Quadratic Equations Question - CBSE_BOARD_PYQ_2026_30_7_1

Question

[Section D]

The sum of the squares of two positive numbers is $29$. The larger number is $1$ more than twice the smaller number. Find the numbers.

OR

A rectangular garden of area $180$ m$^2$ is to be fenced on three sides, leaving one longer side unfenced. If the length of the fencing is $39$ m, find the dimensions of the garden.

Accepted Answer

[Main Question Solution]

Let smaller number be $x$, larger number $y = 2x + 1$. [1.0 Mark]

$x^2 + (2x + 1)^2 = 29 \Rightarrow 5x^2 + 4x - 28 = 0 \Rightarrow (x - 2)(5x + 14) = 0$. [2.0 Marks]

Since $x > 0$, $x = 2$. Larger number $y = 2(2) + 1 = 5$. Required numbers are $2$ and $5$. [2.0 Marks]

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[OR Choice Question Solution]

Let longer side be $x$, smaller side $y$. $xy = 180$ and $x + 2y = 39$. [1.5 Marks]

$x + 2\left(\dfrac{180}{x} ight) = 39 \Rightarrow x^2 - 39x + 360 = 0 \Rightarrow (x - 24)(x - 15) = 0$. [2.0 Marks]

$x = 24 \Rightarrow y = 7.5$ m OR $x = 15 \Rightarrow y = 12$ m. Dimensions: $24$ m $ imes 7.5$ m or $15$ m $ imes 12$ m. [1.5 Marks]

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