CBSE Class 10 Pair of Linear Equations in Two Variables Sure Shot Questions Set 12

Question. The value of \( k \) so that the following system of equations has no solution is \( 3x - y - 5 = 0, 6x - 2y + k = 0 \)
(a) 10
(b) -10
(c) Both 10 and -10
(d) All real values of \( k \) except -10
Answer: (d) All real values of \( k \) except -10

Question. If the pair of equations \( x \sin \theta + y \cos \theta = 1 \) and \( x + y = \sqrt{2} \) has infinitely many solutions, then the value of \( \theta \) is
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Answer: (b) 45°

Question. The number of solutions of the following pair of linear equations is \( x + 2y - 8 = 0, 2x + 4y = 16 \)
(a) No solutions
(b) One solution
(c) Two solutions
(d) Infinitely many solutions
Answer: (d) Infinitely many solutions

Question. The lines represented by the equations \( 5x - 4y + 8 = 0, 7x + 6y - 9 = 0 \) will
(a) intersect at a point
(b) be parallel
(c) be coincident
(d) None of the options
Answer: (a) intersect at a point

Question. The lines represented by the equations \( 9x + 3y + 12 = 0 \) and \( 18x + 6y + 24 = 0 \) will
(a) intersect at a point
(b) be parallel
(c) be coincident
(d) None of the options
Answer: (c) be coincident

Question. The lines represented by the equations \( 6x - 3y + 10 = 0 \) and \( 2x - y + 9 = 0 \) will
(a) intersect at a point
(b) be parallel
(c) be coincident
(d) None of the options
Answer: (b) be parallel

Question. On comparing \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \), the graphical representation of equations \( 3x + 2y = 5, 2x - 3y = 7 \) will be
(a) Intersecting
(b) Coincident
(c) Parallel
(d) None of the options
Answer: (a) Intersecting

Question. On comparing \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \), the graphical representation of equations \( 2x - 3y = 8 \) and \( 4x - 6y - 9 = 0 \) will
(a) Intersecting lines
(b) Coincident lines
(c) Parallel lines
(d) None of the options
Answer: (c) Parallel lines

Question. The pair of linear equations \( \frac{3x}{2} + \frac{5y}{3} = 7 \) and \( 9x + 10y = 14 \) is
(a) consistent
(b) inconsistent
(c) consistent with one solution
(d) consistent with many solutions
Answer: (b) inconsistent

Question. The value of \( k \) for which the system of linear equations \( x + 2y = 3, 5x + ky + 7 = 0 \) is inconsistent is
(a) \( -\frac{14}{3} \)
(b) \( \frac{2}{5} \)
(c) 5
(d) 10
Answer: (d) 10

Question. If a pair of linear equations is consistent, then the line represented by them are
(a) parallel
(b) intersecting or coincident
(c) always coincident
(d) always intersecting
Answer: (b) intersecting or coincident

Question. The value of \( k \) for which the system of equations \( kx + 4y = k - 4, 16x + ky = k \) have infinite number of solutions is
(a) \( k = 2 \)
(b) \( k = 4 \)
(c) \( k = 6 \)
(d) \( k = 8 \)
Answer: (d) \( k = 8 \)

Question. The value of \( k \) for which the system of linear equations \( 3x + y = 1, (2k - 1)x + (k - 1)y = 2k + 1 \) have no solution is
(a) \( k = 2 \)
(b) \( k = 4 \)
(c) \( k = 6 \)
(d) \( k = 8 \)
Answer: (a) \( k = 2 \)

Question. If the equations \( kx - 2y = 3 \) and \( 3x + y = 5 \) represent two intersecting lines at unique point, then the value of \( k \) is
(a) Only 4
(b) Only 5
(c) Only 6
(d) Any number other than -6
Answer: (d) Any number other than -6

Question. The value of \( k \) for which the given system has unique solution \( 2x + 3y - 5 = 0, kx - 6y - 8 = 0 \) is
(a) \( k = 2 \)
(b) \( k \neq 4 \)
(c) \( k = 4 \)
(d) \( k \neq -4 \)
Answer: (d) \( k \neq -4 \)

Question. For what value of \( k \), the following system of equations have infinite solutions: \( 2x - 3y = 7, (k + 2)x - (2k + 1)y = 3(2k - 1) \)?
(a) \( k = 2 \)
(b) \( k = 3 \)
(c) \( k = 4 \)
(d) \( k = 8 \)
Answer: (c) \( k = 4 \)

