Class 10 Maths MCQ Questions of Pair of Linear Equations in Two Variables with Answers

Question

Important Class 10 Maths MCQ Questions of Pair of Linear Equations in Two Variables with Answers?

Answer 1

Two linear equations are 2 variables  It can be tended to graphically and algebraically moreover called the pair of linear equations. Algebraically, students can utilize the Substitution Method, Elimination strategy, or Cross-multiplication technique to solve the pair of linear equations in two variables.

MCQ Questions of Pair of Linear Equations in Two Variables

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

(a) intersecting
(b) parallel
(c) always coincident
(d) intersecting or coincident

2. The sides of a triangle are AB, BC and AC whose equations are respectively y-2 = 0; y = 3x – 7 and 2y + x = 0, The coordinates of points A and B are :

(a) A(3,2), B(-4,2)
(b) A(-4,2), B(3,2)
(c) A(3,2), B(2,-l)
(d) A(-4,2), B(2, -1)

3. 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 who took part in the quiz.

(a) 3
(b) 7
(c) 2
(d) 8

4. 5 pencils and 7 pens together cost Rs. 50 whereas 7 pencils and 5 pens cost Rs. 46. Find the cost of 2 pencils and 3 pens.

(a) Rs. 42
(b) Rs. 39
(c) Rs. 21
(d) Rs. 16

5. A linear equation in two variables has:

(a) 1 solution
(b) 2 solutions
(c) no solution
(d) infinitely many

6. A pair of linear equations is not consistent if:

(a) if has one solution
(b) it has many solutions
(c) graph intersect or coincide
(d) graph is parallel

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

(a) Parallel
(b) Always coincident
(c) Intersecting or coincident
(d) Always intersecting

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

(a) One solution
(b) Two solutions
(c) Infinitely many solutions
(d) No solution

9. The father’s age is six times his son’s 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

(a) 4 and 24
(b) 5 and 30
(c) 6 and 36
(d) 3 and 24

10. Rakshita has only Rs. 1 and Rs. 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs 75, then the number of Rs.1 andRs.2 coins is, respectively

(a) 35 and 15
(b) 35 and 20
(c) 15 and 35
(d) 25 and 25

11. The perimeter of a rectangle is 44 cm. Its length exceeds twice its breadth by 4 cm. Find the area of the rectangle.

(a) 46 \(cm^2\)
(b) 49 \(cm^2\)
(c) 96 \(cm^2\)
(d) 69 \(cm^2\)

12. For what value of k, will the equations, x + 2y + (11 – k) = 0 and 2x + ky + (10 + k) = 0 represent the coincident lines :

(a) k = 12
(b) k = 4
(c) k = 36
(d) k = 2

13. An equation ax + by + c = 0 is a linear equation in 2 variables, where a, b, c are :

(a) natural numbers
(b) whole numbers
(c) integers
(d) real numbers

14. The pair of equations y = 9 and y = – 7 has:

(a) one solution
(b) two solutions
(c) infinitely many
(d) no solution solutions

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

(a) – 10
(b) – 5
(c) – 6
(d) – 15

16. A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Reema paid Rs. 22 for a book kept for six days, while Ruchika paid Rs 16 for the book kept for four days, then the charge for each extra day is:

(a) Rs 5
(b) Rs 4
(c) Rs 3
(d) Rs.2

17. The sum of a two digit number is 8. The number obtained by reversing the digits exceeds the number by 18. Then the given number is :

(a) 53
(b) 35
(c) 26
(d) 62

18. In a triangle, the sum of two angles is equal to the third angle. If the difference between two angles is 30°, find the angles.

(a) 15°, 45°, 75°
(b) 20°, 50°, 80°
(c) 30°, 60°, 90°
(d) 45°, 45°, 90°

19. If x + 5y = 34 and x – 5y = – 6, find the value of 5y – 2x.

(a) – 8
(b) 14
(c) 8
(d) 20

20. Two equations in two variables taken together are called

(a) linear equations
(b) quadratic equations
(c) simultaneous equations
(d) none of these

Answer 2

Answers & Explanations

1. Answer: (d) intersecting or coincident

Explanation: A pair of linear equations is called inconsistent when the lines doesn’t have any solution. It means both the lines are parallel to each other.

A pair of linear equations is called consistent when they have infinite number of solutions or they have a unique solution.

An intersecting line will always have a unique solution.

A coincident line will have infinite number of solutions.

So the line represented by a pair of linear equations in two variables is always intersecting or coincident if the system of equation is consistent.

Let a₁x + b₁y = c₁ & a₂x + b₂y = c₂ be two lines where

a₁ & a₂ are the coefficients of x

b₁ & b₂ are the coefficients of y

c₁ & c₂ are the constants

For the system of linear equations to have infinitely many solutions we must have

\(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)....(i)

The system of linear equations will have unique solution if

\(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\)....(ii)

For the system of linear equations to be consistent either condition (i) or (ii) must be satisfied.

