Solution of Chapter 3. Pair of Linear Equations in Two Variables (RD Sharma - Mathematics Book)

Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a rig on the items kept in the stall, and if the ring covers any object completely you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. Each ride costs Rs 3, and a game of Hoopla costs Rs 4. If she spent Rs 20 in the fair, represent this situation algebraically and graphically.

Aftab tells his daughter, ‘Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Is not this interesting? Represent this situation algebraically and graphically.

The path of a train A is given by the equation 3x + 4y – 12 = 0 and the path of another train B is given by the equation 6x + 8y – 48 = 0. Represent this situation graphically.

Gloria is walking along the path joining (-2, 3) and (2, -2), while Suresh is walking along the path joining (0, 5) and (4, 0). Represent this situation graphically.

On comparing the ratios , and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide:

(i)

(ii)

(iii)

Given the linear equation , write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines (ii) parallel lines

(iii) coincident lines.

The cost of 2 kg of apples and l kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically.

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Solve the following systems of equations graphically:

Show graphically that each one of the following systems of equations has infinitely many solutions:

Show graphically that each one of the following systems of equations has infinitely many solutions:

Show graphically that each one of the following systems of equations has infinitely many solutions:

Show graphically that each one of the following systems of equations has infinitely many solutions:

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution):

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution):

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution):

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution):

Determine graphically the vertices of the triangle, the equations of whose sides are given below:

(i)

(ii)

Determine, graphically whether the system of equations is consistent or in-consistent.

Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not:

(i)

(ii)

Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are:

(i)

(ii)

Solve the following system of linear equations graphically and shade the region between the two lines and x-axis:

(i)

(ii)

(iii)

Draw the graphs of the following equations on the same graph paper:

Find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis.

Draw the graphs of and . Determine the coordinates of the vertices of the triangle formed by these lines and x-axis and shade the triangular area. Calculate the area bounded by these lines and x-axis.

Solve graphically the system of linear equations:

Find the area bounded by these lines and x-axis.

Solve the following system of linear equations graphically:

Shade the region bounded by these lines and y-axis. Also, find the area of the region bounded by the these lines and y-axis.

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system:

(i)

(ii)

(iii)

(iv)

Draw the graphs of the following equations:

Find the vertices of the triangle so obtained. Also, find the area of the triangle.

Solve the following system of equations graphically:

Also, find the area of the region bounded by these two lines and y-axis.

Solve the following system of linear equations graphically:

Determine the vertices of the triangle formed by the lines representing the above equation and the y-axis.

Draw the graphs of the equations

. Determine the coordinates of the vertices of the triangle formed by these lines and y-axis. Calculate the area of the triangle so formed.

Form the pair of linear equations in the following problems, and find their solution graphically:

(i) 10 students of class X took part in Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and a pen.

(iii) Champa went to a ‘sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased.” Help her friends to find how many pants and skirts Champa bought.

Solve the following system of equations graphically:

Shade the region between the lines and the y-axis

(i)

(ii)

Represent the following pair of equations graphically and write the coordinates of points where the lines intersects y-axis

Given the linear equation , write another linear equation in two variables such that the geometrical representation of the pair so formed is

(i) intersecting lines (ii) Parallel lines

(iii) coincident lines

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

= 10

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

where, x + y ≠ 0 and x - y ≠ 0

Solve the following systems of equations:

where, x + y ≠ 0 and x - y ≠ 0

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve the following systems of equations:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

ax + by = a – b

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

3x + 5y = 4

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

(a – 2b)x + (2a + b)y = 3

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

Solve each of the following systems of equations by the method of cross-multiplication:

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

Find the value of k for which the following system of equations has a unique solution:

Find the value of k for which the following system of equations has a unique solution:

Find the value of k for which the following system of equations has a unique solution:

Find the value of k for which the following system of equations has a unique solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

(k+1)x + 9y = k + 1

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

Find the value of k for which each of the following systems of equations have infinitely many solution:

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

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

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

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

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

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

For what value of k the following system of equations will be inconsistent?

