1. If *a*
: *b* : : *c* : *d*, then *ad* = *bc*

**2.** If *a* : *b*
: : *c* : *d***,** then *a* + *b* : *b* : : *c*
+ *d* : *d*

3. If *a*
: *b* : : *c* : *d*, then *a* - *b* : *b* : : *c* - *d* : *d*

4. If *a*
: *b* : : *c* : *d*, then *a* + *b* : *a* - *b* : : *c* + *d* :
*c* - *d*

5. If _{} then *k* =_{}

**NUMBERS**

**1.** *a*^{3} + *b*^{3} +
*c*^{3} – 3*abc *= (*a* + *b* + *c*) (*a*^{2}
+ *b*^{2} + *c*^{2} – *ab* – *bc* – *ca*)

**2.** The
product of *n* consecutive integers is always divisible by *n*! (*n*
factorial)

**3.** The
sum of any number of even numbers is always even

**4.** The sum of even number of odd numbers is
always even

**5.** The sum of odd number of odd numbers is
always odd

**6.** If N is a
composite number such that N = *a ^{p}*

a) the number of factors of N is given by the expression (*p*
+ 1) (*q* + 1) (*r* + 1) ...

b) it can be expressed as the product of two
factors in 1/2 {(*p* + 1) (*q* + 1) (*r* + 1).....} ways

c) if
N is a perfect square, it can be expressed

(i)** **as
a product of two DIFFERENT factors in 1/2 {(*p* + 1) (*q* + 1) (*r*
+ 1) ... -1 } ways

(ii) as
a product of two factors in 1/2 {(*p* + 1) (*q* + 1) (*r* + 1)
... +1} ways

d) sum of all factors of N = _{}

e) the number of
co-primes of N (< N), f (N) = _{}

f) sum
of the numbers in (e) = _{}

g) it
can be expressed as a product of two factors in 2^{n}^{–1},
where ‘*n*’ is the number of different prime factors of the given number N

SIMPLE INTEREST AND COMPOUND INTEREST

I = Interest, P is Principle, A = Amount, *n*
= number of years, *r* is rate of interest

1. Interest
under

a) Simple interest, I =_{}

b) Compound interest, I =
P_{}^{}

2. Amount under

a) Simple interest, A = _{}

b) Compound interest, A =
P_{}

3. Effective
rate of interest when compounding is done *k* times a year

*r _{e}* =

**MIXTURES AND
ALLIGATION**

**1.** If *p*_{1},
*p*_{2 }and *p* are the respective concentrations of the
first mixture, second mixture and the final mixture respectively, and *q*_{1
}and *q*_{2} are the quantities of the first and the second
mixtures respectively, then Weighted Average (*p*)

*p*
= _{}

**2. **If C is the concentration after a dilutions, V is the original volume
and *x* is the volume of liquid. Replaced each time then C = _{}

**QUADRATIC
EQUATIONS**

1. If *a*, *b* and *c* are all rational and *x*
+_{}is an irrational root of *ax*^{2} + *bx* + *c*
= 0, then *x* -_{} is the other root

2. If *a* and *b* are the roots of *ax*^{2} + *bx* + *c* = 0,
then *a* + *b* =_{}and *a**b* = _{}

**3.** When *a*
> 0, *ax*² + *bx* + *c* has a minimum value equal to_{}, at *x* = _{}

**4. **When *a* < 0, *ax*² + *bx*
+ *c* has a maximum value equal to_{}, at *x* = _{}

**PROGRESSIONS**

*a*
is the first term, *d* is the last term and *n* is the number of
terms

**1.** T* _{n}*
=

2. S* _{n}*
=

^{ }

3. T* _{n}* = S

4. S* _{n}*
= A.M ´

__Geometric Progression__** (G.P)**

*a*
is the first term, *r* is the common ratio and *n* is the number of
terms

**5.** T* _{n}*
=

6. S* _{n}*
=

__Harmonic Progression__ (H.P)

7. H.M of *a*
and *b* =_{}

8. A.M > G.M > H.M_{}

9. (G.M)^{2} = (A.M) (H.M)

**10.** Sum of first *n*
natural numbers å*n*
= _{}

11. Sum
of squares of first *n* natural numbers å*n*^{2} = _{}

**12. **Sum of cubes of first *n* natural
numbers å*n*^{3} =_{}= (å*n*)^{2}

^{ }

**GEOMETRY**

**1.** In a triangle ABC, if AD is the angular
bisector, then_{}

**2. **In a triangle ABC, if E and F are the points of AB and AC respectively
and EF is parallel to BC, then _{}

**3.** In a triangle ABC, if AD is the median, then
AB^{2} + AC^{2} = 2(AD^{2} + BD^{2})

**4.** In parallelogram, rectangle, rhombus and
square, the diagonals bisect each other

**5.** Sum of all the angles in a polygon is (2*n*
– 4)90

**6.** Exterior angle of a polygon is _{}

**7.** Interior angle of a polygon is _{}

**8.** Number of diagonals of a polygon is _{}

**9.** The
angle subtended by an arc at the centre is double the angle subtended by the
arc in the remaining part of the circle

