### Question. 1

If f(x+y) = f(x)f(y) and f(5) = 4, then f(10) - f(-10) is equal to

(a) (b) (c) (d)
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### Question. 2

Let f(x) = x2 + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

(a) (b) (c) (d)
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### Question. 3

If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

(a) (b) (c) (d)
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### Question. 4

Let f be a function such that f (mn) = f (m) f (n) for every positive integers m and n. If f (1), f (2) and f (3) are positive integers, f (1) < f (2), and f (24) = 54, then f (18) equals

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### Question. 5

Consider a function f(x+y) = f(x) f(y) where x , y are positive integers, and f(1) = 2. If f (a+1) + f (a+2) + ..... + f(a+n) = 16 (2n - 1) then a is equal to.

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### Question. 6

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals

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### Question. 7

Let f(x)=max{5x, 52 - 2x2}, where x is any positive real number.Then the minimum possible value of f(x) is

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### Question. 8

Let f(x)=min{2x2, 52 - 5x}, where x is any positive real number.Then the maximum possible value of f(x) is [TITA]

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### Question. 9

If f(x + 2) = f(x) + f(x + 1) for all positive integers x, and f(11) = 91, f(15) = 617, then f(10) equals

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### Question. 10

Let f(x) = 2x – 5 and g(x) = 7 – 2x. Then |f(x) + g(x)| = |f(x)| + |g(x)| if and only if

(a) (b) (c) (d)
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### Question. 11

If f(ab) = f(a)f(b) for all positive integers a and b, then the largest possible value of f(1) is

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### Question. 12

If 9x - (1/2) – 22x – 2 = 4x – 32x – 3, then x is

(a) (b) (c) (d)
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### Question. 13

The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

(a) (b) (c) (d)
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### Question. 14

Let f(x) = x2 and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

(a) (b) (c) (d)
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### Question. 15

If f(x) = (5x+2)/(3x5) and g(x) = x2 – 2x – 1, then the value of g(f(f(3))) is:

(a) (b) (c) (d)
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### Question. 16

If f1(x) = x2 + 11x + n and f2(x) = x, then the largest positive integer n for which the equation f1(x) = f2(x) has two distinct real roots, is

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### Question. 17

Suppose, log3x = log12y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6G is equal to

(a) (b) (c) (d)
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### Question. 18 (a) (b) (c) (d)
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### Question. 19 (a) (b) (c) (d)
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### Question. 20

If log 2x = 2 log (x + 1), find the number of real values of x?.

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### Question. 21

The coordinates of two diagonally opposite vertices of a rectangle are (4, 3) and (-4,-3). Find the number of such rectangle(s), if the other two vertices also have integral coordinates.

(a) (b) (c) (d)
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### Question. 22

‘f’ is a real function such that f(x + y) = f(xy) for all real values of x and y. If f(–7) = 7, then the value of f(–49) + f(49) is

(a) (b) (c) (d)
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### Question. 23

Let f(x) = ax^2 + bx + c, where a, b and c are real numbers and a is not equal to 0. If f(x) attains its maximum value at x = 2, then what is the sum of the roots of f(x) = 0?

(a) (b) (c) (d)
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### Question. 24 (a) (b) (c) (d)
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### Question. 25 (a) (b) (c) (d)
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### Question. 26 (a) (b) (c) (d)
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### Question. 27 (a) (b) (c) (d)
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### Question. 28 (a) (b) (c) (d)
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### Question. 29 (a) (b) (c) (d)
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### Question. 30

When ‘2’ is added to each of the three roots of x^3 – Ax^2 + Bx – C = 0, we get the roots of x^3 + Px^2 + Qx – 18 = 0. A, B, C, P and Q are all non-zero real numbers. What is the value of (4A + 2B + C)?

(a) (b) (c) (d)
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### Question. 31

If three positive real numbers a, b and c (c > a) are in Harmonic Progression, then log (a + c) + log (a – 2b + c) is equal to:

(a) (b) (c) (d)
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### Question. 32

There are three coplanar parallel lines. If any p points are taken on each of the lines, then find the maximum number of triangles with the vertices of these points.

(a) (b) (c) (d)
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### Question. 33 (a) (b) (c) (d)
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### Question. 34 (a) (b) (c) (d)
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### Question. 35 (a) (b) (c) (d)
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### Question. 36

The graph of ‘3 – x’ against ‘y + 5’ is as shown below. (All the graphs in this question are drawn to scale and the same scale has been used on each axis. Which of the following shows the graph of y against x?

