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Title: All Mathematics formulas for high school and University students
Description: These are my personal digital notes of mathematics formulas. It is very useful for high school students. It also contains some short tricks and advanced formulas to solve math problems. These notes will definitely help improve your math problem solving skills! Topics covered: ALL high school math topics Complete Algebra, Calculus, Analytical Geometry, Probability, Progression and series, Probability, Trigonometry, 3-D, Vector, Functions, Permutation and combination, binomial theorem, solution of triangle etc...
Description: These are my personal digital notes of mathematics formulas. It is very useful for high school students. It also contains some short tricks and advanced formulas to solve math problems. These notes will definitely help improve your math problem solving skills! Topics covered: ALL high school math topics Complete Algebra, Calculus, Analytical Geometry, Probability, Progression and series, Probability, Trigonometry, 3-D, Vector, Functions, Permutation and combination, binomial theorem, solution of triangle etc...
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MATHS FORMULA - POCKET BOOK
QUADRATIC EQUATION & EXPRESSION
1
...
Conjugate roots :
Irrational roots and complex roots occur in conjugate pairs
i
...
Quadratic expression :
A polynomial of degree two of the form ax2 + bx + c, a ≠ 0 is
called a quadratic expression in x
...
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
...
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
...
b2 4ac
...
e
...
Suppose a, b, c ∈ Q a ≠ 0 then
If D > 0 and D is a perfect square ⇒ roots are rational
(i)
6
...
Roots under particular cases :
For the equation ax2 + bx + c = 0, a ≠ 0
& unequal
(ii)
(i)
If D > 0 and D is not a perfect square ⇒ roots are
(ii)
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If c = 0 ⇒ one root is zero and other is b/a
(iii)
For a quadratic equation their will exist exactly 2 roots real
or imaginary
...
Also in this case a = b = c = 0
...
irrational and unequal
...
PAGE # 1
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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
(i)
(α β) =
(ii)
α2 + β2 = (α + β)2 2αβ =
(iii)
α2 β2 = (α + β)
2
(α + β) − 4αβ = ±
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2c2
a2
b2 + ac
a2
2
(α + β) − 4αβ =
−b b2 − 4ac
a2
9
...
) Ph
...
8
...
) Ph
...
Maximum and Minimum value of quadratic expression :
In a quadratic expression ax
2
LF b I
+ bx + c = a MG x + 2a J
K
MNH
2
−
(v)
OP
PQ
D
4a2 ,
D ≥ 0, a
...
f(k2) > 0, k1 <
If a > 0, quadratic
a
...
f(k2) < 0
expression has minimum value
−b
4ac − b2
at x =
and there is no maximum value
...
2a
4a
12
...
Location of roots :
(i)
If k lies between the roots then a
...
f(k2) < 0
13
...
f(k) > 0,
−b
c
−d
, αβ + βγ + γα =
and αβγ =
a
a
a
where α, β, γ are its roots
...
r
...
f(k) > 0,
−b
>k
2a
(necessary & sufficient)
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PAGE # 6
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
*
1
...
x
Polar representation :
x = r cosθ, y = r sinθ, r = |z| =
(ii)
z1z2 = z 1 z 2
If α = f(z), then α = f( z )
x2 + y2 ,
amplitude or amp(z) = arg(z) = θ = tan1
(i)
z1 − z2 = z 1 z 2
*
−1 ) is called
a complex number, where x is called a real part i
...
x = Re(z)
and y is called an imaginary part i
...
y = Im(z)
...
+zn = z 1 + z 2 +
...
(z)
1
2
n
Fz I
Gz J
H K
1
=
2
(provided z2 ≠ 0)
Where α = f(z) is a function in a complex variable
with real coefficients
...
z + z = 0 or z = z ⇒ z = 0 or z is purely imaginary
*
P(x, y) then its vector representation is z = OP
z= z
⇒
z is purely real
Integral Power of lota :
i=
4
...
|z| =
Complex conjugate of z :
(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
...
) Ph
...
) Ph
...
(iv) |z1 + z2|2 + |z1 z2|2 = 2 [|z1|2 + |z2|2]
(v)
MATHS FORMULA - POCKET BOOK
Square root of a complex no
...
Argument of a complex number :
= ±
Argument of a complex number z is the ∠ made by its radius
vector with +ve direction of real axis
...
7
...
= θ,
and (cosθ + isinθ)n = cos nθ i sin nθ
arg (any real + ve no
...
) = π
arg (z z ) = ± π/2
8
...
z2) = arg z1 + arg z2 + 2 k π
(v)
arg
FG z I
Hz J
K
1
2
F 1I
G zJ
H K
9
...
(n 1)
= arg z, if z is real
(i)
= arg z π, arg z ∈ (0, π ]
(viii) arg (zn) = n arg z + 2 k π
Sum of all roots of z1/n is always equal to zero
(ii)
(vii) arg ( z) = arg z + π, arg z ∈ ( π , 0]
(ix)
eiθ + eiθ = 2cosθ and eiθ eiθ = 2 isinθ
∴
= arg z1 arg z2 + 2 k π
(vi) arg ( z ) = arg z = arg
Product of all roots of z1/n = (1)n1 z
10
...
ω=
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P
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|z|+a
|z|−a
−i
, for b < 0
2
2
(cosθ + isinθ)n = cosθ + isin nθ
= θ π , z ∈ 4 quad
...
(i)
L
M
M
N
I O
N
S
PAGE # 9
−1 + i 3
and 1 + ω + ω2 = 0, ω3 = 1
2
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PAGE # 10
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
11
...
1
= 2cosnθ
zn
(iv) If x = cosα + isinα , y = cos β + i sin β & z = cosγ + isinγ
and given x + y + z = 0, then
13
...
Equation of Circle :
*
*
|z| = r represents circle with centre at origin
...
*
z z + a z + a z + b = 0 represents a general circle
where a ∈ c and b ∈ R
...
* Parametric form z = tz1 + (1 t)z2 where t ∈ R
* Non parametric form
z
z1
z2
z 1
z1 1
z2 1
= 0
...
) Ph
...
z1
z2
z3
z1 1
z2 1 = 0
z3 1
or slope of AB = slope of BC = slope of AC
...
) Ph
...
if k ≠ 1 and a straight line if k = 1
...
(xiv) z1, z2, z3
...
e
...
(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
...
(ix)
If three complex numbers are in A
...
, then they lie on
a straight line in the complex plane
...
(xi)
If z1, z3, z3
...
+ zn2 = nz02
...
) Ph
...
) Ph
...
(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
Factorial notation The continuous product of first n natural numbers is called
factorial
i
...
n or n! = 1
...
3
...
n
n! = n(n 1)! = n(n 1)(n 2)! & so on
the n objects is
4
...
2
...
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
...
(iii)
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nm
Pr (nmCrm × r!)
Circular Permutations When clockwise & anticlockwise orders are treated as
different
...
(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
...
) Ph
...
Permutations (Arrangement of objects) (i)
The number of permutations of n different things taken
(ii)
The number of permutations of n different things taken
r at a time, when m particular things are always to be
excluded (included)
=
Bijection Rule : Number of favourable cases
= Total number of cases
Unfavourable number of cases
...
(n r + 1) =
(n − r)!
or
n!
a! b! c!
(ii)
The number of circular permutations of n different things
taken all together
PAGE # 15
Pr
2r
n
1
Pn
=
(n 1)!
2
2n
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PAGE # 16
MATHS FORMULA - POCKET BOOK
6
...
The number of combinations of n different things taken r at
a time is denoted by nCr or C (n, r)
Cr =
n
(i)
(iv)
n
(v)
n
(vii)
n
Pr
r!
Cr + nCr1 = n+1Cr
⇒
r = s or r + s = n
C0 = Cn = 1
n
n
r
Cr =
(vii) The number of selections taking atleast one out of
a1 + a2 + a3 +
...
an are alike (of kth kind) and k are distinct is
1
(n r + 1) nCr1
r
n1
Cr1
[(a1 + 1) (a2 + 1) (a3 + 1)
...
Restricted combinations -
Division and distribution (i)
The number of combinations of n distinct objects taken r at
a time, when k particular objects are always to be
(i)
(ii)
excluded is nkCr
(iii)
included and s particular things are to be excluded is
8
...
+
(iii)
A T
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(iii)
Cn = 2n 1
n
The total number of ways in which n different objects
are to be divided into r groups of group sizes n1, n2, n3,
...
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(m + n + p)!
m! n! p!
n!
is same is n ! n !
...
1
2
r
Total number of combinations in different cases -
(ii)
D U
(ii)
Crk
(i)
E
The number of ways in which (m + n + p) different
objects can be divided into there groups containing m,
included is nkCrk
nks
Cn = 2n
n
[(a1 + 1) (a2 + 1) (a3 + 1) +
...
+
n
(vi) The total number of selections of at least one out of
a1 + a2 +
...
an are
alike (of nth kind) is
Cr = nCnr
Cr = nCs
Total number of selections of zero or more objects
out of n different objects
=
n
(vi)
7
...
(nr !)k r k1 ! k 2 !
...
) Ph
...
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)
C3
n
(c)
(number of ways of group formation) × k!
(ii)
If m parallel lines in a plane are intersected by a
family of other n parallel lines
...
(e)
The number of solution of the equation x1 + x2 +
...
nr ≤ xr ≤ n'r
number of quadrilaterals nC4
where all x'is are integers is given as
(f)
Coefficient of xn is
n1 +1
+
...
+x
n'2
j
...
+ 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
...
Derangement Theorem (i)
Number of diagonals in a polygon of n sides is
(d)
The coefficient of xn in the expansion of (1 xr) is
equal to n + r 1Cr 1
+x
C3
...
(i)
n1
m
n
10
...
+(−1) 1 OP
= n! M1 −
n!Q
N 1! 2! 3! 4!
(h)
∑ r3
r =1
and number of squares of any size is
∑ r2
...
+(−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)
...
Some Important results (a)
Number of total different straight lines formed by joining
the n points on a plane of which m(
C2 mC2 + 1
...
) Ph
...
) Ph
...
(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 :
Probability of an event
=
(viii) P( A + B ) = 1 P(AB) = P(A) + P(B) 2P(AB)
= P(A + B) P(AB)
No
...
of cases
Note :
(ii)
P(A) + P(Not A) = 1 i
...
P(A) + P( A ) = 1
When a coin is tossed n times or n coins are tossed
Probability of a sure event is one
...
(i)
(ix)
once, the probability of each simple event is
(xi)
Odds for an event :
If P(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
Then odds in favour of A =
1
...
