest power
of x present in the expression and then after removing
common factor which is participating in making
(v)
n
n
lim x − a = nan–1
x→a
x −a
By using standard results (limits) :
g(x) function f(x) is continuous.
(vi)
lim [f(x) + k] = lim f(x) + k
x→a
IK
x→0
(b)
x→0
(c)
x→0
(d)
x→0
lim f(x)
(vii) xlim
→ a log(f(x)) = log x → a
g(x)
(viii) xlim
=
→ a (f(x))
LM lim f(x)OP
Q
N
D U
C
A T
I O
N
S
x
lim tan x = 1 = lim
x→0
x
tan x
lim sinx = 0
lim g(x)
x→a
x→a
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
lim
x
lim sinx = 1 = x → 0
x
sin x
(a)
x→a
FH
1 1
, 2 ,.. by 0.
x x
PAGE # 117
lim cosx = lim
x→0
1
=1
cos x
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 118
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(vi) By substitution :
(a) If x → a, then we can substitute
x=a+t ⇒ t=x–a
If x → a, t → 0.
(b) When x → – ∞ substitute x = – t ⇒ t → ∞
(e)
0
lim sin x = π
x→0
180
x
(f)
−1
x
lim sin x = 1 = lim
−1
x→0
x→0
sin
x
x
(g)
−1
x
lim tan x = 1 = lim
x→0
x→0
tan−1 x
x
(h)
lim a − 1 = log a
x→0
e
x
ex = 1 + x +
x2
x3
+
+ .....
2!
3!
(i)
x
lim e − 1 = 1
x→0
x
e–x = 1 – x +
x2
x3
–
+ .....
2!
3!
(j)
lim log(1 + x) = 1
x→0
x
log(1 + x) = x –
x2
x3
+
– ......
2
3
(k)
1
lim loga (1 + x) =
x→0
loga
x
log(1 – x) = –x –
x2
x3
–
–.....
2
3
(l)
lim (1 + x) − 1 = n
x
ex ln a = ax = 1 + xlogea +
(c)
When x → ∞ substitute t =
(vii) By using some expansion :
x
n
(m)
x→0
sinx = x –
lim sinx = lim cos x = 0
x→∞
x
x
lim
x→∞
1
x
1
x
=1
(p)
FG1 + 1 IJ
H xK
F aI
= lim G1 + J
H xK
lim (1 + x)1/x = e = lim
x→∞
x→0
1/x
lim
= ea
x → 0 (1 + ax)
D U
C
A T
I O
N
S
cosx = 1 –
x2
x4
+
–......
2!
4!
tanx = x +
2
x3
+
x5 + .....
15
3
n(n − 1) 2
x + .....
2!
Sandwich Theorem : In the neighbour hood of x = a
f(x) < g(x) < h(x)
7.
x
lim f(x) = lim h(x) = l, then lim g(x) = l.
x →a
x →a
x →a
x→∞
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
x3
x5
+
–.......
3!
5!
(1 + x)n = 1 + nx +
x
(o)
(x loge a)2
(x loge a)3
+
+ ......
2!
3!
x→∞
sin
(n)
1
⇒ t → 0+
x
⇒
PAGE # 119
l < lim
g(x) < l.
x →a
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 120
MATHS FORMULA - POCKET BOOK
DIFFERENTIATION
1.
MATHS FORMULA - POCKET BOOK
2.
FUNDAMENTAL RULES FOR DIFFERENTIATION :
SOME STANDARD DIFFERENTIATION :
Function
(i)
d
f(x) = 0 if and only if f(x) = constant
dx
Derivative
Function
Derivative
0
xn
nxn – 1
(ii)
d
dx
ccf(x)h
1
x loge a
loge x
1
x
(iii)
d
dx
cf(x) ± g(x)h
ax loge a
ex
ex
sin x
cos x
cos x
–sin x
(iv)
d
dv
du
(uv) = u
+ v
, where u & v are functions
dx
dx
dx
tan x
sec2 x
cot x
–cosec2 x
–cosec x cot x
sec x
sec x tan x
A cons. (k)
loga x
ax
cosec x
1−x
2
,–1<x<1
cos–1x
–
1 − x2
tan–1 x
1
|x| 1 − x
1
1+x
2
2
,|x|>1
,x ∈ R
0, x ∉ I
[x]
cosec–1 x –
1
|x| 1 − x2
cot–1 x
|x|
–
1
1 + x2
,–1|x|>|
If
= a φ(ax + b)
, x ∈R
(vi)
x
, x≠ 0
| x|
d
dx
FG u IJ
H vK
D U
C
A T
I O
N
S
v
=
du
dv
−u
dx
dx
2
v
(quotient rule)
(vii) If y = f(u), u = g(x) [chain rule or differential coefficient of a function of a function]
d
[x] does not exist at any integral Point.
dx
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
d
g(x)
dx
d
d
f(x) = φ(x), then
f (ax + b)
dx
dx
then
NOTE :
d
f(x) ±
dx
,–1<x<1
(v)
sec–1 x
=
d
du
dv
dw
(uvw) = vw
+ uw
+ uv
.
dx
dx
dx
dx
1
1
d
f(x), where c is a constant.
dx
of x. (Product rule)
or
sin–1 x
= c
dy
dy
du
=
×
dx
dx
du
llly If y = f(u), u = g(v), v = h(x), then
PAGE # 121
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 122
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(xii) Differentiation of implicit function : If f (x, y) = 0,
differentiate w.r.t. x and collect the terms containing
dy
dy
du
dv
=
×
×
dx
du
dv
dx
i.e if y = un ⇒
dy
at one side and find
dx
dy
du
= nun–1
dx
dx
dy
.
dx
[The relation f(x, y) = 0 in which y is not expressible
OR
explicitly in terms of x are called implicit functions]
(viii) Differentiation of composite functions
Suppose a function is given in form of fog(x) or
f[g(x)], then differentiate applying chain rule
(xiii) Differentiation of parametric functions : If x = f(t)
and y = g(t), where t is a parameter, then
d
f[g(x)] = f'g(x) . g'(x)
i.e.,
dx
FG 1IJ
H uK
−1
dy
g'(t)
dy
dt
=
=
f'(t)
dx
dx
dt
(ix)
d
dx
(x)
u du
d
|u| =
,
|
u| dx
dx
(xi)
Logarithmic Differentiation : If a function is in the
=
u2
du
, u ≠ 0
dx
u ≠ 0
(xiv) Differentiation of a function w.r.t. another function : Let y = f(x) and z = g(x), then differentiation
of y w.r.t. z is
f1 ( x ) f2 ( x )....
form (f(x))g(x) or g ( x ) g ( x ).... We first take log on
1
2
dy / dx
f'(x)
dy
=
=
g'(x)
dz / dx
dz
both sides and then differentiate.
(a)
loge (mn) = logem + logen
(b)
loge
(c)
loge (m)n = nlogem
(xv) Differentiation of inverse Trigonometric functions
m
= logm – logen
n
m
(e) logan xm = loga x
n
(g) loge e = 1
(h)
(d)
logn m
(f)
aloga
D U
C
A T
I O
N
S
logm n = 1
first try for a suitable substitution to simplify it and
then differentiate. If no such substitution is found
= x
then differentiate directly by using trigonometrical
formula frequently.
loge m
logn m = log n
e
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
x
using Trigonometrical Transformation : To solve
the problems involving inverse trigonometric functions
PAGE # 123
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 124
MATHS FORMULA - POCKET BOOK
3.
MATHS FORMULA - POCKET BOOK
Important Trigonometrical Formula :
(i)
sin2x = 2sinx. cosx =
(ii) cos2x =
(viii) tan2x =
(iii)
1 − tan2 x
1 + tan2 x
(xvi) sin–1 sin (x) = x, for –
2 tan x
1 + tan2 x
π
2
π
2
≤ x ≤
cos–1 (cos x) = x, for 0 ≤ x ≤ π
= 2 cos2 x – 1 = 1 – 2 sin2 x
tan–1 (tan x) = x, for –
π
π
< x <
2
2
(xvii) sin–1 (–x) = –sin–1 x, tan–1 (–x) = – tan–1 x,
2 tan x
1 − tan2 x
cos–1 (–x) = π – cos–1 x
(xviii) sin–1
FG 1 IJ
H xK
= cosec–1 x, cos–1
tan–1
FG 1 IJ
H xK
= cot–1 x, cot–1
sec–1
FG 1IJ
H xK
= cos–1 x, cosec–1
sin3x = 3sinx – 4sin3 x
(vi) cos3x = 4cos3 x – 3cosx
3 tan x − tan3 x
(ix)
tan3x =
(x)
sin–1 x + cos–1 x = π /2
(xi)
sec–1 x + cosec–1 x = π /2
1 − 3 tan2 x
FG 1 IJ
H xK
FG 1 IJ
H xK
= sec–1 x,
= tan–1 x,
FG 1IJ
H xK
= sin–1 x
(xii) tan–1 x + cot–1 x = π /2
(xiii) tan–1 x ± tan–1 y = tan–1
FG x ± y IJ
H 1 m xy K
(xix) sin–1 (cos θ ) = sin–1
F
H
(xiv) sin–1 x ± sin–1 y = sin–1 x 1 − y2 ± y 1 − x2
(xv) cos–1 x ± cos–1 y = cos–1
F xy m
H
1 − x2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
I
K
1 − y2
=
Fcos FG π − θIJ IJ
GH H 2 K K
=
π
– θ
2
tan–1 (cot θ ) = tan–1
FG tan FG π − θIJ IJ
H H 2 KK
=
π
– θ
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
π
– θ
2
cos–1 (sin θ ) = cos–1
I
K
PAGE # 125
Fsin FG π − θIJ I
GH H 2 K JK
PAGE # 126
MATHS FORMULA - POCKET BOOK
4.
