If the system of equations has infinitely many solutions, then : $2x + 3y = 7(a + b)x + (2a - b)y = 21$
- A$a = 1, b = 5$
- ✓$a = 5, b = 1$
- C$a = -1, b = 5$
- D$a = 5, b = -1$
Answer: B.
View full solution →Question types
Linear Equations in Two Variables question types
420 questions across 8 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
420
Questions
8
Question groups
5
Question types
01
M.C.Q (1 Marks)
78 Q→02Assertion (A) & Reason (B) MCQ
8 Q→03Fill In The Blanks[1 Marks ]
15 Q→041 Marks Question
11 Q→052 Marks Questions
6 Q→063 Marks Question
86 Q→075 Marks Questions
209 Q→08Case study (4 Marks)
7 Q→Sample Questions
Linear Equations in Two Variables questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The value of k for which the system of equations x + 2y - 3 = 0 and 5x + ky + 7 = 0 has no solution, is:
- ✓10.
- B6.
- C3.
- D1.
Answer: A.
View full solution →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:
- A25.
- B72.
- C63.
- ✓36.
Answer: D.
View full solution →If am ≠ bl, then the system of equations $ax + by = clx + my = nax + by = clx + my = n.$
- ✓Has a unique solution.
- BHas no solution.
- CHas infinitely many solutions.
- DMay or may not have a solution.
Answer: A.
View full solution →If 2x - 3y = 7 and (a + b)x - (a + b - 3)y = 4a + b represent coincident lines, then a and b satisfy the equation:
- Aa + 5b = 0
- B5a + b = 0
- ✓a - 5b = 0
- D5a - b = 0
Answer: C.
View full solution →Statement-1 (A) : The area of the trapezium formed by the lines $3 x+4 y-12=0$ and $3 x+4 y=6$ is $\frac{9}{2}$ square units.
Statement-2 (R): The system of equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ is inconsistent, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
Statement-2 (R): The system of equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ is inconsistent, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
- AStatement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Statement-1 (A): The system of equations $2 x+y+9=0$ and $x+3 y+7=0$ is consistent having unique solution.
Statement-2 (R): The system of equations $a x+b y+c=0$ and $p x+q y+r=0$ is always consistent with unique solution, if $a q \neq b p$.
Statement-2 (R): The system of equations $a x+b y+c=0$ and $p x+q y+r=0$ is always consistent with unique solution, if $a q \neq b p$.
- ✓Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): The system of linear equations $2 x+3 y=7$ and $k x+\frac{9}{2} y=12$ have no solution, if $k=3$.
Statement -2(R) : The system of linear equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ have no solution, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}$.
Statement -2(R) : The system of linear equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ have no solution, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}$.
- AStatement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- ✓Statement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: C.
View full solution →Statement-1 $(A)$ : The system of linear equations $9 x+3 y+12=0$ and $18 x+6 y+24=0$ have infinitely many solutions.
Statement-2 $(R)$ : The system of linear equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ have infinitely many solutions, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
Statement-2 $(R)$ : The system of linear equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ have infinitely many solutions, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
- ✓Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
- CStatement-1 is True, Statement-2 is False.
- DStatement-1 is False, Statement-2 is True.
Answer: A.
View full solution →Statement-1 (A): If the system of equations $3 x+6 y=10$ and $2 x-k y+5=0$ is inconsistent, then $k=-4$.
Statement-2 $(R)$ : The system of equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ is inconsistent iff $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
Statement-2 $(R)$ : The system of equations $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ is inconsistent iff $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
- AStatement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- BStatement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement-1.
- ✓Statement-1 is true, Statement-2 is false.
- DStatement-1 is false, Statement-2 is true.
Answer: C.
View full solution →The system of equations ax + 3y = 1 - 12x + ay = 2 has __________ for all real values of 4.
If $2^{x+y}=2^{x-y}=\sqrt{8}$, then x = __________ and y = __________.
View full solution →The line 4x + 3y - 12 = 0 cuts the coordinate axes at A and B. The area of $\triangle O A B$ is __________.
