KVPY SX SB Exam 20141 Question Paper

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Question : 18 of 120

Marks: +1, -0

Let

S = \{(a,b): a,b ∊ Z, 0 ≤ a, b≤18\}

. The number of elements (x, y ) in S such that 3x + 4y + 5 is divisible by 19 is

38
19
18
1

Solution:

3x + 4y + 5 = 1 9

5 ≤ (3x + 4y + 5) ≤ 131

5 ≤ 19 ≤ 131

Case (i)

3x + 4y + 5 = 19

3x + 4y = 14

y={14-3x}/4

x = 2

Case (ii)

3x + 4y + 5 = 38

3x + 4y = 33

4y = 33 - 3x

y={33-3x}/4

x = 3, 7, 11

Case (iii)

3x + 4y = 52

x={52-3x}/4

x = 0, 4, 8, 12, 16

Case (iv)

3x + 4y = 71

4y = 71 - 3x

y={ 71 - 3x}/4

x = 1, 5, 9, 13, 17

Case (v)

3x + 4y = 90

y={90-3x}/4

x = 6, 10, 14, 18

Case (vi)

3x + 4y = 109

y={109-3x}/4

x = 15 is only possibility.

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