Mathematics exemplar problems class 10 Solutions for Chapter 3 pair of linear equations in two variables

If x = a, y = b, is the solution of the equations x – y = 2 and x + y = 4, then the values of a and b are, respectively

For what value of k do the equations 3x – y + 8 = 0 and 6x – ky = –16 represent coincident lines?

Which of the following pair of linear equations has a solution x = 2, y = –3? Select all the correct answers.

One equation of a pair of dependent linear equations is –5x + 7y = 2. The second equation can be

The value of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinity many solutions is

The father’sage is six times his son, 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.

Graphically, the pair of equations

6x – 3y + 10 = 0

2x – y + 9 = 0

represents two lines which are

The pair of equations y = 0 and y = –7 has

The pair of equations x + 2y + 5 = 0 and –3x – 6y + 1 = 0 have

The pair of equations x = a and y = b graphically represents lines which are

Aruna has only ₹1 and ₹2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹75, then the number of ₹1 and ₹2 coins are respectively

If the line given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is

If a pair of linear equations is consistent, then the lines will be

For the pair of equations, λx + 3y = –7 and 2x + 6y = 14, to have infinitely many solutions the value of λ should be 1. Is the statement true?

For all real values of c, the pair of equations x – 2y = 8 and 5x – 10y = c have a unique solution. Justify whether it is true or false.

Solve x = 2y ; y = 2x , find if they have

Solve 3x + 1/7y = 3 ; 7x + 3y = 7 , find if they have

Solve –3x– 4y = 12 ; 4y + 3x = 12 and check whether equations are

Solve (3/5)x – y = 1/2 ; (1/5)x – 3y= 1/6 and check whether equations are

Solve 2ax + by –a = 0 ; 4ax + 2by – 2a = 0 and check whether equations are

Solve –2x – 3y = 1 ; 6y + 4x = – 2 , find if they have

Solve x/2 + y + 2/5 = 0 ; 4x + 8y + 5/16 = 0 , find if they have

Solve x + 3y = 11 ; 2x + 6y = 11 and check whether equations are

Solve 2x + 4y = 3 ; 12y + 6x = 6 , find if they have

Solve 3x + y – 3 = 0 ; 2x + 2/3y = 2 , find if they have

The line represented by x = 7 is parallel to x–axis. State whether the statement is true or not.

For which value(s) of λ do the pair of linear equations λx + y = λ² and x + λy = 1 have no solution ?

For which value(s) of λ do the pair of linear equations λx + y = λ² and x + λy = 1 have infinitely many solutions ?

For which value(s) of λ do the pair of linear equations λx + y = λ² and x + λy = 1 have for unique solution ?

Solve: x+y=3.3,\frac{0.6}{3x-2y}=-1,3x-2y\ne0

Solve: \frac{x}{3}+\frac{y}{4}=4, \frac{5 x}{6}-\frac{y}{8}=4.

Solve 4 x+\frac{6}{y}=15,6 x-\frac{8}{y}=14, y \neq 0.

Solve \frac{1}{2x}-\frac{1}{y}=-1,\frac{1}{x}+\frac{1}{2y}=8,x,y\ne0.

Solve \frac{2 x y}{x+y}=\frac{3}{2}, \frac{x y}{2 x-y}=\frac{-3}{10}, x+y \neq 0,2 x-y \neq 0.

Solve 43x + 67y = –24, 67x+ 43y = 24

Find the solution of the pair of equations \frac{x}{10}+\frac{y}{5}-1=0 \text { and } \frac{x}{8}+\frac{y}{6}=15. Hence, find λ if y = λx+5.

Find out whether 3x + y + 4 = 0 and 6x – 2y + 4 = 0 is consistent or not. If consistent, solve them.

Find out whether x + y = 3 and 3x + 3y = 9 is consistent or not. If consistent, find if they are dependent or independent.

Find out whether x – 2y = 6 and 3x – 6y = 0 is consistent or not.

– x + py = 1 and px – y = 1, if the pair of equations has no solution. Find p.