Cn
52
(xiii) If n cards are drawn one after the other with replacement, the probability of each simple event is
3
...
) Ph
...
(52)n
(xiv) P(none) = 1 P (atleast one)
(i)
E
1
...
) Ph
...
(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 =
Probability that all letters are not in right enve-
Note :
1
n!
Probability that no letters are in right envelope
4
...
e
...
+(−1)
M2! 3! 4!
N
n− r
1
(n − r)!
OP
Q
or
⇒
P(E1 ∩ E2 ∩ E3 ∩
...
If events are independent, then
P(A ∩ B) = 0
P(E1 ∩ E2 ∩ E3 ∩
...
P(En)
When events are not mutually exclusive i
...
6
...
Pn are the probabilities of n independent
events A, A2,
...
P(A ∩ B) ≠ 0
∴
P(A ∪ B) = P(A) + P(B) P(A ∩ B)
or
P(A + B) = P(A) + P(B) P(AB)
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or
P(A + B) = P(A) + P(B) P(A) P(B)
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1 [(1 P1) (1 P2)
...
e
...
P(B), P(B) ≠ 0
When events are mutually exclusive
(ii)
If A and B are independent event, then P(A/B) = P(A)
and P(B/A) = P(B)
(ii)
Addition Theorem of Probability :
(i)
No
...
in A ∩ B
...
of pts
...
+ (1)n
2!
3!
4!
n!
(d)
P(A ∩ B)
P(B)
P(B/A) = Probability of occurrence of B, given that A has
1
n!
lopes = 1
(c)
already happened =
P(A1 + A2 +
...
P( A n )
PAGE # 23
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PAGE # 24
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
Total Probability :
(a)
Let A1, A2,
...
(c)
Variance E(x2) (E(x))2 = npq
(d)
Standard deviation = npq
P(A) = P(A ∩ A1) + P(A ∩ A2) +
...
∑
r =1
(b)
(i)
P(Ar) P(A/Ar)
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
p1p2
p1p2 + (1 − p1 ) (1 − p2 )
...
e
...
xn
with probabilities P1, P2,
...
Given that the probability of A & B speaking truth is p1, p2
...
Truth of the statement :
Baye's Rule :
9
...
+ Pn = 1
αp1p2
+ (1 − α) (1 − p1) (1 − p2 ) β
...
) Ph
...
) Ph
...
(iv) If A1, A2,
...
M's between a and b,
then A1 = a + d, A2 = a + 2d,
...
P
...
P
...
, a + (n 1) d
where a is the first term and d is the common difference
(b)
General (nth) term of an A
...
(v)
Tn = a + (n 1)d [nth term from the beginning]
= a + (m n)d
term i
...
an =
Sum of n terms of an A
...
Sn
=
n
n
[2a + (n 1)d] = [a + Tn]
2
2
2
...
Supposition of terms in A
...
(i)
Four terms as a 3d, a d, a + d, a + 3d
(iii)
(e)
Three terms as a - d, a, a + d
(ii)
Five terms as a 2d, a d, a, a + d, a + 2d
A
...
of n numbers A1, A2,
...
M
...
P
...
P
...
) Ph
...
M
...
a and b,
then
a+b
i
...
2
1
(a + an+r), r < n
2 nr
(d)
For an A
...
, A
...
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
...
of middle term
...
) Ph
...
+ A n
ΣA i
Sum of numbers
=
=
n
n
n
A=
E
Sn
Arithmetic mean (A
...
) :
(i)
n
(a + b)
2
Geometric Progression (G
...
)
(a) General G
...
a, ar, ar2 ,
...
P
...
P
...
P
...
e
...
M's inserted between a and b is
(vi) Any term of an A
...
(except first term) is equal to
the half of the sum of term equidistant from the
If an A
...
having m terms, then nth term from end
(c)
b−a
n+1
PAGE # 28
MATHS FORMULA - POCKET BOOK
(f)
MATHS FORMULA - POCKET BOOK
Geometric Mean (G
...
)
(i)
(ii)
ab
If G1, G2,
...
M's between a
and b, then
G1 = ar, G2 = ar ,
...
F bI
= ar , where r = G J
H aK
Σ2n = 2 + 4 + 6 +
...
or first find n A
...
's between
1
1
&
, then their
a
b
reciprocal will be required H
...
Relation Between A
...
, G
...
and H
...
AH = G2
(ii)
n(n + 1)
2
Σn2 = 12 + 22 + 32 +
...
M
...
M
...
) Ph
...
, Hn =
bn + a
na + b
(i)
(b)
N
If H1, H2,
...
M's between a and b,
then H1 =
6
...
+ n =
I O
2ab
a+b
If H is the H
...
between a and b, then H =
d(1 − r n−1)
a
+ r
...
M
...
P
...
G
...
S∞ =
1
1
1
,
,
+
...
P
...
G
...
(c)
3
General H
...
Arithmetico - Geometric Progression (A
...
P
...
M
...
(2n 1) = n2
3
Harmonic Progression (H
...
+ (n times) = na
(f)
5
...
+ n
(e)
If G is the G
...
between two given numbers a
and b, then
G2 = ab ⇒ G =
2
(d)
G
...
= (x1 x2
...
(i)
Ln(n + 1) O
= M
N 2 P
Q
(c)
Geometrical mean of n numbers x1, x2,
...
) Ph
...
MATHS FORMULA - POCKET BOOK
3
...
+ nCr xnr ar +
...
= C1 + C 3 + C5 +
...
+ nCn = n
...
= 0
*
C0 + 2C 1 + 3C2 +
...
+ Cn2 =
*
middle term
C0 C 1 + C2 C3 +
...
+ Cn = 2n
*
(m + 1) term from the end = (n m + 1) from beginning = Tnm+1
th
C02 C 12 + C22 C32 +
...
To determine a particular term in the given expasion :
±
1
xβ
I
J
K
n
, if xn occurs in
Tr+1 (r + 1)th term then r is given by n α r (α + β) = m
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C
A T
I O
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S
Cn + r =
n−2
Cr −2 and so on
...
n Cn are usually denoted by C0, C1 ,
...
(ii)
2
...
C0 ,
C r
...
n
=
n
For the sake of convenience the coefficients
Binomial Theorem for any +ve integral index :
PAGE # 31
R
|
Sc−1h
|
T
c2nh!
cn!h
2
0,
n/2 n
D U
C
A T
I O
N
S
2n
Cn
if n is odd
Cn/2 , if n is even
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=
PAGE # 32
MATHS FORMULA - POCKET BOOK
Note :
2n + 1
2n + 1
2n + 1
C0 +
2n + 1
Cn + 2 +
...
+
C2n + 1 = 2
2n + 1
Cn =
Cn + 1 +
(ii)
2n
(x + y + z)n =
∑
r + s + t =n
n!
xr ys z t
s! r !t !
Generalized (x1 + x2 +
...
+
=
2
3
n+1
n+1
*
4
...
+
2
3
4
n+1
=
=
1
n+1
6
...
rk =n
n!
r1 r2
rk
r1 ! r2 !
...
xk
Total no
...
xn)m is
m+n1
C n1
Greatest term :
(i)
If
(n + 1)a
∈ Z (integer) then the expansion has two
x+a
greatest terms
...
(ii)
If
(n + 1)a
∉ Z then the expansion has only one greatx+a
est term
...
] denotes greatest integer less than or equal to x}
5
...
) Ph
...
) Ph
...
FG π IJ
H nK
sinθ
A T
I O
N
S
cosθ
cosθ
tanθ
tanθ
m sinθ
m cotθ
cotθ
2
6
...
) Ph
...
Allied∠ of (θ)
900 ± θ 1800 ± θ 2700 ± θ 3600 ± θ
T-ratios
2
F a cos ec π IJ
G2
H
nK
x
y'
π
1 2
a cos
4
n
Area of circumcircle = π
I quadrant
All +ve
O
π
a
cot
2
n
a
π
=
cosec
2
n
2
...
4
...
MATHS FORMULA - POCKET BOOK
(iii)
PAGE # 35
tan (A ± B) =
tan A ± tanB
1 m tan A tan B
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PAGE # 36
MATHS FORMULA - POCKET BOOK
(iv) cot (A ± B) =
MATHS FORMULA - POCKET BOOK
7
...
(viii) sin75 =
0
= cos15
2 2
3 −1
(i)
sin2A = 2sinA cosA =
(ii)
cos750 =
tan75 = 2 +
3 = cot15
cot750 = 2
2 tan A
1 + tan2 A
2 tan A / 2
1 + tan2 A / 2
cos2A = cos2A sin2A = 2cos2A 1
0
3 = tan15
2 2
0
= sin150
I O
N
S
= 1 2sin2A =
0
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T-ratios of multiple and submultiple angles :
⇒ sinA = 2sinA/2 cosA/2 =
(xi)
C
sin(A + B)
cos A cos B
0
(x)
D U
tanA + tanB =
= (sin A + cos A)2 1 = 1 (sin A cos A)2
3 +1
(ix)
E
sinC + sinD = 2sin
(ix)
S1 − S3 + S5 − S7 +
...
2
4
6
8
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
2cosA cosB = cos(A + B) + cos(A B)
(iv) 2sinA sinB = cos(A B) cos(A + B)
Generalized tan (A + B + C +
...
) Ph
...
Some useful series :
sinα + sin(α + β) + sin(α + 2β) +
...
+ to nterms
L F n − 1I βO sin nβ
M G 2JP 2
N H KQ
β ≠ 2nπ
cos α +
Maximum and minimum value of the expression :
acosθ + bsinθ
=
Maximum (greatest) Value = a2 + b2
Minimum (Least) value = a2 + b2
(iii)
sin
β
2
cosα
...
cos22α
...
Conditional trigonometric identities :
= 1 , α = (2k+1)π
If A, B, C are angles of triangle i
...
A + B + C = π, then
(i)
L F n − 1IβO sinLnβ O
M G 2 J P M2P
N H K Q N Q , β ≠ 2nπ
sin α +
(viii) cosA/2 =
9
...
e
...
) Ph
...
) Ph
...
Thus the equation reduces to form
cos(θ α) =
General solution of the equations of the form
(i)
sinθ = 0
⇒
θ = nπ,
n∈I
(ii)
cosθ = 0
⇒
θ = (2n + 1)
(iii)
tanθ = 0
⇒
θ = nπ,
(iv) sinθ = 1
⇒
θ = 2nπ +
(v)
cosθ = 1
⇒
θ = 2πn
(vi) sinθ = 1
⇒
θ = 2nπ
(vii) cosθ = 1
⇒
⇒
= cosβ(say)
θ = nπ + (1)nα
now solve using above formula
π
, n∈I
2
3
...
