GEOMETRY - TRIANGLES
1. Area of a triangle with base b and height h = (1/2)*b*h
2. The area of an equilateral triangle with side a is [sqrt(3)/4]*a^2
3. The area of any triangle given the length of its 3 sides a, b and c:is sqrt[s(s-a)(s-b)(s-c)] where s= (a+b+c)/2
SOME USEFUL FACTS
Number Theory
The product of any three consecutive integers is divisible by 6.
Similarly, the product of any four consecutive integers is divisible by 24.
Permutation and Combination
When n dice (n > 1) are rolled simultaneously, the number of outcomes in which all n dice show the same number is 6, irrespective of the value of n.
Similarly, when n fair coins (n > 1) are tossed simultaneously, the number of outcomes in which all n coins turn up as heads or as tails is 1, irrespective of the value of n.
Speed Time
When an object travels the first x hours at p km/hr and the next x hours at q km/hr, the average speed of travel is the arithmetic mean of p and q.
However, when the object travels the first x kms at p km/hr and the next x kms at q km/hr, the average speed is the harmonic mean of p and q.
Number Theory
Any perfect square has an odd number of factors including 1 and the number itself and a composite number has an even number of factors including 1 and itself.
Any perfect square can be expressed in the form 4n or 4n+1.
Profit Loss
If the selling price of 2 articles are equal and 1 of them is sold at a profit of p% and the other at a loss of p%, then the 2 trades will result in a cumulative loss of ((p^2)/100)%.
If the cost of price of 2 articles are equal and 1 of them is sold at a profit of p% and the other at a loss of p%, then the 2 trades will result in no profit or loss.
Progressions
Arithmetic mean of 'n' numbers will always be greater than the geometric mean of those 'n' numbers which will be greater than the Harmonic mean of those 'n' numbers.
Arithmetic mean of 2 numbers = geometric mean of '2' numbers = harmonic mean of '2' numbers if both the numbers are equal.
TIME AND WORK
If A can do a piece of work in x days, then A’s one day’s work=1/x
• If the ratio of time taken by A and B in doing a work is x:y, then, ratio of work done is 1/x :1/y=y:x. And the ratio in which the wages is to be distributed is y:x
• If A can do a work in x days and B can do the same work in y days, then A and B can together do the work in (xy)/(x+y) days
• If “a” men or “b” women can do a piece of work in x days, then “m” men and “n” women can together finish the work in (abx)/(an+bm) days
• If A is x times efficient than B, and working together, they finish the work in y days, then Time taken by A=y(x+1)/(x), Time taken by B=y(x+1)
• If A and B can finish a work in “x” and “ax” days respectively, that is if A is “a” times efficient than B, then working together, they can finish the work in (ax)/(a+1) days
• If A and B working together can complete a work in x days, whereas B working alone can do the same work in y days, ten, A alone will complete the work in (xy)/(y-x) days.
• Pipe A can fill a tank in x hrs and B can empty a tank in y hrs.If both pipes are opened together, the tank will be filled in (xy)/(y-x) hrs
• A pipe can fill a cistern in x hrs but due to leakage in the bottom, it is filled in y hrs, then the time taken by the leak to empty the cistern is (xy)/(y-x) hrs
GEOMETRY STUFF
Prisms
Volume = Base area X Height
Surface = 2b + Ph (b is the area of the base P is the perimeter of the base)
Cylinder
Volume = r*2 h
Surface = 2rh
Pyramid
V = 1/3 bh
b is the area of the base
Surface Area: Add the area of the base to the sum of the areas of all of the triangular faces. The areas of the triangular faces will have different formulas for different shaped bases.
Cones
Volume = 1/3 r*2 x height = 1/3 r2h
Surface = r2 + rs = r2 + r
Sphere
Volume = 4/3 r^3
Surface area = 4r^2
Distance of a Point from a Line
The perpendicular distance d of a point P (x 1, y 1) from the line ax +by +c = 0 is given by:
d =| ax1 +by1 +c|/[ (a² +b²)]
Simple And Compound Interest
1. Simple Interest = PNR/100
where, P --> Principal amount
N --> time in years
R --> rate of interest for one year
2. Compound interest (abbreviated C.I.) = A -P =
where A is the final amount, P is the principal, r is the rate of interest compounded yearly and n is the number of years
3. When the interest rates for the successive fixed periods are r1 %, r2 %, r3 %, ..., then the final amount A is given by A =
4. S.I. (simple interest) and C.I. are equal for the first year (or the first term of the interest period) on the same sum and at the same rate.
5. C.I. of 2nd year (or the second term of the interest period) is more than the C.I. of Ist year (or the first term of the interest period), and C.I. of 2nd year -C.I. of Ist year = S.I. on the interest of the first year.
STATISTICS
1. Mean.
(i) Mean (for ungrouped data) = where x1, x2, x3, ..., xn are the observations and n is the total no. of observations.
(ii) Mean (for grouped data) = , where x1, x2, x3, ..., xn are different variates with frequencies f1, f2, f3, ..., fn respectively.
(iii) Mean for continuous distribution.
Let there be n continuous classes, yi be the class mark and fi be the frequency of the ith class, then
mean = (Direct method)
Let A be the assumed mean, then
mean = A + , where di = yi -A (Short cut method)
If the classes are of equal size, say c, then
mean = A +c x , where ui = (Step deviation method)
ANALYTICAL GEOMETRY
LINES - BASICS:
1. The equation of X axis: y =0
2. The equation of Y axis: x = 0
3. Equation of straight line parallel to X axis: y =a, where a is any constant
4. Equation of straight line parallel to Y axis: x =a, where a is any constant
5. Equation of a straight line through a given point (x1, y1) and having a given slope m: y -y1 = m (x - x1)
6. Equation of a straight line through a given point (0, 0) and having a given slope m: y = m x
7. Equation of a straight line with a slope m and y-intercept c is: y = mx + c
8. Equation of a straight line passing through two points (x1, y1) and (x2, y2) is:
(y -y1)/(y2 - y1) = (x -x1)/(x2 -x1)
9. Equation of a straight line whose x and y intercepts are a and b is:
x/a + y/b = 1
10. The length of the perpendicular drawn from origin (0,0) to the line Ax + By + C =0 is :
C/ sqrt(A^2 + B^2)
11. Length of the perpendicular from (x1, y1) to the line Ax + By + C =0 is:
Ax1 + By1 +C / sqrt(A^2 + B^2)
12. The point of intersection of two lines a1x + b1y +c1 = 0 and a2x + b2y +c2 = 0 is :
([b1*c2 - b2*c1]/[a1*b2 - a2*b1], [c1*a2 - c2*a1]/[a1*b2 - a2*b1])
13. The condition for concurrency of three lines a1x + b1y +c1 = 0, a2x + b2y +c2 = 0 and a3x + b3y +c3 = 0 is (in determinant form)
| a1 b1 c1 |
| a2 b2 c2 | = 0
| a3 b3 c3 |
14. The angle between two lines y = m1x + c1 and y = m2x + c2 is tan inverse of the modulus of :
[(m1 - m2)/(1 + m1*m2)]
15. Condition for parallelism of two lines with slopes m1 and m2 is m1 = m2
16. Condition for perpendicularity of two lines with slopes m1 and m2 is m1*m2=-1
CIRCLES:
17. General equation of a circle with centre (x1, y1) and radius r is:
(x - x1)^2 + (y - y1)^2 = r^2
18. The equation of a circle whose diameter is the line joining the points (x1, y1) and (x2, y2) is :
(x - x1)(x - x2) + (y - y1)(y - y2) = 0
19. The equation of the tangent to the circle x^2 + y^2 = a^2 (where a is the radius of the circle) at the point (x1, y1) on it is :
x*x1 + y*y1 + a^2
20. The condition for y = mx + c to be a tangent to the circle x^2 + y^2 = a^2 is :
c^2 = a^2 (1 + m^2)
Multiplication of 2digit by 2 digit number
ab
x cd
------
pqrs
1. first multiply bd - write down the unit fig at s carry over the tens fig.
2. Multiply axd & bxc add them together and also add the carry over from step 1 write down the units fig at r and carry over the tens fig.
3. Multiply axc and add the carry over from step 2. write down at pq.
TRIGONOMETRY
For angle A = 0, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2):
sin A = (sqrt0)/2, (sqrt1)/2, (sqrt2)/2, (sqrt3)/2, (sqrt4)/2
cos A = (sqrt4)/2, (sqrt3)/2, (sqrt2)/2, (sqrt1)/2, (sqrt0)/2
tan A = 0, (sqrt3)/3, 1, sqrt3, undefined
In any triangle:
sine = (opposite side) / hypotenuse
cosine = (adjacent side) / hypotenuse
tan = (opposite side)/(adjacent side) = (sine/cosine)
Probability - 'The Rules'
1. If two events are mutually exclusive (i.e. they cannot occur at the same time), then the probability of them both occurring at the same time is 0. then: P(A and B) = 0 and P(A or B) = P(A) + P(B)
2. if two events are not-mutually exclusive (i.e. there is some overlap) then: P(A or B) = P(A) + P(B) - P(A and B)
3. If events are independent (i.e. the occurrence of one does not change the probability of the other occurring), then the probability of them both occurring is the product of the probabilities of each occurring. Then: P(A and B) = P(A) * P(B)
4. If A, B and C are not mutually exclusive events, then P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(B and C) - P(C and A) + P(A and B and C)
and = intersection
or = union
sorry for making the formulae clusmsy using this 'and' and 'or'.
