**Question 1:**

The number of ordered pairs of natural numbers (a, b) satisfying the equation 2a + 3b = 100 is:

[1] 13

[2] 14

[3] 15

[4] 16

Valid solutions:

a = 2; b = 32

a = 5; b = 30

.

.

a = 47; b = 2

No. of solutions = **16. Option D**

**Question 2:**

For how many positive integral values of N, less than 40 does the equation 3a – Nb = 5, have no integer solution

[1] 13

[2] 14

[3] 15

[4] 12

If N is a multiple of 3, then the LHS would be divisible by 3 and RHS won’t be. Number of positive integral values less than 40 which are multiple of 3 =

**13. Option A****Question 3:**

What are the number of integral solutions of the equation 7x + 3y = 123 for x,y > 0

[1] 3

[2] 5

[3] 12

[4] Infinite

Valid Solution:

x = 3; y = 34

x = 6; y = 27

.

.

x = 15; y = 6

Number of integral solutions such that x, y > 0 are **5. Option B**

**Question 4:**

The cost of 3 hamburgers, 5 milk shakes, and 1 order of fries at a certain fast food restaurant is $23.50. At the same restaurant, the cost of 5 hamburgers, 9 milk shakes, and 1 order of fries is $39.50. What is the cost of 2 hamburgers, 2 milk shakes, and 2 orders of fries at this restaurant?

[1] 10

[2] 15

[3] 7.5

[4] Cannot be determined

3H + 5M + 1F = 23.50

H + M + F = 7.5

2H + 2M + 2F =

5H + 9M + 1F = 39.50

2H + 2M + 2F = ?

Calculate 2(Equation 1) – (Equation 2)

H + M + F = 2×23.5 – 39.5H + M + F = 7.5

2H + 2M + 2F =

**15. Option B****Question 5:**

How many integer solutions are there for the equation: |x| + |y| =7?

[1] 24

[2] 26

[3] 14

[4] None of these

x can take any integer value from [-7,7].

So, there are 15 valid values of x.

For each of these values, there are 2 corresponding values of y. eg: For x = 3; y can be 4 or -4.

Except when x = 7 or -7; where the only possible value of y is 0.

Total valid values of x = 13×2 + 1 + 1 = **28. Option D**

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