Diagnostic Exam

You wrote this diagnostic exam on our first day in class.

It was adapted from an old Grade 9 math exam that Mr. Gordon gave his students many years ago – a bit shortened to fit available time.

If you wish, you can review these questions again to gain further practice with key skills.

TIP

If you would rather work from paper, use the Print button on this page, or open the handout directly:

Diagnostic – Grade 9 Skills

Part A

Fill in the blanks. Answers only.

Question 1

Evaluate.

a) $3(-4{)}^2$

b) $\frac{5^{300}}{5^{298}}$

c) $5\left(\frac{3}{4}\right)$ (as a fraction)

d) $4-2(-5)$

Answer

a) $48$

b) $25$

c) $\frac{15}{4}$

d) $14$

Question 2

Give the equation of the line with slope $\frac{3}{2}$ and $y$-intercept $-5$.

Answer

$y=\frac{3}{2}x-5$

Question 3

State the slope of the line parallel to $y=5x+3$.

Answer

$m=5$

Question 4

State the slope of a line perpendicular to $y=-2x+7$.

Answer

$m=\frac{1}{2}$

Question 5

Find the slope of the line joining the points $A(2,0)$ and $B(0,-8)$.

Answer

$m=\frac{-8-0}{0-2}=\frac{-8}{-2}=4$

Question 6

Evaluate 15% of 350.

Answer

$52.5$

Question 7

A car depreciates by $5 000 per year and has an initial value of $30 000.

a) What does the independent variable represent?

b) What does the dependent variable represent?

c) Write an equation for this relation.

Answer

a) $x$, the number of years that have passed since the car was bought

b) $y$, the value of the car, in dollars

c) $y=-5000x+30000$

Question 8

State the equation of the horizontal line passing through the point $P(5,7)$.

Answer

$y=7$

Question 9

Reduce $12:24:36$ to lowest terms.

Answer

$1:2:3$

Question 10

Find the $x$-intercept of $7x-3y=-42$.

Answer

$-6$

Let $y=0$: $7x=-42$, so $x=-6$.

Question 11

State the value of each variable.

a)

Three lines meet at a point on a straight line, making angles of 50 degrees and 60 degrees with the straight line and an angle x between them; the two lower lines form an isosceles triangle with equal sides marked, whose base angle is w

$w=$ ______ $x=$ ______

b)

A parallelogram ABCD with a diagonal from B to D; the diagonal makes a 40 degree angle at B with the bottom side and a 70 degree angle at D; the angle at A is z and the angle at C is y

$y=$ ______ $z=$ ______

Answer

a) $w=40^{\circ }$, $x=70^{\circ }$

$x=180^{\circ }-50^{\circ }-60^{\circ }=70^{\circ }$ (opposite angles); the triangle is isosceles, so $w=180^{\circ }-2(70^{\circ })=40^{\circ }$.

b) $y=70^{\circ }$, $z=70^{\circ }$

$y=180^{\circ }-40^{\circ }-70^{\circ }=70^{\circ }$ (angle sum); $z=y=70^{\circ }$ (opposite angles of a parallelogram).

Question 12

A line has slope $\frac{5}{2}$ and passes through the point $P(1,3)$. State two other points on the line.

Answer

$(3,8)$ and $(5,13)$

Question 13

The volume of a rectangular prism is $960$ cm${}^3$. If the length is $10$ cm and the width is $24$ cm, then the height is ______.

Answer

$4$ cm

$h=960÷(10\times 24)=4$

Question 14

A rectangular garden is to be surrounded by $48$ m of fencing.

a) State the dimensions in order to maximize the area.

b) State the maximum area.