STATISTICS – HARMONIC MEAN
The harmonic mean of x1,...,xn is
n / (1/x1 + ... + 1/xn)
As the name implies, it's a mean (between the smallest and largest values). An example of the use of the harmonic mean: Suppose we're driving a car from Amherst (A) to Boston (B) at a constant speed of 60 miles per hour. On the way back from B to A, we drive a constant speed of 30 miles per hour (damn Turnpike). What is the average speed for the round trip?
We would be inclined to use the arithmetic mean; (60+30)/2 = 45 miles per hour. However, this is incorrect, since we have driven for a longer time on the return leg. Let's assume the distance between A and B is n miles. The first leg will take us n/60 hours, and the return leg will take us n/30 hours. Thus, the total round trip will take us (n/60) + (n/30) hours to cover a distance of 2n miles. The average speed (distance per time) is thus:
2n / {(n/60) + (n/30)} = 2 / (1/20) = 40 miles per hour.
The reason that the harmonic mean is the correct average here is that the numerators of the original ratios to be averaged were equal (i.e. n miles at 60 miles/hour versus n miles at 30 miles/hour). In cases where the denominators of two ratios are averaged, we can use the arithmetic mean.
SOME USEFUL SHORTCUTS
* Product of 2 numbers is the produst of their LCM & HCF.
* LCM of a fraction = LCM of numerator/HCF 0f denominator.
*HCF of a fraction = HCF of numer./LCM of denom.
Ratio & Proportion:
* if a/b = c/d = e/f = .....
then, a/b = c/d = e/f =(a+c+e+...)/(b+d+f+...)
* If a/b = c/d,
Then,
i) b/a = d/c
ii) a/c = b/d
iii) (a+b)/ b = (c+d)/d
iv) (a-b)/b = (c-d)/d
v) (a+b)/(a-b) = (c+d)/(c-d)
FORMULAE ON INTEREST
Money in Compound Interest gets doubled in 70/r years (approximately)
ie. P(1+r/100)^N = 2P when N=70/r
DIVISIBILITY RULES
Divisibility by:
2 If the last digit is even, the number is divisible by 2.
3 If the sum of the digits is divisible by 3, the number is also.
4 If the last two digits form a number divisible by 4, the number is also.
5 If the last digit is a 5 or a 0, the number is divisible by 5.
6 If the number is divisible by both 3 and 2, it is also divisible by 6.
7 Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.
8 If the last three digits form a number divisible by 8, then so is the whole number.
9 If the sum of the digits is divisible by 9, the number is also.
10 If the number ends in 0, it is divisible by 10.
11 Alternately add and subtract the digits from left to right. If the result (including 0) is divisible by 11, the number is also.
Example: to see whether 365167484 is divisible by 11, start by subtracting:
3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.
12 If the number is divisible by both 3 and 4, it is also divisible by 12.
13 Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13,then so is the original number.
Friday, December 21, 2007
GRE USEFUL FORMULAE........
GRE USEFUL FORMULAE.......
EXPLAINATION OF THE ABOVE TOPIC (FOR NON-MATHEMATICAL BACKGROUND STUDENTS
Say that x1, x2, x3, x4, x5, ...., xn are n draws from a (random) sample. Then:
Step 1: Compute the mean, i.e. m =[ Sum xi (i=1,..., n) ] / n
Step 2: Compute the squared deviation of each observation from its mean, i.e.
For x1 --------> (x1-m)^2
For x2---------> (x2-m)^2
.....
For xn---------> (xn-m)^2
Step 3: The variance is V= [(x1-m)^2 + (x2-m)^2 + .... + (xn-m)^2 ] / n
Step 4: The s.d. is s.d. = V^(1/2)
Example: Let x1=10, x2= 20 and x3=30
Then:
(1) m=20
(3) V = [ (10-20)^2 + 0 + (30-20)^2] / 3 = 200/3
(4) s.d. = (20/3) ^ (1/2)
SOME USEFUL INFORMATION ABOUT GEOMETRY
- If perimeters of a square and parallelogram are equal, then area of a square is always greater than area of a parallelogram.
- Similarly, if perimeters of a square and circle are same, then area of a circle is greater than area of a square.
NUMBER FACTS
Addition/Subtraction Property for Inequalities
If a <>
If a <>
Multiplication/Division Properties for Inequalities
· when multiplying/dividing by a positive value
If a <>c is positive, then ac <>
If a <>c is positive, then a/c <>
· when multiplying/dividing by a negative value
If a <>c is negative, then ac > bc
If a <>c is negative, then a/c > b/c
Natural (or Counting) Numbers : N = {1, 2, 3, 4, 5, ...}
Whole Numbers : {0, 1, 2, 3, 4, 5, ...}
Integers : Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}
Real Numbers : R = {x | x corresponds to point on the number line}
If a cube of sides n*n*n is painted and then divided into 1*1*1 size cubes then number cubes with NO face painted is given by (n-2)^2
If SD of x1, x2, x3, ... xn is sigma then SD of x1+k , x2+k, x3+k ... Xn+k is also sigma
If SD of x1, x2, x3, ... xn is sigma then SD of x1*k , x2*k, x3*k ... Xn*k is k*sigma
Variance (kx) = k^2 Variance(x)
Binomial probability mass function: P(x) = nCx * p^x * q ^ (n-x)
where x is happening event, n is total number of event, p is probability of happening of event and q is probability of not happening.
A remainder rule to remember:
If a product of 2 integers, x and y is divided by an integer n, then the remainder that you get will be the product of the remainders when x is divided by n and y is divided by n.
R[] ---> remainder function
R[(1046*1047*1048)/11] = R[1046/11]*R[1047/11]*[1048/11] = 1*2*3 = 6
Note: Sometimes the product of the remainders will be greater than the original divisor. In this case you'll have to repeat the process.
Pick's theorem
Pick's theorem provides an elegant formula for the area of a simple lattice polygon: a lattice polygon whose boundary consists of a sequence of connected nonintersecting straight-line segments.
The formula is Area = I +B/2 – 1where I = number of interior lattice points () and
B = number of boundary lattice points ()
For example, the area of the simple lattice polygon in the figure is
31 + 15 /2 – 1 = 37.5.
The interior and boundary lattice points of the fourteen pieces of the Stomachion are indicated on the second figure. Using Pick's theorem the areas of the fourteen pieces can be determined as in the above example; e.g., the blue piece in the upper right-hand corner has area
18 + 14 /2 – 1 = 24
MORE ON PERCENTAGES
ONE VARIABLE INCREASED/ DECREASED PROBLEMS:
PRICE INCREASED AND REDUCTION OF THE CONSUMPTION:
1. Price of sugar is increased 25%. How much percent must a house hold must reduce his consumption of sugar so as not to increase his expenditure?
how much time u require to this problem? just try this short cut less then 5 sec u will get the answer
% REDUCTION= (INCREASE/100+INCREASE)* 100
lets try this with short cut
increase = 25% so reduction = (25/ 100+25 ) * 100
= (25/125) * 100
= 1/5* 100
= 20 %
so house hold have to decrease 20% of their consuption to keep constant .
PRICE DECREASED INCREASE IN CONSUPTION:
PROBLEM : 2
Certain familyhave fixed budget for ice cream purchase for year . but,Ice cream price decreased by 20% due to winter season. find by how much % a consumer must increase his consumpion of ice creame so as not to decrease his expenditure.
Here the short cut
%INCREASE IN CONSUMPTION = (REDUCTION/ 100- REDUCTION) *100
just as mentioned above
decrease icecreame price= 20
= 20/(100-20) *100
=20/80 * 100
=1/4 *100
= 25%
BOTH VARIABLES INCREASED/ DECREASED PROBLEMS
TYPE3:
Petrol tax is increased by 20% and the costumer comsumption also increased by 20%.Find the % increase or decrease in the expenditure
OR
Water tax is increased by 20% and consumption also increased by 20% find what is the net effect in change?
the short cut: [(A+B) + AB]/100
increase A: 20
increase B: 20
= (20+20) + (20*20)/100
= 40+ 400/100
= 40 +4
= 44 % net increase
ONE INCREASED ONTHER DECREASED PROBLEMS:
Shop keeper decreased the price of a article by 20% and then increased the artical by 30% what is the net effect of the artical is it increased or decreased?
SHORT CUT IS SAME AS ABOVE
so first decreased the price so we have to take as negative value for A
decrease A : -20%
increase A : 30%
= (-20 + 20)+ (-20)*20/100
= (0) +(-400)/100
= -4%
so net effect is 4% loss to the shop keeper.
Quantitative Ability – POINTS TO REMEMBER
1. If an equation (i.e. f(x) = 0) contains all positive co-efficients of any powers of x, it has no positive roots.
Eg: x3+3x2+2x+6=0 has no positive roots
2. For an equation, if all the even powers of x have same sign coefficients and all the odd powers of x have the opposite sign coefficients, then it has no negative roots.
3. For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x)
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 is real. This real root could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum.
5.
