NCERT Exemplar for Class 10 Maths - Pair of Linear Equations in Two Variables - Free PDF Download
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Access NCERT Exemplar Solutions for Class 10 Maths Chapter 3 – Pair of Linear Equations in Two Variables
Sample Questions
1. The pair of equations
(A) one solution
(B) two solutions
(C) infinitely many solutions
(D) no solution
Ans: D
Given equations are
The equation
The equations
Comparing with
Therefore, the pair of equations
2. The sum of the digits of a two-digit number is
(A) 25
(B) 72
(C) 63
(D) 36
Ans: D
Consider one digit number be
And, to find the number be
We have:
The sum of the digits of a two-digit number is
i.e.,
And if
Substitute the value of
Substitute the value of
Then ,
Number
Number
Number
Number
EXERCISE 3.1
Choose the correct answer from the given four options:
1. Graphically, the pair of equations
represents two lines which are
(A) intersecting at exactly one point.
(B) intersecting at exactly two points.
(C) coincident.
(D) parallel.
Ans: D
Given, the linear equations are
On comparing with
So,
And
And
Therefore, the lines representing the given equations are parallel.
2. The pair of equations
(A) a unique solution
(B) exactly two solutions
(C) infinitely many solutions
(D) no solution
Ans: D
From the equations
The above equations are of the form
Therefore, the pair of equations
3. If a pair of linear equations is consistent, then the lines will be
(A) parallel
(B) always coincident
(C) intersecting or coincident
(D) always intersecting
Ans: C
If a pair of linear equations is consistent, then it has unique solutions and infinite solutions.
If
, then the graph will be a pair of lines intersecting at a unique point, which is the solution of the pair of equations.
If
, then the graph will be a pair of coincident lines. Each point on the lines will be a solution, and so the pair of equations will have infinitely many solutions
4. The pair of equations
(A) one solution
(B) two solutions
(C) infinitely many solutions
(D) no solution
Ans: D
Given the pair of equations are
Graph the pair of equations

From the above graph
5. The pair of equations
(A) parallel
(B) intersecting at
(C) coincident
(D) intersecting at
Ans: D
Given pair of equations

