Ultimate Quant Marathon Blog For IIM CAT

I could not post due to some engagements. Here are a bunch of problems to compensate

Question 1)

A + B + C + D = D + E + F + G = G + H + I = 17 where each letter represent a number from 1 to 9. Find out number of ordered pairs (D,G) if letter A = 4.
a) 0                       b) 1                 c)2                        d) 3                 e) none of these

Question 2)

The sequence 1, 3, 4, 9, 10, 12….. includes all numbers that are a sum of one or more distinct powers of 3. Then the 50th term of the sequence is
a. 252                    b. 283                     c. 327                      d. 360                  e) none of these

Question 3)

Given that g(h(x)) = 2x² + 3x and h(g(x)) = x² + 4x − 4 for all
real x. WHich of the following could be the value of g(-4)?
a)1                     b) -1                          c) 2                 d) -2                   e) -3

Question 4)

If a, x, b and y are real numbers and ax+by = 4 and ax² +by² = 2 and
ax³ + by³= −3 then find (2x − 1)(2y − 1)
a)4                      b) 3                    c) 5                 d) -3          e) cannot be determined.

Question 5)

K1,K2,K3…K30 are thirty toffees. A child places these toffees on a circle, such that there are exactly n ( n is a positive integer) toffees placed between Ki and Ki+1 and no two toffees overlap each other. Find n
a)4                        b) 5                     c) 9                 d) 12                       e) 13

Question 6)
For the n found in previous  question, which of the two toffees are adjacently
placed on the circle? ( All other conditions remaining same)
a) K11 and K13                    b) K6 and K23                   c) K2 and K10              d) K11 and K18
e) K20 and K28

The Answer key is as

1) C  2)  E  3) B  4) A 5) E  6) D 7) B E 9) C  10) E

Question 1 is easy just use a-1/a= 1 and 1/a-a=1 will give two roots sum them

Question 2  10^j-1o^i=10^i(10^(j-1)-1)

1001=7.11.13=10^3+1

so clearly 10^3+1 divides 10^6-1 and therefore 10^6k-1

hence j-i=6k, k is a natural number

applying other constraints we get option e

Question 3) 1+2+3..30=30.31/2=31.15=465

[465/2]=232 now suppose a subset A of S doe not have sum more than 232 then A’ must have sum more than 232 hence 1/2 of the subsets of S will have sum more than 232

so 2^30/2=2^29

question 4 use the concept of reflection we will get min distance sum as 5root(2)

Question 5) Function is not correctly defined as 0 is not a natural number

so option e)

Question 6)

let the radius be r
and the point of tangency be P and Q and triangle be ABC. P lies on AB and Q on BC

let AP = m and BQ = n

m^2 = 15^2-x^2
n^2 = 20^2-x^2
m = 9 and n = 16 x =12 arcPQ = 6pi

question 7) see n numbers product is n and sum is zero

if n is odd then sum can’t be zero

similalry check for other cases it will easily come out n =4k

question 8 toughest problem of the test

let g(n)=p(n)-n

then g(17)=-7=g(24)

let a,b be integers such that p(n)=n+3

then a-17 divides g(a)-g(17)=3-(-7)=10

similarly for 24

hence we find a-17 and a-24 both divide 10 this means k=a-17 and k+7 both divide 10

this means k=-5 or-2

a=19 b=22

hence ab=418

question 9 can be easily done

question 10) tricky enough problem

look for a series and solve it you will get E

We will start with power plays now. It shall consist of 10 Problems and You are expected to solve them in 25-30 Mins. Each correct answer carries 5 marks. The first two wrong answers carry (-1) marks each, the next two (-2) marks each and so on. Maximum time is 30 mins, but you are expected to solve in 25 mins!

Good Luck !

Please hit the comment button and post  your keys, I will post the official keys and solution in 7 days time!

And We will have a better mechanism hopefully, by the next powerplay!

Power Play I

1) Two different positive numbers a and b differ from their reciprocals by 1. Find a+b

A)   1                   B)  √6            c) √5              D)  3          E) None of these.

2) How many positive integer multiples of 1001 can be expressed in the form 10^j-10^i where i and j are integers such that 0<=i<j<99?

A)  15                    B) 90                     C) 64                     D) 720                     E) 784

3) How many subsets of the set {1,2,3…,30} have the property that the sum of all elements is more than 232?

A) 2^30                   B)    2^29                      C) 2^29  -1            D)  2^29+1           E ) None of these

4) Let the point P be (4,3). Choose a point Q on y=x line and another point R on the line y=0 such that sum of lengths  PQ+QR+PR is minimum. Find this minimum length.

A) 5√2                B)  3+3√2               C) 5                   D) 4                  E) 10

5) Let f be a function defined on odd natural numbers which return natural numbers such that f(n+2)=f(n)+n

and f(1)=0 . Then f(201)?

A) 10000                 B)20000                         C) 40000                  D) 2500                E) None of these

6) A semicircle  is inscribed in a right triangle so that its diameter lies on the hypotenuse and the centre divides the hypotenuse into segments  15 cm and 20 cm long. FInd the length of the arc of the semicricle included between its points of tangency with the legs.

A)  2π                B) 3π                      C)  4π                D)  6π                  E) none of these

7) The product of n numbers is n and their sum is 0. Then n is always divisible by

A)  3                       B) 4                               C)  5                         D)  2       E) None of the foregoing

8 ) Let P(x) be a polynomial with integer coefficients such that P(17)=10 and P(24)=17 . It is further know that P(n)=n+3 has two distinct integer solutions a and b. Find a.b

A) 10                     B) 220                      C)  190                       D)  48                       E) 418

9) Let A and B be two points on the plane. Let S be the set of points P such that PA^2+PB^2 is at most 10. Find the area of S

A)  2π                B) π                      C)  4π                D)  6π                  E) none of these

10) Let T={9^k: k is an ineteger 0<=k<=4000}. Given that 9^4000 has its 3817 digits and its leftmost digit is 9. Find the number of elements in T which also have leftmost digit as 9.

A) 92                          B) 93                    C) 183                          D) 184                       E) 185

Good Luck !