Basics (Functions) — CAT Previous-Year Questions

Consider two sets A = {2, 3, 5, 7, 11, 13} and B = {1, 8, 27}. Let f be a function from A to B such that for every element b in B, there is at least one element a in A such that f(a) = b. Then, the total number of such function f is

A function f maps the set of natural numbers to whole numbers, such that f(xy) = f(x)f(y) + f(x) + f(y) for all x, y and f(p) = 1 for every prime number p. Then, the value of f(160000) is

For any non-zero real number x, let f(x) + 2f(1/x) = 3x. Then, the sum of all possible values of x for which f(x) = 3, is

Suppose f(x, y) is a real-valued function such that f(3x + 2y, 2x - 5y) = 19x, for all real numbers x and y. The value of x for which f(x, 2x) = 27, is

For any real number x, let [x] be the largest integer less than or equal to x. If $\sum _{n=1}^N\left[\frac{1}{5}+\frac{n}{25}\right]$ = 25, then N is

Among 100 students, x₁ have birthdays in January, x₂ have birthdays in February, and so on. If x₀ = max(x₁, x₂, …., x₁₂), then the smallest possible value of x₀ is

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

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 8 f(m + 1) − f(m) = 2, then m equals

Consider a function f satisfying  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

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

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

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

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

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

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

Let f(x) be a function satisfying f(x) × f(y) = f(xy) for all real x, y. Let f(2) = 4, then what is the value of $f\left(\frac{1}{2}\right)?$

A function f(x) satisfies f(1) = 3600, and f(1) + f(2) + ... + f(n) = n²f(n), for all positive integers n > 1. What is the value of f(9)?

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

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

The number of non-negative real roots of 2^x – x – 1 = 0 equals

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

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 of the following?

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In the above table, for suitably chosen constants a, b and c, which one of the following best describes the relation between y and x?

Which of the following expressions yields a positive value for every pair of non-zero real number (x, y)?

Under which of the following conditions is f(x, y) necessarily greater than g(x, y)?

Which of the following expressions is necessarily equal to 1?

Which of the following expressions is indeterminate?

The set of all positive integers is the union of two disjoint subsets

{f(1), f(2) ....f(n),......} and {g(1), g(2),......,g(n),......}, where

f (1) < f(2) <...< f(n) ....., and g(1) < g(2) <...< g(n) ......., and

g(n) = f(f(n)) + 1 for all n ≥ 1. 

What is the value of g(1)?

For all non-negative integers x and y, f(x, y) is defined as below

f(0, y) = y + 1

f(x + 1, 0) = f(x, 1)

f(x + 1,y + 1) = f(x, f(x + 1, y))

Then, what is the value of f(1, 2)?

Which of the following statements is true?

Given that x > y > z > 0. Which of the following is necessarily true?

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

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))).

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

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

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

For what values of ‘a’ is me(a² – 3a, a – 3) < 0?

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

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

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

The maximum possible value of y = min (1/2 – 3x²/4, 5x²/4) for the range 0 < x < 1 is

A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?  

If y = f(x) and f(x) = $\frac{(1–x)}{(1+x)}$, which of the following is true?

Let Y = minimum of {(x + 2), (3 – x)}. What is the maximum value of Y for 0 ≤ x ≤ 1?