✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 3: Pair of Linear Equations in Two Variables
Assertion-Reason Question Bank
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Assertion (A)
If \(217x + 131y = 913\) and \(131x + 217y = 827\), then \(x + y = 5\).
Reason (R)
Adding the two given equations gives \(348x + 348y = 1740 \implies x + y = 5\).
Correct Option: (B)
Both A and R are true and R shows the symmetric addition shortcut.
Both A and R are true and R shows the symmetric addition shortcut.
Correct Answer: (B)
Both statements are true and R explains the method of adding equations.
Both statements are true and R explains the method of adding equations.
Assertion (A)
The line \(x = 0\) is the equation of the y-axis.
Reason (R)
The line \(y = 0\) is the equation of the x-axis.
Correct Option: (B)
Both statements are true basic facts of coordinate geometry, but R does not explain A.
Both statements are true basic facts of coordinate geometry, but R does not explain A.
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
The line \(2x + 3y = 6\) intersects the x-axis at \((3, 0)\).
Reason (R)
On the x-axis, the y-coordinate of every point is \(0\).
Correct Option: (B)
Both statements are true. Setting \(y=0\) gives \(2x = 6 \implies x = 3\), so point is \((3,0)\). R is the general property of the x-axis.
Both statements are true. Setting \(y=0\) gives \(2x = 6 \implies x = 3\), so point is \((3,0)\). R is the general property of the x-axis.
Correct Answer: (B)
Both true, but R is a general property rather than the exact calculation step.
Both true, but R is a general property rather than the exact calculation step.
Assertion (A)
The perimeter of triangle formed by line \(x/a + y/b = 1\) with coordinate axes is \(a + b + \sqrt{a^2 + b^2}\).
Reason (R)
The line \(x/a + y/b = 1\) cuts x-axis at \((a, 0)\) and y-axis at \((0, -b)\).
Correct Option: (C)
Assertion (A) is true (\(O=(0,0), A=(a,0), B=(0,b)\), sides are \(a, b, \sqrt{a^2+b^2}\)). Reason (R) is false because it cuts y-axis at \((0, b)\), not \((0, -b)\).
Assertion (A) is true (\(O=(0,0), A=(a,0), B=(0,b)\), sides are \(a, b, \sqrt{a^2+b^2}\)). Reason (R) is false because it cuts y-axis at \((0, b)\), not \((0, -b)\).
Correct Answer: (C)
Assertion true; Reason false (cuts y-axis at \((0, b)\)).
Assertion true; Reason false (cuts y-axis at \((0, b)\)).
Assertion (A)
If \(kx - y = 2\) and \(6x - 2y = 3\) has a unique solution, then \(k \ne 3\).
Reason (R)
Condition for unique solution is \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\).
Correct Option: (C)
Assertion (A) is true (\(k/6 \ne -1/-2 = 1/2 \implies k \ne 3\)). Reason (R) is false because condition for unique solution is \(a_1/a_2 \ne b_1/b_2\).
Assertion (A) is true (\(k/6 \ne -1/-2 = 1/2 \implies k \ne 3\)). Reason (R) is false because condition for unique solution is \(a_1/a_2 \ne b_1/b_2\).
Correct Answer: (C)
Assertion true; Reason false (condition is \(\ne\)).
Assertion true; Reason false (condition is \(\ne\)).
Assertion (A)
A linear equation in two variables \(ax + by + c = 0\) (\(a, b \ne 0\)) has infinitely many solutions.
Reason (R)
For every real value of \(x\), there exists a corresponding real value of \(y = \frac{-c - ax}{b}\).
Correct Option: (B)
Both A and R are true facts about a single linear equation.
Both A and R are true facts about a single linear equation.
Correct Answer: (B)
Both true, R gives algebraic justification.
Both true, R gives algebraic justification.
Assertion (A)
If \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\), the lines are coincident.
Reason (R)
The condition \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\) represents parallel lines with no solution.
Correct Option: (D)
Assertion (A) is false (it represents parallel lines). Reason (R) is true.
Assertion (A) is false (it represents parallel lines). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If a pair of linear equations is consistent, the lines must always be intersecting.
Reason (R)
A consistent pair of linear equations can have either a unique solution (intersecting) or infinitely many solutions (coincident).
