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Full Grade 9 test

Full Grade 9 mathematics test with mixed topics, dynamically selected questions and a choice of duration. Practise before an assessment or review the whole course at your own pace.

Questions: 24Estimated time: 46 minutes

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Question 1out of 24

A rectangle has perimeter 54 and area 170. Find its side lengths.

Only 10

-10 and 17

10 and 17

10 and -17

11 and 18

Question 2out of 24

Simplify ((x² − 49)/(x − 3)) · ((x − 3)/(x + 7)), where x ≠ 3 and x ≠ −7.

(x − 7)/(x − 3)

x − 7

x² − 49

x + 7

First recognise a difference of squares in the numerator:

x² − a² = (x − a)(x + a).

After substituting this product, the numerators and denominators contain matching factors. Cancel x − b and x + a. The remaining factor is the final answer.

Even after the factors are cancelled, the restrictions from the original expression remain: its denominators cannot equal zero.

Question 3out of 24

The opposite and adjacent sides are known. Which function is best for finding the angle?

Sine, because no hypotenuse is needed

Tangent, because it relates opposite and adjacent sides

No trigonometric ratio applies

Cosine, because it uses two legs

Question 4out of 24

An observer is 50 m from a building. The tangent of the elevation angle is 0.50. Find the height.

13 m

25 m

19 m

26 m

47 m

Question 5out of 24

Which condition is required for a table or mapping to define a function?

One input x has two different outputs

Only positive values may appear

Each input x has exactly one output y

Every output must equal its input

Question 6out of 24

A plane travels 120 m along a path whose angle has sine 0.50. How high does it climb?

59 m

96 m

42 m

92 m

60 m

Question 7out of 24

Add a/x and b/x when a = 6, b = 8.

x/14

14/x

48/x

2/x

Question 8out of 24

Given f(x) = 2x + (-1), find x when f(x) = -7.

-3

-9

Question 9out of 24

State the restrictions for 1/((x − (-4))(x − 2)).

x ≠ -4 and x ≠ 2

x ≠ -2

All real numbers

x = -4 and x = 2

Question 10out of 24

sin α = 0.60 and α is acute. Use sin²α + cos²α = 1 to find cos α.

0.80

0.60

Question 11out of 24

Which statement correctly describes a tangent to a circle?

A secant touches at one point only

A chord never meets the circle

Every radius is a tangent

A tangent touches the circle at exactly one point

Question 12out of 24

When factoring using roots x₁ and x₂, why is the form (x − x₁)(x − x₂) used?

The constant term is ignored

The roots must be multiplied by 2

The factors use the same signs as the roots

The signs in the factors are opposite to the root signs

Question 13out of 24

A conclusion about all students is based on five members of one team. What is the main problem?

Sample size never matters

The result certainly represents everyone

A non-random sample is always better

The sample may be biased and too small

Question 14out of 24

Given y = 3x + (6), find y when x = -1.

-4

Question 15out of 24

Expand (x − (-7))(x − (-8)).

x² + (-15)x + (56)

x² + (15)x + (-56)

x² + (15)x + (56)

x² + (56)x + (15)

To expand the brackets, multiply each term in the first pair of brackets by each term in the second:

(x − a)(x − b) = x² − bx − ax + ab.

Combine the like terms:

(x − a)(x − b) = x² − (a + b)x + ab.

Multiply x · x.
Combine the two terms containing x: −bx − ax.
Multiply the constant terms: (−a)(−b) = ab.

Pay close attention to the signs. If a negative number appears in brackets, for example x − (−3), the expression is equal to x + 3.

Question 16out of 24

Does (-2, 4) satisfy y = x² and y = 4?

Yes

A point satisfies a system of equations only when its coordinates satisfy both equations.

Substitute the point x-coordinate into the first equation and calculate x².
Compare the result with the point y-coordinate.
Also check that the same y-coordinate satisfies the second equation.

If either equality is false, the point does not satisfy the system.

Question 17out of 24

Where does y = 3x² + (-2)x + (-5) cross the y-axis?

(2, -8)

(1, 1)

(0, -5)

(0, -15)

(-4, -9)

Question 18out of 24

An object is 30 m high and the observation angle has tangent 0.50. Find the horizontal distance.

60 m

61 m

47 m

13 m

37 m

Question 19out of 24

Calculate: tan 0° = ?

√3

√2/2

1/2

√3/3

In the trigonometric table, find the tan row and the stated angle column. Their intersection contains the answer.

Question 20out of 24

Does (2, 10) lie on y = 1x + (8)?

Yes

Question 21out of 24

Which chart is most suitable for studying the relationship between two numerical variables?

A single bar

A scatter plot

A pie chart

A box plot only

Question 22out of 24

Give the vertex of y = 3(x − (-6))² + (-5).

(-6, 5)

(-5, -6)

(-5, -5)

(6, -5)

(-6, -5)

(-6, -4)

The vertex form of a parabola is y = a(x − m)² + n.

In this form, the vertex is V(m, n). Coefficient a does not change the vertex coordinates; it determines the opening direction and width.

Do not reverse the sign mentally: if the brackets contain x − (−m), the x-coordinate of the vertex is negative.

Question 23out of 24

Simplify ((x² − 25)/(x − 6)) ÷ ((x − 5)/(x − 6)), where x ≠ 6 and x ≠ 5.

(x + 5)/(x − 6)

x + 5

x − 5

Replace division by a fraction with multiplication by its reciprocal:

A/B ÷ C/D = A/B · D/C.

Then factor the difference of squares:

x² − a² = (x − a)(x + a).

Cancel matching factors. Remember that the divisor cannot equal zero, so in addition to denominator restrictions you must exclude the value that makes the numerator of the second fraction zero.

Question 24out of 24

Calculate the discriminant of 5x² + (3)x + (-5) = 0.

121

128

124

109

137

The discriminant is an expression formed from the coefficients of a quadratic equation ax² + bx + c = 0. It helps determine the number of real solutions.

Formula: D = b² − 4ac.

Identify the coefficients a, b and c, paying attention to their signs.
Calculate b².
Calculate 4ac and subtract it from b².

If D > 0, the equation has two real solutions; if D = 0, it has one; if D < 0, it has no real solutions.

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Full Grade 9 mathematics test

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