CAT Functions, Graphs and Statistics questions are very important to practice while preparing for the CAT exam. Every year these questions frequently appear in the CAT question papers. Check out the below-given practice questions from this topic(Functions, Graphs and Statistics). All these questions are taken from **CAT's previous papers** and separated year-wise. One can practice these questions in a test format or can also download these questions in a PDF format along with the detailed video solutions for every question explained by the CAT toppers. Click on the below link to download the CAT Functions, Graphs and Statistics Questions PDF.

**Note: No Sign-up is required to download the questions PDF**

Year | Weightage |

2022 | 5 |

2021 | 1 |

2020 | 8 |

2019 | 5 |

2018 | 3 |

Let $$0 \leq a \leq x \leq 100$$ and $$f(x) = \mid x - a \mid + \mid x - 100 \mid + \mid x - a - 50\mid$$. Then the maximum value of f(x) becomes 100 when a is equal to

correct answer:-**3**

Let $$f(x)$$ be a quadratic polynomial in $$x$$ such that $$f(x) \geq 0$$ for all real numbers $$x$$. If f(2) = 0 and f( 4) = 6, then f(-2) is equal to

correct answer:-**2**

Suppose for all integers x, there are two functions f and g such that $$f(x) + f (x - 1) - 1 = 0$$ and $$g(x ) = x^{2}$$. If $$f\left(x^{2} - x \right) = 5$$, then the value of the sum f(g(5)) + g(f(5)) is

correct answer:-**12**

Let r be a real number and $$f(x) = \begin{cases}2x -r & ifx \geq r\\ r &ifx < r\end{cases}$$. Then, the equation $$f(x) = f(f(x))$$ holds for all real values of $$x$$ where

correct answer:-**2**

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

correct answer:-**44**

If $$f(x)=x^{2}-7x$$ and $$g(x)=x+3$$, then the minimum value of $$f(g(x))-3x$$ is:

correct answer:-**4**

Let $$f(x)=x^{2}+ax+b$$ and $$g(x)=f(x+1)-f(x-1)$$. If $$f(x)\geq0$$ for all real x, and $$g(20)=72$$. then the smallest possible value of b is

correct answer:-**2**

The number of real-valued solutions of the equation $$2^{x}+2^{-x}=2-(x-2)^{2}$$ is:

correct answer:-**4**

The area of the region satisfying the inequalities $$\mid x\mid-y\leq1,y\geq0$$ and $$y\leq1$$ is

correct answer:-**3**

In a group of 10 students, the mean of the lowest 9 scores is 42 while the mean of the highest 9 scores is 47. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by

correct answer:-**2**

The area, in sq. units, enclosed by the lines $$x=2,y=\mid x-2\mid+4$$, the X-axis and the Y-axis is equal to

correct answer:-**1**

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

correct answer:-**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 these roots is

correct answer:-**4**

In how many ways can a pair of integers (x , a) be chosen such that $$x^{2}-2\mid x\mid+\mid a-2\mid=0$$ ?

correct answer:-**4**

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 (2$$^n$$ - 1) then a is equal to

correct answer:-**3**

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

correct answer:-**10**

The number of the real roots of the equation $$2 \cos (x(x + 1)) = 2^x + 2^{-x}$$ is

correct answer:-**2**

Let S be the set of all points (x, y) in the x-y plane such that $$\mid x \mid + \mid y \mid \leq 2$$ and $$\mid x \mid \geq 1.$$ Then, the area, in square units, of the region represented by S equals

correct answer:-**2**

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

correct answer:-**12**

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

correct answer:-**20**

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

correct answer:-**32**

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

correct answer:-**54**

Let $$f(x) = x^{2}$$ and $$g(x) = 2^{x}$$, for all real x. Then the value of f[f(g(x)) + g(f(x))] at x = 1 is

correct answer:-**3**

If $$f_{1}(x)=x^{2}+11x+n$$ and $$f_{2}(x)=x$$, then the largest positive integer n for which the equation $$f_{1}(x)=f_{2}(x)$$ has two distinct real roots is

correct answer:-**24**

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

correct answer:-**1**

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

correct answer:-**4**

$$f(x) = \frac{5x+2}{3x-5}$$ and $$g(x) = x^2 - 2x - 1$$, then the value of $$g(f(f(3)))$$ is

correct answer:-**1**

Let $$f(x)\neq0$$ for any '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 $$f(\frac{1}{2})$$?

correct answer:-**2**

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

seed(n) = n, if n < 10

seed(n) = 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?

correct answer:-**5**

Find the sum $$\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}} +....+ \sqrt{1+\frac{1}{2007^2}+\frac{1}{2008^2}}$$

correct answer:-**1**

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that $$f(5) = - 3f(2)$$. and that 3 is a root of $$f(x) = 0$$.