View full solution →If the pair of equations ax + 2y = 7 and 3x + by = 16 represent parallel lines, then ab =__________.
If one equation of a pair of dependent linear equations is 5x - 7y + 2 = 0 then the second equation is given by __________.
Solve for x and y:
23x + 24y = 23
24x + 23y = 24
23x + 24y = 23
24x + 23y = 24
Sum of two numbers is 105 and their difference is 45. Find the numbers.
If 2x + y = 13 and 4x - y = 17, find the value of (x - y).
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 number of solutions of the following pair of linear equations:
x+3y -4 = 0
2x +6y = 7
x+3y -4 = 0
2x +6y = 7
Which value(s) of $\lambda,$ do the pair of linear equations $\lambda\text{x}+\text{y}=\lambda^2$ and $\text{x}+\lambda\text{y}=1$have:
Infinitely many solutions?
View full solution →Infinitely many solutions?
On comparing the ratios $\frac{\text{a}_1}{\text{a}_2},\frac{\text{b}_1}{\text{b}_2}$ and $\frac{\text{c}_1}{\text{c}_2},$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.
View full solution →$5x - 4y + 8 = 0$
$7x + 6y - 9 = 0$
Which value(s) of $\lambda,$ do the pair of linear equations $\lambda\text{x}+\text{y}=\lambda^2$ and $\text{x}+\lambda\text{y}=1$have:
a unique solutions?
View full solution →a unique solutions?
On comparing the ratios $\frac{\text{a}_1}{\text{a}_2},\frac{\text{b}_1}{\text{b}_2}$ and $\frac{\text{c}_1}{\text{c}_2},$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.
View full solution →$6x - 3y + 10 = 0$
$2x - y + 9 = 0$
On comparing the ratios $\frac{\text{a}_1}{\text{a}_2},\frac{\text{b}_1}{\text{b}_2}$ and $\frac{\text{c}_1}{\text{c}_2},$ and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.
View full solution →$9x + 3y + 12 = 0$
$18x + 6y + 24 = 0$
Find the value of k for which the following system of equations has a unique solution:
View full solution →$kx + 2y = 5$
$3x + y = 1$
Find the value of k for which the following system of equations has no solution:
View full solution →$2x + ky = 11$
$5x - 7y = 5$
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.
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 notes Rs. 50 and Rs 100 she received.
The cost of 4 pens and 4 pencil boxes is ₹ 100. Three times the cost of a pen is ₹ 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 pencil box.
Write a pair of linear equations which has the unique solution $x = -1, y = 3$. How many such pairs can you write?
View full solution →Solve the following systems of equations:
$\frac{\text{xy}}{\text{x}+\text{y}}=\frac{6}{5}$
$\frac{\text{xy}}{\text{y}-\text{x}}=6$ where $\text{x}+\text{y}\neq0,\text{y}-\text{x}\neq0.$
View full solution →$\frac{\text{xy}}{\text{x}+\text{y}}=\frac{6}{5}$
$\frac{\text{xy}}{\text{y}-\text{x}}=6$ where $\text{x}+\text{y}\neq0,\text{y}-\text{x}\neq0.$
In the following system of equation determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:
View full solution →$x - 2y = 8$
$5x - 10y = 10$
Find the value of k for which each of the following system of equations have infinitely many solutions:
View full solution →$2x + 3y = k$
$(k - 1)x + (k + 2)y = 3k$
Find the values of a and b for which the following system of equations has infinitely many solutions:
View full solution →$2x + 3y = 7$
$(a - 1)x + (a + 2)y = 3a$
Essel World is one of India's largest amusement parks that offers a diverse range of thrilling des, water attractions and entertainment options for visitors of all ages. The park is known for its iconic -Vater Kingdom" section, making it a popular destination for family outings and fun-filled adventure. The ket charges for the park are ₹ 150 per child and ₹ 250 per adult.
On a day, the cashier of the park found that 300 tickets were sold and an amount of ₹$ 55,000$ was collected. Based on the above, answer the following questions:
(i) If the number of children visited be $x$ and the number of adults visited be $y$, then write the given situation algebraically.
