XAT Previous Year Questions Paper - 2021 | Crash Course for XAT - CAT PDF Download

 Page 1


XAT 2021
  Quant
1. If  where m and n are positive real numbers, then which of the following must be true?
A    
B    m = n
C    
D    
E    No values of m and n can satisfy the given equation
A n s w e r : E
Explanation:
 
Squarring on both sides
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
  
2. Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining
eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her
fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?
A    9
B    17
C    11
D    20
E    29
A n s w e r : C
Explanation:
Let the number of eggs bought = 9x+2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x-2
Given, 4x-2 <=5
=> x
Hence the max value of x is 1
Max number of eggs bought = 11
log m + 4 log n = 4 log ( m + 2 n)
+ m
1
= n
1
1
m +
2
n =
2
1
+ m
1
= n
1
2
log m n = 4 log ( m + 2 n)
=
 m n
( m + n)
m +
2
n +
2
m n = 0
= 4
7
  
.
Page 2


XAT 2021
  Quant
1. If  where m and n are positive real numbers, then which of the following must be true?
A    
B    m = n
C    
D    
E    No values of m and n can satisfy the given equation
A n s w e r : E
Explanation:
 
Squarring on both sides
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
  
2. Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining
eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her
fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?
A    9
B    17
C    11
D    20
E    29
A n s w e r : C
Explanation:
Let the number of eggs bought = 9x+2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x-2
Given, 4x-2 <=5
=> x
Hence the max value of x is 1
Max number of eggs bought = 11
log m + 4 log n = 4 log ( m + 2 n)
+ m
1
= n
1
1
m +
2
n =
2
1
+ m
1
= n
1
2
log m n = 4 log ( m + 2 n)
=
 m n
( m + n)
m +
2
n +
2
m n = 0
= 4
7
  
.
3. Mohan has some money ( ?M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that
offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both
interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is ?830. It is known
that one of the interest rates is 10% and that Mohan deposited more than ?1000 in each saving scheme at the start. What is the
value of M?
A    7500
B    6000
C    To solve this, the other interest rate must also be given.
D    4500
E    12000
A n s w e r : B
Explanation:
Let the total amount be 3x
Case 1: 
Smaller amount = x, rate of interest = 10
Larger amount = 2x, rate of interest = 5
Total amount received at the end of two years( smaller amount) = . CI = 0.21x
Total amount received at the end of two years( larger amount) =    CI = 0.205x
Given, 0.21x + 0.205x = 830
=> x = 2000
2x= 4000
Case 2:
Smaller amount = x, rate of interest = 20
Larger amount = 2x, rate of interest = 10
Total amount received at the end of two years( smaller amount) = . CI = 0.44x
Total amount received at the end of two years( larger amount) =  CI = 0.42x
Given, 0.44x+0.42x = 830
=> x = 965.11 which is not valid since it should be greater than 1000
4. A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the
shop to buy a unit each. Each one has a pre- determined choice of the colour and will not buy a unit of any other colour. All the
three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to
satisfy all the three customers?
A    
B    
C    
D    
E    
A n s w e r : C
x 1 + = ( 100
10
)
2
1.21 x
2 x 1 + = ( 100
5
)
2
2.205 x
x 1 + = ( 100
20
)
2
1.44 x
2 x 1 + = ( 100
10
)
2
2.42 x
5
4
9
7
3
2
9
8
3
1
  
.
Page 3


XAT 2021
  Quant
1. If  where m and n are positive real numbers, then which of the following must be true?
A    
B    m = n
C    
D    
E    No values of m and n can satisfy the given equation
A n s w e r : E
Explanation:
 
Squarring on both sides
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
  
2. Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining
eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her
fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?
A    9
B    17
C    11
D    20
E    29
A n s w e r : C
Explanation:
Let the number of eggs bought = 9x+2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x-2
Given, 4x-2 <=5
=> x
Hence the max value of x is 1
Max number of eggs bought = 11
log m + 4 log n = 4 log ( m + 2 n)
+ m
1
= n
1
1
m +
2
n =
2
1
+ m
1
= n
1
2
log m n = 4 log ( m + 2 n)
=
 m n
( m + n)
m +
2
n +
2
m n = 0
= 4
7
  
