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

Question. The pair of linear equations \( 2kx + 5y = 7 \), \( 6x - 5y = 11 \) has a unique solution, if
(a) \( k \neq -3 \)
(b) \( k \neq \frac{2}{3} \)
(c) \( k \neq 5 \)
(d) \( k \neq \frac{2}{9} \)
Answer: (a)
Given the pair of linear equations are \( 2kx + 5y - 7 = 0 \) and \( 6x - 5y - 11 = 0 \)
On comparing with \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \)
we get, \( a_1 = 2k, b_1 = 5, c_1 = -7 \) and \( a_2 = 6, b_2 = -5, c_2 = -11 \)
For unique solution, \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
\( \frac{2k}{6} \neq \frac{5}{-5} \Rightarrow \frac{k}{3} \neq -1 \Rightarrow k \neq -3 \)

Question. A and B together can do a piece of work in 12 days, B and C together in 15 days. If A is twice as good a workman as C , then in how many days will B alone do it?
(a) 10 days
(b) 15 days
(c) 20 days
(d) 25 days
Answer: (c)
Let \( A \) alone complete the work in \( x \) days. \( C \) alone will take \( 2x \) days to complete it. Let \( B \) alone complete the work in \( y \) days. According to the question,
\( \frac{1}{x} + \frac{1}{y} = \frac{1}{12} \) ...(1)
and \( \frac{1}{y} + \frac{1}{2x} = \frac{1}{15} \) ...(2)
Let \( \frac{1}{x} = u \) and \( \frac{1}{y} = v \), then Eq. (1) and (2) become
\( u + v = \frac{1}{12} \) ...(3)
and \( \frac{1}{2}u + v = \frac{1}{15} \) ..(4)
Subtracting Eq. (4) from Eq. (3), we get
\( u - \frac{1}{2}u = \frac{1}{12} - \frac{1}{15} \Rightarrow \frac{u}{2} = \frac{5-4}{60} = \frac{1}{60} \Rightarrow u = \frac{2}{60} = \frac{1}{30} \)
Putting the value of \( u \) in Eq. (3), we get
\( \frac{1}{30} + v = \frac{1}{12} \Rightarrow v = \frac{1}{12} - \frac{1}{30} = \frac{5-2}{60} = \frac{3}{60} = \frac{1}{20} \)
Now, we have \( u = \frac{1}{30} \) and \( v = \frac{1}{20} \Rightarrow x = 30, y = 20 \)
Hence, \( B \) alone will complete the work in 20 days.

Question. When a man travels equal distance at speed \( x \) km/h and \( y \) km/h, his average speed is 4 km/h. But when he travels at these speed for equal time, his average speed is 4.5 km/h. The difference of the two speed is
(a) 2 km/h
(b) 4 km/h
(c) 3 km/h
(d) 5 km/h
Answer: (c)
Suppose the equal distance \( = D \) km. Then, time taken with \( x \) and \( y \) speed are \( D/x \) h and \( D/y \) h, respectively.
Case I: Average speed \( = \frac{\text{Total distance}}{\text{Total time}} \Rightarrow 4 = \frac{2D}{\frac{D}{x} + \frac{D}{y}} \Rightarrow \frac{2xy}{x+y} = 4 \) ...(1)
Case II: Average speed \( = \frac{x+y}{2} = 4.5 \) km/h \( \Rightarrow x+y = 9 \) ...(2)
On putting this value in Eq. (1), we get \( \frac{2xy}{9} = 4 \Rightarrow xy = 18 \) ...(3)
Now, difference between two speed \( = x - y \)
Now, \( (x-y)^2 = (x+y)^2 - 4xy = (9)^2 - 4(18) = 81 - 72 = 9 \Rightarrow x-y = 3 \) ...(4)
On adding Eq. (2) and (4), we get \( 2x = 12 \Rightarrow x = 6 \).
On putting the value of \( x \) in Eq. (4), we get \( 6 - y = 3 \Rightarrow y = 3 \).
Hence, \( x = 6 \) km/h and \( y = 3 \) km/h and, their difference \( = 6 - 3 = 3 \) km/h.