Correct Option: (D)
Assertion (A) is false (consistent lines can be coincident too). Reason (R) is true.
Assertion (A) is false (consistent lines can be coincident too). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The point \((3, 2)\) lies on the line \(2x + 3y = 12\).
Reason (R)
Substituting \(x = 3, y = 2\) into \(2x + 3y\) gives \(2(3) + 3(2) = 6 + 6 = 12\).
Correct Option: (A)
Direct substitution verifies that LHS = RHS = 12. Reason explains Assertion.
Direct substitution verifies that LHS = RHS = 12. Reason explains Assertion.
Correct Answer: (A)
Substitution proves point lies on line. Reason explains Assertion.
Substitution proves point lies on line. Reason explains Assertion.
Assertion (A)
The system of equations \(x + y = 3\) and \(3x + 3y = 9\) is dependent and consistent.
Reason (R)
A pair of linear equations having infinitely many solutions is dependent and consistent.
Correct Option: (A)
\(1/3 = 1/3 = 3/9 = 1/3\). Coincident lines are termed dependent and consistent. Reason explains Assertion.
\(1/3 = 1/3 = 3/9 = 1/3\). Coincident lines are termed dependent and consistent. Reason explains Assertion.
Correct Answer: (A)
Coincident lines are dependent and consistent. Reason explains Assertion.
Coincident lines are dependent and consistent. Reason explains Assertion.
Assertion (A)
The system \(x + 2y = 3\) and \(2x + 4y = 6\) has infinitely many solutions.
Reason (R)
The graph of \(2x + 4y = 6\) passes through origin \((0, 0)\).
Correct Option: (C)
Assertion (A) is true (\(1/2 = 2/4 = 3/6\)). Reason (R) is false because \(2(0)+4(0)=0 \ne 6\).
Assertion (A) is true (\(1/2 = 2/4 = 3/6\)). Reason (R) is false because \(2(0)+4(0)=0 \ne 6\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The value of \(k\) for which \(x + 2y = 3\) and \(5x + ky + 7 = 0\) has a unique solution is \(k \ne 10\).
Reason (R)
Condition for unique solution is \(\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \implies \frac{1}{5} \ne \frac{2}{k} \implies k \ne 10\).
Correct Option: (B)
Both A and R are true and R provides the algebraic condition.
Both A and R are true and R provides the algebraic condition.
Correct Answer: (B)
Both true, R provides the condition.
Both true, R provides the condition.
Assertion (A)
The area of triangle formed by lines \(x = 0, y = 0\) and \(x + y = 6\) is \(36\) sq units.
Reason (R)
Area of right-angled triangle formed by axes with \(x + y = 6\) is \(\frac{1}{2} \times 6 \times 6 = 18\) sq units.
Correct Option: (D)
Assertion (A) is false (area is \(18\) sq units, not \(36\)). Reason (R) is true.
Assertion (A) is false (area is \(18\) sq units, not \(36\)). Reason (R) is true.
Correct Answer: (D)
Assertion false (area is 18); Reason true.
Assertion false (area is 18); Reason true.
Assertion (A)
The system \(2x - 3y = 4\) and \(4x - 6y = 9\) is consistent.
Reason (R)
Since \(2/4 = -3/-6 = 1/2 \ne 4/9\), the lines are parallel and inconsistent.
Correct Option: (D)
Assertion (A) is false (system is inconsistent). Reason (R) is true.
Assertion (A) is false (system is inconsistent). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The pair of equations \(y = 0\) and \(y = -7\) has no solution.
Reason (R)
The lines \(y = 0\) and \(y = -7\) are intersecting lines.
Correct Option: (C)
Assertion (A) is true (parallel horizontal lines \(y=0\) and \(y=-7\) never intersect). Reason (R) is false because parallel lines do not intersect.
Assertion (A) is true (parallel horizontal lines \(y=0\) and \(y=-7\) never intersect). Reason (R) is false because parallel lines do not intersect.
Correct Answer: (C)
Assertion true; Reason false (lines are parallel).
Assertion true; Reason false (lines are parallel).
Assertion (A)
If the lines given by \(3x + 2ky = 2\) and \(2x + 5y + 1 = 0\) are parallel, then \(k = 15/4\).