Some Useful Substitutions :
Part B
Part A
Expression
Formula
Result
a2 + x2
3x – 4x3
x = sinθ
3sinθ – 4sin3 θ
sin3θ
x = cosθ
4cos3 θ – 3cosθ
x = a tan θ
4x3 – 3x
cos3θ
a− x
a+ x
or
a+ x
a− x
a2 – x2
x = a sin θ or x = a cos θ
2x
1+x
2
2x
x = tanθ
x = tanθ
3 tan θ − tan3 θ
1 − 3 tan2 θ
2 tan θ
1 + tan2 θ
2 tan θ
a+x
or
a−x
sin2θ
1 − tan θ
1 – 2x2
x = sinθ
1 – 2sin2 θ
cos2θ
2x2 – 1
x = cosθ
2cos2 θ – 1
cos2θ
1 – x2
x = sinθ
1 – sin2 θ
cos2 θ
x = cosθ
1 – cos2 θ
sin2 θ
x = secθ
sec θ – 1
tan θ
x = cosecθ
cosec2 θ – 1
cot 2 θ
x = tanθ
1 + tan2 θ
sec2 θ
x = cotθ
1 + cot2 θ
cosec2 θ
x
2
– 1
1 + x2
2
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
I O
N
S
a−x
a+x
x = a cos θ
x = a sec θ or x=acosec θ
x 2 – a2
a2 + x2
tan2θ
2
A T
tan3θ
x = tanθ
1−x
C
x = a tan θ or x = a cot θ
Substitution
1 − 3x 2
D U
Substitution
Expression
3x − x3
E
MATHS FORMULA - POCKET BOOK
a2 − x2
5.
a2 − x2
or
x2 = a2 cos θ
a2 + x2
Successive differentiations or higher order derivatives :
(a)
If y = f(x) then
dy
= f'(x) is called the first derivadx
tive of y w.r.t. x
2
⇒
d2 y
dx
2
=
d
dx
FG dy IJ
H dx K
=
d
f'(x)
dx
c
h
is called the second derivative of y w.r.t. x
PAGE # 127
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 128
MATHS FORMULA - POCKET BOOK
llly
d3 y
dx3
=
d2
dx2
cf'(x)h
MATHS FORMULA - POCKET BOOK
FG
H
If y = sin (ax + b), then yn = an sin ax + b +
(f)
etc......
Thus, This process can be continued and we can
If y = cos (ax + b), then yn = an cos
obtain derivatives of higher order
Note : To obtain higher order derivative of parametric
i.e. if x = 2t, y = t2
dn
(i)
⇒
⇒
(b)
dxn
dy
= t
dx
d2 y
dx 2
=
d
dx
(ii)
FG dy IJ
H dx K
If y = (ax + b)
=
m
IJ
K
FGax + b + nπ IJ
H
2K
nth Derivatives of Some Functions :
6.
functions we use chain rule
nπ
2
d
dt
1
(t) = 1.
=
dx
dx
t
(iii)
m ∉ I, then
dn
dx
n
dn
dx
n
ex j
n
= n!
csin xh
FG
H
nπ
= sin x + 2
FG
H
IJ
K
(cos x) = cos x +
πn
2
IJ
K
yn = m(m–1) (m–2) ..... (m–n+1) (ax + b)m–n .an
(c)
If m ∈I, then
(iv)
dn
dx n
(emx ) = mn emx
ym = m! am and ym+1 = 0
(d)
(−1)n n!
1
, then yn =
an
If y =
(ax + b)n+1
ax + b
(v)
dn
dx n
NOTE :
n−1
(e)
If y = log (ax + b), then y n =
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
(−1)
(n − 1)!
(ax + b)n
(log x) = (– 1)n–1 (n–1)! x–n
If u = g(x) is such that g'(x) = K (constant)
an
then
PAGE # 129
dn
dx
n
LM d
MNdu
n
c h
f g(x)
= Kn
n
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
OP
PQ
f(u)
u= g(x)
PAGE # 130
MATHS FORMULA - POCKET BOOK
7.
Differentiation of Infinite Series : method is illustrated
MATHS FORMULA - POCKET BOOK
9.
Differentiation of Determinant :
with the help of example
x
if y = x
x− −∞
then function becomes y = xy now taking log
∆ =
on both sides
R1
R2
R3
= |C1 C2 C3|
i.e logy = y log x, differentiating both sides w.r.t. x
we get
dy
=
dx
⇒
8.
1
y
R'1
∆' = R 2
R3
dy
1
dy
= y
+ logx
dx
dx
x
y
x
FG 1 − log xIJ
K
Hy
R1
'2
R
+
R3
R1
R
2
+
R'3
= |C'1 C2 C3| + |C1 C'2 C3| + |C1 C2 C'3|
y2
=
x(1 − y log x)
L-hospital rule :
if as x → a f(x) & g(x) either both → 0 or both → ∞, then
lim
f(x)
f'(x)
= lim
x →a g'(x)
g(x)
(a)
it can be applied only on 0/0 or ∞/∞ form
x →a
(b) Numerator & denominator are differentiated separately
not
(c)
u
formulae.
v
If R.H.S. exist or d'not exist because value → ∞, then
L.H rule can be applied.
But if value fluctuate on R.H.S. then L.H. rule can't
be applied.
If it is applied continuously then at each step 0/0 or
∞/∞ should be checked.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 131
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 132
MATHS FORMULA - POCKET BOOK
APPLICATION OF DERIVATIVES
MATHS FORMULA - POCKET BOOK
7.
Length of intercepts made on axes by the tangent :
TANGENT AND NORMAL :
1.
Geometrically f'(a) represents the slope of the tangent to
the curve y = f(x) at the point (a, f(a))
x – intercept = x1
If the tangent makes an angle ψ (say) with +ve x direction
2.
R|
| y
– S F dy I
|| GH dx JK
T
1
(x , y )
1
1
U|
|V
||
W
then
f'(x) =
FG dy IJ
H dx K
(x1 , y1)
FG dy IJ
H dx K
⇒
8.
(x1 , y1)
Length of perpendicular from origin to the tangent :
= 0.
⇒
5.
FG dy IJ
H dx K
6.
(x1 , y1)
→ ∞
(x1 , y1)
y – y1 =
FG dy IJ
H dx K
(x1 , y1)
I O
N
S
Slope of the normal
=–
=–
1
Slope of the tan gent
FG dx IJ
H dy K
(x 1 , y 1 )
10. If normal makes an angle of φ with +ve direction of x-axis,
(x – x1)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
9.
= ± 1.
Equation of the tangent to the curve y = f(x) at a point
(x1, y1) is
C
=
(x1 ,y1 )
2
(x1 , y1 )
If the tangent line makes equal angle with the axes, then
FG dy IJ
H dx K
FG dy IJ
H dx K
F dy I
1+G J
H dx K
y1 − x1
π
If the tangent is perpendicular to x-axis, ψ =
2
4.
D U
(x , y )
1
1
If the tangent is parallel to x-axis, ψ = 0
3.
E
FG dy IJ
H dx K
y – intercept = y1 – x1
= tan ψ = slope of the tangent.
then
PAGE # 133
dy
= – cot φ .
dx
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 134
MATHS FORMULA - POCKET BOOK
11. If the normal is parallel to x-axis ⇒
FG dy IJ
H dx K
12. If the normal is perpendicular to x-axis ⇒
MATHS FORMULA - POCKET BOOK
17. Angle of intersection of the two curves :
= 0.
(x1 , y1)
FG dy IJ
H dx K
tanθ = ±
(x1 , y1)
= 0.
FG dy IJ − FG dy IJ
H dx K H dx K
F dy I F dy I
1−G J G J
H dx K H dx K
1
2
1
where
FG dy IJ
H dx K
is the slope of first
1
2
13. If normal is equally inclined from both the axes or cuts
equal intercept then
FG dy IJ
H dx K
FG dy IJ of second. If both curves intersect orthogoH dx K
F dy I F dy I
nally then GH dx JK GH dx JK = –1
curve &
= ± 1.
2
14. The equation of the normal to the curve y = f(x) at a point
(x1, y1) is
1
y – y1 = –
FG dy IJ
H dx K
1
2
(x – x1)
18. Length of tangent, normal, subtangent & subnormal :
(x1 , y1 )
15. Length of intercept made on axes by the normal :
x – intercept = x1 + y1
FG dy IJ
H dx K
(x1 ,y1 )
y – intercept = y1 + x1
FG dx IJ
H dy K
(x1 ,y1 )
y 1+
1+
16. Length of perpendicular from origin to normal :
x1
=
F dy I
+y G J
H dx K
F dy I
1+G J
H dx K
D U
C
A T
I O
N
S
FG dy IJ
H dx K
2
Length of sub-tangent =
y
dy / dx
Length of sub-normal = y
dy
dx
1
(x1 ,y1 )
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
2
dy
dx
Length of tangent =
Length of normal = y
FG dy IJ
H dx K
PAGE # 135
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 136
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
MONOTONICITY, MAXIMA & MINIMA :
(ii)
1.