ü For a cubic equation ax3+bx2+cx+d=o
· Sum of the roots = - b/a
· Sum of the product of the roots taken two at a time = c/a
· Product of the roots = -d/a
ü For a bi-quadratic equation ax4+bx3+cx2+dx+e = 0
· Sum of the roots = - b/a
· Sum of the product of the roots taken three at a time = c/a
· Sum of the product of the roots taken two at a time = -d/a
· Product of the roots = e/a
6. If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign coefficients, then the equation has no real roots in each case (except for x=0 in the second case)
7. Consider the two equations
a1x+b1y=c1
a2x+b2y=c2
Then,
ü If a1/a2 = b1/b2 = c1/c2, then we have infinite solutions for these equations.
ü If a1/a2 = b1/b2 <> c1/c2, then we have no solution.
ü If a1/a2 <> b1/b2, then we have a unique solution.
8. Roots of x2 + x + 1=0 are 1, w, w2 where 1 + w + w2=0 and w3=1
9. |a| + |b| = |a + b| if a*b>=0
else, |a| + |b| >= |a + b|
10. The equation ax2+bx+c=0 will have max. value when a<0>0. The max. or min. value is given by (4ac-b2)/4a and will occur at x = -b/2a
11.
ü If for two numbers x + y=k (a constant), then their PRODUCT is MAXIMUM if x=y (=k/2). The maximum product is then (k2)/4.
ü If for two numbers x*y=k (a constant), then their SUM is MINIMUM if
x=y (=root(k)). The minimum sum is then 2*root (k).
12. 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.
13. For any 2 numbers a, b where a>b
ü a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic means respectively)
ü (GM)^2 = AM * HM
14. For three positive numbers a, b, c
ü (a + b + c) * (1/a + 1/b + 1/c)>=9
15. For any positive integer n
ü 2<= (1 + 1/n)^n <=3
16. a2 + b2 + c2 >= ab + bc + ca
If a=b=c, then the case of equality holds good.
17. a4 + before + c4 + d4 >= 4abcd (Equality arises when a=b=c=d=1)
18. (n!)2 > nn
19. If a + b + c + d=constant, then the product a^p * b^q * c^r * d^s will be maximum if a/p = b/q = c/r = d/s
20. If n is even, n(n+1)(n+2) is divisible by 24
21. 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)
22. e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity
Note: 2 <>
23. log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [Note the alternating sign . .Also note that the logarithm is with respect to base e]
24. (m + n)! is divisible by m! * n!
25. When a three digit number is reversed and the difference of these two numbers is taken, the middle number is always 9 and the sum of the other two numbers is always 9.
26. Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y)
27.
ü The sum of first n natural numbers = n(n+1)/2
ü The sum of squares of first n natural numbers is n(n+1)(2n+1)/6
ü The sum of cubes of first n natural numbers is (n(n+1)/2)2/4
ü The sum of first n even numbers= n (n+1)
ü The sum of first n odd numbers= n2
28. If a number ‘N’ is represented as a^x * b^y * c^z… where {a, b, c, …} are prime numbers, then
ü the total number of factors is (x+1)(y+1)(z+1) ....
ü the total number of relatively prime numbers less than the number is
N * (1-1/a) * (1-1/b) * (1-1/c)....
ü the sum of relatively prime numbers less than the number is
N/2 * N * (1-1/a) * (1-1/b) * (1-1/c)....
ü the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x * y *...)
29.
ü Total no. of prime numbers between 1 and 50 is 15
ü Total no. of prime numbers between 51 and 100 is 10
ü Total no. of prime numbers between 101 and 200 is 21
30.
ü The number of squares in n*m board is given by m*(m+1)*(3n-m+1)/6
ü The number of rectangles in n*m board is given by n+1C2 * m+1C2
31. If ‘r’ is a rational no. lying between 0 and 1, then, r^r can never be rational.
32. Certain nos. to be remembered
ü 210 = 45 = 322 = 1024
ü 38 = 94 = 812 = 6561
ü 7 * 11 * 13 = 1001
ü 11 * 13 * 17 = 2431
ü 13 * 17 * 19 = 4199
ü 19 * 21 * 23 = 9177
ü 19 * 23 * 29 = 12673
33. Where the digits of a no. are added and the resultant figure is 1 or 4 or 7 or 9, then, the no. could be a perfect square.
34. If a no. ‘N’ has got k factors and a^l is one of the factors such that l>=k/2, then, a is the only prime factor for that no.
35. To find out the sum of 3-digit nos. formed with a set of given digits
This is given by (sum of digits) * (no. of digits-1)! * 1111…1 (i.e. based on the no. of digits)
Eg) Find the sum of all 3-digit nos. formed using the digits 2, 3, 5, 7 & 8.
Sum = (2+3+5+7+8) * (5-1)! * 11111 (since 5 digits are there)
= 25 * 24 * 11111
=6666600
36. Consider the equation x^n + y^n = z^n
As per Fermat’s Last Theorem, the above equation will not have any solution whenever n>=3.
37. Further as per Fermat, where ‘p’ is a prime no. and ‘N’ is co-prime to p, then,
N^(p-1) – 1 is always divisible by p.
38. 145 is the 3-digit no. expressed as sum of factorials of the individual digits i.e.
145 = 1! + 4! + 5!
39.
ü Where a no. is of the form a^n – b^n, then,
· The no. is always divisible by a - b
· Further, the no. is divisible by a + b when n is even and not divisible by
a + b when n is odd
ü Where a no. is of the form a^n + b^n, then,
· The no. is usually not divisible by a - b
· However, the no. is divisible by a + b when n is odd and not divisible by
a + b when n is even
40. The relationship between base 10 and base ‘e’ in log is given by
log10N = 0.434 logeN
41. WINE and WATER formula
Let Q - volume of a vessel, q - qty of a mixture of water and wine be removed each time from a mixture, n - number of times this operation is done and A - final qty of wine in the mixture, then,
A/Q = (1-q / Q)^n
42. Pascal’s Triangle for computing Compound Interest (CI)
The traditional formula for computing CI is
CI = P*(1+R/100)^N – P
Using Pascal’s Triangle,
Number of Years (N)
-------------------
1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
… 1 .... .... ... ... ..1
Eg: P = 1000, R=10 %, and N=3 years. What is CI & Amount?
Step 1:
Amount after 3 years = 1 * 1000 + 3 * 100 + 3 * 10 + 1 * 1 = Rs.1331
The coefficients - 1,3,3,1 are lifted from the Pascal's triangle above.
Step 2:
CI after 3 years = 3*100 + 3*10 + 3*1 = Rs.331 (leaving out first term in step 1)
If N =2, we would have had,
Amt = 1 * 1000 + 2 * 100 + 1 * 10 = Rs.1210
CI = 2 * 100 + 1* 10 = Rs.210
43. Suppose the price of a product is first increased by X% and then decreased by Y% , then, the final change % in the price is given by:
Final Difference% = X - Y - XY/100
Eg) The price of a T.V set is increased by 40 % of the cost price and then is decreased by 25% of the new price. On selling, the profit made by the dealer was Rs.1000. At what price was the T.V sold?
Applying the formula,
Final difference% = 40 – 25 - (40*25/100) = 5 %.
So if 5 % = 1,000
Then, 100 % = 20,000.
Hence, C.P = 20,000
& S.P = 20,000+ 1000= 21,000
44. Where the cost price of 2 articles is same and the mark up % is same, then, marked price and NOT cost price should be assumed as 100.
45.
ü Where ‘P’ represents principal and ‘R’ represents the rate of interest, then, the difference between 2 years’ simple interest and compound interest is given by P * (R/100)2
ü The difference between 3 years’ simple interest and compound interest is given by (P * R2 *(300+R))/1003
46.
ü If A can finish a work in X time and B can finish the same work in Y time then both of them together can finish that work in (X*Y)/ (X+Y) time.
ü If A can finish a work in X time and A & B together can finish the same work in S time then B can finish that work in (XS)/(X-S) time.
ü If A can finish a work in X time and B in Y time and C in Z time then all of them working together will finish the work in (XYZ)/ (XY +YZ +XZ) time
ü If A can finish a work in X time and B in Y time and A, B & C together in S time then
· C can finish that work alone in (XYS)/ (XY-SX-SY)
· B+C can finish in (SX)/(X-S); and
· A+C can finish in (SY)/(Y-S)
47. In case ‘n’ faced die is thrown k times, then, probability of getting atleast one more than the previous throw = nC5/n5
48.
ü When an unbiased coin is tossed odd no. (n) of times, then, the no. of heads can never be equal to the no. of tails i.e. P (no. of heads=no. of tails) = 0
ü When an unbiased coin is tossed even no. (2n) of times, then,
P (no. of heads=no. of tails) = 1-(2nCn/22n)
49. Where there are ‘n’ items and ‘m’ out of such items should follow a pattern, then, the probability is given by 1/m!
Eg)1. Suppose there are 10 girls dancing one after the other. What is the probability of A dancing before B dancing before C?
Here n=10, m=3 (i.e. A, B, C)
Hence, P (A>B>C) = 1/3!
= 1/6
Eg)2. Consider the word ‘METHODS’. What is the probability that the letter ‘M’ comes before ‘S’ when all the letters of the given word are used for forming words, with or without meaning?
P (M>S) = 1/2!
= 1/2
50. CALENDAR
ü Calendar repeats after every 400 years.