For what value of a, the system of equations is inconsistent

ax + 3y = a - 3

12x + ay = a

Find the value of k for which the system

has (i) a unique solution, and (ii) no solution.

Prove that there is a value of c (≠ 0) for which the system

Has infinitely many solutions. Find this value.

Find the values of k for which the system

Will have (i) a unique solution and (ii) no solution. Is there a value of k for which the system has infinitely many solutions?

For what value of k, the following system of equations will represent the coincident lines?

Obtain the condition for the following system of linear equations to have a unique solution

Determine the values of a and b so that the following system of linear equations have infinitely many solutions:

3x + (b – 1)y – 2 = 0

Find the values of a and b for which the following system of linear equations has infinite number of solutions:

Find the values of p and q for which the following system of linear equations has infinite number of solutions:

Find the values of a and b for which the following system of equations has infinitely many solutions:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

5 pens and 6 pencils together cost Rs 9 and 3 pens and 2 pencils cost Rs 5. Find the cost of 1 pen and 1 pencil.

7 audio cassettes and 3 video cassettes cost Rs 1110, while 5 audio cassettes and 4 video cassettes cost Rs 1350. Find the cost of an audio cassette and a video cassette.

Reena has pens and pencils which together are 40 in number. If she has 5 more pencils and 5 less pens, then number of pencils would become 4 times the number of pens. Find the original number of pens and pencils.

4 tables and 3 chairs, together, cost Rs 2,250 and 3 tables and 4 chairs cost Rs 1950. Find the cost of 2 chairs and 1 table.

3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost Rs 324. Find the total cost of 1 bag and 10 pens.

5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77. Find the total cost of 1 book and 2 pens.

A and B each have a certain number of mangoes. A says to B, “if you give 30 of your mangoes, I will have twice as many as left with you.” B replies, “if you give me 10, I will have thrice as many as left with you.” How many mangoes does each have?

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs 2000. But if he sells the T.V. at 10% gain and the fridge at 5% loss. He gains Rs 1500 on the transaction. Find the actual prices of T.V. and fridge.

The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, he buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

One says, “Give me a hundred, friend! I shall then become twice as rich as you.” Tell me what is the amount of their respective capital?

A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

The sum of two numbers is 8. If their sum is four times their difference, find the number.

The sum of digits of a two digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number?

A number consists of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number.

The sum of digits of a two digit number is 15. The number obtained by reversing the order of digits of the given number exceeds the given number by 9. Find the given number.

The sum of two-digit number and the number formed by reversing the order of digits is 66. If the two digits differ by 2, find the number. How many such numbers are there?

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.

A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number.

A two-digit number is 4 times the sum of its digits and twice the product of the digits. Find the number.

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

The difference between two numbers is 26 and one number is three times the other. Find them.

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3. Find the number.

The numerator of a fraction is 4 less than the denominator. If the numerator is decreased by 2 and denominator is increased by 1, then the denominator is eight times the numerator. Find the fraction.

A fraction becomes 9/11 if 2 is added to both numerator and the denominator, it becomes 5/7 if 2 is subtracted from both numerator and the denominator. Find the fraction.

A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator. If 1 is added to both the numerator and denominator, it becomes 1/2. Find the fraction.

If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes 1/2 if we only add 1 to the denominator. What is the fraction?

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes 6/5. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes 2/5. Find the fraction.

When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes 1/4. And, when 6 is added to numerator and the denominator is multiplied by 3, it becomes 2/3. Find the fraction.

The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.

If 2 is added to the numerator of a fraction, it reduces to 1/2 and if 1 is subtracted from the denominator, it reduces to 1/3. Find the fraction.

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction.

The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.

A father is three times as old as his son. After twelve years, his age will be twice as that of his son then. Find their present ages.

Ten years later, A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B?

A is elder to B by 2 years. A’s father F is twice as old as A and B is twice as old as his sister S. If the ages of the father and sister differ by 40 years, find the age of A.

