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Set 1: Linear Equations

1. x + y = 100. What is the number of solutions, when

a)  x and y are integers

b) 

x and y are whole numbersc)  x and y are natural numbers

d)  x and y are natural numbers and x is even

2. 2x + 3y = 100. What is the number of solutions, when

a)  x and y are whole numbers

b)  x and y are natural numbers

c)  x and y lie in the range [-100,100]

3. 2x - 3y = 100. What is the number of solutions, whena)  x and y are whole numbers

b)  x and y are whole numbers less than 200

4. In a written test having 100 questions, each question was awarded +4 marks for

answering correctly, –2 marks for answering wrongly and –1 mark for leaving the question

un-attempted. If a student scored 50 marks, in how many different ways (different number

of corrects, wrongs and un-attempted) could he have scored the marks?

Set 2: Quadratic and Higher Degree Polynomials

Directions for Questions 1 and 2:

Let f  (x) = ax 2+ bx + c, where a, b and c are certain constants and a ≠ 0. It is known that f (5) =

-3 f (2) and that 3 is a root of f  (x) = 0.

1. What is the other root of f  (x) = 0?

(1) -7 (2) - 4 (3) 2 (4) 6 (5) cannot be determined

2. What is the value of a + b + c?  

(1) 9 (2) 14 (3) 13 (4) 37 (5) cannot be determined

3. A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x =

0 is 1. What is the value of f(x) at x = 10?

(1) - 159 (2) - 110 (3) - 180 (4) - 105 (5) – 119

4. If the roots of the equation x3- ax 

2 +bx - c =0 are three consecutive integers, then what is

the smallest possible value of b?(1)-1/√3 (2)-1 (3)0 (4)1 (5) 1/√3 

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Set 3: Modulus Equations and Inequalities

1. Find the value / values of x, given that

a)  |x – 5| = 4

b) 

|x – 1| + |x – 8| = 6c)  |x – 1| + |x – 8| = 7

d)  |x – 1| + |x – 8| = 8

e)  |x – 1| + |x – 8| = 14

f)  |x – 5| < 4

g)  |x – 5| > 4

2. Find the minimum value and also the value at which the minimum occurs

a)  |x – 1|

b) 

|x – 1| + |x – 8|c)  |x – 1| + |x – 5| + |x – 8|

d)  |x – 1| + |x – 2| + |x – 3| + |x – 4| + |x – 5|

e)  |x – 1| + |x – 2| … |x – 10|

f)  |2x + 3| + |3x – 5|

Set 4: Maximum and Minimum Values

1. For positive values of x, y and z, find the minimum value of

a) 

x + 1/x

b)   xyz 

 y x z  z  x y z  y x   )()()(   222

 

c)  (x+y+z)(1/x+1/y+1/z)

2. For real values of x

a)  Find out the minimum value for x2 + 6x + 7

b)  Find out the maximum value for 1/(x2 + 6x + 10)

3. Given that a, b and c are positive real numbers

a)  If a + b + c = 12 then find maximum value of abc

b)  If a + b + c = 12 then find maximum value of (a^2)(b^3)c

c)  If 2a + 3b + 2c = 12, find the maximum product of a^3*b^2*c

4. For real values of x

a)  Find the minimum value of max(2x + 1, 3 – x)

b) 

Find the maximum value of min(2x + 1, 3 – x)