Answer

a) $12$ m by $12$ m

b) $144$ m${}^2$

Question 15

Match each equation to the corresponding graph.

$$y=-3x+2y=3x+2y=-3x-2y=3x-2$$

a)

A small sketch of a line rising steeply from left to right and crossing the y-axis above the origin

b)

A small sketch of a line falling steeply from left to right and crossing the y-axis above the origin

c)

A small sketch of a line falling steeply from left to right and crossing the y-axis below the origin

d)

A small sketch of a line rising steeply from left to right and crossing the y-axis below the origin

Answer

a) $y=3x+2$

b) $y=-3x+2$

c) $y=-3x-2$

d) $y=3x-2$

Question 16

Simplify.

a) $\left(27x^5\right)\left(3x^2\right)$

b) $\frac{27x^5}{3x^2}$

c) ${\left(3x^5\right)}^2$

Answer

a) $81x^7$

b) $9x^3$

c) $9x^{10}$

Part B

Give full solutions to all problems.

Question 17

Find the intersection of the following two lines by graphing them on a grid.

a) $2x-y=8$

b) $y=x-5$

Intersection point: ______

Answer

a) $x$-intercept: $y=0$, $2x=8$, $x=4$. $y$-intercept: $x=0$, $-y=8$, $y=-8$.

b) slope $1$, $y$-intercept $-5$

Intersection point: $(3,-2)$

A grid from -10 to 10 on both axes showing line a) 2x - y = 8 through (4, 0) and (0, -8), line b) y = x - 5 through (5, 0) and (0, -5), and their intersection circled at (3, -2)

Question 18

A family is moving from Brampton to Ottawa. Cheap Movers charges $1750 for 1000 kg or $2125 for 2500 kg.

a) Find the equation representing the cost of moving using Cheap Movers.

b) Explain the meaning of the slope in this problem.

c) Explain the meaning of the $y$-intercept in this problem.

Answer

a) Let $x$ be the mass moved, in kg, and $y$ be the cost, in dollars. The points are $(1000,1750)$ and $(2500,2125)$.

$$\begin{array}{rl}m& =\frac{\Delta y}{\Delta x}=\frac{2125-1750}{2500-1000}\\ & =\frac{375}{1500}=0.25\end{array}$$

Sub $(2500,2125)$ into $y=0.25x+b$:

$$\begin{array}{rl}2125& =0.25(2500)+b\\ 2125& =625+b\\ 2125-625& =625+b-625\\ 1500& =b\end{array}$$

$\therefore$ $y=0.25x+1500$

b) The cost per kilogram moved: each extra kilogram costs $0.25.

c) The fixed cost of hiring the movers ($1500), charged before anything is moved.

Question 19

The length of a rectangle is 3 m more than twice the width. If the perimeter is 42 m, find both the length and the width.

Answer

Let $x$ be the width, in metres. Then the length is $2x+3$ metres.

$$\begin{array}{rl}P& =2l+2w\\ 42& =2(2x+3)+2x\\ 42& =4x+6+2x\\ 42& =6x+6\\ 42-6& =6x+6-6\\ 36& =6x\\ \frac{36}{6}& =\frac{6x}{6}\\ 6& =x\end{array}$$

Length: $2x+3=2(6)+3=15$

Check: $2(15)+2(6)=30+12=42$ ✓

$\therefore$ the width is 6 m and the length is 15 m.

Question 20

Solve the following equations.

a) $5x+3=2x+16.5$

b) $8(2x+4)-3=9(x-3)+15x$

c) $\frac{4x}{3}+\frac{3x-2}{4}=\frac{x}{2}-10$

Answer

a)

$$\begin{array}{rl}5x+3& =2x+16.5\\ 5x+3-2x-3& =2x+16.5-2x-3\\ 3x& =13.5\\ \frac{3x}{3}& =\frac{13.5}{3}\\ x& =4.5\end{array}$$

b)

$$\begin{array}{rl}16x+32-3& =9x-27+15x\\ 16x+29& =24x-27\\ 16x+29-16x+27& =24x-27-16x+27\\ 56& =8x\\ \frac{56}{8}& =\frac{8x}{8}\\ 7& =x\end{array}$$

c) Multiply both sides by the lowest common denominator, 12.

$$\begin{array}{rl}12\left[\frac{4x}{3}+\frac{3x-2}{4}\right]& =12\left[\frac{x}{2}-10\right]\\ 4(4x)+3(3x-2)& =6(x)-120\\ 16x+9x-6& =6x-120\\ 25x-6& =6x-120\\ 25x-6-6x+6& =6x-120-6x+6\\ 19x& =-114\\ \frac{19x}{19}& =\frac{-114}{19}\\ x& =-6\end{array}$$

Question 21

A rectangular floor measuring 15 m by 12 m is to be covered in tiles measuring 0.3 m by 0.3 m. A single box contains 500 tiles and costs $149.95. Find the total cost if taxes are 14%.