A function is said to be monotonic function in a domain if it is
either monotonic increasing or monotonic decreasing in that
domain
2.
At a point function f(x) is monotonic increasing if f'(a) > 0
At a point function f(x) is monotonic decreasing if f'(a) < 0
3.
6.
Working rule for finding local maxima & Local Minima :
(i)
Find the differential coefficient of f(x) w.r.to x, i.e.
f'(x) and equate it to zero.
(ii) Solve the equation f'(x) = 0 and let its real roots
(critical points) be a, b, c ......
(iii) Now differentiate f'(x) w.r.to x and substitute the
critical points in it and get the sign of f"(x) for each
critical point.
(iv) If f"(a) < 0, then the value of f(x) is maximum at
x = 0 and if f"(a) > 0, then the value of f(x) is
minimum at x = a. Similarly by getting the sign of f"(x)
for other critical points (b, c, ......) we can find the
points of maxima and minima.
7.
Absolute (Greatest and Least) values of a function in
a given interval :
(i)
A minimum value of a function f(x) in an interval [a,
b] is not necessarily its greatest value in that interval.
Similarly a minimum value may not be the least value
of the function.
(ii) If a function f(x) is defined in an interval [a, b], then
greatest or least values of this function occurs either
at x = a or x = b or at those values of x for which
f'(x) = 0.
Thus greatest value of f(x) in interval [a, b]
= max [f(a), f(b), f(c), f(d)]
Least value of f(x) in interval [a, b]
= min. [f(a), f(b), f(c), f(d)]
Where x = c, x = d are those points for which f'(x) = 0.
In an interval [a, b], a function f(x) is
Monotonic increasing if f'(x) ≥ 0
Monotonic decreasing if f'(x) ≤ 0
constant if f'(x) = 0 ∀ x ∈ (a, b)
Strictly increasing if f'(x) > 0
Strictly decreasing if f'(x) < 0
4.
Maximum & Minimum Points :
Maxima : A function f(x) is said to be maximum at x =
a, if there exists a very small +ve number h, such that
f(x) < f(a), ∀ x ∈ (a – h, a + h), x ≠ a.
Minima : A function f(x) is said to be minimum at x = b,
if there exists a very small +ve number h, such that
f(x) > f(b), ∀ x ∈ (b – h, b + h), x ≠ b.
Remark :
5.
(a)
The maximum & minimum points are also known as
extreme points.
(b)
A function may have more than one maximum &
minimum points.
Conditions for Maxima & Minima of a function :
(i)
Necessary condition : A point x = a is an extreme
point of a function f(x) if f'(a) = 0, provided f'(a)
exists.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
Sufficient condition :
(a) The value of the function f(x) at x = a is
maximum if f'(a) = 0 and f"(a) < 0.
(b) The value of the function f(x) at x = a is
minimum if f'(a) = 0 and f"(a) > 0.
PAGE # 137
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 138
MATHS FORMULA - POCKET BOOK
8.
ROLLE'S THEOREM & LAGRANGES THEOREM:
1.
Rolle's Theorem : If f(x) is such that
(a) It is continuous on [a, b]
(b) It is differentiable on (a, b) and
(c) f(a) = f(b), then there exists at least one point
c ∈ (a, b) such that f'(c) = 0.
2.
Mean value theorem [Lagrange's theorem] :
(i)
If f(x) is such that
(a) It is continuous on [a, b]
(b) It is differentiable on (a, b), then
there exists at least one c ∈ (a, b) such that
Some Geometrical Results :
In Usual Notations
Area of equilateral
and its perimeter
MATHS FORMULA - POCKET BOOK
Results
3
(side)2.
4
3 (side)
Area of square
(side)2
Perimeter
4(side)
Area of rectangle
l × b
Perimeter
2(l × b)
Area of trapezium
1
(sum of parallel sides)
2
f(b) − f(a)
= f'(c)
b−a
× (distance between them)
Area of circle
πr2
Perimeter
2πr
Volume of sphere
4 3
πr
3
Surface area of sphere
4πr2
Volume of cone
1 2
πr h
3
Surface area of cone
πrl
Volume of cylinder
πr2h
Curved surface area
2πrh
Total surface area
2πr(h + r)
Volume of cuboid
l × b × h
Surface area of cuboid
2(lb + bh + hl)
Area of four walls
2(l × b) h
Volume of cube
l3
Surface area of cube
6l2
Area of four walls of cube
4l2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
(ii)
If for c in lagrange's theorem (a < c < b) we can say
that c = a + θ h where 0 < θ < 1 and h = b – a
the theorem can be written as
f(a + h) = f(a) + h f'(a + θ h), 0 < θ < 1, h = b – a
PAGE # 139
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 140
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
INDEFINITE INTEGRATION
1.
d
F(x) = f(x), then
If
dx
(i)
Here
zm r
zch
(iii)
(iv)
2.
zch
f x dx = f(x)
f' x dx = f(x) + c, c ∈ R
z
k f x dx = k f(x) dx
z
(v)
ch ch
(f x ± g x ) dx =
zch zch
f x dx ± g x dx
FUNDAMENTAL FORMULAE :
Function
Integration
z
zc
n+1
x
+ c, n ≠ –1
n+1
n
x dx
h
n
ax + b dx
c
h
+ c, n ≠ –1
z
1
dx
x
log|x| + c
z
z
z
z
1
dx
ax + b
1
(log|ax + b|) + c
a
ex dx
ex + c
ax dx
ax
+ c
loge a
sinx dx
–cos x + c
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
sin x + c
dx
sec2 x dx
tan x + c
cos ec2x dx
– cot x + c
sec x tan x dx
sec x + c
cos ec x cot xdx
–cosec x + c
tanx dx
–log|cos x| + c = log|sec x| + c
cot x dx
log|sin x| + c = –log|cosec x| + c
sec x dx
log|sec x + tan x|+c = log tan
cos ec x dx
x
log|cosec x – cot x|+c = log tan +c
2
z
z
z
n+1
1 ax + b
.
a
n+1
z cos x
z
z
z
z
z
z
z
z
dx is the notation of integration, f(x) is the
z ch
z ch
Integration
z
f x dx = F(x) + c
integrand, c is any real no. (integrating constant)
d
(ii)
dx
Function
z
z
dx
sin–1 x + c = –cos–1x + c
1 − x2
dx
a −x
2
2
dx
1 + x2
dx
a2 + x2
dx
|x| x2 − 1
dx
|x| x2 − a2
PAGE # 141
sin–1
x
x
+ c = –cos–1 + c
a
a
tan–1x + c = –cot–1x + c
x
1
x
−1
tan–1
+ c =
cot–1 a + c
a
a
a
sec–1x + c = –cosec–1x + c
1
x
sec–1
+ c =
a
a
x
−1
cosec–1 a + c
a
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
FG π + x IJ +c
H 4 2K
PAGE # 142
MATHS FORMULA - POCKET BOOK
3.
MATHS FORMULA - POCKET BOOK
SOME RECOMMENDED SUBSTITUTION :
INTEGRATION BY SUBSTITUTION :
z
By suitable substitution, the variable x in
Function
ch
1
Substitution
a −x ,
a2 − x2
x2 + a2 ,
x + a2
2
into another variable t so that the integrand f(x) is changed
into F(t) which is some standard integral. Some following
suggestions will prove useful.
Function
f ax + b dx
f x f' x dx
fφx
f' x
f x
φ x dx
dx
z d c hi c h
z cc hh
f x
n
f' x dx
ax + b = t
x −a ,
φ(x) = t
f(x) = t
c
f t dt
c
dx
f(x) = t
+ c, n ≠ – 1
C
A T
I O
N
S
h
, x 2 – a2
x = a sec θ
c
a− x
,
a+ x
PAGE # 143
x = a tan2 θ
h
1
c
x a− x
x = a sin2 θ
h
x−a
,
x
h
x x−a ,
2[f(x)]1/2 + c
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
x = a tanθ or x = a sinhθ
a− x
,
x
x
,
x−a
x−α
,
β−x
E
c
x a−x ,
f' x
fx
h
x
,
a− x
log|f(x)| + c
n+1
x2 − a2
1
x a+ x . x a+ x
z ch
cf(x)h
, x2 + a2
a+ x
,
x
x
,
a+ x
n+1
f(x) = t
x = a sin θ or a cos θ
or x = a cosh θ
+c
2
2
2
1
F(ax + b) + c
a
dfcxhi
, a2 – x2
1
Integration
2
f(x) = t
2
1
2
zc h
zch ch
z d c hi c h
z cc hh
Substitution
f x dx is changed
1
c
x a− x
h
a+x
a−x
cx − αh cβ − xh ,(β > α)
x = a sec2 θ
x = a cos 2θ
x = α cos2 θ + β sin2 θ
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 144
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
IMPORTANT RESULTS USING STANDARD SUBSTITUTIONS :
Function
INTEGRATION OF FUNCTIONS USING ABOVE STANDARD
RESULTS :
Integration
Function
z
x−a
1
log
x+a
2a
1
x −a
2
2
=
z
1
a −x
2
2
dx
z
z
+ c
x
−1
coth–1 a + c when x > a
a
a+ x
1
log
a− x
2a
z
x −a
2
2
log{|x +
= cosh–1
z
dx
x +a
2
2
FG x IJ + c
H aK
= sinh–1
FG IJ
H K
FG x IJ + c
H aK
z
z
x − a dx
1
1
x x2 − a2 – a2 log {|x +
2
2
x − a |} + c
x2 + a2 dx
1
1
x x2 + a2 + a2 log {|x +
2
2
x2 + a2 |} + c
2
D U
C
A T
I O
N
S
px + q
ax2 + bx + c
dx or
(px + q) (ax2 + bx + c) dx
z ax
z
2
dx or
LMF
MNGH
a x+
b
2a
IJ
K
2
+
4ac − b2
2
4a
OP
PQ
then use appropriate formula
Express : px + q
= λ
d
(ax2 + bx + c) + µ
dx
evaluate λ & µ by equat
constant, the integral reduces to
known form
1
1 2
x a2 − x2 +
a sin–1
2
2
2
ax + bx + c
+ c
a − x dx
2
px + q
2
Express : ax2 + bx + c =
ing coefficient of x and
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
z
x2 + a2 |} + c
x
a
ax2 + bx + c
dx or
(ax2 + bx + c) dx
z
z
2
dx or
1
z
x − a |} + c
2
log{|x +
ax + bx + c
+ c
x
1
= tanh–1 a + c, when x < a
a
dx
1
2
Method
2
2
P(x)
2
+ bx + c
dx ,
Apply division rule and express it
in form Q(x) +
polynomial of degree
The integral reduces to known
2 or more
form
PAGE # 145
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
ch
R x
where P(x) is a
I O
N
S
ax + bx + c
2
PAGE # 146
MATHS FORMULA - POCKET BOOK
z
1
a sin x + b cos2 x + c
2
MATHS FORMULA - POCKET BOOK
dx Divide numerator & denominator
z
2
by cos x,
zc
or
z
1
a sin x + b cos x
h
2
dx
dx
dx
a sin x + b cos x + c
Replace sin x =
2 tan x / 2
1 + tan2 x / 2
z
dx
,
then add & sub x2. Thus the form
reduces to the known form.