If the equations are consistent then they are either intersecting or coincident.

2. Answer: (b) A(-4,2), B(3,2)

Explanation: Equation of line AB : y – 2 = 0 Equation of line BC: y = 3 x -7 Equation of line AC: 2y + x = 0 Coordinates of point B can be obtained by solving AB and BC \(\Rightarrow\) (3,2).

Coordinate of point A are (- 4, 2)
A (- 4,2) and B(3,2)

3. Answer: (a) 3

Explanation: No. of boys = x
and No. of girls = y
x + y =10
y = x + 4
x = 3
y = 7
\(\Rightarrow\) Number of boys = 3

4. Answer: (c) Rs. 21

Explanation: Let, cost of 1 pencil = x and cost of 1 pen = y – 2

Cost of 2 pencil and 3 pens = 6 +15 = Rs. 21

5. Answer: (d) infinitely many

Explanation: A linear equation in two variables has infinite number of solutions.

6. Answer: (d) graph is parallel

7. Answer: (c) Intersecting or coincident

Explanation: If a pair of linear equations is consistent the two lines represented by these equations definitely have a solution, this implies that either lines are intersecting or coincident.

8. Answer: (d) No solution

Explanation: The graph of equations will be parallel lines. So the equations have no solution.

9. Answer: (c) 6 and 36

Explanation: Let the age of father be x and of son is y.

Then according to question,

x = 6y …..(i)

Four years hence age of son will be y + 4 and age of father will be x + 4

Then according to question,

x + 4 = 4 (y + 4)

x – 4y = 12 …..(ii)

Solving equations (i) and (ii) we get:

y = 6 and x = 36

10. Answer: (d) 25 and 25

Explanation: Let her number of Rs.1 coins are x

Let the number of Rs.2 coins are y

Then

By the given conditions

x + y = 50 …..(i)

1 × x + 2 × y = 75

⇒ x + 2y = 75 …..(ii)

Solving equations (i) and (ii) we get:

(x + 2y) – (x + y) = 75 – 50

⇒ y = 25

Therefore, x = 50 – 25 = 25

So the number of coins are 25, 25 each.

11. Answer: (c) 96 \(cm^2\)

Explanation: Let, Length = x cm
Breadth = y cm
=> 2(x + y) = 44
=> x + y =221, y = 6 cm ,x = 16 cm
x = 2y+4
=> Area = 16 x 6 = 96 \(cm^2\)

12. Answer: (b) k = 4

Explanation: \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

\(\frac{1}{2}=\frac{2}{k}=\frac{11-k}{10+k}\)

k= 4

13. Answer: (d) real numbers

14. Answer: (d) no solution solutions

Explanation: y = 9 and y = – 7 are lines parallel to X-axis, thus these are parallel lines and hence no solution.

15. Answer: (d) – 15

Explanation: Given:

Equation 1: kx – 5y = 2

Equation 2: 6x + 2y = 7

Both the equations are in the form of :

a₁x + b₁y = c₁ & a₂x + b₂y = c₂ where

a₁ & a₂ are the coefficients of x

b₁ & b₂ are the coefficients of y

c₁ & c₂ are the constants

For the system of linear equations to have no solutions we must have

\(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}\)....(i)

According to the problem:

a₁ = k

a₂ = 6

b₁ = – 5

b₂ = 2

c₁ = 2

c₂ = 7

Putting the above values in equation (i) and solving we get:

\(\frac{k}{6}=\frac{-5}{2}\)

\(\Rightarrow k=\frac{-5\times 6}{2}\Rightarrow k=-15\)

Also we find

\(\frac{-15}{6}=\frac{-5}{2}\neq \frac{2}{7}\)

The value of k for which the system of equations has no solution is k = – 15

16. Answer: (c) Rs 3

Explanation: Let Rs. x be the fixed charge and Rs. y be the charge for each extra day.

Then by the given conditions

x + 4y = 22 …..(i)

x + 2y = 16 …..(ii)

Subtracting equation (ii) from (i), we get:

y = Rs. 3

17. Answer: (b) 35

Explanation: Let, the digit at units’ place = x and digit at ten’s place = y => x + y = 8 and lOx + y = 18 + (10y + x)
No. = 35.

18. Answer: (c) 30°, 60°, 90°

Explanation: (x) + (y) + (x + y) 180°
x + y = 90°
x – y = 30°
=> x = 60°, y = 30°
=> Angles are 30°, 60° and 90°

19. Answer: (a) – 8

Explanation: x + 5y = 34
x – 5y = – 6
x= 14
Solving these,
x = 14; y = 4

20. Answer: (c) simultaneous equations

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