List of Maths Formulas for CAT

8/3/2019 List of Maths Formulas for CAT

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(1) If an equation (i:e f(x)=0 ) contains all positive co-efficients of any powers of x , it has nopositive roots then.eg: x^4+3x^2+2x+6=0 has no positive roots .

(2) For an equation , if all the even powers of x have some sign coefficients and all the odd

powers of x have the opposite sign coefficients , then it has no negative roots .(3)Summarising DESCARTES RULE OF SIGNS:

For an equation f(x)=0 , the maximum number of positive roots it can have is the number of signchanges in f(x) ; and the maximum number of negative roots it can have is the number of signchanges in f(-x) .Hence the remaining are the minimum number of imaginary roots of the equation(Since we alsoknow that the index of the maximum power of x is the number of roots of an equation.)

(4) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to

be 2-3i and if there are three possible roots of the equation , we can conclude that the last root isreal . This real roots could be found out by finding the sum of the roots of the equation andsubtracting (2+3i)+(2-3i)=4 from that sum. (More about finding sum and products of roots nexttime )

07/10/2002 THEORY OF EQUATIONS

(1) For a cubic equation ax^3+bx^2+cx+d=o

sum of the roots = b/asum of the product of the roots taken two at a time = c/aproduct of the roots = -d/a

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0

sum of the roots = b/asum of the product of the roots taken three at a time = c/asum of the product of the roots taken two at a time = -d/aproduct of the roots = e/a

(3) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficientsor if f(x) = 0 has only odd powers of x and all these have the same signcoefficients then the equation has no real roots in each case(except for x=0 in the second case.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root ,then even 2-root(3) is a root .(All these are very useful in finding number of positive , negative , real ,complex etc roots of anequation )


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Todays Section:-08/10/2002-

(1) If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM ifx=y(=k/2). The maximum product is then (k^2)/4 .

(2) If for two numbers x*y=k(=constant), then their SUM is MINIMUM ifx=y(=root(k)). The minimum sum is then 2*root(k) .

(3) |x| + |y| >= |x+y| (|| stands for absolute value or modulus )(Useful in solving some inequations)

(4) Product of any two numbers = Product of their HCF and LCM .Hence product of two numbers = LCM of the numbers if they are prime to each other .

1) For any regular polygon , the sum of the exterior angles is equal to 360 degreeshence measure of any external angle is equal to 360/n. ( where n is the number of sides)

(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.

(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique siesequal).

(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum ofthe other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .

(5) Area of a regular hexagon : root(3)*3/2*(side)*(side)

1) For any 2 numbers a>b

a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasarespectively)

(2) (GM)^2 = AM * HM

(3) For three positive numbers a, b ,c

(a+b+c) * (1/a+1/b+1/c)>=9

(4) For any positive integer n

2


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(5) a^2+b^2+c^2 >= ab+bc+caIf a=b=c , then the equality holds in the above.

(6) a^4+b^4+c^4+d^4 >=4abcd

(7) (n!)^2 > n^n (! for factorial)This is for 21/10/2002

(1) If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximumif a/p = b/q = c/r = d/s .

(2)Consider the two equations

a1x+b1y=c1a2x+b2y=c2

Then ,If a1/a2 = b1/b2 = c1/c2 , then we have infinite solutions for these equations.If a1/a2 = b1/b2 c1/c2 , then we have no solution for these equations.( means not equal to )If a1/a2 b1/b2 , then we have a unique solutions for these equations..

(3) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.

(4) Problems on clocks can be tackled as assuming two runners going round a circle , one 12times as fast as the other . That is ,the minute hand describes 6 degrees /minutethe hour hand describes 1/2 degrees /minute .

Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .

(5) The hour and the minute hand meet each other after every 65(5/11) minutes after beingtogether at midnight.(This can be derived from the above) .

1)If n is even , n(n+1)(n+2) is divisible by 24(2)If n is any integer , n^2 + 4 is not divisible by 4

(3)Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meetingpoint of the diagonals can be found out by solving for[(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]


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(4)Area of a triangle1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c))where s=a+b+c/2=a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius ofthe triangle .

(5)In any trianglea=b*CosC + c*CosBb=c*CosA + a*CosCc=a*CosB + b*CosA

(6)If a1/b1 = a2/b2 = a3/b3 = .. , then each ratio is equal to(k1*a1+ k2*a2+k3*a3+..) / (k1*b1+ k2*b2+k3*b3+..) , which is also equalto(a1+a2+a3+./b1+b2+b3+.)