1 − tan2 x / 2
1 + tan2 x / 2
4.
INTEGRATION BY PARTS :
when integrand involves more than one type of functions
the formula of integration by parts is used to integrate the
product of the functions i.e.
Express : num. = λ(deno.) +
z
(i)
d
(deno.) Evaluate λ & µ. Thus
dx
integral reduces to known form.
a sin x + b cos x + c
dx
p sin x + q cos x + r
Express : Num. = λ(deno.) +
= (1st fun)
d
(deno.) + ν Evaluate λ, µ, ν.
dx
Thus integral reduces to known
form.
x 2 ± a2
x 4 + kx 2 + a 4
dx
(ii)
Divide numerator & denominator
Fx ± a I
GH x JK
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
C
A T
I O
N
S
h c
du
dx
IK OP dx
Q
υ dx
h
1st fun. . 2nd fun. dx
z
2nd fun. dx –
z LMNFGH
IJ ez 2nd fun.dxjOP dx
K
Q
d
1st fun.
dx
Rule to choose the first function : first fun. should
be choosen in the following order of preference (ILATE).
[The fun. on the left is normally chosen as first
function]
L – Logarithmic function
= t, the
A – Algebraic function
T – Trigonometric function
integral becomes one of standard
forms.
E
zc
z LMN FHz
I – Inverse trigonometric function
2
by x2 and put
z
u. υ dx = u. υ dx –
or
µ
z
Divide num & deno. by 2a2 and
x + kx2 + a2
4
µ
z
Divide numerator & denominator
by 2 and then add & sub. a2.
Thus the form reduces as above.
then put tan x/2 = t and
replace 1 + tan2 x/2 = sec2 x/2
a sin x + b cos x
dx
c sin x + d cos x
dx
x + kx 2 + a 4
then put tanx = t & solve.
cos x =
z
x2
4
E – Exponential function
PAGE # 147
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 148
MATHS FORMULA - POCKET BOOK
(iii)
z
z
z
(a)
(b)
(c)
MATHS FORMULA - POCKET BOOK
5.
ch ch
ex f x + f' x dx = ex f(x) + c
INTEGRATION OF RATIONAL ALGEBRAIC FUNCTIONS
USING PARTIAL FRACTION :
ch ch
mf x + f' x dx = e
e
mx
LM f' cxh OP dx
MN c h m PQ
emx f x +
f(x) + c
ch
emx f x
=
Every Rational fun. may be represented in the form
where P(x), Q(x) are polynomials.
If degree of numerator is less than that of denominator,
the rational fun. is said to be proper other wise it is
improper. If deg (num.) ≥ deg(deno.) apply division rule
+ c.
m
ch
gcxh
f x
(iv)
i.e.
z
ch ch
xf' x + f x dx = x f(x) + c.
z
eax sinbx dx and
e ax
=
a2 + b2
e
ax
e
=
h
eax sin bx + c dx
c
px2 + qx + r
,
x−a x−b x−c
hc
hc
hc
e ax
a +b
2
z
e
c
A
+
x−a
≠ b
px2 + qx + r
h
cos bx + c dx
cacos bx + b sinbxh
A
B
C
+
+
x−a
x −b
x−c
h
cx − ah cx − bh , a
c
ax
A
B
+
x−a
x −b
h
px2 + qx + r
[a sin (bx + c) – b cos(bx + c)] + k1
2
Types of partial
fractions
a, b, c are distinct
(a sin bx – b cos bx) + k and
cos bx dx and
2
c
px + q
x−a x−b , a ≠ b
2
a +b
z
c
Types of proper
rational functions
ax
2
(vi)
z
c h , for integrating rcxh , resolve the
gcxh
gcxh
r x
= q(x) +
fraction into partial factors. The following table illustrate
the method.
NOTE : Breaking (iii) & (iv) integral into two integrals.
Integrate one integral by parts and keeping other integral
as it is by doing so we get the result (integral).
(v)
c h,
Qcxh
Px
mx
he
x − a x2 + bx + c , where
j
B
+
cx − ah
2
C
x−b
Bx + C
A
+ 2
x−a
x + bx + c
x2 + bx + c can
not be factorised
px3 + qx2 + rx + s
+ k
ex
2
+ ax + b x2 + cx + d ,
je
j
Ax + B
x + ax + b
2
+
Cx + D
x + cx + d
2
2
and
eax
a2 + b2
c
h
c
a cos bx + c + b sin bx + c
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
h
where x + ax + b,
x2 + cx + d can not
be factorised
+ k1.
PAGE # 149
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 150
MATHS FORMULA - POCKET BOOK
6.
MATHS FORMULA - POCKET BOOK
INTEGRATION OF IRRATIONAL ALGEBRAIC FUNCTIONS :
(i)
If integrand is a function of x & (ax + b)1/n then put
(ax + b) = tn
(ii)
If integrand is a function of x, (ax + b)1/n and
(viii) To evaluate
1
or
quadratic
where p = (L.C.M. of m & n).
z
(iv) To evaluate
z
(v) To evaluate
z
or
or
put
linear linear
dx
quad. linear
linear = t
form :
put linear = t2
7.
linear. quadratic
zc
h
put linear = 1/t
dx
z
z
dx
pure quad. pure quad
put
dx
pure quad. pure quad
then is the resulting integral, put
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
A T
I O
N
S
linear
quad. quad
dx
into partial fractions and
dx
linear quad.
1 − cos 2 mx
2
(i)
sin2 mx =
(ii)
cos2 mx =
(iii)
sin mx = 2sin
(iv)
sin3 mx =
(v)
cos3 mx =
(vi)
(vii)
(viii)
(ix)
(x)
tan2 mx
cot2 mx
2 cos A
2 sin A
2 sin A
1 + cos 2mx
2
x dx
2
linear . quadratic
(vii) To evaluate
C
z
linear . quadratic
pure quad = t
D U
FG linear IJ
H quadratic K
2
(vi) To evaluate
E
quad. quad
z
INTEGRATION USING TRIGONOMETRICAL IDENTITIES :
(A) To evaluate trigonometric functions transform the
function into standard integrals using trigonometric
identities as
dx
h
or
split the integral into two, each of which is of the
dx
zc
dx
and if the quadratic not under the square root can
be resolved into real linear factors, then resolve
(ax + b)1/m then put (ax + b) = tp
(iii) To evaluate
z
put x =
1
and
t
pure quad = u
PAGE # 151
mx
mx
cos
2
2
3 sin mx − sin 3mx
4
3 cos mx + cos 3mx
4
= sec2 mx – 1
= cosec2 mx – 1
cos B = cos (A + B) + cos (A – B)
cos B = sin (A + B) + sin (A – B)
sin B = cos (A – B) – cos (A + B)
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 152
MATHS FORMULA - POCKET BOOK
(B)
z
MATHS FORMULA - POCKET BOOK
sinm x cosm xdx .
(i)
(ii)
(iii)
(iv)
if m is odd put cos x = t
if m is even put sin x = t
if m & n both odd put sin x or cos x as t
if m & n both even use the formula of sin2x & cos2x
(v)
if m & n rational no. &
z
z
z
m+n−2
is –ve integer
2
put tan x = t
8.