**10.** Angles in the
same segment are equal

**11. **The angle subtended by the diameter of the
circle is 90°

**MENSURATION**

1. Plane figures

2.
Solids

**(PERMUTATIONS
& COMBINATIONS, PROBABILITY)**

**1. ***n* (A È B) = *n *(A) + *n* (B)
– *n* (A Ç B)

**2.** If A and B are two tasks that must be performed such
that A can be performed in '*p*' ways and for each possible way of
performing A, say there are '*q*' ways of performing B, then the two tasks
A and B can be performed in *p* ´ *q* ways

**3.** The number of ways of dividing (*p* + *q*)
items into two groups containing *p* and *q* items respectively is _{}

**4. **The number of ways of dividing 2*p*
items into two equal groups of *p* each is _{}, when the two groups have distinct identity and _{}, when the two groups do not have distinct identity

**5.** * ^{n}*C

**6. **The total number of ways in which a
selection can be made by taking some or all out of (*p* + *q* + *r*
+ .....) items where *p* are alike of one kind, *q* alike of a second
kind, *r* alike of a third kind and so on is {(*p* + 1) (*q* +
1) (*r* + 1) ....} -1

**7.** P(Event) = _{} and 0 £ P(Event) £ 1

**8. **P(A Ç B) = P(A) ´ P(B), if A and B are independent events

**9. **P(A È B) = 1, if A and B are exhaustive events

**10.** Expected Value =
_{}[Probability (E_{i})]´ [Monetary value associated with event E_{i}]

**HIGHER MATHS – II**

**(STATISTICS, NUMBER SYSTEMS, INEQUALITIES & MODULUS, SPECIAL
EQUATIONS)**

**1.** G.M. = (*x*_{1
}× *x*_{2 }× ...... .*x _{n}*)

**2. **_{}

**3.** For any two positive numbers *a*, *b*

(i) A.M. ³ G.M. ³ H.M. (ii) (G.M.)^{2} = (A.M.) (H.M.)

**4. **Range = Maximum value – Minimum value

**5. **Q.D. =_{}(i.e., one-half the range of quartiles)

**6.** If *a*
> *b*, _{}, for any two positive numbers *a* and *b*

**7. ***|x* + *y| *£ |*x*| + |*y*|, for any two real
numbers *x* and *y*

**8.** If for
two positive values *a* and *b*; *a* + *b* = constant (*k*),
then the maximum value of the product *ab* is obtained for *a* = *b
*= _{}

**9. **If for two positive values *a* and *b*;
*ab* = constant (*k*), then the minimum value of the sum (*a* + *b*)
is obtained for *a* = *b*

= _{}

**HIGHER MATHS – III**

**(CO-ORDINATE
GEOMETRY, FUNCTIONS & GRAPHS, TRIGONOMETRY)**

**1. **If a point P(*x*, *y*) divides the line segment joining A(*x*_{1},
*y*_{1}) and B(*x*_{2}, *y*_{2}) in the
ratio *m* : *n*, then *x* =_{}and *y* =_{}, positive sign for internal division and negative sign for
external division

**2. **The area of a triangle with the vertices at
(0, 0), (*x*_{1}, *y*_{1}) and (*x*_{2},
*y*_{2}) is D = _{}.

**3.** The
coordinates of the centroid C(*x*, *y*) of a triangle ABC formed by
joining the points

A(*x*_{1}, *y*_{1}); B(*x*_{2}, *y*_{2})
and C(*x*_{3}, *y*_{3}) are given by _{}

**4. **The slope of line with points (*x*_{1},
*y*_{1}) and (*x*_{2}, *y*_{2}) lying on
it is *m* = _{}

**5.** If *m*_{1}
and *m*_{2} are the slopes of two lines L_{1} and L_{2}
respectively, then the angle ‘q’ between them is given by tanq = _{}

**6.** The
equation of the *x*-axis is ** y = 0** and that of

**7.** The
equation of a line parallel to *x*-axis is of the form ** y = b**
and that of a line parallel to

**8. **Point slope form of a line: *y – y _{1}*
=

**9. **Two point form of a line:_{}

**10. **Slope intercept form of a line: *y *= *mx*
+ *b*

**11. **Intercept form of a line : _{}

**12.** Two lines *a*_{1}*x* + *b*_{1}*y*
+ *c*_{1} = 0 and *a*_{2}*x* + *b*_{2}*y*
+ *c*_{2} = 0 are

(i) parallel if _{}or *m*_{1}= *m*_{2}

(ii)
perpendicular if *a*_{1} *a*_{2} + *b*_{1}
*b*_{2} = 0 or *m*_{1} *m*_{2} = -1

**13.** The distance
between two parallel lines of the form *ax* + *by* +*c*_{1}
= 0 and *ax* + *by* + *c*_{2} = 0 is given by _{}

**14.** If *ax*
+ *by* + *c* = 0 is the equation of a line, then the perpendicular
distance of a point (*x*_{1}, *y*_{1}) from the line
is given by _{}

**15. **sine rule : _{}= 2R, where R is the circumradius of triangle ABC

**16.** cosine rule :
cosA =_{}, similarly cosB and cosC can be defined