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
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]
3. (m + n)! is divisible by m! * n!
4. 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.
5. Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y)
6. To Find Square of a 3-Digit Number
Let the number be XYZ
Steps
a.Last digit = Last digit of Sq(Z)
b. Second last digit = 2*Y*Z + any carryover from STEP 1
c. Third last digit 2*X*Z+ Sq(Y) + any carryover from STEP 2
d. Fourth last digit is 2*X*Y + any carryover from STEP 3
e. Beginning of result will be Sq(X) + any carryover from Step 4
Eg) Let us find the square of 431
Step
a. Last digit = Last digit of Sq(1) = 1
b. Second last digit = 2*3*1 + any carryover from STEP 1=6+0=6
c. Third last digit 2*4*1+ Sq(3) + any carryover from STEP 2 = 8+9+0 = 17 i.e. 7 with carryover of 1
d. Fourth last digit is 2*4*3 + any carryover from STEP 3 = 24+1 = 25 i.e. 5 with carryover of 2
e. Beginning of result will be Sq(4) + any carryover from Step 4 = 16+2 = 18
THUS SQ(431) = 185761
7. If the answer choices provided are such that the last two digits are different, then, we need to carry out only the first two steps only.
-> The sum of first n natural numbers = n(n+1)/2
-> The sum of squares of first n natural numbers is n(n+1)(2n+1)/6
-> The sum of cubes of first n natural numbers is (n(n+1)/2)2/4
-> The sum of first n even numbers= n (n+1)
-> The sum of first n odd numbers= n2
8. If a number ‘N’ is represented as a^x * b^y * c^z… where {a, b, c, …} are prime numbers, then
-> the total number of factors is (x+1)(y+1)(z+1) ....
-> the total number of relatively prime numbers less than the number is N * (1-1/a) * (1-1/b) * (1-1/c)....
-> the sum of relatively prime numbers less than the number is N/2 * N * (1-1/a) * (1-1/b) * (1-1/c)....
-> the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x * y *...)
-> Total no. of prime numbers between 1 and 50 is 15
-> Total no. of prime numbers between 51 and 100 is 10
-> Total no. of prime numbers between 101 and 200 is 21
-> The number of squares in n*m board is given by m*(m+1)*(3n-m+1)/6
-> The number of rectangles in n*m board is given by n+1C2 * m+1C2
9. If ‘r’ is a rational no. lying between 0 and 1, then, r^r can never be rational.
10.The number of squares in n*m board is given by m*(m+1)*(3n-m+1)/6
-> The number of rectangles in n*m board is given by n+1C2 * m+1C2
11. If ‘r’ is a rational no. lying between 0 and 1, then, r^r can never be rational.
12. Certain nos. to be remembered
-> 2^10 = 4^5 = 32^2 = 1024
-> 3^8 = 9^4 = 81^2 = 6561
-> 7 * 11 * 13 = 1001
-> 11 * 13 * 17 = 2431
-> 13 * 17 * 19 = 4199
-> 19 * 21 * 23 = 9177
-> 19 * 23 * 29 = 12673
13. 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.
14. 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.
15. 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
16. 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.
17. 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.
18. 145 is the 3-digit no. expressed as sum of factorials of the individual digits i.e.
145 = 1! + 4! + 5!
19.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
20. 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
21. The relationship between base 10 and base ‘e’ in log is given by log10N = 0.434 logeN
22. WINE and WATER Mixture 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
23. 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
24. 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
25. 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.
26. -> 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
-> The difference between 3 years’ simple interest and compound interest is given by (P * R2 *(300+R))/1003
27. -> 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.
-> 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.
-> 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
-> 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)
28. In case ‘n’ faced die is thrown k times, then, probability of getting atleast one more than the previous throw = nC5/n5
29.-> 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
-> When an unbiased coin is tossed even no. (2n) of times, then, P (no. of heads=no. of tails) = 1-(2nCn/22n)
30. 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
31. CALENDAR
-> Calendar repeats after every 400 years.
-> Leap year- it is always divisible by 4, but century years are not leap years unless they are divisible by 400.
-> Century has 5 odd days and leap century has 6 odd days.
-> 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.
-> January 1, 1901 was a Tuesday.
32. -> 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)
-> For any regular polygon, the sum of interior angles =(n-2)*180 degrees
So measure of one angle is (n-2)/n *180
-> If any parallelogram can be inscribed in a circle, it must be a rectangle.
-> If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i.e. oblique sides equal).
33. 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)
34. -> 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.
-> 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.
-> Area of a Rhombus = Product of Diagonals/2
35. 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]
36. Area of a triangle
-> 1/2*base*altitude
-> 1/2*a*b*sin C (or) 1/2*b*c*sin A (or) 1/2*c*a*sin B
-> 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 of the triangle
37. In any triangle
-> a=b*cos C + c*cos B
-> b=c*cos A + a*cos C
-> c=a*cos B + b*cos A
-> a/sin A=b/sin B=c/sin C=2R, where R is the circumradius
-> cos C = (a^2 + b^2 - c^2)/2ab
-> sin 2A = 2 sin A * cos A
-> cos 2A = cos^2 (A) - sin^2 (A)
38. The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1
39. Appollonius Theorem
In a triangle ABC, if AD is the median to side BC, then
AB2 + AC2 = 2(AD2 + BD2) or 2(AD2 + DC2)
40. -> In an isosceles triangle, the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.
-> In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.
41. The quadrilateral formed by joining the angular bisectors of another quadrilateral is always a rectangle.
42. Let W be any point inside a rectangle ABCD, then,
WD2 + WB2 = WC2 + WA2
43. 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))
44. -> 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)
-> Distance between 2 points (x1, y1) and (x2, y2) is given by Sq((x1-x2)2+ (y1-y2)2)
45. 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.
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