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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: 33Estimated time: 61 minutes

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Question 1out of 33

When is a line perpendicular to a plane?

It lies in the plane

It is perpendicular to two intersecting lines in the plane

It is perpendicular to one arbitrary line only

It is parallel to the plane

Question 2out of 33

A sphere has radius 12 cm. Using π ≈ 3.14, find volume.

6453.38 cm³

188.88 cm³

3197.86 cm³

7234.56 cm³

9.99 cm³

Question 3out of 33

A pyramid has base area 18 cm² and height 3 cm. Find its volume.

38 cm³

32 cm³

47 cm³

18 cm³

3 cm³

Question 4out of 33

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

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

It is never an extremum

It is always a maximum

The function must equal zero there

Question 5out of 33

A rectangle has perimeter 60 cm. What should one side be for maximum area?

4 cm

22 cm

13 cm

35 cm

15 cm

Question 6out of 33

For three observations, the pairs (observed value; predicted value) are: (7; 9), (26; 25), (36; 34). Calculate the root mean square error (RMSE). Give the exact value.

√3

√6

For each pair find the residual eᵢ = yᵢ − predictionᵢ. Square the residuals, calculate the mean of their squares, and take the square root: RMSE = √((e₁² + e₂² + e₃²) / 3).

Question 7out of 33

The normal vectors of two planes are n₁ = (1; 0; 0) and n₂ = (3; 4; 0). Find the cosine of the acute angle between the planes.

3/4

4/3

4/5

3/5

The angle between two planes equals the acute angle between their normal vectors. Use cos α = |n₁ · n₂| / (|n₁||n₂|): calculate the dot product and the lengths of both vectors.

Question 8out of 33

A cylinder has radius 4 cm and height 3 cm. Find axial-section area.

1 cm²

78 cm²

93 cm²

37 cm²

24 cm²

Question 9out of 33

Why does strong correlation not necessarily mean causation?

Third variables never matter

It always proves direct causation

Only negative correlations need caution

A strong association may be caused by a third variable

Question 10out of 33

Independent events A and B have probabilities 80% and 80%. Find P(A and B) as a decimal.

0.64

0.1

0.84

0.99

0.27

Question 11out of 33

What shape is an axial section of a right cone?

An isosceles triangle with base equal to the diameter

A rectangle

A circle

A scalene triangle in every case

Question 12out of 33

What should be considered when deciding whether a linear model is suitable?

Ignore residuals

Use slope alone

Consider R², residuals, context and whether extrapolation is justified

Any line is suitable

Question 13out of 33

How can an outlier affect statistical analysis?

It changes only sample size

It always proves causation

It can strongly change the mean, standard deviation and fitted line

It never affects analysis

Question 14out of 33

Find the tangent equation to f(x) = 2x² at x = -1.

y = -4x + 2

y = -4x − 2

y = 4x − 2

y = 2x − 1

Question 15out of 33

For acute α with tan α = 1.50, approximate cos α using 1 + tan²α = 1/cos²α.

0.40

0.55

1.50

Question 16out of 33

For f(x) = 4x² + (-4)x, solve f′(x) = -36.

-2

-4

-10

-1

Question 17out of 33

A linear regression model has R² = 0.25, its line is decreasing, sₓ = 5, and sᵧ = 15. Find the slope of the regression line.

-1.5

-0.17

1.5

-0.5

First find the correlation coefficient: |r| = √R². Determine its sign from the direction of the regression line. Then use the slope formula a = r · sᵧ / sₓ. Keep the sign of r.

Question 18out of 33

A cone has radius 9 cm and height 36 cm. Using π ≈ 3.14, find volume.

3052.08 cm³

2651.74 cm³

2955.89 cm³

1884.82 cm³

1182.44 cm³

Question 19out of 33

An open-top box has square base side x and volume V = 600. Which function gives surface area?

S(x) = 4x + 600/x

S(x) = x² + 600/x

S(x) = 600x²

S(x) = x² + 2400/x

Question 20out of 33

Some school students were surveyed. What are the surveyed group and all school students called?

Both are populations

The terms are interchangeable

The surveyed students are the sample; all school students are the population

Both are samples

Question 21out of 33

What does “axial section” mean?

It never meets the axis

An axial section passes through the axis of rotation

It is always parallel to the axis

It is defined by volume

Question 22out of 33

What does a cross-section of a solid mean?

The intersection of a solid with a plane

Every face of a solid

The shadow of a solid only

The volume of a solid

Question 23out of 33

What can be stated if f′(a) = 0?

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

It is always a maximum

The function is constant everywhere

The function value is zero

Question 24out of 33

Why must endpoints also be checked on a closed interval?

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

The derivative is undefined at every endpoint

Endpoints never matter

Only endpoints can be extrema

Question 25out of 33

What shape is an axial section of a cylinder?

A rectangle with sides 2r and h

A circle of radius h

A square of side r only

An equilateral triangle

Question 26out of 33

A cone has radius 3 cm and slant height 4 cm. Using π ≈ 3.14, find lateral area.

93.22 cm²

60.51 cm²

37.68 cm²

51.09 cm²

41.61 cm²

Question 27out of 33

A rectangular prism measures 5 cm, 4 cm and 12 cm. Find total surface area.

336 cm²

157 cm²

19 cm²

241 cm²

256 cm²

Question 28out of 33

Compare correlations r₁ = 0.20 and r₂ = 0.30. Which is stronger?

They are equal

Signs alone determine strength

The first is stronger

The second correlation is stronger

Question 29out of 33

A box has 8 red and 6 blue balls. Two are drawn with replacement. Find the probability both are red.

6/14

16/49

8/14

7/13

Question 30out of 33

f(5) = 3 and f(8) = 18. Find the average rate of change on [5, 8].

-8

-7

-5

The average rate of change of a function on [a, b] shows the average change in the function value when x increases by one unit. Use:

(f(b) − f(a)) / (b − a).

First find the difference between the function values, then divide it by the interval length. Pay attention to the signs of negative numbers.

Question 31out of 33

A box has 6 red and 3 blue balls. Two are drawn without replacement. Find the probability both are red.

5/12

5/8

3/9

6/9

Question 32out of 33

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.

12 cm

17 cm

13 cm

5 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 33out of 33

Position is s(t) = 2t² + (-6)t. Find velocity at t = 4.

44 m/s

-7 m/s

22 m/s

10 m/s

70 m/s

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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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