(7)In any trianglea/SinA = b/SinB =c/SinC=2R , where R is the circumradius

cosC = (a^2 + b^2 c^2)/2ab

sin2A = 2 sinA * cosAcos2A = cos^2(A) sin^2 (A)

1) x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .+ a^(n-1) ) Very useful for finding multiples.For example (17-14=3 will be a multiple of 17^3 14^3)

(2) e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ..to infinity(2a) 2 < e < 3

(3) log(1+x) = x (x^2)/2 + (x^3)/3 (x^4)/4 to infinity [ Note the alternating sign ..Also note that the ogarithm is with respect to base e ]

(4) In a GP the product of any two terms equidistant from a term is always constant .

(5) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

(6) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the

internal opposite angle.(7) (m+n)! is divisible by m! * n! .

_________________I have miles to go before I sleep


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-02/11/2002-

(1) If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal to the

sum of the other pair .(2)The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratioof the GP .

(3)The equation whose roots are the reciprocal of the roots of the equationax^2+bx+c is cx^2+bx+a

(4) The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)is((a+c+e)/3 , (b+d+f)/3) .

(5) The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .(6) Area of a parallelogram = base * height

(7)APPOLLONIUS THEOREM:

In a triangle , if AD be the median to the side BC , thenAB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .

_________________

1) for similar cones , ratio of radii = ratio of their bases.(2) The HCF and LCM of two nos. are equal when they are equal .

(3) Volume of a pyramid = 1/3 * base area * height

(4) In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisectorfrom vertex to base bisects the base.

(5) In any triangle the angular bisector of an angle bisects the base in the ratio of theother two sides.

(6) the quadrilateral formed by joining the angular bisectors of another quadrilateral isalways a rectangle.

(7) Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1

( |a|+|b| = |a+b| if a*b>=0else |a|+|b| >= |a+b|


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(9) 2


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(20) Let x be certain base in which the representation of a number is abcd , then the decimalvalue of this number is a*x^3 + b*x^2 + c*x + d

5) For a cyclic quadrilateral , area = root( s* (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

Here are some neat shortcuts on Simple/Compound Interest.Shortcut #1:-We all know the traditional formula to compute interestCI = P*(1+R/100)^N P

The calculation get very tedious when N>2 (more than 2 years). The method suggested below iselegant way to get CI/Amount after N years.

You need to recall the good ol Pascals Triange in following way:

Code:Number of Years (N)-1 12 1 2 13 1 3 3 14 1 4 6 4 1. 1 . . 1

Example: P = 1000, R=10 %, and N=3 years. What is CI & Amount?

Step 1: 10% of 1000 = 100, Again 10% of 100 = 10 and 10% of 10 = 1We did this three times bcos N=3.

Step 2:Now Amount after 3 years = 1 * 1000 + 3 * 100 + 3 * 10 + 1 * 1 = Rs.1331/-The coefficents 1,3,3,1 are lifted from the pascals triangle above.

Step 3:CI after 3 years = 3*100 + 3*10 + 3*1 = Rs.331/- (leaving out first term in step 2)

If N =2, we would have had, Amt = 1 * 1000 + 2 * 100 + 1 * 10 = Rs. 1210/-CI = 2 * 100 + 1* 10 = Rs. 210/-

This method is extendable for any N and it avoids calculations involving higher powers on Naltogether!

A variant to this short cut can be applied to find depreciating value of some property. (Example,A property worth 100,000 depreciates by 10% every year, find its value after N years).


8/19

Shortcut #2:-(i) When interest is calculated as CI, the number of years for the Amount to double (two timesthe principal) can be found with this following formula:P * N ~ 72 (approximately equal to).

Exampe, if R=6% p.a. then it takes roughly 12 years for the Principal to double itself.Note: This is just a approximate formula (when R takes large values, the error % in formulaincreases).

(ii) When interest is calculated as SI, number of years for amt to double can be found as:N * R = 100 . BTW this formula is exact!

Adding to what Peebs said, this shortcut does work for any P/N/R.

Basically if you look closely at this method, what I had posted is actually derived from the

Binomial expansion of the polynomial (1+r/100)^n but in a more edible format digestableby us!

BTW herez one shortcut on recurring decimals to fractions Its more easier to explain with anexample..

Eg: 2.384384384 .

Step 1: since the 3 digits 384 is recurring part, multiply 2.384 by 1000 = so we get 2384.

Next 2 is the non recurring part in the recurring decimal so subtract 2 from 2384 = 2382.

If it had been 2.3848484.., we would have had 2384 23 = 2361. Had it been 2.384444.. NRwould be 2384 238 = 2146 and so on.

We now find denominator part .

Step 3: In step 1 we multiplied 2.384384 by 1000 to get 2384, so put that first.