ü Leap year- it is always divisible by 4, but century years are not leap years unless they are divisible by 400.
ü Century has 5 odd days and leap century has 6 odd days.
ü In a normal year 1st January and 2nd July and 1st October fall on the same day. In a leap year 1st January 1st July and 30th September fall on the same day.
ü January 1, 1901 was a Tuesday.
51.
ü For any regular polygon, the sum of the exterior angles is equal to 360 degrees, hence measure of any external angle is equal to 360/n (where n is the number of sides)
ü For any regular polygon, the sum of interior angles =(n-2)*180 degrees
So measure of one angle is (n-2)/n *180
ü If any parallelogram can be inscribed in a circle, it must be a rectangle.
ü If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i.e. oblique sides equal).
52. For an isosceles trapezium, sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides (i.e. AB+CD = AD+BC, taken in order)
53.
ü For any quadrilateral whose diagonals intersect at right angles, the area of the quadrilateral is
0.5*d1*d2, where d1, d2 are the length of the diagonals.
ü For a cyclic quadrilateral, area = root((s-a) * (s-b) * (s-c) * (s-d)), where
s=(a + b + c + d)/2
Further, for a cyclic quadrilateral, the measure of an external angle is equal to the measure of the interior opposite angle.
ü Area of a Rhombus = Product of Diagonals/2
54. Given the coordinates (a, b); (c, d); (e, f); (g, h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for
[(a + e)/2, (b + f)/2] = [(c + g)/2, (d + h)/2]
55. Area of a triangle
ü 1/2*base*altitude
ü 1/2*a*b*sin C (or) 1/2*b*c*sin A (or) 1/2*c*a*sin B
ü root(s*(s-a)*(s-b)*(s-c)) where s=(a+b+c)/2
ü a*b*c/(4*R) where R is the circumradius of the triangle
ü r*s ,where r is the inradius of the triangle
56. In any triangle
ü a=b*cos C + c*cos B
ü b=c*cos A + a*cos C
ü c=a*cos B + b*cos A
ü a/sin A=b/sin B=c/sin C=2R, where R is the circumradius
ü cos C = (a^2 + b^2 - c^2)/2ab
ü sin 2A = 2 sin A * cos A
ü cos 2A = cos^2 (A) - sin^2 (A)
57. The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1
58. Appollonius Theorem
In a triangle ABC, if AD is the median to side BC, then
AB2 + AC2 = 2(AD2 + BD2) or 2(AD2 + DC2)
59.
ü In an isosceles triangle, the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
ü In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
60. The quadrilateral formed by joining the angular bisectors of another quadrilateral is always a rectangle.
61. Let W be any point inside a rectangle ABCD, then,
WD2 + WB2 = WC2 + WA2
62. Let a be the side of an equilateral triangle, then, if three circles are drawn inside this triangle such that they touch each other, then each circle’s radius is given by a/(2*(root(3)+1))
63.
ü Distance between a point (x1, y1) and a line represented by the equation
ax + by + c=0 is given by |ax1+by1+c|/Sq(a2+b2)
ü Distance between 2 points (x1, y1) and (x2, y2) is given by
Sq((x1-x2)2+ (y1-y2)2)
64. Where a rectangle is inscribed in an isosceles right angled triangle, then, the length of the rectangle is twice its breadth and the ratio of area of rectangle to area of triangle is 1:2.
If there are three sets A, B, and C, then
P(AuBuC) = P(A) + P(B) + P(C) – P(AnB) – P(AnC) – P(BnC) + P(AnBnC)
Number of people in exactly one set =
P(A) + P(B) + P(C) – 2P(AnB) – 2P(AnC) – 2P(BnC) + 3P(AnBnC)
Number of people in exactly two of the sets =
P(AnB) + P(AnC) + P(BnC) – 3P(AnBnC)
Number of people in exactly three of the sets =
P(AnBnC)
Number of people in two or more sets =
P(AnB) + P(AnC) + P(BnC) – 2P(AnBnC)
GRE CHALLENGING PROBLEMS...
Questions
1
_ Set 1
Start off with these simple series of numbers. Number
series questions measure your ability to reason without
words. To answer these questions, you must determine
the pattern of the numbers in each series before you will
be able to choose which number comes next. These
questions involve only simple arithmetic. Although
most number series items progress by adding or subtracting,
some questions involve simple multiplication
or division. In each series, look for the degree and
direction of change between the numbers. In other
words, do the numbers increase or decrease, and by
how much?
1. Look at this series: 2, 4, 6, 8, 10, . . . What number
should come next?
a. 11
b. 12
c. 13
d. 14
2. Look at this series: 58, 52, 46, 40, 34, . . . What
number should come next?
a. 26
b. 28
c. 30
d. 32
3. Look at this series: 40, 40, 47, 47, 54, . . . What
number should come next?
a. 40
b. 44
c. 54
d. 61
4. Look at this series: 544, 509, 474, 439, . . . What
number should come next?
a. 404
b. 414
c. 420
d. 445
5. Look at this series: 201, 202, 204, 207, . . . What
number should come next?
a. 205
b. 208
c. 210
d. 211
6. Look at this series: 8, 22, 8, 28, 8, . . . What
number should come next?
a. 9
b. 29
c. 32
d. 34
7. Look at this series: 80, 10, 70, 15, 60, . . . What
number should come next?
a. 20
b. 25
c. 30
d. 50
8. Look at this series: 36, 34, 30, 28, 24, . . . What
number should come next?
a. 20
b. 22
c. 23
d. 26
9. Look at this series: 22, 21, 23, 22, 24, 23, . . .
What number should come next?
a. 22
b. 24
c. 25
d. 26
–QUESTIONS–
2
10. Look at this series: 3, 4, 7, 8, 11, 12, . . . What
number should come next?
a. 7
b. 10
c. 14
d. 15
11. Look at this series: 31, 29, 24, 22, 17, . . . What
number should come next?
a. 15
b. 14
c. 13
d. 12
12. Look at this series: 21, 9, 21, 11, 21, 13, . . .
What number should come next?
a. 14
b. 15
c. 21
d. 23
13. Look at this series: 53, 53, 40, 40, 27, 27, . . .
What number should come next?
a. 12
b. 14
c. 27
d. 53
14. Look at this series: 2, 6, 18, 54, . . . What number
should come next?
a. 108
b. 148
c. 162
d. 216
15. Look at this series: 1,000, 200, 40, . . . What
number should come next?
a. 8
b. 10
c. 15
d. 20
16. Look at this series: 7, 10, 8, 11, 9, 12, . . . What
number should come next?
a. 7
b. 10
c. 12
d. 13
17. Look at this series: 14, 28, 20, 40, 32, 64, . . .
What number should come next?
a. 52
b. 56
c. 96
d. 128
18. Look at this series: 1.5, 2.3, 3.1, 3.9, . . . What
number should come next?
a. 4.2
b. 4.4
c. 4.7
d. 5.1
19. Look at this series: 5.2, 4.8, 4.4, 4, . . . What
number should come next?
a. 3
b. 3.3
c. 3.5
d. 3.6
20. Look at this series: 2, 1, _
1
2_, _
1
4_, . . . What number
should come next?
a. _
13
_
b. _
18
_
c. _
28
_
d. _1
1
6 _
–QUESTIONS–
3
_ Set 2 (Answers begin on page 101.)
This set contains additional, and sometimes more
difficult, number series questions. Again, each question
has a definite pattern. Some of the number series
may be interrupted by a particular number that
appears periodically in the pattern. For example, in
the series 14, 16, 32, 18, 20, 32, 22, 24, 32, the number
32 appears as every third number. Sometimes, the
pattern contains two alternating series. For example,
in the series 1, 5, 3, 7, 5, 9, 7, the pattern is add 4, subtract
2, add 4, subtract 2, and so on. Look carefully for
the pattern, and then choose which pair of numbers
comes next. Note also that you will be choosing from
five options instead of four.
21. 84 78 72 66 60 54 48
a. 44 34
b. 42 36
c. 42 32
d. 40 34
e. 38 32
22. 3 8 13 18 23 28 33
a. 39 44
b. 38 44
c. 38 43
d. 37 42
e. 33 38
23. 20 20 17 17 14 14 11
a. 8 8
b. 11 11
c. 11 14
d. 8 9
e. 11 8
24. 18 21 25 18 29 33 18
a. 43 18
b. 41 44
c. 37 18
d. 37 41
e. 38 41
25. 9 11 33 13 15 33 17
a. 19 33
b. 33 35
c. 33 19
d. 15 33
e. 19 21
26. 2 8 14 20 26 32 38
a. 2 46
b. 44 50
c. 42 48
d. 40 42
e. 32 26
27. 28 25 5 21 18 5 14
a. 11 5
b. 10 7
c. 11 8
d. 5 10
e. 10 5
28. 9 12 11 14 13 16 15
a. 14 13
b. 18 21
c. 14 17
d. 12 13
e. 18 17
29. 75 65 85 55 45 85 35
a. 25 15
b. 25 85
c. 35 25
d. 85 35
e. 25 75
–QUESTIONS–
4
30. 1 10 7 20 13 30 19
a. 26 40
b. 29 36
c. 40 25
d. 25 31
e. 40 50
31. 10 20 25 35 40 50 55
a. 70 65
b. 60 70
c. 60 75
d. 60 65
e. 65 70
32. 40 40 31 31 22 22 13
a. 13 4
b. 13 5
c. 4 13
d. 9 4
e. 4 4
33. 17 17 34 20 20 31 23
a. 26 23
b. 34 20
c. 23 33
d. 27 28
e. 23 28
34. 2 3 4 5 6 4 8
a. 9 10
b. 4 8
c. 10 4
d. 9 4
e. 8 9
35. 61 57 50 61 43 36 61
a. 29 61
b. 27 20
c. 31 61
d. 22 15
e. 29 22
36. 9 16 23 30 37 44 51
a. 59 66
b. 56 62
c. 58 66
d. 58 65
e. 54 61
37. 8 22 12 16 22 20 24
a. 28 32
b. 28 22
c. 22 28
d. 32 36
e. 22 26
38. 6 20 8 14 10 8 12
a. 14 10
b. 2 18
c. 4 12
d. 2 14
e. 14 14
39. 11 16 21 26 31 36 41
a. 47 52
b. 46 52
c. 45 49
d. 46 51
e. 46 52
40. 8 11 21 15 18 21 22
a. 25 18
b. 25 21
c. 25 29
d. 24 21
e. 22 26
ANSWERS:
1. b. This is a simple addition series. Each number
increases by 2.