Question. 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 the cost of the saree and the list price (price before discount) of the sweater is
(a) 300, 400
(b) 400, 300
(c) 400, 600
(d) 600, 400
Answer: (d)
Let, the cost price of the saree and the list price of the sweater be ₹\( x \) and ₹\( y \), respectively.
Case I: Sells a saree at 8% profit + Sells a sweater at 10% discount = ₹1008
\( (100+8)\% \text{ of } x + (100-10)\% \text{ of } y = 1008 \Rightarrow 108\% \text{ of } x + 90\% \text{ of } y = 1008 \Rightarrow 1.08x + 0.9y = 1008 \) ...(1)
Case II: Sold the saree at 10% profit + Sold the sweater at 8% discount = ₹1028
\( (100+10)\% \text{ of } x + (100-8)\% \text{ of } y = 1028 \Rightarrow 110\% \text{ of } x + 92\% \text{ of } y = 1028 \Rightarrow 1.1x + 0.92y = 1028 \) ...(2)

Question. If \( 3|x| + 5|y| = 8 \) and \( 7|x| - 3|y| = 48 \), then the value of \( x + y \) is
(a) 5
(b) -4
(c) 4
(d) The value does not exist
Answer: (d)
Let \( |x| = a \) and \( |y| = b \). Then, given equations becomes
\( 3a + 5b = 8 \) ...(1)
and \( 7a - 3b = 48 \) ...(2)
Now, on multiplying Eq. (1) by 3 and Eq. (2) by 5, we get
\( 9a + 15b = 24 \) ...(3)
\( 35a - 15b = 240 \) ...(4)
On adding Eq. (3) and (4), we get
\( 44a = 264 \)
\( a = 6 \)
Now, on substituting \( a = 6 \) in Eq. (1), we get
\( 18 + 5b = 8 \)
\( 5b = -10 \)
\( b = -2 \)
Thus, we get \( a = 6 \) and \( b = -2 \).
But \( b = -2 \Rightarrow |y| = -2 \), which is not possible. Hence, the value of \( x + y \) does not exist.

Question. A fraction becomes 4/5 when 1 is added to each of the numerator and denominator. However, If we subtract 5 from each of them, it becomes 1/2. Then, numerator of the fraction is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (b)
Let the fraction be \( \frac{x}{y} \).
Then, according to question
\( \frac{x+1}{y+1} = \frac{4}{5} \)
\( 5x + 5 = 4y + 4 \)
\( 5x - 4y = -1 \) ...(1)
and \( \frac{x-5}{y-5} = \frac{1}{2} \)
\( 2x - 10 = y - 5 \)
\( 2x - y = 5 \) ...(2)
On multiplying Eq. (1) by 2 and Eq. (2) by 5 and then subtracting Eq. (2) from Eq. (1), we get
\( 10x - 8y = -2 \)
\( 10x - 5y = 25 \)
\( -3y = -27 \)
\( y = 9 \)
Substituting the value of \( y \) in Eq. (1), we get
\( 5x - 4 \times 9 = -1 \)
\( 5x = -1 + 36 \)
\( x = 7 \)
Hence, Fraction = \( \frac{7}{9} \)
Therefore, numerator of this fraction is 7.