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

[CAT 2008]

correct answer:-**2**

Let $$f(x) = ax^2 + bx + c$$, where a, b and c are certain constants and $$a \neq 0$$ ?

It is known that f(5) = - 3f(2). and that 3 is a root of f(x) = 0.

What is the value of a + b + c?

[CAT 2008]

correct answer:-**5**

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

correct answer:-**1**

Which of the following best describes $$a_n + b_n$$ for even n?

correct answer:-**2**

If p = 1/3 and q = 2/3 , then what is the smallest odd n such that $$a_n+b_n < 0.01$$?

correct answer:-**4**

For general n, consider any two members of S that are friends. How many other members of S will be common friends of both these members?

correct answer:-**4**

For general n, how many enemies will each member of S have?

correct answer:-**4**

The graph of y - x (on the y axis) against y + x (on the x axis) is as shown below. (All graphs in this question are drawn to scale and the same scale and the same scale has been used on each axis.)

Which of the following shows the graph of y against x?

correct answer:-**4**

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

correct answer:-**5**

If $$a_1 = 1$$ and $$a_{n+1} - 3a_n + 2 = 4n$$ for every positive integer n, then $$a_{100}$$ equals

correct answer:-**3**

In the X-Y plane, the area of the region bounded by the graph of |x+y| + |x-y| = 4 is

correct answer:-**3**

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?

correct answer:-**4**

If $$f(x)=x^3-4x+p$$ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true

[CAT 2004]

correct answer:-**2**

Let $$ y = \frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+…}}}}$$. Then y equals?

correct answer:-**4**

Let $$f(x) = ax^2 - b|x|$$ , where a and b are constants. Then at x = 0, f(x) is

[CAT 2004]

correct answer:-**4**

If $$\frac{a}{b+c}=\frac{b}{a+c} =\frac{c}{b+a} =r$$, then r cannot take any value except

correct answer:-**3**

How many of the following products are necessarily zero for every x:

$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$

correct answer:-**3**

Which of the following is necessarily true?

correct answer:-**2**

When the curves $$y = log_{10}x$$ and $$y = x^{-1}$$ are drawn in the x-y plane, how many times do they intersect for values $$x \geq 1$$ ?

correct answer:-**2**

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

correct answer:-**4**

The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at

correct answer:-**2**

Consider the following two curves in the x-y plane:

$$y = x^3 + x^2 + 5$$

$$y = x^2 + x + 5$$

Which of following statements is true for $$-2 \leq x \leq 2$$ ?

correct answer:-**4**

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 is impossible to find any two positive real numbers x and y for which

correct answer:-**4**

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?

correct answer:-**2**

The area bounded by the three curves |x+y| = 1, |x| = 1, and |y| = 1, is equal to:

correct answer:-**2**

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)?

correct answer:-**2**

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)?

correct answer:-**2**

correct answer:-**3**

correct answer:-**4**

correct answer:-**2**

What is the value of the product, $$f(2) f^2(2)f^3(2) f^4(2)f^5(2)$$?

correct answer:-**3**

r is an integer 2. Then, what is the value of $$f^{r-1}(-r) + f^r(-r) + f^{r+1}(-r)$$?

correct answer:-**2**

Which of the following is necessarily greater than 1?

correct answer:-**4**

Which of the following expressions is necessarily equal to 1?

correct answer:-**1**

Which of the following expressions is indeterminate?

correct answer:-**2**

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

correct answer:-**4**

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

correct answer:-**1**

For two positive integers a and b define the function h(a,b):as the greatest common factor (G.C.F) of a, b. Let A be a set of n positive integers. G(A), the GCF of the elements of set A is computed by repeatedly using the function h.

The minimum number of times h is required to be used to compute G is:

correct answer:-**2**

correct answer:-**4**

correct answer:-**2**

correct answer:-**2**

correct answer:-**3**

Which of the following statements is true?

correct answer:-**4**

What is the value of f(G(f(1, 0)), f(F(f(1, 2)), G(f(1, 2))))?

correct answer:-**3**

Which of the following expressions yields $$x^2$$ as its result?

correct answer:-**3**

The robot reaches point (6, 6) when a sequence of three instructions is executed, the first of which is a GOTO(x, y) instruction, the second is WALKX(2) and the third is WALKY(4). What are the values of x and y?

correct answer:-**3**

The robot is initially at (x, y), x > 0 and y < 0. The minimum number of instructions needed to be executed to bring it to the origin (0,0) if you are prohibited from using the GOTO instruction is:

correct answer:-**1**

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

correct answer:-**4**

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

correct answer:-**2**

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?

correct answer:-**2**

What is the value of k for which the following system of equations has no solution:

2x-8y = 3 and kx +4y = 10

correct answer:-**3**

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

correct answer:-**4**

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

correct answer:-**4**