(ii) (a) How many children visited the amusement park that day?
OR
(b) How many adults visited the amusement park that day?
(iii) How much amount will be collected if 250 children and 100 adults visit the amusement park?
View full solution → On a day, the cashier of the park found that 300 tickets were sold and an amount of ₹$ 55,000$ was collected. Based on the above, answer the following questions:
(i) If the number of children visited be $x$ and the number of adults visited be $y$, then write the given situation algebraically.
(ii) (a) How many children visited the amusement park that day?
OR
(b) How many adults visited the amusement park that day?
(iii) How much amount will be collected if 250 children and 100 adults visit the amusement park?
A coaching institute of Mathematics conducts classes in two batches I and 11 and fees for rich and poor children are different. In batch I, there are 20 poor and 5 rich children, whereas in batch II, there are 5 poor and 25 rich children. The total monthly collection of fees from batch I is ₹9,000 and from batch II is ₹ 26,000 . Assume that each poor child pays ₹ $x$ per month and each rich child pays ₹ y per month.
Based on the aborv information, ansuyy the followents puestions:
(i) Represent the information given ahorv in terms of $x$ and $y$.
(ii) Find the monthly for puid by a poor child.
(iii) Find the difference in the monthly fov paid ty a poor child and a rikh chill.
(iv) If there are 10 poor and 20 rich children in Mutch 11 , what is the tolat mowhly collectious of fwa fome batch 11 .
View full solution →Based on the aborv information, ansuyy the followents puestions:
(i) Represent the information given ahorv in terms of $x$ and $y$.
(ii) Find the monthly for puid by a poor child.
(iii) Find the difference in the monthly fov paid ty a poor child and a rikh chill.
(iv) If there are 10 poor and 20 rich children in Mutch 11 , what is the tolat mowhly collectious of fwa fome batch 11 .
It is common that governments revise travel fares from time to time based on various factors such as inflation (a general increase in prices and fall in the purchasing value of money) on different types of vehicles like auto rickshaws, taxis, radio cabs etc. The auto charges in a city comprise of a fixed charge together with the charge for the distance covered. Study the following situations:
(i) If the fixed charges of autorickshaw be ₹ $x$ and the running charges be ₹ $y$ per km, the pair of linoar equations representing the travel in city $A$ is
(a) $x+10 y=75, x+5 y=145$
(b) $x+10 y=75, x+15 y=110$
(c) $x+8 y=91, x+14 y=145$
(d) $x+8 y=145, x+14 y=91$
(ii) If the fixed charges of autorikshaw be ₹$ x$ and the running charges be ₹$ y$ per km, the pair of linoar equations representing the travel in City $B$ is
(a) $x+10 y=75, x+5 y=145$
(b) $x+10 y=75, x+15 y=110$
(c) $x+8 y=91, x+14 y=145$
(d) $x+8 y=145, x+14 y=91$
(iii) The amount paid by a person travelling 100 km in city $A$ is
(a) ₹ 310 $\qquad$ (b) ₹ 510 $\qquad$ (c) ₹ 705 $\qquad$ (d) ₹ 710
(iv) The amount paid by a person travelling 60 km in city $B$ is
(a) ₹ 370 $\qquad$ (b) ₹ 578 $\qquad$ (c) ₹ 559 $\qquad$ (d) ₹ 610
View full solution →| Name of the City | Distance travelled (km) | ( Amount paid (₹)) |
| City A | 10 | 75 |
| 15 | 110 | |
| City B | 8 | 91 |
| 14 | 145 |
(a) $x+10 y=75, x+5 y=145$
(b) $x+10 y=75, x+15 y=110$
(c) $x+8 y=91, x+14 y=145$
(d) $x+8 y=145, x+14 y=91$
(ii) If the fixed charges of autorikshaw be ₹$ x$ and the running charges be ₹$ y$ per km, the pair of linoar equations representing the travel in City $B$ is
(a) $x+10 y=75, x+5 y=145$
(b) $x+10 y=75, x+15 y=110$
(c) $x+8 y=91, x+14 y=145$
(d) $x+8 y=145, x+14 y=91$
(iii) The amount paid by a person travelling 100 km in city $A$ is
(a) ₹ 310 $\qquad$ (b) ₹ 510 $\qquad$ (c) ₹ 705 $\qquad$ (d) ₹ 710
(iv) The amount paid by a person travelling 60 km in city $B$ is
(a) ₹ 370 $\qquad$ (b) ₹ 578 $\qquad$ (c) ₹ 559 $\qquad$ (d) ₹ 610
Ravish is planning to buy a house whose layout is given below. The design and the measurement has been made such that areas of two bedrooms and kitchen together is $95 m^2$.