From the graph, the lines are
6. For what value of
(A)
(B)
(C)
(D)
Ans: C
Given equations
The above equations are of the form
So,
since, lines are coincident
therefore,
7. If the lines given by
(A)
(B)
(C)
(D)
Ans: C
Given lines
On comparing the linear equations with
Since, the lines are parallel.
8. The value of
(A)
(B)
(C)
(D) no value
Ans: D
Given a pair of equations
On comparing the linear equations with
since,
Then,
Here, the value of
9. One equation of a pair of dependent linear equations is
(A)
(B)
(C)
(D)
Ans: D
Given pair of dependent linear equation is
From the equation
Then, from the dependent linear condition
where,
Substituting
The required second line becomes
10. A pair of linear equations which has a unique solution
(A)
(B)
(C)
(D)
Ans: (B, D)
The pair of linear equations which have a unique solution
If
Substitute
Substitute
Substitute
Substitute
By substituting the values in
11. If
(A)
(B)
(C)
(D)
Ans: C
Given
Substitute the values of
Add equations (1) and (2)
Substitute the value of
Therefore, the value of
12. Aruna has only Rs
(A)
(B)
(C)
(D)
Ans: D
Consider,
From the question,
The total number of coins that she has is
Total number of coins
Also, Amount of money with her
=(Number of Rs
Subtract Eq. (i) from Eq. (ii), then
So,
Substituting
Therefore, Aruna has
13. The father's age is six times his son's age. Four years hence, the age of the father will be four times his son's age. The present ages, in years, of the son and the father are, respectively
(A)
(B)
(C)
(D)
Ans: C
Consider the present age of father be '
From the given question, Four years hence, the age of the father will be four times his son's age then
And the father's age is six times his son's age,
Substituting the value of Eq. (ii) in Eq. (i) then
Therefore,
Substituting
Therefore, the present age of the father is
Sample Question
Short Answer Questions with Reasoning
1. Is it true to say that the pair of equations
Ans: The given equations are in the form
So,
Consider,
Since,
Therefore, the pair of equations
Hence, the given statement “Is it true to say that the pair of equations
2. Do the equations
Ans: The equations
The above equations are of the form
So,
So that
Here,
Thus, The given equations
Hence, the given statement “The statement is “the equations
3. Is the pair of equations
Ans: Equations
The above equations are of the form
So,
Then,
Therefore,
EXERCISE 3.2
1. Do the following pair of linear equations have no solution? Justify your answer.
(i)
(ii)
(iii)
Ans:
(i) Given Linear equations
The above equations are of the form
So,
And
From the above
Therefore, the given pair of linear equations
(ii) Given, pair of Linear equations
Rewrite the equations as
The above equations are of the form
So,
And
Here,
Therefore, the given pair of linear equations
(iii) Given, the pair of Linear equations
The above equations are of the form
So,
And
From the above
Therefore, the given pair of linear equations
2. Do the following equations represent a pair of coincident lines? Justify your answer.
(i)
(ii)
(iii)
Ans:
(i)Given, the pair of linear equations are
The above equations are of the form
So,
And
From the above
Therefore, the given pair of linear equations has a unique solution.
A pair of equations
(ii) Given, the pair of linear equations are
The above equations are of the form
So,
Then,
Hence,
Therefore, the given pair of linear equations
(iii) Given, the pair of linear equations are
The above equations are of the form
So,
Then
Hence,
Therefore, the given pair of linear equations
3. Are the following pair of linear equations consistent? Justify your answer.
(i)
(ii)
(iii)
(iv)
Ans:
(i) Given, Pair of linear equations are
The above equations are of the form
Then
So,
Hence,
Therefore, the pair of linear equations
(ii) Given, the pair of linear equations are
The above equations are of the form
So,
Then,
And
Hence,
Therefore, the pair of linear equations are
(iii) Given , Pair of linear equations are
The above equations are of the form
So,
Then,
Hence,
Therefore, the pairs of linear equations
(iv) Given , pairs of linear equations are
The above equations are of the form
So,
Then,
Hence,
Therefore, Pair of linear equations
4. For the pair of equations
Ans: Pair of linear equations are
The above equations are of the form
So,
If
Since,
And
Therefore,
So, the value of
Therefore, the given statement “For the pair of equations
5. For all real values of
Ans: The statement for all real values of
Given the pair of equations
The above equations are of the form
So,
From the above
The statement that for all real values of
6. The line represented by
Ans: The line represented by
In the given statement the line represented by
SAMPLE QUESTIONS
1. For which values of
Ans:
Given: A pair of linear equation
If
From the given Equations we can derive the below as,
If
Therefore,
So we can write that,
By substituting the value of q in eq (2) we get:
Hence, For
2. Solve the following pair of linear equations:
Ans: A pair of linear equations
Multiplying Equation (1) by 21 and Equation (2) by 47 then,
Subtracting Equation (3) from Equation (4), we get
Therefore,
Substituting value of
3. Draw the graphs of the pair of linear equations
Ans:
(1) Considering the equation
The line will cut the
It will cut the
(2) Considering the equation
The line will cut the
It will cut the
Therefore, the graph of the two lines is shown below by taking

