Important Formulae_for Website_for CAT2012 Students

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RATIO PROPORTION VARIATION

1. Ifa : b : : c : d, then ad= bc

2. Ifa : b : : c : d, then a + b : b : : c + d: d

3. Ifa : b : : c : d, then a-b : b : : c-d: d4. Ifa : b : : c : d, then a + b : a-b : : c + d: c-d

5. If then k=

NUMBERS

1. a3 + b3 + c3 3abc = (a + b + c) (a2 + b2 + c2 ab bc ca)

2. The product ofn 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 = ap. bq. cr.... where a, b, c are prime factors of N andp, q, r

.... are positive integers, then

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) it can be expressed as a product of two co-primes in 2n1 ways, where n is the number of differentprime factors of the given number N


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f) the number of co-primes of N (< N), f (N) =

g) sum of the numbers in (e) =

SIMPLE INTEREST AND COMPOUND INTEREST

I = Interest, P is Principle, A = Amount, n = number of years, ris 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

re =

MIXTURES AND ALLIGATION1. Ifp1,p2 andp are the respective concentrations of the first mixture, second mixture and the final mixture

respectively, and q1 and q2 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


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C =

QUADRATIC EQUATIONS

1. Ifa, b and c are all rational andx + is an irrational root ofax2 + bx + c = 0, thenx- is the other

root

2. Ifa and b are the roots ofax2 + bx + c = 0, then a + b = and ab =

3. When a > 0, ax + bx + c has a minimum value equal to , atx =

4. When a < 0, ax + bx + c has a maximum value equal to , atx =

PROGRESSIONS

Arithmetic Progression (A.P)

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

1. Tn = a + (n 1)d

2. Sn = =

3. Tn = Sn Sn-1

4. Sn = A.M n

Geometric Progression (G.P)

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

5. Tn = arn-1

6. Sn = =

Harmonic Progression (H.P)


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7. H.M ofa and b =

8. A.M > G.M > H.M

9. (G.M)2 = (A.M) (H.M)

10. Sum of first n natural numbers n =

11. Sum of squares of first n natural numbers n2 =

12. Sum of cubes of first n natural numbers n3 = = (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 AB2 + AC2 = 2(AD2 + BD2)

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

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

6. Exterior angle of a polygon is

7. Interior angle of a polygon is

8. Number of diagonals of a polygon is


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Figure Perimeter Area

Triangle = a + b + c

Right angled

triangle

a + b +

Equilateral

triangle 3a

Isoceles

triangle2a + b

Circle 2pr pr2

Sector of acircle + 2r

(q is in degrees)

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

of the circle10. Angles in the same segment are equal11. The angle subtended by the diameter of the circle is 90

MENSURATION

1. Plane figures


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Square 4a a2

Rectangle 2(l + b) lb

Trapezium a + b + c + d(a + b)h

Parallelogram 2(a + b) bh orabsinq

2. Solids

FigureLateral Surface

Area

Total Surface

Area

Volume

Cube 4a2

6a2

a3

Cuboid 2h(l + b) 2(lb + bh + lh) lbh


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Cylinder 2prh 2pr(r + h) pr2h

Cone prl pr(l + r)pr2h

Sphere 4pr2pr3

Hemisphere 2pr2 3pr2pr3

Right prism

(i) Equilateral

triangular

prism

(ii) Square

prism

(iii) Hexagonal

Prism

3ah

4ah

6ah

3ah +

2a(2h + a)

a2h

Frustum of a

cone

pl(R + l)

l= p(R

2 + r2

+ Rl+

rl)

ph

(R2 + Rr+ r2)

Frustum of a

Pyramid

Perimeter

of

L.S.A + A1 + A2


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base Slant

height

Torus 4p2ra 2p2r2a

HIGHER MATHS I

(PERMUTATIONS & COMBINATIONS, PROBABILITY)

1. n (A B) = n (A) + n (B) n (A 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 inpq ways

3. The number of ways of dividing (p + q) items into two groups containingp and q items respectively is

4. The number of ways of dividing 2p items into two equal groups ofp each is , when the two groups

have distinct

identity and , when the two groups do not have distinct identity

5. nCr=nCn r

6. The total number of ways in which a selection can be made by taking some or all out of (p + q + r+ .....)

items wherep are alike of one kind, q alike of a second kind, ralike of a third kind and so on is {(p + 1)

(q + 1) (r+ 1) ....} -1

7. P(Event) = and 0 P(Event) 1


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

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

10. Expected Value = [Probability (Ei)] [Monetary value associated with event Ei]

HIGHER MATHS II(STATISTICS, NUMBER SYSTEMS, INEQUALITIES & MODULUS, SPECIAL EQUATIONS)

1. G.M. = (x1 x2 ...... .xn)1/n

2.

3. For any two positive numbers a, b

(i) A.M. G.M. 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. Ifa > b, , for any two positive numbers a and b

7. |x +y| |x| + |y|, for any two real numbersx andy

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 =


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HIGHER MATHS III

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

1. If a point P(x,y) divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m : n, then x =

andy

= , positive sign for internal division and negative sign for external division

2. The area of a triangle with the vertices at (0, 0), (x1,y1) and (x2,y2) is D =

3. The coordinates of the centroid C(x,y) of a triangle ABC formed by joining the points A(x1,y1); B(x2,y2)

and C(x3,y

3) are

given by

4. The slope of line with points (x1,y1) and (x2,y2) lying on it is m =

5. Ifm1 and m2 are the slopes of two lines L1 and L2 respectively, then the angle q between them is given

by

tanq =

6. The equation of thex-axis isy = 0 and that ofy-axis isx = 0

7. The equation of a line parallel to x-axis is of the formy = b and that of a line parallel toy-axis is of the

formx = a (a and b are some constants)8. Point slope form of a line:y y1 = m (x x1)

9. Two point form of a line:

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

11. Intercept form of a line :


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12. Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are

(i) parallel if orm1= m2

(ii) perpendicular ifa1a2 + b1b2 = 0 orm1m2 = -1

13. The distance between two parallel lines of the form ax + by +c1 = 0 and ax + by + c2 = 0 is given by

14. Ifax + by + c = 0 is the equation of a line, then the perpendicular distance of a point (x1,y1) 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

Errata: Formula for Area of a Parallelogram has been corrected on 28/10/2011

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Important Formulae_for Website_for CAT2012 Students