Step 4: next since all digits of the decimal part of recurring decimal is recurring, subtract 1 fromstep 3. Had the recurring decimal been 2.3848484, we need to subtract 10. If it had been2.3844444, we needed to have subtracted 100 ..and so on

Hence here, DR = 1000 1 = 999

Hence fraction of the Recurring decimal is 2382/999!!

Some more examples .

1.56787878 = (15678 156) / (10000 100) = 15522/9900


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23.67898989 = (236789 2367) / (10000 100) = 234422/9900

124.454545 = (12445 124) / (100 1) = 12321/99

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10/19

This is a special notes for all CAT and MBA aspirants.. by IIM Topper.

QUANT THEORY

1) nPr= n!/(n-r)!

2) nPn = n!

3) nCr= n!/(n-r)!r!

4) nCn = 1

5) nP0 = 1

6) nC0 = 1

7) AP An = a + (n-1)dSn = n/2[2a + (n-1)d]

GP An = ar(n-1)

Sn = a(rn 1 )/ (r-1)

S = a/(1-r)

9) 1 mile = 1760 yards

10) 1 yard = 3 feet

11) 1 mile2 = 640 acres

12) I gallon = 4 quarts

13) 1 quart = 2 pints

14) 1 pint = 2 cups

15) 1 cup = 8 ounces

16) 1 pound = 16 ounces

17) 1 ounce = 16 drams

18) 1 kg = 2.2 pounds


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19) 30-60-90 triangle 1:3:2 sides

20) 45-45-90 triangle 1:1:2 sides

21) a3>b3 a>b

22) If A than B => not B than not A

23) Zero divided by any nonzero integer is zero.

24) Division by 0 is undefined.

25)


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26) The standard deviation is a statistic that tells you how tightly all the various examples areclustered around the mean in a set of data. When the examples are pretty tightly bunchedtogether and the bell-shaped curve is steep, the standard deviation is small. When the examplesare spread apart and the bell curve is relatively flat, that tells you have a relatively large standard

deviation.27) n(A U B U C) = n(A) + n (B)+ n(C) n(A n B) n(A n C) n(B n C) + n(A n B n C)

28) n(Aonly) = n(A) n(A n C) n(A n B) + n(A U B U C)

29) Dividend = Divisor * Quotient + Remainder


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30) LCM * HCF = Product of 2 numbers.

31) 1 + 2 + 3 ..n = n * n+1 / 2

32) Sum of squares of 1st n natural numbers = n (n+1)(2n+1) / 6

33) Sum of cubes of 1st n natural numbers = [n (n+1)/2]2

34)

Squares and Cubes

Number ( x ) Square ( x 2 ) Cube ( x 3 )1 1 12 4 83 9 27

4 16 645 25 1256 36 2167 49 -8 64 -9 81 -10 100 -11 121 -12 144 -13 169 -14 196 -

15 225 -16 256 -17 28918 32419 36121 44122 48423 52924 57625 625

35)Fractions and Percentage:

Fraction Decimal Percentage1 / 2 0.5 501 / 3 0.33 33 1/32 / 3 0.66 66 2/3


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1 / 4 0.25 253 / 4 0.75 751 / 5 0.2 202 / 5 0.4 403 / 5 0.6 60

4 / 50.

8 801 / 6 0.166 16 2/35 / 6 0.833 83 2 / 31 / 8 0.125 12 1 / 23 / 8 0.375 37 1 / 25 / 8 0.625 62 1 / 27 / 8 0.875 87 1 / 21 / 9 0.111 112 / 9 0.222 221 / 10 0.1 101 / 20 0.05 5

1 / 100 0.01 1

36) Average speed = Total distance / Total Time

When equal distances are covered in different speed then we take the harmonic mean

Av Speed = 2ab / a + b

Different distances in same time we take AM

Av Speed is = a + b / 2

37) Simple Interest: SI = PRT / 100, A = P + SI

38) 1 Nickel = 5 cents

1 dime = 10 cents

1 quarter = 25 cents

1 half = 50 cents

1 dollar = 100 cents

39) Equilateral triangle, Area = (3 * a2)/4

40) Area of trapezium = (Height * Sum of parallel sides)

41) Arc Length = (/ 360) 2 r


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42) Area of sector = (/ 360) r2

43) Equal chords are equidistant from the center.