Question. The value of \( m \) for which the pair of linear equations \( 2x + 3y - 7 = 0 \) and \( (m - 1)x + (m + 1)y = (3m - 1) \) has infinitely many solutions is
(a) 5
(b) 8
(c) -5
(d) 8
Answer: (a) 5

Question. For what values of \( p \), the pair of equations \( 4x + py + 8 = 0 \) and \( 2x + 2y + 2 = 0 \) have unique solution?
(a) \( p = 4 \)
(b) \( p \neq 4 \)
(c) \( p = 7 \)
(d) \( p \neq 7 \)
Answer: (b) \( p \neq 4 \)

Question. What type of straight lines will be represented by the system of equations \( 2x + 3y = 5 \) and \( 4x + 6y = 7 \)?
(a) Intersecting
(b) Parallel
(c) Coincident
(d) None of the options
Answer: (b) Parallel

Question. For what value of \( p \), the following pair of linear equations have infinitely many solutions? \( (p - 3)x + 3y = p, px + py = 12 \)
(a) 4
(b) 6
(c) 9
(d) 11
Answer: (b) 6

Question. The value of \( k \) for which the following pair of linear equations have infinitely many solutions: \( 2x + 3y = 7, (k - 1)x + (k + 2)y = 3k \) is
(a) 2
(b) 4
(c) 7
(d) 9
Answer: (c) 7

Question. The value(s) of \( k \) for which the pair of linear equations \( kx + y = k^2 \) and \( x + ky = 1 \) have infinitely many solutions is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

B. Assertion-Reason Type Questions

In the following questions, a statement of assertion (A) is followed by a statement reason (R). Choose 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 (A): Pair of linear equations \( x + y = 14, x - y = 4 \) is consistent.
Reason (R): By comparing \( \frac{a_1}{a_2} \) and \( \frac{b_1}{b_2} \) if we get \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), then given system of equations is consistent.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): For \( k = 6 \), the system of linear equations \( x + 2y + 3 = 0 \) and \( 3x + ky + 6 = 0 \) is inconsistent.
Reason (R): The system of linear equations \( a_1 x + b_1 y + c_1 = 0 \) and \( a_2 x + b_2 y + c_2 = 0 \) is inconsistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \).
Answer: (c) Assertion (A) is true but reason (R) is false.

2. Solving a Pair of Linear Equations by Algebraic Methods

The solution of a pair of linear equations can be obtained by using any one of the following algebraic methods:
(i) Substitution Method
(ii) Elimination Method

A. To find the solution of a pair of linear equations using Substitution Method, we follow these steps:

Step 1: From one equation, find the value of one variable (say y) in terms of other variable, i.e. x.
Step 2: Substitute the value of variable obtained in step 1 in the other equation to get an equation in one variable.
Step 3: Solve the equation obtained in step 2 to get the value of one variable.
Step 4: Substitute the value of variable so obtained in any given equation to find the value of other variable.

B. To find the solution of a pair of linear equations using Elimination Method, we follow these steps:

Step 1: Multiply the given equations to make the coefficient of one of the unknown variables (either x or y), numerically equal.
Step 2: Add the equations obtained after multiplication, if the numerically equal coefficients are opposite in signs or else subtract them.
Step 3: Solve the resulting linear equation in one unknown variable to find its value.
Step 4: Substitute this value in any one of the equations and find the value of the other variable.

Question. The solution of the system of equations \( x + y = 5, x - y = 2 \) using substitution method is:
(a) \( x = \frac{7}{2}, y = \frac{3}{2} \)
(b) \( x = \frac{3}{5}, y = \frac{1}{2} \)
(c) \( x = \frac{3}{5}, y = \frac{1}{4} \)
(d) \( x = \frac{2}{5}, y = \frac{5}{2} \)
Answer: (a) \( x = \frac{7}{2}, y = \frac{3}{2} \)

Question. The solution of given system of equations: \( x + y = a + b, ax - by = a^2 - b^2 \) is
(a) \( x = 2a, y = b \)
(b) \( x = a, y = 2b \)
(c) \( x = a, y = b \)
(d) \( x = \frac{1}{a}, y = \frac{1}{b} \)
Answer: (c) \( x = a, y = b \)

Question. When \( 3x + 2y = \frac{11}{3} \) and \( -7x + 5y = \frac{31}{3} \) are solved by elimination method, we get
(a) \( x = \frac{5}{19}, y = \frac{111}{37} \)
(b) \( x = \frac{9}{85}, y = \frac{160}{27} \)
(c) \( x = \frac{-4}{71}, y = \frac{5}{28} \)
(d) \( x = -\frac{7}{87}, y = \frac{170}{87} \)
Answer: (d) \( x = -\frac{7}{87}, y = \frac{170}{87} \)