Function
Integration
z
1 n ax
n
x e –
I
a
a n–1
z
z
z
z
z
xneaxdx , n ∈ N
z
cos ecnx dx
–
sinm x cosn x dx
C
A T
I O
n−2
cos ecn−2 x cot x
+
I
n − 1 n–2
n−1
c
cm + nh
h
cosn−1 x sinm+1 x + n − 1 Im, n−2
–sinm–1x cosn+1x + (m – 1) Im–2,n
NOTE : These formulae are specifically useful when m & n
are both even nos.
xn−1eax dx
xn sin x dx
–xn cos x + nxn–1sin x – n(n – 1) In–2
sinn x dx
–
cosn x dx
n−1
cosn−1 x sin x
+
In–2
n
n
n−1
sinn−1 cos x
+
In–2
n
n
n−1
tann x dx
ctanxh
cotn x dx
–
– In–2
n−1
n−1
ccot xh
n−1
– In–2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
n−2
secn−2 x tan x
+
I
n − 1 n–2
n−1
INTEGRATION BY SUCCESSIVE REDUCTION (REDUCTION
FORMULA) :
where In–1 =
E
secn x dx
N
S
PAGE # 153
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 154
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
DEFINITE INTEGRATION
z
f x dx = – f x dx
z
f x dx =
b
II.
zch
a
ch
a
1.
b
Definite Integration :
z
If
z
b
b
ch
f x dx = F(x) + c, then
III.
c
zch
ch
a
a
ch
ch
f x dx = F x + c
b
a
zch
b
f x dx +
f x dx where a < c < b
c
This property is mainly used for modulus function,
greatest integer function & breakable function
= F(b) – F(a) is called definite integral
a
zch zc
a
of f(x) w.r.t. x from x = a to x = b Here a is called lower
limit and b is called upper limit.
a
IV.
b
a
h
f a + b − x dx or
f x dx =
zch
f x dx =
0
b
zc
a
h
f a − x dx
0
Remarks :
*
To evaluate definite integral of f(x). First obtain the
indefinite integral of f(x) and then apply the upper
and lower limit.
*
a
z
V.
ch
f x dx =
−a
For integration by parts in definite integral we use
following rule.
b
z
uv dx =
a
*
{uz v.dx}
a
z FGH z
b
b
–
a
R|2 f x dx
S| z c h
T 0
=
0
du
. v. dx dx
dx
When we use method of substitution. We note that
while changing the independent variable in a definite
integral, the limits of integration must also we changed
accordingly.
zch
2a
f x dx =
VI.
0
VII.
c h
f x + f −x dx
0
a
IJ
K
z ch
a
ch
, if f cxh is an odd function
, if f x is an even function
R|
S|2 fcxh dx
|T 0
z
a
0
c h ch
, if f c2a − xh = − f cxh
, if f 2a − x = f x
If f(x) is a periodic function with period T, Then
PROPERTIES OF DEFINITE INTEGRAL :
zch
nT
zch
b
f x dx =
I.
a
z
b
D U
C
A T
I O
N
S
f x dx = n f x dx
ch
f t dt
0
a
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
zch
T
0
and further if a ∈ R+, then
PAGE # 155
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 156
MATHS FORMULA - POCKET BOOK
a+nT
zch
zch zch
a
f x dx =
nT
f x dx ,
0
nT
b +nT
MATHS FORMULA - POCKET BOOK
(ii)
zch
T
f x dx = (n – m) f x dx ,
mT
If the function φ(x) and ψ(x) are defined on [a, b]
and differentiable at a point x ∈ (a, b), and f(x, t)
is continuous, then,
0
d
dx
zch zch
b
f x dx =
a+ nT
f x dx
a
ψ(x)) –
VIII. If m and M are the smallest and greatest values of
a function f(x) on an interval [a, b], then
3.
zch
b
m(b – a) <
LM
MN
ψ (x)
z
f (x, t) dt
φ(x)
RSdφ(x) UV f(x,
T dx W
f x dx < M(b – a)
z
ch
f x dx
z
z ch
|f x dx|
a
=
z
b
If f(x) < g(x) on [a, b], then
ch
f x dx ≤
a
zch
b
cosn x dx =
A T
sinn x dx
if n is odd
if n is even
π /2
For integration
z
sinm x cosn x dx follow the following
0
Leibnitz's Rule :
steps
(i)
(a)
If m is odd put cos x = t
(b)
If n is odd put sin x = t
(c)
If m and n are even use sin2x = 1– cos2x
If f(x) is continuous and u(x), v(x) are differentiable
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
z
R| n − 1 . n − 3 ..... 2 .1,
S|n −n1 nn−−32 13 π
|T n . n − 2 ...... 2 . 2 ,
a
z
v(x)
f(t) dt =
or cos2x = 1 – sin2x and then use
u(x)
I O
N
S
π /2
π /2
z
d
d
f{v(x)}
{v(x)} – f{u(x)}
{u(x)}.
dx
dx
D U
φ(x)).
0
Differentiation Under Integral Sign :
d
functions in the interval [a, b], then,
dx
E
φ(x)
RSdψ (x) UV f(x,
T dx W
g x dx
(ii)
2.
f (x, t) dt +
b
<
a
X.
z
π /2
π /2
(i)
a
b
ψ (x)
Reduction Formulae :
a
IX.
OP
PQ =
0
PAGE # 157
n
sin x dx or
z
cosn x dx
0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 158
MATHS FORMULA - POCKET BOOK
∞
(iii)
z
e
− ax
cos bx dx =
0
MATHS FORMULA - POCKET BOOK
4.
a
Summation of series by Definite integral or limit as a
sum :
a2 + b2
zch
b
∞
(iv)
z
lim h[f(a) + f(a + h) + f(a + 2h) +.....
f x dx = h→0
(i)
b
e−ax sinbx dx =
a
a + b2
2
0
+f(a + (n – 1)h]
where nh = b – a.
∞
(v)
z
e −ax xndx =
0
n!
FI
∑ GH JK
zch
1
n
a +1
n
(ii)
lim
n→∞
r
1
f
n r =1 n
f x dx
=
0
π /2
(vi)
z
sinn x cosm x dx
[i.e. exp. the given series in the form
0
LM m − 1 . m − 3 .... 2 . 1
MM m + n m + n − 2 3 + n 1 + n
MM mm +− 1n. mm+ n− 3− 2.... 2 1+ n. n n− 1. nn −− 32 .... 23
MMm − 1. m − 3 .... 1 . n − 1. n − 3.... 1 . π
Nm + n m + n − 2 2 + n n n − 2 2 2
=
OP
PP
PP
PP
Q
; if m is odd and n may be even or odd
; if m is enen and n is odd
; if m is even and n is even
replace
1
FrI
∑ n f GH nJK
r
1
by x and
by dx and the limit of the
n
n
zch
1
sum is
f x dx ]
0
These formulae can be expressed as a single formula :
π /2
z
5.
sinm x cos n x dx
Key Results :
π /2
0
z
*
[(m − 1) (m − 3)....] [(n − 1) (n − 3) .....]
(m − n) (m + n − 2) ....
=
0
π /2
to be multiplied
by
π /2
logsin x dx =
π
when m and n are both even
2
z
*
0
z
logcos x dx =
0
c h
f csin xh + fccos xh dx =
f sin x
π /2
zc
0
−π
log2
2
c h
h c h
f cos x
f sin x + f cos x dx
integers.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 159
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 160
MATHS FORMULA - POCKET BOOK
c
π /2
z
=
c
h c
π /2
h
f tan x + f cot x
0
=
f tan x
c
zc
f cos ec x
h c
h
z
f cos ec x + f sec x
sinmx sin nx dx =
R|0
S| π
T2
z
z
π
1
dx =
2
a2 − x2
z
z
x2 a2 − x2 dx =
z
x2
0
FG
H
π 2
a2 − x2
−
dx = a3 4 3
a+ x
a
cos mx . cos nx dx
*
0
z
2a
*
z
*
0
If n ∈ N, then
*
*
C
A T
I O
N
S
a2 − x2
n
j
z
x−a
z
x−a
π b−a
dx =
a+x
2
b
(i)
x
a − x2
2
dx = a
dx
a
(ii)
x
a2 − x2
πa
3a
+
6
8
2
2
dx =
c h
2. 4. 6...... 2n
2n+1
dx = 3.5 . 7..... 2n + 1 a
c
h
b−x
=π
c
h
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
if a > 0
If a < b then
zc
a
(iii)
0
E
ze
a
π 2
a
4
IJ
K
πa2
2
2ax − x2 dx =
0
a
πa4
16
0
a
0
j
.
2a2
a
*
a
*
dx =
3/2
0
z
0
0
c h dx
= π/4.
fctan xh + fccot xh
f cot x
if m = n
0
*
z
1
x dx
a2 + x2
0
if m, n are different + ve int egers
a2 − x2 dx =
a
h
dx=
*
0
a
*
0
ze
a
c h dx
f csec xh + f ccos ecxh
f sec x
π /2
0
=
z
π /2
π /2
*
h
dx =
MATHS FORMULA - POCKET BOOK
PAGE # 161
x − a b − x dx = π b − a 2
2
hc
h
c
h
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 162
MATHS FORMULA - POCKET BOOK
z
*
b
(iv)
a
MATHS FORMULA - POCKET BOOK
dx
=
cx − ahcb − xh
x
π
If f(x) is continuous on [a, b] then there exists a
2
ab
zch
b
ab > 0
point c ∈ (a, b) s.t
f x dx = f(c) [b – a]. The no.
a
*
If a > 0 then
zch
b
z
a
a+x
π +2
dx =
2
a−x
z
a− x
a
π −2
a + x dx = 2
a
(i)
c
0
a
(ii)
c
0
z
z
0
a
(iv)
h
0
a+ x
a− x
dx =
(i)
a+ x
a − x dx =
z
x e −ax dx =
0
∞
(ii)
z
e −r
0
∞
(iii)
z
0
C
A T
I O
N
S
zd
2k
h
*
i
x − x dx = k, where k ∈ I,
Q x – [x] is a periodic function with period 1.