(a) (b) (c) (d)
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### Question. 37

A function f(x) is defined for all real values of x as f(x) = ax^2 + bx + 1. It is also known that f (5) = f(k) = 0, where k is not equal to 5. If a < 0, then which of the following is definitely correct?

(a) (b) (c) (d)
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### Question. 38

In the X-Y plane two distinct lines are drawn parallel to the line 3y – 4x = 15, each at a distance of 3 units from the given straight line. What are the lengths of the line segments of these two lines lying inside the circle x^2 + y^2 = 25?

(a) (b) (c) (d)
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### Question. 39

A function f(x) is defined for all real values of x as 2f(x) + f (1 – x) = x2 . What is the value of f(5)?

(a) (b) (c) (d)
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### Question. 40

[x] = Greatest integer less than or equal to x {x} = x – [x] How many real values of x satisfy the equation 5[x] + 3{x} = 6 + x?

(a) (b) (c) (d)
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### Question. 41

Let f (x) be a function satisfying f (x) f (y) = f (xy) for all real x, y. If f (2) = 4, then what is the value of f(1/2)?

(a) (b) (c) (d)
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### Question. 42

Suppose, the seed of any positive integer n is defined as follows :

seed (n) = n, if n < 10 = seed (s (n)), otherwise, where s (n) indicates the sum of digits of n. For example, seed (7) = 7, seed (248) = seed (2 + 4 + 8) = seed (14) = seed (1 + 4) = seed (5) = 5 etc. How many positive integers n, such that n < 500, will have seed (n) = 9?

(a) (b) (c) (d)
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### Question. 43

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?

(a) (b) (c) (d)
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### Question. 44

Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx² , where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 2 66(2/3). However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit

How many units should Mr. David produce daily?

(a) (b) (c) (d)
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### Question. 45

Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx² , where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 2 66(2/3). However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit

What is the maximum daily profit, in rupees, that Mr. David can realize from his business?

(a) (b) (c) (d)
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### Question. 46

Let f(x) = max (2x + 1, 3– 4x), where x is any real number. Then the minimum possible value of f(x) is

(a) (b) (c) (d)
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### Question. 47

The graph of y – x against y + x is as shown below. (All graphs in this question are drawn to scale and the same scale is used on each axis) Which of the following shows the graph of y against x?

(a) (b) (c) (d)
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### Question. 48

Let g(x) be a function such that g(x + 1) + g(x – 1) = g(x) for every real x. Then for what value of p is the relation g(x + p) = g(x) necessarily true for every real x?

(a) (b) (c) (d)
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### Question. 49 Which of the following is necessarily true?

(a) (b) (c) (d)
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### Question. 50  (a) (b) (c) (d)
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### Question. 51

Let f (x) = ax² – b | x |, where a and b are constants. Then at x = 0, f (x) is

(a) (b) (c) (d)
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### Question. 52

If f (x) = x³ – 4x + p, and f (0) and f (1) are of opposite signs, then which of the following is necessarily true?

(a) (b) (c) (d)
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### Question. 53

On January 1, 2004 two new societies, S1 and S2 , are formed, each with n members. On the first day of each subsequent month, S1 adds b members while S2 multiplies its current number of members by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b = 10.5n, what is the value of r?

(a) (b) (c) (d)
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### Question. 54

Consider the following two curves in the x-y plane; y = x³ + x² + 5; y = x² + x + 5

Which of the following statements is true for -2 ≤ x ≤ 2 ?

(a) (b) (c) (d)
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### Question. 55

When the curves y = log10 x and y = x–1 are drawn in the x-y plane, how many times do they intersect for values x ≥ 1 ?

(a) (b) (c) (d)
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### Question. 56 (a) (b) (c) (d)
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### Question. 57

Let g (x) = max (5–x, x + 2). The smallest possible value of g (x) is

(a) (b) (c) (d)
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### Question. 58

Functions m and M are defined as follows:

m(a, b, c) = min (a + b, c, a)

M(a, b, c) = max (a + b, c, a)

If a = – 2, b = – 3 and c = 2 what is the maximum between [m(a,b,c) M(a,b,c)]/ 2 + and [m(a,b,c) M(a,b,c)]/2 ?