44) (x+y) 8 = 8C8x8 + 8C7x7y + 8C6x6y2 + 8C5x5y3 + + 8C2x2y6 + 8C1xy7 + 8C0y8

45) Sometimes we get so involved with the nitty-gritties of mathematics that we start functioninglike automatons and stop thinking. Dont fall prey to this trap. For example, what is theprobability that a number amongst the first 1000 positive integers is divisible by 8? Dont startcounting the multiples of 8! The figure of 1000 is a red herring. Use a little common sense. Thenumbers will be 8,16,24,32So, 1 in every 8 numbers is a multiple of 8, even if you considerthe first million integers. So Probability is 1/8

46) The number of integers from A to B inclusive is = B -A +1

47) Average of consecutive numbers:

Eg from 13 to 77 = (13+77)/2

48) Slope = (change in y)/(change in x)

49) 00 = undefined

50)

51) Sum of interior angles of a polygon with n sides = (n-2)*180

52) Degree measure of one angle in a regular polygon with n sides

= {(n-2)*180 }/n

53) When multiplying ordividing both sides of an inequality by a negative number, theinequality sign reverses.

x < y => -(-x) > y => x > -y

54) Fraction > (fraction)2 for all positive fractions

55) Fraction > (fraction ) for all positive fractions


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56) If n is a positive integer, (n6)/2 = (n12 / 4)

57) If z1, z2, z3 zn are consecutive positive integers and their average is an odd integer => nis odd => sum of series is odd

58) In a triangle with sides of measure a, b and c SHAPE \* MERGEFORMAT , a-b a + b = odd59) Before confirming try and back solve and make sure that u have answered what has beenasked.

60) When the question mentions prime number, remember to think of 2 too.

61) In a triangle, if the sum of two angles = third angle, then it is a right angled triangle.

62) Do not transport information from another statement unless considering both collectively.

63) A-b = odd => a + b = odd64)

65)


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66) If a DS question simply asks whether a, b, c and d are consecutive integers; use your brain. Ithas just asked u to answer if they are consecutive, not if they are consecutive in order.

67) Measure of an angle of a cyclic polygon = 180 360/n , where n is the number of sides ofthe polygon.

68) Sometimes, mistakes might also be committed by simply misreading the statement. Eg

Both Tim and Harry received an acre of land more than Neel => t = n + 1, h = n + 1

Tim and Harry received an acre more than Neel => t + h = n+1

69)

Let Triangle ABC be equilateral with each side of measure a and AC ^ BD

AB = BD = AD = a

a = b = c = 600

AC = (a2 + a/2 2)

= 3*a/2

Area = 3 * a2/4

Perimeter = 3a

Radius of circle O = a/3 = AC * 2/3

Radius of circle O = 3a/6 = AC * 1/3


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70) Two

circles will touch each or intersect each other if the distance between their centers d is such that

R r d R + r, where R and r are the radii of the two circles

71) Remainder of less than two means not just one; it also means remainder of zero.

72) Do not make unwarranted assumptions. 12 midnight to 12 noon does not mention what days,and hence you cannot find out the time period.

73) Standard deviation of a set is always negative and equals zero only if all elements of the setare equal.

74) If the difference between the largest and the smallest divisor of a number is X, the number isX + 1

75) Always remember the special watch out cases in DS questions. If the question mentionsmean of a set, the mean can be ZERO also.

76) If area of a rectangle is known, diagonal is known, perimeter can be found

a2 + b2 = diagonal2

a2 + b2 + 2ab = diagonal2+ 2ab

(a + b)2 = diagonal2+ 2*area

77) (y2) = |y| => y if y is positive, -y if y is negative

78) angle = mod [(60H - 11M) /2 ]

H = value of hour handM = value of minute hand

eg, if time is 2:30, then H =2 and M =30


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79) Every number raised to power 5 has the number itself as unit digit

80) If a + b + c = Z, than the largest of a, b, and c cannot be greater than the mean of the othertwo.

81) The rule that one side of a triangle cannot be > sum of other two, only applies to sides, notangles

82) FINALLY, MAKE SURE OF WHAT THE QUESTION SAYS INTEGER MEANSINTEGRAL LENGTH. And, DIVIDING A WIRE INTO PIECES, DOES NOTNECESSARILY IMPLY THAT THEY WILL BE INTEGRAL LENGTHS. Similarly, that aboat covers a distance upstream in 3 hours, states only the time, even if it has been mentionedthat it covers a a distance 12 km downstream in 2 hours.

83) x2 = 9*y2 does not necessarily imply that x2 > y2. (Hint : consider x=y=0)

84) When we say multiples between 16 and 260, and inclusive/exclusive is not mentioned, take16 and 260 to be exclusive.

85) The statement implies :

The hourly wage for each employee ranges from $5 an hour to $20 an hour.

minimum average = (20 + 5 + 5 + 5 + 5)/5

maximum average = (5 + 20 + 20 + 20 + 20)/5

Just mug up these notes and you will be able to crack any MBA exam likeCAT,XAT,XLRI,FMS and GMAT.

List of Maths Formulas for CAT

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