Solve \frac{x}{a}+\frac{y}{b}=a+b, \frac{x}{a_{2}}+\frac{y}{b_{2}}=2, a, b \neq 0.

The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father.

Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio becomes 4:5. find the numbers.

Two years ago Salim was thrice as old his daughter and six years later he will be four years older than twice her age. How old are they now?

Which of the following is a line passing through the point representing the solution of the pair of linear equations x + y = 2 and 2x – y = 1?

Draw the graph of the pair of equations 2x + y = 4 and 2x – y = 4 and find the area of this triangle.

The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)°, ∠C = (x + y)°, ∠D = (3y – 10)°. Find x and y, and hence the values of four angles.

There are some students in two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from A to B. But if 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the numbers of student in the two halls.

The angles of a triangle are x, y and 40°. The difference between the two angles x and y is 30°. Find x and y.

A shopkeeper given 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 ₹22 for a book kept for six days, while Anand paid ₹16 for the book kept for days. Find the fixed charges and the charge for each extra day.

In a competitive examination, 1 mark is awarded for each correct answer, while ½ mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly ?

Solve s – t = 3 and (s/3) + (t/2) = 6 of linear equations by the substitution method and find s and t.

3x – y – 5 = 0 and 6x – 2y – p = 0, if the lines represented by these equations are parallel. Find p.

For which values of a and b, will the following pair of linear equations have infinitely many solutions?

x + 2y = 1

(a – b)x + (a + b)y = a + b – 2

2x + 3y – 5 = 0 and px – 6y – 8 = 0, if the pair of equations has a unique solution. Find p.

For which value(s) of k will the pair of equations kx + 3y = k – 3 ; 12x + ky = k have no solution?

– 3x + 5y = 7 and 2px – 3y = 1, if the lines represented by these equations are intersecting at a unique point. Find p.

2x+ 3y= 7 and 2p + py = 28 – qy, if the pair of equations have infinitely many solutions. Find p and q.

Two straight paths are represented by the equations x – 3y = 2 and –2x + 6y = 5. Check whether the paths cross each other or not.

Write a pair of linear equations which has the unique solution x = – 1, y =3. How many such pairs can you write?

If 2x + y = 23 and 4x – y = 19, find the values of 5y – 2x and y/x – 2.

Find the values of x and y in the following rectangle

Find out whether 2x- 2y = 6 and 3x-2y = 0 is consistent or not.

Find out whether x – y = 1 and x – y = 0 is consistent or not.

Find out whether 2x – y = 1 and x – 2y = 2 is consistent or not.

If (x + 1) is a factor of 2x³ + ax² + 2bx + 1, then find the values of a and b given that 2a – 3b = 4.

Graphically, solve the following pair of equations:

2x + y = 6

2x – y + 2 = 0

Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.

Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8

Find the area of the quadrilateral and graph formed by the lines and the x–axis of equations x = 3, x = 5 and 2x – y – 4 = 0.

The cost of 4 pens and 4 pencil boxes is Rs 100. Three times the cost of a pen is Rs 15 more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box.

Determine, algebraically, the vertices of the triangle formed by the lines

3x – y = 3

2x – 3y = 2

x + 2y = 8

A person, rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of the stream.

A two–digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.

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 station A to B costs ₹ 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B costs ₹ 3810. Find the full first class fare from station A to B, and also the reservation charges for a ticket.

A shopkeeper sells a saree at 8% profit and a sweater at 10% discount; thereby getting a sum of ₹1008. If she had sold the saree at 10% profit and the sweater at 8% discount. She would have got ₹1028, then find the cost price of the saree and the list price (price before discount) of the sweater.

Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received ₹1860 as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received ₹20 more as annual interest. How much money did she invest in each scheme ?

Vijay had some bananas, and he divided them into two lots A and B. He sold the first lot at the rate of ₹2 for 3 bananas and the second lot at the rate of ₹1 per banana, and got a total of ₹400. If he had sold the first lot at the rate of ₹1 per banana, and the second lot at the rate of ₹4 for 5 bananas, his total collection would have been ₹460. Find the total number of bananas he had.