Question 1 :
State whether the following pair of linear equations has unique solution, no solution, or infinitely many solutions : $x – 3y – 3 = 0 ; 3x – 9y – 2 = 0$
Question 2 :
In the following pair of equations: 2x + y = 6 and 2x – y + 2 = 1, 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.
Question 3 :
The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
Question 4 :
Two linear equations are in the same two variables x and y. Equations like these are called a pair of linear equations in one variable. TRUE or FALSE?
Question 5 :
The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
Question 6 :
The ratio of incomes of two persons is 9 : 7 and the ratio of their expenditures is 4 : 3. If each of them manages to save Rs. 2000 per month, find their monthly incomes.
Question 7 :
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?
Question 8 :
An equation which can be put in the form ax + by + c = 0,where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables x and y. TRUE or FALSE?
Question 9 :
Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Question 10 :
In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠A + ∠ B). Find the three angles.
Question 11 :
2 women and 5 men can together finish an embroidery work 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 work, and also that taken by 1 man alone.
Question 12 :
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed are parallel lines.
Question 13 :
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.
Question 14 :
Solve the following pair of linear equations by the elimination method and the substitution method : $3x – 5y – 4 = 0 ~and ~9x = 2y + 7$
Question 15 :
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.
Question 16 :
On comparing the ratios $\frac{a_1}{a_2]$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the lines representing a pair of linear equations are consistent or inconsistent: $5x – 3y = 11 ; – 10x + 6y = –22$
Question 17 :
Solve the following pair of linear equations: 21x + 47y = 110 and 47x + 21y = 162.
Question 18 :
10 students of Class X took part in a 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. Find the solution graphically.
Question 20 :
A pair of linear equations is inconsistent, if it has ___________.
Question 21 :
For which values of a and b does the following pair of linear equations have an infinite number of solutions? $2x + 3y = 7 ; (a – b) x + (a + b) y = 3a + b – 2$
Question 22 :
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis.
Question 23 :
If the lines are represented by the equation $a_1x + b_1y + c_1 =0$ and $a_2x + b_2y + c_2 =0$, then the lines are intersecting when _____________.
Question 24 :
Solve the following pair of equations by reducing them to a pair of linear equations : $\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2 ; \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = -1$.
Question 25 :
Places A and B are 100 km apart on a highway. One car starts from 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 the two cars?
Question 26 :
The sum of the digits of a two-digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is ___________.
Question 27 :
The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs. 155. How much does a person have to pay for travelling a distance of 25 km?
Question 28 :
Solve the following pair of linear equations by the elimination method and the substitution method : $3x + 4y = 10 ~and ~2x – 2y = 2$
Question 29 :
Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co-ordinates ofthe vertices of the triangle formed by these lines and the y axis.
Question 30 :
Use elimination method to find all possible solutions of the following pair of linear equations :$2x + 3y =8 , 4x + 6y =7$
Question 31 :
Is the pair of equations x + 2y – 3 = 0 and 6y + 3x – 9 = 0 consistent?
Question 33 :
Let a pair of linear equations in two variables be $a_{1}x+b_{1}y+c_{1}=0$ and $a_{2}x+b_{2}y+c_{2}=0$. If $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, then the pair of linear equations is _______.
Question 34 :
The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs. 155. What are the fixed charges and the charge per km?
Question 35 :
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.
Question 36 :
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.
Question 37 :
On comparing the ratios $\frac{a_1}{a_2]$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the lines representing a pair of linear equations intersect at a point, are parallel or coincident: $9x + 3y + 12 = 0 ; 18x + 6y + 24 = 0$
Question 38 :
Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
Question 39 :
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes $\frac{1}{2}$ if we only add 1 to the denominator. What is the fraction?
Question 40 :
Akhila goes to a fair with Rs. 20 and wants to have rides on the Giant Wheel and play Hoopla. The number of times she played hoopla is half the number of times she went on giant wheel. Which of these represent this situation algebraically ?
Question 41 :
For what values of k will the following pair of linear equations have infinitely many solutions? $kx + 3y – (k – 3) =0 ; 12x + ky – k =0$
Question 42 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19b5c273b230584979988.png' />
In the above given graph of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0, calculate the area of the triangle formed by the lines so drawn and the x-axis.
Question 43 :
Solve the following pair of equations by reducing them to a pair of linear equations : $\frac{5}{x-1} + \frac{1}{y-2} = 2 ; \frac{6}{x-1} - \frac{3}{y-2} = 1$.
Question 44 :
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed are coincident lines.
Question 45 :
For which values of p and q, will the following pair of linear equations have infinitely many solutions? 4x + 5y = 2 and $\left(2p+7q\right)x+\left(p+8q\right)y=2q-p+1$.
Question 47 :
The pair of equations 5x – 15y = 8 and $3x-9y=\frac{24}{5}$ has __________.
Question 48 :
On comparing the ratios $\frac{a_1}{a_2]$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the lines representing a pair of linear equations intersect at a point, are parallel or coincident: $5x – 4y + 8 = 0 ; 7x + 6y – 9 = 0$
Question 49 :
Other than algebraical methods, how can the pair of linear equations be solved?
Question 50 :
Is x = 1, y = 7 a solution of $2x + 3y = 5$?