If f(x) is a periodic fun. with period T, then
*
10 a a
3
a+ T
zch
z c
FG π + 1IJ a
H2 K
a
π /4
2 2
x
dx =
π
2a a
h
log 1 + tan x dx =
*
0
π
log2
8
a > 0
π
(r > 0)
2r
e−ax − e −bx
dx = loge(b/a) (a, b > 0)
x
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
fun. f(x) on the interval [a, b]. The above result is
called the first mean value theorem for integrals.
f x dx is independent of a.
∞
E
a
If a > 0, n ∈ N, then
*
1
f x dx is called the mean value of the
b−a
0
a
(iii)
f(c) =
PAGE # 163
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 164
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
DIFFERENTIAL EQUATIONS
1.
(B)
dy
= f(x) g(y)
dx
This can be integrated as
the form
Order of a differential equation : The order of a differential
equation is the order of the highest derivative occurring in
it.
2.
z
Degree of a differential equation : The degree of a
differential equation is the degree of the highest order
derivative occurring in it when the derivatives are made
free from the radical sign.
d2 y
Eg. (i)
dx
2
+
(ii)
F d yI
GH dx JK
3
(iii)
2
+
3
(C)
F dy I
1+G J
H dx K
FG1 + dy IJ
H dx K
dy
=
f(y)
+ 5y = 0
D U
Equations Reducible to Homogeneous form and
variable separable form
*
Form
z
dy =
z
C
A T
I O
N
S
ax + by + c
dy
=
Ax + By + C
dx
........... (1)
a
b
≠
A
B
This is non Homogeneous
Put
x = X + h and y = Y + k in (1)
where
dy
= f(x) or
dx
dy
dY
=
Put ah + bk + c = 0, Ah + Bk + C = 0,
dX
dx
find h, k
dY
aX + bY
=
. This is homogeneous.
dX
AX + BY
Solve it and then put X = x – h, Y = y – k we shall
get the solution.
f(x) dx
Then
dx to get its solution.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
(D)
∴
z
FG IJ
H K
x
or yn f y .
Such an equation can be solved by putting y = vx or
x = vy. After substituting y = vx or x = vy. The given
equation will have variables separable in v and x.
dy
= f(y)
dx
z
Homogeneous Equations : It is a differential equation
FG IJ
H K
2
(A) Differential equation of the form
or
f(x) dx + c
y
of degree n if it can be written as xn f x
2
SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST
ORDER AND FIRST DEGREE :
Integrate both sides i.e.
z
f(x, y)
dy
=
, where f(x, y) and g(x,
g(x, y)
dx
y) are homogeneous functions of x and y of the same
degree. A function f(x, y) is said to be homogeneous
order of (i) 2 (ii) 1 & (iii) 3,
degree of (i) 1 (ii) 2 & (iii) 2
3.
dy
=
g(y)
of the form
dy
+ 5y = 0
dx
dy
+
y = x
dx
Variable Separable Form : Differential equation of
PAGE # 165
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 166
MATHS FORMULA - POCKET BOOK
*
Form
where
ax + by + c
dy
=
+ By + C
Ax
dx
MATHS FORMULA - POCKET BOOK
..... (1),
*
In x :
dx
+ Rx = S, where R, S are functions of y
dy
alone or constant.
a
b
=
= k say
A
B
its solution x e
k (Ax + By) + c
dy
=
∴
Ax + By + C
dx
where
dy
dz
A + B
=
dx
dx
⇒
Put
Ax + By = z
⇒
dz
kz + c
= A + B
dx
z+c
e
z
R.dy
z
R dy
=
z
S. e
z
R.dy
dy + c
is called the integrating factor (I.F.) of
the equation.
(F) Equation reducible to linear form :
This is variable separable form and can be solved.
*
dy
Form
= f(ax + by + c)
dx
*
⇒
Put
ax + by = z
∴
dz
= a + b f(z)
dx
a + b
Differential equation of the form
Linear equation :
*
In y :
dy
dz
=
dx
dx
y–n
Put
where
e
z
P dx
P dx
y–n
+ 1
+ 1
= Q
= z
Note : In general solution of differential equation we can
take integrating constant c as tan–1 c, ec, log c etc.
according to our convenience.
alone or constant.
z
dy
+ py–n
dx
⇒ The given equation will be linear in z and can be
solved in the usual manner.
dy
+ Py = Q, where P, Q are function of x
dx
its solution ye
+ Py = Qyn
where P and Q are functions of x or constant is called
Bernoulli's equation. On dividing through out by yn, we
get
This is variable separable form and can be solved.
(E)
dy
dx
=
z
Qe
z
P dx
dx + c
is called the integrating factor (I.F.) of
the equation.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 167
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 168
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
4.
Vectors in terms of position vectors of end points -
VECTORS
1.
AB = OB – OA = Position vector of B – position vector of A
i.e. any vector = p.v. of terminal pt – p.v. of initial pt.
Types of vectors :
(a) Zero or null vector : A vector whose magnitude is
zero is called zero or null vector.
r
a
Vector a
(b) Unit vector :
a$ = |a| =
Magnitude of a
(c)
2.
5.
Multiplication of a vector by a scalar :
r
r
If a is a vector and m is a scalar, then m a is a vector and
r
magnitude of m a = m|a|
r
$
and if a = a1 $i + a2 j + a3 k$
Equal vector : Two vectors a and b are said to be
equal if |a| = |b| and they have the same direction.
Triangle law of addition : AB + BC = AC
r
$
then m a = (ma1) $i + (ma2) j + (ma3) k$
6.
Distance between two points :
Distance between points A(x1, y1, z1) and B(x2, y2, z2)
c = a + b
C
→
= Magnitude of AB
–c =
–b
–a +
=
–
b
7.
A
3.
–
a
Position vector of a dividing point :
r
(i)
If A( a ) & B( b ) be two distinct pts, the p.v. c of the
point C dividing [AB] in ratio m1 : m2 is given by
r
r
m1b + m2a
r
c = m +m
1
2
B
Parallelogram law of addition : OA + OB = OC
a + b = c
B
C
–
b
D
–
a
(x2 − x1 )2 + (y2 − y1)2 + (z2 − z1 )2
1
[p.v. of A + p.v. of B]
2
(ii)
p.v. of the mid point of [AB] is
(iii)
If point C divides AB in the ratio m1 : m2 externally,
then p.v. of C is c =
A
m1 b − m2 a
m1 − m2
where OC is a diagonal of the parallelogram OABC
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 169
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 170
MATHS FORMULA - POCKET BOOK
(iv) p.v. of centriod of triangle formed by the points A( a ),
MATHS FORMULA - POCKET BOOK
10. Coplanar and non coplanar vector :
(v)
If a , b , c be three non coplanar non zero vector
(i)
r
r
a+b+c
B( b ) and C ( c ) is
3
then x a + y b + z c = 0
⇒ x = 0, y = 0, z = 0
p.v. of the incentre of the triangle formed by the points
r
r
r
A( α ), B( β ) and C( γ ) is
(ii)
If a , b , c be three coplanar vectors, then a vector
c can be expressed uniquely as linear combination of
aα + bβ + cγ
where a = |BC|, b = |CA|, c = |AB|
a+b+c
8.
remaining two vectors i.e. c = λ a + µ b
(iii)
Some results :
(i)
b and c i.e. r = x a + y b + z c
If D, E, F are the mid points of sides BC, CA & AB
respectively, then AD + BE + CF = 0
Any vector r can be expressed uniquely as inner combination of three non coplanar & non zero vectors a ,
11. Products of vectors :
(ii)
If G is the centriod of ∆ABC, then GA + GB + GC = 0
(I)
Scalar or dot product of two vectors :
(iii)
If O is the circumcentre of a ∆ABC, then
(i)
a . b = |a| |b| cosθ
OA + OB + OC = 3 OG = OH where G is centriod and
H is orthocentre of ∆ABC.
(ii)
Projection of a in the direction of b =
a. b
| b|
(iv) If H is orthocentre of ∆ABC, then
a. b
& Projection of b in the direction of a =
| a|
HA + HB + HC = 3 HG = OH
9.
Collinearity of three points :
(i)
Three points A, B and C are collinear if AB = λ AC for
some non zero scalar λ.