(a) (b) (c) (d)
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### Question. 59

Functions m and M are defined as follows:

m(a, b, c) = min (a + b, c, a)

M(a, b, c) = max (a + b, c, a)

If a and b, c are negative, then what gives the minimum of a and b?

(a) (b) (c) (d)
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### Question. 60

Functions m and M are defined as follows:

m(a, b, c) = min (a + b, c, a)

M(a, b, c) = max (a + b, c, a)

What is m (M(a–b, b,c), m (a + b,c,b) , –M (a,b,c)) for a = 2, b= 4, c = 3?

(a) (b) (c) (d)
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### Question. 61

Suppose, for any real number x, [x] denotes the greatest integer less than or equal to x. Let L (x, y) = [x] + [y] + [x + y] and R(x,y)= [2x] + [2y]. Then it’s impossible to find any two positive real numbers x and y for which

(a) (b) (c) (d)
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### Question. 62 (a) (b) (c) (d)
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### Question. 63  (a) (b) (c) (d)
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### Question. 64 If x = –1 what will f5 (x) be

(a) (b) (c) (d)
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### Question. 65

Graphs of some functions are given. Mark the correct options from the following: (a) (b) (c) (d)
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### Question. 66

Graphs of some functions are given. Mark the correct options from the following: (a) (b) (c) (d)
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### Question. 67

Graphs of some functions are given. Mark the correct options from the following: (a) (b) (c) (d)
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### Question. 68

Which of the following equations will best fit for the given data ? (a) (b) (c) (d)
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### Question. 69

If f(0, y) y 1, and f (x 1, y) f (x,f (x, y)) = + += then, what is the value of f (1, 2) ?

(a) (b) (c) (d)
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### Question. 70 (a) (b) (c) (d)
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### Question. 71 Which of the following is necessarily false ?

(a) (b) (c) (d)
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### Question. 72 If f (x, y) = g (x, y) then

(a) (b) (c) (d)
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### Question. 73

Certain relation is defined among variable A & B.

Using the relation answer the questions given below :

@ (A, B) = average of A and B

\ (A, B) = product of A and B

x (A, B) = the result when A is divided by B

The average of A, B and C is given by

(a) (b) (c) (d)
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### Question. 74

Certain relation is defined among variable A & B.

Using the relation answer the questions given below :

@ (A, B) = average of A and B

\ (A, B) = product of A and B

x (A, B) = the result when A is divided by B

The sum of A and B is given by

(a) (b) (c) (d)
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### Question. 75

There is a set of 'n' natural numbers. The function 'H' is such that it finds the highest common factor between any two numbers. What is the minimum number of times that the function has to be invoked to find the H.C.F. of the given set of numbers?

(a) (b) (c) (d)
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### Question. 76

Any function has been defined for a variable x, where range of x (–2,2). (a) (b) (c) (d)
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### Question. 77

Any function has been defined for a variable x, where range of x (–2,2). (a) (b) (c) (d)
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### Question. 78

Any function has been defined for a variable x, where range of x (–2,2). (a) (b) (c) (d)
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### Question. 79

Any function has been defined for a variable x, where range of x (–2,2). (a) (b) (c) (d)
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### Question. 80

If x & y are real numbers, the functions are defined as f (x, y) = | x + y |,F (x, y) = –f (x, y) and G (x, y) = –F (x, y) . Now with the help of this information answer the following questions.

What will be the final value given by the function G (f (G (F (f (2, –3),0) – 2),0), -1)?

(a) (b) (c) (d)
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### Question. 81

If x & y are real numbers, the functions are defined as f (x, y) = | x + y |,F (x, y) = –f (x, y) and G (x, y) = –F (x, y) . Now with the help of this information answer the following questions.

If y = x, which of the following will give x² as the final value ?

(a) (b) (c) (d)
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### Question. 82

If x & y are real numbers, the functions are defined as f (x, y) = | x + y |,F (x, y) = –f (x, y) and G (x, y) = –F (x, y) . Now with the help of this information answer the following questions.

Which of the following will be necessarily true?

(a) (b) (c) (d)
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### Question. 83

The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1. a ∇ b = (ab)a+b if ab is positive else a ∇ b = 1.