Six years hence a man’s age will be three times the age of his son and three years ago he was nine times as old as his son. Find their present ages.

Ten years ago, a father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be then. Find their present ages.

The present age of a father is three years more than three times the age of the son. Three years hence father’s age will be 10 years more than twice the age of the son. Determine their present ages.

A father is three times as old as his son. In 12 years time, he will be twice as old as hi son. Find the present ages of father and the son.

Father’s age is three times the sum of ages of his two children. After 5 years his age will be twice the sum of ages of two children. Find the age of father.

Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju as twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

Points A and B are 70 km a part on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours, but if they travel towards each other, they meet in one hour. Find the speed of the two cars.

A sailor goes 8 km downstream in 40 minutes and returns in 1 hour. Determine the speed of the sailor in still water and the speed of the current.

The boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of stream and that of the boat in still water.

A boat goes 24 km upstream and 28 km downstream in 6 hrs. It goes 30 km upstream and 21 km downstream in hrs. Find the speed of the boat in still water and also speed of the stream.

While covering distance of 30 km. Ajeet takes 2 hours more than Amit. If Ajeet doubles his speed, he would take 1 hour less than Amit. Find their speeds of walking.

A man walks a certain distance with certain speed. If he walks 1/2 km an hour faster, he takes 1 hour less. But, if he walks 1 km an hour slower, her takes 3 more hours. Find the distance covered by the man and his original rate of walking.

Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he travels 160 km by train and the rest by car. He takes 12 minutes more if the travels 240 km by train and the rest by car. Find the speed of the train and car respectively.

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

Places A and B are 80 km apart form each other on a highway. A car starts from A and other from B at the same time. If they move in the same direction, they meet in 8 hours and if they move in opposite directions, they meet in 1 hour and 20 minutes. Find the speeds of the cars.

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the oat in still water and the speed of the stream.

Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus respectively.

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

A takes 3 hours more than B to walk a distance of 30 km. But, if A doubles his pace (speed) he is ahead of B by hours. Find the speeds of A and B.

Abdul travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But if he travels 260 km by train and 240 km by taxi he takes 6 minutes longer. Find the speed of the train and that of the taxi.

A train covered a certain distance at a uniform speed. If the train could have been 10 km/hr. faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/hr, it would have taken 3 hours more than the scheduled time. Find the distance covered by train.

Places A and B are 100 km apart on a highway. One car starts form A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of two cars?

If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units. If, however the length is reduced by 1 unit and the breadth increased by 2 units, the area increases by 33 square units. Find the area of the rectangle.

The area of a rectangle remains the same if the length is increased by 7 metres and the breadth is decreased by 3 metres. If the length is decreased by 7 metres and breadth is increased by 4 metres, the area is decreased by 21 sq. metres. Find the dimensions of the rectangle.

In a rectangle, if the length is increased by 3 metres and breadth is decreased by 4 metres, the area of the rectangle is reduced by 67 square metres. If length is reduced by 1 metre and breadth is increased by 4 metres, the area is increased by 89 sq. metres. Find the dimensions of the rectangle.

The incomes of X and Y are in the ratio of 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves Rs 1250, find their incomes.

A and B each has some money. If A gives Rs 30 to B, then B will have twice the money left with A. But, if B gives Rs 10 to A, then a will have thrice as much as is left with B. How much money does each have?

There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is same. If 20 candidates are sent from B to A, the number of students in A is double the number of students in B. Find the number of students in each room.

2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long would it take one man and one boy to do it?

In a

. Also, . Find the three angles.

In a cyclic quadrilateral ABCD , . Find the four angles.

A railway half ticket costs half the full fare and the reservation charge is the same on half ticket as on full ticket. One reserved first class ticket from Mumbai to Ahmedabad costs Rs 216 and one full and one half reserved first class tickets cost Rs 327. What is the basic first class full fare and what is the reservation charge?

