logo
iconCall Us
Test NameNo. of QuestionsMarks (of each)TimeTake Test
Test-110340 MinutesTake Test
Test-210340 MinutesTake Test
Test-310340 MinutesTake Test
Test-410340 MinutesTake Test
Test-510340 MinutesTake Test
Test-610340 MinutesTake Test
Test-710340 MinutesTake Test
Test-810340 MinutesTake Test

Tips To Solve CAT Functions And Graphs

- Fundamentals of this concept are useful in solving the questions of the other topics by assuming the unknown values as variables. Make sure to cover other inter-related concepts of CAT syllabus. All the inter-related concepts need to be covered to have a good foundation in concepts.

- Be careful of silly mistakes in this topic, as that is how students generally lose marks here. The number of equations needed to solve the given problem equals the number of variables. A linear equation is an equation which gives a straight line when plotted on a graph.

- If you are confused, enrolling in CAT online coaching will help you a long way.

- Linear equations can be of one variable or two variables, or three variables.

  • Let a, b, c and d be constants, and x, y, and z are variables. A general form of a single variable linear equation is ax + b = 0.
  • A general form of two-variable linear equation is ax + by = c.
  • A general form of three-variable linear equation is ax + by + cz = d.

CAT Functions And Graphs PDF

To help CAT aspirants in their preparation, we have made a comprehensive formula PDF containing all the important linear equations that are essential. This PDF includes all the necessary formulas, techniques, and examples required to solve linear equations efficiently. Click on the link below to download the Linear equations formula PDF.

1. Linear Equations Formulae: Solving Linear Equations

For equations of the form ax + by = c and mx + ny = p, find the LCM of b and n.

Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1.

2. Linear Equations Formulae: Straight Lines

Equations with 2 variables: Consider two equations ax + by = c and mx + ny = p. Each of these equations represents two lines on the x-y coordinate plane. The solution of these equations is the point of intersection.

If am=bncp: This means that both the equations have the same slope but different intercepts, and hence are parallel to each other. There is no point of intersection and no solution.

If ambn: They have different slopes and hence must intersect at some point, resulting in a unique solution.

If am=bn=cp: The two lines have the same slope and intercept. Hence, they are the same lines. As they have infinite points common between them, there are infinitely many solutions possible.

Question 1.

Coachify Logo

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

A
B
C
D

Question 2.

Coachify Logo

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

A
B
C
D

Question 3.

Coachify Logo

Suppose for all integers x, there are two functions f and g such that f(x) + f(x - 1) - 1 = 0 and g(x) = x2. If f(x2 - x) = 5, then the value of the sum f(g(5)) + g(f(5)) is

A
B
C
D

Question 4.

Coachify Logo

Let r be a real number and f(x) = { 2x-r if x \ge  r

                                                          { r if x<r

Then, the equation f(x) = f(f(x)) holds for all real values of x where

A
x ≠ r
B
x ≥ r
C
x > r
D
x ≤ r

Question 5.

Coachify Logo

f(x) = x2+2x15x27x18\frac{x^2+2x-15}{x^2-7x-18}is negative if and only if

A
X < -5 or 3 < x < 9
B
-2 < x < 3 or x > 9
C
-5 < x < -2 or 3 < x < 9
D
x < -5 or -2 < x <

Question 6.

Coachify Logo

If f(x) = x2 – 7x and g(x) = x + 3, then the minimum value of f(g(x)) – 3x is

A
-12
B
-16
C
-15
D
20

Question 7.

Coachify Logo

Among 100 students, x1x_{1}have birthdays in January, x2x_{2} have birthdays in February, and so on. Ifx0x_{0} = max(x1, x2, …., x12), then the smallest possible value ofx0x_{0} is

A
10
B
8
C
12
D
9

Question 8.

Coachify Logo

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

A
20
B
40
C
0
D
10

Question 9.

Coachify Logo

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0 and y ≤ 1 is

A
B
C
D

Question 10.

Coachify Logo

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
16
B
1
C
4
D
0

Question 11.

Coachify Logo

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

A
0
B
15.9375
C
3
D
14.0625

Question 12.

