(D) Let the speed of the current be $x\,km/h$.Speed downstream = $(9 + x)\,km/h$.Speed upstream = $(9 - x)\,km/h$.Time taken downstream = $\frac{12}{9

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(D) Let the speed of the current be $x\,km/h$.Speed downstream = $(9 + x)\,km/h$.Speed upstream = $(9 - x)\,km/h$.Time taken downstream = $\frac{12}{9 + x}$.Time taken upstream = $\frac{12}{9 - x}$.Total time = $3\,hours$,so $\frac{12}{9 + x} + \frac{12}{9 - x} = 3$.Dividing by $3$: $\frac{4}{9 + x} + \frac{4}{9 - x} = 1$.$4(9 - x) + 4(9 + x) = (9 + x)(9 - x)$.$36 - 4x + 36 + 4x = 81 - x^2$.$72 = 81 - x^2$.$x^2 = 81 - 72 = 9$.$x = 3\,km/h$ (since speed cannot be negative).

Class 10 Mathematics Quadratic Equations Mix Examples - Quadratic Equations

Difficult

The speed of a motorboat in still water is $9\,km/h$. In a river,it covers $12\,km$ downstream and the same distance upstream in a total time of $3\,hours$. Find the speed of the current of the river. (The speed of the current of the river is less than the speed of the motorboat in still water.) (in $,km/h$)

A
$5$
B
$1$
C
$7$
D
$3$

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Mix Examples - Quadratic Equations

In the centre of a rectangular lawn of dimensions $50\, m \times 40\, m$,a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be $1184\, m^{2}$ [see $Fig.$]. Find the length and breadth of the pond. (in $m$)

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In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $AB = 3x - 2$,$AC = 5x + 4$,and $BC = 6x + 2$,find the value of $x$.

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The hypotenuse of a garden in the shape of a right triangle is $1$ metre more than twice the shortest side. If the longer side containing the right angle is $7$ metres more than the shorter side,find the sides of the garden.

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State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has at most two roots.
$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.

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The perimeter of a bigger square is $12\,m$ more than the perimeter of a smaller square. Three times the area of the smaller square is $11\,m^2$ more than the area of the bigger square. Find the length of the side of the bigger square.

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In the centre of a rectangular lawn of dimensions $50\, m \times 40\, m$,a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be $1184\, m^{2}$ [see $Fig.$]. Find the length and breadth of the pond. (in $m$)

In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $AB = 3x - 2$,$AC = 5x + 4$,and $BC = 6x + 2$,find the value of $x$.

The hypotenuse of a garden in the shape of a right triangle is $1$ metre more than twice the shortest side. If the longer side containing the right angle is $7$ metres more than the shorter side,find the sides of the garden.

State whether the following statements are true or false. Justify your answers.$(i)$ Every quadratic equation has at most two roots.$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.

The perimeter of a bigger square is $12\,m$ more than the perimeter of a smaller square. Three times the area of the smaller square is $11\,m^2$ more than the area of the bigger square. Find the length of the side of the bigger square.

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State whether the following statements are true or false. Justify your answers.
$(i)$ Every quadratic equation has at most two roots.
$(ii)$ If the coefficient of $x^{2}$ and the constant term of a quadratic equation have opposite signs,then the quadratic equation has real roots.