CBSE Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Model Questions - Assertion and Reasoning
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CBSE Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Model Questions - Assertion and Reasoning
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SECTION F — Assertion and Reasoning Type Questions
[1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
(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.
Q1.Assertion (A): The pair of linear equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ has a unique solution. Reason (R): A pair of linear equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ has a unique solution if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
Answer: (d) Assertion (A) is false, but Reason (R) is true.
Justification:
* For the given equations, $a_1 = 1$, $b_1 = 2$, $c_1 = -4$ and $a_2 = 2$, $b_2 = 4$, $c_2 = -12$.
* Checking the ratios: $\frac{a_1}{a_2} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{2}{4} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{-4}{-12} = \frac{1}{3}$.
* Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the pair of linear equations is parallel and has no solution (making Assertion (A) false).
* However, the condition given in Reason (R) ($\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$) is the standard correct condition for a unique solution (making Reason (R) true).
Q2.Assertion (A): The system of equations $2x + 3y = 7$ and $4x + 6y = 14$ has infinitely many solutions. Reason (R): When $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the pair of linear equations is dependent and consistent, representing coincident lines.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* For the given equations, $a_1 = 2$, $b_1 = 3$, $c_1 = -7$ and $a_2 = 4$, $b_2 = 6$, $c_2 = -14$.
* Checking the ratios: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{-7}{-14} = \frac{1}{2}$.
* Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system represents coincident lines and has infinitely many solutions (making Assertion (A) true).
* Reason (R) correctly states the mathematical condition and properties for a system of linear equations to have infinitely many solutions, and it directly explains why Assertion (A) is true.
Q3.Assertion (A): The equations $3x - y = 5$ and $6x - 2y = 10$ represent parallel lines and have no solution. Reason (R): For parallel lines, the condition is $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
Answer: (c) Assertion (A) is true, but Reason (R) is false.
Justification:
* For the given equations, $a_1 = 3$, $b_1 = -1$, $c_1 = -5$ and $a_2 = 6$, $b_2 = -2$, $c_2 = -10$.
* Checking the ratios: $\frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{-1}{-2} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{-5}{-10} = \frac{1}{2}$.
* Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are actually coincident (not parallel) and have infinitely many solutions, making the first part of Assertion (A) technically false? Wait—let's re-read carefully: Assertion says "represent parallel lines and have no solution". Actually, coincident lines represent dependent consistent equations with infinitely many solutions. Let's check: are they parallel? No, they are coincident. So Assertion (A) is false!
* Let's check Reason (R): The condition given for parallel lines is $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, which is correct.
* Therefore, Assertion (A) is false, but Reason (R) is true. Hence, the correct option is (d).
Q4.Assertion (A): If the system $kx + 2y = 5$ and $3x + y = 1$ has a unique solution, then $k \neq 6$. Reason (R): For a unique solution, $\frac{a_1}{a_2} = \frac{k}{3}$ must not equal $\frac{b_1}{b_2} = \frac{2}{1}$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* For the given system of equations, $a_1 = k$, $b_1 = 2$, $c_1 = -5$ and $a_2 = 3$, $b_2 = 1$, $c_2 = -1$.
* The condition for a unique solution is $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.
* Substituting the values: $\frac{k}{3} \neq \frac{2}{1} \implies k \neq 6$.
* Thus, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).
Q5.Assertion (A): Every consistent pair of linear equations always intersects at a single unique point. Reason (R): A consistent pair of linear equations can either have a unique solution (intersecting lines) or infinitely many solutions (coincident lines).
Answer: (d) Assertion (A) is false, but Reason (R) is true.
Justification:
* A consistent pair of linear equations is defined as a pair that has at least one solution. This includes both unique solutions (intersecting lines) and infinitely many solutions (coincident lines). Therefore, Assertion (A) is false because consistent equations do not always intersect at a single unique point (coincident lines intersect at infinitely many points).
* Reason (R) is completely true, as consistent systems encompass both unique and infinitely many solutions.
Q6.Assertion (A): The pair of linear equations $(k-1)x + y = 2$ and $(k+1)x + (k+1)y = 3k$ has infinitely many solutions when $k = 2$. Reason (R): Setting $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ for these equations yields $k = 2$.
Answer: (d) Assertion (A) is false, but Reason (R) is true.
Justification:
* For the given equations, $a_1 = k-1$, $b_1 = 1$, $c_1 = -2$ and $a_2 = k+1$, $b_2 = k+1$, $c_2 = -3k$.
* The condition for infinitely many solutions is $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$:
$\frac{k-1}{k+1} = \frac{1}{k+1} = \frac{-2}{-3k}$
* From $\frac{k-1}{k+1} = \frac{1}{k+1}$, we get $k-1 = 1 \implies k = 2$.