Coachify Logo

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

A
B
C
D

Question 13.

Coachify Logo

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

A
B
C
D

Question 14.

Coachify Logo

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

A
B
C
D

Question 15.

Coachify Logo

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

A
B
C
D

Question 16.

Coachify Logo

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

A
B
C
D

Question 17.

Coachify Logo

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

A
B
C
D

Question 18.

Coachify Logo

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

A
1
B
0
C
√2
D
√32

Question 19.

Coachify Logo

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

A
2
B
13
C
6
D
23

Question 20.

Coachify Logo

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
16
B
18
C
36
D
40

Question 21.

Coachify Logo

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

A
B
C
D

Question 22.

Coachify Logo

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

A
5/2 < x < 7/2
B
x \le 5/2 or x \ge 7/2
C
x<5/2 or x \ge 7/2
D
5/2 \le x \le 7/2

Question 23.

Coachify Logo

What is the value of a + b + c?

A
9
B
14
C
13
D
Cannot be determined

Question 24.

Coachify Logo

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(1/2) ?

A
0
B
1/4
C
1/2
D
1

Question 25.

Coachify Logo

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)?

A
80
B
240
C
200
D
180

Question 26.

Coachify Logo

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 has been used on each axis). Then, which of the options given shows the graph of y against x.

a)  

b)

c)

d)

A
B
C
D

Question 27.

Coachify Logo

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

A
1/3
B
1/2
C
2/3
D
5/3

Question 28.

Coachify Logo

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

A
8
B
12
C
16
D
20

Question 29.

Coachify Logo

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
5
B
3
C
2
D
6

Question 30.

Coachify Logo

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

A
maximized whenever a > 0, b > 0
B
maximized whenever a > 0, b < 0
C
minimized whenever a > 0, b > 0
D
minimized whenever a > 0, b < 0

Question 31.

Coachify Logo

How many of the following products are necessarily zero for every xf1(x)f2(x), f2(x)f3(x), f2(x)f4(x)?

A
0
B
1
C
2
D
3

Question 32.

Coachify Logo

Which of the following is necessarily true?

A
f4(x) = f1(x) for all x
B
f1(x) = –f3(–x) for all x
C
f2(–x) = f4(x) for all x
D
f1(x) + f3(x) = 0 for all x

Question 33.

Coachify Logo

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

A
0
B
1
C
2
D
3

Question 34.

Coachify Logo

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 of x ≥ 1?

A
Never
B
Once
C
Twice
D
more than Twice

Question 35.

Coachify Logo

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

A
4.0
B
4.5
C
1.5
D
None of the Above

Question 36.

Coachify Logo

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

A
x = 2.3
B
x = 2.5
C
x = 2.7
D
None of the above

Question 37.

Coachify Logo

Consider the following two curves in the X-Y planey = x3 + x2 + 5y = x2 + x + 5Which of the following statements is true for −2 ≤ x ≤ 2?

A
The two curves intersect once
B
The two curves intersect Twice
C
The two curves do not intersect
D
The two curves intersect thrice

Question 38.

Coachify Logo

If f ( x ) = \log(\1+x/1-x) , then f(x) + f(y) =

A
f(x + y)
B
f(1 + xy)
C
(x + y) f(1 + xy)
D
f(x+y1+xy)f(\frac{x+y}{1+xy})

Question 39.

Coachify Logo

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?

A
L(x, y) = R(x, y)
B
L(x, y) ≠ R(x, y)
C
L(x, y) < R(x, y)
D
L(x, y) > R(x, y)

Question 40.

Coachify Logo

​​​​​​​

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?

A
y = a + bx
B
y = a + bx + cx2
C
y = ea + bx
D
None of the above

Question 41.

Coachify Logo

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

A
f(x, y) – g(x, y)
B
f(x, y) – (g(x, y))^2
C
g(x, y) – (f(x, y))^2
D
f(x, y) + g(x, y)

Question 42.

Coachify Logo

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

A
Both x and y are less than –1
B
Both x and y are positive
C
Both x and y are negative
D
y > x

Question 43.

Coachify Logo

Which of the following expressions is necessarily equal to 1?