.
3. Mohan has some money ( ?M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that
offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both
interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is ?830. It is known
that one of the interest rates is 10% and that Mohan deposited more than ?1000 in each saving scheme at the start. What is the
value of M?
A    7500
B    6000
C    To solve this, the other interest rate must also be given.
D    4500
E    12000
A n s w e r : B
Explanation:
Let the total amount be 3x
Case 1: 
Smaller amount = x, rate of interest = 10
Larger amount = 2x, rate of interest = 5
Total amount received at the end of two years( smaller amount) = . CI = 0.21x
Total amount received at the end of two years( larger amount) =    CI = 0.205x
Given, 0.21x + 0.205x = 830
=> x = 2000
2x= 4000
Case 2:
Smaller amount = x, rate of interest = 20
Larger amount = 2x, rate of interest = 10
Total amount received at the end of two years( smaller amount) = . CI = 0.44x
Total amount received at the end of two years( larger amount) =  CI = 0.42x
Given, 0.44x+0.42x = 830
=> x = 965.11 which is not valid since it should be greater than 1000
4. A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the
shop to buy a unit each. Each one has a pre- determined choice of the colour and will not buy a unit of any other colour. All the
three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to
satisfy all the three customers?
A    
B    
C    
D    
E    
A n s w e r : C
x 1 + = ( 100
10
)
2
1.21 x
2 x 1 + = ( 100
5
)
2
2.205 x
x 1 + = ( 100
20
)
2
1.44 x
2 x 1 + = ( 100
10
)
2
2.42 x
5
4
9
7
3
2
9
8
3
1
  
.
Explanation:
Number of white phones = 2
Number of black phones = 2
Number of red phones = 1
customer 1 will have 3 choices
customer 2 will have 3 choices
customer 3 will have 3 choices
Hence total choices = 3 x 3 x 3 = 27
The cases not possible = BBB, RRR,WWW, RRB,RBR,BRR, RRW,RWR, WRR
Possible cases = 18
Probability = 18/27 = 2/3
         
5. At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in
degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?
A    45
B    105
C    90
D    0
E    75
A n s w e r : C
Explanation:
The difference between the hour and minute hand of a clock is given by . Here H is the current hour and m represents
the number of completed minutes in the current hour.
In the given time frame of 2: 30 to 3: 00 pm.
At 2 : 30 pm the angle = 
At 3: 00 pm the angle = 
The function of  constantly increases as the value of m increases from 31, 32................ 59.
Because of the modulus function, the net value of the function remains positive
Between 2: 30 to 2: 59 the angle is constantly increasing. The minimum value is 2: 30 which is equal to 105 degrees which is greater
than the 90 degrees when the time is 3: 00.
Hence 90 degrees is the minimum angle.
6. One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in
the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always
start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?
A    
B    
C    
D    
30 H - 5.5 m | |
30 · 2 - 5.5 · 30 = | | 105 deg r e e s
30 · 3 - 5.5 · 0 = | | 90 deg r e e s
30 · H - 5.5 · m = | |
7
1
9
1
3
1
9
7
4
  
.
Page 4


XAT 2021
  Quant
1. If  where m and n are positive real numbers, then which of the following must be true?
A    
B    m = n
C    
D    
E    No values of m and n can satisfy the given equation
A n s w e r : E
Explanation:
 
Squarring on both sides
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
  
2. Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining
eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her
fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?
A    9
B    17
C    11
D    20
E    29
A n s w e r : C
Explanation:
Let the number of eggs bought = 9x+2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x-2
Given, 4x-2 <=5
=> x
Hence the max value of x is 1
Max number of eggs bought = 11
log m + 4 log n = 4 log ( m + 2 n)
+ m
1
= n
1
1
m +
2
n =
2
1
+ m
1
= n
1
2
log m n = 4 log ( m + 2 n)
=
 m n
( m + n)
m +
2
n +
2
m n = 0
= 4
7
  