(2 # 1)/(1 ∇ 2) =

(a) (b) (c) (d)
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### Question. 84

The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1. a ∇ b = (ab)a+b if ab is positive else a ∇ b = 1. (a) (b) (c) (d)
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### Question. 85

The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1. a ∇ b = (ab)a+b if ab is positive else a ∇ b = 1.

((X # – Y)/(– X ∇ Y)) = 3/8, then which of the following must be true?

(a) (b) (c) (d)
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### Question. 86

The following functions have been defined :

la (x, y, z) = min (x + y, y + z)

le (x, y, z) = max (x – y, y – z)

ma (x, y, z) = (½) [le (x, y, z) + la (x, y, z)]

Given that x > y > z > 0, which of the following is necessarily true?

(a) (b) (c) (d)
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### Question. 87

The following functions have been defined :

la (x, y, z) = min (x + y, y + z)

le (x, y, z) = max (x – y, y – z)

ma (x, y, z) = (½) [le (x, y, z) + la (x, y, z)]

What is the value of ma (10, 4, le (la (10, 5, 3), 5, 3)) ?

(a) (b) (c) (d)
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### Question. 88

The following functions have been defined :

la (x, y, z) = min (x + y, y + z)

le (x, y, z) = max (x – y, y – z)

ma (x, y, z) = (½) [le (x, y, z) + la (x, y, z)]

For x = 15, y = 10 and z = 9, find the value of : le (x, min (y, x – z), le (9, 8, ma (x, y, z)))

(a) (b) (c) (d)
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### Question. 89

A,S, M and D are functions of x and y, and they are defined as follows :

A (x, y) = x + y

S (x, y) = x – y

M (x, y) = xy

D (x, y) = x / y

where y ≠ 0.

What is the value of S[M(D(A(a, b),2), D(A(a, b),2)),M(D(S(a,b),2), D(S(a, b), 2))] ?

(a) (b) (c) (d)
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### Question. 90

A,S, M and D are functions of x and y, and they are defined as follows :

A (x, y) = x + y

S (x, y) = x – y

M (x, y) = xy

D (x, y) = x / y

where y ≠ 0.

What is the value of M(M(A(M(x, y), S (y, x)), x), A (y,x)) for x =2 , y = 3 ?

(a) (b) (c) (d)
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### Question. 91

Largest value of min (2 + x², 6 – 3x), when x > 0 is

(a) (b) (c) (d)
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### Question. 92

le (x, y) = least of (x, y)

mo (x) = |x|

me (x, y) = maximum of (x, y)

For what values of a le (a² – 3a, a – 3) < 0 ?

(a) (b) (c) (d)
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### Question. 93

le (x, y) = least of (x, y)

mo (x) = |x|

me (x, y) = maximum of (x, y)

Find the value of me (a mo(le (a,b)), mo (a me (mo (a) mo (b)))), + + at a = –2 and b = –3.

(a) (b) (c) (d)
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### Question. 94

le (x, y) = least of (x, y)

mo (x) = |x|

me (x, y) = maximum of (x, y)

Which of the following must always be correct for a, b > 0

(a) (b) (c) (d)
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### Question. 95

le (x, y) = least of (x, y)

mo (x) = |x|

me (x, y) = maximum of (x, y)

For what values of a is me (a² -  3a, a - 3) < 0?

(a) (b) (c) (d)
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### Question. 96 What is value of (gofofogogof )(x)(fogofog)(x) ?

(a) (b) (c) (d)
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### Question. 97 What is the value of fo(fog) o (gof) (x)?

(a) (b) (c) (d)
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### Question. 98 fog (x) =

(a) (b) (c) (d)
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### Question. 99 For what value of x; f (x) = g (x – 3) ?

(a) (b) (c) (d)
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### Question. 100

If md (x) = | x |,

mn (x, y) = minimum of x and y and

Ma (a, b, c, ....) = maximum of a, b, c, ....

Given that a > b then the relation Ma [md (a), mn (a, b)] = mn [a, md (Ma (a, b))] does not hold if

(a) (b) (c) (d)
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### Question. 101

If md (x) = | x |,

mn (x, y) = minimum of x and y and

Ma (a, b, c, ....) = maximum of a, b, c, ...

Value of Ma [md (a), mn (md(b), a), mn (ab, md(ac))] where a = –2, b = –3, c = 4 is

(a) (b) (c) (d)
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