A
(f(x, y, z) – m(x, y, z))/(g(x, y, z) – h(x, y, z))
B
(m(x, y, z) – f(x, y, z))/(g(x, y, z) – n(x, y, z))
C
(j(x, y, z) – g(x, y, z))/h(x, y, z)
D
(f(x, y, z) – h(x, y, z))/f(x, y, z)

Question 44.

Coachify Logo

Which of the following expressions is indeterminate?

A
(f(x, y, z) – h(x, y, z))/(g(x, y, z) – j(x, y, z))
B
(f(x, y, z) + h(x, y, z) + g(x, y, z) + j(x, y, z))/(j(x, y, z) + h(x, y, z) – m(x, y, z) – n(x, y, z))
C
(g(x, y, z) – j(x, y, z))/(f(x, y, z) – h(x, y, z))
D
(h(x, y, z) – f(x, y, z))/(n(x, y, z) – g(x, y, z))

Question 45.

Coachify Logo

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

A
1
B
2
C
3
D
4

Question 46.

Coachify Logo

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

A
1
B
2
C
3
D
4

Question 47.

Coachify Logo

Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as

1. if f(x) = 3 f(–x);

2. if f(x) = –f(–x);

3. if f(x) = f(–x); and

4. if 3 f(x) = 6 f(–x), for x ≥ 0.

A
1
B
2
C
3
D
4

Question 48.

Coachify Logo

For a real number x, let

f(x) = 1/(1 + x),                   if x is non-negative

      = 1+ x,                          if x is negative

f n(x) = f(f n – 1(x)), n = 2, 3, ....

What is the value of the product, f(2)f2(2)f3(2)f4(2)f5(2)?

 

A
1/3
B
3
C
1/18
D
None of these

Question 49.

Coachify Logo

r is an integer > 2. Then, what is the value of f r – 1(–r) + f r(–r)+ f r + 1 (–r)?

A
-1
B
0
C
1
D
None of these

Question 50.

Coachify Logo

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)?

A
ZERO
B
TWO
C
ONE
D
Cannot be determined

Question 51.

Coachify Logo

For all non-negative integers x and y, f(x, y) is defined as belowf(0, y) = y + 1f(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)?

A
Two
B
Four
C
Three
D
Cann't be determined

Question 52.

Coachify Logo

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

A
if F1(x) = –F(x)
B
if F1(x) = F(–x)
C
if F1(x) = -F(–x)
D
if none of the above is true

Question 53.

Coachify Logo

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

A
if F1(x) = –F(x)
B
if F1(x) = F(-x)
C
if F1(x) = –F(-x)
D
if none of the above is true

Question 54.

Coachify Logo

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

A
if F1(x) = –F(x)
B
if F1(x) = F(-x)
C
if F1(x) = –F(-x)
D
if none of the above is true

Question 55.

Coachify Logo

In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straightline segments, shown as solid lines, in the domain x ∈ (−2, 2).

A
if F1(x) = –F(x)
B
if F1(x) = F(-x)
C
if F1(x) = –F(-x)
D

Question 56.

Coachify Logo

Which of the following statements is true?

A
F(f(x, y)) ⋅ G(f(x, y)) = –F(f(x, y)) ⋅ G(f(x, y))
B
F(f(x, y)) ⋅ G(f(x, y)) > –F(f(x, y)) ⋅ G(f(x, y))
C
F(f(x, y)) ⋅ G(f(x, y)) ≠ G(f(x, y)) ⋅ G(f(x, y))
D
F(f(x, y)) + G(f(x, y)) + f(x, y) = f(–x, –y)

Question 57.

Coachify Logo

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

A
3
B
2
C
1
D
0

Question 58.

Coachify Logo

Which of the following expressions yields x2 as its result?

A
F(f (x, − x))⋅G(f (x, − x))
B
F(f (x, x))⋅G(f (x, x))⋅ 4
C
-F(f(x, x)) ∙ G(f(x, x)) ÷ log2 16
D
f (x, x)⋅ f (x, x)

Question 59.

Coachify Logo

 

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

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2[le(x, y, z) + la(x, y, z)]

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

A
la(x, y, z) < le(x, y, z)
B
ma(x, y, z) < la(x, y, z)
C
ma(x, y, z) < le(x, y, z)
D
None of these

Question 60.