Reason (R)
For parallel lines, \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \implies \frac{3}{2} = \frac{2k}{5} \implies 4k = 15 \implies k = \frac{15}{4}\).
Correct Option: (A)
Parallel lines condition \(a_1/a_2 = b_1/b_2 \implies 3/2 = 2k/5 \implies 4k = 15 \implies k = 15/4\). Reason explains Assertion.
Parallel lines condition \(a_1/a_2 = b_1/b_2 \implies 3/2 = 2k/5 \implies 4k = 15 \implies k = 15/4\). Reason explains Assertion.
Correct Answer: (A)
Reason correctly explains Assertion.
Reason correctly explains Assertion.
Assertion (A)
The pair \(x = a\) and \(y = b\) graphically represents lines intersecting at \((a, b)\).
Reason (R)
\(x = a\) is a line parallel to y-axis and \(y = b\) is a line parallel to x-axis.
Correct Option: (B)
Both statements are true and R describes the nature of two lines.
Both statements are true and R describes the nature of two lines.
Correct Answer: (B)
Both true, R describes line positions.
Both true, R describes line positions.
Assertion (A)
The solution of \(2x + y = 7\) and \(4x - y = 5\) is \((2, 3)\).
Reason (R)
Substituting \(x = 2, y = 3\) into \(4x - y\) gives \(4(2) - 3 = 8 - 3 = 6\).
Correct Option: (C)
Assertion (A) is true (\(2(2)+3=7\) and \(4(2)-3=5\)). Reason (R) is false because \(8 - 3 = 5\), not \(6\).
Assertion (A) is true (\(2(2)+3=7\) and \(4(2)-3=5\)). Reason (R) is false because \(8 - 3 = 5\), not \(6\).
Correct Answer: (C)
Assertion true; Reason false (\(8-3=5\)).
Assertion true; Reason false (\(8-3=5\)).
Assertion (A)
The lines \(2x + 3y = 7\) and \(2x + 3y = 9\) intersect at point \((1, 2)\).
Reason (R)
Lines with equal slope coefficients \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\) are parallel and never intersect.
Correct Option: (D)
Assertion (A) is false (parallel lines cannot intersect). Reason (R) is true.
Assertion (A) is false (parallel lines cannot intersect). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The graph of \(y = 5\) is a line parallel to the x-axis.
Reason (R)
The graph of \(x = 3\) is a line parallel to the y-axis.
Correct Option: (B)
Both statements are true geometrical facts.
Both statements are true geometrical facts.
Correct Answer: (B)
Both true, but R does not explain A.
Both true, but R does not explain A.
Assertion (A)
The system \(x + 2y = 5\) and \(3x + 6y = 15\) has a unique solution.
Reason (R)
If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the system has infinitely many solutions.
Correct Option: (D)
Assertion (A) is false (\(1/3 = 2/6 = 5/15 = 1/3\), so infinitely many solutions, not unique). Reason (R) is true.
Assertion (A) is false (\(1/3 = 2/6 = 5/15 = 1/3\), so infinitely many solutions, not unique). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(a_1/a_2 \ne b_1/b_2\), the system \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is consistent.
Reason (R)
A system of linear equations is consistent if it has at least one solution.
Correct Option: (B)
Both statements are true. Unique solution means at least one solution, so consistent.
Both statements are true. Unique solution means at least one solution, so consistent.
Correct Answer: (B)
Both true, R gives the definition of consistency.
Both true, R gives the definition of consistency.
Assertion (A)
The pair of equations \(x + y = 0\) and \(x - y = 0\) has no solution.
Reason (R)
The unique solution of \(x + y = 0\) and \(x - y = 0\) is \(x = 0, y = 0\).
Correct Option: (D)
Assertion (A) is false (has unique solution \((0,0)\)). Reason (R) is true.
Assertion (A) is false (has unique solution \((0,0)\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The pair of linear equations \(x + 2y - 4 = 0\) and \(2x + 4y - 12 = 0\) is inconsistent (has no solution).
Reason (R)
The lines are parallel because \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}\) (i.e., \(\frac{1}{2} = \frac{2}{4} \ne \frac{-4}{-12}\)).