.
3. Mohan has some money ( ?M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that
offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both
interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is ?830. It is known
that one of the interest rates is 10% and that Mohan deposited more than ?1000 in each saving scheme at the start. What is the
value of M?
A    7500
B    6000
C    To solve this, the other interest rate must also be given.
D    4500
E    12000
A n s w e r : B
Explanation:
Let the total amount be 3x
Case 1: 
Smaller amount = x, rate of interest = 10
Larger amount = 2x, rate of interest = 5
Total amount received at the end of two years( smaller amount) = . CI = 0.21x
Total amount received at the end of two years( larger amount) =    CI = 0.205x
Given, 0.21x + 0.205x = 830
=> x = 2000
2x= 4000
Case 2:
Smaller amount = x, rate of interest = 20
Larger amount = 2x, rate of interest = 10
Total amount received at the end of two years( smaller amount) = . CI = 0.44x
Total amount received at the end of two years( larger amount) =  CI = 0.42x
Given, 0.44x+0.42x = 830
=> x = 965.11 which is not valid since it should be greater than 1000
4. A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the
shop to buy a unit each. Each one has a pre- determined choice of the colour and will not buy a unit of any other colour. All the
three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to
satisfy all the three customers?
A    
B    
C    
D    
E    
A n s w e r : C
x 1 + = ( 100
10
)
2
1.21 x
2 x 1 + = ( 100
5
)
2
2.205 x
x 1 + = ( 100
20
)
2
1.44 x
2 x 1 + = ( 100
10
)
2
2.42 x
5
4
9
7
3
2
9
8
3
1
  
.
Explanation:
Number of white phones = 2
Number of black phones = 2
Number of red phones = 1
customer 1 will have 3 choices
customer 2 will have 3 choices
customer 3 will have 3 choices
Hence total choices = 3 x 3 x 3 = 27
The cases not possible = BBB, RRR,WWW, RRB,RBR,BRR, RRW,RWR, WRR
Possible cases = 18
Probability = 18/27 = 2/3
         
5. At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in
degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?
A    45
B    105
C    90
D    0
E    75
A n s w e r : C
Explanation:
The difference between the hour and minute hand of a clock is given by . Here H is the current hour and m represents
the number of completed minutes in the current hour.
In the given time frame of 2: 30 to 3: 00 pm.
At 2 : 30 pm the angle = 
At 3: 00 pm the angle = 
The function of  constantly increases as the value of m increases from 31, 32................ 59.
Because of the modulus function, the net value of the function remains positive
Between 2: 30 to 2: 59 the angle is constantly increasing. The minimum value is 2: 30 which is equal to 105 degrees which is greater
than the 90 degrees when the time is 3: 00.
Hence 90 degrees is the minimum angle.
6. One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in
the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always
start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?
A    
B    
C    
D    
30 H - 5.5 m | |
30 · 2 - 5.5 · 30 = | | 105 deg r e e s
30 · 3 - 5.5 · 0 = | | 90 deg r e e s
30 · H - 5.5 · m = | |
7
1
9
1
3
1
9
7
4
  
.
E    
A n s w e r : A
Explanation:
Let us assume there are 9 buses.
3 of them stop at C and 6 go non-stop
Given, One-third of the passengers, in the buses stopping at City C, continue to City B, while the rest alight at City C
=> Since all buses have equal capacity. we can say 2 will elite at C and 1 will proceed to B.
Hence required proportion = 1/7
7. Rajesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a
straight line between any two consecutive points. The following are known: (i) AB and CD intersect at a right angle at E, and (ii)
BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long. If AD is parallel to BC, then what is the total distance (in km)
that Rajesh covers in travelling from A to D?
A    10.2
B    12
C    11.5
D    5.5
E    18
A n s w e r : C
Explanation:
Given, CE=0.5, BC = 1.3 and ED=2.5
Triangle CEB is a right-angled triangle => EB = 1.2
Triangles ECB is similar to triangle EDA
EB/EC = AE/ED  => AE = 6
Hence total distance travelled = AB + BC + CD = 7.2 + 1.3 + 3.5 = 11.5km
        
8. Let  if  and 1 if x = 1, -1. Let  if  and 3 if x = 1.
What is the minimum possible values of  ?
A    
B    -1
C    
D    
E    1
A n s w e r : D
Explanation:
9
4
f( x) = x -1
2
x +1
2
x =
?
1, -1, g( x) = x-1
x+1
x =
?
1,
g( x)
f( x)
2
1
4
1
3
1
=
g x ( )
f x ( )
· x -1
2
x +1 (
2
)
= x+1
x-1 ( )
x+1 ( )
2
x +1 (
2
)
  
.
Page 5


XAT 2021
  Quant
1. If  where m and n are positive real numbers, then which of the following must be true?
A    
B    m = n
C    
D    
E    No values of m and n can satisfy the given equation
A n s w e r : E
Explanation:
 
Squarring on both sides
Since m, n are positive real numbers, no value of m and n satisfy the above equations.
  