2. b. This is a simple subtraction series. Each
number is 6 less than the previous number.
3. c. This is an alternation with repetition series
in which each number repeats itself and
then increases by 7.
4. a. This is a simple subtraction series. Each
number is 35 less than the previous number.
5. d. In this addition series, 1 is added to the first
number; 2 is added to the second number; 3
is added to the third number; and so forth.
6. d. This is a simple addition series with a random
number, 8, interpolated as every other
number. In the series, 6 is added to each
number except 8, to arrive at the next
number.
7. a. This is an alternating addition and subtraction
series. In the first pattern, 10 is subtracted
from each number to arrive at the
next. In the second, 5 is added to each number
to arrive at the next.
8. b. This is an alternating number subtraction
series. First, 2 is subtracted, then 4, then 2,
and so on.
9. c. In this simple alternating subtraction and
addition series; 1 is subtracted, then 2 is
added, and so on.
10. d. This alternating addition series begins with
3; then 1 is added to give 4; then 3 is added
to give 7; then 1 is added, and so on.
11. a. This is a simple alternating subtraction
series, which subtracts 2, then 5.
12. c. In this alternating repetition series, the random
number 21 is interpolated every other
number into an otherwise simple addition
series that increases by 2, beginning with
the number 9.
13. b. In this series, each number is repeated, then
13 issubtracted to arrive at the next number.
14. c. This is a simple multiplication series. Each
number is 3 times more than the previous
number.
15. a. This is a simple division series. Each number
is divided by 5.
99
16. b. This is a simple alternating addition and
subtraction series. In the first pattern, 3 is
added; in the second, 2 is subtracted.
17. b. This is an alternating multiplication and
subtracting series: First, multiply by 2 and
then subtract 8.
18. c. In this simple addition series, each number
increases by 0.8.
19. d. In this simple subtraction series, each number
decreases by 0.4.
20. b. This is a simple division series; each number
is one-half of the previous number.
–ANSWERS–
100
_ Set 2 (Page 4)
21. b. In this simple subtraction series, each number
is 6 less than the previous number.
22. c. In this simple addition series, each number
is 5 greater than the previous number.
23. e. This is a simple subtraction with repetition
series. It begins with 20, which is repeated,
then 3 is subtracted, resulting in 17, which
is repeated, and so on.
24. d. This is a simple addition series with a random
number, 18, interpolated as every third
number. In the series, 4 is added to each
number except 18, to arrive at the next
number.
25. a. In this alternating repetition series, a random
number, 33, is interpolated every third
number into a simple addition series, in
which each number increases by 2.
26. b. This is a simple addition series, which
begins with 2 and adds 6.
27. a. This is an alternating subtraction series with
the interpolation of a random number, 5, as
every third number. In the subtraction series,
3 is subtracted, then 4, then 3, and so on.
28. e. This is a simple alternating addition and
subtraction series. First, 3 is added, then 1 is
subtracted, then 3 is added, 1 subtracted,
and so on.
29. b. This is a simple subtraction series in which a
random number, 85, is interpolated as every
third number. In the subtraction series, 10 is
subtracted from each number to arrive at the
next.
30. c. Here, every other number follows a different
pattern. In the first series, 6 is added to each
number to arrive at the next. In the second
series, 10 is added to each number to arrive at
the next.
31. e. This is an alternating addition series, in
which 10 is added, then 5, then 10, and
so on.
32. a. This is a subtraction series with repetition.
Each number repeats itself and then
decreases by 9.
33. e. This is an alternating subtraction series with
repetition. There are two different patterns
here. In the first, a number repeats itself;
then 3 is added to that number to arrive at
the next number, which also repeats. This
gives the series 17, 17, 20, 20, 23, and so on.
Every third number follows a second pattern,
in which 3 is subtracted from each
number to arrive at the next: 34, 31, 28.
34. d. This is an alternating addition series with a
random number, 4, interpolated as every
third number. In the main series, 1 is added,
then 2 is added, then 1, then 2, and so on.
35. e. This is an alternating repetition series, in
which a random number, 61, is interpolated
as every third number into an otherwise
simple subtraction series. Starting with the
second number, 57, each number (except
61) is 7 less than the previous number.
36. d. Here is a simple addition series, which
begins with 9 and adds 7.
37. c. This is an alternating repetition series, with a
random number, 22, interpolated as every
third number into an otherwise simple addition
series. In the addition series, 4 is added
to each number to arrive at the next number.
38. d. This is an alternating addition and subtraction
series. In the first pattern, 2 is added to
each number to arrive at the next; in the
alternate pattern, 6 is subtracted from each
number to arrive at the next.
39. d. In this simple addition series, each number
is 5 more than the previous number.
40. b. This is an alternating addition series, with a
random number, 21, interpolated as every
third number. The addition series alternates
between adding 3 and adding 4. The number
21 appears after each number arrived at
by adding 3.
GRE CHALLENGING PROBLEMS....
_ Set 3
This set will give you additional practice dealing with
number series questions.
41. 44 41 38 35 32 29 26
a. 24 21
b. 22 19
c. 23 19
d. 29 32
e. 23 20
42. 6 10 14 18 22 26 30
a. 36 40
b. 33 37
c. 38 42
d. 34 36
e. 34 38
43. 34 30 26 22 18 14 10
a. 8 6
b. 6 4
c. 14 18
d. 6 2
e. 4 0
44. 2 44 4 41 6 38 8
a. 10 12
b. 35 32
c. 34 9
d. 35 10
e. 10 52
45. 32 29 26 23 20 17 14
a. 11 8
b. 12 8
c. 11 7
d. 32 29
e. 10 9
46. 14 14 26 26 38 38 50
a. 60 72
b. 50 62
c. 50 72
d. 62 62
e. 62 80
47. 8 12 9 13 10 14 11
a. 14 11
b. 15 12
c. 8 15
d. 15 19
e. 8 5
48. 4 7 26 10 13 20 16
a. 14 4
b. 14 17
c. 18 14
d. 19 13
e. 19 14
49. 3 8 10 15 17 22 24
a. 26 28
b. 29 34
c. 29 31
d. 26 31
e. 26 32
50. 17 14 14 11 11 8 8
a. 8 5
b. 5 2
c. 8 2
d. 5 5
e. 5 8
51. 13 29 15 26 17 23 19
a. 21 23
b. 20 21
c. 20 17
d. 25 27
e. 22 20
–QUESTIONS–
6
52. 16 26 56 36 46 68 56
a. 80 66
b. 64 82
c. 66 80
d. 78 68
e. 66 82
53. 7 9 66 12 14 66 17
a. 19 66
b. 66 19
c. 19 22
d. 20 66
e. 66 20
54. 3 5 35 10 12 35 17
a. 22 35
b. 35 19
c. 19 35
d. 19 24
e. 22 24
55. 36 31 29 24 22 17 15
a. 13 11
b. 10 5
c. 13 8
d. 12 7
e. 10 8
56. 42 40 38 35 33 31 28
a. 25 22
b. 26 23
c. 26 24
d. 25 23
e. 26 22
57. 11 14 14 17 17 20 20
a. 23 23
b. 23 26
c. 21 24
d. 24 24
e. 24 27
58. 17 32 19 29 21 26 23
a. 25 25
b. 20 22
c. 23 25
d. 25 22
e. 27 32
59. 10 34 12 31 14 28 16
a. 25 18
b. 30 13
c. 19 26
d. 18 20
e. 25 22
60. 32 31 32 29 32 27 32
a. 25 32
b. 31 32
c. 29 32
d. 25 30
e. 29 30
–QUESTIONS–
7
_ Set 4
This set contains additional number series questions,
some of which are in Roman numerals. These items differ
from Sets 1, 2, and 3 because they ask you to find the
number that fits somewhere into the middle of the
series. Some of the items involve both numbers and letters;
for these questions, look for a number series and
a letter series. (For additional practice in working letter
series questions, see Set 5.)
