CBSE Class 10 Mathematics Pairs of Linear Equations in Two Variables MCQs Set 04

Question: The value of k for which the system of equations 2x + y – 3 = 0 and 5x + ky + 7 = 0 has no solution is
a) 2
b) 5
c) 5/2
d) 3/7
Answer: c

Question: Anmol’s age is six times his son’s age. Four years hence, the age of Anmol will be four times his son’s age. The present age in years, of the father and the son are respectively
a) 24 and 4
b) 30 and 5
c) 36 and 6
d) 24 and 32
Answer: b

Question: If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is
a) 3/2
b) 15/4
c) 2/5
d) –5/4
Answer: b

Question: If we add 1 to the numerator and denominator of a fraction, it becomes 1/2. It becomes 1/3 if we only add 1 to the denominator. The fraction is
a) 3/4
b) 2/5
c) 3/5
d) 1/4
Answer: c

Question: If x = a and y = b is the solution of the equations x – y = 2 and x + y = 4, then the values of a and b are respectively
a) 3 and 5
b) 5 and 3
c) 3 and 1
d) –1 and –3
Answer: c

Question: The value of g for which the system of equations 5gx – 2y = 1 and 10x + y = 3 has a unique solution is
a) = 4
b) ≠ 4
c) = –4
d) ≠ –4
Answer: d

Question: The pair of equations x + 2y – 3 = 0 and 4x + 5y = 8 has
a) no solution
b) infinitely many solutions
c) a unique solution
d) exactly two solutions
Answer: c

Question: One equation of a pair of dependent linear equations is 3x – 4y = 7. The second equation can be
a) –6x + 8y = 14
b) –6x + 8y + 14 = 0
c) 6x + 8y = 14
d) –6x – 8y – 14 = 0
Answer: b

Question: The sum of the digits of a two-digit number is 12. If 18 is subtracted from it, the digits of the number get reversed. The number is
a) 57
b) 75
c) 84
d) 48
Answer: b

Question: If a pair of linear equations is consistent, then the lines will be
a) always intersecting
b) always coincident
c) intersecting or coincident
d) parallel
Answer: b

Question: A pair of linear equations which has a unique solution x = 1, y = –3 is
a) x – y = 4; 2x + 3y = 5
b) 2x – y = –5; 5x – 2y = 11
c) 3x + y = 0; x + 2y = –5
d) x + y = –2; 4x + 3y = 52
Answer: b

Question: If x = a, y = b is the solution of the equation x + y = 3 and x – y = 5, then the values of a and b are, respectively
a) 4 and –1
b) 1 and 2
c) –1 and 4
d) 2 and 3
Answer: c

Question: The value of c for which the pair of equations 4x – 5y + 7 = 0 and 2cx – 10y + 8 = 0 has no solution is
a) 8
b) –8
c) 4
d) –4
Answer: a

Question: The pair of equations x = a and y = b graphically represents lines which are
a) coincident
b) parallel
c) intersecting at (a, b)
d) intersecting at (b, a)
Answer: c

Question: If the lines given by 2x – 5y + 10 = 0 and kx + 15y – 30 = 0 are coincident, then the value of k is
a) –6
b) 6
c) 1/3
d) –1/3
Answer: c

Question: Gunjan 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
a) 25 and 25
b) 15 and 35
c) 35 and 15
d) 35 and 20
Answer: a

Question: The pair of equations 6x – 4y + 9 = 0 and 3x – 2y + 10 = 0 has
a) a unique solution
b) no solution
c) exactly two solutions
d) infinitely many solutions
Answer: c

Question: A pair of linear equations in two variables may not have infinitely many solutions.
Answer: T

Question: An equation of the form ax + by + c = 0, where a, b and c are real numbers is called a linear equation in two variables.
Answer: F

Question: A pair of linear equations in two variables is said to be consistent if it has no solution.
Answer: F

Question: A linear equation in two variables always has infinitely many solutions.
Answer: T

Question: The pair of equations 4x -5y = 8 and 8x -10 y = 3 has a unique solution.
Answer: F

Question: A pair of linear equations cannot have exactly two solutions.
Answer: T

Question: If two lines are parallel, then they represent a pair of inconsistent linear equations.
Answer: T

Question: A pair of intersecting lines represent a pair of linear equations in two variables having a unique solution.
Answer: T

Question: How many solutions does the pair of equations.
x +2y = 3 and 1/2 x + y 3/2 – = 0 have?
Answer: Infinite

Question: Is the pair of equations 3x -5y = 6 and 4x -6 y = 7 consistent? Justify your Answer.
Answer: Yes, since a1/a2≠b1/b2

Question: Do the equations 5x +7y = 8 and 10x +14 y = 4 represent a pair of coincident lines? Justify your Answer.
Answer: No, because a1/a2≠b1/b2 ≠c1/c2 ¹ , so the equations represent parallel lines.

Question: Write the number of solutions of the following pair of linear equations:
3x -7y =1 and 6x -14 y -3 = 0
Answer: 0

Question: For the pair of equations lx +3y = -7, 2x -6 y =14 to have infinitely many solutions, the value of λ should be 1. Is the statement true? Give reason.
Answer: False, it should be –1

Question: Is the pair of equations x – y = 5 and 2y – x =10 inconsistent? Justify your Answer.
Answer: No, since a1/a2≠b1/b2 , so it has a unique solution

Question: State whether the following statements are true or false. Justify your Answer: .
(i) The equations x/2+y+(1/5) and 4x+8y+(8/5)=0 represent a pair of coincident lines.
(ii) For all real values of k, except –6, the pair of equations kx -3y = 5 and 2x + y = 7 has a unique solution.
Answer: (i) True (ii) False

Question:
(i) Draw the graphs of the equations y = 3, y = 5 and 2x – y – 4 = 0. Also, find the area of the quadrilateral formed by the lines and the y-axis.
(ii) A motorboat 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.
Answer:
(i) x =1, y = 4, Areas = 8 sq. units, 2 square units; ratio = 4 : 1
(ii) Speed of the bus is 60 km/h; Speed of the train is 90 km/h

Question: State whether the following statements are true or false. Justify your Answer: .
(i) The pair of equations 3x – 4 y =1 and 4x +3y =1 has a unique solution.
(ii) For the pair of equations 4x +ly = –3 and 6x +9 y + 4 = 0 to have no solution, the value of l should not be
Answer: (i) True (ii) True

Question:
(i) For what 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
(ii) Solve for x and y
x/a + y/b =a + b , x/a2 + y/b2 = 2, a, b ≠ 0
Answer:
(ii) x=20 y=30 ∠A=130° ∠B=100° ∠C=50° ∠D=80°
(i) x =7 and y =9, values –1 and 30/7

Question:
(i) If 2x + y = 23 and 4x – y =19, find the values of 3y – 4x and y/x+3.
(ii) 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 value of the four angles.
Answer: (i) a= 3, b=1(ii) x= a2 , y= b2

Question:
(i) Graphically solve the 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.
(ii) Saksham travels 360 km to his home partly by train and partly by bus. He takes four and a half hours if he travels 90 km by bus and the remaining by train. If he travels 120 km by bus and remaining by train, he takes 10 minutes longer. Find the speed of the train and the bus separately.
Answer:
(i) Area of trapezium=8 sq. units
(ii) Speed of the boat in still water = 10 km/h; Speed of the stream = 4 km/h