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LOGICAL PUZZLE
Level :: 1
(10 points)
A Math teacher, in order to keep two of her students
occupied thought of two consecutive numbers in the
range 1 to 10, and told one of the students Bohr one
of the numbers and the other student, Heisenberg
the other.
Bohr and Heisenberg then had the following
conversation:
Heisenberg: I don't know your number.
Bohr:
I don't know your number, either.
Heisenberg: Now I know!
How many such sets of numbers can you find? Can
you find all the sets which could have been thought
up by the Math teacher?
4 sets : (2,3) (3,4) (6,7)(7,8)
LOGICAL PUZZLE
Level:: 2
(20 points)
Ozama, McKhan and Hillary live on an island
inhabited by 3 groups:
Truthtellers(always speaks truth), Falsifiers(always
lies) and Alternators(alternately tell truth and lie).
• One of them is obviously the president.
• Ozama says: “the President belongs to a group
different from the group to which the other two
belong” and “Hillary is not the president”
• McKhan says “the President is a falsifier” and
“Ozama is not the President”
• Hillary says ..
There are many combinations possible for this
one.
GEOMETRY
Level :: 2
(20 points)
In triangle ABC, let D be the midpoint of BC. IF angle
ADB=45 and angle ACD=30, determine angle BAD.
(all angle measurements in degrees)
Answer =30
Drop a perpendicular from B to AD and complete
the triangle formed to get an equilateral
triangle(can be proved)
BASIC COMBINATORICS
Level :: 1
(10 points)
What is the probability that a randomly chosen
positive divisor of 10^99 is an integral multiple of
10^88?
Basically,
Favourable cases = factors of 10^11 =
(2^11)(5^11)
=12*12
Total cases = factors of 10^99 = (2^99)(5^99)
=100*100
P= 144/10000
BASIC COMBINATORICS
Level :: 2
(20 points)
Find the number of all 5 digit numbers each of which
contains the block 15 and is divisible by 15.
Example 34515, 12150, not 12750 (no block of 15)
Make cases
Answer = 479
WHAT’S MY MISTAKE
Level :: 1
(10 points)
4*4 = 4 + 4 + 4 + 4
4*4= 4 + 4 +… 4 times.
x*x = x + x + … x times.
Differentiating,
2*x = 1 + 1 + 1 … x times.
:. 2*x = x
Hence 2 = 1!!!
Differentiating constants leads to 0
WHAT’S MY MISTAKE
Level :: 2
(10 + 10 = 20 points)
a) -1 = - 1
or, sqrt(-1) = sqrt(-1)
or, sqrt(-1/1) = sqrt(1/-1)
or, sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
or, i/1 = 1/i
or, i2 = 1
or, -1 = 1
or, 2 = 0
or, 1 = 0
or, 2 =1 !!
Fundamental operations of mathematics are not
applicable when you are dealing with imaginary
numbers and we know that sqrt(-1) is an
imaginary number
b)
(n+1)^2 = n^2 + 2n + 1
or, (n+1)^2 - (2n+1) = n^2
subtracting n(2n+1),
or, (n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)
or, (n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2
= n^2 - n(2n+1) + 1/4(2n+1)^2
or, (n - 1/2)^2 = (n + 1/2)^2
or, (n - 1/2) = (n + 1/2)
or, -1/2 = +1/2
or, 1 = 0
or, 2 = 1 !!
if (-3)^2=(3)^2
it doesn’t imply -3=3
ALGEBRA
Level :: 1
( 5 + 5 =10 points)
Give the next 2 terms of the following series:
(a) 77,49,36,18,___,___
(b) 61,52,63,94,145,___,____
Product of digits = 7*7=49…Implies 1*8=08 -> 0
Difference in terms is in AP
ALGEBRA
Level :: 2
(20 points)
Find all real numbers for which the equation has
exactly 3 distinct real solutions in x.
x2 + (a-2)x + 1= 3|x|
Make four cases
One with |x|=+x and D>0 & |x|=-x and
One with |x|=+x and D<0 & |x|=-x and
One with |x|=-x and D>0 & |x|=+x and
One with |x|=-x and D<0 & |x|=+x and
D<0
D>0
D<0
D>0
Answer a=-1,7
ARE YOU SMARTER THAN A 5th GRADER
Level :: 1
(10 points)
Fill in the boxes:
Complex, Imaginary , Irrational , Fraction
ARE YOU SMARTER THAN A 5th GRADER
Level :: 2
(4*5 = 20 points)
1. Find the next number in the series:
2,3,10,12,13,20……
All begin with “t” Hence 21
2. Chose the correct option : A polynomial of
degree n has n (Real/ Complex) roots.
Complex roots(include real)
3. State True or False(with reason):
The proper factors of 12 are: -6,-4,-3,-2,1,1,2,3,4,6.
False
4. State True or False(with reason):
The multiples of 12 are: -24,-12,0,12,24.
True
5. What do these Symbols mean in Algebra or
number theory?
Q rational, Z integer, N natural, R real?
PHYSICS IN MATH
Level :: 1
(10 points)
Sir Isaac Newton is dying of thirst at B. Usian Bolt
standing at A must save him by running upto the
river Pregolya and fetching water for Newton at B. A
is 4m from the river whereas B is 12m far. Points A
and B are 17 m apart from each other. What is the
minimum distance Bolt will have to run to save the
man without whom this question would not have
been possible?
(HINT: Physics in Mathematics, what else?)
______________________(river)
A
B
According to Fermat’s Laws, light travels the
minimum distance. Hence Bolt should trace lights
path. Consider the river as the mirror and then find
the distance between A and the image of B in the
mirror to get answer as root(481)
PHYSICS IN MATH
Level :: 2
(20 points)
Descartes places a Plane Mirror in a plane passing
through origin such that (1/√3 i -1/√3 j + 1/√3
k) is a unit vector perpendicular to the mirror. A
source of light is kept at a point on the X Axis such
that an incident ray strikes the mirror at origin. Find
the unit vector directed along the reflected ray.
(HINT: Imagine in a 3D plane)
Use both the Snells laws
1)incident ray, normal and reflected ray lie in the
same plane
2) i=r