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NUMBER SYSTEMS TEST
Total Number of Questions: 15
Total Time = 20 mins
Marking Scheme:
A correct answer fetches you +4 marks and a wrong answer will lead to a deduction of 1 mark.
No marks will be added or deducted for un-attempted questions.
1. What is the remainder when P2 is divided by 12, where P is a prime number?
A: P is form of 4k+3 where k is positive
B: P is form 2k-1, where k is integer greater than 1
(1)
A alone but not B
(2)
B alone but not A
(3)
Both A and B
(4)
Neither A nor B
2. There is a 5 digit natural number N1 = abcde whose digits a, b, c, d and e are in an arithmetic
progression and there is another 4 digit number N2 = ABCD whose digits A, B, C, D are in a
geometric progression.
Which of the following statements is invalid?
(1)
Sum of digits of N2 is 15
(2)
N1 has 12 possible values
(3)
N1 + N2 has 24 possible values
(4)
N1 is never divisible by 4
(5)
None of these
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3. A is the set of first 40 natural numbers. B is a subset of A such that there exist at least 2 pairs of
elements in B whose difference is 12. If n is the number of elements in B, then which of the
following is correct?
(i) N≥14
4.
(ii) n≥24
(iii) n≥26
(iv) n≥28
(v) n≥30
A rod of length ‘L’ cm is broken into two parts of lengths L1 cm and L2 cm, where L1 = 2a × 3b
and L2 = 3a × 2b. If the mid-point of the rod is at a distance of 90 cm from the ends, then find
the value of L1 × L2 (Here, a and b are positive integers).
(i) 7776
(ii) 8176
(iii) 8076
(iv) 7716
(v) Data insufficient
5. Let P be a natural number that leaves a remainder 3, when divided by 7 and let Q be another
natural number that leaves a remainder 1, when divided by 5. How many ordered pairs (P, Q)
exist such that the difference between P and Q is greater than 177 and the sum of P and Q is less
than 203?
(1) 12
(2) 14
(3) 16
(4) 10
(5) 20
6. Nangru is standing on a point X such that the point X lies on the first quadrant of a Cartesian
Co-ordinate system. After travelling to five different points in the first quadrant, he came back
to the point X. The ‘x’ and ‘y’ co-ordinates of each point to which he traveled is a prime number.
What is the minimum possible area of the region enclosed by Nangru? Assume that during the
course of his travel he neither retraced nor crossed the path through which he has already
traveled once. Also, the distance between the two consecutive points which he had traveled
is an integer.
(1) 3 square units
(2) 4.5 square units
(3) 2 square units
(4) 5.5 square units
(5) 4 square units
7. Find the digit sum of the number (ab3cdefghi1)43, where a, b…. h and i are nine distinct single
digit natural numbers. (Digit sum of 926 = 9 + 2 + 6 = 17 = 1 + 7 = 8).
(1) 8
(2) 2
(3) 4
(4) 7
(5) 5
8. How many integers exist such that not only are they multiples of 20082008 but also are factors
of 20082020?
(1) 12
(2) 481
(3) 587
(4) 200812
(5) 637
9. A number when divided by 100 leaves a quotient (Q) and a remainder (R). How many three-digit
natural numbers are there such that Q + R is divisible by 11?
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(1) 9
(2) 99
(3) 80
(4) 81
(5) 90
10. The HCF of (n + 3) and (7n + 48) is ‘k’, where ‘n’ is a natural number. How many values of ‘k’ are
possible?
(1) 4
(2) 5
(3) 1
(4) 2
(5) 3
11. How many three-digit numbers are there such that no two adjacent digits of the number are
consecutive?
(1) 592
(2) 516
(3) 552
(4) 600
(5) 596
12. How many natural numbers ‘k’ exist such that (2k + 3) divides (24k2 + 201)?
(1) 4
(2) 0
(3) 6
(4) 7
(5) 8
13. There are 5 distinct real numbers. All triplets are selected and the numbers are added. The
different sums that are generated are: (– 8, 1, 3, 5, 7, 8, 10, 16, 19 and 23). The smallest number
among the 5 numbers is
(1) – 10
(2) – 9
(3) – 8
(4) – 7
(5) – 6
14. In base 7: when a four digit number ‘abcd’ is added to a positive integer ‘x’ resultant number is
another four digit number ‘dcba’ where a, b, c and d are distinct integers. The maximum
possible value of ‘x’ is
(1) 3006
(2) 5412
(3) 5502
(4) 4323
(5) 2920
15. What is the remainder when n! + (n! + 1) + (n! – 2) + (n! + 3) ..... + (n! – 2006) is divided by 1003
for n = 1003?
(1) 1
(2) 0
(3) 2006
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(4) 2005
(5) None of these
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