Question. Which of the following pair of equations are inconsistent?
(a) \( 3x - y = 9, x - \frac{y}{3} = 3 \)
(b) \( 4x + 3y = 24, -2x + 3y = 6 \)
(c) \( 5x - y = 10, 10x - 2y = 20 \)
(d) \( -2x + 3y = 3, -4x + 2y = 10 \)
Answer: (d)
On comparing the above equations with standard from of pair of linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \), we get
(a) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) as \( \frac{3}{1} = \frac{-1}{-1/3} = \frac{-9}{-3} \), consistent
(b) \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) as \( \frac{4}{-2} \neq \frac{3}{3} \), consistent
(c) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) as \( \frac{5}{10} = \frac{-1}{-2} = \frac{10}{20} \), consistent
(d) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) as \( \frac{-2}{-4} = \frac{3}{2} \neq \frac{3}{10} \), inconsistent

Question. The value of \( a \) for which the lines \( x = 1, y = 2 \) and \( a^2x + 2y - 20 = 0 \) are concurrent, is
(a) 1
(b) 8
(c) -4
(d) -2
Answer: (c)
Given lines are, \( x = 1, y = 2 \) and \( a^2x + 2y - 20 = 0 \). Since the lines are concurrent, \( x = 1, y = 2 \) is a solution of given equation:
\( a^2(1) + 2(2) - 20 = 0 \)
\( a^2 + 4 - 20 = 0 \)
\( a^2 - 16 = 0 \)
\( a^2 = 16 \Rightarrow a = \pm 4 \)

Question. 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. Total number of bananas, he had is
(a) 200
(b) 300
(c) 400
(d) 500
Answer: (d)
Let, the number of bananas in lots A and B be \( x \) and \( y \), respectively.
Case I: \( \frac{2}{3}x + y = 400 \Rightarrow 2x + 3y = 1200 \) ...(1)
Case II: \( x + \frac{4}{5}y = 460 \Rightarrow 5x + 4y = 2300 \) ...(2)
On multiplying Eq. (1) by 4 and Eq. (2) by 3 and then subtracting them, we get,
\( 8x + 12y = 4800 \)
\( 15x + 12y = 6900 \)
\( -7x = -2100 \Rightarrow x = 300 \)
Now, on putting the value of \( x \) in Eq. (1), we get
\( 2 \times 300 + 3y = 1200 \)
\( 600 + 3y = 1200 \)
\( 3y = 600 \Rightarrow y = 200 \)
Total number of bananas = \( x + y = 300 + 200 = 500 \).

Question. Shashi is decided fixed distance to walk on a tread mill. First day, she walks at a certain speed. Next day, she increases the speed of the tread mill by 1 km/h, she takes 6 min less and if she reduces the speed by 1 km/h, then she takes 9 min more. What is the distance that she has decided to walk everyday?
(a) 4 km
(b) 6 km
(c) 5 km
(d) 3 km
Answer: (d)
Let the speed on first day be \( x \) km/h and time be \( y \) h. Distance = \( xy \).
Case I: \( (x + 1)(y - 0.1) = xy \) (Since 6 min = 0.1 h)
\( -0.1x + y = 0.1 \) ...(1)
Case II: \( (x - 1)(y + 0.15) = xy \) (Since 9 min = 0.15 h)
\( 0.15x - y = 0.15 \) ...(2)
On adding Eq. (1) and (2), we get
\( 0.05x = 0.25 \Rightarrow x = 5 \)
On substituting \( x = 5 \) in Eq. (1), we get
\( y = 0.1 + 0.5 = 0.6 \)
Distance \( xy = 5 \times 0.6 = 3 \) km.

Question. A vessel contain a mixture of 24 L milk and 6 L water and second vessel contains a mixture of 15 L milk and 10 L water, then how much mixture of milk and water should be taken from the first and the second vessel separately and kept in a third vessel so that the third vessel may contain a mixture of 25 L milk and 10 L water.
(a) 15 L and 15 L
(b) 20 L and 10 L
(c) 20 L and 15 L
(d) None of these
Answer: (c)
Let \( x \) L be taken from 1st vessel and \( y \) L from 2nd vessel.
Milk from 1st: \( \frac{24}{30}x = \frac{4}{5}x \). Milk from 2nd: \( \frac{15}{25}y = \frac{3}{5}y \).
Total milk: \( \frac{4}{5}x + \frac{3}{5}y = 25 \Rightarrow 4x + 3y = 125 \) ...(1)
Water from 1st: \( \frac{6}{30}x = \frac{x}{5} \). Water from 2nd: \( \frac{10}{25}y = \frac{2}{5}y \).
Total water: \( \frac{x}{5} + \frac{2}{5}y = 10 \Rightarrow x + 2y = 50 \) ...(2)
Multiplying Eq. (2) by 4 and subtracting Eq. (1):
\( 5y = 200 - 125 = 75 \Rightarrow y = 15 \)
Substituting \( y = 15 \) in Eq. (2): \( x + 30 = 50 \Rightarrow x = 20 \).
Hence, 20 L and 15 L.