(ii)
π
2
If two equations are given then find the common values of θ between 0 & 2π and then add 2nπ to this
common solution (value)
...
a + b2
θ = (2n + 1)π
(viii) sinθ = sinα
c
2
⇒
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
and put
a
2
a +b
2
= cosα,
b
2
a + b2
= sinα
...
) Ph
...
) Ph
...
If y = sin x, then x = sin1 y, similarly for other inverse Tfunctions
...
(ii)
Range (R)
1 ≤ x ≤ 1
π
2
cos1 x
1 ≤ x ≤ 1
0 ≤ θ ≤ π
tan1 x
∞ < x < ∞
cot1 x
∞ < x < ∞
sin (sin1 x) = x provided 1 ≤ x ≤ 1
0 < θ < π
sec1 x
x ≤ 1, x ≥ 1
0 ≤ θ ≤ π, θ ≠
x ≤ 1, x ≥ 1
π
π
, θ ≠ 0
≤ θ ≤
2
2
sin
x
cosec1 x
≤ θ ≤
π
2
cos (cos1 x) = x provided 1 ≤ x ≤ 1
Domain (D)
1
≤ θ < 0
or 0 < θ ≤
Domain and Range of Inverse T-functions :
Function
π
2
cosec1 (cosec θ ) = θ provided
tan (tan1 x) = x provided ∞ < x < ∞
π
2
cot (cot1 x) = x provided ∞ < x < ∞
sec (sec1 x) = x provided ∞ < x ≤ 1 or 1 ≤ x < ∞
cosec (cosec1 x) = x provided ∞ < x ≤ 1
π
π
< θ <
2
2
or 1 ≤ x < ∞
(iii)
π
2
sin1 ( x) = sin1 x,
cos1 ( x) = π cos1 x
tan1 ( x) = tan1 x
cot1 ( x) = π cot1 x
cosec1 ( x) = cosec1 x
3
...
) Ph
...
) Ph
...
Value of one inverse function in terms of another
inverse function :
(i)
sin1 x = cos1
1
= sec1
(ii)
1−x
1−x
2
2
= tan1
(iii)
tan1 x = sin1
= sec1
(iv) sin1
(v)
cos1
(vi) tan1
1
1
= cosec1
x
FG 1 I
H xJ
K
FG 1 I
H xJ
K
FG 1 I
H xJ
K
1 − x2
x
1 + x2
1 − x2
1 − x2
= cot1
x
1
1 + x2
D U
C
A T
I O
N
S
1 − x2
x
(i)
x
1
(iii)
1 − x2
(iv)
= cot1
(v)
1
x
F x + y I ; if x > 0, y > 0, xy < 1
G 1 − xy J
H K
F x + y I ; if x > 0, y > 0, xy > 1
tan x + tan y = π + tan G
H 1 − xy J
K
F x−yI
tan x tan y = tan G 1 + xy J ; if xy > 1
H
K
F x − y I ; if x > 0, y < 0, xy < 1
tan x tan y = π + tan G
H 1 + xy J
K
F x + y + z − xyz I
tan x + tan y + tan z = tan G
K
H 1 − xy − yz − zx J
L
O
sin x ± sin y = sin Mx 1 − y ± y 1 − x P ;
N
Q
tan1x + tan1y = tan1
(ii)
(vi)
1 + x2
, x ≥ 0
x
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
if x,y ≥ 0 & x2 + y2 ≤ 1
2
L
M
N
O
P
Q
2
2
(vii) sin1x ± sin1y = π sin1 x 1 − y ± y 1 − x ;
= cosec1 x , ∀ x ∈ ( ∞ , 1] ∪ [1, ∞ )
if x,y ≥ 0 & x2 + y2 > 1
R cot x
S− π + cot x
|
T
−1
=
−1
L
M
N
O
P
Q
2
2
(viii) cos1x ± cos1y = cos1 xy m 1 − x 1 − y ;
= sec1 x, ∀ x ∈ ( ∞ , 1] ∪ [1, ∞ )
if x,y > 0 & x2 + y2 ≤ 1
L
M
N
O
P
Q
2
2
(ix) cos1x ± cos1y = π cos1 xy m 1 − x 1 − y ;
for x > 0
for x < 0
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Formulae for sum and difference of inverse trigonometric function :
, 0 ≤ x ≤ 1
= cos1
1
1 + x2 = cosec
= cot1
5
...
) Ph
...
MATHS FORMULA - POCKET BOOK
PROPERTIES & SOLUTION OF TRIANGLE
Inverse trigonometric ratios of multiple angles
(i)
2sin1x = sin1(2x
(ii)
2cos1x = cos1(2x2 1), if 1 ≤ x ≤ 1
1 − x2 ), if 1 ≤ x ≤ 1
Properties of triangle :
F 2x I = sin F 2x IJ = cos FG 1 − x IJ
G1 − x J
G1 + x K
H K
H
H1 + x K
1
...
In any ∆ABC, we write BC = a, AB = c, AC = b
2
(iii)
2tan1x = tan1
1
2
2
1
A
2
A
(iv) 3sin1x = sin1(3x 4x3)
(v)
c
3cos x = cos (4x 3x)
1
1
B
F 3x − x I
G 1 − 3x J
H
K
B
3
(vi) 3tan1x = tan1
b
3
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
...
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
...
) Ph
...
) Ph
...
MATHS FORMULA - POCKET BOOK
Projection formula :
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A
6
...
(i)
(a)
sin
A
=
2
1
1
1
∆ = 2 ab sin C = 2 bc sin A = 2 ca sin B
(ii)
Half angled formula - In any ∆ABC :
7
...
) Ph
...
10
...
) Ph
...
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
(vi)
A
B
C
sin
sin
2
2
2
(vii)
(vi) cos A + cos B + cos C = 1 +
r
R
1
1
1
1
r1 + r2 + r3 = r
1
2
r1
+
(ix)
A
B
, r2 = s tan
, r3 = s tan
2
2
r1 = s tan
(iii)
r1 = 4R sin
A
B
cos
sin
2
2
D U
C
A T
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
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1
A
B
C
sin
cos
,
2
2
2
r3 = 4R cos
+
A
B
C
cos
cos
,
2
2
2
r2 = 4R cos
2
r3
a cos
∆
∆
∆
, r2 =
, r3 =
s−a
s −b
s−c
(ii)
1
1
1
1
1
+
+
=
2Rr
bc
ca
ab
12
...
) Ph
...
(ii)
d = h (cotα cotβ)
Angle of elevation and depression :
h
If an observer is at O and object is at P then ∠ XOP is
called angle of elevation of P as seen from O
...
2
...
) Ph
...
) Ph
...
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
...
(i)
If AB = CD, AD = BC, then ABCD is a parallelogram
...
(iii) If AB = BC = CD = AD, then ABCD is a rhombus
...
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Section formula :
(i)
Internally :
Use of Distance Formula :
(a) In Triangle :
Calculate AB, BC, CA
(i)
If AB = BC = CA, then ∆ is equilateral
...
(iii) If sum of square of any two sides is equal to
the third, then ∆ is right triangle
...
e
...
Here points are collinear
...
For circumcentre of a triangle :
Circumcentre of a triangle is equidistant from vertices
i
...
PA = PB = PC
...
(i)
Circumcentre of an acute angled triangle is inside the triangle
...
(iii) Circumcentre of an obtuse angled triangle is
outside the triangle
...
) Ph
...
Area of Polygon :
Area of polygon having vertices (x 1, y1), (x2, y2), (x3, y3)
...
Area of Triangle :
The area of triangle ABC with vertices A(x1, y1), B(x2, y2)
and C(x3, y3)
...
(ii) If in a triangle point arrange in anticlockwise then
value of ∆ be +ve and if in clockwise then ∆ will be
ve
...
) Ph
...
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
...
Points must be taken in order
...
=
M
x
(Determinant method)
y1
x1
=
M
Rotational Transformation :
If coordinates of any point P(x, y) with reference to new
axis will be (x', y') then
y1 1
x2
6
...
y2
x3
I
J
K
y1
ax + by + c = 0 is
PAGE # 57
c2
...
) Ph
...
(x)
making an angle θ , 0 ≤ θ ≤ π , θ ≠
a
b
(xi)
(vi) Slope of two parallel lines are equal
...
2
...
(vi) Slope Intercept form : Equation of a line with slope m
and making an intercept c on the y-axis is y = mx + c
...
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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
...
2
1
Slope of the line ax + by + c = 0, b ≠ 0 is
x
y
+
= 1
...
Thus m = tan θ
...
) Ph
...
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
...
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
...
lines through (x1,y1) making an angle α
with y = mx + c is
MATHS FORMULA - POCKET BOOK
12
...
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 =
If y = m1x + c & y = m2x + c represents two straight lines
length of perpendicular from (x1, y1) on ax + by + c = 0
7
...
then m1 + m2 =
a2 + b2
Distance between two parallel lines ax + by + ci = 0,
i = 1, 2 is
9
...
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
...
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
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−2h
a
, m1m2 =
...
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
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PAGE # 62
MATHS FORMULA - POCKET BOOK
CIRCLE
1
...
e
...
Concentric circles : Two circles having same centre C(h, k)
but different radii r1 & r2 respectively are called concentric
circles
...
Position of a point w
...
t
...
of x, −1 coeff
...
2
...
(ii) x2 + y2 = r2 , where (0, 0) origin is circle centre and r is
the radius
...
Chord length (length of intercept) = 2 r2 − p2
8
...
of a
diameter of a circle, then its equation is
(x x1) (x x2) + (y y1) (y y2) = 0
Parametric equations :
(i)
The parametric equations of the circle x2 + y2 = r2 are
x = rcosθ, y = r sinθ ,
4
...
10
...
e
...
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
...
) Ph
...
) Ph
...
Equation of tangent, T = 0 :
(i)
MATHS FORMULA - POCKET BOOK
15
...
1
Equation of tangent to the circle x2 + y2 = a2 at any
point (x1, y1) is xx1 + yy1 = a2
The line y = mx ± a 1 + m2 is a tangent to the circle
±am
1 + m2
,
±a
1 + m2
I
J
K
(iv) Equation of tangent at (a cos
16
...
θ ) to the
19
...
r
...
the circle S = 0 is T = 0
20
...
Equation of normal :
F −a l −a mI
are G n , n J
H
K
2
Equation of normal to the circle
lx + my + n = 0 w
...
t the circle x + y = a
2
x2 + y2 + 2gx + 2fy + c = 0 at any point P(x1, y1) is
y1 + f
y y1 = x + g (x x1)
1
(ii)
I
J
J
J
K
18
...