61. Look at this series: 8, 43, 11, 41, __, 39, 17, . . .
What number should fill in the blank?
a. 8
b. 14
c. 43
d. 44
62. Look at this series: 15, __, 27, 27, 39, 39, . . .
What number should fill the blank?
a. 51
b. 39
c. 23
d. 15
63. Look at this series: 83, 73, 93, 63, __, 93, 43, . . .
What number should fill the blank?
a. 33
b. 53
c. 73
d. 93
64. Look at this series: 4, 7, 25, 10, __, 20, 16, 19, . . .
What number should fill the blank?
a. 13
b. 15
c. 20
d. 28
65. Look at this series: 72, 76, 73, 77, 74, __, 75, . . .
What number should fill the blank?
a. 70
b. 71
c. 75
d. 78
66. Look at this series: 70, 71, 76, __, 81, 86, 70, 91, . . .
What number should fill the blank?
a. 70
b. 71
c. 80
d. 96
67. Look at this series: 664, 332, 340, 170, __, 89, . . .
What number should fill the blank?
a. 85
b. 97
c. 109
d. 178
68. Look at this series: 0.15, 0.3, __, 1.2, 2.4, . . .
What number should fill the blank?
a. 4.8
b. 0.006
c. 0.6
d. 0.9
69. Look at this series: _
1
9_, _
1
3_, 1, __, 9, . . . What number
should fill the blank?
a. _
23
_
b. 3
c. 6
d. 27
70. Look at this series: U32, V29, __, X23, Y20, . . .
What number should fill the blank?
a. W26
b. W17
c. Z17
d. Z26
–QUESTIONS–
8
71. Look at this series: J14, L16, __, P20, R22, . . .
What number should fill the blank?
a. S24
b. N18
c. M18
d. T24
72. Look at this series: F2, __, D8, C16, B32, . . .
What number should fill the blank?
a. A16
b. G4
c. E4
d. E3
73. Look at this series: V, VIII, XI, XIV, __, XX, . . .
What number should fill the blank?
a. IX
b. XXIII
c. XV
d. XVII
74. Look at this series: XXIV, XX, __, XII, VIII, . . .
What number should fill the blank?
a. XXII
b. XIII
c. XVI
d. IV
75. Look at this series: VI, 10, V, 11, __, 12, III, . . .
What number should fill the blank?
a. II
b. IV
c. IX
d. 14
ANSWERS:
41. e. This is a simple subtraction series, in which
3 is subtracted from each number to arrive
at the next.
42. e. This simple addition series adds 4 to each
number to arrive at the next.
43. d. This is a simple subtraction series, in which
4 is subtracted from each number to arrive
at the next.
44. d. Here, there are two alternating patterns, one
addition and one subtraction. The first
starts with 2 and increases by 2; the second
starts with 44 and decreases by 3.
45. a. In this simple subtraction series, the numbers
decrease by 3.
46. b. In this simple addition with repetition
series, each number in the series repeats
itself, and then increases by 12 to arrive at
the next number.
47. b. This is an alternating addition and subtraction
series, in which the addition of 4 is
alternated with the subtraction of 3.
48. e. Two patterns alternate here, with every
third number following the alternate pattern.
In the main series, beginning with 4, 3
is added to each number to arrive at the
next. In the alternating series, beginning
with 26, 6 is subtracted from each number
to arrive at the next.
49. c. This is an alternating addition series that
adds 5, then 2, then 5, and so on.
50. d. In this simple subtraction with repetition
series, each number is repeated, then 3 is
subtracted to give the next number, which is
then repeated, and so on.
51. b. Here, there are two alternating patterns,
with every other number following a different
pattern. The first pattern begins with 13
and adds 2 to each number to arrive at the
next; the alternating pattern begins with 29
and subtracts 3 each time.
52. c. Here, every third number follows a different
pattern from the main series. In the main
series, beginning with 16, 10 is added to
each number to arrive at the next. In the
alternating series, beginning with 56, 12 is
added to each number to arrive at the next.
53. a. This is an alternating addition series with
repetition, in which a random number, 66,
is interpolated as every third number. The
regular series adds 2, then 3, then 2, and so
on, with 66 repeated after each “add 2” step.
54. c. This is an alternating addition series, with a
random number, 35, interpolated as every
third number. The pattern of addition is to
add 2, add 5, add 2, and so on. The number
35 comes after each “add 2” step.
55. e. This is an alternating subtraction series,
which subtracts 5, then 2, then 5, and so on.
56. c. This is an alternating subtraction series in
which 2 is subtracted twice, then 3 is subtracted
once, then 2 is subtracted twice, and
so on.
57. a. This is a simple addition series with repetition.
It adds 3 to each number to arrive at
the next, which is repeated before 3 is added
again.
58. c. Here, there are two alternating patterns.
The first begins with 17 and adds 2; the second
begins with 32 and subtracts 3.
59. a. Two patterns alternate here. The first pattern
begins with 10 and adds 2 to each
number to arrive at the next; the alternating
pattern begins with 34 and subtracts 3 each
time.
60. a. This is an alternating repetition series. The
number 32 alternates with a series in which
each number decreases by 2.
–ANSWERS–
102
_ Set 4 (Page 8)
61. b. This is a simple alternating addition and
subtraction series. The first series begins
with 8 and adds 3; the second begins with
43 and subtracts 2.
62. d. In this simple addition with repetition
series, each number in the series repeats
itself, and then increases by 12 to arrive at
the next number.
63. b. This is a simple subtraction series in which
a random number, 93, is interpolated as
every third number. In the subtraction
series, 10 is subtracted from each number to
arrive at the next.
64. a. Two series alternate here, with every third
number following a different pattern. In the
main series, 3 is added to each number to
arrive at the next. In the alternating series, 5
is subtracted from each number to arrive at
the next.
65. d. This series alternates the addition of 4 with
the subtraction of 3.
66. a. In this series, 5 is added to the previous
number; the number 70 is inserted as every
third number.
67. d. This is an alternating division and addition
series: First, divide by 2, and then add 8.
68. c. This is a simple multiplication series. Each
number is 2 times greater than the previous
number.
69. b. This is a multiplication series; each number
is 3 times the previous number.
70. a. In this series, the letters progress by 1; the
numbers decrease by 3.
71. b. In this series, the letters progress by 2, and
the numbers increase by 2.
72. c. The letters decrease by 1; the numbers are
multiplied by 2.
73. d. This is a simple addition series; each number
is 3 more than the previous number.
74. c. This is a simple subtraction series; each
number is 4 less than the previous number.
75. b. This is an alternating addition and subtraction
series. Roman numbers alternate with
Arabic numbers. In the Roman numeral
pattern, each number decreases by 1. In the
Arabic numeral pattern, each number
increases by 1.
CHALLENGING GRE PROBLEMS......
Set 5 (Answers begin on page 104.)
Another type of sequence question involves a series of
letters in a pattern.Usually, these questions use the letters’
alphabetical order as a base. To make matters more
complicated, sometimes subscript numbers will be
thrown into the letter sequence. In these series, you will
be looking at both the letter pattern and the number
pattern. Some of these questions ask you to fill the
blank in the middle of the series; others ask you to add
to the end of the series.
76. QPO NML KJI _____ EDC
a. HGF
b. CAB
c. JKL
d. GHI
77. JAK KBL LCM MDN _____
a. OEP
b. NEO
c. MEN
d. PFQ
78. B2CD _____ BCD4 B5CD BC6D
a. B2C2D
b. BC3D
c. B2C3D
d. BCD7
79. ELFA GLHA ILJA _____ MLNA
a. OLPA
b. KLMA
c. LLMA
d. KLLA
80. P5QR P4QS P3QT _____ PQV
a. PQW
b. PQV2
c. P2QU
d. PQ3U
81. CMM EOO GQQ _____ KUU
a. GRR
b. GSS
c. ISS
d. ITT
82. QAR RAS SAT TAU _____
a. UAV
b. UAT
c. TAS
d. TAT
83. DEF DEF2 DE2F2 _____ D2E2F3
a. DEF3
b. D3EF3
c. D2E3F
d. D2E2F2
84. SCD TEF UGH ____ WKL
a. CMN
b. UJI
c. VIJ
d. IJT
85. FAG GAF HAI IAH ____
a. JAK
b. HAL
c. HAK
d. JAI
86. BCB DED FGF HIH ___
a. JKJ
b. HJH
c. IJI
d. JHJ
87. ZA5 Y4B XC6 W3D _____
a. E7V
b. V2E
c. VE5
d. VE7
–QUESTIONS–
10
_ Set 6
The next two sets contain verbal classification questions.
For these questions, the important thing (as the
name “verbal classification” indicates) is to classify the
words in the four answer choices. Three of the words
will be in the same classification; the remaining one will
not be. Your answer will be the one word that does
NOT belong in the same classification as the others.
