CBSE Class 10 Mathematics Pairs of Linear Equations in Two Variables MCQs Set 15

Question. One equation of a pair of dependent linear equations is \( -5x + 7y = 2 \), the second equation can be:
(a) \( 10x + 14y + 4 = 0 \)
(b) \( -10x - 14y + 4 = 0 \)
(c) \( -10x + 14y + 4 = 0 \)
(d) \( 10x - 14y = -4 \)
Answer: (d) \( 10x - 14y = -4 \)

Question. The pair of linear equations \( 2x - 3y = 1 \) and \( 3x - 2y = 4 \) has:
(a) One solution
(b) Two solutions
(c) No solution
(d) Many solutions
Answer: (a) One solution

Question. Two lines are given to be parallel. The equation of one of the lines is \( 4x + 3y = 14 \). The equation of the second line can be
(a) \( 3x + 4y = 14 \)
(b) \( 8x + 6y = 28 \)
(c) \( 12x + 9y = 42 \)
(d) \( -12x = 9y \)
Answer: (d) \( -12x = 9y \)

Question. Match the Column:
(1) \( 2x + 5y = 7, 3x + 4y = 7 \)            (A) Inconsistent pair of equations
(2) \( 2x + 5y = 7, 4x + 10y = 7 \)          (B) Consistent pair of equations
(3) \( 2x + 5y = 7, 4x + 10y = 14 \)        (C) Dependent consistent pair of equations
(a) 1 – A, 2 – C, 3 – B
(b) 1 – B, 2 – C, 3 – A
(c) 1 – B, 2 – A, 3 – C
(d) 1 – C, 2 – A, 3 – B
Answer: (c) 1 – B, 2 – A, 3 – C

Question. \( y = a + \frac{b}{x} \) where \( a, b \) are real numbers, if \( y = 1 \) when \( x = -1 \) and \( y = 5 \) when \( x = -5 \), then \( a + b \) equals
(a) – 1
(b) 0
(c) 11
(d) 10
Answer: (c) 11

Question. For what value of \( k \), the pair of equations \( 2x + 3y + 5 = 0 \) and \( kx + 4y = 10 \), has a unique solution?
(a) \( k = \frac{8}{3} \)
(b) \( k \neq \frac{8}{3} \)
(c) \( k = 3 \)
(d) \( k \neq 3 \)
Answer: (b) \( k \neq \frac{8}{3} \)

Question. The value of \( a \) the following pair of linear equations \( ax - 3y = 1, -12x + ay = 2 \) has infinitely many solution is
(a) 6
(b) –6
(c) \( \pm 6 \)
(d) 36
Answer: (b) –6

Question. The values of \( x \) and \( y \) satisfying the two equations \( 32x + 33y = 34, 33x + 32y = 31 \) respectively are:
(a) –1, 2
(b) –1, 4
(c) 1, –2
(d) –1, –4
Answer: (a) –1, 2

Question. For what value of \( k \) the following pair of linear equations has unique solution? \( 7x + 8y = k, 9x - 4y = 12 \)
(a) 9
(b) 6
(c) 8
(d) any value of \( k \)
Answer: (d) any value of \( k \)

Question. For what value of \( k \) the following pair of linear equation has unique solution? \( kx + 3y = 3, 12x + ky = 6 \)
(a) 6
(b) – 6
(c) \( \pm 6 \)
(d) 36
Answer: (d) 36

Question. Find whether the following pair of equations has no solution, unique solution or infinitely many solutions. \( 5x - 8y + 1 = 0; 3x - \frac{24}{5}y + \frac{3}{5} = 0 \)
(a) No solution
(b) unique solution
(c) infinitely many solution
(d) None of the options
Answer: (c) infinitely many solution

Question. Harsh correctly solved a pair of linear equations in two variables and found their only point of Intersection as (3, –2). One of the lines was \( x - y = 5 \). Which of the following could have been the other line?
I. \( 3x - 3y = 15 \)
II. \( 2x - 3y = 12 \)
III. \( 2x - 3y = 14 \)
(a) only I
(b) only II
(c) only I and II
(d) only II and III
Answer: (b) only II

Question. Two linear equations in variables \( x \) and \( y \) are given below. \( a_1x + b_1y + c = 0, a_2x + b_2y + c = 0 \). Which of the following pieces of information is independently sufficient to determine if a solution exists or not for this pair of linear equations?
I. \( \frac{a_1}{b_1} = \frac{a_2}{b_2} = 1 \)
II. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)
III. \( \frac{a_1}{a_2} = \frac{b_1}{b_1} \neq 1 \)
IV. \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
(a) IV
(b) I and IV
(c) II and IV
(d) I and III
Answer: (a) IV

Question. The ratio of a two-digit number and the sum of its digits is 7:1. How many such two-digit numbers are possible?
(a) 1
(b) 4
(c) 9
(d) (infinitely many)
Answer: (b) 4

