Graphical method — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Graphical method MCQs with step-by-step solutions (10 questions). Part of Pair of Linear Equations in Two Variables. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Graphical method · easy · theory
If two linear equations have the condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, what does this tell us geometrically?
A. The lines are parallel and never intersect
B. The lines are coincident (the same line) and have infinite solutions ✓ Correct
C. The lines intersect at exactly one point
D. The lines are perpendicular to each other
Solution: When all three ratios are equal, the equations represent the same line (coincident). Every point on this line satisfies both equations, giving infinitely many solutions. Geometrically, the two equations are just scalar multiples of each other.
Q2 — Graphical method · easy · theory
A system of two linear equations has a unique solution. What must be true about the geometric representation?
A. The two lines are parallel
B. The two lines are the same (coincident)
C. The two lines intersect at exactly one point ✓ Correct
D. The lines are perpendicular at the origin
Solution: A unique solution corresponds to the point where the two lines intersect. This occurs when $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, meaning the slopes are different and the lines are neither parallel nor coincident.
Q3 — Graphical method · easy · theory
For the system $2x + 3y = 7$ and $4x + 6y = 14$, which method would be MOST appropriate to recognize the relationship without full solving?
A. Graphical method to plot both lines
B. Observe the ratio condition to recognize these are proportional equations (coincident lines) ✓ Correct
C. Substitution method to find x first
D. Cross-multiplication to get the unique solution
Solution: Notice that the second equation is exactly 2 times the first equation: $4x + 6y = 2(2x + 3y)$ and $14 = 2(7)$. The ratio condition $\frac{2}{4} = \frac{3}{6} = \frac{7}{14} = \frac{1}{2}$ immediately tells us this system has infinitely many solutions without solving anything.
Q4 — Graphical method · easy · theory
If the ratio $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, what can we conclude about the system without solving?
A. The system has no solution (inconsistent)
B. The system has infinitely many solutions
C. The system has exactly one unique solution ✓ Correct
D. We cannot determine without knowing the constant terms
Solution: When the ratios of coefficients of $x$ and $y$ are different ($\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$), the lines have different slopes, meaning they are not parallel and not coincident. Therefore, they must intersect at exactly one point, guaranteeing a unique solution.
Q5 — Graphical method · medium · theory
A system of equations is said to be inconsistent. What does this mean geometrically?
A. The two lines intersect at the origin
B. The two lines have no point in common (they are parallel) ✓ Correct
C. The two lines are perpendicular
D. The two lines pass through the same three points
Solution: An inconsistent system has no solution, meaning the equations cannot be satisfied simultaneously. Geometrically, this occurs when the lines are parallel: they run side by side but never meet. Algebraically, this is detected by the ratio condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
Q6 — Graphical method · medium · theory
Consider the system $kx + 3y = 7$ and $6x + 9y = 21$. For what value of $k$ does this system have infinitely many solutions?
A. $k = 2$ ✓ Correct
B. $k = 3$
C. $k = 6$
D. $k = 9$
Solution: For infinitely many solutions, the equations must be proportional: $\frac{k}{6} = \frac{3}{9} = \frac{7}{21}$. Simplifying: $\frac{3}{9} = \frac{1}{3}$ and $\frac{7}{21} = \frac{1}{3}$. So $\frac{k}{6} = \frac{1}{3}$, giving $k = 2$. This makes the second equation exactly 3 times the first.
Q7 — Graphical method · medium · theory
For what condition on the parameters does the system $x + ky = 5$ and $2x + 4y = 10$ have infinitely many solutions?
A. $k = 1$
B. $k = 2$ ✓ Correct
C. $k = \frac{1}{2}$
D. $k = 5$
Solution: For infinitely many solutions: $\frac{1}{2} = \frac{k}{4} = \frac{5}{10}$. From $\frac{5}{10} = \frac{1}{2}$ and $\frac{1}{2} = \frac{k}{4}$, we get $k = 2$. The second equation becomes $2(x + 2y = 5)$, so the equations are proportional and represent the same line.
Q8 — Graphical method · hard · theory
Given a system where the ratio of a coefficients equals the ratio of b coefficients but not the ratio of c coefficients, explain why checking these ratios (without solving) guarantees no solution exists.
A. Because equal slopes with different y-intercepts mean parallel lines that never intersect, making the system inconsistent ✓ Correct
B. Because the determinant is always positive, which precludes solutions
C. Because the system is always dependent when ratios are equal
D. Because constant terms with unequal ratios cancel out both variables
Solution: When the ratios of x and y coefficients are equal, the lines have the same slope. But if the constant ratio is different, the y-intercepts differ. This means parallel lines with no intersection, making the system inconsistent.
Q9 — Graphical method · hard · theory
A student claims: 'If a system has infinitely many solutions, then one equation must be a scalar multiple of the other.' Is this claim true and why?
A. True. Infinitely many solutions occur when both equations represent the same line, which means one must be a scalar multiple of the other ✓ Correct
B. False. Infinitely many solutions can occur even when equations are completely different
C. True, but only if the constant term is zero in both equations
D. False, because infinitely many solutions means the lines are perpendicular
Solution: For infinitely many solutions, all three ratios (coefficients of x, coefficients of y, and constants) must be equal. This means the first equation equals k times the second equation for some non-zero k, representing the same line.
Q10 — Graphical method · hard · theory
Two lines intersect at point (2, 3). Based solely on this geometric fact, can we determine whether the system is consistent or inconsistent?
A. No, because knowing the intersection point tells us nothing about consistency
B. Yes, it is consistent because the lines have at least one common point at (2, 3) ✓ Correct
C. It depends on whether the lines are horizontal or vertical
D. Inconsistent systems cannot have intersecting lines, so this is impossible
Solution: A consistent system has at least one solution. If two lines intersect at a point, that point is a solution. Therefore, the system is consistent. This demonstrates the connection between geometry and algebra.