Director circle is a concentric circle whose
circle x2 + y2 = a2 is x cos θ + y sin θ = a
...
Equation of a chord whose middle pt
...
Family of Circles :
S + λS' = 0 represents a family of circles passing through
the pts
...
Equation of pair of tangents SS1 = T2
S + λ L = 0 represent a family of circles passing through
the point of intersection of S = 0 & L = 0
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
...
) Ph
...
) Ph
...
Equation of tangent at point of contact of circle is
S 1 S2 = 0
26
...
Equation of Common Chord is S S1 = 0
...
The point of concurrency of the three radical axis of
three circles taken in pairs is called radical centre of
three circles
...
Orthogonality condition :
If two circles S ≡ x2 + y2 + 2gx + 2fy + c = 0
23
...
2
2
r1 + r1 − d2
cosθ =
, where d = C1C2
2r1r2
24
...
Then following cases arise as
(i)
C1 C2 > r1 + r2 ⇒ do not intersect or one outside the
other, 4 common tangents
...
(iii)
|r1 r2| < C1 C2 < r1 + r2 ⇒ Intersection at two real
points, 2 common tangents
...
(v)
C1 C2 < |r1 + r2| ⇒ one inside the other, no tangent
...
) Ph
...
) Ph
...
Standard Parabola :
Imp
...
R
...
) Ph
...
) Ph
...
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
...
e
...
) Ph
...
r
...
parabola
y2 = 4ax
...
2
= 4ay
Fa
Gm
H
2
F− a
G m
H
,
2
,
2a
m
slope (m)
I
J
K
2a
m
y = mx +
I
J
K
a
m
y = mx
a
m
c =
a
m
c =
a
m
(2am, am2)
y = mx am2
c = am2
(2am, am )
y = mx + am
c = am2
2
2
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
...
(a(G
...
)2, a(2A
...
))
6
...
D = 0
...
Parametric
coordinates't'
y 2=4ax
Equation of the form ax2 + bx + c = y represents
parabola
...
) Ph
...
MATHS FORMULA - POCKET BOOK
Note :
(i)
In circle normal is radius itself
...
(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
...
of
Normal
Point
Normals
parabola
at (x1, y1)
't'
at 't'
y2 = 4ax
yy1 =
−y1
(xx1) (at2, 2at)
2a
y+tx = 2at+at3
y2 = 4ax
yy1 =
y1
(xx1)
2a
ytx = 2at+at3
x2 = 4ay
x2 = 4ay
(ii)
yy1 =
yy1 =
2a
x1
2a
x1
(xx1)
(2at, at2)
x+ty = 2at+at3
(xx1)
(2at, at2)
Slope form
Point of
Equations
contact
of normal
x2 = 4ay
F − 2a , a I
G m mJ
H
K
2
F 2a , − a I
Gm m J
H
K
2
y = mx+2a+
a
2
m
Equation of focal chord of parabola y2 = 4ax at t1 is
y =
(ii)
c = 2a+
a
y = mx2a
a
2
m
m
c = 2a
2t1
2
t1
−1
a
m2
PAGE # 73
(x a)
If focal chord of y2 = 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)
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10
...
If point lies on x-axis, then one normal will be x-axis
itself
...
xty = 2at+at3
parabola
C
(ii)
at t2 then t2 = t1
Eqn
...
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))
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PAGE # 74
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
11
...
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
...
m
(vi) Intercepted portion of a tangent between the point
of tangency and directrix will make right angle at
focus
...
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
...
Its equation is y =
(viii) Circle drawn on any focal chord as diameter will touch
directrix
...
e
...
r
...
the
parabola y2 = 4ax is
F n , −2amI
Gl l J
H
K
14
...
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
...
(iv) Any light ray travelling parallel to axis of the parabola
will pass through focus after reflection through
parabola
...
) Ph
...
) Ph
...
Standard Ellipse (e < 1)
Rx
|
Sa
|
T
2
Ellipse
2
Imp
...
) Ph
...
) Ph
...
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
...
If
a2
+
y2
x2
a2
(iii)
The point lies outside, on or inside the ellipse if
S1 =
*
+
a
2
y1
2
5
...
yy1
b2
y
(i)
A T
I O
N
S
±b2
a2m2 + b2
I
J
...
y1
x1
= 1
...
) Ph
...
Equation of tangent in different forms :
2
b2
F
G
H
x
y
cos φ +
sin φ = 1
...
r
...
an ellipse :
2
x1
2
= 1, then c2 = a2m2 + b2
...
*
+
Point of contact :
x2 + y2 = a2
...
a2
y2
a2m2 + b2 always represents
the tangents to the ellipse
...
x2
Slope form : If the line y = mx + c touches the
PAGE # 79
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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
...
2
(iii)
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
Sum of square of intercept made by auxillary circle on
any two perpendicular tangents of an ellipse will be
constant
...
Equation of chord of contact of the tangents drawn from
the external point (x1, y1) to an ellipse is given by
xx1
...
e
...
The co-ordinates of the point of contact are
F
GH
±a2
a2 + b2m2
,
±mb2
a2 + b2m2
I
J
...
x
cos
a
(ii)
The locus of foot of perpendicular drawn from either
focii to any tangent lies on auxillary circle
...
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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
...
) Ph
...
i
...
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
...
Equation of chord joining the points (a cos θ , b sin θ ) and
= 1
...
a2
+
mid point is (x1, y1) is T = S1
...
The equation of a chord of an ellipse
S
PAGE # 82
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
12
...
r
...
the ellipse
x2
a2
y2
+
= 1 is given by
b2
xx1
yy1
+
a2
b2
(c)
= 0 i
...
T = 0
...
H
K
2
y2
= 1 is
b2
(d)
2
13
...
(i)
x2
+
2
a
y2
2
b
F ± b IJ
...
S'P = CQ2
The pole of the line l x + my + n = 0 w
...
t
...
15
...
e
...
16
...
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
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PAGE # 83
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PAGE # 84
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
HYPERBOLA
1
...
terms
x2
a2
or
x2
a2
y2
+
= 1
b2
y2
b2
(0, 0)
(0, 0)
2a
2b
2b
(±ae, 0)
2a
(0, ±be)
x = ±a/e
= 1
Centre
Length of
transverse axis
Length of
conjugate axis
Foci
Equation of
directrices
y = ± b/e
Eccentricity
e =
Length of L
...
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
G a J
H
K
2
Focal radii
2
2
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
D U
C
A T
I O
N
S
Fa + b I
GH b JK
2
e =
2
2
y = b, y = b
x = 0
Conjugate Hyperbola
y = 0
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Hyperbola
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PAGE # 86
MATHS FORMULA - POCKET BOOK
2
...
(y − k)2
b2
x2
a2
= 1
...
according as c2 <, =, > a2m2 b2
...
xx1
2
a
yy1
2
b
C
A T
I O
N
S
y2
b2
= 1 are y = mx ±
a2m2 − b2 and the
a2m
a2m2 − b2
,±
b2
a2m2 − b2
I
J
...
y1
x1
Parametric form : The equation of normal at
(a sec θ , b tan θ ) to the hyperbola
x2
= 1
...
) Ph
...
The line y = mx + c does not intersect, touches, intersect
the hyperbola
5
...
r
...
a hyperbola :
x2
x
y
sec φ
tan φ = 1
...
Parametric form : The equation of tangent to the
hyperbola
a2
PAGE # 87
y2
b2
= 1 is ax cos θ + by cot θ = a2 + b2
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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
F±
GH
a2 − b2m2
a2
a2 − b2m2
or c2 =
m(a2 + b2 )2
a2
9
...
a2
yy1
b2
2
2
a2
A T
I O
N
S
I
J
K
y
sin
b
Fφ
G
H
1
+ φ2
2
I
J
K
= cos
Fφ
G
H
1
I
J
K
+ φ2
...
y2
b2
b2
a2
The equations of asymptotes of the hyperbola
a2
a2
= 1 is
b2
x2
x2
F − a l , b mI
G n nJ
K
H
2
y2
15
...
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m1m2 =
y2
b2
= 1 are y = ±
b
x
...
= 1
*
The asymptote of a hyperbola passes through the
centre of the hyperbola
...
D U
− φ2
2
14
...
The equation of chord of the hyperbola
E
1
13
...
2
Fφ
G
H
12
...
r
...
the hyperbola
is given by T = 0
...
r
...
is condition of normality
...
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
...
) Ph
...
x
y
= c2 is x + y = 2
...
*
xx1 yy1 = x12 y12
A hyperbola and its conjugate hyperbola have the
same asymptotes
...
Equation of normal at t on xy = c2 is
The bisector of the angles between the asymptotes
are the coordinate axes
...
(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
...
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
...
F 2c t t
Gt +t
H
1 2
1
2
,
2c
t1 + t 2
I
J
K
Parametric equation of the hyperbola xy = c2 are
x = ct, y =
*
c
, where t is a parameter
...
) Ph
...
) Ph
...
MATHS FORMULA - POCKET BOOK
3
...
M
...
xn)1/n
Arithmetic mean :
(i)
n
For ungrouped data (individual series)
or
x =
n
(ii)
x1 + x2 +
...
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
G∑
J
G
J
= antilog G
G ∑f J
J
H
K
n
Σ fixi
i
i =1
f
f
f
G
...
= x11 x22
...
M
...
n
i
i
i =1
n
i=1
i
be n observations and fi be their corresponding
frequencies
(b)
Σfidi
short cut method : x = A + Σf ,
i
i= 1
4
...
where A = assumed mean, di = xi A = deviation
for each term
2
...
xn
...
axn is a x where a is any number different from
zero
...
) Ph
...
M
...
i
Relation between A
...
, G
...
M
...
e
...
C
i
n
If each of the n given observation be doubled, then
their mean is doubled
(iii)
1
∑x
i=1
In a statistical data, the sum of the deviation of items
from A
...
is always zero
...
M
...
M
...
M
...
M
...
M
...
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PAGE # 94
MATHS FORMULA - POCKET BOOK
6
...
of observations
...
(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
...
(iii)
Discrete series : First find cumulative frequencies of
the variables arranged in ascending or descending
order and
Median =
(c)
F n + 1I
G 2 J
H K
For continuous frequency distribution : first find the
model class i
...
the class which has maximum frequency
...
f1 = Frequency of the model class
...
× i
l = Lower limit of the median class
...
N = Sum of all frequencies
...
