- Equations
- Progressions
- Functions
- Maxima Minima
- Logarithms

All these algebra questions are with detailed explanations.

**Question 1:**

For the given pair (x, y) of positive integers, such that 4x-17y=1 and x<1000 how many integer values of y satisfy the given conditions?

[1] 56

[2] 57

[3] 58

[4] 59

Valid Solutions:

x = 13, y = 3

x = 30, y = 7

x = 47, y = 11

.

.

x = 999, y = 235

No. of terms =\(\frac{{999 - 13}}{{17}} + 1 = \) = 58 + 1 = **59. Option D**

**Question 2:**

One year payment to the servant is Rs. 90 plus one turban. The servant leaves after 9 months and receives Rs. 65 and turban. Then find the price of the turban

[1] Rs.10

[2] Rs.15

[3] Rs.7.5

[4] Cannot be determined

Payment for 9 months should be ¾(90 + t)

Payment for 9 months is given to us as 65 + t

Equating the two values we get

¾(90 + t) = 65 + t

270 + 3t = 260 + 4tt =

**10 Rs. Option A**

**Question 3:**

In CAT 2007 there were 75 questions. Each correct answer was rewarded by 4 marks and each wrong answer was penalized by 1 mark. In how many different combination of correct and wrong answer is a score of 50 possible?

[1] 14

[2] 15

[3] 16

[4] None of these

4c – w + 0n= 50

Adding the two equations we get

5c + n = 125Values of both c & n will be whole numbers in the range [0, 50]

c (max) = 25; when n = 0

c (min) = 13; when n = 60 {Smallest value of ‘c’ which will take the marks from correct questions greater than or equal to 50}

No. of valid combinations will be for all value of ‘c’ from 13 to 25 =

**13. Option D**

**Question 4:**

How many integer solutions exist for the equation 8x – 5y = 221 such that x ´ y < 0

[1] 4

[2] 5

[3] 6

[4] 8

Valid Solutions:

x = 32; y = 7

x = 37; y = 15

x = 42; y = 23

But we need the solutions where one variable is negative whereas the other one is positive. so, we will move in the other direction.

x = 27; y = -1

x = 22; y = -9

x = 17; y = -17

x = 12; y = -25

x = 7; y = -33

x = 2; y = -41

So, number of integer solutions where x ´ y < 0 is **6. Option C**

**Question 5:**

How many integer solutions exists for the equation 11x + 15y = -1 such that both x and y are less than 100?

[1] 15

[2] 16

[3] 17

[4] 18

x = 4; y = -3

x = 19; y = -14

.

.

x = 94; y = -69

So, there are 7 solutions of positive values of ‘x’.

x = -11; y = 8

x = -26; y = 19

.

.

x = __; y = 96

So, there are 9 solutions for positive values of ‘y’.

Total number of integer solutions = 7 + 9 = **16. Option B**

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