* However, testing $k = 2$ with the third ratio: $\frac{1}{2+1} = \frac{1}{3}$, but $\frac{2}{3(2)} = \frac{2}{6} = \frac{1}{3}$. Wait, let's re-verify: if $k=2$, $\frac{a_1}{a_2} = \frac{1}{3}$, $\frac{b_1}{b_2} = \frac{1}{3}$, and $\frac{c_1}{c_2} = \frac{2}{6} = \frac{1}{3}$. All ratios are equal to $\frac{1}{3}$! Let's re-evaluate: if all three are equal, then $k=2$ does give infinitely many solutions. Wait, let's check carefully. Is $k=2$ correct? Let's check: $a_1 = 2-1 = 1$, $a_2 = 2+1 = 3 \implies \frac{1}{3}$. $b_1 = 1$, $b_2 = 2+1 = 3 \implies \frac{1}{3}$. $c_1 = 2$, $c_2 = 3(2) = 6 \implies \frac{2}{6} = \frac{1}{3}$. All ratios are equal, so $k=2$ actually does give infinitely many solutions. Therefore, Assertion (A) is true and Reason (R) is true, making the option (a)! Let's correct this: Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation.
Q7.Assertion (A): The lines given by $3x + py = 7$ and $9x + 3y = 14$ are parallel if $p = 1$. Reason (R): Parallel lines satisfy the ratio condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* For the given equations, $a_1 = 3$, $b_1 = p$, $c_1 = -7$ and $a_2 = 9$, $b_2 = 3$, $c_2 = -14$.
* The condition for parallel lines is $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
* Substituting the values: $\frac{3}{9} = \frac{p}{3} \neq \frac{-7}{-14}$
* Simplifying the first part: $\frac{1}{3} = \frac{p}{3} \implies p = 1$.
* Checking the third ratio: $\frac{c_1}{c_2} = \frac{-7}{-14} = \frac{1}{2}$, and since $\frac{1}{3} \neq \frac{1}{2}$, the condition is fully satisfied.
* Thus, both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).
Q8.Assertion (A): The substitution method can be used to solve any consistent system of linear equations in two variables. Reason (R): The substitution method involves expressing one variable in terms of the other from one equation and substituting it into the second equation.
Answer: (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: The substitution method (along with elimination and graphical methods) is an algebraic technique that can be successfully applied to find the solution(s) for any consistent system of linear equations in two variables (whether it has a unique solution or infinitely many solutions).
* Reason (R) is true: It correctly defines the fundamental procedure and steps involved in the substitution method.
* However, Reason (R) simply defines how the method works; it does not explain why the substitution method is applicable to all consistent systems. Therefore, Reason (R) is not the correct explanation of Assertion (A).
Q9.Assertion (A): The elimination method yields the same solution set as the substitution method for a linear system. Reason (R): Elimination works by multiplying equations by suitable non-zero constants to equalize coefficients of one variable so it can be subtracted out.
Answer: (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: Both the elimination and substitution methods are equivalent algebraic techniques designed to solve systems of linear equations, and they will always yield the exact same solution set.
* Reason (R) is true: It accurately describes the fundamental mechanism and procedure of the elimination method.
* However, Reason (R) describes how the elimination method operates rather than explaining why different algebraic methods produce the same solution set. Thus, Reason (R) is not the correct explanation of Assertion (A).
Q10.Assertion (A): The pair of equations $2x + 3y = 7$ and $4x + 3y = 11$ has the unique solution $x = 2, y = 1$. Reason (R): Substituting $x = 2$ and $y = 1$ into both equations satisfies both mathematical statements simultaneously.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Checking Assertion (A):
Equation 1: $2(2) + 3(1) = 4 + 3 = 7$ (True)
Equation 2: $4(2) + 3(1) = 8 + 3 = 11$ (True)
Checking ratios: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$ and $\frac{b_1}{b_2} = \frac{3}{3} = 1$. Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system has a unique solution. Thus, Assertion (A) is true.
* Checking Reason (R): Reason (R) correctly defines the meaning of a solution to a system of equations and directly explains why $x = 2, y = 1$ solves the given pair.
Q11.Assertion (A): Equations like $\frac{2}{x} + \frac{3}{y} = 13$ are not linear equations initially, but they can be reduced to linear equations by substitution. Reason (R): Substitution variables like $u = \frac{1}{x}$ and $v = \frac{1}{y}$ transform non-linear fractional forms into standard linear equations in terms of $u$ and $v$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: Equations with variables in the denominator are non-linear in their original form, but they can be successfully reduced to linear equations using appropriate substitution.
* Reason (R) is true and explains (A): Setting $u = \frac{1}{x}$ and $v = \frac{1}{y}$ converts the non-linear fractional terms directly into standard linear forms (e.g., $2u + 3v = 13$), which directly explains how and why the reduction works.
Q12.Assertion (A): If the digits of a two-digit number are $x$ (tens) and $y$ (units), the number can be algebraically expressed as $10x + y$. Reason (R): Reversing the digits of this number results in the expression $10y + x$.
Answer: (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: A two-digit number with tens digit $x$ and units digit $y$ is correctly written in expanded/algebraic form as $10x + y$.
* Reason (R) is true: Reversing the digits makes $y$ the tens digit and $x$ the units digit, yielding $10y + x$.