Question. \( x = 3, y = 4 \) is a solution of the linear equation.
(a) \( 2x + 3y - 17 = 0 \)
(b) \( 3x + 2y - 17 = 0 \)
(c) \( 2x - 3y + 17 = 0 \)
(d) \( 2x + 3y + 17 = 0 \)
Answer: (b) \( 3x + 2y - 17 = 0 \)

Question. Find the conditions to be satisfied by coefficients for which the following pair of equations \( ax + by + c = 0, dx + ey + f = 0 \) represent coincident lines.
(a) \( ab = ed; bf = ce \)
(b) \( ae = bd; bc = ef \)
(c) \( ad = bc; bf = ce \)
(d) \( ae = bd; bf = ce \)
Answer: (d) \( ae = bd; bf = ce \)

Question. In a \( \Delta ABC \), if \( \angle C = 50^\circ \) and \( \angle A \) exceeds \( \angle B \) by \( 44^\circ \), then \( \angle A = \)
(a) \( 43^\circ \)
(b) \( 40^\circ \)
(c) \( 67^\circ \)
(d) \( 87^\circ \)
Answer: (d) \( 87^\circ \)

Question. If \( 3^{x - y} = 9 \) and \( x - 2y = 6 \) represent a system of the equations, then the value of \( x + y \) is
(a) –2
(b) –6
(c) –4
(d) None of the options
Answer: (b) –6

Question. The value of \( y \) when \( \frac{1}{y} + \frac{1}{x} = 3 \) and \( \frac{1}{y} - \frac{1}{x} = 7 \), is
(a) \( \frac{1}{5} \)
(b) \( -\frac{1}{3} \)
(c) \( -\frac{1}{5} \)
(d) \( \frac{1}{3} \)
Answer: (a) \( \frac{1}{5} \)

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) 3 and 1

Question. Aruna 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 and Rs. 2 coins are, respectively.
(a) 35 and 15
(b) 35 and 20
(c) 15 and 35
(d) 25 and 25
Answer: (d) 25 and 25

Question. 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 age, (in years) of the son is
(a) 4
(b) 5
(c) 6
(d) 3
Answer: (c) 6

Question. If \( x + 4y = 27, x + 2y = 21 \) then the value of \( x - y \) is
(a) 5
(b) 2
(c) 12
(d) 18
Answer: (c) 12

Question. Graphically the pair of equations \( 6x - 3y + 10 = 0, 2x - y + 9 = 0 \) represent two lines which are
(a) intersecting exactly at one point
(b) intersecting exactly at two point
(c) coincident
(d) parallel
Answer: (d) parallel

Question. It is given that there is no solution to the system of equations \( x + 2y = 3, ax + by = 4 \). Which one of the following is true?
(a) \( a \) has a unique value
(b) \( b \) has a unique value
(c) \( a \) can have more than one value
(d) \( a \) has exactly two different values
Answer: (c) \( a \) can have more than one value

Question. Solve the following equations for \( x \) and \( y \): \( mx - ny = m^2 + n^2, x + y = 2m \)
Answer: From second equation \( y = 2m - x \).
Substituting in first equation: \( mx - n(2m - x) = m^2 + n^2 \)

\( \implies mx - 2mn + nx = m^2 + n^2 \)

\( \implies x(m + n) = m^2 + 2mn + n^2 \)

\( \implies x(m + n) = (m + n)^2 \)

\( \implies x = m + n \).
Then \( y = 2m - (m + n) = m - n \).
The solution is \( x = m + n, y = m - n \).

Question. Find the value of \( k \) for which the system of equations \( x + 2y = 5 \) and \( 3x + ky + 15 = 0 \) has no solution.
Answer: For no solution, \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).
Given \( x + 2y = 5 \) and \( 3x + ky = -15 \).

\( \implies \frac{1}{3} = \frac{2}{k} \neq \frac{5}{-15} \)

\( \implies \frac{1}{3} = \frac{2}{k} \implies k = 6 \).
Since \( \frac{1}{3} \neq -\frac{1}{3} \), \( k = 6 \) is the required value.

Question. Solve: \( 99x + 101y = 499, 101x + 99y = 501 \)
Answer: Adding both equations:
\( 200x + 200y = 1000 \implies x + y = 5 \) ...(i)
Subtracting the first from the second:
\( 2x - 2y = 2 \implies x - y = 1 \) ...(ii)
Adding (i) and (ii): \( 2x = 6 \implies x = 3 \).
Substituting in (i): \( 3 + y = 5 \implies y = 2 \).
The solution is \( x = 3, y = 2 \).

Question. Show that the system of equations is consistent and dependent: \( x - 5y = 6, 2x - 10y = 12 \).
Answer: Comparing ratios:
\( \frac{a_1}{a_2} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{6}{12} = \frac{1}{2} \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the lines are coincident. Thus, the system is consistent and dependent.

Question. Find whether the following pair of linear equations is consistent or inconsistent: \( x + 3y = 5; 2x + 6y = 8 \)
Answer: Comparing ratios:
\( \frac{a_1}{a_2} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{5}{8} \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the lines are parallel. Thus, the system is inconsistent.