(ii)
The necessary and sufficient condition for three points
(iii)
Component of r ⊥
with p.v. a , b , c to be collinear is that there exist
three scalars l, m, n all non zero such that
l a + m b + n c = 0, l + m + n = 0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
FG IJ
H Ka
F r. a IJ
to a = r – G
H|a| K a
r. a
Component of r on a =
|a|2
(iv)
$i . $i = $j . $j = k$ . k$ = 1
(v)
$i . $j = $j . k$ = k$ . $i = 0
PAGE # 171
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 172
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(vii) a × ( b × c ) = ( a × b ) × c
(vi) If a and b are like vectors, then a . b = | a || b | and
(viii) a × ( b + c ) = ( a × b ) + ( a × c )
If a and b are unlike vectors, then a . b = –| a || b |
(vii) a , b are ⊥ ⇔ a . b = 0
(ix)
$i × $i = $j × $j = k$ × k$ = 0 , $i × $j = k$ ,
(x)
$j
$
× k$ = $i , k$ × $i = j
Area of triangle :
(viii) ( a . b ). b is not defined
(ix)
( a ± b )2 = a2 ± 2 a . b + b2
(x)
| a + b | = | a| + | b |
⇒ a || b
(xi)
| a + b |2 = |a|2 + |b|2
⇒ a ⊥ b
(xii) | a + b | = | a – b |
⇒
a ⊥ b
(a)
1
2
(b)
If a , b , c are p.v. of vertices of ∆ABC,
(xiii) work done by the force :
work done = F . d ,
is displacement vector.
then =
where F is force vector and d
(xi)
(II) Vector or cross product of two vectors :
a × b = |a| |b| sinθ n$
(ii)
if a , b are parallel
(iii)
⇔
If a & b are two adjacent sides of a parallelogram, then area = | a × b |
(b)
a × b
1
|( a × b ) + ( b × c ) + ( c × a )|
2
Area of parallelogram :
(a)
(i)
AB × AC
= 0
If a and b are two diagonals of a parallelogram,
then area =
a × b = –( b × a )
1
|a × b |
2
(xii) Moment of Force :
(iv)
(v)
n$ = ±
a×b
Moment of the force F acting at a point A about O is
| a × b|
Moment of force = OA × F = r × F
$
$
let a = a1 $i + a2 j + a3 k$ & b = b1 $i + b2 j + b3 k$ , then
a. a a. b
(xiii) Lagrange's identity : | a × b |2 =
a. b b. b
$i
$j k$
a a2 a3
a × b = 1
b1 b2 b3
(vi)
a × a =0
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
(III) Scalar triple product :
r
r
(i)
If a = a1 $i + a2 $j + a3 k$ , b = b1 $i + b2 $j + b3 k$ and
r
c = c1 $i + c2 $j + c3 k$ then
PAGE # 173
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 174
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(d)
r r
r
r r r
( a × b ). c = [ a b c ] =
a1 a2 a3
b1 b2 b3
c1 c 2 c 3
r r r
and [ a b c ] = volume of the parallelopiped whose
r r r
coterminus edges are formed by a , b , c
r r r
r r r
r r r
(ii) [ a b c ] = [ b c a ] = [ c a b ],
r r r
r r r
r r r
but [ a b c ] = – [ b a c ] = – [ a c b ] etc.
r r r
r r
(iii) [ a b c ] = 0 if any two of the three vectors a , b ,
r
c are collinear or equal.
r r
r
r
r r
(iv) ( a × b ). c = a .( b × c ) etc.
(v)
→
→
→
→
r
r
r r r
c + a ] = 2[ a b c ]
r
r
c – a] = 0
r
r
r r r
c × a ] = [ a b c ]2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
r
r
b – c,
a b
c
r
Direction cosines of r = ai$ + bj$ + ck$ are r , r , r .
|r | |r | |r |
(i)
→
(ii)
Incentre formula : The position vector of the incentre
r
r
r
aa + bb + cc
of ∆ ABC is
.
a+b+c
(iii)
Orthocentre formula : The position vector of the
r
r
r
a tan A + b tanB + c tan C
orthocentre of ∆ ABC is
tan A + tanB + tan C
→
[ AB AC AD ] = 0
r r
r
r
(xii) (a) [ a + b b + c
r r
r
r
(b) [ a – b b – c
r r
r
r
(c) [ a × b b × c
r
r
a × b,
12. Application of Vector in Geometry :
1
| AB × AC . AD |
6
r r r r
(x) Four points with p.v. a , b , c , d will be coplanar if
r r r
r r r
r r r
r r r
[d b c ] + [d c a] + [d a b] = [a b c ]
(xi) Four points A, B, C, D are coplanar if
(ix) Volume of tetrahedron ABCD is
r r r
a , b , c are coplanar, then so are
r r
r
× c , c × a and
r r
r
r r
r
r
+ b , b + c , c + a and a – b ,
r
– a are also coplanar.
(IV) Vector triple Product :
r
r
r r r
r
If a , b , c be any three vectors, then ( a × b ) × c
r
r
r
and a × ( b × c ) are known as vector triple product
and is defined as
r
r r r
r
r
r r r
( a × b ) × c = ( a . c ) b – (b . c ) a
r
r
r
r r r
r r r
and a × ( b × c ) = ( a . c ) b – ( a . b ) c
r
r
r
r
r
r
Clearly in general a × ( b × c ) ≠ ( a × b ) × c but
r
r
r
r
r
r
r r
( a × b ) × c = a × ( b × c ) if and only if a , b
r
& c are collinear
[ $i $j k$ ] = 1
r r r
r r r
(vi) If λ is a scalar, then [λ a b c ] = λ[ a b c ]
r r r
r r r
r
r r r
(vii) [ a + d b c ] = [ a b c ] + [ d b c ]
r r r
r r r
(viii) a , b , c are coplanar ⇔ [ a b c ] = 0
If
r
b
r
a
r
c
(iv) Vector equation of a straight line passing through a
r
fixed point with position vector a and parallel to a
r
r
r r
given vector b is r = a + λb .
PAGE # 175
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 176
MATHS FORMULA - POCKET BOOK
(v)
MATHS FORMULA - POCKET BOOK
(viii) The equation of the plane passing through a point
r
r
r
having position vector a and parallel to b and c is
r
r r
r
rrr
rrr
r = a + λb + µc or [ r bc ] = [ abc ], where λ and µ are
scalars.
(ix) Vector equation of a plane passing through a point
r
r
r
r
rrr
abc is r = 1 − s − t a + sbt + c
r r r r r r r
rrr
or r. b × c + c × a + a × b = [ abc ].
The vector equation of a line passing through two
r
r
points with position vectors a and b is
r r
r r
r =a+ λb−a .
e
j
(vi) Shortest distance between two parallel lines : Let l1
and l2 be two lines whose equations are l 1 :
r
r
r r
r r
r = a1 + λb1 and l2 : r = a2 + µb2 respectively.
c
e
Then, shortest distance
cb
1
PQ =
(x)
h c
× b2 . a2 − a1
|b1 × b2 |
c
h=
b1 b2 a2 − a1
h
constant.
The perpendicular distance of a point having position
r
r r
vector a from the plane r.n = d is given by
r r
|a.n − d|
r
p=
.
|n|
r r
(xii) An angle θ between the planes r1.n1 = d1 and
r
r
r
| a2 − a1 × b|
r
r r
r
and r = a2 + µb is given by d =
.
|b|
If the lines
r
r r
r = a1 + λb1 and
h
r
r r
r = a2 + µb2 intersect,
n1 . n2
r r
r2 .n2 = d2 is given by cos θ = ± |n ||n | .
1
2
then the shortest distance between them is zero.
c
(xiii) The equation of the planes bisecting the angles
r r
between the planes r1 .n1 = d1
r r
r r
|r.n1 − d1| |r.n2 − d2|
r r
=
r
r
and r2 .n2 = d2 are
|n1|
|n2|
r r
r r
(xiv) The plane r .n = d touches the sphere | r − a | = R,
r r
|a.n − d|
r
if
= R.
|n|
h
Therefore, [ b1 b2 a2 − a1 ] = 0
⇒
r
r r r
[ a2 − a1 b1b2 ] = 0
c
h
⇒
car
2
r
r
r
− a1 . b1 × b2
h e
j = 0.
r
(vii) Vector equation of a plane normal to unit vector n and
at a distance d from the origin is
r
r . n$ = d.
r
If n is not a unit vector, then to reduce the equation
r r
r
r . n = d to normal form we divide both sides by | n |
r
r n
r
d
d
r = r
r
.
to obtain
or r . n$ = r .
|n|
|n|
|n|
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
h
(xi)
shortest distance between two parallel lines : The
r
r r
shortest distance between the parallel lines r = a1 + λb
c
j
The equation of any plane through the intersection
r r
r r
of planes r . n1 = d1 and r . n2 = d2 is
r r
r . n1 + λn2 = d1 + λd2, where λ is an arbitrary
c
|b1 × b2 |
h
(xv)
If the position vectors of the extremities of a diamr
r
eter of a sphere are a and b , then its equation is
r r r
r r
r
r r
r
( r − a ).( rr − b ) = 0 or | r |2 – r. a − b + a.b = 0.
e
PAGE # 177
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
j
PAGE # 178
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
THREE DIMENSIONAL GEOMETRY
*
Coordinates of the centroid of a triangle are
FG x
H
1
1.
Points in Space :
(i)
Origin is (0, 0, 0)
(ii)
Equation of x-axis is y = 0, z = 0
(iii)
Equation of y-axis is z = 0, x = 0
*
FG x
H
Equation of YOZ plane is x = 0
y1
1 y
2
Where ∆x =
2 y
3
Distance formula :
Distance between two points A(x1, y1, z1) and B(x2,
y2, z2) is given by
AB =
(ii)
(x 2 − x1 )2 + (y 2 − y1 )2 + (z 2 − z1 )2
(iii)
3.
Distance of a point p(x, y, z) from coordinate axes
OX, OY, OZ is given by
z2 + x2 and
*
4.
Section formula :
*
Internally are
*
Externally are
FG mx + nx
H m+n
FG mx − nx
H m−n
1
,
my 2 + ny1 mz 2 + nz1
,
m+n
m+n
2
1
,
my2 − ny1
m−n
C
A T
I O
N
S
2
∆2x + ∆2y + ∆2z
and so.