Correct Option: (A)
\(\frac{a_1}{a_2} = 1/2\), \(\frac{b_1}{b_2} = 2/4 = 1/2\), \(\frac{c_1}{c_2} = -4/-12 = 1/3\). Since \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\), the lines are parallel and have no solution. Reason explains Assertion.
\(\frac{a_1}{a_2} = 1/2\), \(\frac{b_1}{b_2} = 2/4 = 1/2\), \(\frac{c_1}{c_2} = -4/-12 = 1/3\). Since \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\), the lines are parallel and have no solution. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The solution of \(x - y = 2\) and \(x + y = 4\) is \(x = 3, y = 1\).
Reason (R)
Adding the equations gives \(2x = 6 \implies x = 3\), and substituting \(x = 3\) gives \(y = 1\).
Correct Option: (A)
Solving by elimination gives \(x = 3, y = 1\). Reason shows step-by-step solution. Reason explains Assertion.
Solving by elimination gives \(x = 3, y = 1\). Reason shows step-by-step solution. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The equations \(3x - y + 8 = 0\) and \(6x - ky = -16\) represent coincident lines if \(k = 2\).
Reason (R)
For coincident lines, \(\frac{a_1}{a_2} \ne \frac{b_1}{b_2} = \frac{c_1}{c_2}\).
Correct Option: (C)
Assertion (A) is true (\(3/6 = -1/-k = 8/16 = 1/2 \implies k = 2\)). Reason (R) is false because coincident lines require ALL ratios to be equal (\(a_1/a_2 = b_1/b_2 = c_1/c_2\)).
Assertion (A) is true (\(3/6 = -1/-k = 8/16 = 1/2 \implies k = 2\)). Reason (R) is false because coincident lines require ALL ratios to be equal (\(a_1/a_2 = b_1/b_2 = c_1/c_2\)).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The equations \(x + 2y + 5 = 0\) and \(-3x - 6y + 1 = 0\) have no solution.
Reason (R)
The ratio condition for no solution is \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).
Correct Option: (C)
Assertion (A) is true (\(1/-3 = 2/-6 = -1/3 \ne 5/1\)). Reason (R) is false because no solution requires \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\).
Assertion (A) is true (\(1/-3 = 2/-6 = -1/3 \ne 5/1\)). Reason (R) is false because no solution requires \(a_1/a_2 = b_1/b_2 \ne c_1/c_2\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The line \(x = 3\) is parallel to the x-axis.
Reason (R)
The line \(x = 3\) is a vertical line perpendicular to x-axis and parallel to y-axis.
Correct Option: (D)
Assertion (A) is false (\(x=3\) is parallel to y-axis, not x-axis). Reason (R) is true.
Assertion (A) is false (\(x=3\) is parallel to y-axis, not x-axis). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The pair of equations \(2x + 3y = 5\) and \(4x + 6y = 10\) has infinitely many solutions.
Reason (R)
The lines are coincident because \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2}\).
Correct Option: (A)
\(2/4 = 3/6 = 5/10 = 1/2\). All ratios are equal, so lines coincide and have infinitely many solutions. Reason explains Assertion.
\(2/4 = 3/6 = 5/10 = 1/2\). All ratios are equal, so lines coincide and have infinitely many solutions. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The lines \(x - 2y = 0\) and \(3x + 4y - 20 = 0\) intersect at a unique point \((4, 2)\).
Reason (R)
The system has a unique solution because \(\frac{a_1}{a_2} \ne \frac{b_1}{b_2}\) (i.e., \(\frac{1}{3} \ne \frac{-2}{4}\)).
Correct Option: (A)
\(a_1/a_2 = 1/3\) and \(b_1/b_2 = -2/4 = -1/2\). Since \(1/3 \ne -1/2\), lines intersect at a unique point. Substituting \((4, 2)\): \(4 - 4 = 0\) and \(3(4) + 4(2) - 20 = 12 + 8 - 20 = 0\). Reason explains Assertion.
\(a_1/a_2 = 1/3\) and \(b_1/b_2 = -2/4 = -1/2\). Since \(1/3 \ne -1/2\), lines intersect at a unique point. Substituting \((4, 2)\): \(4 - 4 = 0\) and \(3(4) + 4(2) - 20 = 12 + 8 - 20 = 0\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.