Let f (x) = Ax² + Bx + C where A, B , C are real numbers. Prove that if f (x) is an integer then the numbers 2A, A + B and C are all integers. Conversely prove that if the numbers 2A, A + B and C are all integers then f ( x ) is an integer whenever x is an integer.

True / False

The function f (x) =  is not one to one.

a) True

b) False

Fill in the blank

Let A be the set of n distinct elements then the total number of distinct functions from A to A is ……… and out of these …… are onto

a) n!, 1

b) nⁿ, n!

c) nⁿ, 1

d) none of the above

If a, b, c are distinct positive numbers then the expression
( b + c – a ) ( c + a – b ) ( a + b – c ) –abc is

a) Positive

b) Negative

c) Non–positive

d) None of these

If b > a then the equation ( x – a ) ( x – b )1 = 0 has

a) Both roots in [ a, b ]

b) Both roots in ( , a )

c) Both roots in (  )

d) One root in ( , a ) and other in ( )

Find the integral solutions of the following system of inequality
 

a) x = 1

b) x = 2

c) x = 3

d) x = 4

Find the integral solutions of the following system of inequality
 

a) Ø

b) x = 1

c) x = 2

d) x = 3

mn squares of equal size are arranged to form a rectangle of dimension m by n, where m and n are natural numbers. Two squares will be called neighbours if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in the neighbouring squares. Show that this is possible only if all the numbers used are equal.

Consider the system of linear equations
 
 
 
Find the value of θ for which the systems of equations have non-trivial solutions.

Let λ and α be real. Find the set of all values of λ for which the system of linear equations
 
 
 
has a non-trivial solution. For λ = 1 find the value of α.

Let a, b, c, ε R and α, β be roots of  such that  and  then show that .

The set of all solutions of the equation

Let a, b, c, d be real numbers in geometric progression. If u, v, w satisfy the system of equations

 
 
 
Then show that the roots of the equation
 
 
and  are reciprocal of each other.

Find the range of values of t for which  

a) (−, −)

b) ( ,  )

c) (− , −  ) U ( ,  )

d) (−,  )

Let ℝ be the real line. Consider the following subsets of the plane ℝ x ℝ
 
 
Which one of the following is true?

a) T is an equivalence relation on ℝ but S is not

b) Neither S nor T is equivalence relation on ℝ

c) Both S and T are equivalence relation on ℝ

d) S is an equivalence relation on ℝ but T is not

If  be a function defined as  where  . Show that f is invertible and its inverse is

a)  

b)  

c)  

d)  

The largest interval lying in  for which the function  is defined is

a)  

b)  

c)  

d)  

Let W denote the words in the English dictionary. Define the relation R by  the words x and y have one letter in common}. Then R is

a) Reflexive, symmetric and not transitive

b) Reflexive, symmetric and transitive

c) Reflexive, not symmetric and transitive

d) Not reflexive, symmetric and transitive

Let  be a relation on the set  The relation is

a) Reflexive and symmetric only

b) An equivalence relation

c) Reflexive only

d) Reflexive and transitive only

Let  be a relation on the set  The relation is

a) a function

b) transitive

c) not symmetric

d) reflexive

A function f from the set of natural numbers to integers defined by
 is

a) One–one but not onto

b) Onto but not one–one

c) One–one and onto both

d) Neither one–one nor onto

Domain of definition of the function  is

a)

b)

c)  

d)

The function  is

a) An even function

b) An odd function

c) A periodic function

d) Neither an even nor an odd function