102. Which word does NOT belong with the
others?
a. leopard
b. cougar
c. elephant
d. lion
103. Which word does NOT belong with the
others?
a. couch
b. rug
c. table
d. chair
104. Which word does NOT belong with the
others?
a. tape
b. twine
c. cord
d. yarn
105. Which word does NOT belong with the
others?
a. guitar
b. flute
c. violin
d. cello
106. Which word does NOT belong with the
others?
a. tulip
b. rose
c. bud
d. daisy
107. Which word does NOT belong with the
others?
a. tire
b. steering wheel
c. engine
d. car
108. Which word does NOT belong with the
others?
a. parsley
b. basil
c. dill
d. mayonnaise
109. Which word does NOT belong with the
others?
a. branch
b. dirt
c. leaf
d. root
110. Which word does NOT belong with the
others?
a. unimportant
b. trivial
c. insignificant
d. familiar
111. Which word does NOT belong with the
others?
a. book
b. index
c. glossary
d. chapter
–QUESTIONS–
13
112. Which word does NOT belong with the
others?
a. noun
b. preposition
c. punctuation
d. adverb
113. Which word does NOT belong with the
others?
a. cornea
b. retina
c. pupil
d. vision
114. Which word does NOT belong with the
others?
a. rye
b. sourdough
c. pumpernickel
d. loaf
115. Which word does NOT belong with the
others?
a. inch
b. ounce
c. centimeter
d. yard
116. Which word does NOT belong with the
others?
a. street
b. freeway
c. interstate
d. expressway
117. Which word does NOT belong with the
others?
a. dodge
b. flee
c. duck
d. avoid
118. Which word does NOT belong with the
others?
a. heading
b. body
c. letter
d. closing
ANSWERS
_ Set 5
76. a. This series consists of letters in a reverse
alphabetical order.
77. b. This is an alternating series in alphabetical
order. The middle letters follow the order
ABCDE. The first and third letters are
alphabetical beginning with J. The third letter
is repeated as a first letter in each subsequent
three-letter segment.
78. b. Because the letters are the same, concentrate
on the number series, which is a simple 2, 3,
4, 5, 6 series, and follows each letter in
order.
79. d. The second and forth letters in the series,
L and A, are static. The first and third letters
consist of an alphabetical order beginning
with the letter E.
80. c. The first two letters, PQ, are static. The
third letter is in alphabetical order, beginning
with R. The number series is in
descending order beginning with 5.
81. c. The first letters are in alphabetical order
with a letter skipped in between each segment:
C, E, G, I, K. The second and third
letters are repeated; they are also in order
with a skipped letter: M, O, Q, S, U.
82. a. In this series, the third letter is repeated as
the first letter of the next segment. The
middle letter, A, remains static. The third
letters are in alphabetical order, beginning
with R.
83. d. In this series, the letters remain the same:
DEF. The subscript numbers follow this
series: 1,1,1; 1,1,2; 1,2,2; 2,2,2; 2,2,3.
84. c. There are two alphabetical series here. The
first series is with the first letters only:
STUVW. The second series involves the
remaining letters: CD, EF, GH, IJ, KL.
85. a. The middle letters are static, so concentrate
on the first and third letters. The series
involves an alphabetical order with a reversal
of the letters. The first letters are in
alphabetical order: F, G, H, I, J. The second
and fourth segments are reversals of the
first and third segments. The missing segment
begins with a new letter.
86. a. This series consists of a simple alphabetical
order with the first two letters of all segments:
B, C, D, E, F, G, H, I, J, K. The third
letter of each segment is a repetition of the
first letter.
87. d. There are three series to look for here. The
first letters are alphabetical in reverse: Z, Y,
X, W, V. The second letters are in alphabetical
order, beginning with A. The number
series is as follows: 5, 4, 6, 3, 7.
–ANSWERS–
104
_ Set 6
102. c. A leopard, cougar, and lion all belong to the cat
family; an elephant does not.
103. b. The couch, table, and chair are pieces of furniture;
the rug is not.
104. a. The yarn, twine, and cord are all used for tying.
The tape is not used in the same way.
105. b. The guitar, violin, and cello are stringed instruments;
the flute is a wind instrument.
106. c. Tulip, rose, and daisy are all types of flowers. A
bud is not.
107. d. Tire, steering wheel, and engine are all parts of
a car.
108. d. Parsley, basil, and dill are types of herbs.Mayonnaise
is not an herb.
109. b. A branch, leaf, and root are all parts of a tree.
The dirt underneath is not a part of the tree.
110. d. The first three choices are all synonyms.
111. a. An index, glossary, and chapter are all parts of
a book. Choice a does not belong because the
book is the whole, not a part.
112. c. The noun, preposition, and adverb are classes
of words that make up a sentence. Punctuation
belongs in a sentence, but punctuation is
not a class of word.
113. d. The cornea, retina, and pupil are all parts of
the eye.
114. d. Rye, sourdough, and pumpernickel are types of
bread. A loaf is not a bread type.
115. b. An ounce measures weight; the other choices
measure length.
116. a. Freeway, interstate, and expressway are all highspeed
highways; a street is for low-speed traffic.
117. b. Dodge, duck, and avoid are all synonyms
meaning evade. Flee means to run away from.
118. c. Heading, body, and closing are all parts of a letter;
the letter is the whole, not a part
GRE CHALLENGING PROBLEMS...
_ Set 8 (Answers begin on page 123.)
Here’s another set of classification questions. Remember,
you are looking for the word that does NOT belong
in the same group as the others. Sometimes, all four
words seem to fit in the same group. If so, look more
closely to further narrow your classification.
119. Which word does NOT belong with the
others?
a. core
b. seeds
c. pulp
d. slice
120. Which word does NOT belong with the
others?
a. unique
b. beautiful
c. rare
d. exceptional
121. Which word does NOT belong with the
others?
a. biology
b. chemistry
c. theology
d. zoology
122. Which word does NOT belong with the
others?
a. triangle
b. circle
c. oval
d. sphere
123. Which word does NOT belong with the
others?
a. excite
b. flourish
c. prosper
d. thrive
124. Which word does NOT belong with the
others?
a. evaluate
b. assess
c. appraise
d. instruct
125. Which word does NOT belong with the
others?
a. eel
b. lobster
c. crab
d. shrimp
126. Which word does NOT belong with the
others?
a. scythe
b. knife
c. pliers
d. saw
127. Which word does NOT belong with the
others?
a. two
b. three
c. six
d. eight
128. Which word does NOT belong with the
others?
a. peninsula
b. island
c. bay
d. cape
129. Which word does NOT belong with the
others?
a. seat
b. rung
c. cushion
d. leg
–QUESTIONS–
15
130. Which word does NOT belong with the
others?
a. fair
b. just
c. equitable
d. favorable
131. Which word does NOT belong with the
others?
a. defendant
b. prosecutor
c. trial
d. judge
132. Which word does NOT belong with the
others?
a. area
b. variable
c. circumference
d. quadrilateral
133. Which word does NOT belong with the
others?
a. mayor
b. lawyer
c. governor
d. senator
134. Which word does NOT belong with the
others?
a. acute
b. right
c. obtuse
d. parallel
135. Which word does NOT belong with the
others?
a. wing
b. fin
c. beak
d. rudder
136. Which word does NOT belong with the
others?
a. aorta
b. heart
c. liver
d. stomach
–QUESTIONS–
16
_ Set 9 (Answers begin on page 108.)
In the next three sets, you will be looking for the essential
part of something. Each question has an underlined
word followed by four answer choices. You will choose
the word that is a necessary part of the underlined
word. A good way to approach this type of question is
to say the following sentence: “A ______ could not
exist without ______.” Put the underlined word in the
first blank. Try each of the answer choices in the second
blank to see which choice is most logical.
For questions 137 through 151, find the word that
names a necessary part of the underlined word.
137. book
a. fiction
b. pages
c. pictures
d. learning
138. guitar
a. band
b. teacher
c. songs
d. strings
139. shoe
a. sole
b. leather
c. laces
d. walking
140. respiration
a. mouth
b. circulation
c. oxygen
d. carbon monoxide
141. election
a. president
b. voter
c. November
d. nation
142. diploma
a. principal
b. curriculum
c. employment
d. graduation
143. swimming
a. pool
b. bathing suit
c. water
d. life jacket
144. school
a. student
b. report card
c. test
d. learning
145. language
a. tongue
b. slang
c. writing
d. words
146. desert
a. cactus
b. arid
c. oasis
d. flat
147. lightning
a. electricity
b. thunder
c. brightness
d. rain
–QUESTIONS–
17
148. monopoly
a. corrupt
b. exclusive
c. rich
d. gigantic
149. harvest
a. autumn
b. stockpile
c. tractor
d. crop
150. gala
a. celebration
b. tuxedo
c. appetizer
d. orator
151. pain
a. cut
b. burn
c. nuisance
d. hurt
–QUESTIONS–
18
_ Set 10 (Answers begin on page 109.)
Remember, you are looking for the essential part of
something. If you had trouble with Set 9, go back
through the items and study each answer explanation.
Then work through this set of more difficult necessary
part questions.
For questions 152 through 166, find the word that
names a necessary part of the underlined word.
152. infirmary
a. surgery
b. disease
c. patient
d. receptionist
153. facsimile
a. picture
b. image
c. mimeograph
d. copier
154. domicile
a. tenant
b. dwelling
c. kitchen
d. house
155. culture
a. civility
b. education
c. agriculture
d. customs
156. bonus
a. reward
b. raise
c. cash
d. employer
157. antique
a. rarity
b. artifact
c. aged
d. prehistoric
158. itinerary
a. map
b. route
c. travel
d. guidebook
159. orchestra
a. violin
b. stage
c. musician
d. soloist
160. knowledge
a. school
b. teacher
c. textbook
d. learning
161. dimension
a. compass
b. ruler
c. inch
d. measure
162. sustenance
a. nourishment
b. water
c. grains
d. menu
163. ovation
a. outburst
b. bravo
c. applause
d. encore
–QUESTIONS–
19
164. vertebrate
a. backbone
b. reptile
c. mammal
d. animal
165. provisions
a. groceries
b. supplies
c. gear
d. caterers
166. purchase
a. trade
b. money
c. bank
d. acquisition
ANSWERS:
_ Set 8
119.d. The core, seeds, and pulp are all parts of an
apple. A slice would be a piece taken out of an
apple.
