Cross-multiplication — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Cross-multiplication MCQs with step-by-step solutions (4 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 — Cross-multiplication · medium · theory
Two lines are described as $L_1: a_1x + b_1y = c_1$ and $L_2: a_2x + b_2y = c_2$. If the determinant $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \neq 0$, what is guaranteed about the system?
A. The lines are parallel and have no solution
B. The lines are coincident with infinitely many solutions
C. The lines intersect at exactly one unique point ✓ Correct
D. The solution involves only positive integers
Solution: A non-zero determinant of the coefficient matrix means the coefficient matrix is invertible, which is equivalent to the condition $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. This guarantees the lines are not parallel and not coincident, so they intersect at exactly one point (unique solution).
Q2 — Cross-multiplication · medium · theory
The cross-multiplication formula for solving $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is: $\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$. What does the denominator $a_1b_2 - a_2b_1$ represent?
A. The determinant of the coefficient matrix, which must be non-zero for a unique solution ✓ Correct
B. The product of all coefficients in the system
C. The sum of the constant terms divided by 2
D. A measure of how large the coefficients are
Solution: The denominator $a_1b_2 - a_2b_1$ is the determinant of the coefficient matrix $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$. If this is zero, the system does not have a unique solution (lines are parallel or coincident). This is why cross-multiplication only works when the determinant is non-zero.
Q3 — Cross-multiplication · hard · theory
Prove why the cross-multiplication method works conceptually. Which principle underlies this technique?
A. Cramer's rule, using determinants of the coefficient and augmented matrices ✓ Correct
B. The graphical intersection of two lines always gives integer coordinates
C. Substitution is equivalent to multiplication of equation coefficients
D. The order of equations determines the sign of the solution
Solution: Cross-multiplication is an application of Cramer's rule, which states that for a system with non-zero determinant, the solution can be expressed using determinants. This determinant-based approach is both theoretically sound and practically efficient for hand calculation.
Q4 — Cross-multiplication · hard · theory
Why does the condition that the determinant of the coefficient matrix is non-zero guarantee that a linear system has a unique solution?
A. Because a non-zero determinant ensures the coefficient matrix is invertible, allowing unique solution ✓ Correct
B. Because it ensures the lines are always perpendicular
C. Because it makes all solutions positive
D. Because it prevents constant terms from affecting the solution
Solution: A non-zero determinant means the coefficient matrix is invertible in linear algebra terms. This is equivalent to the slopes of the lines being different, guaranteeing exactly one intersection point.