Given the relation f: x (2x−1)/(x2+3x+2).

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Question 1:
Given the relation $f: x \to \frac{2x-1}{x^2+3x+2}$ .

i) Find the zero of the relation.

The relation is zero when the numerator is equal to zero.
$2x - 1 = 0$
Solve for $x$ .
$2x = 1$ $x = \frac{1}{2}$
The zero of the relation is $\boxed{\frac{1}{2}}$ .

ii) Find the value of $x$ which makes the relation undefined.
The relation is undefined when the denominator is equal to zero.
$x^2 + 3x + 2 = 0$
Factor the quadratic equation.
$(x+1)(x+2) = 0$
Solve for $x$ .
$x+1 = 0 \quad \text{or} \quad x+2 = 0$ $x = -1 \quad \text{or} \quad x = -2$
The values of $x$ which make the relation undefined are $\boxed{-1, -2}$ .

Question 2:
Calculate the (i) Area (ii) Perimeter of the given foundation drawing.

i) Area
Divide the L-shaped figure into two rectangles.
We can split it into a large bottom rectangle and a smaller top-left rectangle.
The bottom rectangle has dimensions $22 \text{ cm} \times 8 \text{ cm}$ .
The top-left rectangle has a width of $22 \text{ cm} - 14 \text{ cm} = 8 \text{ cm}$ and a height of $19 \text{ cm} - 8 \text{ cm} = 11 \text{ cm}$ .
Calculate the area of each rectangle.
Area of bottom rectangle $= 22 \text{ cm} \times 8 \text{ cm} = 176 \text{ cm}^2$ .
Area of top-left rectangle $= 8 \text{ cm} \times 11 \text{ cm} = 88 \text{ cm}^2$ .
Add the areas to find the total area.
Total Area $= 176 \text{ cm}^2 + 88 \text{ cm}^2 = 264 \text{ cm}^2$ .
The area is $\boxed{264 \text{ cm}^2}$ .

ii) Perimeter
Identify all the outer side lengths of the figure.
The given lengths are $19 \text{ cm}$ , $8 \text{ cm}$ , $11 \text{ cm}$ , $14 \text{ cm}$ , $8 \text{ cm}$ , and $22 \text{ cm}$ .
Sum all the outer side lengths.
Perimeter $= 19 \text{ cm} + 8 \text{ cm} + 11 \text{ cm} + 14 \text{ cm} + 8 \text{ cm} + 22 \text{ cm}$
Perimeter $= 82 \text{ cm}$ .
The perimeter is $\boxed{82 \text{ cm}}$ .

Question 3:
Find the value of $x$ and $y$ in the diagram below if O is the centre of the circle.
Use the property of angles in a cyclic quadrilateral.
The sum of opposite angles in a cyclic quadrilateral is $180^\circ$ .
The angles $x$ and $3x$ are opposite angles in the cyclic quadrilateral.
$x + 3x = 180^\circ$ $4x = 180^\circ$
Solve for $x$ .
$x = \frac{180^\circ}{4}$ $x = 45^\circ$
Use the property that the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.
The angle at the center is $y$ , and the angle at the circumference subtended by the same arc is $x$ .
$y = 2x$
Substitute the value of $x$ to find $y$ .
$y = 2(45^\circ)$ $y = 90^\circ$
The value of $x$ is $\boxed{45^\circ}$ and the value of $y$ is $\boxed{90^\circ}$ .

Question 4:

a) y varies directly as x. If $y = 18$ when $x = 6$ , find:

i) the equation connecting y and x

ii) the value of y when $x = 15$

i) The equation connecting y and x
For direct variation, the relationship is $y = kx$ , where $k$ is the constant of proportionality.
Use the given values $y=18$ and $x=6$ to find $k$ .
$18 = k(6)$ $k = \frac{18}{6}$ $k = 3$
The equation connecting $y$ and $x$ is $\boxed{y = 3x}$ .

ii) The value of y when $x = 15$
Use the equation $y = 3x$ found in part (i).
Substitute $x=15$ into the equation.
$y = 3(15)$ $y = 45$
The value of $y$ when $x=15$ is $\boxed{45}$ .

b) y varies inversely as x. If $y = 4$ when $x = 3$ , find y when $x = 12$ .
For inverse variation, the relationship is $y = \frac{k}{x}$ , where $k$ is the constant of proportionality.
Use the given values $y=4$ and $x=3$ to find $k$ .
$4 = \frac{k}{3}$ $k = 4 \times 3$ $k = 12$
The equation connecting $y$ and $x$ is $y = \frac{12}{x}$ .
Find $y$ when $x=12$ .
$y = \frac{12}{12}$ $y = 1$
The value of $y$ when $x=12$ is $\boxed{1}$ .

Given the relation f: x (2x-1)/(x2+3x+2).

Step-by-step solution

The relation is zero when the numerator is equal to zero.

Solve for $x$ .

The relation is undefined when the denominator is equal to zero.

Factor the quadratic equation.

Solve for $x$ .

Divide the L-shaped figure into two rectangles.

Calculate the area of each rectangle.

Add the areas to find the total area.

Identify all the outer side lengths of the figure.

Sum all the outer side lengths.

Use the property of angles in a cyclic quadrilateral.

Solve for $x$ .

Use the property that the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Substitute the value of $x$ to find $y$ .

For direct variation, the relationship is $y = kx$ , where $k$ is the constant of proportionality.

Use the given values $y=18$ and $x=6$ to find $k$ .

Use the equation $y = 3x$ found in part (i).

Substitute $x=15$ into the equation.

For inverse variation, the relationship is $y = \frac{k}{x}$ , where $k$ is the constant of proportionality.

Use the given values $y=4$ and $x=3$ to find $k$ .

Find $y$ when $x=12$ .

Still stuck on this one?

More Mathematics Questions

Given the relation f: x (2x-1)/(x2+3x+2).

Step-by-step solution

The relation is zero when the numerator is equal to zero.

Solve for xxx.

The relation is undefined when the denominator is equal to zero.

Factor the quadratic equation.

Solve for xxx.

Divide the L-shaped figure into two rectangles.

Calculate the area of each rectangle.

Add the areas to find the total area.

Identify all the outer side lengths of the figure.

Sum all the outer side lengths.

Use the property of angles in a cyclic quadrilateral.

Solve for xxx.

Use the property that the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Substitute the value of xxx to find yyy.

For direct variation, the relationship is y=kxy = kxy=kx, where kkk is the constant of proportionality.

Use the given values y=18y=18y=18 and x=6x=6x=6 to find kkk.

Use the equation y=3xy = 3xy=3x found in part (i).

Substitute x=15x=15x=15 into the equation.

For inverse variation, the relationship is y=kxy = \frac{k}{x}y=xk​, where kkk is the constant of proportionality.

Use the given values y=4y=4y=4 and x=3x=3x=3 to find kkk.

Find yyy when x=12x=12x=12.

Still stuck on this one?

More Mathematics Questions

Solve for $x$ .

Solve for $x$ .

Solve for $x$ .

Substitute the value of $x$ to find $y$ .

For direct variation, the relationship is $y = kx$ , where $k$ is the constant of proportionality.

Use the given values $y=18$ and $x=6$ to find $k$ .

Use the equation $y = 3x$ found in part (i).

Substitute $x=15$ into the equation.

For inverse variation, the relationship is $y = \frac{k}{x}$ , where $k$ is the constant of proportionality.

Use the given values $y=4$ and $x=3$ to find $k$ .

Find $y$ when $x=12$ .