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Mode = l 1 +
frequency
...
observation
FG nIJ + value of
H 2K
F n + 1IJ ] observation
...
f0 = Frequency of the class preceding model
f2 = Frequency of the class succeeding model
i = Size of the model class
...
Relation between Mean, Mode & Median :
(i)
In symmetrical distribution : Mean = Mode = Median
(ii)
In Moderately symmetrical distribution : Mode = 3 Median 2 Mean
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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
...
1
...
(a) Individual series (ungrouped data)
Mean deviation =
(b)
Σ|x − S|
n
2
or
σ=
F I
G J
H K
Σd2
Σd
−
N
N
2
Variance Square of standard direction
i
...
variance = (S
...
)2 = (σ)2
Coefficient of variance = Coefficient of S
...
× 100
Σ f | x − s|
Σf |x − s|
=
Σf
N
=
Note : Mean deviation is the least when measured from the
median
...
F I
G J
H K
Σfd2
Σfd
−
N
N
Where d = x A = Derivation from assumed mean A
f = Frequency of item (term)
N = Σf = Total frequency
...
Continuous series (grouped data)
...
D
...
M
...
) Ph
...
) Ph
...
Trace of a matrix : Sum of the elements in the principal
diagonal is called the trace of a matrix
...
1
...
trace (A ± B) = trace A ± trace B
trace kA = k trace A
trace A = trace AT
trace In = n when In is identity matrix
...
trace On = O
The number of rows is written first and then number of columns
...
On is null matrix
...
5
...
Types of matrices : A matrix A = (aij)m×n
A matrix A = (aij)mxn over the field of complex numbers is
said to be
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
...
A diagonal matrix
if m = n and aij = 0 for i ≠ j
...
e
...
= ann = k (cons
...
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
...
) Ph
...
e
...
(ii)
or AT = A
E
= (Ka)m×n where K is constant
...
) Ph
...
(B + C) = AB + AC
[Distributive law]
DETERMINANT :
1
...
e
...
cofactor of an element aij is denoted by Cij or Fij and is equal
to (1)i+j Mij
or
A' or A T is obtained by interchanging rows into columns or
columns into rows
(AT)T = A
(ii)
Determinant : if A is a square matrix then determinant of
matrix is denoted by det A or |A|
...
if i ≠ j
and a11 F21 + a12 F22 + a13 F23 = 0
(A ± B)T = AT ± BT
(iii)
if i = j
Note : |A| = a11F11 + a12 F12 + a13 F13
2
...
IT = I
Some special cases of square matrices : A square matrix
is called
or
AAT = In = ATA
(i)
Orthogonal matrix : if
(ii)
Idempotent matrix : if A2 = A
(iii)
Involutory matrix : if A = I
c1
c2
c3
b1
= a2 b
3
b2
= a1 b
3
c2
a2
b1 a
c3
3
c1
a1
+ b2 a
c3
3
c2
a2 b2
+ c1 a b
c3
3
3
c1
a1 b1
c3 c2 a3 b3
Properties :
(i)
|AT| = |A|
(iv) Nilpotent matrix : if ∃ p ∈ N such that Ap = 0
(ii)
Hermitian matrix : if Aθ = A i
...
aij = a ji
By interchanging two rows (or columns), value of determinant differ by ve sign
...
) Ph
...
) Ph
...
(i)
(ii)
(b)
An orthogonal matrix is 1 or 1
(c)
A unitary matrix is of modulus unity
...
(e)
An identity matrix is one i
...
|In| = 1, where In is a
unit matrix of order n
...
e
...
(h)
3
...
Skew symmetric matrix of odd order is zero
...
(i)
(ii)
N
S
1
| A|
(vii) If A & B are invertible square matrices then
a1m1 + b1m2
a2m1 + b2m2
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(A1)1 = A
(vi) |A1| = |A|1 =
(AB)1 = B1 A1
3
...
e
...
E
(adj KA) = Kn1(adj A)
Inverse of a matrix :
Multiplication of two second order determinants is defined as
follows
...
(adj AT) = (adjA)T
(iv) adj(adjA) = |A|n2
Multiplication of two determinants :
a1 b1
l1 m1
a1l1 + b1l2
× l m = a l +b l
a2 b2
2 1
2 2
2
2
(adjAB) = (adjB) (adjA)
(iii)
(a)
|adj A| = |A|n1
(iii)
(vi) Determinant of :
A(adj A) = (adjA) A = |A|In
There exists at least one square submatrix of order r
which is non singular
...
) Ph
...
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
...
The homogeneous system Ax = B, B = 0 of n equations
in n variables is
(i)
5
...
n
xi = 0 is called trivial solution
...
n,
(for determinant method)
(b) |A| = 0, (adj A) B = 0
(for matrix method)
NOTE : A homogeneous system of equations is never
inconsistent
...
Inconsistent (with no solution) if |A| = 0 and
at least one of the det (Ai) is non 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
...
if A is non singular, x = A1 B
...
Consistent (with infinitely many solutions), if
|A| = 0 and (adj A) B is a null matrix
...
) Ph
...
) Ph
...
(i)
loga a = 1
(iii)
Rx, x > 0
|
|x| = S−x, x < 0
|0, x = 0
T
loga 1 = 0
(ii)
Modulus function :
aloga b = b
if k > 0,
Properties :
|x| ≠ ± x
|xy| = |x||y|
(i)
(ii)
(iii)
(iv) loga b1 + loga b2 +
...
bn)
(v)
x
|x|
=
y
|y|
|x| = a
⇒ no solution
|x| > a
⇒ x < a or x > a
|x| ≤ a
|x| < a
⇒ a ≤ x ≤ a
⇒ No solution
...
]denotes greatest integer function equal
or less than x
...
e
...
2] = 4, [4
...
) Ph
...
1
loga b = log a
b
= loga b = log1/a b
(x)
logb a to be defined a > 0, b > 0, b ≠ 1
(ii)
D U
FG I
H J
K
1
(viii) loga b
or
n
loga b
m
(vii) logam bn =
Logarithmic Function :
⇒
= loga b loga c
(ix)
(i)
E
FG bI
H cJ
K
logc b
loga b = log a
c
⇒ x=±a
|x| = a
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
...
(vii) If a > 0
2
...
) Ph
...
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
...
Here set A is called domain and set of all f images of the
elements of A is called range
...
Table : Domain and Range of some standard functions Functions
or
f(x)
=
|x|
,
x
= 0,
A T
I O
N
S
Range
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}
Reciprocal function
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
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Domain
Polynomial function
Signum function :
R−1
|
|0
S
f(x) = sgn (x) = |
|1
T
D U
Range = For all values of x, all possible values of f(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
...
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PAGE # 110
MATHS FORMULA - POCKET BOOK
Trigonometric
MATHS FORMULA - POCKET BOOK
7
...
U
S2 2 V
T
W
R
cot x
R- {0, ± π , ± 2 π ,
...
U
S2 2 V
T
W
R - (-1,1)
cosec x
R- {0, ± π , ± 2 π }
R - (-1,1)
Inverse
Domain
Kinds of functions :
Range
or
-1
x
(-1, 1]
Graphically-no horizontal line intersects with the graph
of the function more than once
...
(v)
LM −π , π OP
N 2 2Q
8
...
) Ph
...
Inverse function : f1 exists iff f is one-one & onto both
(0, π )
sec-1 x
D U
Many one function : f : A → B is a many one function
if there exist x, y ∈ A s
...
x ≠ y
[0, π ]
x
I O
N
S
i
...
if to each y ∈ B ∃ x ∈ A s
...
f(x) = y
(iv) Into function : f is said to be into function if R(f) < B
tan-1 x
E
Onto function (surjection) - f : A → B is onto if
R (f) = B
[-1,1]
cos
-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
f1 : B → A, f1(b) = a
9
...
RU
SV
TW
If
a is +ve it moves towards right
...
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
...
) Ph
...
Zero function i
...
f(x) = 0 is the only function which
is even and odd both
...
(iii)
(g)
(h)
If f(x) is odd (even) function then f'(x) is even (odd)
function provided f(x) is differentiable on R
...
(i)
A given function can be expressed as sum of even
& odd function
...
e
...
H
...
y is present and
mode is taken on R
...
S
...
(v)
Replace x by ax (a > 0), then divide all the value on xaxis by a
...
Even function if f(x) = f(x) and
(ii)
Odd function if f(x) = f(x)
...
12
...
14
...
There may be infinitely many such T which satisfy the above
equality
...
T is called period of f(x)
...
Properties of even & odd function :
f(x + T) = f(x),
(a)
The graph of an even function is always symmetric
about y-axis
...
(c)
A T
(ii)
If the period of f(x) is T1 & g(x) has T2 then the period
of f(x) ± g(x) will be L
...
M
...
(iii)
If period of f(x) & g(x) are same T, then the period of
af(x) + bg(x) will also be T
...
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Period of (x/n + a) = nT
Product of an even or odd function is an odd function
...
(e)
D U
(i)
Product of two even or odd function is an even
function
...
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
...
Even and odd function : A function is said to be
(i)
f(x) =
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PAGE # 114
MATHS FORMULA - POCKET BOOK
Function
15
...
tan 4x
cos 2πx
1
π
|cos x|
sin x + cos x
4
2 cos
4
FG x − π I
H 3 J
K
π/2
6π
sin3 x + cos3 x
2π/3
sin x + cos x
2π
sin x
sin5x
2π
tan2 x cot2 x
π
x [x]
1
[x]
1
3
4
NON PERIODIC FUNCTIONS :
2
3
x, x , x , 5
cos x2
x + sin x
x cos x
cos x
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PAGE # 115
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PAGE # 116
MATHS FORMULA - POCKET BOOK
LIMIT
1
...
5
...
MATHS FORMULA - POCKET BOOK
4
...
Methods of evaluation of limits :
f(x)
0
is of
form
g(x)
0
then factorize num
...
separately and cancel the
lim
Factorisation method : If x → a
(i)
Theorems on limits :
lim (k f(x)) = k lim f(x),
x→a
(i)
x→a
(ii)
x→a
(iii)
x→a
(iv)
lim
k is a constant
...
g(x) = lim f(x)
...
0
Rationalization method : If we have fractional powers
on the expression in num, deno or in both, we rationalize
the factor and simplify
...
& deno
...
