Board 2024 Class 12 Maths Q5 Solution: Find 24/x + 24/y from equal matrices

The question

If

\begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix},

then the value of $\left(\frac{24}{x} + \frac{24}{y}\right)$ is :

(A) $7$

(B) $6$

(C) $8$

(D) $18$

Exam answer

Equating corresponding elements, $x + y = 6$ and $xy = 8$ ⁠.

Why?

Two matrices of the same order are equal only when every pair of corresponding elements is equal.

Row 1, column 1 gives $x + y = 6$ ⁠. Row 2, column 2 gives $xy = 8$ ⁠. The other two pairs ( $2 = 2$ and $5 = 5$ ⁠) already match.

\begin{aligned} \frac{24}{x} + \frac{24}{y} &= \frac{24y + 24x}{xy} \\ &= \frac{24(x + y)}{xy} \\ &= \frac{24 \times 6}{8} = 18 \end{aligned}

Why?

Adding the fractions over the common denominator $xy$ brings in exactly $x + y$ and $xy$ ⁠, the two values we know. So we never need $x$ and $y$ separately.

Give each fraction the denominator $xy$ ⁠:

\begin{aligned} \frac{24}{x} &= \frac{24 \times y}{x \times y} = \frac{24y}{xy} \\[4pt] \frac{24}{y} &= \frac{24 \times x}{y \times x} = \frac{24x}{xy} \end{aligned}

Take out the common factor 24 on top, then put in $x + y = 6$ and $xy = 8$ ⁠:

\begin{aligned} 24y + 24x &= 24(x + y) \\[4pt] \frac{24(6)}{8} &= \frac{144}{8} \\ &= \frac{144 \div 8}{8 \div 8} = 18 \end{aligned}

Check: $x + y = 6$ and $xy = 8$ give $x$ and $y$ as $2$ and $4$ (in either order), since $2 + 4 = 6$ and $(2)(4) = 8$ ⁠. Then

\frac{24}{2} + \frac{24}{4} = 12 + 6 = 18

Answer: (D) $18$

Where this comes from

Chapter 3, Matrices: Equality of matrices.

Question type: Equal matrices

How to spot it: click to see

Type 2. Equal matrices: match entry with entry

Spot it: two matrices joined by an equals sign, with unknowns such as $x$ or $a - b$ inside.

Here: The two matrices are joined by an equals sign, so match the corners: $x + y = 6$ and $xy = 8$ ⁠. Then $\frac{24}{x} + \frac{24}{y} = \frac{24(x + y)}{xy} = 18$ ⁠, with no need for $x$ and $y$ themselves.

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Q5: Find 24x+24y\frac{24}{x} + \frac{24}{y}x24​+y24​ from equal matrices

Q5: Find $\frac{24}{x} + \frac{24}{y}$ from equal matrices