Functions & Graphs — CAT Previous-Year Questions

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Answer the next 2 questions based on the information given below.

Let a₁ = p and b₁ = q, where p and q are positive quantities.

Define:
a_n = pb_n−1     b_n = qb_n−1,  for even n > 1 and 
a_n = pa_{n − 1}   b_n = qa_{n − 1},  for odd n > 1.

Answer the next 2 questions based on the information given below:

A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at  a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at  a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Answer the following question based on the information given below.

f₁(x) = x                  0 ≤ x ≤ 1
        = 1                   x ≥ 1
        = 0                   otherwise

f₂(x) = f₁(–x)           for all x
f₃(x) = –f₂(x)           for all x
f₄(x) = f₃(–x)           for all x

Answer the following question based on the information given below.

f₁(x) = x                  0 ≤ x ≤ 1
        = 1                   x ≥ 1
        = 0                   otherwise

f₂(x) = f₁(–x)           for all x
f₃(x) = –f₂(x)           for all x
f₄(x) = f₃(–x)           for all x

Answer the following question based on the information given below.

New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

Answer the following question based on the information given below.

New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

Answer the following question based on the information given below.

A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E1, the east end point of OR to N2, the north end point of IR; from N1, the north end point of OR to W2, the west end point of IR; from W1, the west end point of OR, to S2, the south end point of IR; and from S1, the south end point of OR to E2, the east endpoint of IR. Traffic moves at a constant speed of 30π km/hr on the OR road, 20π km/hr on the IR road, and km/hr on all the chord roads.

Answer the following question based on the information given below.

A city has two perfectly circular and concentric ring roads, the outer ring road (OR) being twice as long as the inner ring road (IR). There are also four (straight line) chord roads from E1, the east end point of OR to N2, the north end point of IR; from N1, the north end point of OR to W2, the west end point of IR; from W1, the west end point of OR, to S2, the south end point of IR; and from S1, the south end point of OR to E2, the east endpoint of IR. Traffic moves at a constant speed of 30π km/hr on the OR road, 20π km/hr on the IR road, and km/hr on all the chord roads.

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Answer the following question based on the information given below.

For real numbers x, y, let

f(x, y) = Positive square-root of (x + y), if (x + y)^0.5 is real

= (x + y)²,    otherwise

g(x, y) = (x + y)², if (x + y)^0.5 is real

= –(x + y), otherwise

Answer the following question based on the information given below.

For real numbers x, y, let

f(x, y) = Positive square-root of (x + y), if (x + y)^0.5 is real

= (x + y)²,    otherwise

g(x, y) = (x + y)², if (x + y)^0.5 is real

= –(x + y), otherwise

Answer the following question based on the information given below.

For three distinct positive real numbers x, y and z, let

f(x, y, z) = min(max(x, y), max(y, z), max(z, x))

g(x, y, z) = max(min(x, y), min(y, z), min(z, x))

h(x, y, z) = max(max(x, y), max(y, z), max(z, x))

j(x, y, z) = min(min(x, y), min(y, z), min(z, x))

m(x, y, z) = max(x, y, z)

n(x, y, z) = min(x, y, z)

Answer the following question based on the information given below.

For three distinct positive real numbers x, y and z, let

f(x, y, z) = min(max(x, y), max(y, z), max(z, x))

g(x, y, z) = max(min(x, y), min(y, z), min(z, x))

h(x, y, z) = max(max(x, y), max(y, z), max(z, x))

j(x, y, z) = min(min(x, y), min(y, z), min(z, x))

m(x, y, z) = max(x, y, z)

n(x, y, z) = min(x, y, z)

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.

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.

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.

Answer the following question based on the information given below.

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, ....

Answer the following question based on the information given below.

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, ....

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Answer the next 4 questions based on the following information.

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).

Choose the answer as

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

Answer the next 4 questions based on the following information.

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).

Choose the answer as

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

Answer the next 4 questions based on the following information.

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).

Choose the answer as

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

Answer the next 4 questions based on the following information.