(vi)
(b)
lim
(viii) x → a (f(x))g(x) =
L lim f(x)O
M
P
N
Q
C
A T
I O
N
S
x→0
(d)
x→0
x
lim tan x = 1 = lim
x→0
x
tan x
lim sinx = 0
lim g(x)
x→a
x→a
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x→0
(c)
I
K
lim
lim
(vii) x → a log(f(x)) = log x → a f(x)
E
x→0
x→a
F
H
lim
x
lim sinx = 1 = x → 0
x
sin x
(a)
lim [f(x) + k] = lim f(x) + k
x→a
1 1
, 2 ,
...
x x
PAGE # 117
lim cosx = lim
x→0
1
=1
cos x
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PAGE # 118
MATHS FORMULA - POCKET BOOK
(e)
−1
x
lim sin x = 1 = lim
x→0
x→0
sin−1 x
x
(g)
(vi) By substitution :
(a) If x → a, then we can substitute
x=a+t ⇒ t=xa
If x → a, t → 0
...
2!
3!
(i)
x
lim e − 1 = 1
x→0
x
ex = 1 x +
x2
x3
+
...
2
3
(k)
1
lim loga (1 + x) =
x→0
loga
x
log(1 x) = x
x2
x3
...
2!
3!
x3
x5
+
...
2!
4!
tanx = x +
=1
2
x3
+
x5 +
...
2!
Sandwich Theorem : In the neighbour hood of x = a
f(x) < g(x) < h(x)
(1 + x)n = 1 + nx +
x
7
...
x →a
x →a
x →a
x
x→∞
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1
⇒ t → 0+
x
⇒
PAGE # 119
l < lim g(x) < l
...
) Ph
...
MATHS FORMULA - POCKET BOOK
2
...
(k)
loga x
ax
cosec x
sin1 x
1
1−x
2
,1
cos1x
1 − x2
tan1 x
1
|x| 1 − x
1
1+x
2
2
,|x|>1
,x ∈ R
0, x ∉ I
[x]
cosec1 x
1
|x| 1 − x2
cot1 x
|x|
1
1 + x2
,1|x|>|
D U
C
A T
I O
N
S
d
g(x)
dx
If
d
d
f(x) = φ(x), then
f (ax + b)
dx
dx
= a φ(ax + b)
, x ∈R
(vi)
x
, x≠ 0
| x|
d
dx
FG u I
H vJ
K
=
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
...
) Ph
...
dx
dx
dx
dx
then
NOTE :
=
,1
(v)
sec1 x
d
f(x), where c is a constant
...
(Product rule)
or
1
= c
dy
dy
du
=
×
dx
dx
du
llly If y = f(u), u = g(v), v = h(x), then
PAGE # 121
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PAGE # 122
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
(xii) Differentiation of implicit function : If f (x, y) = 0,
differentiate w
...
t
...
e if y = un ⇒
dy
at one side and find
dx
dy
du
= nun1
dx
dx
OR
[The relation f(x, y) = 0 in which y is not expressible
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)
...
e
...
r
...
another function : Let y = f(x) and z = g(x), then differentiation
of y w
...
t
...
form (f(x))g(x) or g ( x ) g ( x )
...
(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
x
using Trigonometrical Transformation : To solve
the problems involving inverse trigonometric functions
logm n = 1
first try for a suitable substitution to simplify it and
then differentiate
...
loge m
logn m = log n
e
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dy
...
) Ph
...
MATHS FORMULA - POCKET BOOK
Important Trigonometrical Formula :
(i)
sin2x = 2sinx
...
) Ph
...
) Ph
...
MATHS FORMULA - POCKET BOOK
Some Useful Substitutions :
Part B
Part A
Expression
Substitution
Expression
Substitution
Formula
Result
a2 + x2
3x 4x3
x = sinθ
3sinθ 4sin3 θ
sin3θ
4x3 3x
x = cosθ
4cos3 θ 3cosθ
cos3θ
a− x
a+ x
or
a+ x
a− x
x = a tan θ
a2 x2
x = a sin θ or x = a cos θ
3x − x3
1 − 3x 2
2x
1+x
2
2x
1−x
2
x = tanθ
x = tanθ
x = tanθ
3 tan θ − tan3 θ
1 − 3 tan2 θ
2 tan θ
1 + tan2 θ
2 tan θ
2
1 − tan θ
tan3θ
a+x
or
a−x
sin2θ
x = sinθ
1 2sin2 θ
x = cosθ
2cos2 θ 1
cos2θ
1 x2
x = sinθ
1 sin2 θ
1 cos2 θ
x = secθ
sec θ 1
cosec2 θ 1
cot 2 θ
x = tanθ
1 + tan2 θ
sec2 θ
x = cotθ
1 + cot2 θ
1
1 + x2
2
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A T
a2 − x2
or
x2 = a2 cos θ
a2 + x2
cosec2 θ
x
C
x = a sec θ or x=acosec θ
tan θ
x = cosecθ
a2 − x2
sin2 θ
2
D U
a2 + x2
cos2 θ
x = cosθ
x = a cos θ
cos2θ
2x2 1
a−x
a+x
x 2 a2
tan2θ
1 2x2
E
x = a tan θ or x = a cot θ
I O
N
S
5
...
r
...
x
2
⇒
d2 y
dx
2
=
d
dx
F dy I
G dx J
H K
=
c
h
d
f'(x)
dx
is called the second derivative of y w
...
t
...
) Ph
...
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
functions we use chain rule
6
...
e
...
=
dx
dx
t
(iii)
If y = (ax + b)m m ∉ I, then
dn
dx
n
dn
dx
n
ex j
n
= n!
csin xh
F
G
H
nπ
= sin x + 2
F
G
H
I
J
K
(cos x) = cos x +
πn
2
I
J
K
yn = m(m1) (m2)
...
an
(c)
If m ∈I, then
(iv)
ym = m! am and ym+1 = 0
(d)
(−1)n n!
1
If y =
, then yn =
an
(ax + b)n+1
ax + b
(v)
dn
dx n
dn
dx n
NOTE :
(e)
If y = log (ax + b), then yn =
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n−1
(−1)
(n − 1)!
(ax + b)n
(emx ) = mn emx
(log x) = ( 1)n1 (n1)! xn
If u = g(x) is such that g'(x) = K (constant)
an
then
PAGE # 129
dn
dx
n
c h
f g(x)
Ld
Mdu
M
N
n
= Kn
n
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O
P
P
Q
f(u)
u= g(x)
PAGE # 130
MATHS FORMULA - POCKET BOOK
7
...
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
...
r
...
x
we get
dy
=
dx
⇒
8
...
v
If R
...
S
...
H rule can be applied
...
H
...
then L
...
rule can't
be applied
...
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PAGE # 132
MATHS FORMULA - POCKET BOOK
APPLICATION OF DERIVATIVES
MATHS FORMULA - POCKET BOOK
7
...
Geometrically f'(a) represents the slope of the tangent to
the curve y = f(x) at the point (a, f(a))
2
...
(x , y )
1
1
(x1 , y1)
Length of perpendicular from origin to the tangent :
= 0
...
FG dy IJ
H dx K
6
...
If normal makes an angle of φ with +ve direction of x-axis,
(x x1)
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9
...
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
F dy I
G dx J
H K
F dy I
1+G J
H dx K
y1 − x1
π
If the tangent is perpendicular to x-axis, ψ =
2
4
...
U
|
|
V
|
|
W
If the tangent is parallel to x-axis, ψ = 0
3
...
dx
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MATHS FORMULA - POCKET BOOK
11
...
If the normal is perpendicular to x-axis ⇒
MATHS FORMULA - POCKET BOOK
17
...
(x1 , y1)
FG dy IJ
H dx K
(x1 , y1)
tanθ = ±
= 0
...
If normal is equally inclined from both the axes or cuts
equal intercept then
F dy I − F dy I
G dx J G dx J
H K H K
F dy I F dy I
1−G J G J
H dx K H dx K
14
...
) Ph
...
Length of perpendicular from origin to normal :
E
2
18
...
Length of intercept made on axes by the normal :
x1
1
2
1
y y1 =
F dy I
G dx J
H K
FG dy I of second
...
1
where
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PAGE # 136
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
MONOTONICITY, MAXIMA & MINIMA :
1
...
(ii)
A function is said to be monotonic function in a domain if it is
either monotonic increasing or monotonic decreasing in that
domain
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
...
Working rule for finding local maxima & Local Minima :
(i)
Find the differential coefficient of f(x) w
...
to x, i
...
f'(x) and equate it to zero
...
(iii) Now differentiate f'(x) w
...
to x and substitute the
critical points in it and get the sign of f"(x) for each
critical point
...
Similarly by getting the sign of f"(x)
for other critical points (b, c,
...
7
...
Similarly a minimum value may not be the least value
of the function
...
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
...
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
...
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
...
The maximum & minimum points are also known as
extreme 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
...
) Ph
...
(b) The value of the function f(x) at x = a is
minimum if f'(a) = 0 and f"(a) > 0
...
) Ph
...
Some Geometrical Results :
In Usual Notations
Area of equilateral
and its perimeter
Results
3
(side)2
...
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
...
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
f(b) − f(a)
= f'(c)
b−a
× (distance between them)
Area of circle
Perimeter
2πr
Volume of sphere
4 3
πr
3
Surface area of sphere
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
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A T
f(a + h) = f(a) + h f'(a + θ h), 0 < θ < 1, h = b a
1 2
πr h
3
Surface area of cone
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
4πr2
Volume of cone
E
(ii)
πr2
I O
N
S
PAGE # 139
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PAGE # 140
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
Function
INDEFINITE INTEGRATION
1
...
(integrating constant)
d
(ii)
dx
(iii)
(iv)
2
...
a
n+1
+ c, n ≠ 1
1
dx
x
log|x| + c
1
dx
ax + b
1
(log|ax + b|) + c
a
ex dx
ex + c
ax dx
sinx dx
cos x + c
D U
C
A T
I O
N
S
sin x + c
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
ax
+ c
loge a
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z
z
z
n+1
dx
Integration
dx
sin1 x + c = cos1x + c
1 − x2
dx
2
a −x
2
dx
1 + x2
dx
a2 + x2
dx
|x| x2 − 1
dx
|x| x2 − a2
PAGE # 141
sin1
x
x
+ c = cos1 + c
a
a
tan1x + c = cot1x + c
x
1
x
−1
tan1
+ c =
cot1 a + c
a
a
a
sec1x + c = cosec1x + c
1
x
sec1
+ c =
a
a
x
−1
cosec1 a + c
a
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F π + x I +c
G 4 2J
H K
PAGE # 142
MATHS FORMULA - POCKET BOOK
3
...
) Ph
...
x a+ x
f t dt
f(x) = t
x + a2
2
x
,
a+ x
z ch
f(x) = t
1
2
, a2 x 2
or x = a cosh θ
+c
2
a2 − x2
x −a ,
2
1
F(ax + b) + c
a
dfcxhi
2
x2 + a2 ,
2
f(x) = t
1
a −x ,
2
into another variable t so that the integrand f(x) is changed
into F(t) which is some standard integral
...