1
6
x1
x2
x3
x4
y1
y2
y3
y4
z1
z2
z3
z4
1
1
1
1
Direction cosines and direction ratios of a line :
*
If
α , β , γ are the angles which a directed line
segment makes with the +ve direction of the coordinate
axes, then l = cos α , m = cos β , n = cos γ are
IJ
K
mz − nz I
,
J
m−n K
2
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
D U
Volume of tetrahedron =
x2 + y2
The coordinates of a point which divides the join of (x1,
y1, z1) and (x2, y2, z2) in the ratio
m : n
E
z1 1
z2 1
z3 1
x12 + y12 + z12
y 2 + z2 ,
IJ
K
x1 − x2
y1 − y2
z1 − z2
Condition of collinearity x − x = y − y = z − z
2
3
2
3
2
3
*
Distance between origin (0, 0, 0) & point (x, y, z)
=
+ x2 + x3 + x 4 y1 + y2 + y3 + y 4 z1 + z 2 + z 3 + z 4
,
,
4
4
4
Area of triangle is given by ∆ =
*
(vii) Equation of XOY plane is z = 0
(i)
1
Note :
(vi) Equation of ZOX plane is y = 0
2.
IJ
K
Coordinates of centroid of a tetrahedron
(iv) Equation of z-axis is x = 0, y = 0
(v)
+ x 2 + x3 y1 + y2 + y 3 z1 + z 2 + z3
,
,
3
3
3
called direction cosines of the line and cos2 α + cos2
β + cos2 γ = 1 i.e. l 2 + m2 + n2 = 1, where 0 ≤
α, β, γ
1
PAGE # 179
≤ π
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 180
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
x − x1
y − y1
z − z1
x2 − x1 = y2 − y1 = z2 − z1
If l , m, n are direction cosines of a line and a, b,
c are proportional to l , m, n respectively, then a, b,
c are called direction ratios of the line and
*
m
n
l
=
=
=±
a
b
c
l 2 + m2 + n2
a2 + b2 + c 2
1
=±
a + b2 + c2
2
.
*
Direction cosines of x-axis are 1, 0, 0, similarly direction
cosines of y-axis and z-axis are respectively 0, 1, 0
and 0, 0, 1.
*
If l , m, n are d.c.s of a line OP and (x, y, z) are
coordinates of P then x = l r, y = mr and z = nr where
r = OP.
Direction cosines of PQ = r, where P is (x1, y1, z1) and
Q(x2, y2, z2) are
*
The angle θ between the lines whose d.c.'s are l 1,
m1, n1 and l 2, m2, n2 is given by
*
cos θ = l 1 l 2 + m1m2 + n1n2.
The lines are || if
The lines are ⊥ if l 1 l 2 + m1m2 + n1n2 = 0
The angle θ between the lines whose d.r.s are a1, b1,
c1 and a2, b2, c2 is given by
*
cos θ = ±
x2 − x1
y2 − y1
z 2 − z1
,
,
r
r
r
*
The lines are || if
If a, b, c are direction no. of a line, then
a2 + b2 + c2 need not to be equal to 1.
A T
a1
b1
c1
=
=
a2
b2
c2 and
= (x2 – x1) l + (y2 – y1)m + (z2 – z1)n, where
p(x1, y1, z1) and Q(x2, y2, z2).
*
Straight line in space :
*
Equation of a straight line passing through a fixed
point and having d.r.'s a, b, c is
Two straight lines in space (not in same plane) which
are neither parallel nor intersecting are called skew
lines.
*
Shortest distance between two skew lines,
x − x1
y − y1
z − z1
=
=
and
m1
n1
l1
I O
x − x2
y − y2
z − z2
=
=
is given
m2
n2
l2
form)
Equation of a line passing through two points is
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
a22 + b22 + c22
L = l (x2 – x1) + m(y2 – y1) + n(z2 – z1)
*
D U
a12 + b12 + c12
l , m, n
x − x1
y − y1
z − z1
=
=
(is the symmetrical
a
c
b
E
a1a2 + b1b2 + c1c2
The lines are ⊥ if a1a2 + b1b2 + c1c2 = 0
Length of the projection of PQ upon AB with d.c.,
*
Note : Direction cosines of a line are unique but the
direction ratios of line are not unique.
If P(x1, y1, z1) & Q(x2, y2, z2) be two points and L be a
line with d.c.'s l , m, n, then projection of [PQ] on
5.
m1
n1
l1
l2 = m2 = n2 and
N
S
PAGE # 181
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 182
MATHS FORMULA - POCKET BOOK
x2 − x1
s.d. = ±
*
l1
l2
*
y2 − y1 z2 − z1
m1
n1
m2
n2
Normal form of the equation of plane is l x + my +
nz = p, where l , m, n are the d.c.'s of the normal
to the plane and p is the length of perpendicular from
the origin.
(m1n2 − m2n1)2 + (n1 l2 − l1n2 )2 + (l1m2 − m1 l2 )2
*
ax + by + cz + k = 0 represents a plane || to the
plane ax + by + cz + d = 0 and ⊥ to the line
Two straight lines are coplanar if they are intersecting
or parallel
condition
6.
MATHS FORMULA - POCKET BOOK
x2 − x1
y2 − y1
z2 − z1
l1
l2
m1
m2
n1
n2
x
y
z
=
=
.
a
c
b
*
= 0
Equation of plane through three non collinear points
is
x
x1
x2
x3
Plane : A plane is a surface such that if two points are
taken in it, straight line joining them lies wholly in the
surface.
*
Ax + By + Cz + D = 0 represents a plane whose
normal has d.c.s proportional to A, B, C.
*
Equation of plane through origin is given by Ax + By
+ Cz = 0.
*
Equation of plane passing through a point (x1, y1, z1)
is A(x – x1) + B(y – y1) + C(z – z1) = 0, where A,
B, C are d.r.'s of a normal to the plane.
*
Equation of plane through the intersection of two
planes
or
*
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
C
A T
I O
N
S
= 0
a1a2 + b1b2 + c1c2
a12 + b12 + c12 a22 + b22 + c22
where θ is the angle between the normals.
Equation of plane which cuts off intercepts a, b, c
respectively on the axes x, y and z is
plane are ⊥ if a1a2 + b1b2 + c1c2 = 0
a1
b1
c1
plane are || if a = b = c = 0.
2
2
2
x
y
z
+
+
= 1.
b
a
c
D U
= 0
y − y1
z − z1
y2 − y1 z2 − z1
y3 − y1 z3 − z1
cos θ = ±
Q ≡ a2x + b2y + c2z + d2 = 0 is P + λ Q = 0.
E
y3
z 1
z1 1
z2 1
z3 1
The angle between the two planes is given by
P ≡ a1x + b1y + c1z + d1 = 0 and
*
x − x1
x2 − x1
x3 − x1
y
y1
y2
PAGE # 183
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 184
MATHS FORMULA - POCKET BOOK
If AP be the ⊥ from A to the given plane, then it
is || to the normal, so that its equation is
*
MATHS FORMULA - POCKET BOOK
7.
Line and Plane :
If ax + by + cz + d = 0 represents a plane and
x − x1
y − y1
z − z1
=
=
represents a straight line, then
m
n
l
x−α
y−β
z−γ
=
=
= r (say)
a
c
b
Any point P on it is (ar + α , br + β , cr + γ )
Length of the ⊥ from P(x1, y1, z1) to a plane
*
ax + by + cz + d = 0 is given by
p=
*
The line is ⊥ to the plane if
*
The line is || to the plane if a l + bm + cn = 0.
*
The line lies in the plane if
a l + bm + cn = 0 and ax1 + by1 + cz1 + d = 0
ax1 + by1 + cz1 + d
The angle θ between the line and the plane is given
by
*
a2 + b2 + c 2
Distance between two parallel planes
(ax + by + cz + d1 = 0, ax + by + cz + d2 = 0) is
given by
*
sin θ =
d2 − d1
*
al + bm + cn
a2 + b2 + c2
l2 + m2 + n2
General equation of the plane containing the line
a + b2 + c2
2
x − x1
y − y1
z − z1
=
=
is
n
m
l
Two points A(x1, y1, z1) and B(x2, y2, z2) lie on the
same or different sides of the plane
A(x – x1) + B(y – y1) + C(z – z1) = 0. where
ax + by + cz + d = 0, according as the expression
A l + Bm + Cn = 0.
ax1 + by1 + cz1 + d and ax2 + by2 + cz2 + d are of
same or different sign.
*
a
b
c
=
=
m
n
l
*
*
Length of the perpendicular from a point (x1, y1, z1)
to the line
Bisector of the angles between the planes
a1x + b1y + c1z + d1 = 0
p2 = (x1 – α )2 + (y1 – β )2 + (z1 – γ )2 – [ l (x1 – α )
and a2x + b2y + c2z + d2 = 0 are
a1x + b1y + c1z + d1
a12 + b12 + c12
+ m(y1 – β ) + n(z1 – γ )]2
a2 x + b2 y + c2 z + d2
= ±
x−α
y−β
z−γ
=
=
is given by
m
n
l
a22 + b22 + c22
if a1a2 + b1b2 + c1c2 is –ve then origin lies in the acute
angle between the planes provided d1 and d2 are of
same sign.
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 185
, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510
E
D U
C
A T
I O
N
S
PAGE # 186