(i) The pair of linear equations in two variables describing this situation is
(a) $2 x+y=19, x+y=13$
(b) $x+2 y=19, x+y=13$
(c) $2 x+y=13, x+y=13$
(d) $2 x+y=13, x+y=19$
(ii) The perimeter and area of the house are respectively
(a) $54 m, 180 m^2$
(b) $180 m, 54 m^2$
(c) $27 m, 90 m^2$
(d) $108 m, 180 m^2$
(iii) The value of $x y$ is
(a) 42 $\qquad$ (b) 48 $\qquad$ (c) 49 $\qquad$ (d) 13
(iv) The value of $x-y$ is
(a) 13 $\qquad$ (b) 1 $\qquad$ (c) -1 $\qquad$ (d) .42
View full solution →(i) The pair of linear equations in two variables describing this situation is
(a) $2 x+y=19, x+y=13$
(b) $x+2 y=19, x+y=13$
(c) $2 x+y=13, x+y=13$
(d) $2 x+y=13, x+y=19$
(ii) The perimeter and area of the house are respectively
(a) $54 m, 180 m^2$
(b) $180 m, 54 m^2$
(c) $27 m, 90 m^2$
(d) $108 m, 180 m^2$
(iii) The value of $x y$ is
(a) 42 $\qquad$ (b) 48 $\qquad$ (c) 49 $\qquad$ (d) 13
(iv) The value of $x-y$ is
(a) 13 $\qquad$ (b) 1 $\qquad$ (c) -1 $\qquad$ (d) .42
A test consists of 'True' or 'False' questions. One mark awarded for every correct answer while $\frac{1}{4}$ mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of the questions he attempted by guessing. He answered 120 questions and scored 95 marks.
(i) If answer to all questions he attempted by guessing were wrong, then the number of questions he answered correctly is
(a) 24 $\qquad$ (b) 96 $\qquad$ (c) 100 $\qquad$ (d) 90
(ii) The number of questions he guessed, is
(a) 20 $\qquad$ (b) 96 $\qquad$ (c) 20 $\qquad$ (d) 90
(iii) If answers to all questions he attempted by guessing were wrong and answered 80 correctly, then how many marks he got?
(a) 40 $\qquad$ (b) 45 $\qquad$ (c) 70 $\qquad$ (d) 35
(iv) If answer to all questions he attempted by guessing were wrong, then the number of questions answered correctly to score 95 marks is
(a) 100 $\qquad$ (b) 105 $\qquad$ (c) 90 $\qquad$ (d) 95
View full solution →(i) If answer to all questions he attempted by guessing were wrong, then the number of questions he answered correctly is
(a) 24 $\qquad$ (b) 96 $\qquad$ (c) 100 $\qquad$ (d) 90
(ii) The number of questions he guessed, is
(a) 20 $\qquad$ (b) 96 $\qquad$ (c) 20 $\qquad$ (d) 90
(iii) If answers to all questions he attempted by guessing were wrong and answered 80 correctly, then how many marks he got?
(a) 40 $\qquad$ (b) 45 $\qquad$ (c) 70 $\qquad$ (d) 35
(iv) If answer to all questions he attempted by guessing were wrong, then the number of questions answered correctly to score 95 marks is
(a) 100 $\qquad$ (b) 105 $\qquad$ (c) 90 $\qquad$ (d) 95
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