Function
Substitution
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
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PAGE # 144
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
IMPORTANT RESULTS USING STANDARD SUBSTITUTIONS :
Function
Integration
z
x−a
1
log
x+a
2a
INTEGRATION OF FUNCTIONS USING ABOVE STANDARD
RESULTS :
1
2
2
x −a
=
z
1
2
a −x
2
dx
Function
z
z
+ c
x
−1
coth1 a + c when x > a
a
a+ x
1
log
a− x
2a
+ c
z
2
2
x −a
log{|x +
= cosh1
z
dx
2
2
x +a
z
F xI + c
G aJ
H K
log{|x +
= sinh1
2
F I
G J
H K
x
a
FG x IJ + c
H aK
x − a |} + c
x2 + a2 dx
1
1
x x2 + a2 + a2 log {|x +
2
2
x2 + a2 |} + c
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S
ax + bx + c
dx or
px + q
ax2 + bx + c
dx or
(px + q) (ax2 + bx + c) dx
z ax
x − a dx
2
px + q
2
LF
MG
MH
N
a x+
b
2a
I
J
K
2
+
4ac − b2
2
4a
O
P
P
Q
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
x x2 − a2 a2 log {|x +
2
2
2
ax2 + bx + c
dx or
+ c
1
1 2
x a2 − x2 +
a sin1
2
2
2
1
Express : ax2 + bx + c =
ing coefficient of x and
a − x dx
2
z
z
E
z
x2 + a2 |} + c
dx or
(ax2 + bx + c) dx
z
z
x − a |} + c
2
ax + bx + c
z
x
1
= tanh1 a + c, when x < a
a
dx
1
2
Method
2
P(x)
2
+ bx + c
dx ,
Apply division rule and express it
ch
R x
where P(x) is a
in form Q(x) +
polynomial of degree
The integral reduces to known
2 or more
2
form
PAGE # 145
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2
ax + bx + c
PAGE # 146
MATHS FORMULA - POCKET BOOK
z
1
2
a sin x + b cos2 x + c
zc
or
z
z
dx Divide numerator & denominator
by cos x,
2
1
a sin x + b cos x
h
2
dx
dx
dx
a sin x + b cos x + c
then put tanx = t & solve
...
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
...
z
Express : num
...
) +
(i)
Express : Num
...
) +
= (1st fun)
d
(deno
...
dx
Thus integral reduces to known
form
...
) Ph
...
2nd fun
...
dx
z LMNFGH
I ez 2nd fun
...
dx
Rule to choose the first function : first fun
...
[The fun
...
E
zc
z LMN FHz
I Inverse trigonometric function
2
by x2 and put
z
u
...
υ dx
or
µ
z
Divide num & deno
...
Thus the form
reduces to the known form
...
) Evaluate λ & µ
...
a sin x + b cos x + c
dx
p sin x + q cos x + r
Divide numerator & denominator
by 2 and then add & sub
...
Thus the form reduces as above
...
) Ph
...
ex f x + f' x dx = ex f(x) + c
ch ch
mx
mf x + f' x dx = e
e
L f' cxh O dx
Mc h m P
P
M
Q
N
emx f x +
mx
INTEGRATION OF RATIONAL ALGEBRAIC FUNCTIONS
USING PARTIAL FRACTION :
Every Rational fun
...
If degree of numerator is less than that of denominator,
the rational fun
...
If deg (num
...
) apply division rule
+ c
...
e
...
z
eax sinbx dx and
e ax
=
a2 + b2
e
ax
e
=
c
h
eax sin bx + c dx
2
a +b
2
z
e
cacos bx + b sinbxh
eax
a2 + b2
c
h
c
C
A T
I O
N
S
c
h
hc
hc
h
A
B
C
+
+
x−a
x −b
x−c
≠ b
A
+
x−a
c
px2 + qx + r
j
Bx + C
A
+ 2
x−a
x + bx + c
he
x − a x2 + bx + c , where
ex
h
B
+
cx − ah
2
C
x−b
x2 + bx + c can
not be factorised
+ k
a cos bx + c + b sin bx + c
A
B
+
x−a
x −b
h
cx − ah cx − bh , a
cos bx + c dx
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px2 + qx + r
,
x−a x−b x−c
hc
px2 + qx + r
ax
Types of partial
fractions
a, b, c are distinct
(a sin bx b cos bx) + k and
cos bx dx and
e ax
and
E
c
[a sin (bx + c) b cos(bx + c)] + k1
2
c h , for integrating rcxh , resolve the
gcxh
gcxh
r x
px + q
x−a x−b , a ≠ b
2
a +b
z
c
= q(x) +
Types of proper
rational functions
ax
2
(vi)
z
ch
gcxh
f x
fraction into partial factors
...
NOTE : Breaking (iii) & (iv) integral into two integrals
...
(v)
c h,
Qcxh
Px
px3 + qx2 + rx + s
2
je
j
+ ax + b x2 + cx + d ,
Ax + B
2
x + ax + b
+
Cx + D
2
x + cx + d
where x + ax + b,
x2 + cx + d can not
be factorised
2
+ k1
...
) Ph
...
MATHS FORMULA - POCKET BOOK
INTEGRATION OF IRRATIONAL ALGEBRAIC FUNCTIONS :
(i)
(ii)
(viii) To evaluate
If integrand is a function of x & (ax + b)1/n then put
(ax + b) = tn
If integrand is a function of x, (ax + b)1/n and
1
or
quadratic
where p = (L
...
M
...
(iv) To evaluate
(v) To evaluate
or
or
z
z
z
put
linear linear
dx
quad
...
quadratic
zc
h
linear = t
form :
put linear = t2
dx
h
7
...
pure quad
dx
pure quad
...
) Ph
...
quad
linear quad
...
quadratic
z
linear
dx
tan2 mx
cot2 mx
2 cos A
2 sin A
2 sin A
1 + cos 2mx
2
x dx
then is the resulting integral, put
D U
z
(ii)
dx
put
pure quad = t
E
F linear I
G quadratic J
H
K
(i)
2
z
z
INTEGRATION USING TRIGONOMETRICAL IDENTITIES :
(A) To evaluate trigonometric functions transform the
function into standard integrals using trigonometric
identities as
linear
...
quad
2
(vi) To evaluate
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)
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PAGE # 152
MATHS FORMULA - POCKET BOOK
(B)
z
MATHS FORMULA - POCKET BOOK
z
z
z
sinm x cosm xdx
...
&
m+n−2
is ve integer
2
put tan x = t
8
...
1 n ax
n
x e
I
a
a n1
where In1 =
n−2
cos ecn−2 x cot x
+
I
n − 1 n2
n−1
sinm1x cosn+1x + (m 1) Im2,n
Integration
xn−1eax dx
xn sin x dx
xn cos x + nxn1sin x n(n 1) In2
sinn x dx
cosn x dx
n−1
cosn−1 x sin x
+
In2
n
n
n−1
sinn−1 cos x
+
In2
n
n
tann x dx
ctanxh
cotn x dx
n−1
In2
n−1
ccot xh
n−1
n−1
In2
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n−2
secn−2 x tan x
+
I
n − 1 n2
n−1
INTEGRATION BY SUCCESSIVE REDUCTION (REDUCTION
FORMULA) :
Function
E
secn x dx
N
S
PAGE # 153
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PAGE # 154
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
z
z
z
b
ch
III
...
z
a
*
{uz v
...
PROPERTIES OF DEFINITE INTEGRAL :
zch
b
f x dx =
a
z
b
a
D U
C
A T
I O
N
S
f x dx =
0
zc
a
h
f a − x dx
0
c h
f x + f −x dx
ch
, if f cxh is an odd function
, if f x is an even function
R
|2 fcxh dx
S
| 0
|
T
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
zch
nT
zch
T
f x dx = n f x dx
ch
f t dt
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0
2a
VI
...
v
...
We note that
while changing the independent variable in a definite
integral, the limits of integration must also we changed
accordingly
...
ch
f x dx =
zch
a
f a + b − x dx or
b
For integration by parts in definite integral we use
following rule
...
To evaluate definite integral of f(x)
...
b
a
a
Remarks :
*
zch
b
f x dx +
This property is mainly used for modulus function,
greatest integer function & breakable function
= F(b) F(a) is called definite integral
of f(x) w
...
t
...
*
zch
c
ch
a
f x dx = F x + c
a
b
b
f x dx = F(x) + c, then
zch
a
ch
a
Definite Integration :
If
f x dx =
b
II
...
f x dx = f x dx
z
DEFINITE INTEGRATION
0
0
and further if a ∈ R+, then
PAGE # 155
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PAGE # 156
MATHS FORMULA - POCKET BOOK
zch
a+nT
zch zch
a
f x dx =
f x dx ,
nT
(ii)
zch
nT
0
MATHS FORMULA - POCKET BOOK
T
f x dx = (n m) f x dx ,
mT
0
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,
zch zch
b +nT
d
dx
b
f x dx =
a+ nT
f x dx
a
ψ(x))
VIII
...
zch
b
m(b a) <
z
b
a
ch
f x dx
|f x dx|
a
=
z
a
2
...
zch
b
z
π /2
cosn x dx =
sinn x dx
0
a
z ch
If f(x) < g(x) on [a, b], then
f (x, t) dt
π /2
(i)
b
X
...
L
M
M
N
R n − 1
...
2
...
n − 2
...
2 ,
|
T
g x dx
z
if n is odd
if n is even
π /2
a
(ii)
Differentiation Under Integral Sign :
For integration
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
d
functions in the interval [a, b], then,
dx
z
v(x)
f(t) dt =
or cos2x = 1 sin2x and then use
u(x)
z
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0
PAGE # 157
z
π /2
π /2
d
d
f{v(x)}
{v(x)} f{u(x)}
{u(x)}
...
) Ph
...
a
2
a + b2
0
+f(a + (n 1)h]
where nh = b a
...
a
∞
(iv)
MATHS FORMULA - POCKET BOOK
e −ax xndx =
0
z
n!
n
a +1
(ii)
lim
n→∞
FI
∑G J
HK
zch
1
n
r
1
f
n r =1 n
f x dx
=
0
π /2
(vi)
sinn x cosm x dx
[i
...
exp
...
m − 3
...
1
Mm+n m+n−2 3 +n 1+n
M
M m − 1
...
1
...
n − 3
...
m − 3
...
n − 1
...
1
...