2. Mr. Jose buys some eggs. After bringing the eggs home, he finds two to be rotten and throws them away. Of the remaining
eggs, he puts five-ninth in his fridge, and brings the rest to his mother’s house. She cooks two eggs and puts the rest in her
fridge. If her fridge cannot hold more than five eggs, what is the maximum possible number of eggs bought by Mr. Jose?
A    9
B    17
C    11
D    20
E    29
A n s w e r : C
Explanation:
Let the number of eggs bought = 9x+2
number of eggs left after throwing away 2 = 9x
number of eggs kept in fridge = 5x
number of eggs brought to his mothers' house = 4x
number of eggs left after cooking 2 which are kept in fridge = 4x-2
Given, 4x-2 <=5
=> x
Hence the max value of x is 1
Max number of eggs bought = 11
log m + 4 log n = 4 log ( m + 2 n)
+ m
1
= n
1
1
m +
2
n =
2
1
+ m
1
= n
1
2
log m n = 4 log ( m + 2 n)
=
 m n
( m + n)
m +
2
n +
2
m n = 0
= 4
7
  
.
3. Mohan has some money ( ?M) that he divides in the ratio of 1:2. He then deposits the smaller amount in a savings scheme that
offers a certain rate of interest, and the larger amount in another savings scheme that offers half of that rate of interest. Both
interests compound yearly. At the end of two years, the total interest earned from the two savings schemes is ?830. It is known
that one of the interest rates is 10% and that Mohan deposited more than ?1000 in each saving scheme at the start. What is the
value of M?
A    7500
B    6000
C    To solve this, the other interest rate must also be given.
D    4500
E    12000
A n s w e r : B
Explanation:
Let the total amount be 3x
Case 1: 
Smaller amount = x, rate of interest = 10
Larger amount = 2x, rate of interest = 5
Total amount received at the end of two years( smaller amount) = . CI = 0.21x
Total amount received at the end of two years( larger amount) =    CI = 0.205x
Given, 0.21x + 0.205x = 830
=> x = 2000
2x= 4000
Case 2:
Smaller amount = x, rate of interest = 20
Larger amount = 2x, rate of interest = 10
Total amount received at the end of two years( smaller amount) = . CI = 0.44x
Total amount received at the end of two years( larger amount) =  CI = 0.42x
Given, 0.44x+0.42x = 830
=> x = 965.11 which is not valid since it should be greater than 1000
4. A small store has five units of a new phone model in stock: two white, two black, and one red. Three customers arrive at the
shop to buy a unit each. Each one has a pre- determined choice of the colour and will not buy a unit of any other colour. All the
three customers are equally likely to have chosen any of the three colours. What is the probability that the store will be able to
satisfy all the three customers?
A    
B    
C    
D    
E    
A n s w e r : C
x 1 + = ( 100
10
)
2
1.21 x
2 x 1 + = ( 100
5
)
2
2.205 x
x 1 + = ( 100
20
)
2
1.44 x
2 x 1 + = ( 100
10
)
2
2.42 x
5
4
9
7
3
2
9
8
3
1
  
.
Explanation:
Number of white phones = 2
Number of black phones = 2
Number of red phones = 1
customer 1 will have 3 choices
customer 2 will have 3 choices
customer 3 will have 3 choices
Hence total choices = 3 x 3 x 3 = 27
The cases not possible = BBB, RRR,WWW, RRB,RBR,BRR, RRW,RWR, WRR
Possible cases = 18
Probability = 18/27 = 2/3
         