The two lines intersect at the point
The distance of the point
Now, the distance of the point
Hence, the area of the triangle
EXERCISE 3.3
1. For which value(s) of λ, do the pair of linear equations
(i) No Solution?
(ii) Infinitely many solutions?
(iii) a unique solution
Ans: The given equations are as follows:
Comparing the above equations with
(i) For no solution
(ii) For infinitely many solutions
Hence, the system of linear equations has infinitely many solutions at
(iii) For a unique solution
2. For which value(s) of
Have no solution?
Ans: Comparing the above equations with
Hence for
3. For which values of a and b will the following pair of linear equations have infinitely many solutions?
Ans: Given pair of linear equations are
Rewrite the above equation, we get:
On comparing with
The system has infinitely many solutions if
Choose the first two parts, we get:
Taking last two parts,
Now, solve the equation (iii) and (iv), we get
Put the value of
So, the value
Hence, required values of
4. Find the value(s) of p in (i) to (iv) and p and q in (v) for the following pair of equations:
(i)
(ii)
(iii)
(iv)
(v)
Ans: (i) Given pair of linear equations are
and
On comparing with
Since, the lines represented by these equations are parallel, then
Taking last two parts, we get
Hence, the given pair of linear equations are parallel for all real values of
(ii) Given pair of linear equation are
On comparing with
Since, the pair of linear equations has no solution.
i.e., both lines are parallel to each other
Taking first two parts, we get
Taking last two parts, we get
Hence, the given pair of linear equations has no solution for
(iii) Given, pair of linear equations are
On comparing with
Since, the lines are intersecting at a unique point,
i.e. it has a unique solution.
Hence, the lines represented by these equations are intersecting at a unique point for all real values of p except
(iv) Given pair of linear equations are
On comparing with
Since, the pair of linear equations has a unique solution,
Hence, the pair of linear equations has a unique solution for all values of
(v) Given pair of linear equation are
Rewrite the equations, we get:
On comparing with
Since, the pair of equations has infinitely many solutions
i.e., both lines are coincident.
Taking first and third parts, we get
Again, taking last two parts, we get
Hence, the pair of equations has infinitely many solutions for the values of
5. Two straight paths are represented by the equations
Ans:
Given linear equations are
and
On comparing with
Here,
and
i.e.,
Thus, lines are parallel.
Hence, two straight paths represented by the given equations never cross each other, because they are parallel to each other.
6. Write a pair of linear equations which has the unique solution
Ans: Let the equations are
If
Given,
Then it must satisfy the given equation.
and
So, many values of
Hence, infinitely many pairs of linear equations are possible.
7. If
Ans: Given equation are
and
On adding both equations, we get
Put the value of
We have,
and
Hence, the values of
8. Find the values of x and y in the following rectangle (see figure).

Ans:

By property of rectangle,
Lengths are equal i.e.,
Breadth are equal i.e.,
On multiplying eq. (ii) by 3 and then subtracting eq. (i), we get
On putting
Hence, the required values of
9. Solve the following pairs of equations:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Ans: Given pair of linear equations are
Now, multiplying eq. (i) by 2 and then adding with eq. (ii), we get
Now, put the value of
Hence, the required values of
(ii) Given, pair of linear equations are
and
Now, adding eq. (i) and eq. (ii), we get
Now, put the value of
Hence, the required values of
(iii) Given pair of linear equations are
and
Let
and
On multiplying eq. (iii) by 8 and eq. (iv) by 6 and then adding both of them, we get
Now, put the value of
Hence, the required values of
(iv) Given pair of linear equations are
and
Put
and
On multiplying eq. (iv) by 2 and then adding with eq. (iii), we get
Now, put the value of u in eq. (iv), we get
Hence, the required values of
(v) Given pair of linear equations are
and
On multiplying eq. (i) by 43 and eq. (ii) by 67 and then subtracting eq (i) from eq (ii), we get
Now, put the value of
Hence, the required values of
(vi) Given pair of linear equations are
and
On multiplying eq. (i) by
Now, put the value of
Hence, the required values of
(vii) Given pair of equations are
and
Now, put
then the above equations become
and
On adding both equations, we get
Now, put the value of
and
Hence, the required values of
10. Find the solution of the pair of equations
Ans: Given pair of equations are
and
On multiplying eq. (i) by 2 and then subtracting from eq. (ii), we get
Put the value of
Now,
Put the values of
Hence, the solution of the pair of equations are
11. By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.
(i)
(ii)
(iii)
Ans: (i) Given pair of equations are
On comparing with
Here,
and,
So, the given pair of linear equations are intersecting at one point, therefore these lines have a unique solution.
Hence, given pairs of linear equations are consistent.
We have,
x | 0 | -1 | -2 |
Y | -4 | -1 | 2 |
And
x | -1 | 0 | 1 |
Y | -1 | 2 | 5 |
Plot the points
Plot the point
(ii) Given pair of equation is
and
On comparing with
Here,
and
Hence, the lines represented by the given equations are parallel.
Therefore, it has no solution. So, the given pair of lines is inconsistent.
(iii) Given pair of equation are
and
On comparing with
Here,
and
So, the given pair of lines is coincident.
Therefore, these lines have infinitely many solutions.
Hence, the given pair of linear equations are consistent.
Now,
x | 0 | 3 | 3 |
Y | 3 | 0 | 1 |
and
x | 0 | 1 | 3 |
Y | 3 | 2 | 0 |
Plotting the points we get the graph of lines.