Answer

Area of the floor: $15\times 12=180$ m${}^2$

Area of one tile: $0.3\times 0.3=0.09$ m${}^2$

Number of tiles needed: $180÷0.09=2000$

Number of boxes needed: $2000÷500=4$

Total cost:

$$\begin{array}{rl}& =4\times 149.95\times 1.14\\ & =599.80\times 1.14\\ & =683.772\end{array}$$

$\therefore$ the total cost is $683.77.

Question 22

Evaluate the following.

a) $2(-3)+4(5{)}^2-(-10)$

b) $\left(2\frac{3}{4}-\frac{3}{8}\right)÷\frac{1}{2}$

Answer

a)

$$\begin{array}{rl}& =-6+4(25)+10\\ & =-6+100+10\\ & =104\end{array}$$

b)

$$\begin{array}{rl}& =\left(\frac{11}{4}-\frac{3}{8}\right)÷\frac{1}{2}=\left(\frac{22}{8}-\frac{3}{8}\right)÷\frac{1}{2}\\ & =\frac{19}{8}\times \frac{2}{1}=\frac{38}{8}=\frac{19}{4}\end{array}$$

Question 23

Simplify the following.

a) $5(3x-7y)+2x-(-x+8y)$

b) $5x^3\left(2x^2-y^3\right)+6x^2\left(x^3+3x{y}^3\right)$

c) $\frac{\left(2x^{2}y\right)\left(-12x^4{y}^8\right)}{-8x^5{y}^9}$

Answer

a) $=15x-35y+2x+x-8y=18x-43y$

b) $=10x^5-5x^3{y}^3+6x^5+18x^3{y}^3=16x^5+13x^3{y}^3$

c) $=\frac{-24x^6{y}^9}{-8x^5{y}^9}=3x$

Question 24

The mass of a baby was taken at the end of each week for 12 consecutive weeks. These values have been plotted on the grid below.

A scatter plot of a baby's mass in grams against the number of weeks, from week 1 at 4000 g rising steadily to week 12 at 8200 g

a) Draw in the line of best fit and extend it to cross the vertical axis.

b) State the initial mass (the $y$-intercept).

c) Choose another point on your line of best fit. It does not need to be one of the plotted data points.

d) Using the points in (b) and (c), find the slope (rate).

e) Write an equation for this relation.

f) What does the slope represent in this problem?

g) Use your equation in (e) to predict the mass after 15 weeks.

Answer

Answers for (b) to (g) depend on the line of best fit drawn. The line here passes through $(0,3500)$ and $(10,7500)$; a line through $(0,3500)$ and $(10,7000)$ is also reasonable and gives a rate of 350 g per week and a prediction of 8750 g.

a)

The same scatter plot with a line of best fit drawn through (0, 3500) and (10, 7500), both circled, extended to cross the vertical axis

b) $3500$ g, or $(0,3500)$

c) $(10,7500)$

d) $m=\frac{\Delta y}{\Delta x}=\frac{7500-3500}{10-0}=\frac{4000}{10}=400$ (grams per week)

e) $y=400x+3500$

f) The mass the baby gains each week: about 400 g per week.

g) $y=400(15)+3500=6000+3500=9500$

$\therefore$ the mass of the baby after 15 weeks will be about 9500 g.

Curriculum connection

B1.3

identify the relationship between the slopes of parallel and perpendicular lines, and use this relationship to solve related problems;

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B1.4

develop the formula for the slope of a line (i.e., $m=\frac{y_2-y_1}{x_2-x_1}$), and use this formula to determine the equations of lines, given information about the lines (e.g., a graph of a line, a table of values, the coordinates of two points);

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