5. At any point of time, let x be the smaller of the two angles made by the hour hand with the minute hand on an analogue clock (in
degrees). During the time interval from 2:30 p.m. to 3:00 p.m., what is the minimum possible value of x?
A    45
B    105
C    90
D    0
E    75
A n s w e r : C
Explanation:
The difference between the hour and minute hand of a clock is given by . Here H is the current hour and m represents
the number of completed minutes in the current hour.
In the given time frame of 2: 30 to 3: 00 pm.
At 2 : 30 pm the angle = 
At 3: 00 pm the angle = 
The function of  constantly increases as the value of m increases from 31, 32................ 59.
Because of the modulus function, the net value of the function remains positive
Between 2: 30 to 2: 59 the angle is constantly increasing. The minimum value is 2: 30 which is equal to 105 degrees which is greater
than the 90 degrees when the time is 3: 00.
Hence 90 degrees is the minimum angle.
6. One third of the buses from City A to City B stop at City C, while the rest go non-stop to City B. One third of the passengers, in
the buses stopping at City C, continue to City B, while the rest alight at City C. All the buses have equal capacity and always
start full from City A. What proportion of the passengers going to City B from City A travel by a bus stopping at City C?
A    
B    
C    
D    
30 H - 5.5 m | |
30 · 2 - 5.5 · 30 = | | 105 deg r e e s
30 · 3 - 5.5 · 0 = | | 90 deg r e e s
30 · H - 5.5 · m = | |
7
1
9
1
3
1
9
7
4
  
.
E    
A n s w e r : A
Explanation:
Let us assume there are 9 buses.
3 of them stop at C and 6 go non-stop
Given, One-third of the passengers, in the buses stopping at City C, continue to City B, while the rest alight at City C
=> Since all buses have equal capacity. we can say 2 will elite at C and 1 will proceed to B.
Hence required proportion = 1/7
7. Rajesh, a courier delivery agent, starts at point A and makes a delivery each at points B, C and D, in that order. He travels in a
straight line between any two consecutive points. The following are known: (i) AB and CD intersect at a right angle at E, and (ii)
BC, CE and ED are respectively 1.3 km, 0.5 km and 2.5 km long. If AD is parallel to BC, then what is the total distance (in km)
that Rajesh covers in travelling from A to D?
A    10.2
B    12
C    11.5
D    5.5
E    18
A n s w e r : C
Explanation:
Given, CE=0.5, BC = 1.3 and ED=2.5
Triangle CEB is a right-angled triangle => EB = 1.2
Triangles ECB is similar to triangle EDA
EB/EC = AE/ED  => AE = 6
Hence total distance travelled = AB + BC + CD = 7.2 + 1.3 + 3.5 = 11.5km
        
8. Let  if  and 1 if x = 1, -1. Let  if  and 3 if x = 1.
What is the minimum possible values of  ?
A    
B    -1
C    
D    
E    1
A n s w e r : D
Explanation:
9
4
f( x) = x -1
2
x +1
2
x =
?
1, -1, g( x) = x-1
x+1
x =
?
1,
g( x)
f( x)
2
1
4
1
3
1
=
g x ( )
f x ( )
· x -1
2
x +1 (
2
)
= x+1
x-1 ( )
x+1 ( )
2
x +1 (
2
)
  
.
This function is definitely greater than 0
=>  which is quadratic in x
Disctiminant should be greater than 0
=> y>=1/2
When x =1, f(x)/g(x) = 1/3
Hence either the value should be greater than 1/2 or should be equal to 1/3
9. Swati can row a boat on still water at a speed of 5 km/hr. However, on a given river, it takes her 1 hour more to row the boat 12
km upstream than downstream. One day, Swati rows the boat on this river from X to Y, which is N km upstream from X. Then
she rows back to X immediately. If she takes at least 2 hours to complete this round trip, what is the minimum possible value of
N?
A    3
B    4.8
C    2
D    3.6
E    2.1
A n s w e r : B
Explanation:
Let the speed of the stream be x
The value of x satisfying the above equation is 1
Now,
=> 
10. Rahul has just made a  magic square, in which, the sum of the cells along any row, column or diagonal, is the same
number N. The entries in the cells are given as expressions in x, y, and Z. Find N?
A    12
B    36
C    21
D    
40
E    24
l e t y =
x+1 ( )
2
x +1 (
2
)
x y - 1 +
2
( ) 2 y x + y - 1 = ( ) 0
4 y -
2
4 y - 1 = ( )
2
0
= 5- x
12
+ 5+ x
12
1
+ 5+1
N
= 5-1
N
2
= 12
2 N+3 N
2
N = 4.8
3 × 3
  
.