We observe that the lines represented by (i) and (ii) are coincident.
12. Draw the graph of the pair of equations
Ans:
Consider the line
x | 0 | 2 |
y | 4 | 0 |
Now, consider the line
x | 0 | 2 |
y | -4 | 0 |
Graphical representation of both lines are as follows:

Here, both lines and
Hence, the vertices of a
Hence, the required area of the triangle is 8 sq units.
13. Write an equation of a line passing through the point representing the solution of the pair of linear equations
Ans: Given pair of linear equation are
and
On comparing with
Hence, both lines intersect at a point.
Consider the line
x | 0 | 2 | 1 |
Y | 2 | 0 | 1 |
The line
x | 0 | 1/2 | 1 |
Y | -1 | 0 | 1 |

The given lines intersect at
Hence, infinite lines can pass through the intersection point of linear equations
14. If
Ans: Let
Putting
Now,
Putting the value of
Putting the value of
Hence, the values of
15. The angles of a triangle are
Ans: According to the given question
Since, sum of all sides of a triangle is
Putting the value of
The value of
16. Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older than twice her age. How old are they now?
Ans: Let Salim and his daughter’s age be
Now, according to question
Two years ago, Salim was thrice as old as his daughter
Again according to the given question
six years later, Salim will be four years older than her daughter’s age.
Now, putting the value of
Now, putting the value of
Thus, Salim and his daughter’s age are
17. The age of the father is twice the sum of the ages of his two children. After
Ans: Let the present age (in year) of father and his two children be
According to the question
And after
On putting the value of
Thus, the father’s age is
18. Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes
Ans: Let the two numbers be
Two numbers are in the ratio 5 : 6 (given)
If
Putting the value of
The value of
19. There are some students in the two examination halls
Ans: Let the number of students in halls
Now, by according to the question,
On subtracting equation
On putting
Hence,
20. A shopkeeper gives books on rent for reading. She takes a fixed charge for the
first two days, and an additional charge for each day thereafter. Latika paid
Ans: Let Latika take a fixed charge for the first two day is
Latika paid
For Latika, Number of days = 6 = 2 (fixed days) + 4 (additional days).
Anand paid
For Anand, Number of days = 4 = 2 (fixed days) + 2 (additional days).
Now, subtracting equation
On putting the value of y in equation
Hence, the fixed charge
and the charge for each extra day
21. In a competitive examination, one mark is awarded for each correct answer while
Ans: Total number of questions
Let x be the number of correct answers.
The number of wrong answers
The marks scored by Jayanti is
Now, according to the given question
Thus, Jayanti answered correctly
22. The angles of a cyclic quadrilateral
Find
Ans: Since, the sum of two opposite angles in a cyclic quadrilateral is
Now, equating equation
Negative sign will get eliminated
Now, putting value of
Thus, four angle of a cyclic quadrilateral are:
SAMPLE QUESTIONS
1. Draw the graphs of the lines
Ans: The graph of
Thus, the line
The graph of
Thus, the line

The figure enclosed by
The vertices of
The length of
The breadth of
Area of a rectangle
Therefore, area of rectangle
2. Determine, algebraically, the vertices of the triangle formed by the lines
Ans: The common solution of the two equations forming its two sides is the vertex of the triangle. So, solving the given equations pairwise gives us the required vertices of the triangle.
The three pairs of equations are:
Solving the first pair of equations
Thus, one vertex of the triangle is
Solving the second pair of equations
Thus, other vertex of the triangle is
Solving the third pair of equations
Thus, the third vertex of triangle is
Therefore, the three vertices of the triangle are
3. Jamila sold a table and a chair for
Ans: Assume that the cost price of the table is
The selling price of table when sold at a profit of
The selling price of chair when sold at a profit of
When the table is sold at a profit of
When the chair is sold at a profit of
From
By adding and subtracting these equations,
i.e.
and
From
Therefore, the cost price of a table is
4. It can take
Ans: Assume the time taken by the pipe of larger diameter to fill the pool as
In
Thus, in
In
In
Using both pipes, the pool can be filled in
Let
Then
And
Multiply equation
Substituting the value of
Therefore, the pipe of larger diameter alone can fill the pool in
EXERCISE 3.4:
1. Graphically, solve the following pair of equations:
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the
Ans: Table for the line
Table for the line
Graph of the lines is as follows:

So, the pair of equations intersect graphically at
Let
And
Therefore, the required ratio
2. Determine, graphically, the vertices of the triangle formed by the lines
Ans: Table for the line
Table for the line
Table for the line
Graph of the lines is as follows:

The lines
Thus,
Therefore, the vertices of
3. Draw the graphs of the equations
Ans: The given equation of lines are
Table for the line
X | 0 | 2 |
Y | -4 | 0 |
Plot (0, -4) and (2, 0) and join these points to get the graph of

From the graph, ABCD is the quadrilateral formed by the given lines and x-axis.
AB = OB - OA = 5 - 3 = 2,AD = 2 and BC = 6
Area of quadrilateral
Therefore, the area of the quadrilateral is 8 sq units.
4. The cost of
Ans: Let the cost of one pen be x and the cost of one pencil box be y .
and
Add equation (1) and equation (2),
Substitute x=10 in equation (1),
Therefore, the cost of one pen and one pencil box are ₹10 and ₹15, respectively.
5. Determine, algebraically, the vertices of the triangle formed by the lines
Ans: The common solution of the two equations forming its two sides is the vertex of the triangle.
Consider, the two lines
Multiply
Put
Thus, vertex
Now, consider the two lines
Multiply
Put
Thus, vertex
Now, consider the lines
Multiply
Put
Thus, vertex A is
Therefore, the vertices of the
6. Ankita travels 14km to her home partly by rickshaw and partly by bus. She takes half an hour, if she travels 2km by rickshaw, and the remaining distance by bus. On the other hand, if she travels 4km by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.
Ans: Let the speed of rickshaw and bus be
The time taken to travel
As speed
$\Rightarrow t_{1}=\frac{2}{x}hours
And the time taken to travel the remaining distance i.e.
Given that
And the time taken to travel the remaining distance i.e.
Given that
(ii)Let
And
Multiply equation(iii) by 2 and subtract equation(iv) from it.
Put
As,
Therefore, the speeds of rickshaw and bus are
7. A person, rowing at the rate of
Ans: Let the speed of the stream be
Speed of a person rowing in still water
Speed of a person rowing downstream
Speed of person rowing upstream
The time taken by the person to cover
The time taken by the person to cover
As
Therefore, the speed of the stream is
8. A motorboat can travel
Ans: Let the speed of the motorboat in still water and speed of stream be
The downstream speed of the motorboat
And the upstream speed of the motorboat
The time taken to travel
And time taken to travel
Given that
Time taken to travel
And time taken to travel
Given that
Then (1) and (2) will become:
And
Multiply (4) by 28 and subtract from (3), we get:
Put
Now,
And
Add (5) and (6) :
Put
Therefore, the speed of the boat in still water is
9. A two-digit number is obtained by either multiplying the sum of the digits by
Ans: Let the two-digit number be
Case I: Multiplying the sum of digits by 8 and then subtracting
Case II: Multiplying the difference of digits by 16 and then adding
Multiply (i) by 3 and subtract from (ii):
Put
Therefore, the required two-digit number
10. A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the stations A to B costs ₹2530. Also, one reserved first class ticket and one reserved first class half ticket from
Ans: Let the cost of full and half first class fare be
Let the reservation charges be ₹
Case I: The cost of one reserved first class ticket from the stations
Case II: The cost of one reserved first class ticket and one reserved first class half ticket from stations
Multiply (1) by 4 and subtract from
Put
Therefore, full first class fare from stations
11. A shopkeeper sells a saree at
Ans: Let the cost price of saree and list price of sweater be
Case II: He sells the saree at
Put
Put
Therefore, the cost price of a saree and the list price of a sweater are
12. Susan invested a certain amount of money in two schemes
Ans: Let the amount of investments in schemes
Case I: Interest at the rate of
Case II: Interest at the rate of
Multiply (i) by 9 and (ii) by 8 and subtract them.
Therefore, she invested ₹ 12000 and ₹ 10000 in schemes
13. Vijay had some bananas, and he divided them into two lots
Ans: Let the number of bananas in lots
Case I: Cost of first lot at the rate of
Case II: Cost of first lot at the rate ₹ 1 per banana + Cost of second lot at the rate of ₹ 4 for 5 bananas = amount received