120.b. Unique, rare, and exceptional are all synonyms.
Beautiful has a different meaning.
121. c. Biology, chemistry, and zoology are all branches
of science. Theology is the study of religion.
122. a. A circle, oval, and sphere are all circular shapes
with no angles. A triangle is a different kind of
shape with angles and three straight sides.
123. a. Flourish, prosper, and thrive are all synonyms;
excite does not mean the same thing.
124.d. Evaluate, assess, and appraise are all synonyms;
instruct does not mean the same thing.
125. a. The lobster, crab, and shrimp are all types of
crustaceans; an eel is a fish.
126. c. The scythe, knife, and saw are all cutting tools.
Pliers are tools, but they are not used for cutting.
127.b. Two, six, and eight are all even numbers; three
is an odd number.
128. c. A peninsula, island, and cape are all landforms;
a bay is a body of water.
129. c. Seat, rung, and leg are all parts of a chair. Not
all chairs have cushions.
130.d. Fair, just, and equitable are all synonyms meaning
impartial. Favorable means expressing
approval.
131. c. Defendant, prosecutor, and judge are all persons
involved in a trial. A trial is not a person.
132.b. Area, circumference, and quadrilateral are all
terms used in the study of geometry. Variable is
a term generally used in the study of algebra.
133.b. The mayor, governor, and senator are all persons
elected to government offices; the lawyer
is not an elected official.
134.d. Acute, right, and obtuse are geometric terms
describing particular angles. Parallel refers to
two lines that never intersect.
135. c. The wing, fin, and rudder are all parts of an
airplane.
136. a. The heart, liver, and stomach are all organs of
the body. The aorta is an artery, not an organ.
–ANSWERS–
107
_ Set 9 (Page 17)
137.b. The necessary part of a book is its pages; there
is no book without pages. Not all books are
fiction (choice a), and not all books have pictures
(choice c). Learning (choice d) may or
may not take place with a book.
138.d. A guitar does not exist without strings, so
strings are an essential part of a guitar. A band
is not necessary to a guitar (choice a). Guitar
playing can be learned without a teacher
(choice b). Songs are byproducts of a guitar
(choice c).
139. a. All shoes have a sole of some sort.Not all shoes
are made of leather (choice b); nor do they all
have laces (choice c).Walking (choice d) is not
essential to a shoe.
140. c. A person or animal must take in oxygen for
respiration to occur. A mouth (choice a) is not
essential because breathing can occur through
the nose. Choices b and d are clearly not essential
and can be ruled out.
141.b. An election does not exist without voters. The
election of a president (choice a) is a byproduct.
Not all elections are held in November (choice
c), nor are they nationwide (choice d).
142.d. A diploma is awarded at graduation, so graduation
is essential to obtaining a diploma.
Employment may be a byproduct (choice c). A
principal and a curriculum (choices a and b)
may play a role in the awarding of some diplomas,
but they are not essential.
143. c. Water is essential for swimming—without
water, there is no swimming. The other choices
are things that may or may not be present.
144. a. Without students, a school cannot exist; therefore,
students are the essential part of schools.
The other choices may be related, but they are
not essential.
145.d. Words are a necessary part of language. Slang is
not necessary to language (choice b). Not all
languages are written (choice c).Words do not
have to be spoken in order to be part of a language
(choice a).
146.b. A desert is an arid tract of land. Not all deserts
are flat (choice d). Not all deserts have cacti or
oases (choices a and c).
147. a. Lightning is produced from a discharge of electricity,
so electricity is essential. Thunder and
rain are not essential to the production of lightning
(choices b and d). Brightness may be a
byproduct of lightning, but it is not essential
(choice c).
148.b. The essential part of a monopoly is that it
involves exclusive ownership or control.
149.d. To harvest something, one must have a crop,
which is the essential element for this item.
Autumn (choice a) is not the only time crops
are harvested. There may not be enough of a
crop to stockpile (choice b), and you can harvest
crops without a tractor (choice c).
150. a. A gala indicates a celebration, the necessary
element here. A tuxedo (choice b) is not
required garb at a gala, nor is an appetizer
(choice c). A gala may be held without the benefit
of anyone speaking (choice d).
151.d. Pain is suffering or hurt, so choice d is the
essential element. Without hurt, there is no
pain. A cut (choice a) or a burn (choice b) may
cause pain, but so do many other types of
injury. A nuisance (choice c) is an annoyance
that may or may not cause pain.
–ANSWERS–
108
_ Set 10 (Page 19)
152. c. An infirmary is a place that takes care of the
infirm, sick, or injured.Without patients, there
is no infirmary. Surgery (choice a) may not be
required for patients. A disease (choice b) is
not necessary because the infirmary may only
see patients with injuries. A receptionist (choice
d) would be helpful but not essential.
153.b. A facsimile must involve an image of some sort.
The image or facsimile need not, however, be a
picture (choice a). A mimeograph and a copier
machine (choices c and d) are just a two of the
ways that images may be produced, so they do
not qualify as the essential element for this item.
154.b. A domicile is a legal residence, so dwelling is the
essential component for this item. You do not
need a tenant (choice a) in the domicile, nor do
you need a kitchen (choice c). A house (choice
d) is just one form of a domicile (which could
also be a tent, hogan, van, camper, motor
home, apartment, dormitory, etc.).
155.d. A culture is the behavior pattern of a particular
population, so customs are the essential
element. A culture may or may not be civil or
educated (choices a and b). A culture may be an
agricultural society (choice c), but this is not the
essential element.
156. a. A bonus is something given or paid beyond
what is usual or expected, so reward is the
essential element. A bonus may not involve a
raise in pay or cash (choices b and c), and it may
be received from someone other than an
employer (choice d).
157. c. An antique is something that belongs to, or
was made in, an earlier period. It may or may
not be a rarity (choice a), and it does not have
to be an artifact, an object produced or shaped
by human craft (choice b). An antique is old
but does not have to be prehistoric (choice d).
158.b. An itinerary is a proposed route of a journey. A
map (choice a) is not necessary to have a
planned route. Travel (choice c) is usually the
outcome of an itinerary, but not always. A
guidebook (choice d) may be used to plan the
journey but is not essential.
159. c. An orchestra is a large group of musicians, so
musicians are essential. Although many orchestras
have violin sections, violins aren’t essential
to an orchestra (choice a). Neither a stage
(choice b) nor a soloist (choice d) is necessary.
160.d. Knowledge is understanding gained through
experience or study, so learning is the essential
element. A school (choice a) is not necessary for
learning or knowledge to take place, nor is a
teacher or a textbook (choices b and c).
161.d. A dimension is a measure of spatial content. A
compass (choice a) and ruler (choice b) may
help determine the dimension, but other
instruments may also be used, so these are not
the essential element here. An inch (choice c) is
only one way to determine a dimension.
162. a. Sustenance is something, especially food, that
sustains life or health, so nourishment is the
essential element.Water and grains (choices b
and c) are components of nourishment, but
other things can be taken in as well. A menu
(choice d) may present a list of foods, but it is
not essential to sustenance.
163. c. An ovation is prolonged, enthusiastic applause,
so applause is necessary to an ovation. An outburst
(choice a) may take place during an ovation;
“bravo” (choice b) may or may not be
uttered; and an encore (choice d) would take
place after an ovation.
164. a. All vertebrates have a backbone. Reptiles
(choice b) are vertebrates, but so are many
other animals. Mammals (choice c) are vertebrates,
but so are birds and reptiles. All vertebrates
(choice d) are animals, but not all
animals are vertebrates.
–ANSWERS–
109
165.b. Provisions imply the general supplies needed,
so choice b is the essential element. The other
choices are byproducts, but they are not
essential.
166.d. A purchase is an acquisition of something. A
purchase may be made by trade (choice a) or
with money (choice b), so those are not essential
elements. A bank (choice c) may or may not
be involved in a purchase.
Thursday, December 20, 2007
Logout Script in Scrapbook
Well here is a small trick i used to tease few of my friends..
Copy the HTML Code below and paste it in your friends scrapbook.
Now each time some one including your friends visits that scrapbook he would automatically log out of Orkut.
Annoying nah... :p
Well here is the procedure to be safe from this trick
How to delete this scrap
if you want to avoid being logged off again when you see the scrap, you can block flash in your browser.
For firefox, download the following plugin :
https://addons.mozilla.org/en-US/firefox/addon/433
In opera, you can disable the flash plugin.
To delete the scrap with the logout script, first open your scrapbook(where the logout script is there), open any page(except the scrapbook) in a new window(don't close your scrapbook window).
When you open the new page of orkut, it'll ask you to log in. Enter your details and login to orkut. After login, go to your scrapbook page(which you had kept open), and delete the scrap which was causing the logout to occur.
see simple... :)
Hope you like this..
and leave your comments about this little trick..