Coachify Logo

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

le(x, y, z) = max (x − y, y − z)

ma(x, y, z) = 1/2 [le(x, y, z) + la(x, y, z)]

 

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

A
7
B
6.5
C
8
D
7.5

Question 61.

Coachify Logo

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
5
B
12
C
9
D
4

Question 62.

Coachify Logo

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 ,  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
60
B
140
C
25
D
70

Question 63.

Coachify Logo

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 , 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
a^2 + b^2
B
ab
C
a^2 - b^2
D
a/b

Question 64.

Coachify Logo

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

A
1
B
2
C
32
D
4

Question 65.

Coachify Logo

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
1
B
0
C
5
D
3

Question 66.

Coachify Logo

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
mo(le(a, b)) ≥ (me(mo(a), mo(b))
B
mo(le(a, b)) > (me(mo(a), mo(b))
C
mo(le(a, b)) < (le(mo(a), mo(b))
D
mo(le(a,b)) = le(mo(a), mo(b))

Question 67.

Coachify Logo

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
a > 3
B
0 < a < 3
C
a < 0
D
a = 3

Question 68.

Coachify Logo

le(x, y) = Least of (x, y)
mo(x) = |x|
me(x, y) = Maximum of (x, y)

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

A
a > 3
B
0 < a < 3
C
a < 0
D
Both b and c

Question 69.

Coachify Logo

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
2
B
6
C
8
D
2

Question 70.

Coachify Logo

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
a < 0, b < 0
B
a > 0, b > 0
C
a > 0, b < 0, |a| < |b|
D
a > 0, b < 0, |a| > |b|

Question 71.

Coachify Logo

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

fog(x) is equal to

A
1
B
gof(x)
C
(15x+9)/(16x-5)
D
1/x

Question 72.

Coachify Logo

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then 

For what value of x; f (x) = g(x −3)?

A
-3
B
1/4
C
-4
D
None of these

Question 73.

Coachify Logo

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

What is the value of (gofofogogof)(x) × (fogofog)(x)?

A
x
B
x^2
C
(5x+3)/(4x-1)
D
(x+3)(5x+3)/(4x-5)(4x-1)

Question 74.

Coachify Logo

If f (x) = 2x + 3 and g(x) = (x-3)/2 , then

What is the value of fo(fog)o(gof)(x)?

A
x
B
x^2
C
2x+3
D
(x+3)/(4x-5)

Question 75.

Coachify Logo

A function f(x) is said to be even if f(–x) = f(x), and odd if f(–x) = –f(x). Thus, for example, the function given by f(x) = x2 is even, while the function given by f(x) = x3 is odd. Using this definition, answer the following questions.

The function given by f(x) = |x|3 is

A
even
B
odd
C
nither
D
both

Question 76.

Coachify Logo

A function f(x) is said to be even if f(–x) = f(x), and odd if f(–x) = –f(x). Thus, for example, the function given by f(x) = x2 is even, while the function given by f(x) = x3 is odd. Using this definition, answer the following questions.

The sum of two odd functions

A
is always an even functions
B
is always an odd function
C
is sometimes odd and sometimes even
D
may be neither odd nor even

Question 77.

Coachify Logo

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

A
1/3
B
1/2
C
5/27
D
5/16

Question 78.

Coachify Logo

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?

A
y = (2x+3)/(3x+4)
B
y = (2x+3)/(3x-4)
C
y = (3x+4)/(4x-5)
D
None of these

Question 79.

Coachify Logo

If y = f(x) and f(x) = (1-x)/(1+x) , which of the following is true?

A
f(2x) = f(x) - 1
B
x = f(2y) - 1
C
f(1/x) = f(x)
D
x = f(y)

Question 80.

Coachify Logo

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

A
1.0
B
1.5
C
3.1
D
2.5

Explore Our Learning Zones

Access a vast collection of previous year questions to sharpen your exam preparation, all for free!

Whatsapp
Logo
CAT Batch Discount?
YoutubeInstagramTelegramWhatsappPhone