Relations and Functions

The sum of squares of all the real solutions of the equation (x+1) (2x2+5x+3 )=4- (2x+3) (x2+2x+1 ) is equal to .

Let [ ] denote the greatest integer function. If the domain of the function f(x)= -1 ( x+[x]3 ) is [ , ), then 2+ 2 is equal to :

1. 2
2. 5
3. 10
4. 13

Let A=\1, 4, 7\ and B=\2, 3, 8\. Then the number of elements, in the relation R=\((a1, b1), (a2, b2)) ((A B) (A B)) : a1+b2 divides a2+b1\ is .

If the domain of the function f(x)= (0.6) ( | 2x-5x2-4 | ) is (- , a] \b\ [c, d) (e, ), then the value of a+b+c+d+e is .

Let A=\2, 3, 4, 5, 6\. Let R be a relation on the set A A given by (x, y)R(z, w) if and only if x divides z and y w. Then the number of elements in R is .

If the domain of the function f(x) = (10x2 - 17x + 7) (18x2 - 11x + 1 ) is (- , a) (b, c) (d, ) - \e\, then 90(a + b + c + d + e) equals:

1. 170
2. 316
3. 177
4. 307

The sum of all the real solutions of the equation (x+3) (6x2 + 28x + 30 ) = 5 - 2 (6x+10) (x2 + 6x + 9 ) is equal to :

1. 1
2. 4
3. 0
4. 2

If the domain of the function f(x) = -1 ( 5 - x3 + 2x ) + 1 e(10 - x) is (- , ] [ , ) - \ \, then 6( + + + ) is equal to

1. 66
2. 68
3. 70
4. 67

Let f(x) = [x]2 - [x + 3] - 3, x R, where [.] is the greatest integer function. Then

1. f(x) = 0 for finitely many values of x
2. f(x) < 0 only for x [-1, 3)
3. 02 f(x) dx = -6
4. f(x) > 0 only for x [4, )

Let the domain of the function f(x) = 3 5 (7 - 2 (x2 - 10x + 85 ) ) + -1 ( | 3x - 717 - x | ) be ( , ]. Then + is equal to :

1. 12
2. 8
3. 10
4. 9

The number of elements in the relation R = \(x, y) : 4x2 + y2 < 52, x, y Z \ is

1. 86
2. 67
3. 89
4. 77

Let the domain of the function f(x) = -1 ( 4x + 53x - 7 ) be [ , ] and the domain of g(x) = 2 (2 - 6 27(2x + 5) ) be ( , ). Then |7( + ) + 4( + )| is equal to .

If the range of the function f(x) = 5-xx2-3x+2, x 1, 2, is (- , ] [ , ), then 2 + 2 is equal to :

1. 188
2. 192
3. 190
4. 194

Let the domains of the functions f(x) = 4 3 7 (8 - 2 (x2 + 4x + 5 ) ) and g(x) = -1 ( 7x + 10x - 2 ) be ( , ) and [ , ], respectively. Then 2 + 2 + 2 + 2 is equal to :

1. 15
2. 13
3. 16
4. 14

If the domain of the function f(x) = e ( 2x-35+4x ) + -1 ( 4+3x2-x ) is [ , ), then 2 + 4 is equal to

1. 4
2. 3
3. 7
4. 5

If the domain of the function f(x) = 7 (1 - 4 (x2 - 9x + 18 ) ) is ( , ) ( , ), then + + + is equal to

1. 17
2. 15
3. 16
4. 18

Let A = \1, 2, 3, , 100\ and R be a relation on A such that R = \(a, b) : a = 2b + 1\. Let (a1, a2), (a2, a3), (a3, a4), , (ak, ak+1) be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to :

1. 6
2. 8
3. 7
4. 5

If the domain of the function f(x) = 1 10 + 3x - x2 + 1 x + |x| is (a, b), then (1 + a)2 + b2 is equal to :

1. 29
2. 30
3. 25
4. 26

Let S = N \0\. Define a relation R from S to R by : R = \(x, y) : e y = x e ( 25 ), x S, y R \. Then, the sum of all the elements in the range of R is equal to :

1. 32
2. 109
3. 52
4. 53

If the domain of the function 5(18x - x2 - 77) is ( , ) and the domain of the function (x-1) ( 2x2 + 3x - 2x2 - 3x - 4 ) is ( , ), then 2 + 2 + 2 is equal to:

1. 186
2. 179
3. 195
4. 174

Let f : R R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a-1 )x + b, a 1. If f(x + y) = f(x) + f(y) + 1 - 27xy, then the value of 28 i=15|f(i)| is

1. 735
2. 675
3. 715
4. 545

The number of real solution(s) of the equation x2+3x+2= \|x-3|,|x+2|\ is :

1. 2
2. 3
3. 1
4. 0

Let f(x)= e x and g(x)= x4-2x3+3x2-2x+22x2-2x+1. Then the domain of f g is

1. (0, )
2. [1, )
3. R
4. [0, )

The product of all solutions of the equation e5( ex)2+3=x8, x>0, is :

1. e2
2. e
3. e6/5
4. e8/5

Let f(x) = 17 - 5x be a function defined on R. Then the range of the function f(x) is equal to :

1. [ 18, 15 ]
2. [ 17, 16 ]
3. [ 17, 15 ]
4. [ 18, 16 ]

If S = \a R : |2a - 1| = 3[a] + 2\a\\, where [t] denotes the greatest integer less than or equal to t and \t\ represents the fractional part of t, then 72 a S a is equal to .

If the domain of the function -1 ( 3x-222x-19 )+ e ( 3x2-8x+5x2-3x-10 ) is ( , ], then 3 +10 is equal to:

1. 95
2. 100
3. 97
4. 98

Let the sum of the maximum and the minimum values of the function f(x)= 2x2-3x+82x2+3x+8 be mn, where (m, n)=1. Then m+n is equal to :

1. 217
2. 182
3. 201
4. 195

If the domain of the function f(x)= x2-25 (4-x2 )+ 10 (x2+2x-15 ) is (- , ) [ , ), then 2+ 3 is equal to :

1. 140
2. 175
3. 125
4. 150

If the domain of the function f(x)= -1 ( 2-|x|4 )+ \ e(3-x) \-1 is [- , )-\ \, then + + is equal to :

1. 11
2. 12
3. 9
4. 8