* However, Reason (R) describes a separate property (reversing the digits) rather than explaining why the original number is expressed as $10x + y$. Thus, Reason (R) is not the correct explanation of Assertion (A).
Q13.Assertion (A): If two angles are supplementary and the larger exceeds the smaller by $18^\circ$, their measures are $99^\circ$ and $81^\circ$. Reason (R): The sum of two supplementary angles is always $180^\circ$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Let the smaller angle be $x$ and the larger be $y$.
* From the conditions: $y - x = 18^\circ$ and $y + x = 180^\circ$ (using Reason R).
* Solving the system: $2y = 198^\circ \implies y = 99^\circ$, and $x = 81^\circ$. Thus, Assertion (A) is true.
* Reason (R) correctly provides the geometric definition of supplementary angles which is essential for setting up the second equation, making it the correct explanation.
Q14.Assertion (A): Opposite angles of a cyclic quadrilateral are supplementary. Reason (R): If $\angle A$ and $\angle C$ are opposite angles in a cyclic quadrilateral $ABCD$, then $\angle A + \angle C = 180^\circ$, which helps form linear equations in $x$ and $y$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: A fundamental theorem of circles states that the opposite angles of any cyclic quadrilateral sum to $180^\circ$.
* Reason (R) is true and explains (A): Reason (R) restates this theorem in mathematical notation and links it to how it is applied in algebra problems to formulate linear equations involving variables like $x$ and $y$.
Q15.Assertion (A): If a boat rows upstream with speed $(u - v)$ and downstream with $(u + v)$, the time taken upstream is always greater than downstream for the same distance (assuming $u > v > 0$). Reason (R): Effective speed against the stream is less than the effective speed with the stream, making the travel time longer for the same distance.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: Since $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$, a lower speed (upstream, $u - v$) results in a greater travel time compared to a higher speed (downstream, $u + v$) for the same fixed distance.
* Reason (R) is true and explains (A): It correctly explains that the reduced effective speed against the stream is the direct physical cause for the increased travel time.
Q16.Assertion (A): The area of a triangle formed by the line $3x + 4y = 12$ with the coordinate axes is $6\text{ square units}$. Reason (R): The intercepts on the $x$-axis and $y$-axis are $4$ and $3$ respectively, and the area of a right-angled triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Finding intercepts: For $y = 0$, $3x = 12 \implies x = 4$ (base). For $x = 0$, $4y = 12 \implies y = 3$ (height).
* Calculating area: $\text{Area} = \frac{1}{2} \times 4 \times 3 = 6\text{ sq. units}$.
* Both Assertion (A) and Reason (R) are true, and Reason (R) provides the exact method used to find the area.
Q17.Assertion (A): The linear equations $2x - y = 0$ and $3x + 2y = 0$ intersect exclusively at the origin $(0, 0)$. Reason (R): A pair of homogeneous linear equations where $c_1 = 0$ and $c_2 = 0$ with non-zero determinant coefficients always passes through the origin as a trivial solution.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Checking cross-multiplication/determinant coefficients: $(2)(2) - (-1)(3) = 4 + 3 = 7 \neq 0$.
* Since the determinant is non-zero, the only solution to this system of homogeneous equations is the trivial solution $x = 0, y = 0$. Thus, Assertion (A) is true, and Reason (R) correctly explains it.
Q18.Assertion (A): If two lines are parallel on a Cartesian plane, the pair of equations is consistent. Reason (R): Parallel lines never intersect each other, meaning there is no common point that satisfies both equations simultaneously, rendering the system inconsistent.
Answer: (d) Assertion (A) is false, but Reason (R) is true.
Justification:
* Assertion (A) is false: A system of linear equations represented by parallel lines has no solution, which by definition makes the system **inconsistent**, not consistent.
* Reason (R) is true: Parallel lines do not intersect, meaning there is no common solution, which correctly defines an inconsistent system.
Q19.Assertion (A): If two persons' incomes are in the ratio $9:7$ and expenditures in $4:3$, their savings can be represented by linear expressions of a common multiplier $x$ and $y$. Reason (R): Savings are calculated mathematically as $\text{Savings} = \text{Income} - \text{Expenditure}$.
Answer: (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: Incomes can be taken as $9x$ and $7x$, and expenditures as $4y$ and $3y$ (or using a single variable multiplier with separate multipliers for ratios), forming linear equations for savings.
* Reason (R) is true: It correctly states the formula for savings.
* However, Reason (R) is a general definition of savings and does not explain why incomes and expenditures given in ratios can be represented using algebraic variables. Thus, Reason (R) is not the correct explanation.
Q20.Assertion (A): If the graphs of two linear equations intersect at a unique point, that point represents the unique solution of the system. Reason (R): The coordinates of any point of intersection on a graph satisfy the equations of both intersecting lines.
Answer: (a) Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
Justification:
* Assertion (A) is true: The graphical solution to a pair of linear equations is the point of intersection of their respective lines.
* Reason (R) is true and explains (A): A point lying on both lines means its coordinates satisfy both equations simultaneously, which is the precise definition of a solution to a system of equations.
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