Question. The ratio of the areas of the two triangles formed by the lines representing the equations \( 2x + y = 6 \) and \( 2x - y + 2 = 0 \) with the X-axis and the lines with the Y-axis is
(a) 1:2
(b) 2:1
(c) 4:1
(d) 1:4
Answer: (c)

FILL IN THE BLANK

Question. If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is ..........
Answer: consistent

Question. An equation whose degree is one is known as a .......... equation.
Answer: linear

Question. If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is ..........
Answer: inconsistent

Question. A pair of linear equations has .......... solution (s) if it is represented by intersecting lines graphically.
Answer: unique

Question. Two distinct natural numbers are such that the sum of one number and twice the other number is 6. The two numbers are ..........
Answer: 4 and 1

Question. The number of common solutions for the system of linear equations \( 5x + 4y + 6 = 0 \) and \( 10x + 8y = 12 \) is ..........
Answer: zero

Question. If \( 2x + 3y = 5 \) and \( 3x + 2y = 10 \), then \( x - y = .......... \)
Answer: 5

TRUE/FALSE

Question. \( 3x - y = 3 \), \( 9x - 3y = 9 \) has infinite solution.
Answer: True

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

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

Question. If a pair of linear equation is given by \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) and \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \). In this case, the pair of linear equations is consistent.
Answer: False

Question. For all real values of \( c \), the pair of equations \( x - 2y = 8, 5x + 10y = c \) have a unique solution.
Answer: True

Question. \( 3x + 2y = 5, 2x - 3y = 7 \) are consistent pair of equation.
Answer: True

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: True

Question. In a \( \triangle ABC \), \( \angle C = 3 \angle B = 2(\angle A + \angle B) \), then angles are \( 20^{\circ}, 40^{\circ}, 100^{\circ} \).
Answer: False

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If the system of equations \( 2x + 3y = 7 \) and \( 2ax + (a + b)y = 28 \) has infinitely many solutions, then \( 2a - b = 0 \)
Reason : The system of equations \( 3x - 5y = 9 \) and \( 6x - 10y = 8 \) has a unique solution.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (c)

Question. Assertion : \( 3x + 4y + 5 = 0 \) and \( 6x + ky + 9 = 0 \) represent parallel lines if \( k = 8 \).
Reason : \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) represent parallel lines if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a)

Question. Assertion : The value of \( q = \pm 2 \), if \( x = 3, y = 1 \) is the solution of the line \( 2x + y - q^2 - 3 = 0 \).
Reason : The solution of the line will satisfy the equation of the line.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a)

Question. Assertion : For \( k = 6 \), the system of linear equations \( x + 2y + 3 = 0 \) and \( 3x + ky + 6 = 0 \) is inconsistent.
Reason : The system of linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) is inconsistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (c)

Question. Assertion : If the pair of lines are coincident, then we say that pair of lines is consistent and it has a unique solution.
Reason : If the pair of lines are parallel, then the pair has no solution and is called inconsistent pair of equations.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d)

Question. Assertion : The value of \( k \) for which the system of equations \( kx - y = 2 \) and \( 6x - 2y = 3 \) has a unique solution is \( 3 \).
Reason : The system of linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) has a unique solutions if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d)