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).

Choose the answer as

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

Directions: Answer the questions based on the following information.
Let x and y be real numbers and let
f (x,y) = |x + y| , F(f (x,y)) = −f (x,y)
and G(f (x, y)) = −F(f (x, y))

Directions: Answer the questions based on the following information.
Let x and y be real numbers and let
f (x,y) = |x + y| , F(f (x,y)) = −f (x,y)
and G(f (x, y)) = −F(f (x, y))

Directions: Answer the questions based on the following information.
Let x and y be real numbers and let
f (x,y) = |x + y| , F(f (x,y)) = −f (x,y)
and G(f (x, y)) = −F(f (x, y))

Direction: Answer the questions based on the following information.

For these questions 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) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Direction: Answer the questions based on the following information.

For these questions 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) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Direction: Answer the questions based on the following information.

For these questions 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) = $\frac{1}{2}$ [le(x, y, z) + la(x, y, z)]

Direction: Answer the questions based on the following information.

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) = $\frac{x}{y}$, y ≠ 0

Direction: Answer the questions based on the following information.

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) = $\frac{x}{y}$, y ≠ 0

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Directions for next 4 questions: Answer the questions based on the following information.

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

Directions for next 4 questions: Answer the questions based on the following information.

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

Directions for next 4 questions: Answer the questions based on the following information.

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

Directions for next 4 questions: Answer the questions based on the following information.

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

Answer the next 2 questions based on the following information:

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Answer the next 2 questions based on the following information:

If
md(x) = x ,
mn(x,y) = minimum of x and y and
Ma(a,b,c,...) = maximum of a,b,c…

Answer the next 4 questions based on the information given below:

If f (x) = 2x + 3 and g(x) =  $\frac{x-3}{2},$ then

Answer the next 4 questions based on the information given below:

If f (x) = 2x + 3 and g(x) =  $\frac{x-3}{2},$ then

Answer the next 4 questions based on the information given below:

If f (x) = 2x + 3 and g(x) =  $\frac{x-3}{2},$ then

Answer the next 4 questions based on the information given below:

If f (x) = 2x + 3 and g(x) =  $\frac{x-3}{2},$ then

Answer the next 2 questions based on the information given below:

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.

Answer the next 2 questions based on the information given below:

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.

Answer the following questions based on the information given below:

ABC forms an equilateral triangle in which B is 2 km from A. A person starts walking from B in a direction parallel to AC and stops when he reaches a point D directly east of C. He, then, reverses direction and walks till he reaches a point E directly south of C.

Use the following information:

Prakash has to decide whether or not to test a batch of 1000 widgets before sending them to the buyer. In case he decides to test, he has two options: (a) Use test I ; (b) Use test II. Test I cost Rs. 2 per widget. However, the test is not perfect. It sends 20% of the bad ones to the buyer as good. Test II costs Rs. 3 per widget. It brings out all the bad ones. A defective widget identified before sending can be corrected at a cost of Rs. 25 per widget. All defective widgets are identified at the buyer’s end and penalty of Rs. 50 per defective widget has to be paid by Prakash.

Use the following information:

Prakash has to decide whether or not to test a batch of 1000 widgets before sending them to the buyer. In case he decides to test, he has two options: (a) Use test I ; (b) Use test II. Test I cost Rs. 2 per widget. However, the test is not perfect. It sends 20% of the bad ones to the buyer as good. Test II costs Rs. 3 per widget. It brings out all the bad ones. A defective widget identified before sending can be corrected at a cost of Rs. 25 per widget. All defective widgets are identified at the buyer’s end and penalty of Rs. 50 per defective widget has to be paid by Prakash.

Answer the following questions based on the information given below:

There were a hundred schools in a town. Of these, the number of schools having a play – ground was 30, and these schools had neither a library nor a laboratory. The number of schools having a laboratory alone was twice the number of those having a library only. The number of schools having a laboratory as  well as a library was one fourth the number of those having a laboratory alone. The number of schools  having either a laboratory or a library or both was 35.