In a . If 3y - 5x = 30, prove that the triangle is right angled.

The car hire charges in a city computerise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is Rs 89 and for a journey of 20 km, the charge paid is Rs 145. What will a person have to pay for travelling a distance of 30 km?

A part of monthly hostel charges in a college are fixed and the remaining depend on the number of days one has taken food in the mess. When a student a takes food for 20 days, he has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charge and the cost of food per day.

Half the perimeter of a garden, whose length is 4 more than its width is 36 m. Find the dimensions of the garden.

The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

2 Women and 5 men can together finish a piece of embroidery in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the embroidery, and that taken by 1 man alone.

A wizard having powers of mystic in candations and magical medicines seeing a cock, fight going on, spoke privately to both the owners of cocks. To one he said; if your bird wins, than you give me your stake-money, but if you do not win, I shall give you two third of that. Going to the other, he promised in the same way to give three fourths. From both of them his gain would be only 12 gold coins. Find the stake of money each of the cock-owners have.

Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many nte Rs 50 and Rs 100 she received.

Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awanded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row there would be 2 rows more. Find the number of students in the class.

One says, “give me hundred, friend! I shall then become twice as rich as you” The other replies, “If you give me fifty, I shall be five times as rich as you.” Tell me what is the amount of their respective capital?

In a cyclic quadrilateral ABCD , . Find the four angles.

Write the value of k for which the system of equations x + y – 4 = 0 and 2x + ky – 3 = 0 has no solution.

Write the value of k for which the system of equations

2x – y = 5

6x + ky = 15

has infinitely many solutions.

Write the value of k for which the system of equations 3x – 2y = 0 and kx + 5y = 0 has infinitely many solutions.

Write the values of k for which the system of equations x + ky = 0, 2x – y = 0 has unique solution.

Write the set of values of a and b for which the following system of equations has infinitely many solutions.

2x + 3y = 7

2ax + (a + b) y = 28

For what value of k, the following pair of linear equations has infinitely many solutions?

10x + 5y – (k – 5) = 0

20x + 10y – k = 0

Write the number of solutions of the following pair of linear equations:

x + 2y– 8 = 0

2x + 4y = 16

Write the number of solutions of the following pair of linear equations:

x + 3y– 4 = 0

2x + 6y = 7

The value of k for which the system of equations

kx – y = 2

6x – 2y = 3

has a unique solution, is

The value of k for which the system of equations

2x + 3y = 5

4x + ky = 10

has infinite number of solutions, is

The value of k for which the system of equations x + 2y – 3 = 0 and 5x + ky + 7 = 0 has no solution, is

The value of k for which the system of equations 3x + 5y = 0 and kx + 10y = 0 has a non-zero solution, is

If the system of equations

2x + 3y = 7

(a + b) x + (2a – b) y = 21

has infinitely many solutions, then

If the system of equations

3x + y = 1

(2k – 1)x + (k – 1)y = 2k + 1

is inconsistent, then k =

If am ≠ bl, then the system of equations

ax + by = c

lx + my = n

If the system of equations

2x + 3y = 7

2ax + (a + b)y = 28

has infinitely many solutions, then

The value of k for which the system of equations

x + 2y = 5

3x + ky + 15 = 0

has no solution is

If 2x – 3y = 7 and (a + b) x – (a + b – 3) y = 4a + b represent coincident lines, then a and b satisfy the equation

If a pair of linear equations in two variables is consistent, then the lines represented by two equations are

The area of the triangle formed by the line with the coordinate axes is

The area of the triangle formed by the lines y = x, x = 6 and y = 0 is

If the system of equations 2x + 3y = 5, 4x + ky = 10 has infinitely many solutions, then k =

If the system of equations kx – 5y = 2, 6x + 2y = 7 has no solution, then k =

The area of the triangle formed by the lines x = 3, y = 4 and x = y is

The area of the triangle formed by the lines 2x + 3y = 12, x – y – 1 = 0 and x = 0 (as shown in Fig. 3.23) , is