## QBM033

Q1. I have a d-digit positive integer where each digit from 1 to d appears exactly once, and in which each digit (except for the leftmost) differs from one further to the left by +1 or -1.
For example, if d = 3, the only numbers satisfying the conditions are, in increasing order, 123, 213, 231 and 321.
Find a formula in terms of d for the number of integers satisfying these conditions.
Q2. Determine the smallest integer that is a square and whose decimal representation starts with 2005.
Q3. A 3x4 rectangle has sides 3,4,3,4 and diagonals 5,5. Its "weight" is said to be 3+4+3+4+5+5 = 24. Find a quadrilateral with all sides and diagonals having integer length and whose weight is smaller than 24.
Q4. Let ABC be an isosceles triangle (AB = AC) with BAC = 20°. Point D is on side AC such that DBC = 60°. Point E is on side AB such that ECB = 50°. Find, with proof, the measure of EDB.
Q5. It's given that the number Z = n^{4} + a is not prime for any natural number n. Then which of the following is true about a
i. There is no such value for a

ii. There is only value for a

iii. Infinity many natural numbers are possible.

iv.None of the above.
Directions Q. 6 to 7: are based on the following information:
There are three different cable channels namely Ahead, Luck and Bang. In a survey it was found that 85% of viewers respond to Bang, 20% to Luck, and 30% to Ahead. 20% of viewers respond to exactly two channels and 5% to none.
Q6. What percentage of the viewers responded to all three?
(1) 10

(2) 12

(3) 14

(4) None of these
Q7. Assuming 20% respond to Ahead and Bang, and 16% respond to Bang and Luck, what is the percentage of viewers who watch only Luck?
(1) 20

(2) 10

(3) 16

(4) None of these
Q8. Pipe A can fill a tank in "a" hours. On account of a leak at the bottom of the tank it takes thrice as long to fill the tank. How long will the leak at the bottom of the tank take to empty a full tank, when pipe A is kept closed?
A. (3/2) a hours

B. (2/3) a hours

C. (4/3) a hours

D.(3/4) a hours
Q9. The sum of the reciprocals of two real numbers is âˆ’1, and the sum of their cubes is 4. What are the numbers ?
Q10. In a zoo, there are rabbits and pigeons. If their heads are counted, these are 90 while their legs are 224. Find the number of pigeons in the zoo.
a) 70

b) 68

c) 72

d) 22
Q11. LCM of two distinct natural numbers is 211. What is their HCF?
a) 37

b) 211

c) 1

d) data insufficient
Q12. a six digit number is formed by writing 3 consecutive two digit number
side by side in ascending order. If the number so formed is divisible by
2,3,4,5,6,8, then what is the hundreds digit of the number?
Q13. If a^{2} = b^{2}, then a = b is
1) always true

2) sometimes true

3) always false

4) none of these
Q14. A man buys shares at a discount of Rs. X. Later he sells all but 10 of the shares he purchased at a premium of Rs. X. If his investment was Rs. 4500 and proceeds from the sale were Rs. 6250, how many shares did he buy originally? [Assume face value of shares as Rs. 100]
(a)50

(b)40

(c)60

(d)90
Q15. Dhruv claims that he in 1994, his age is equal to the sum of the digits of his year of birth. What is the sum of the digits of Dhruva's age in 1994?
1) 3

2) 5

3) 7

4) 17
Q16. ABCD is a square whose side is 2 cm each; taking AB and AD as axes, the equation of the circle circumscribing the square is:
(A) x2 + y2 = (x + y)

(B) x2 + y2 = 2(x + y)

(C) x2 + y2 = 4

(D) x2 + y2 = 16

END

## Few Answers

Q1 ANS: C(d - 1,0) + C(d - 1,1) + C(d - 1,2) + ... + C(d - 1,d - 2) + C(d - 1,d - 1), where C(d-1,k) is the number of ways to select a subset of size k from a set of size d - 1. The above sum is the desired

Q4. EDK = 70° and EDB = 30°.

Q6. Ans: 1

Q7. Ans: 4

Q8. ANS: b

Q9. ANS: Therefore the real solutions are x = (1 Â± )/2, y = (1 )/2.

Q10. ANS:b

Q11. Ans:(c)

Q15. Answer 3

Q16. ANS: b