] [(n − 1) (n − 3)
...
to be multiplied
zch
Key Results :
0
=
r
1
by x and
by dx and the limit of the
n
n
0
5
...
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PAGE # 159
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PAGE # 160
MATHS FORMULA - POCKET BOOK
z
π /2
=
=
c
h c
h
c
zc
f cos ec x
h c
f cos ec x + f sec x
sinmx sin nx dx =
R0
|π
S
|2
T
0
c h dx
= π/4
...
) Ph
...
4
...
2n
2n+1
dx = 3
...
7
...
) Ph
...
cos nx dx
...
f(x) on the interval [a, b]
...
2 2
x
dx =
zd
2k
*
Q x [x] is a periodic function with period 1
...
with period T, then
*
zch
z c
a+ T
f x dx is independent of a
...
) Ph
...
The no
...
t
a
a+x
π +2
dx =
2
a−x
a
(ii)
zch
b
ab > 0
b
0
D U
If f(x) is continuous on [a, b] then there exists a
If a > 0 then
a
E
MATHS FORMULA - POCKET BOOK
PAGE # 163
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PAGE # 164
MATHS FORMULA - POCKET BOOK
MATHS FORMULA - POCKET BOOK
DIFFERENTIAL EQUATIONS
1
...
2
...
d2 y
Eg
...
(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
f(x) dx
dY
aX + bY
=
...
dX
AX + BY
Solve it and then put X = x h, Y = y k we shall
get the solution
...
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F I
G J
H K
x
or yn f y
...
After substituting y = vx or x = vy
...
∴
z
Homogeneous Equations : It is a differential equation
F I
G J
H K
2
(A) Differential equation of the form
Integrate both sides i
...
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 :
dy
= f(y)
dx
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
...
dy
=
g(y)
of the form
dy
+ 5y = 0
dx
dy
y = x
+
dx
Variable Separable Form : Differential equation of
PAGE # 165
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PAGE # 166
MATHS FORMULA - POCKET BOOK
*
Form
where
ax + by + c
dy
=
Ax + By + C
dx
MATHS FORMULA - POCKET BOOK
...
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
...
e
z
R
...
F
...
(F) Equation reducible to linear form :
This is variable separable form and can be solved
...
In y :
Put
dy
+ Py = Q, where P, Q are function of x
dx
where
e
z
P dx
z
P dx
=
z
Qe
z
P dx
dy
+ pyn
dx
yn
+ 1
+ 1
= Q
= z
Note : In general solution of differential equation we can
take integrating constant c as tan1 c, ec, log c etc
...
alone or constant
...
Linear equation :
*
dy
dx
where P and Q are functions of x or constant is called
Bernoulli's equation
...
F
...
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PAGE # 168
MATHS FORMULA - POCKET BOOK
VECTORS
1
...
Vectors in terms of position vectors of end points -
AB = OB OA = Position vector of B position vector of A
i
...
any vector = p
...
of terminal pt p
...
of initial pt
...
r
a
Vector a
$
(b) Unit vector :
a = || =
a
Magnitude of a
(c)
2
...
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
$
$
i
and if a = a1 $ + 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
...
Distance between two points :
Distance between points A(x1, y1, z1) and B(x2, y2, z2)
C
→
= Magnitude of AB
=
b
7
...
a
Position vector of a dividing point :
r
(i)
If A( a ) & B( b ) be two distinct pts, the p
...
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
(iii)
a
p
...
of the mid point of [AB] is
If point C divides AB in the ratio m1 : m2 externally,
then p
...
of C is c =
A
where OC is a diagonal of the parallelogram OABC
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1
[p
...
of A + p
...
of B]
2
(ii)
b
D
(x2 − x1 )2 + (y2 − y1)2 + (z2 − z1 )2
PAGE # 169
m1 b − m2 a
m1 − m2
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PAGE # 170
MATHS FORMULA - POCKET BOOK
(iv) p
...
of centriod of triangle formed by the points A( a ),
MATHS FORMULA - POCKET BOOK
10
...
v
...
remaining two vectors i
...
c = λ a + µ b
(iii)
Some results :
(i)
b and c i
...
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
...
b = |a| |b| cosθ
OA + OB + OC = 3 OG = OH where G is centriod and
H is orthocentre of ∆ABC
...
b
| b|
(iv) If H is orthocentre of ∆ABC, then
a
...
Collinearity of three points :
(i)
Three points A, B and C are collinear if AB = λ AC for
some non zero scalar λ
...
v
...
) Ph
...
$ = k
...
a I
to a = r G
H|a| J a
K
r
...
k = k
...
) Ph
...
b = | a || b | and
(viii) a × ( b + c ) = ( a × b ) + ( a × c )
If a and b are unlike vectors, then a
...
b = 0
$
$
$
j × k = $, k × $ = $
j
i
i
Area of triangle :
(viii) ( a
...
b is not defined
(ix)
( a ± b )2 = a2 ± 2 a
...
v
...
d ,
is displacement vector
...
a a
...
b b
...
) Ph
...
) Ph
...
c = [ a b c ] =
MATHS FORMULA - POCKET BOOK
(d)
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 r
(iv) ( a × b )
...
( b × c ) etc
...
|r | |r | |r |
(i)
→
(ii)
Incentre formula : The position vector of the incentre
r
r
r
aa + bb + cc
of ∆ ABC is
...
) Ph
...
Application of Vector in Geometry :
1
| AB × AC
...
v
...
(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 ) a
r
r
r
r r r
r r r
and a × ( b × c ) = ( 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
[$ $ k] = 1
i j $
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
...
) Ph
...
(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
...
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
Then, shortest distance
PQ =
cb
1
(x)
h c
× b2
...
n2
r r
r2
...
1
2
(xiii) The equation of the planes bisecting the angles
r r
between the planes r1
...
n1 − d1| |r
...
n2 = d2 are
|n1|
|n2|
r r
r r
(xiv) The plane r
...
n − d|
r
if
= R
...
b1 × b2
h e
j = 0
...
n = d
...
n = d to normal form we divide both sides by | n |
r
r n
r
d
d
$
r
...
n = r
...
) Ph
...
The perpendicular distance of a point having position
r
r r
vector a from the plane r
...
n − d|
r
p=
...
n1 = d1 and
then the shortest distance between them is zero
...
|b|
If the lines
h
The equation of any plane through the intersection
r r
r r
of planes r
...
n2 = d2 is
r r
r
...
( r − b ) = 0 or | r |2 r
...
b = 0
...
) Ph
...
*
Coordinates of the centroid of a triangle are
Fx
G
H
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
*
Equation of YOZ plane is x = 0
FG x
H
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
...
Section formula :
*
Internally are
*
Externally are
F mx + nx
G m+n
H
F mx − nx
G m−n
H
1
,
C
A T
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N
S
*
If
and so
...
) Ph
...
I
J
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
...
l 2 + m2 + n2 = 1, where 0 ≤
α, β, γ
1
PAGE # 179
≤ π
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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
2
a + b2 + c2
*
The angle θ between the lines whose d
...
's are l 1,
m1, n1 and l 2, m2, n2 is given by
...
*
cos θ = l 1 l 2 + m1m2 + n1n2
...
c
...
Direction cosines of PQ = r, where P is (x1, y1, z1) and
Q(x2, y2, z2) are
*
The lines are || if
The lines are ⊥ if l 1 l 2 + m1m2 + n1n2 = 0
*
The angle θ between the lines whose d
...
s are a1, b1,
c1 and a2, b2, c2 is given by
cos θ = ±
x2 − x1
y2 − y1
z 2 − z1
,
,
r
r
r
*
m1
n1
l1
l2 = m2 = n2 and
The lines are || if
If a, b, c are direction no
...
2
2
2
a1 + b1 + c1
a2 + b2 + c2
2
2
2
a1
b1
c1
= b = c and
a2
2
2
The lines are ⊥ if a1a2 + b1b2 + c1c2 = 0
Length of the projection of PQ upon AB with d
...
,
*
Note : Direction cosines of a line are unique but the
direction ratios of line are not unique
...
c
...
L = l (x2 x1) + m(y2 y1) + n(z2 z1)
Straight line in space :
*
Equation of a straight line passing through a fixed
point and having d
...
's a, b, c is
5
...
*
Shortest distance between two skew lines,
x − x1
y − y1
z − z1
=
=
and
m1
n1
l1
x − x1
y − y1
z − z1
=
=
(is the symmetrical
a
c
b
*
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x − x2
y − y2
z − z2
=
=
is given
m2
n2
l2
form)
Equation of a line passing through two points is
N
S
PAGE # 181
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PAGE # 182
MATHS FORMULA - POCKET BOOK
x2 − x1
s
...
= ±
*
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
...
's of the normal
to the plane and p is the length of perpendicular from
the origin
...
MATHS FORMULA - POCKET BOOK
l1
l2
y2 − y1
m1
m2
x
y
z
=
=
...
*
Ax + By + Cz + D = 0 represents a plane whose
normal has d
...
s proportional to A, B, C
...
*
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
...
's of a normal to the plane
...
) Ph
...
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
...
a
c
b
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
...
) Ph
...
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 )
p=
*
The line is || to the plane if a l + bm + cn = 0
...
where
ax + by + cz + d = 0, according as the expression
A l + Bm + Cn = 0
...
*
*
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
2
2
2
a1 + b1 + c1
= ±
x−α
y−β
z−γ
=
=
is given by
m
n
l
+ m(y1 β ) + n(z1 γ )]2
a2 x + b2 y + c2 z + d2
a2 + b2 + c2
2
2
2
if a1a2 + b1b2 + c1c2 is ve then origin lies in the acute
angle between the planes provided d1 and d2 are of
same sign
...
) Ph
...
) Ph
Title: All Mathematics formulas for high school and University students
Description: These are my personal digital notes of mathematics formulas. It is very useful for high school students. It also contains some short tricks and advanced formulas to solve math problems. These notes will definitely help improve your math problem solving skills! Topics covered: ALL high school math topics Complete Algebra, Calculus, Analytical Geometry, Probability, Progression and series, Probability, Trigonometry, 3-D, Vector, Functions, Permutation and combination, binomial theorem, solution of triangle etc...
Description: These are my personal digital notes of mathematics formulas. It is very useful for high school students. It also contains some short tricks and advanced formulas to solve math problems. These notes will definitely help improve your math problem solving skills! Topics covered: ALL high school math topics Complete Algebra, Calculus, Analytical Geometry, Probability, Progression and series, Probability, Trigonometry, 3-D, Vector, Functions, Permutation and combination, binomial theorem, solution of triangle etc...