1. Solve |x-16| > ( x^2 – 7x + 24 )
2. Find the max. 19/39 , 16/51, 10/31, 11/34
3. Find the min. 443/21, 780/37, 275/13, 360/17
4. If x and y are +ve integers and x^2 + y^2 = 1800 then what is the max value of x+y?
5.The tens digit of (23)^24 * (25)^26??
1) A 5-digit no. is taken. Sum of the first four digits (excl. the unit digit) equals sum of all the five digits. Which of the following will not divide this number necessarily ?
a) 10 b) 2 c) 4 d) 5 e) 3
2) A 128 digit no. is formed by writing the first X natural no.s in front of each other as 12345678910111213……….Find the remainder when this no. is divided by 8 ?
a) 6 b) 7 c) 2 d) 0
3) From a number M subtract 1. Take the reciprocal of the result to get the value ‘N’. Then which of the following is necessarily true?
a) (M^N) 3
c ) 1 < (M^N) < 3
d) 1< (M^N) < 5
4) Find the 28383rd term of the series : 123456789101112…..
a) 3 b) 4 c) 9 d) 7
5) If you form a subset of integers chosen between 1 to 3000 (both inclusive), such that no two integers add up to a multiple of 9, what should be the maximum no. of elements of the subset:
a) 1668 b) 1332 c) 1333 d) 1334
1. If n is even , n(n+1)(n+2) is divisible by 24
If n is any integer , n^2 + 4 is not divisible by 4
2. 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]
3. Area of a triangle
1/2*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 of the triangle .
In any triangle
a=b*CosC + c*CosB
b=c*CosA + a*CosC
c=a*CosB + b*CosA
4. 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 equal to
(a1+a2+a3+…………./b1+b2+b3+……….)
5. (7)In any triangle
a/SinA = b/SinB =c/SinC=2R , where R is the circumradius
6. 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)
7. e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ……..to infinity
2 < e =0
else |a|+|b| >= |a+b|
25. 2<= (1+1/n)^n <=3
26. WINE and WATER formula:
If Q be the volume of a vessel
q qty of a mixture of water and wine be removed each time from a mixture
n be the number of times this operation be done
and A be the final qty of wine in the mixture
then ,
A/Q = (1-q/Q)^n
27.
Area of a regular hexagon = (root(3) * 3 * (side)^2)/2
28.
(1+x)^n ~ (1+nx) if x<<<1
29. Some Pythagorean triplets:
3,4,5 (3^2=4+5)
5,12,13 (5^2=12+13)
7,24,25 (7^2=24+25)
8,15,17 (8^2 / 2 = 15+17 )
9,40,41 (9^2=40+41)
11,60,61 (11^2=60+61)
12,35,37 (12^2 / 2 = 35+37)
16,63,65 (16^2 /2 = 63+65)
20,21,29(EXCEPTION)
30. Apollonius theorem could be applied to the 4 triangles formed in a parallelogram.
31.
Area of a trapezium = 1/2 * (sum of parallel sides) * height = median * height
where median is the line joining the midpoints of the oblique sides.
32.
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 .
33.
Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .
34. Let W be any point inside a rectangle ABCD .
Then
WD^2 + WB^2 = WC^2 + WA^2
Let a be the side of an equilateral triangle. Then if three circles be drawn inside
this triangle touching each other then each has a radius = a/(2*(root(3)+1))
35.
Let ‘x’ be certain base in which the representation of a number is ‘abcd’ , then the decimal value of this number is a*x^3 + b*x^2 + c*x + d
36.
When you multiply each side of the inequality by -1, you have to reverse the direction of the inequality.
37.
To find the squares of numbers from 50 to 59
For 5X^2 , use the formulae
(5X)^2 = 5^2 +X / X^2
E.g. (55^2) = 25+5 /25
=3025
(56)^2 = 25+6/36
=3136
(59)^2 = 25+9/81
=3481
38. a+b+(ab/100)
This is used for successive discounts types of sums.
Like in 1999 population increases by 10% and then in 2000 by 5%
So the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999
and if there is a decrease then it will be preceded by a negetive sign and likewise.
1. To find the number of factors of a given number, express the number as a product of powers of prime numbers.
In this case, 48 can be written as 16 * 3 = (2^4 * 3)
Now, increment the power of each of the prime numbers by 1 and multiply the result.
In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)
Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 – 2 = 8 factors.
2. 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 first n even numbers= n (n+1).
The sum of first n odd numbers= n^2
3. To find the squares of numbers near numbers of which squares are known
To find 41^2, Add 40+41 to 1600 = 1681
To find 59^2, Subtract 60^2-(60+59) = 3481
4. If an equation (i: e f(x) =0) contains all positive co-efficient of any powers of x , it has no positive roots then.
e.g.: x^4+3x^2+2x+6=0 has no positive roots.
5. 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) .
Hence the remaining is the minimum number of imaginary roots of the equation (Since we also know that the index of the maximum power of x is the number of roots of an equation.)
6. For a cubic equation ax^3+bx^2+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
7. For a bi-quadratic equation ax^4+bx^3+cx^2+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
8. If for two numbers x+y=k (=constant), then their PRODUCT is MAXIMUM if
x=y (=k/2). The maximum product is then (k^2)/4
9. If for two numbers x*y=k (=constant), then their SUM is MINIMUM if
x=y (=root (k)). The minimum sum is then 2*root (k).
10. 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.
11. 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 in
Square =90
Pentagon =108
Hexagon =120
Heptagon =128.5
Octagon =135
Nonagon =140
Decagon = 144
12. If any parallelogram can be inscribed in a circle, it must be a rectangle.
13. If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sides equal).
14. 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) .
15. Area of a regular hexagon : root(3)*3/2*(side)*(side)
16. For any 2 numbers a>b
a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively)
(GM)^2 = AM * HM
17. For three positive numbers a, b ,c
(a+b+c) * (1/a+1/b+1/c)>=9
18. For any positive integer n
2<= (1+1/n)^n = ab+bc+ca
If a=b=c , then the equality holds in the above.
a^4+b^4+c^4+d^4 >=4abcd
20. (n!)^2 > n^n (! for factorial)
21. 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 .
22. Consider the two equations
a1x+b1y=c1
a2x+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..
23. For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is
0.5*d1*d2, where d1,d2 are the lengths of the diagonals.
24. Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is ,
the minute hand describes 6 degrees /minute
the hour hand describes 1/2 degrees /minute .
Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .
25. The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight.
1. What is the remainder when (3^444) + (4^333) is divided by 5?
2. What is the remainder when (5555^2222 )+ (2222^ 5555) is divided by 7?
3. (20^ 2004) + (16^2004) – (3^ 2004) − 1 is divisible by:
(a) 317 (b) 323 (c) 253 (d) 91
1.Find the no. of zeros at the right end of 300!
2. If 500! is divisible by 1000^n, then what is the max. integral value of n?