Question. Solving \( 3x - 5y - 4 = 0 \) and \( 9x = 2y + 7 \) by the elimination method, we get the values of \( x \) and \( y \) as
(a) \( x = \frac{9}{13}, y = -\frac{5}{13} \)
(b) \( x = \frac{11}{24}, y = \frac{15}{23} \)
(c) \( x = \frac{17}{25}, y = \frac{16}{9} \)
(d) None of the options
Answer: (a) \( x = \frac{9}{13}, y = -\frac{5}{13} \)

Question. Solution of the simultaneous linear equations: \( \frac{2x}{y} - \frac{y}{2} = -\frac{1}{6} \) and \( \frac{x}{2} + \frac{2y}{3} = 3 \) is
(a) \( x = 2, y = -3 \)
(b) \( x = -2, y = 3 \)
(c) \( x = 2, y = 3 \)
(d) \( x = -2, y = -3 \)
Answer: (c) \( x = 2, y = 3 \)

Question. The value of \( x \) satisfying both the equations \( 4x - 5 = y \) and \( 2x - y = 3 \), when \( y = -1 \) is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (a) 1

Question. Which of the following is not a solution of the pair of equations \( 3x - 2y = 4 \) and \( 6x - 4y = 8 \)?
(a) \( x = 2, y = 1 \)
(b) \( x = 4, y = 4 \)
(c) \( x = 6, y = 7 \)
(d) \( x = 5, y = 3 \)
Answer: (d) \( x = 5, y = 3 \)

Question. If \( 2x + 5y - 1 = 0, 2x + 3y - 3 = 0 \), then
(a) \( x = 1, y = -3 \)
(b) \( x = 3, y = -1 \)
(c) \( x = 2, y = 5 \)
(d) \( x = 5, y = -3 \)
Answer: (b) \( x = 3, y = -1 \)

Question. If \( x + 2y - 3 = 0, 3x - 2y + 7 = 0 \), then
(a) \( x = -1, y = 2 \)
(b) \( x = 1, y = 2 \)
(c) \( x = 2, y = 3 \)
(d) \( x = -2, y = -3 \)
Answer: (a) \( x = -1, y = 2 \)

Question. If \( 2x = 5y + 4, 3x - 2y + 16 = 0 \), then
(a) \( x = 2, y = -2 \)
(b) \( x = 3, y = -3 \)
(c) \( x = 4, y = 5 \)
(d) \( x = -8, y = -4 \)
Answer: (d) \( x = -8, y = -4 \)

Question. If \( \frac{4}{x} + 3y = 8; \frac{6}{x} - 4y = -5 \), then
(a) \( x = 2, y = 2 \)
(b) \( x = 1, y = -1 \)
(c) \( x = 2, y = -2 \)
(d) \( x = 3, y = -3 \)
Answer: (a) \( x = 2, y = 2 \)

Question. If \( 2x + 3y = 11 \) and \( 2x - 4y = -24 \), then the value of ‘m’ for which \( y = mx + 3 \) is
(a) 0
(b) 1
(c) -1
(d) -2
Answer: (c) -1

Question. If \( 2x + 3y = 11 \) and \( x - 2y = -12 \), then the value of ‘m’ for which \( y = mx + 3 \) is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (b) -1

Question. If \( 7(y + 3) - 2(x + 2) = 14, 4(y - 2) + 3(x - 3) = 2 \), then
(a) \( x = 1, y = 4 \)
(b) \( x = 3, y = 5 \)
(c) \( x = 5, y = 1 \)
(d) None of the options
Answer: (c) \( x = 5, y = 1 \)

Question. If \( \frac{4}{x} + 5y = 7; \frac{3}{x} + 4y = 5 \), then
(a) \( x = \frac{1}{3}, y = -1 \)
(b) \( x = 8, y = 3 \)
(c) \( x = 4, y = 7 \)
(d) \( x = 5, y = 9 \)
Answer: (a) \( x = \frac{1}{3}, y = -1 \)

Question. If \( 3 - (x - 5) = y + 2, 2(x + y) = 4 - 3y \), then
(a) \( x = \frac{13}{4}, y = \frac{9}{10} \)
(b) \( x = \frac{7}{16}, y = \frac{5}{8} \)
(c) \( x = \frac{4}{9}, y = \frac{9}{12} \)
(d) \( x = \frac{26}{3}, y = \frac{-8}{3} \)
Answer: (d) \( x = \frac{26}{3}, y = \frac{-8}{3} \)