Multiply (1) by 4 and (2) by 3 and subtract them.
Put
The total number of bananas = Number of bananas in lot
Therefore, he had 500 bananas.
Why NCERT Exemplar for Class 10 Maths - Pair of Linear Equations in Two Variables
The NCERT Maths Exemplar for Class 10 is useful for students who want to do well in their Class 10 board exams. Aside from that, you can use reference books to help you answer the questions, however, these reference books can occasionally lead you astray and confuse you. To achieve well in the Class 10 board exams, it is best to stick to the NCERT books for Class 10 and practice various tasks.
Refer to the NCERT Exemplar Solutions for Class 10 for the answers to the NCERT Maths exemplar questions. For a better comprehension of the subject's subjects, these solutions are offered in a concise, quick, and correct manner. These solutions can be used to effectively prepare for the Maths topic of the Class 10th board exams.
Concepts Included in Class 10 Maths - Pair of Linear Equations in Two Variables
Equations
Linear Equations
General Form of Linear Equation in two Variables
Geometrical Representation of a linear equation
Comparing the ratios of coefficients of a linear equation
Algebraic Solution
Substitution Method
Elimination Method
Cross-Multiplication Method
Number of Exercises in NCERT Exemplar for Class 10 Maths - Pair of Linear Equations in Two variables
Exercise 3.1 (13 Questions)
Exercise 3.2 (5 Questions)
Exercise 3.3 (22 Questions)
Exercise 3.4 (13 Questions)
Benefits of solving Problems from NCERT Exemplar for Class 10 Maths
Mathematics is a subject that demands not only a thorough comprehension of the material but also many practice sessions to master it and get high grades. As a result, just finishing NCERT isn't enough to learn this subject; you'll also need the necessary study tools. As a result, to grasp the material quickly, you need to use NCERT Exemplar for Class 10 Maths. Take a look at the advantages of using these NCERT Maths examples to help you prepare for your board test. Some of these advantages are as follows:
NCERT exemplars adhere to the CBSE curriculum and cover the entire syllabus for the Class 10 board exam. This study material will also assist you in forming a foundation for studying for board examinations, as the questions in this textbook are based on the subject's prescribed syllabus.
Aids in board preparation- The NCERT Maths example contains advanced-level questions that appear in board exams. Solving NCERT example problems can assist you in honing your math skills. This will undoubtedly assist you in completing difficult questions and MCQs.
Clears topic fundamentals- The NCERT books not only cover the whole test syllabus but also offer all of the fundamentals on all topics in plain language. To avoid rehashing the same themes, make sure your concepts are crystal clear. To create a clear concept, you must refer to the NCERT Maths Exemplar for a complete and extensive study. As a result, you will be required to update the formulas, terms, and execution during the revision.
NCERT Exemplars are essentially practice books that contain additional key questions at a higher level and allow you to learn in greater depth. Conceptual sums are included in these examples, which are important for CBSE and competitive exams. These NCERT Exemplar Class 10 Maths Solutions will provide you with MCQs, skill-development questions, and new topics. All of the themes under each subject are covered in NCERT Exemplars.
FAQs on NCERT Exemplar for Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables (Book Solutions)
1. Will solving Exemplar exercises for Class 10 Maths benefit me?
These NCERT exemplars are written by subject experts following extensive research into the difficulties surrounding the subject. They offer in-depth information. You'll be able to get accurate and trustworthy information on the topics. Each topic in the exemplar is described in detail. By answering questions from the NCERT Exemplar, you will check your basic grasp and thoroughly understand the in-depth features of the material. Students can solve the Exemplar questions to practice for the final board exams and score high marks in the exams.











