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Full Grade 12 test

Full Grade 12 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: 17Estimated time: 31 minutes

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Question 1out of 17

A box has 4 red and 5 blue balls. Two are drawn without replacement. Find the probability both are red.

3/8

4/9

5/9

1/6

Question 2out of 17

A plane passes through the midpoints of the three edges of a cube that meet at the same vertex. The edge of the cube is 20 cm. Find the area of the cross-section.

50√3 cm²

25√3 cm²

100√3 cm²

200 cm²

The vertices of the cross-section are the midpoints of the three cube edges meeting at one vertex. Each side of the cross-section is a midline in a triangular half of a cube face, so its length is a/√2. The cross-section is an equilateral triangle. Calculate its area using S = s²√3/4.

Question 3out of 17

What can be stated if f′(a) = 0?

The function value is zero

The tangent is horizontal, but the point need not be an extremum

The function is constant everywhere

It is always a maximum

Question 4out of 17

Which variable in a linear model is called dependent?

A variable with no measurements

The response being explained or predicted

The variable chosen alphabetically first

Always the larger number

Question 5out of 17

An oblique segment from point A to a plane is 13 cm long, and its orthogonal projection onto the plane is 12 cm. Find the distance from point A to the plane.

13 cm

12 cm

5 cm

17 cm

The distance from a point to a plane is the length of the perpendicular. The perpendicular, the oblique segment and its projection form a right triangle, with the oblique segment as the hypotenuse. Apply the Pythagorean theorem: if the oblique length is l and the projection is p, then h = √(l² − p²).

Question 6out of 17

Is every point where f′(x) = 0 necessarily an extremum?

The function must equal zero there

It is always a maximum

It is a candidate; the derivative sign or other analysis is needed

It is never an extremum

Question 7out of 17

A pyramid has base area 6 cm² and height 10 cm. Find its volume.

20 cm³

10 cm³

50 cm³

18 cm³

39 cm³

Question 8out of 17

How are position and velocity related in a motion problem?

Time is the derivative of velocity

Position is always velocity squared

Velocity is the derivative of position with respect to time

Velocity is the area of position

Question 9out of 17

Which sequence is appropriate when analysing a function with derivatives?

Find critical points, make a derivative sign chart, then evaluate

Use only f(0)

Ignore the domain

Differentiate the answer choices

Question 10out of 17

Why must endpoints also be checked on a closed interval?

Because an absolute optimum on a closed interval may occur at an endpoint

Only endpoints can be extrema

Endpoints never matter

The derivative is undefined at every endpoint

Question 11out of 17

How are solutions in a stated interval selected from a general trigonometric solution?

Ignore interval endpoints

List one arbitrary period

Keep only general-solution values that belong to the stated interval

Convert all values to positive

Question 12out of 17

A cone has radius 4 cm and height 33 cm. Using π ≈ 3.14, find volume.

134.32 cm³

552.64 cm³

48.39 cm³

187.07 cm³

252.9 cm³

Question 13out of 17

Which sum differentiation rule is correct?

(f + g)' = f + g

(f + g)' = f' + g'

(f + g)' = 0

(f + g)' = f'g'

Question 14out of 17

An oblique segment of length 25 has an orthogonal projection of length 7 on a plane. Find the sine of the angle α between the segment and the plane.

7/25

24/7

7/24

24/25

The oblique segment, its orthogonal projection and the perpendicular to the plane form a right triangle. First find the perpendicular length using the Pythagorean theorem. The sine of the angle between the segment and the plane is the perpendicular length divided by the oblique length.

Question 15out of 17

How is an event probability calculated in a probability tree?

Multiply every branch in the whole tree

Multiply probabilities along a path and add probabilities of favourable disjoint paths

Ignore conditional probabilities

Add along every path

Question 16out of 17

A rectangle has perimeter 76 cm. What should one side be for maximum area?

56 cm

10 cm

27 cm

66 cm

19 cm

Question 17out of 17

Two trend models are y = 3x and y = x + 22. For what value of x do both models give the same prediction?

-11

Set the two model predictions equal: a₁x + b₁ = a₂x + b₂. Collect the x terms on one side, the constants on the other, and solve the resulting linear equation.

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Full Grade 12 mathematics test

This complete Grade 12 mathematics test helps students review the main school topics in one mixed set of tasks. Questions are selected from different tests for this grade, so each attempt can be different.

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